Citation
Surface tension and computer simulation of polyatomic fluids

Material Information

Title:
Surface tension and computer simulation of polyatomic fluids
Creator:
Haile, James Mitchell, 1946-
Publication Date:
Language:
English
Physical Description:
xxxi, 376 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Fluids ( jstor )
Interfacial tension ( jstor )
Liquids ( jstor )
Mathematical tables ( jstor )
Molecular dynamics ( jstor )
Molecules ( jstor )
Perturbation theory ( jstor )
Quadrupoles ( jstor )
Simulations ( jstor )
Spherical harmonics ( jstor )
Liquids ( lcsh )
Surface tension ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Includes bibliographical references (leaves 357-375).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by James Mitchell Haile.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
04102852 ( OCLC )
ocm04102852
00208276 ( ALEPH )
Classification:
QD541 .H13 ( lcc )

Downloads

This item has the following downloads:

surfacetensionco00hail.pdf

surfacetensionco00hail_0395.txt

surfacetensionco00hail_0220.txt

surfacetensionco00hail_0035.txt

surfacetensionco00hail_0023.txt

surfacetensionco00hail_0222.txt

surfacetensionco00hail_0151.txt

surfacetensionco00hail_0168.txt

surfacetensionco00hail_0397.txt

surfacetensionco00hail_0062.txt

surfacetensionco00hail_0150.txt

surfacetensionco00hail_0401.txt

surfacetensionco00hail_0121.txt

surfacetensionco00hail_0247.txt

surfacetensionco00hail_0097.txt

surfacetensionco00hail_0028.txt

surfacetensionco00hail_0079.txt

surfacetensionco00hail_0361.txt

surfacetensionco00hail_0252.txt

surfacetensionco00hail_0358.txt

surfacetensionco00hail_0106.txt

surfacetensionco00hail_0086.txt

surfacetensionco00hail_0320.txt

surfacetensionco00hail_0290.txt

surfacetensionco00hail_0283.txt

surfacetensionco00hail_0291.txt

surfacetensionco00hail_0084.txt

surfacetensionco00hail_0261.txt

surfacetensionco00hail_0339.txt

surfacetensionco00hail_0076.txt

surfacetensionco00hail_0344.txt

surfacetensionco00hail_0038.txt

surfacetensionco00hail_0263.txt

surfacetensionco00hail_0295.txt

surfacetensionco00hail_0212.txt

surfacetensionco00hail_0343.txt

surfacetensionco00hail_0226.txt

surfacetensionco00hail_0350.txt

surfacetensionco00hail_0321.txt

surfacetensionco00hail_0317.txt

surfacetensionco00hail_0099.txt

surfacetensionco00hail_0173.txt

surfacetensionco00hail_0324.txt

surfacetensionco00hail_0330.txt

surfacetensionco00hail_0093.txt

surfacetensionco00hail_0403.txt

surfacetensionco00hail_0120.txt

surfacetensionco00hail_0036.txt

surfacetensionco00hail_0057.txt

surfacetensionco00hail_0179.txt

surfacetensionco00hail_0169.txt

surfacetensionco00hail_0132.txt

surfacetensionco00hail_0346.txt

surfacetensionco00hail_0157.txt

surfacetensionco00hail_0292.txt

surfacetensionco00hail_0149.txt

surfacetensionco00hail_0279.txt

surfacetensionco00hail_0045.txt

surfacetensionco00hail_0311.txt

surfacetensionco00hail_0349.txt

surfacetensionco00hail_0203.txt

surfacetensionco00hail_0378.txt

surfacetensionco00hail_0387.txt

surfacetensionco00hail_0082.txt

surfacetensionco00hail_0209.txt

surfacetensionco00hail_0373.txt

surfacetensionco00hail_0065.txt

surfacetensionco00hail_0177.txt

surfacetensionco00hail_0380.txt

surfacetensionco00hail_0000.txt

surfacetensionco00hail_0033.txt

surfacetensionco00hail_0119.txt

surfacetensionco00hail_0166.txt

surfacetensionco00hail_0388.txt

surfacetensionco00hail_0107.txt

surfacetensionco00hail_0064.txt

surfacetensionco00hail_0255.txt

surfacetensionco00hail_0022.txt

surfacetensionco00hail_0341.txt

surfacetensionco00hail_0348.txt

surfacetensionco00hail_0153.txt

surfacetensionco00hail_0366.txt

surfacetensionco00hail_0137.txt

surfacetensionco00hail_0030.txt

surfacetensionco00hail_0152.txt

surfacetensionco00hail_0309.txt

surfacetensionco00hail_0124.txt

surfacetensionco00hail_0264.txt

surfacetensionco00hail_0357.txt

surfacetensionco00hail_0027.txt

surfacetensionco00hail_0298.txt

surfacetensionco00hail_0056.txt

surfacetensionco00hail_0198.txt

surfacetensionco00hail_0131.txt

surfacetensionco00hail_0277.txt

surfacetensionco00hail_0325.txt

surfacetensionco00hail_0336.txt

surfacetensionco00hail_0078.txt

surfacetensionco00hail_0319.txt

surfacetensionco00hail_0399.txt

surfacetensionco00hail_0383.txt

surfacetensionco00hail_0199.txt

surfacetensionco00hail_0193.txt

surfacetensionco00hail_0221.txt

surfacetensionco00hail_0014.txt

surfacetensionco00hail_0355.txt

surfacetensionco00hail_0181.txt

surfacetensionco00hail_0390.txt

surfacetensionco00hail_0278.txt

surfacetensionco00hail_0109.txt

surfacetensionco00hail_0337.txt

surfacetensionco00hail_0008.txt

surfacetensionco00hail_0037.txt

surfacetensionco00hail_0142.txt

surfacetensionco00hail_0208.txt

surfacetensionco00hail_0335.txt

surfacetensionco00hail_0302.txt

surfacetensionco00hail_0236.txt

surfacetensionco00hail_0147.txt

surfacetensionco00hail_0175.txt

surfacetensionco00hail_0313.txt

surfacetensionco00hail_0130.txt

surfacetensionco00hail_0196.txt

surfacetensionco00hail_0011.txt

surfacetensionco00hail_0135.txt

surfacetensionco00hail_0249.txt

surfacetensionco00hail_0276.txt

surfacetensionco00hail_0360.txt

surfacetensionco00hail_0053.txt

surfacetensionco00hail_0105.txt

surfacetensionco00hail_0155.txt

surfacetensionco00hail_0262.txt

surfacetensionco00hail_0323.txt

surfacetensionco00hail_0081.txt

surfacetensionco00hail_0145.txt

surfacetensionco00hail_0334.txt

surfacetensionco00hail_0017.txt

surfacetensionco00hail_0267.txt

surfacetensionco00hail_0071.txt

surfacetensionco00hail_0068.txt

surfacetensionco00hail_0154.txt

surfacetensionco00hail_0195.txt

surfacetensionco00hail_0187.txt

surfacetensionco00hail_0326.txt

surfacetensionco00hail_0040.txt

surfacetensionco00hail_0061.txt

surfacetensionco00hail_0101.txt

surfacetensionco00hail_0048.txt

surfacetensionco00hail_0308.txt

surfacetensionco00hail_0329.txt

surfacetensionco00hail_0286.txt

surfacetensionco00hail_0316.txt

surfacetensionco00hail_0365.txt

surfacetensionco00hail_0225.txt

surfacetensionco00hail_0189.txt

surfacetensionco00hail_0016.txt

surfacetensionco00hail_0235.txt

surfacetensionco00hail_0372.txt

surfacetensionco00hail_0354.txt

surfacetensionco00hail_0306.txt

surfacetensionco00hail_0381.txt

surfacetensionco00hail_0041.txt

surfacetensionco00hail_0128.txt

surfacetensionco00hail_0171.txt

surfacetensionco00hail_0069.txt

surfacetensionco00hail_0402.txt

surfacetensionco00hail_0077.txt

surfacetensionco00hail_0269.txt

surfacetensionco00hail_0002.txt

surfacetensionco00hail_0148.txt

surfacetensionco00hail_0376.txt

surfacetensionco00hail_0046.txt

surfacetensionco00hail_0170.txt

surfacetensionco00hail_0034.txt

surfacetensionco00hail_0367.txt

surfacetensionco00hail_0052.txt

surfacetensionco00hail_0250.txt

surfacetensionco00hail_0012.txt

surfacetensionco00hail_0184.txt

surfacetensionco00hail_0003.txt

surfacetensionco00hail_0029.txt

surfacetensionco00hail_0238.txt

surfacetensionco00hail_0333.txt

surfacetensionco00hail_0205.txt

surfacetensionco00hail_0043.txt

surfacetensionco00hail_0356.txt

surfacetensionco00hail_0114.txt

surfacetensionco00hail_0060.txt

surfacetensionco00hail_0239.txt

surfacetensionco00hail_0406.txt

surfacetensionco00hail_0201.txt

surfacetensionco00hail_0375.txt

surfacetensionco00hail_0074.txt

surfacetensionco00hail_0161.txt

surfacetensionco00hail_0186.txt

surfacetensionco00hail_0364.txt

surfacetensionco00hail_0058.txt

surfacetensionco00hail_0156.txt

surfacetensionco00hail_0129.txt

surfacetensionco00hail_0265.txt

surfacetensionco00hail_0307.txt

surfacetensionco00hail_0331.txt

surfacetensionco00hail_0318.txt

surfacetensionco00hail_pdf.txt

surfacetensionco00hail_0345.txt

surfacetensionco00hail_0347.txt

surfacetensionco00hail_0039.txt

surfacetensionco00hail_0191.txt

surfacetensionco00hail_0190.txt

surfacetensionco00hail_0122.txt

surfacetensionco00hail_0213.txt

surfacetensionco00hail_0232.txt

surfacetensionco00hail_0112.txt

surfacetensionco00hail_0224.txt

surfacetensionco00hail_0368.txt

surfacetensionco00hail_0407.txt

surfacetensionco00hail_0092.txt

surfacetensionco00hail_0013.txt

surfacetensionco00hail_0393.txt

surfacetensionco00hail_0176.txt

surfacetensionco00hail_0398.txt

surfacetensionco00hail_0180.txt

surfacetensionco00hail_0228.txt

surfacetensionco00hail_0103.txt

surfacetensionco00hail_0192.txt

surfacetensionco00hail_0134.txt

surfacetensionco00hail_0111.txt

surfacetensionco00hail_0163.txt

surfacetensionco00hail_0136.txt

surfacetensionco00hail_0385.txt

surfacetensionco00hail_0245.txt

surfacetensionco00hail_0230.txt

surfacetensionco00hail_0054.txt

surfacetensionco00hail_0197.txt

surfacetensionco00hail_0024.txt

surfacetensionco00hail_0294.txt

surfacetensionco00hail_0202.txt

surfacetensionco00hail_0332.txt

surfacetensionco00hail_0229.txt

surfacetensionco00hail_0243.txt

surfacetensionco00hail_0009.txt

surfacetensionco00hail_0303.txt

surfacetensionco00hail_0227.txt

surfacetensionco00hail_0178.txt

surfacetensionco00hail_0310.txt

surfacetensionco00hail_0216.txt

surfacetensionco00hail_0182.txt

surfacetensionco00hail_0259.txt

surfacetensionco00hail_0391.txt

surfacetensionco00hail_0299.txt

surfacetensionco00hail_0353.txt

surfacetensionco00hail_0044.txt

surfacetensionco00hail_0219.txt

surfacetensionco00hail_0405.txt

surfacetensionco00hail_0281.txt

surfacetensionco00hail_0240.txt

surfacetensionco00hail_0223.txt

surfacetensionco00hail_0104.txt

surfacetensionco00hail_0063.txt

surfacetensionco00hail_0001.txt

surfacetensionco00hail_0340.txt

surfacetensionco00hail_0204.txt

surfacetensionco00hail_0031.txt

surfacetensionco00hail_0284.txt

AA00011132_00001.pdf

surfacetensionco00hail_0327.txt

surfacetensionco00hail_0118.txt

surfacetensionco00hail_0258.txt

surfacetensionco00hail_0050.txt

surfacetensionco00hail_0384.txt

surfacetensionco00hail_0248.txt

surfacetensionco00hail_0055.txt

surfacetensionco00hail_0110.txt

surfacetensionco00hail_0377.txt

surfacetensionco00hail_0280.txt

surfacetensionco00hail_0108.txt

surfacetensionco00hail_0275.txt

surfacetensionco00hail_0102.txt

surfacetensionco00hail_0115.txt

surfacetensionco00hail_0206.txt

surfacetensionco00hail_0116.txt

surfacetensionco00hail_0051.txt

surfacetensionco00hail_0231.txt

surfacetensionco00hail_0304.txt

surfacetensionco00hail_0409.txt

surfacetensionco00hail_0215.txt

surfacetensionco00hail_0095.txt

surfacetensionco00hail_0246.txt

surfacetensionco00hail_0083.txt

surfacetensionco00hail_0408.txt

surfacetensionco00hail_0004.txt

surfacetensionco00hail_0256.txt

surfacetensionco00hail_0088.txt

surfacetensionco00hail_0159.txt

surfacetensionco00hail_0207.txt

surfacetensionco00hail_0162.txt

surfacetensionco00hail_0254.txt

surfacetensionco00hail_0026.txt

surfacetensionco00hail_0251.txt

surfacetensionco00hail_0389.txt

surfacetensionco00hail_0241.txt

surfacetensionco00hail_0113.txt

surfacetensionco00hail_0288.txt

surfacetensionco00hail_0214.txt

surfacetensionco00hail_0091.txt

surfacetensionco00hail_0282.txt

surfacetensionco00hail_0080.txt

surfacetensionco00hail_0297.txt

surfacetensionco00hail_0089.txt

surfacetensionco00hail_0141.txt

surfacetensionco00hail_0371.txt

surfacetensionco00hail_0273.txt

surfacetensionco00hail_0362.txt

AA00011132_00001_pdf.txt

surfacetensionco00hail_0233.txt

surfacetensionco00hail_0338.txt

surfacetensionco00hail_0019.txt

surfacetensionco00hail_0096.txt

surfacetensionco00hail_0047.txt

surfacetensionco00hail_0133.txt

surfacetensionco00hail_0271.txt

surfacetensionco00hail_0032.txt

surfacetensionco00hail_0006.txt

surfacetensionco00hail_0400.txt

surfacetensionco00hail_0270.txt

surfacetensionco00hail_0293.txt

surfacetensionco00hail_0025.txt

surfacetensionco00hail_0070.txt

surfacetensionco00hail_0015.txt

surfacetensionco00hail_0042.txt

surfacetensionco00hail_0020.txt

surfacetensionco00hail_0018.txt

surfacetensionco00hail_0066.txt

surfacetensionco00hail_0072.txt

surfacetensionco00hail_0075.txt

surfacetensionco00hail_0174.txt

surfacetensionco00hail_0090.txt

surfacetensionco00hail_0322.txt

surfacetensionco00hail_0005.txt

surfacetensionco00hail_0098.txt

surfacetensionco00hail_0100.txt

surfacetensionco00hail_0300.txt

surfacetensionco00hail_0305.txt

surfacetensionco00hail_0067.txt

surfacetensionco00hail_0363.txt

surfacetensionco00hail_0143.txt

surfacetensionco00hail_0359.txt

surfacetensionco00hail_0287.txt

surfacetensionco00hail_0379.txt

surfacetensionco00hail_0123.txt

surfacetensionco00hail_0217.txt

surfacetensionco00hail_0140.txt

surfacetensionco00hail_0242.txt

surfacetensionco00hail_0386.txt

surfacetensionco00hail_0396.txt

surfacetensionco00hail_0268.txt

surfacetensionco00hail_0374.txt

surfacetensionco00hail_0200.txt

surfacetensionco00hail_0296.txt

surfacetensionco00hail_0007.txt

surfacetensionco00hail_0146.txt

surfacetensionco00hail_0289.txt

surfacetensionco00hail_0394.txt

surfacetensionco00hail_0312.txt

surfacetensionco00hail_0094.txt

surfacetensionco00hail_0127.txt

surfacetensionco00hail_0301.txt

surfacetensionco00hail_0073.txt

surfacetensionco00hail_0382.txt

surfacetensionco00hail_0185.txt

surfacetensionco00hail_0234.txt

surfacetensionco00hail_0392.txt

surfacetensionco00hail_0125.txt

surfacetensionco00hail_0370.txt

surfacetensionco00hail_0274.txt

surfacetensionco00hail_0164.txt

surfacetensionco00hail_0352.txt

surfacetensionco00hail_0188.txt

surfacetensionco00hail_0266.txt

surfacetensionco00hail_0194.txt

surfacetensionco00hail_0126.txt

surfacetensionco00hail_0138.txt

surfacetensionco00hail_0117.txt

surfacetensionco00hail_0167.txt

surfacetensionco00hail_0272.txt

surfacetensionco00hail_0021.txt

surfacetensionco00hail_0260.txt

surfacetensionco00hail_0085.txt

surfacetensionco00hail_0257.txt

surfacetensionco00hail_0165.txt

surfacetensionco00hail_0342.txt

surfacetensionco00hail_0160.txt

surfacetensionco00hail_0158.txt

surfacetensionco00hail_0218.txt

surfacetensionco00hail_0172.txt

surfacetensionco00hail_0144.txt

surfacetensionco00hail_0210.txt

surfacetensionco00hail_0369.txt

surfacetensionco00hail_0351.txt

surfacetensionco00hail_0328.txt

surfacetensionco00hail_0237.txt

surfacetensionco00hail_0183.txt

surfacetensionco00hail_0253.txt

surfacetensionco00hail_0139.txt

surfacetensionco00hail_0010.txt

surfacetensionco00hail_0059.txt

surfacetensionco00hail_0244.txt

surfacetensionco00hail_0211.txt

surfacetensionco00hail_0049.txt

surfacetensionco00hail_0315.txt

surfacetensionco00hail_0285.txt

surfacetensionco00hail_0087.txt

surfacetensionco00hail_0404.txt

surfacetensionco00hail_0314.txt


Full Text










SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS










By

JAMES MITCHELL HAILE


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY







UNIVERSITY OF FLORIDA


1976




SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS
By
JAMES MITCHELL HAILE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976


ACKNOWLEDGEMENTS
It is a pleasure to express my gratitude to those who have freely
contributed to this work through their instruction, guidance, and advice.
Keith Gubbins initiated the research reported herein and enthu
siastically stimulated and supported its development, as well as my own
professional growth. Dr. Gubbins consistently provided an extraordinary
environment for learning, provided a strong example of the scientific
method, and exposed me to numerous knowledgeable scientists and engineers
on both sides of the Atlantic. In addition, he expended considerable
effort in obtaining the financial support, travel and computer funds
which made this work possible.
Bill Streett generously allowed me to spend several months in
his laboratory at the U.S. Military Academy and patiently taught me
molecular dynamics. He obtained the copious amounts of computer time
used in the molecular dynamics work reported here and kept the program
running in my absence. Many of the ideas for presenting the molecular
dynamics results came to light in discussions with Colonel Streett.
Further, I am grateful to Colonel and Mrs. Streett for the hospitality
extended to me during my visits to West Point.
John O'Connell, University of Florida, continually inspired me
through open-ended questioning concerning classical and statistical
thermodynamics, science, engineering, and, most importantly, the
character of life.
ii


Chris Gray, University of Guelph, instructed me in spherical
trigonometry, spherical harmonic expansions, Racah algebra, etc.,
thereby developing in me a healthy respect for the physicist's view
of applied science.
I have benefited greatly from countless discussions with my
colleague Chorng-Horng Twu on various aspects of thermodynamics,
statistical mechanics, numerical methods, and Chinese cooking.
S5ren Toxvaerd, University of Copenhagen, contributed much
valuable advice on the theory and associated calculations for fluid
interfaces. Thanks are also due Dr. Toxvaerd for providing a copy
of his computer program for calculating the vapor-liquid interfacial
density profile for Lennard-Jones fluids.
Peter Egelstaff allowed me to spend several months in the
stimulating atmosphere of the Physics Department at the University
of Guelph. I am grateful to the faculty and staff for their hospi
tality and for the large amount of NOVA 2 computer time made available
to me. I am especially thankful to Dan Litchinsky for useful advice
on NOVA 2 software and to Ross McPherson for timely hardware support
on the NOVA. I am also indebted to Shien-Shion Wang for spending many
hours in teaching me the Monte Carlo method.
Dick Dale and Ron Franklin of the Engineering Information Office,
University of Florida, gave timely and enthusiastic photographic tech
nical assistance in producing the filmed animation of molecular dynamics
simulations. Larry Mixon in the Northeast Regional Data Center, Univer
sity of Florida, provided valuable software support in developing the
filmed animation technique.
iii


I am grateful to Dr. J. W. Dufty for serving on the supervisory
committee. I would also like to remember Dr. T. M. Reed who was an
original member of the committee and who strongly encouraged me in the
initial phases of the research reported here.
P.S.Y. Cheung, T. Keyes, C. G. Gray and R. L. Henderson, W. B. Streett,
and S. Toxvaerd kindly provided manuscripts of their work prior to publica
tion.
Mrs. J. Ojeda, University of Florida, performed the remarkably
excellent typing of the manuscript.
Finally, I am grateful to Tricia who, in addition to all the
\
usual annoyances with which wives of Ph.D. students are plagued,
quietly endured our being separated for the greater part of the last
year and a half of this work.
The three dimensional drawings presented in Chapter 7 were done
on a Gould 5100 electrostatic plotter driven by the IBM 370/165 at the
Northeast Regional Data Center, University of Florida. The associated
software was the SYMVU Computer Graphics Program, Version 1.0, of the
Laboratory for Computer Graphics and Spatial Analysis, Harvard University.
I thank the Petroleum Research Fund (administered by the American
Chemical Society) and the National Science Foundation for financial
support of this study.
iv


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF TABLES ix
LIST OF FIGURES xv
KEY TO SYMBOLS xxii
ABSTRACT xxix
CHAPTERS:
1 INTRODUCTION 1
1.1 Theory of Surface Properties 6
1.2 Computer Simulation Methods 9
1.3 Outline of Dissertation 10
2 THEORY OF SURFACE TENSION 13
2.1 General Expressions for Surface Tension of
Polyatomic Fluids 13
2.2 General First Order Perturbation Theory for
Surface Tension 18
2.3 Perturbation Theory for Surface Tension using
a Pople Reference 23
2.4 Fowler Model Expressions for Perturbation Terms
Y2A Y2B Y3A Y3B in PPle ExPansion 29
2.5 Superficial Excess Internal Energy from the
Pad Perturbation Theory for Surface Tension 32
3 NUMERICAL CALCULATIONS OF SURFACE TENSION 34
F F F F
3.1 Evaluation of Y2A, Y3g> Y3A> and y3b 35
3.2 Surface Tension Calculations for Model Fluids.... 40
v


TABLE OF CONTENTS (Continued)
CHAPTERS: Page
3.3 Calculation of the Superficial Excess Internal
Energy for Model Fluids 46
3.4 Surface Tension Calculations for Real Fluids 48
3.5 Correlation of Surface Tension for Pure Poly
atomic Liquids.... 58
4 VAPOR-LIQUID DENSITY-ORIENTATION PROFILES 72
4.1 First Order Perturbation Theory for p(z.U)^) 72
4.2 Calculations of p(z^co^) for Overlap and Dis
persion 78
5
6
MONTE CARLO SIMULATION OF MOLECULAR FLUIDS ON. A
MINICOMPUTER 94
5.1 Introduction 94
5.2 Monte Carlo Method for Nonspherical Molecules 96
5.3 Description of the Minicomputer System 101
5.4 Monte Carlo Program for the NOVA 102
5.5 Comparison of NOVA Results with Full-Size
Computer Results 107
5.6 Conclusions '. 112
MOLECULAR DYNAMICS METHOD FOR AXIALLY SYMMETRIC
MOLECULES 114
6.1Introduction 114
6.2Expressions for the Force and Torque for Axially
Symmetric Molecules 117
6.3 Method of Solution of the Equations of Motion
and the Molecular Dynamics Algorithm 129
6.4 Evaluation of Pair Correlation Functions 136
6.5Equilibrium Properties from the g^ ^ n/r12^
146


TABLE OF CONTENTS (Continued)
CHAPTERS: Page
7 MOLECULAR DYNAMICS RESULTS 158
7.1 Potential Models 158
7.2 Equilibrium Properties 171
7.3 Spherical Harmonic Coefficients, g (r,) 191
7.4 Angular Pair Correlation Function 215
7.5 Site-Site Pair Correlation Functions 238
7.6 Filmed Animation of Molecular Motions 250
8 CONCLUSIONS 259
8.1 Theory for Surface Tension of Polyatomic Fluids... 259
8.2 Theory for the Interfacial Density-Orientation
Profile of Polyatomic Fluids 261
8.3Computer Simulation of Polyatomic Fluids 262
APPENDICES:
A EXPRESSIONS FOR THE ANGLE AVERAGES IN EQUATIONS (3-4)
TO (3-7) 267
B COORDINATE TRANSFORMATION AND INTEGRATION OVER EULER
ANGLES TO OBTAIN EQUATIONS (2-89) AND (2-90) 270
B.l Choice of Euler Angles 270
B.2 Evaluation of Integral I 273
C MODELS FOR ANISOTROPIC POTENTIALS OF LINEAR MOLECULES... 277
D EXPRESSIONS FOR y^, Y^g. AND y^g FOR VARIOUS
ANISOTROPIC POTENTIALS FOR AXIALLY SYMMETRIC MOLECULES.. 283
E THE INTEGRALS KY(U';nn'n") AND LY(£;nn') 288
F EXPRESSIONS FOR THE SPHERICAL HARMONIC COEFFICIENTS
g£ £ m(r12> IN EQUATION (6-77) 292
G THE INTEGRAL I USED TO CALCULATE THE ANGULAR COR
RELATION PARAMETER G2 FOR QUADRUPOLES 294
vii


TABLE OF CONTENTS (Continued)
APPENDICES: Page
H VALUES FOR THE g (r. J COEFFICIENTS 297
1 zm ^
I VALUES FOR THE J INTEGRALS 328
n
J VALUES OF THE SITE-SITE CORRELATION FUNCTIONS 349
K VALUES OF THE INTEGRAL H^2,6^ 335
LITERATURE CITED 357
BIBLIOGRAPHY 365
BIOGRAPHICAL SKETCH 376
viii


2
3
4
5
6
7
8
9
10
11
12
13
Page
5
47
57
62
100
106
109
150
151
152
153
154
155
LIST OF TABLES
Examples of Macroscopic and Microscopic Interfacial
Properties Related by Equations of the Form (1-1)
Test of the Gibbs-Helmholtz Equation in the Fowler
Model Perturbation Theory for Lennard-Jones plus
Quadrupole Fluids
Potential Parameter Values used in Calculating Surface
Tension.
Values for the Parameters a^ and a2 in the Surface
Tension Correlation of Equation (3-61)
Equilibrium Properties in the Form of Ensemble Averages...
Approximate Number of Monte Carlo Configurations
Generated per Hour on the NOVA 2
Comparison of NOVA and CDC Results for Property Values
of Lennard-Jones + Quadrupole Model Fluid. kT/e = 0.719,
pa3 = 0.80, Q/(ea5)!/2 = 1
Expressions for in Terms of g (r,J
a J^a)^ ^1 zm 1Z
for Various Model Potentials
Expressions
8l1H2(r12)
for the Configurational Energy in Terms
for Various Model Potentials
Expressions for the Pressure in Terms of g^ ^ (r-^)
Various Model Potentials 1.2
of
for
Expressions for the Fowler Model Surface Tension in
Terms of g0 (r,) for Various Model Potentials
Expressions for the Fowler Model Surface Excess Internal
Energy in Terms of g (r ) for Various Model
Potentials 1.2?
Expressions for the Mean Squared Force in Terms of
g (r,) for Various Model Potentials
a/. JO-ID J.Z


Table
Page
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Expressions for the Mean Squared Torque in Terms of
g £ m^r12^ ^r ^ar*ous Mdel Potentials 156
Expressions for the Angular Correlation Functions in
Terms of g^ ^ m^r12^ ^or ^ar^ous Model Potentials 157
Primary Orientations for Pairs of Linear Molecules 162
Property Values of a Lennard-Jones plus Quadrupole
Fluid Obtained in this Work and Compared with those
given by Berne and Harp 177
Equilibrium Properties for Lennard-Jones plus Quadru
pole Fluid at pa3 = 0.85, Q/(ea3)-*-/2 = 2/2 181
Equilibrium Properties for Lennard-Jones plus Quadru
pole Fluid at po3 = .931, Q/(ea5)1/2 = 0.707 182
Equilibrium Properties for Lennard-Jones plus Quadru
pole Fluid at po3 = 0.85, Q/(ea5)1/2 = 1.0 183
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = 0.85,
0 /(ea5)1/2 = 1/2 188
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = .931,
Q/iec5)1/2 = 0.707 189
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = 0.85,
Q/( ea5)1/2 = 1.0 190
Equilibrium Properties for Lennard-Jones plus Overlap
Fluid at pa3 = 0.85, = 0.10 192
Equilibrium Properties for Lennard-Jones plus Overlap
Fluid at pa3 = 0.85, = 0.30 193
Effect of Potential Model and State Condition on the
Fii'st Peak Height of the g Coefficients 201
l zm
Comparison of Molecular Dynamics Results for Equilibrium
Properties with Values Obtained from
£^£^1
J Integrals for Lennard-Jones plus Quadrupole Fluid
with po3 = 0.85, kT/e = 1.277 Q/(ea5)]/2 = 0.5 210
x


Table
Page
28 Comparison of Molecular Dynamics Results for Equilib
rium Properties with Values Obtained from
J Integrals for Lennard-Jones plus Quadrupole
Fluid with pa3 = 0.85, kT/e = 1.294, Q/(ea5)1^ = 1.0... 211
29 Comparison of Molecular Dynamics Results for Equilib
rium Properties with Values Obtained from
l^m
J Integrals for Lennard-Jones plus Overlap Fluid
with po3 = 0.85, kT/e = 1.291, 6 = 0.10 212
30 Comparison of Molecular Dynamics Results for Equilib
rium Properties with Values Obtained from
J Integrals for Lennard-Jones plus Overlap Fluid
with po3 = 0.85, kT/e = 1.287, 6 = 0.30 213
31 Range of Values for Orientational Contributions to
Property Integrands for Quadrupole and Overlap Fluids... 243
Cl Expressions for the Expansion Coefficients E for
Various Interaction Potentials for Linear Molecules 279
C2
C3
C4
D1
D2
D3
D4
El
E2
Expressions for Anisotropic Potential Models in the
Intermolecular Frame of Figure 32 280
Expressions for Anisotropic Potential Models in the
Intermolecular Frame, using y rather than
Derivatives of Various Anisotropic Potentials for
Evaluating the Force and Torque from Equations (6-25)
and (6-34)
281
282
y
Expressions for yj: for Various Anisotropic Potentials
for Linear Molecules 284
F
Expressions for y^ for Multipole Potentials for
Linear Molecules 285
F
Expressions for yB for Various Anisotropic Potentials
for Axially Symmetric Molecules 286
F
Expressions for y^B for Multipole Potentials for
Linear Molecules 287
The Integrals K^(££'£";nn'n") for Pure Fluids 289
The Integrals L^(£;nn') for Pure Fluids 290
xi


Table
Page
E3 The Constants in Equation El 291
G1 The Integral I ^ for Pure Fluids 296
HI Values of goOO^r12^ ~ 400^r12^ fr Fennard-dones plus
Quadrupole Fluid at kT/e = 1.277, pPd = 0.85,
Q/( ea5)1/2 = 0.5 298
H2 Values of g420^r12^ ~ §442^r12^ for the Fluid of
Table HI 300
H3 Values of 8443(1^) g660^r12^ for the Fluid of
Table HI 302
H4 Values of g000^r12^ 4Q0^r12^ fr Lennard-Jones plus
Quadrupole Fluid with kT/e = 0.765, pad = 0.931, and
Q/(eo5 d/2 = 0.707 304
H5 Values of g420^r12^ g442^r12^ for the Fluid of
Table H4 306
H6 Values of g^Cr^) 8660^12^ for the Fluid of
Table H4 308
H7 Values of g000^r12^ g400^r12^ fr Lennard-Jones plus
Quadrupole Fluid with kT/e = 1.294, pad = 0.85, and
Q/(eo5)1/2 = 1.0 310
H8 Values of g420^r12^ g442^r12^ fr t*ie Fdudd f
Table H7 312
H9 Values of 8443(^2) 8640^12^ for the Fluid of
Table H7 314
H10 Values of goo0^r12^ g400^r12^ fr Lennard-Jones plus
Anisotropic Overlap Fluid with kT/e = 1.291, pa^ = 0.85,
and = 0.10 316
Hll Values of g420^r12^ g442^r12^ fr t^ie Fdudd f
Table H10 318
H12 Values of 8443(^2) g660^r12^ for the Fluid of
Table H10 320
H13 Values of gQ00^r12^ ~ g400^r12^ fr Lennard-Jones plus
Anisotropic Overlap Fluid with kT/e = 1.287, pad = 0.85,
and 6 = 0.30 322
xii


Table
Page
H14
Values of g49n(r19)
Table H13........7.
- §442^r12^ for t*ie FFuid oF
324
H15
Values of g,,.(r J
Table H13........7.
- §660^12^ for the Fluid of
11
The Integrals J^
Quadrupole Fluid.
Q/Ceo5)1/2 0.5...
222
- J for a Lennard-Jones plus
pa3n= 0.85, kT/e = 1.277,
12
The Integrals J^^
n
440
- J for the Fluid of Table 11...
n
330
13
441
The Integrals J
n
- J^O for the Fluid of Table 11...
n
331
14
620
The Integrals J^
- J^ 0 for the Fluid of Table 11...
n
332
15
The Integrals J^^
Quadrupole Fluid.
QCea5)1/2 0.707..
222
- J for a Lennard-Jones plus
pa3n= 0.931, kT/e = 0.765,
16 The Integrals J^*3 for the Fluid of Table 15 334
n n
17 The Integrals J^^ J^^ for the Fluid of Table 15 335
n n
18 The Integrals J^ for the Fluid of Table 15 336
n n
000 222
19 The Integrals J J for a Lennard-Jones plus
Quadrupole Flui§. pa^n= 0.85, kT/e = 1.294,
Q/Ceo5)1/2 = 1.0 337
400 440
110 The Integrals J J for the Fluid of Table 19 338
n n
111 The Integrals for the Fluid of Table 19 339
n n
112 The Integrals J^ for the Fluid of Table 19 340
n n
000 222
113 The Integrals J J for a Lennard-Jones plus
Anisotropic Overlap Fluid. pp3 = 0.85, kT/e = 1.291,
6 = 0.10 341
400 440
114 The Integrals J J for the Fluid of Table 113.... 342
n n
115 The Integrals J^^ for the Fluid of Table 113.... 343
n n
116 The Integrals J^ for the Fluid of Table 113.... 344
0 n n
000 222
117 The Integrals J J for a Lennard-Jones plus
Anisotropic Overlap Fluid. p3 = 0.85, kT/e = 1.287,
= 0.30 345
xiii


Table
Page
400 440
118 The Integrals J J for the Fluid of Table 117.... 346
n n
119 The Integrals J441 J6 for the Fluid of Table 117.... 347
n n
120 The Integrals J^ j^O for Fluid Qf Table 117.... 348
n n
J1 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with Z/o = 0.3292, kT/e = 1.277,
pa3 = 0.85, Q/(CO5)1/2 = 0.5 350
J2 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with i/a = 0.2955, kT/e = 0.765,
pa3 = 0.931, Q/(eo5)l/2 = 0.707 351
J3 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with i/o = 0.3292, kT/e = 1.294,
pa3 = 0.85, Q/(ea3)l/2 = i.o 352
J4 Site-Site Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid with £/o = 0.3292, kT/e = 1.287,
pa3 = 0.85, 6 = 0.10 353
J5 Site-Site Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid with i/o 0.3292, kT/e = 1.287,
po3 = 0.85, = 0.30 354
K1 The Integral for pure Fluids 356
xiv


LIST OF FIGURES
Figure Page
1 Relation of Theory, Experiment, and Computer Simula
tion in the Study of Liquids 2
2 Variation of Fluid Density with Position through a
Planar Vapor-Liquid Interface 4
3 Two Possible Values for the Maximum Zj Value for a Pair
of Molecules in the Fowler Model Interface 22
4 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.85,
kT/e = 1.273 42
5 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.45,
kT/e = 2.934 43
6 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Quadrupole Model Potential. pa^ = 0.85,
kT/e = 1.273 44
7 Fowler Model Surface Tension for Fluids of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Various Anisotropic Potentials. pc?3 = 0.85,
kT/e = 1.273 45
8 Corresponding States Plot for Surface Tension of
Simple Liquids 52
9 Surface Tension for CO2 Comparing Perturbation Theory
Calculations with Experimental Values 55
10 Surface Tensions for C2H2 and HBr Comparing Perturba
tion Theory Calculations with Experimental Values 56
11 Test of Surface Tension Correlation for CO2 64
12 Test of Surface Tension Correlation for Acetic Acid 65
13 Test of Surface Tension Correlation for Methanol 66
xv


Figure Page
14 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Polyatomic Liquids 68
15 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Polyatomic Liquids 69
16 Test of Surface Tension Correlation for n-Hexane and
n-Octane 70
17 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Hydrocarbons 71
18 Interfacial Density Profile for Lennard-Jones Fluid 80
19 Interfacial Density-Orientation Surface for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 83
20 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 84
21 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 85
22 Difference in Normal and Tangential Components of Stress
Tensor for Lennard-Jones Fluid 87
23 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, = 0.10 89
24 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = 0.10 90
25 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = -0.10 91
xv i


Figure Page
26 Interfacial Density-Orientation Profiles for a Fluid
of Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = -0.10 92
27 Simplified Schematic Flow Diagram of Fortran Monte
Carlo Program Developed for NOVA 2 2.05
28 Comparison of CDC and NOVA 2 Monte Carlo Results for
the Center-Center Pair Correlation Function for a
Lennard-Jones plus Quadrupole Fluid 210
29 Comparison of CDC and NOVA 2 Monte Carlo Results for
the Angular Pair Correlation Function for a Lennard-
Jones plus Quadrupole Fluid for Molecular Pairs in
the Tee Orientation Ill
30 Methods of Specifying the Orientation of an Axially
Symmetric Molecule 119
31 Orientation Angles for Axially Symmetric Molecules in
an Arbitrary Space Fixed Frame 121
32 Orientation Angles for Axially Symmetric Molecules in
the Intermolecular Frame 122
33 Geometry of a Pair of Diatomic Molecules 142
34 Pair Potential for Lennard-Jones plus Dipole Model
Fluid at Primary Pair Orientations 161
35 Pair Potential for Lennard-Jones plus Quadrupole Model
Fluid at Primary Pair Orientations 165
36 Surface of the Lennard-Jones plus Quadrupole Pair
Potential for the Tee Orientation as a Function of
the Quadrupole Strength 1 166
37 Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. p/(e3)l/2 = 1.0, Q/(ea3)!/2 = 2.75 167
38 Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. y/(ea3)l/2 = 1.75, Q/(ea3)1/2 = 1.0 168
39 Pair Potential for Lennard-Jones plus Anisotropic
Overlap Model Fluid at Primary Pair Orientations 270
xvii


Figure
Page
40
41
42
43
44
45
46
47
48
49
50
51
52
53
Mean-Squared Displacement of Molecular Centers of
Mass for Lennard-Jones plus Quadrupole Fluid
Fluctuation in Temperature for Lennard-Jones plus
Quadrupole Fluid
Fluctuation in the Ratio of Translational to Rotational
Kinetic Energy for Lennard-Jones plus Quadrupole Fluid 176
Effect of Quadrupole Moment on the Center-Center Pair
Correlation Function at pa3 = 0.85 194
Spherical Harmonic Coefficients g for Lennard-Jones
q y^2m
plus Quadrupole Fluid at po = 0.85, kT/e = 1.294,
Q/(ea5)1/2 = 1.0 196
Spherical Harmonic Coefficients g/n for the Fluid of
. . 4 X/ oITl
Figure 44 4 197
Spherical Harmonic Coefficients g,. for the Fluid of
,, 44m
Figure 44 198
Spherical Harmonic Coefficients g for the Fluid of
Figure 44 2 199
Effect of Anisotropic Overlap Parameter on the Center-
Center Pair Correlation Function at pa3 = 0.85 203
Spherical Harmonic Coefficients gn for Lennard-Jones
2.)
plus Anisotropic Overlap Fluid at pa3 = 0.85, kT/e = 1.287,
6 = 0.30 204
Spherical Harmonic Coefficients g. for the Fluid of
Figure 49 ?T 205
Spherical Harmonic Coefficients g,, for the Fluid of
Figure 49 T 206
Spherical Harmonic Coefficients g,. for the Fluid of
Figure 49 .2 207
2
Integrands tg22C)(r) ~ 2g221^ + 28222^r^r and
[g220^ + 4/38221^ + 1/38222^r^r_3 for G2 and Ua
respectively, for Lennard-Jones plus Quadrupole Fluid 214
xviii


jgu
54
55
56
57
58
59
60
61
62
63
64
65
Page
217
218
220
222
223
226
227
228
229
231
232
234
Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for the Tee Orienta
tion (0^ = 90, 02 = 0,

Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for the Cross and
Parallel Orientations (0^ = = 90)
Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for a Skewed Orienta
tion (0^ = @2 = Angular Pair Correlation Function for the Lennard-Jones
plus Anisotropic Overlap Fluid of Figure 49 for the Tee
Orientation (0^ = 90, 2 = 0,

Angular Pair Correlation Function for the Lennard-Jones
plus Anisotropic Overlap Fluid of Figure 49 for the Endon
Orientation (0^ = @2 =

Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0 = 90, = 0,
with Q* = 0.5, T* = 1.277, p* = .85 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0 = 90,

with Q* = 1.0, T* = 1.294, p* = .85 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones .plus Quadrupole Fluid for 0.. = 90, 0 = 90,
with Q* = 0.5, T* = 1.277, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 01 = 90, 0 = 90,
with Q* = 1.0, T* = 1.294, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0, = 45, 0 = 45,
with Q* = 0.5, T* = 1.277, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0, =45, 0 = 45,
with Q* = 1.0, T* = 1.294, p* = .85 7 7
Comparison of Peak Heights in the Angular Pair Correla
tion Function with Well Depths in the Pair Potential
for the Lennard-Jones plus Quadrupole Fluid with
Q/Cea5)1/2 = 1.0
XIX


Figure Page
66 Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Anisotropic Overlap Fluid for
0X 90, 67 Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Anisotropic Overlap Fluid for
= 90,

o
68 Integrand for Internal Energy, r u(12)g(12), for
Lennard-Jones plus Quadrupole Fluids for the Tee
Orientation 239
69 Integrand for Pressure, r^ g(12), for Lennard-
Jones plus Quadrupole Fluids for the Tee Orientation 240
2
70 Integrand for Internal Energy, r u(12)g(12), for Lennard-
Jones plus Anisotropic Overlap Fluids for Parallel
Orientation 241
71Integrand for Pressure, r^ ^ g(12), for Lennard-
Jones plus Anisotropic Overlap Fluids for Parallel
Orientation 242
72 Site-Site Pair Correlation Function for Lennard-Jones
plus Quadrupole Fluids 245
73 Possible Square Packing of Lennard-Jones plus Quadru
pole Molecules for Interpreting gag(r) 247
74 Site-Site Pair Correlation Function for Lennard-Jones
plus Anisotropic Overlap Fluids 249
75Box Representing the Molecular Dynamics System with the
Volume Element Sampled for the Filmed Animation Indicated. 253
76 Initial FCC Lattice Configuration of Lennard-Jones
Molecules in the Volume Element Sampled in the Filmed
Animation 256
77 Frame from the Filmed Animation of Lennard-Jones Mole
cules Corresponding to the Sixth Time-Step in the
Molecular Dynamics Calculation 257
78 Frame from the Filmed Animation of Lennard-Jones Mole
cules Corresponding to the 101st Time-Step in the
Molecular Dynamics Calculation 258
xx


Page
Figure
B1
Rotations
Defining the Euler Angles
40x1
271
B2
Rotations
Interface
in the Triangle 123 in the
to Define Values for z
max
Fowler Model
... 274
xxi


KEY TO SYMBOLS
c(£1
A
. c
A.
x
A(ziz12)
B(ZlZl2)
D
£
i
mn
£*
mn
Fi
F2A,F2B
F3AF3B
n
Roman Upper Case
Helmholtz free energy
Configurational Helmholtz free energy
The ith term in the perturbation expansion for
Helmholtz free energy
Function defined in Equation (4-26)
Function defined in Equation (4-27)
Residual contribution to constant volume heat capacity
Clebsch-Gordan coefficient
Diffusion coefficient
Representation coefficient
Complex conjugate of representation coefficient
Force on molecule 1
Function defined in Equation (2-32)
Functions defined in Equation (2-62) and (2-63),
respectively
Functions defined in Equations (2-77) and (2-78),
respectively
Angular correlation parameter
Integral defined in Equation (3-34)
xxii


I =
I.(r)
^nn' Z
K =
K
KY
L =
L =
LY =
N =
"
P =
P.
l
P(x)
o(i)
Q
*
T
Moment of inertia
Functions defined in Equations (6-65) and (6-69)
Integral defined in Equations (2-40) and (B7)
Integral defined in Equation (Gl)
Integral defined in Equation (3-16)
Integral defined in Equation (6-89)
Coefficient of ellipticity for plane polarized light
Integral defined in Equation (3-25)
Integral defined in Equation (3-18)
Functions defined in Equations (4-30) and (4-31),
respectively
Angular momentum operator
Integral defined in Equation (3-24)
Integral defined in Equation (3-17)
Number of molecules
Avogadro's number
Pressure
t h
The i component of the local polarization vector
Legendre polynomial of order i
Probability of the i^ state occurring, defined in
Equation (5-8)
Quadrupole moment
5 1/2
Reduced quadrupole moment = Q/(ea )
General multipole moment
Temperature
Critical temperature
xxiii


tr
*
T
= Reduced temperature, T/T^
= Reduced temperature, kT/e
U
s
= Superficial excess internal energy
V
= Volume
V
c
= Critical volume
Y£m
*
Y*m :
= Spherical harmonic
= Complex conjugate of
Z
= Configurational integral
Roman Lower Case
a. =
i
= Semiempirical parameters defined in Equations (3-50)
to (3-53)
c =
= Constant defined in Equations (4-21) and (4-22)
c =
= Cosine of angle d). .
ij
c. =
1
= Cosine of angle 0^
c(Y) =
= Cosine of angle y
c(zlz2r12) =
= Interfacial direct correlation function
d =
5 Hard sphere diameter defined in Equation (3-39)
d,dft
5 Maximum allowable step lengths for translational and
rotational motion, respectively, of molecules in one
Monte Carlo generated configuration
£ '
Interfacial pair distribution function for spherical
molecules
f(zl£l2£l3)
Interfacial triplet distribution function for spherical
molecules
g(r12) =
Radial distribution function
xxiv


gc(rl2)
= Center-center pair correlation function
gaB^r12)
= Site-site pair correlation function
g(zlz2r12)
= Interfacial pair correlation function
g(r12wla)2)
= Angular pair correlation function
g(12)
= g(r12J1t02)
gi.1i,2m(rl2)
= Coefficients in the expansion of g(12) in spherical
harmonics of the molecular orientations
fi. =
1
= Unit vector aligned along the axis of linear molecule i
k =
= Boltzmann's constant
i =
= Molecular bond length between atoms
m :
= Molecular mass
n :
= Index of refraction
n =
= Constant, values given in Equations (4-21) and (4-22)
n =
= Exponent of r in repulsive part of Mie potential,
Equation (3-32)
n =
s
= Power of r in various model potentials
PN
= Normal component of the stress tensor
pt =
= Tangential component of the stress tensor
r. =
i
= Vector location of center of mass of molecule i
-12
= Vector separation of centers of mass of molecules 1 and 2
rl2
*
r12
: Magnitude of r^2
Reduced distance, r^^/o
r =
m
: Value of r where pair potential is a minimum
s. =
i
Sine of angle 0^
t =
Time
u(ri2) =
Spherically symmetric intermolecular pair potential
XXV


u(r12lu2)
= Orientation dependent intermolecular pair potential
u (12)
= u(r12c10)2)
ao{z12>
= Isotropic, reference contribution to u(12)
ua(12)
"*(r12)
= Anisotropic contribution to u(12)
= Reduced potential, u(r^2)/e
u
s
= Superficial excess internal energy per unit of surface area
V
= Molecular translational velocity
X12
= x-component of r^2
y12 '
= y-component of r^2
y(ri2) ;
= Function defined in Equation (3-38)
Z1 :
= z-coordinate location of molecule 1
Z12 :
= z-component of r^2
z =
max
*
zi =
= Maximum value of z-, defined in Equations (2-42) and (B6)
= Reduced coordinate, z^/o
5 =
Roman Script
= Interfacial area
U =
= Total intermolecular potential
E =
= Coefficients in the spherical harmonic expansion of the
anisotropic pair potential
Greek Upper Case
r. =
i
: Surface adsorption of component i
A =
Triplet of indices, for example
52 =
Volume in angle space
xxvi


2 = Angular velocity
= The l^k component of Q_
a
a.
i
a. ,B.
3
Y
Y
Y
F
*
Y
6
. .
ij
e
0(x)
0.
Greek Lower Case
= Adjustable parameter in Equations (4-26) and (4-27)
= Interior angle in the triangle formed by r^2 ri3 r23
at molecule i
= Constants in the predictor-corrector algorithm; values
given in Equations (6-45) and (6-46), respectively
= Azimuthal and polar angles, respectively, for molecular
orientation in the space fixed frame
= 1/kT
= Angle between the axes of a pair of linear molecules
= Surface tension
= Fowler model surface tension
~ The ith term in the perturbation expansion for surface
tension
2
= Surface tension reduced by potential parameters, yO /e
= Surface tension reduced by critical constants,
Y(Vc/NA)2/3(kTc)"1
= Dimensionless anisotropic overlap parameter
= Kronecker delta
= Interinolecular potential energy parameter
= Unit step function
- Polar angle for molecular orientation in the inter-
molecular frame
xxvii


Angular displacement vector
Polar angle for r^ in spherical coordinates
Dimensionless anisotropic polarizability
Wavelength of incident light in light scattering
experiments
Perturbation parameter
Dipole moment
3 1/2
Reduced dipole moment, y/(eo )
Random numbers
Fluid number density
Interfacial number density profile
Interfacial number density-orientation profile
First order term in perturbation expansion for p(z^OJ^)
Intermolecular potential distance parameter
Standard deviation of property i
Torque on molecule 1
Azimuthal angle for molecular orientation in the inter
molecular frame
Azimuthal angle for in spherical coordinates
Function defined in Equation (3-19)
Set of variables specifying the orientation of molecule i
xxviii


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS
By
James Mitchell Haile
December, 1976
Chairman: Keith E. Gubbins
Major Department: Chemical Engineering
A third order thermodynamic perturbation theory for the surface
tension of polyatomic liquids has been developed using a Pople reference
fluid. The expansion includes terms containing the unknown pair and
triplet interfacial correlation functions, gQand g^Cz^r^X23)
for the reference fluid. These are removed by using the Fowler approxima
tion for the interface. The resulting theory has been tested against
Monte Carlo results for the Fowler model surface tension of a fluid
whose molecules interct with a Lennard-Jones plus dipole potential.
The behavior of the theory compared with similation parallels that
of bulk fluid properties; namely, the second order theory agrees with
the Monte Carlo results for small values of the dipole moment up to
3 1/2
y/(ea ) 0.6. For larger dipole strengths neither the second nor
third order theories agree with the computer simulation results. How
ever, when the third order expansion is recast in the form of a simple
[1,2] Pad approximant, the theory agrees with the Monte Carlo results
up to dipole moments as large as u/(ecr ) = 1.75. This Pad! theory
has been used to calculate the surface tension of pure dipolar and
xx ix


quadrupolar liquids. The theory agrees with experimental values of
surface tension in the neighborhood of the triple point; however, the
Fowler model does not give the correct temperature dependence of the
surface tension. The theory has also been used as a basis for develop
ing useful correlations of surface tensions of pure polyatomic liquids.
A first order perturbation theory has been developed for the
interfacial density-orientation profile p(z^o^) for polyatomic fluids.
Upon introduction of a Pople reference, the first order term
vanishes for multipolar anisotropies, but does not vanish for anisotropic
overlap or dispersion potential models. Calculations of p(z^aj^) for
axially symmetric molecules interacting with each of the latter potentials
have been performed, using a Lennard-Jones reference fluid and the inter
facial pair correlation function model used by Toxvaerd. The calculations
indicate that the axially symmetric molecules have preferred orientations
in the interfacial region.
A method has been devised for using a NOVA 2 minicomputer with
32K words of core and external disc storage to perform Monte Carlo
simulations of 128 nonspherical molecules. The simulations generally
require several days of continuous calculation; however, several
equilibrium property values and values for the angular pair correlation
function gCr^co^t^) at five to seven specific orientations may be obtained
at a fraction of the cost of doing the calculations on a full size machine.
Results from the minicomputer compare within the statistical precision
with results previously obtained on CDC and IBM machines.
The method of molecular dynamics has been used to study systems
of 256 molecules interacting with Lennard-Jones plus quadrupole and
xxx


Lennard-Jones plus anisotropic overlap potentials. The equilibrium
properties determined include configurational internal energy,
pressure, Fowler model surface tension, Fowler model surface excess
internal energy, mean squared force, and mean squared torque. In
addition, the coefficients g^ ^ m^ri2^ :''n an exPansin fr the
angular pair correlation function gCr^^ci^) in terms of products
of spherical harmonics of the molecular orientations are determined.
Site-site pair correlation functions g D(r) are also found. Relations
dp
are developed between the gg (r ) coefficients and the above listed
A/
equilibrium properties for several anisotropic potential models. Study
is made of orientational structure in quadrupolar and overlap fluids
via g(r^2W-^W2^ as obtained from the recombined spherical harmonic
expansion. A method of producing filmed animations of molecular
motions from molecular dynamics data is described.
xxxi


CHAPTER 1
INTRODUCTION
The development of rigorous methods for prediction of properties
of liquids must come from an understanding of how molecules are dis
tributed and interact with one another. Such fundamental understanding
of liquids may be pursued from three directions: theory, experiment,
and computer simulation. Molecular theory is the domain of statistical
mechanics; molecular level experimentation usually involves light,
x-ray, or neutron scattering, while computer simulation embodies the
Monte Carlo and molecular dynamics techniques. Computer simulation
adds a powerful new dimension to the study of matter which in no way
supplants the other two methods of study. As indicated in Figure 1,
simulation complements both theory and experiment. Thus, comparison
of simulation with laboratory experiment provides information about
the molecular interactions in the liquid. On the other hand, com
parison of simulation with theory provides a stringent test of the
theory.
In spite of current gains being made in the study of liquids,
fundamental understanding of fluid-fluid interfacial phenomena has
been slow to develop. The difficulty with study of interfacial
properties arises from the inhomogeneity, nonuniformity and anisotropy
of the interfacial region between homogeneous fluid phases. In general,
microscopic fluid properties vary through the interface from their
values in one bulk phase to their values in the other bulk phase, as
1


THEORY
EXPERIMENT
COMPUTER
SIMULATION
Figure 1. Relation of Theory, Experiment, and Computer
Simulation in the Study of Liquids


3
in Figure 2. These properties include density, refractive index,
dielectric constant, etc. The oft measured (macroscopic) properties
of interfaces are, in many cases, related to integrals over the
microscopic properties [1]:
X
OO
f
M(z)]dz
(1-1)
where X represents a macroscopic property, M is a microscopic property,
subscript B indicates, usually, a bulk phase value and k is a pro
portionality constant. Examples of specific properties having the
general form of (1-1) are given in Table 1. Equation (1-1) indicates
that prediction of macroscopic interfacial properties and comparison
with experiment is not a completely satisfactory test of theory. Of
more value is understanding of the microscopic properties and their
relations to macroscopic properties of interest. In the case of
theoretical study, such an approach must lie in statistical mechanics.
The prediction of macroscopic property values in a variety of
situations, e.g., over a large portion of the phase diagram and in
multicomponent systems, remains a vital engineering concern. In the
case of interfacial problems, the macroscopic property of interest is
the interfacial tension. The ability to predict interfacial tensions
is required in the equipment and process design and operation of numerous
fluid-fluid contacting operations, such as distillation, solvent extrac
tion, liquid membrane separation techniques, and tertiary oil recovery.
\
Interfacial tension is also important in understanding various biological
processes such as blood oxygenation and eye lubrication. Of particular


4
z
Figure 2.
Variation of Fluid Density with Position through
a Planar Vapor-Liquid Interface


5
TABLE 1
Examples of Macroscopic and Microscopic
Interfacial Properties Related by Equations
of the Form (1-1)
X = k
[M M(z)]dz
D
r.
i
[Pi(z) p]dz +
[pi(z) p^]dz
(1-2)
Y =
[pN PT(z)]dz
(1-3)
K =
X + D
00
P (z)
P (z)
z
X
P ()
p ()
CO
z
X
dz
(1-4)
t
Reference [2]
r.
i
pi
PI
pi
Y
N
References [3,4]'
surface adsorption of component i
density of component i
bulk vapor phase density
bulk liquid phase density
surface tension
normal component of stress tensor
tangential component of stress tensor
PT(z)
K
P.
i
n
A
= coefficient of ellipticity for light plane polarized at
45 to the plane of incidence, when incident at Brewster's
angle
= ith component of local polarization vector
= bulk phase index of refraction
= wavelength of incident beam


6
importance is the determination of how molecular characteristics of
the fluid contribute to the interfacial tension. An important goal
in design considerations is the improved efficiency of fluid-fluid
contacting operations by modifying the system (e.g., by introduction
of additives) in order to lower the interfacial tension. This, in
turn, leads to consideration of how molecules absorb and orient
themselves in the interfacial region.
1.1 Theory of Surface Properties
Readily used methods currently available for predicting inter
facial tension have been reviewed [5,6]. These methods include
corresponding states approaches, techniques based on regular solution
theory and scaled particle theory, and more empirical methods. These
methods are generally applicable to spherical and nearly spherical
molecules, nonpolar polyatomics and their mixtures. These methods
fail to accurately predict interfacial tensions for strongly polar,
quadrupolar or associating liquids.
Almost all rigorous theoretical work on interfacial phenomena
has been for fluids with spherically symmetric molecular interactions.
Two rigorous relations have been developed for surface tension. The
Kirkwood-Buff equation relates the surface tension y to the inter
facial density profile p(z^) and the interfacial pair correlation
function g(z1z2r12) [2,7]
Y = T
dz,
d12
du(r12)
dr
12
P S
*12-
J12
12
(1-5)


7
The Kirkwood-Buff equation is valid for spherical molecules and
assumes the intermolecular potential to be a sum of pair potentials.
This relation is intractable as it stands because of the unknown
function gCz^z^r^^)- Consequently, the Kirkwood-Buff relation has
been studied using various simplifying models for the interfacial
region. The simplest such model is due to Fowler [8] and assumes
an abrupt transition from liquid to vapor phase. Further, the vapor
phase density is assumed to be negligible compared to the liquid
density. Thus, the model may be expressed as:
p(z1)p(z2) g(z1z2ri2^
0(-z1)0(-z2) pLgL(r12)
(1-6)
where 0 is the unit step function,
0(x)
1 if x ^ 0
0 if x < 0
(1-7)
subscript L indicates a bulk liquid property and the negative z
direction is into the liquid. The resulting Fowler-Kirkwood-Buff
expression for surface tension is:
0
4 du(r12} ,
rl2 dr12 gL r12
(1-8)
As could be expected, the Fowler-Kirkwood-Buff theory works well near
the fluid triple point but gives increasing errors in surface tension
as the temperature is raised towards the critical point. Recent


8
evidence indicates that the good agreement at the triple point is
due to cancellation of errors [6].
The second rigorous relation for surface tension is a
generalized van der Waals equation which gives the surface tension
in terms of the density gradient dp(z^)/dz^ and the interfacial
direct correlation function cCz^^r^) [9]:
CO oo oo

f
dz..
dz
1
2
CO
-CO
dx
12
dy
dp(z^) dp(z2)
12 dz.
dz,
c(z1z2r12)(x^2+ y^2) (1-9)
This relation is more general than the Kirkwood-Buff expression since
no assumption of pairwise additive potential is made in its derivation.
It has the further advantage that the direct correlation function c(r^2)
is generally of shorter range than the pair correlation function g(r^2).
The generalized van der Waals equation (1-9) has not been as thoroughly
studied as the older Kirkwood-Buff formula.
The Kirkwood-Buff equation has recently been generalized to
nonspherical molecules [10]. From both a practical standpoint and a
desire to gain understanding of interfacial phenomena, there is strong
need for using this new relation as a basis for developing predictive
methods for surface tension of polar and quadrupolar systems.
Evaluation of the interfacial density profile p(z^) is required
in order to determine the surface tension from the rigorous expressions
(1-5) and (1-9). Study of the interfacial density profile is also
of interest per se. Nearly all studies thus far have been for
spherical molecules. A variety of approaches have been used: a)
van der Waals theory [11], b) constant chemical potential through


9
the interface [12,13], c) first Born-Green-Yvon equation [14], d)
constant normal pressure through the interface [15,16], e) minimiza
tion of system free energy obtained by perturbation theory [17,18],
and f) computer simulation [19,20,21]. Determination of surface
tension for nonspherical molecules will require knowledge of the
interfacial density-orientation profile p(z^U)^), where oo^ is a set
of Euler angles specifying the orientation of molecule 1. Further,
there is considerable interest in determining how modification of
molecular orientation affects 'the surface tension.
1.2 Computer Simulation Methods
In the computer simulation approach to the study of liquids
either the Monte Carlo or molecular dynamics technique may be used.
These methods provide detailed information on equilibrium and time
dependent properties and on liquid structure for molecules interacting
with known force laws. In addition to use of simulation results as
a standard for evaluating theories of liquids, there is now serious
interest in using simulation as a vehicle for evolving and evaluating
realistic potential models for liquids. Much of the simulation work
has been performed for simple, spherical molecules. Only in the past
few years has simulation of nonspherical molecules been undertaken.
There is need of exploiting these simulation methods as fully as
possible since they provide a wealth of detailed information for
study.
Much of the effort in the history of computer simulation has
been expended in developing the methods, demonstrating their usefulness
and potential applicability and, in general, making simulation a


10
respectable research tool. There is currently a need to explore
methods for improving the efficiency of these calculations with a
view towards conserving computer resources and making the simulation
techniques accessible to more users. One possible approach is the
use of a minicomputer for performing simulations of liquids.
1.3 Outline of Dissertation
This work is divided into two parts. Part I includes
Chapters 2-4 and is concerned with the theory of vapor-liquid
interfaces for nonspherical molecules. In Chapter 2 a statistical
mechanical perturbation theory is developed for the surface tension
of polyatomic fluids. The general first order perturbation term is
obtained and the second and third order terms are found when a Pople
reference is used. Specific relations for the perturbation terms
are given for several anisotropic potential models: dipole, quadrupole,
overlap and dispersion. The corresponding perturbation terms are also
derived when the Fowler" approximation for the interface is introduced.
In Chapter 3 numerical calculations are reported for surface
tension using the perturbation theory in the Fowler model. The results
are compared with experiment and computer simulation. Semiempirical
methods based on perturbation theory are explored for correlating
surface tensions of polyatomic and polar substances.
Chapter 4 develops a perturbation theory for determining the
vapor-liquid interfacial density-orientation profile of fluids with
anisotropic potentials. Calculations are presented for overlap and
dispersion model interactions. The calculations predict that these
nonspherical molecules exhibit preferred orientations in the inter
facial region.


11
Part II of this work describes computer simulation studies
of linear molecules. Chapter 5 reports development of a method for
performing Monte Carlo calculations for linear molecules on a NOVA 2
minicomputer.
Chapter 6 describes the molecular dynamics method for linear
molecules, including: derivation of expressions for efficient evalua
tion of the force and torque exerted on a molecule due to various
anisotropic potential interactions, the method used to solve Newton's
equations of motion and the molecular dynamics algorithm for linear
molecules, evaluation of the coefficients in a spherical harmonic
expansion of the angular pair correlation function, and development
of relations between these coefficients and various fluid equilibrium
properties.
Chapter 7 reports the results of the molecular dynamics
calculations for equilibrium property values obtained for Lennard-
Jones plus quadrupole and Lennard-Jones plus anisotropic overlap
fluids. Comparisons of the equilibrium property values are made
with perturbation theory predictions in the case of the quadrupole
fluid calculations. Study is made of the local structure in these
fluids using the molecular dynamics determined angular pair cor
relation functions. A method for producing filmed animations of
molecular motions from molecular dynamics calculations is presented.
Chapter 8 draws conclusions from this study.


PART I
THEORETICAL STUDY OF FLUID INTERFACES


CHAPTER 2
THEORY OF SURFACE TENSION
2.1 General Expressions for Surface Tension of Polyatomic Fluids
There are two rigorous expressions for the surface tension of
polyatomic fluids. One is the generalized van der Waals equation (1-9).
The second is a generalization of the Kirkwood-Buff expression (1-5)
which has been previously derived [10,22]. In this section the deriva
tion for the general Kirkwood-Buff equation is summarized and its
simplified form obtained when the Fowler approximation is made.
2.1.1 Generalized Kirkwood-Buff Formula
Consider a planar interface between vapor and liquid phases with
a coordinate system oriented such that the xy plane is in the plane of
the interface and the +z direction points into the vapor. The two phase
system has N molecules .in volume V at temperature T and interfacial area
S. Thermodynamically, the surface tension y is related to the Helmholtz
free energy A of the two phase system by [23]:
Y =
9A
9 S
NVT
(2-1)
c
Since only the configurational part of the free energy A depends on
the interfacial area, (2-1) may be written as:
Y = kT
L £n Z
NVT
(2-2)


14
where
A = kT £n Z
(2-3)
is the statistical mechanical definition of the configurational
Helmholtz free energy, k = Boltzmann's constant, and Z is the con
figurational integral. For nonspherical molecules:
Z =
r N N -3(rNwN)
dr dw e
(2-4)
where 3 = 1/kT and U is the full intermolecular potential. For non
spherical molecules U depends on the orientations ) of the molecules,
as well as the positions of their centers of mass _r. The orientations
0) are usually specified by a set of Euler angles OJ = body-fixed reference frame on the molecule and a space-fixed frame
located external to the system. Substituting (2-4) into (2-2):
Y = -
kT
3_

f W \
, N N -3U(r w )
dr dto e -
3 S
.
(2-5)
NVT
The differentiation in (2-5) cannot be done immediately since the
N
integration limits on the integrals over r_ depend on 5. We, therefore,
follow Green [24] and change _r variables using the transformation:
cl/2 ,
x = b x
Ql/2 ,
y = S y
V ,
2 = s 2
(2-6)
Performing the differentiation in (2-5) and changing back to the old
variables gives:


15

f MM
, N N -gU(r u )
dr di) e -
(~lu N N.
8u(r w )

.35
NVT
(2-7)
Assuming the potential to be pairwise additive,
(2-7) becomes
U = j i u (r .co ,io.)
ij i j
i (2-8)
Y
d£.ld£2djldJ2
f (£1£2JiJ2'>
8u (12)]
l 95 J
NVT
(2-9)
where u(12) = uCr^^j^^ and tde de^inition of the angular pair dis
tribution function has been used:
f (£1£2jij2)
N(N-l)
Z
d£3'
diNd3'
dor
ant N Nn
-pu(r to )
e
(2-10)
Integrating (2-9) over x^, and transforming r\, to £^2 §dves
dzl d12 dtaidJ2
f ^Z1-12C1 3u(12)
9S
NVT
(2-11)
P 3u(12)
as
can be evaluated for nonspherical molecules by considering:
3u(12) 9-12 3u(12)
as 6 as ar12
1 3u(12) 3u(12)
= 2 [ri2 177 3z12 -alj
(2-12)
(2-13)
Hence, (2-11) becomes:


16
OO
1
Y = 4
dzl
d£1? dc1dco2 f (z1£1 ?oj1c9)
3u(12) 3u(12)
12 3r10 ~i2 3z,
-OO
12 12
Defining the angle average by:
(2-14)
<>
lU2 tt2
( )dJ1do2
where
fi =
dw
(2-15)
(2-16)
Equation (2-14) can be written:
Y = 7-
dz,
d12
3u(12)
3r
- 3z
3u(12)
12
12
]>
8z12 V2
(2-17)
Equation (2-17) is one form of the general Kirkwood-Buff equation.
Another form may be obtained by transforming (2-17) to spherical
coordinates. Since [25]
3u(12)
3z
12
= cos 0
X12,y12
12
3u(12)
3r
12
sin 0
12
012^12
12
3u(12)
30
12
r12^12
then
(2-18)
3u(12)
r, ^ 3z
3u(12)
12 3r
12
12 3z
12
2P2(cos ei2)r12 9r
+ 3 sin 012 cos 012 9Q
12
3u(12)
12
(2-19)
where P2(x) is the second order Legendre polynomial, P2(x) = y (3x 1)
Substituting (2-19) into (2-17) gives:


17
Y = YA +
(2-20)
ft
2
dz,
dri2 P2(cos 012) r12
V -
3ft
B 4
dz.
d12 sIn 012 cos S12 <£(zl12lw2) <2'22)
Equations (2-20,21,22) give the general Kirkwood-Buff equation. The
equation applies to general shaped molecules, the only assumption being
a pairwise additive potential.
In the case of spherical molecules, the potential u(12) goes to
u(r,) so that the derivative in the Y term vanishes and Y, reduces to
12 B A
the Kirkwood-Buff equation (1-5).
2.1.2 Fowler-Kirkwood-Buff Equation for Nonspherical Molecules
The general expression for the surface tension (2-20) is in
tractable as it stands due to the unknown distribution function
f (z^r^co (02)' Varius simplifying assumptions can be made for f to
enable calculations to be performed. The simplest assumption is that
due to Fowler [8]:
2
f(z1r12(a10)2) = 0(-Zl) 0(-z2) gL(12) (2-23)
where 0 is the unit step function as in (1-7), p is the bulk liquid
1j
density, and g is the bulk liquid angular pair correlation function.
Li
Introducing the Fowler model (2-23) into (2-21) allows the integrations
over z^ and co^2 to be performed giving:


18
ttpt
Ya =
dr
12 12
<8,(12)
3u(12).
9r.oo,co
12 12
(2-24)
where the superscript F indicates Fowler model. Similarly, (2-22)
reduces to:
YB 32
3tr2 2
PL
3 9u(12)
drl2 ri2 gL(12) 90. w.w
(2-25)
0
12
12
Further, the Yr, term can be shown to vanish by symmetry arguments. One
D
such argument is that 9u(12)/90^2 is proportional to the 0^ component
of the force on molecule 2 due to molecule 1. This should vanish by
symmetry when averaged over U)^ and Thus, the Fowler-Kirkwood-Buff
equation for nonspherical molecules is:
TTp
Y =
, 4 9u(12)^
dr. r.
12 12
0
3r12 l2
(2-26)
For the special case of spherical molecules, (2-26) reduces to
the usual FKB expression (1-8).
2.2 General First Order Perturbation Theory
for Surface Tension
2.2.1 Rigorous First Order Term
The difficulty with using the general Kirkwood-Buff expression
(2-20) lies with the, in general, unknown interfacial distribution
function f (z^_r^2u)^lJJ2^ This problem may be avoided by use of per
turbation theory, if the interfacial pair distribution function is
available (or may be approximated) for some reference substance.
Perturbation theory for fluid properties may be developed by con
sidering the anisotropic pair potential u(12) to be the sum of a
(known) reference term uq and a perturbing term u^:


19
u(12;A) = uq(12) + Au (12) (2-27)
where A is a perturbation parameter such that when A = 0, (2-27) gives
the reference potential and when A = 1, (2-27) gives the full potential.
Expanding the two-phase system Helmholtz free energy (2-3) in powers of
A and setting A = 1 gives:
A = A + A. + '
o 1
(2-28)
where
A1 2
dld2
3u(12;A)
9A
(2-29)
A=0
and fQ is the interfacial angular pair distribution function for the
reference system.
The corresponding expansion for surface tension is obtained by
applying (2-1) to (2-28):
Y Y0 + Yi +
where = ~2
9_
9S
-ld-2 Fl(zl£l2)
NVT
and V2!^) E 9u(12;A)
9A
A=0
(2-30)
(2-31)
(2-32)
Applying Green's method as in section (2.1), integrating over x^ and
y^, and transforming _r^ to _r 2 (2-31) becomes:


20
^1 2
dz.
d£l2 S
8F1(z112)
3S
NVT
(2-33)
S SF^ (z^r\^2)/3S may be evaluated by considering:
9Fi(zili?) 3jrn 9 3F (z.,r..9) 3F (z r ) 3z
5^ = s _^£ .KJ- + s i__L_L£__i (2-34)
9 S
3 S
812
3z^
9S
12 3F1(Z1^12)
912
3 9F1(z1-12)
2 Z12 9z12
(2-35)
Thus, (2-33) becomes:
8Fl(z!12>
3z,
dz.
8F1(z1-12) 3 9F1(z1-12)
drl2 Z1 3z, + 2 Z12 3z12
3F(z,r._)
-12
lv 1-12;
912
(2-36)
The second two terms in (2-36) can be integrated by parts and shown
to cancel, leaving:
Y1 = '
d12
dzl Z1
3u(12;A)
3A
A=0
9fo(zA2V2)
3z,
lu2
(2-37)
Equation (2-37) is valid for spherical or nonspherical reference fluids
the only assumption has been a pairwise additive potential. Note that
if the reference fluid is taken to be a Pople reference, defined by
(2-48) below, then y^ vanishes.


21
2.2.2 First Order Term in the Fowler Model
To make (2-37) amenable to calculation, the Fowler approximation
(2-23) may be introduced:
^1 2
2(2
d12
dzl Z1 3i7 9(-zl) 6(-z2)<:
3u(12;A)
3A
n8L^12^>(ja1a)
A=0 12
(2-38)
Substituting (2-15) and changing the order of integration:
2(2
d12
dco^ da>2
3u(12;A)
3A
(2-39)
A=0
where I =
z
dzl Z1 3¡~ 0(_Z1) 0(Z2) goL(12)
(2-40)
The integral I may be evaluated by parts to give:
I
z
g T (12) z
oL max
(2-41)
where
max
-rl2 cos 012
lf 012 > 2
if 012 < I
(2-42)
and 0^2 is the spherical coordinate polar angle for _r^> as shown in
Figure 3. Equation (2-39) becomes, therefore:
2
Yl = +
2(2
dr
12
d(j0^d(jJ2
3u(12;A)
3A
A=0
g T (12) z
oL max
(2-43)


22
z
z
a. 12 < 1T^2
b. 012 > tt/2
Figure 3. Two Possible Values for the Maximum z^ Value (zmax)
for a Pair of Molecules in the Fowler Model Interface


23
Reintroducing the angle average (2-15) and noting that g(12) in the
bulk liquid is independent of (2-43) can be written as:
Performing the integration over U)^ gives:
(2-45)
Equation (2-45) is the general result for the first order perturbation
term in the Fowler approximation.
2.3 Perturbation Theory for Surface Tension
using a Pople Reference
When the general expansion for surface tension (2-30) is in
creased to higher order, the second order term is found to include a
term containing the reference four-body distribution function. Higher
order terms in the expansion contain even higher order multibody terms.
These complicated terms can be made to vanish up to at least the second
order term in the expansion by using the isotropic reference potential
first suggested by Pople [26]:
u0(ri2) =
(2-46)
With this choice of reference, (2-27) becomes:
u(12;A) = u (r ) + Au (12)
o 1/ a
(2-47)


24
(2-46) and (2-47) together give the simplification:
9u(12;A)
9A
>
A=0 u
i2
= = 0
a Ulw2
Then (2-29) and (2-37) have:
(2-48)
Ax = Y1 = 0 (2-49)
and the second and third order terms (A^A^.y^y^) simplify. Thus, in
the Pople expansion, (2-28) and (2-30) become, to third order:
AC = + A2 + A3 (2-50)
Y = Yo + Y2 + Y3 (2-51)
2.3.1 Derivation of Second Order Terms, y01 and y
Zn Z D
The second order term in (2-50) for the bulk phase liquid is [27]:
^ldr2 (2-52)
where g^(12) is the first order term in a perturbation theory expansion
of the angular pair correlation function. When the expansion is about
a Pople reference, g^(12) is given by [28]:
§1(12) = Bua(12) go(r12) 6P
dr [ + ] x
3 a 03^ a (jl)^
go(r12r13r23)
(2-53)


25
where §0(r^2rl3r23^ '*'s t*ie triplet correlation function for the reference.
When the anisotropic potential contains only spherical harmonics of order
it ^ 0, such as multipoles, then the angle averages in the second term in
(2-53) vanish. Such potential models as anisotropic overlap and dispersion
contain L = 0 spherical harmonics, in which case the second term must be
included.
Equation (2-52) may be written, therefore, as
A2 A2A + A2B
(2-54)
where
P2B
2 A
dld2 go(r12) (2-55)
Bp
3 r
2B
dr dr dr, g (r10riarQ)
1 2 3 &o 12 13 23 a a a a W1W2U3
(2-56)
Since _r^, and _r^ are each integrated over in (2-56), the indices are
dummy indices and (2-56) may, hence, be written as:
PB'
2B
dr dr dr g (r r r_) (2-57)
1 2 3 o 12 13 23 a a
For a fluid nonuniform in the z-direction, (2-55) and (2-57)
generalize to:
2A 4
Lld-2 Po(zl)Po(z2> 0)lM2
(2-58)


26
2B 2
dr dr dr p (z,.)p (z)p (z) g (z..r r,^)
1 2 3 o 1 o 2 o 3 o 11213 a a 00^0)2^2
(2-59)
The corresponding terms in the expansion for surface tension (2-51)
may be found by applying (2-1) to (2-58) and (2-59):
Y
2A 4
2B 2
3_
as
a_
as
d-ld-2 F2A(z1£12)
- NVT
dr1dr2dr3 F^z^)
(2-60)
(2-61)
NVT
where
F2A Po(zl)Po(z2) go(zl12) Ul(02
(2-62)
F2B E Po(z1)Po(z2)Po(z3) 8o(zl£l213) Equations (2-60) and (2-61) have the same form as (2-31); thus they may
be evaluated in a similar manner to give, analogous to (2-37):
r -!
2A 4
dr
12
dz, z
3F2A(Z1-El2)
1 1 3z,
(2-64)
Y
2B 2
d12d13
dz, z
9F2B(Z1-12^
1 1 az,
(2-65)
Substituting (2-62) and (2-63) into (2-64) and (2-65), respectively,
and using the relations:


27
f
o
(zlil2> '
p0(zi)p0(z2^ go^Zl-l2^
(2-66)
fo(zl-12-13) Po(z1)Po(z2)Po(z3) go(zl-12-13)
(2-67)
gives,
2A
3
4
dr
12 a w to2
dz, z
3fo(zlil2)
1 1 3z,
(2-68)
2B
dr,dr10
-12 -13 a a oi
dz^ z^
3fo(zlll2ll3)
3z,
(2-69)
The derivation of (2-68) and (2-69) only assumes a pairwise
additive potential and use of a Pople reference fluid. If the aniso
tropic potential contains only £ ^ 0 spherical harmonics, then 2$
vanishes because of (2-53).
2.3.2 Derivation of Third Order Terms, Y_. and Y
j_i-3 A L3 B
The third order term in (2-50) for the bulk phase liquid is:
A
3
d£]dJr2
g9 (12)>
0)^2
(2-70)
where g2 is the second order term in the perturbation theory expansion
for g(12) [28J. If only anisotropic potentials containing £ / 0 spherical
harmonics are considered, g2(12) simplifies. Using arguments anologous
to those for simplifying the AD term in (2-56), and generalizing to a
fluid which is nonuniform in the z-direction, (2-70) becomes (for £ ^ 0
harmonics):


28
A3 A3A + A3B
A3A
B
12 J
dridr0 Pn^l^o^2?) g_(z,r J
I 2 o 1 o 2 o 112 a W1W2
A3B 6
dld2d3 P0(zl)po(z2)po(z3) go(z1^12^13)
M
a a a
(2-71)
(2-72)
(2-73)
Applying (2-1) to (2-71) gives:
Y3 Y3A + Y3B
(2-74)
where
Y3A 12
3B
r
6
9
9S

9
9S

dridr2 F3A(Zl£12)
(2-75)
NVT
dld2d3 F3B(zl1213}
(2-76)
NVT
and
F3A Wl0,2
(2-77)
F (z r r ) = p (z )p (z )p (z ) g (z r r )
3B 11211 o 1 o 2 o 1 o 11211 a a a oi-^co^to^
(2-78)
Again, (2-75) and (2-76) have the same form as (2-31) and,
consequently, they may be evaluated in the same manner to give


29
analogous to (2-37) (the Jacobian of the transformation djr^dj:^dr_3 = J
dr^dr^dr.-^ is unity):
Y3A 10
£
12
dr19
12 a
dz. z
3fo(zlil2)
1 1 3z,
(2-79)
3B
dr dr19
12 13 a a a Ia12W3
dzl Z1
9fo(zl-12-13)
9z,
(2-80)
where f^z^r.^) and fQ (zjjr^JL^) are 8^ven by (2-66) and (2-67), respec
tively.
The derivation of (2-79) and (2-80) assumes: a) pairwise additive
potential, b) use of a Pople reference fluid, and c) the anisotropic
potential contains only terms with spherical harmonics of order £ ^ 0.
2.4 Fowler Model Expressions for Perturbation
Terms y^, Jnb_l_JL3aj_L3d in Pople Expansion
The Fowler approximation for the spherically symmetric Pople
reference may be expressed as:
fo(z1^12} = 0(-zi)0(-z2) pL.8oL(r12) (281)
fo^Zl1213^ = 0(-z1^0(-z2^9(-z3^ PL g0l/r12^ (2-82)
Putting (2-81) into (2-68), (2-69), and (2-79) for Y2A> ^3 and YoA>
respectively, and using (2-82) in (2-80) for y all give terms analo-
gous to (2-39) for y^ in the Fowler model. In each case, the integration
over z^ may be done by parts giving, analogous to (2-44):


30
Bp
2 00
2A
2B
Bp
0
3
dr r2 g (r )
12 12 a wi)2 oL 12
doo.. z (2-83)
12 max
dr12 rl2
dr r..
13 13 a a W1W2W3
dco, 0 dco1 g (r10r _r) z
12 13 oL 12 13 23 max
(2-84)
Y
D2 2 00
F ^ PL
3A 12
0
dr r2 g (r10)
12 12 a 0)^2 6oL 12
dco z (2-85)
12 max
Y
2 3
F 6 PS
3B 6
dr12 rl2
dr rf_
13 13 a a a W1W2W3
dw dco g (r r r ) z
12 13 oL 12 13 23 max
(2-86)
In the case of the two-body terms, Y2A an^ Y^> values fr
Zmax are §lven by (2-42); see Figure 3. Hence, the integration over
co
12 can be performed giving:
Y
2 00
,F *BpL t
2A 4
0
dr 0 r2 g (r )
12 12 &oL 12 a co.^
(2-87)
Y
0 2 2
F 716 PL '
3A 12
dr r2 g (r )
12 12 oL 12 a )i)2
(2-88)
The integrations over un and co, in the three-body terms, Yon
1Z J Zd
and Yon cannot be performed immediately since z and g T (r,,r,,,,r~)
Jij IT13X OL 1Z 1 j Z j
depend on these angles. A portion of this angle dependence may be
integrated out, however, if the angles ^2^12^13^13 are transformed to
a set of angles which include Euler angles specifying the orientation


31
of the triangle whose vertices are the molecules 1, 2 and 3. Details
of the coordinate transformation and integration over the Euler angles
are reserved for Appendix B. The resulting expressions are:
Y
2 3 00
F *>L '
2B 2
dr12 r12
dr13 rl3
r12+ r13
dr23 r23 SoL(r12r13r23)
r12" r13 ^
x (r + r + r__)
a a wiU)2a)3 12 13 23
(2-89)
22 3
* 6 PL '
3B
dr12 r12
dr13 r13
r12+ r13
dr23 r23 goL(r12r13r23>
ri2- r13l
X (r + r + r) (2-90)
a a a 12 13 23
F F F F
Computationally convenient forms for y y yD, and You are
zA 3A zB 3B
derived from (2-87), (2-88), (2-89), and (2-90), respectively, in the
F F
next chapter. Equations for each of these perturbation terms y YOI>,
zA zB
F F
^3A ^3B ^or sPec^^^c anisotropic potentials are tabulated in Appendix D.
In the case of bulk fluid properties, Stell et al. [29] have
obtained improved results over the perturbation theory by resumming
the series (2-50) in the form of a Pad approximant:
(2-91)
The analogous form for the surface tension expansion (2-51) is:


32
Y = Y
o
+
1
Y
2
(2-92)
Calculations based on (2-92) in the Fowler model are given in the next
chapter.
2.5 Superficial Excess Internal Energy from the
Pad Perturbation Theory for Surface Tension
The surface excess internal energy is related to the temperature
dependence of the surface tension by the classical Gibbs-Helmholtz equa
tion:
u
s
= Y T
(2-93)
where u^ is the surface excess internal energy, Us> per unit of surface
area 5:
(2-94)
The Fowler approximation does not predict the correct temperature depen
dence of the surface tension for fluids of spherical molecules. Further,
the Fowler model may be used to obtain an equation for ug in terms of
the pair distribution function for the bulk liquid [2], and this second
equation is inconsistent with (2-93) [30,31]; Freeman and McDonald show
that these two expressions give quite different results for use in the
case of Lennard-Jones liquids [31]. To determine the validity of (2-93)
for the Pad perturbation theroy for surface tension presented above,


33
we combine (2-92) and (2-93) to obtain:
u = u + u (2-95)
s so sa
where u is
so
the isotropic reference contribution,
and
u
sa
T
3Yo 3Yo
_ T
3T 9T
(2-96)


CHAPTER 3
NUMERICAL CALCULATIONS OF SURFACE TENSION
The Fowler model perturbation theory developed in Chapter 2
has been used to calculate surface tensions for pure polyatomic
fluids. The intermolecular potential used for the fluids is of
the form:
u(r12a)lW2) = Uo(r12) + Ua(rl2)lJ2) (3_1)
where uq is the Lennard-Jones potential,
Uo(ri2} = 4e[(a/r12)12 (a/r^)6] (3-2)
and u is the dipole, quadrupole, anisotropic overlap or anisotropic
dispersion potential. Equations for these potentials are given in
Appendix C. In these calculations the superposition approximation
is made for gQ^12*13r23) :
go(rl2rl3r23) So(rl2) go(r23) go(rl3)
(3-3)
and Verlet's molecular dynamics results are used for gQ(r) [32].
In Section 3.1 forms amenable to calculation are presented
In Sections 3.2 and 3.3 calculations
, F F F J F
fr Y2A Y2B Y3A and Y3B*
34


35
of surface tension and surface excess internal energy are presented
for various model fluids which obey (3-1); comparisons with Monte Carlo
calculations are made for surface tension of polar liquids. In Section
3.4 results for real fluids are given and compared with experimental
measurements. In Section 3.5 the perturbation theory is used as a
basis for developing a correlation of surface tension for polyatomic
liquids.
Irl Evaluation of y^, y^A and
Rewriting (2-87,88,89,90) in dimensionless form:
k ? co
F* F a TT PL
Y2A y2A e 4 T*
F* F (3 tt PL
r3A Y3A e 12 t*2
0
*2
dr* r*3 g T(r* )
12 12 oL 12 a 0)^0)^
*
dr. r. g (r )
12 12 6oL 12
F* F o'
k 7 co
2 pT
IT L
Y2B Y2B e 2 *
* *
dr12 r12
* *
dr13 r13
* *
r12+ r13
. *
dr23 r23
* i
r12" r13'
lu2
(3-4)
(3-5)
goL(rl2r13r23) (r12+ rl3+ r23)ru)) (3 6)
12 3
*3
CO
I
oo
a2
-TT2 V
* *
dr12 r12 .
/
JU 4-
dr13 r13
£
6 *2 J
* k
-f- r
12 13
* *
dr23.r23
k k i
I r12~ r13'
* * * * *
X goL rl2r] 3r23^ ^rl2+ r13+ r23^
a)lJ2W3
k
where p
kT/e,
*
u
a
u /e, and r
a
r/a.
(3-7)


36
The angle averages in (3-4,5,6,7) have been evaluated [33] and
are tabulated in Appendix A. Substituting those expressions into
(3-4,5,6,7) gives:
yZ I I, y£(A;ss)
2A
(3-8)
A
ss
y\l- I l yllw-.ss')
AA'
SS
, 2B
(3-9)
Y31= I I Y^(AAA";ss's")
J AA'A" sss" J
(3-10)
F*
r3B I I y**(M'A";ssV)
J AA'A" ss's" J
(3-11)
In each of these equations A = Z^Z^Z; in (3-10) A' = Z'^Z'^Z' A" = ,
while in (3-9) and (3-11) A' = A" = SL^Z^Z". In these equations,
F* i\ r. tzx. t 11 T r
Y2a(A;ss ) (2£,+1)(2£0+l) Jn +n ,-l(P ,T ) ^
*2
L (2£ + 1)
* *
16T '~l-~/ s s'
x Eg(A;n1n2) Eg,(A;n1n2)/e2
nln2
(3-12)
*3
^2B(AA' ss') g T* ni
x l (-) Es(£10£1;n10) Es, (^O^jn^/e (3-13)


37
*2
Yo*(AA,A";ss's") = ,/9 -r7 (2£+l)(2£'+l)(2£"+l)
J 96tt 7 T
x Jn . i(P*>T*)
n +n ,+n
s s s
£" £' £
0 0 0
£J £.[ £
*2 £2 £2
£" £' £
n1n2n1
_ __ M __ M
2nl 2
£"
*1
£1
^2
*2
*2
il
-1'
2
-2
-2-
(3-14)
Y~*(AA'A";ss's")
Jd
Es(A;nin2) E^CA'jnjnp /e~
2 p*3 £_+£'+£"
L / \ Z o r
6 t*2 \£'
x ^ o o "
f(2£+l)(2£'+l)]
1/2
££J (2£1+1)(2£2+1)(2£^+1)
x
£ £' £"
£' 5 £
J3 x,2
K1 (££'£";n n'n")
s s s
n1 +n+nl
x I (-) Es(A;nin2) Es,(A,;n[nJ)
nln2n3
x EslI(A";n2n)/e-
(3-15)
In these equations, is the Kronecker delta, n = -n,
£" £' £
0 0 0
is a 3j symbol,
[ £ £' £
|m m' m
tO
is a 6j symbol [34,35], and


38
£" j?' a
i l l
*2 £2 £2
is a 9j symbol [36]. The Eg are coefficients in a
r v a
spherical harmonic expansion of the anisotropic potential u The
superscript on Eg indicates a complex conjugate. Details of the
expansion and equations for Eg for specific interactions are given in
Appendix C.
£ ^
In (3-12) and (3-14) Jn(p ,T ) is the single integral:
. -k k
Jn(p ,T )
* *-(n-2) .
dr, r,v g(ri2^
12 12
(3-16)
Values of the J integrals are tabulated elsewhere and have been fitted
n
k k
to an empirical equation in p and T [37].
y
In (3-13) L (£^;ngn ,) is the triple integral:
L^(£;nn') =
* *-(n-l)
drl2 rl2
* *-(n'-l)
dr13 rl3
* *
rl2+r13
r12 r13'
X dr23 r23^r12+ rl3+ r23^ goL^r12rl3r23^ P£^cos l^ 17')
t h
Here is the £ order Legendre polynomial and is the interior angle
y
at molecule 1 in the triangle formed by r^j r^3> anc* r23 Values of L
are tabulated in Appendix E and have been fitted to an empirical equation
k k
in p and T .
In (3-15) K1 (££'£";ngns,n^,,) is the triple integral:


39
KY(££,£,,;nn'n")
dr* r*-(n-X)
12 12
* *-(n'-l)
drl3 ri3
* *-(r\"-l) * *
x dr r ^ o (x r r )
23 23 goLv 12 13 23;
* *
ri2+ rl3
* ,
r12_ r13'
(ri2+ r13+ r23) (t1a2a3)
(3-18)
The function ^is given by a spherical harmonic expansion:
^££'£,"(aIa2a3^ = ^ C(££'£";mm'm") Ym(u>12)
mm'm"
* Y£'m'(W13) Y£"m"(W23)
(3-19)
where C(££'£";mm'm") is a Clebsch-Gordan coefficient [33]. The are
spherical harmonics in the convention of Rose [38]. In (3-18) and (3-19)
the cu are the interior angles at molecule i in the triangle formed by
r12 rl3 and r23 Expressions for for multipole interactions
are given in Appendix A.
Note that y!^(AA A"; ss s") given by (3-14) vanishes if (£+£'+£")
is odd or if the molecules are linear and either (£^+£3+£^) or
F*
(£+£'+£") is odd. yOT) (AA';ss ) vanishes unless the anisotropic potential
Z Z Z ZJd
contains £=0 spherical harmonics.
F* F* F* F*
Specific expressions for YA, Y,, Y,,. and Y_ have been evaluated
2A 2B 3A 3B
from (3-12), (3-13), (3-14), and (3-15), respectively, for various aniso
tropic potentials. The results are tabulated in Appendix D.
F
The contributions to y given in (3-12,13,14,15) are simply
related to the corresponding terms in the free energy expansion [33]:


40
^(A;as') - f
n +n .-1
s s
^ n +n ,
s s
A2A(-A;SS'-)
kT
(3-20)
YF2B(M,;ss,) - V
** L,
L (£.,;n n ,)
1 s s' J
A2b(AA';ss')
kT
(3-21)
a a
Y3A(AA'A";sss") = -
n +n ,+n -l
s s s
n +n ,+n '
s s s
A (AA'A";ss's")
-j A
NkT
(3-22)
Y3B(AAA";ss's") = -
p*T* KY(££*r;nsns,nsll) (AA A"; ss s")
8 K (££'£";nsns,ns)
NkT
(3-23)
The L and K integrals in (3-21) and (3-23) are defined by:
L(£;nn') =
H *-(n-l)
d 12 12
* *-(n'-l)
d 13 13
* *
ri2+ rl3
k k
dr23 r23
* a .
rl2_ r13'
a a a
X go(r12r13r23) VC0S al)
(3-24)
K(££'£";nn'n")
dr* r*-(n-1)
12 12
, *-(n'-l)
d 13 13
A A
r + r
12 13
r12 r13'
x dr23 r23 go^rl2r13r23^ ^££'£"^aia2a3^ ^ 25^
3.2 Surface Tension Calculations for Model Fluids
In this section model fluid calculations using the Pad per
turbation theory for molecules obeying potentials of the form (3-1)
are presented. For these calculations, the reference fluid surface
F
tension YQ was obtained from the Fowler model expression for a
Lennard-Jones fluid:


41
YV/e = 3ttP*2[J5 2JU] (3-26)
where the integrals are defined by (3-16) and are tabulated elsewhere
[37]. Figures 4 and 5 show the effect of dipolar forces (A E £ = 112)
on the surface tension as predicted by the second order theory, the third
order theory, and the Pad! theory (2-92). The points on these figures
are Monte Carlo calculations of the Fowler model surface tension for
the Lennard-Jones plus dipole fluids [39]. Figure 6 shows similar
results for quadrupoles (A = 224). The second order and third order
terms used in these calculations were determined from the expressions
given in Appendix D.
The results in Figures 4, 5, and 6 are similar to those found
from the corresponding theories for the Helmholtz free energy [29,40].
Including the third order term extends the range of application of the
expansion somewhat; however, the third order term overcorrects the
second order theory for |j* or Q* > 1. The Pad! theory, on the other
hand, interpolates between the second and third order theories. In
the case of the polar fluids, the Pad agrees well with Monte Carlo
results up to y* 1.75.
In Figure 7 comparison is made of the effects of various aniso
tropies on the surface tension. The dipole and quadrupole curves are
the Pad! results from Figures 4 and 6, respectively. As in the case
of bulk fluid thermodynamic properties, for a given value of the
multipole strength (y or Q ), the quadrupole potential is found to
have a larger effect on surface tension than the dipole potential.


42
Figure 4. Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.85, kT/e = 1.273


U)
Figure 5. Fowler Model Surface Tension for a Fluid of Axially Symmetric Molecules Interacting
with Lennard-Jones plus Dipole Model Potential. pa^ = 0.45, kT/e =-* 2,934


44
Q/(cr5)1/2
Figure 6. Fowler Model Surface Tension for a Fluid of
Axially Symmetric Molecules Interacting with
Lennard-Jones plus Quadrupole Model Potential.
pa3 = 0.85, kT/e = 1.273


0-2/e
45
Figure 7. Fowler Model Surface Tension for Fluids of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Various Anisotropic Potentials. p= 0.85,
kT/e = 1.273


46
The anisotropic overlap and dispersion results are from the second order
F F
theory, using expressions for Yo and Yor> given in Appendix D.
ZA ZB
3.3 Calculation of the Superficial Excess Internal
Energy for Model Fluids
To obtain the surface excess internal energy from the Pad
expression (2-96), the temperature derivatives of the anisotropic
contributions to the surface tension are required. Equations (3-12),
(3-13), (3-14), and (3-15) give, respectively:
3y^(A;ss')
* = Yo*(A;ss')
9T 2A
9£n J
n +n ,
s s -1
3T
(3-27)
9T
ZD
9£n L3(£,;n n ,)
1 s s
9T
(3-28)
3Y^a(AA,A";ss's")
= Y~(AA'A";ss's")
3T 3A
9£n J
n +n ,+n ,
s s s -1
9T
(3-29)
9Y^(AA'A";ss's") _
* = Yor (AA' A"; ss s")
9T
9£n K^(££'£";n n ,n ,,
s s s 2
9T
(3-30)
In Table 2 values for ug from (2-95) and (2-96) are compared
with computer simulation results for Lennard-Jones plus quadrupole
model fluids (see Chapters 6 and 7). The reference fluid surface
excess internal energy, was obtained from the Fowler model
expression:


47
TABLE 2
Test of the Gibbs-Helmholtz Equation in the Fowler Model
Perturbation Theory for Lennard-Jones Plus Quadrupole Fluids
Q/(ea5)1/2
pa3
kT/e
Pad
F 2,
u o /e
s
MD
0.5
0.85
1.277
1.913
1.923 .016
0.707
0.931
0.765
2.670
2.656 .012
1.0
0.85
1.294
2.505
2.475 .019


48
UgQcr2/e = 2ttP*2 [j5 Ju] (3-31)
Y
where the J integrals are defined by (3-16). The values for u were
n sa
determined in the simulation by evaluating the ensemble average given
in Chapter 5.
In view of the demonstrated inapplicability of Equation (2-93) in
the Fowler approximation for Lennard-Jones fluids [31], the agreement
between the theory and computer simulation shown in Table 2 is
surprising. Since the inconsistency in the Fowler expressions for y and
u is not limited to spherical potentials [6], the results in Table 2
s
must be fortuitous. The agreement may, in part, be attributed to the
high density state conditions considered, wherein the Fowler approxima
tion is more accurate. Much of the agreement, however, must be due to
cancellation of errors between the y and dy/dT terms in (2-93).
3.4 Surface Tension Calculations for Real Fluids
The Pad perturbation theory developed in Chapter 2 has been used
to predict pure liquid surface tensions for CO^, C^H^, and HBr. In these
calculations the reference fluid was taken to be a Mie (n,6) fluid. The
anisotropies considered were the multipole interactions up through the
quadrupole-quadrupole term, as well as anisotropic dispersion and
overlap contributions.
3.4.1 Mie (n,6) Reference Contribution to Surface Tension
The Mie (n,6) fluid was taken as the reference in the perturbation
theory calculations since Twu has determined values for e, O, and n by


49
fitting perturbation theory calculations of liquid densities and pressures
to experimental values along the orthobaric line for the fluids considered
here [37]. It is felt that the test of the Pad! theory for surface ten
sion is strengthened by using these independently determined potential
parameters.
The Mie (n,6) potential u^n,^(r) is given by [41]:
u(n>6)(r)
ne
(n-6)
6(n-6)
-1
(3-32)
To determine the surface tension for this potential, the Helmholtz
free energy of the nonhomogeneous, two phase, (n,6) fluid is expanded to
first order in powers of n ^ about the Lennard-Jones (12,6) fluid free
energy. The surface tension is obtained by applying the thermodynamic
definition (2-1). Then, the Fowler approximation is made in order to
obtain a form amenable to calculation.
The expansion of the (n,6) free energy about the (12,6) may be
done in two ways. In one method, the values of £ and a are taken to be
the same for the two fluids. The expression for Yq resulting from
this expansion is:
F(n,6) 2 F(12,6) 2
---£ --a Y £ 48,p*2 [. 2U2-6> J<12-6)} + 6
1 1_
n 12
(3-33)
In (3-33) is the single integral:
h26)(p\t*) e
* *-(n'_2) (12,6). *, ...
dr r nr gv 7(r ) (3-34)
0


50
Values of this integral for n' = 11 are tabulated in Appendix K and
* *
have been fitted to an empirical equation in p and T .
In the second method of doing the expansion, the values of £
and r^ are taken to be the same for the (n,6) and (12,6) fluids. Here
r^ is the value of r where u(r) = -£. The expression for
resulting from this second expansion is:
F(n,6) 2
y a
o
F(12,6) 2
Yo 0 *2 r T (12,6)
48fTp [ (l-£n2) *
1_
12
1 .(12,6) , (12,6) .
2 J5 + 6 H11 J
(3-35)
When the (n,6) fluid is used as the reference, the second and third
tion 1
(n, 6)
order terms and y^ in the perturbation theory involve integrals over
the (n,6) pair correlation function g^^/(r). These (n,6) correlation
functions can be related to Lennard-Jones g^^^^(r) functions (for which
there are molecular dynamics results [32]) in the following way [42,43]:
and
g(n,6)(r) s g(12,6),rep = e~0u( n6)rep yHS(d(n,6)}
g0(126)(r) g126)rep e-BuU26)rep yHS(d(12,6)}
(3-36)
(3-37)
where the superscripts rep and HS indicate the repulsive and hard sphere
potential contributions, respectively. The function y is defined by:
, s. 3u(r) s
y (r) = e g (r)
(3-38)


51
In (3-36) and (3-37) the hard sphere diameter d is taken to
be [42]:
d
[1 e-^Wjdr
Further, assuming
(3-39)
yHS(d(n-6)) yHV12-6))
(3-40)
(3-36), (3-37), and (3-40) give:
g(n,6)(r) g(12,6)(r) e-6[u(n-6)>rel> uU2,6),rPj (3-41)
where (u(n-6> rcp ua2,6>,rep vanlshes £or r > r .
in
Using (3-41) with the Mie (n,6) potential in the integrals J^,,
Equation (3-16), and K(££'£";nn'n"), Equation (3-24), Twu has found the
resulting values to be negligibly different from those values obtained
for the Lennard-Jones (12,6) potential [37], at least for values of n
close to 12. Hence, in the calculations reported here, the Lennard-Jones
(12,6) pair correlation function has been used in evaluating the terms y^
and in the surface tension expansion.
In the calculations reported here, Equation (3-35) was used to
determine the reference fluid contribution. Values for the Lennard-Jones
term y^(12,6) were obtained from a corresponding states plot of surface
tension for simple fluids, Figure 8. For the temperature range
0.6 kT/e < 1.24, the curve in Figure 8 obeys:


52
Figure 8. Corresponding States Plot for Surface Tension
of Simple Liquids


53
(12 6) 2. *2 *
Yq a /e = 0.8950T 3.5177T + 3.0166
(3-42)
3.4.2 Anisotropic Contribution to Real Fluid Surface Tension
The calculations presented here include anisotropic dispersion,
overlap, and multipole contributions up to the quadrupole-quadrupole
term:
u =
(202) + u,. (022) 4- Uj. (224) + u (202) + u (022)
dis dis dis over over
+ u (202) + u .. (022) +u, (224)
over-dis over-dis dis-Q
+ u .(112) + u (123) + u .(213) + u .(224)
mul mul mul mul
(3-43)
wherein the subscripts dis, over, Q, and mul refer to dispersion, overlap,
quadrupole, and multipole, respectively. Appropriate parts of this model
have been used by Flytzani-Stephanopoulos e_t ail. [33] and Twu [37] in
studies of bulk fluid thermodynamic properties. The multipole contribu
tions to surface tension for this model potential are obtained by combining
(3-8), (3-9), (3-10), and (3-11) with (3-12), (3-13), (3-14), and (3-15),
respectively:
Y2A Y2A(112) + 2 Y2A(123) + Y2A(224)
Y3A 3 13jU12;112;224) + 6 y!j* (112; 123; 213)
T-* C,&
+ 6 y^A(123; 123; 224) + y^(224 ; 224; 224)
T7& TT&
Y3B = Y3B(H2;112;112) + 3 Y3B(H2;123;123)
+ 3 Y^b(123;123;224) + Y3*(224;224;224)
(3-44)
(3-45)
(3-46)


54
Expressions for the terms on the right side of (3-44), (3-45), and (3-46)
F*
are given in Appendix D. The term is zero for multipoles since only
terms with l 0 spherical harmonics occur in the multipole potentials.
The dispersion and overlap anisotropies in (3-43) have been
included in only the second order term y^ in calculating the surface
tension from the Pad perturbation theory (2-92). The inclusion of
dispersion and overlap contributions to the third order term y^ requires
evaluation of difficult multibody terms. Expressions for the y and
Aci.
y2g terms for anisotropic overlap and dispersion are given in Appendix D.
3.4.3 Results for Real Fluids
Figure 9 compares the Pad! predictions of surface tension for
CC>2 with experiment, while Figure 10 shows a similar comparison for
C2H2 and HBr. The experimental values for surface tension of C02 and
C2H2 were taken from the compilation by Jasper [48]. The corresponding
values of saturated liquid densities for C02 and C2H2 were taken from
Vargaftik [49]. Experimental values of surface tension and saturated
liquid density for HBr were obtained from Pearson and Robinson [50].
Values of the potential parameters for C02> C2H2, and HBr were taken
from Twu [37] and are listed in Table 3.
The deviations in surface tension between theory and experiment
are less than 10% for C02 and less than 12% for C2H2 and HBr for the
temperatures shown in the accompanying figures. The consistent deviations
between theory and experiment, especially for C2H2 and HBr, suggest that
adjustment of the potential parameters would improve the agreement. It
is not a very informative test of the theory, however, to adjust potential


7ct2/
55
0.8 1.0 1.2
kT/e
Figure 9. Surface Tension for CO2 comparing Perturbation
Theory Calculations (points) with Experimental
Values (line)


y o~21$
56
Figure 10. Surface Tension for C2H2 and HBr comparing
Perturbation Theory Calculations (points)
with Experimental Values (lines)


57
TABLE 3
Potential Parameter Values Used in
Calculating Surface Tension
Fluid
e/k
(K)
0
(A)
n
M(1018)
(esu cm)
Q(1024)
, 2.
(esu cm )
6
K
co2
244.31+
3.687
16

-4.30 [51]
-0.1
0.257
c2h2
253.66
3.901
16

5.01 [52]
0.3
0.270
HBr
248.47
3.790
12
0.788 [51]
4.0 [51]


All values for e, o, n, 6 and K are taken from Twu [37].


58
parameters using surface tension in order to calculate surface tension.
The deviations between theory and experiment shown in Figures 9 and 10
do not increase much with temperature, and, in fact, for CC^ the devia
tions decrease. This is in contrast to the results found for simple
Lennard-Jones fluids wherein surface tension calculated in the Fowler
model show rapidly increasing disagreement with experiment as the
temperature is increased [2,31]. Further, the Fowler model values of
surface tension for Lennard-Jones fluids are generally larger than the
experimental values at the same temperature. It may be that, in addi
tion to the questionable adequacy of potential parameter values used
here, the Pade approximant in some way corrects for errors introduced
in using the Fowler model for the interface. There is some evidence
for this from the Pad values for the surface excess internal energy
presented in Section 3.3.
3.5 Correlation of Surface Tension for
Pure Polyatomic Liquids
The perturbation theory for surface tension presented in Chapter 2
has been used as a basis for correlating surface tensions of a large number
of polyatomic liquids. The perturbation theory gives the surface tension
in the Fowler approximation as:
Y Y0 + Y2A + Y2B + Y3A + Y3B (3-47)
A simple correlation for y may be obtained by making the van der Waals-
type assumption that the reference fluid radial distribution function is
a constant:


59
?oL(r12} = C
(3-48)
Using (3-48) in (2-87), (2-88), (2-89), and (2-90), together with a
similar approximation for the triplet correlation function, gQ(r^2r13r23^
(3-47) becomes, in reduced form:
*
Y
, *2 *2 *3 *4
. a p a'p a'p a'p
= y + 1 L + -2- -L- + 3 L +
o * *2 *2
(3-49)
where
a' = I
al 4
=
2 2
a a?
dr r ]f
12 12 a a)lt02
(3-50)
CO
I
CO
* *
A A
dr12 r12
dr13 r13
0 0 | r
A A
A A
-f- r
12 13
AAA A
dr0- r0
A 23a 23 a a W1W2W3
12 rl3'
x (r 4- r 4- r )
k 12 13 23;
(3-51)
a. = -
7T
12
A A3 AT
dr r
12 12 a ^1^2
A A
dr12 rl2
A A
dr13 r13
r* + r*
12 13
A A
,dr23,r23
r r I
12 131
(3-52)
AAA AAA
X (r10 + r..., + r~)
a a' av a)lJ2a>3 12 13 23
(3-53)
Equation (3-49) can be written in an equivalent form using the
critical constants, T^, V and p^ as reducing parameters rather than
the potential parameters, e and O:


60
y +V£ + ^ + !A + V
roR T T 2 2
R R
R R
where
Yr = y(vc/na)2/3 (ut.)'1
(3-54)
(3-55)
PR = P/Pc (3-56)
Tr = T/Tc (3-57)
In transforming (3-49) into (3-54), the proportionality of the potential
parameters to the critical constants is obtained by the usual correspond
ing states method [41]- Here, however, the potential is for polyatomic
fluids and, therefore, contains parameters in addition to the energy and
distance parameters £ and O, e.g., the multipole moments, y, Q, and
anisotropic polarizability, K, and overlap parameters 6. For such an
intermolecular potential, if the usual derivation of corresponding states
theory is followed [41], one finds:
* c *
Tc = ~~ = c^y ,Q ,6,k, )
* 3 *
Pc = pca = c2(y ,Q ,6,k,)
p a3
* C *
~ ^ (y > Q
(3-58)
(3-59)
(3-60)
where c^, c^, and c^ are constants only if the anisotropic potential para-
* *
meters y Q o, K, are kept fixed. However, these "constants" can
be absorbed in the terms a^, a2, etc., as has been done in (3-54). Thus,
2
= a|c2/c^, etc. The transformation from the potential parameters £
and a as reducing parameters to critical constants as reducing parameters
may then be made in the usual way [41].


61
Equation (3-54) has been used as a basis for correlating surface
tension by treating the parameters a^ as semiempirical constants. Various
truncated forms of (3-54), including its Pad form analogous to (2-92),
have been tested against experimental surface tensions for numerous poly
atomic liquids. The form giving the best comparison with experiment was
found to be that terminated after the y term:
ZB
YR YoR + T ^1 + a2PR)
R
(3-61)
Values for a-^ and a^ for use in (3-61) have been determined for numerous
polyatomic liquids by least squares fitting to available experimental
data. The resulting values are listed in Table 4. In these calculations,
the reference contribution was obtained from a fit of reduced surface
tensions of the inert gases and methane, analogous to the curves in
Figure 8.
Y = 2.4724T^ 7.5918TD + 5.0748 (3-62)
OK K K
which applies for 0.4 < T £ 0.95.
R
The validity of the correlation (3-61) may be tested by determining
2
whether experimental data give a linear relation between (y y )T /p
R oR R R
and p as implied in (3-61). Such a test has been conducted for several
K
polyatomic liquids, and results for carbon dioxide, acetic acid, and
methanol are shown in Figures 11, 12, and 13, respectively. These
figures are typical results for small to moderately large polyatomics
and indicate a satisfactory correlation. The corresponding comparisons


62
TABLE 4
Values for the Parameters and a^ in the
Surface Tension Correlation of Equation (3-61)
Substance
Range
a1(10^)
a2(10 )
References
T
r
P Y
Paraffins
Ethane
.43-
.59
6.880
-2.246
49
48
Propane
.50-
.77
14.17
-5.606
53
48
n-Butane
.50-
.57
8.548
-2.469
49
49
i-Butane
.52-
.60
7.749
-2.082
49
49
n-Pentane
.54-
.67
6.504
-1.498
49
49
i-Pentane
.59-
.66
6.250
-1.451
49
49
n-Hexane
.54-
.93
-2.398
1.918
49
49
n-Heptane
.54-
.95
-1.860
1.910
49
49
n-Octane
.48-
.90
1.335
0.7617
49
49
i-Octane
.50-
.67
8.536
-1.942
49
49
n-Nonane
. 46-
.63
7.340
-1.230
49
49
n-Decane
.44-
.60
8.699
-1.576
49
49
n-Dodecane
.41-
.57
11.42
-2.262
49
49
n-Tridecane
.40-
.55
12.06
-2.360
49
49
n-Tetradecane
.41-
.54
12.21
-2.339
49
49
n-Pentadecane
.40-
.53
12.83
-2.452
49
49
n-Hexadecane
.41-
.55
11.59
-1.893
49
48
n-Heptadecane
.41-
.54
11.10
-1.720
49
48
n-Octadecane
.40-
.52
10.25
-1.439
49
48
n-Nonadecane
.41-
.52
11.15
-1.596
49
48
n-Eicosane
.40-
.51
11.15
-1.539
49
48
Cycloparaffins
Cyclopentane
.55-
.61
5.689
-1.339
49
48
Methylcyclo-
.53-
.59
5.973
-1.288
49
48
pentane
Ethylcyclo-
.50-
.55
10.65
-2.832
49
48
pentane
Cyclohexane
.51-
.62
6.249
-1.351
49
48
Methylcyclo-
.48-
.65
2.885
-0.4730
49
49
hexane
t
Reduced temperature
range over
which a.
and a were
fitted.
Sources of experimental data for liquid densities p and surface
tensions y.


63
TABLE 4 (Continued)
Substance
2 2
Range a (10 ) a-UO ) References
Tr P Y
Olefins
Propylene
.53-.67
7.143
-2.173
49
48
1-Butene
.48-.70
4.752
-0.9880
49
48
2-Butene
.51-.65
3.606
-0.8401
49
48
1-Hexene
.54-.64
8.630
-2.142
49
49
1-Octene
.47-.56
8.047
-1.876
49
49
Cyclopentene
.56-.62
5.445
-1.195
49
49
Aromatics
Benzene
.48-.93
0.4634
0.5432
53
11
Toluene
.46-.63
8.130
-2.080
49
49
Ethylbenzene
.44-.60
9.602
-2.384
49
49
Isopropylbenzene
.46-.57
8.175
-1.851
49
49
Alcohols
Methanol
.53-.92
11.60
-4.971
49
49
n-Propanol
.55-.68
25.56
-8.854
49
48
i-Propanol
.56-.60
33.13
-10.98
49
49
n-Butanol
.48-.54
20.35
-6.624
49
49
Organic Halides
Methyl Chloride
.68-.73
1.215
-0.1476
53
48
Ethyl Bromide
.56-.60
15.00
-4.565
53
48
Carbon Tetra-
.49-.89
0.9412
0.4869
49
49
chloride
Chlorobenzene
.43-.88
2.138
-0.000452
49
49
Oxides
Carbon monoxide
.61-.68
8.502
-2.748
49
48
Carbon dioxide
.71-.95
-1.587
1.395
49
49
Water
.44-.58
34.75
-11.90
54
48
Others
Acetic Acid
.49-.86
5.984
-2.906
49
49
Acetone
.54-.69
6.178
-1.783
49
49
Ammonia
.49-.58
6.878
-2.269
53
48
Aniline
Carbon disulfide
.39-.65
.51-.58
13.38
5.968
-3.976
-1.869
49
49
Chlorine
.47-.57
4.103
-1.340
49
48
Diethyl ether
.59-.95
-0.4972
1.227
49
49
Ethyl acetate
.52-.90
0.08686
1.058
49
49


64
Figure 11. Test of Surface Tension Correlation (line)
for CC>2


65
Figure 12. Test of Surface Tension Correlation (line) for
Acetic Acid


66
Figure 13. Test of Surface Tension Correlation (line) for
Methanol


67
of predicted surface tensions with experimental values for these and
several other liquids are shown in Figures 14 and 15.
The linear relation suggested by (3-61) is not obeyed by experi
mental data for long chain hydrocarbons, however. Typical plots are
shown in Figure 16. In spite of the poor correlation for these
substances, the predicted surface tensions using (3-61) were usually
within 3% of the experimental values as shown in Figure 17.
Generally, the correlation of (3-61) reproduced the experimental
data for substances tested here within 3% for values of T < 0.92. In
K
order for the correlation to apply in the critical region, we must have:
C.P. C.P.
0
(3-63)
hence, from (3-61):
(3-64)
Then
yr YoR+ aT (1 V
K
(3-65)
in the critical region. This relation was not obeyed by many of the
liquids in Table 4.


68
Figure 14. Comparison of Surface Tension Calculated from
the Correlation (lines) with Experimental Values
(points) for Several Polyatomic Liquids


Full Text

PAGE 1

685)$&( 7(16,21 $1' &20387(5 6,08/$7,21 2) 32/<$720,& )/8,'6 %\ -$0(6 0,7&+(// +$,/( $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( &281&,/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

$&.12:/('*(0(176 ,W LV D SOHDVXUH WR H[SUHVV P\ JUDWLWXGH WR WKRVH ZKR KDYH IUHHO\ FRQWULEXWHG WR WKLV ZRUN WKURXJK WKHLU LQVWUXFWLRQ JXLGDQFH DQG DGYLFH .HLWK *XEELQV LQLWLDWHG WKH UHVHDUFK UHSRUWHG KHUHLQ DQG HQWKXn VLDVWLFDOO\ VWLPXODWHG DQG VXSSRUWHG LWV GHYHORSPHQW DV ZHOO DV P\ RZQ SURIHVVLRQDO JURZWK 'U *XEELQV FRQVLVWHQWO\ SURYLGHG DQ H[WUDRUGLQDU\ HQYLURQPHQW IRU OHDUQLQJ SURYLGHG D VWURQJ H[DPSOH RI WKH VFLHQWLILF PHWKRG DQG H[SRVHG PH WR QXPHURXV NQRZOHGJHDEOH VFLHQWLVWV DQG HQJLQHHUV RQ ERWK VLGHV RI WKH $WODQWLF ,Q DGGLWLRQ KH H[SHQGHG FRQVLGHUDEOH HIIRUW LQ REWDLQLQJ WKH ILQDQFLDO VXSSRUW WUDYHO DQG FRPSXWHU IXQGV ZKLFK PDGH WKLV ZRUN SRVVLEOH %LOO 6WUHHWW JHQHURXVO\ DOORZHG PH WR VSHQG VHYHUDO PRQWKV LQ KLV ODERUDWRU\ DW WKH 86 0LOLWDU\ $FDGHP\ DQG SDWLHQWO\ WDXJKW PH PROHFXODU G\QDPLFV +H REWDLQHG WKH FRSLRXV DPRXQWV RI FRPSXWHU WLPH XVHG LQ WKH PROHFXODU G\QDPLFV ZRUN UHSRUWHG KHUH DQG NHSW WKH SURJUDP UXQQLQJ LQ P\ DEVHQFH 0DQ\ RI WKH LGHDV IRU SUHVHQWLQJ WKH PROHFXODU G\QDPLFV UHVXOWV FDPH WR OLJKW LQ GLVFXVVLRQV ZLWK &RORQHO 6WUHHWW )XUWKHU DP JUDWHIXO WR &RORQHO DQG 0UV 6WUHHWW IRU WKH KRVSLWDOLW\ H[WHQGHG WR PH GXULQJ P\ YLVLWV WR :HVW 3RLQW -RKQ 2n&RQQHOO 8QLYHUVLW\ RI )ORULGD FRQWLQXDOO\ LQVSLUHG PH WKURXJK RSHQHQGHG TXHVWLRQLQJ FRQFHUQLQJ FODVVLFDO DQG VWDWLVWLFDO WKHUPRG\QDPLFV VFLHQFH HQJLQHHULQJ DQG PRVW LPSRUWDQWO\ WKH FKDUDFWHU RI OLIH LL

PAGE 3

&KULV *UD\ 8QLYHUVLW\ RI *XHOSK LQVWUXFWHG PH LQ VSKHULFDO WULJRQRPHWU\ VSKHULFDO KDUPRQLF H[SDQVLRQV 5DFDK DOJHEUD HWF WKHUHE\ GHYHORSLQJ LQ PH D KHDOWK\ UHVSHFW IRU WKH SK\VLFLVWnV YLHZ RI DSSOLHG VFLHQFH KDYH EHQHILWHG JUHDWO\ IURP FRXQWOHVV GLVFXVVLRQV ZLWK P\ FROOHDJXH &KRUQJ+RUQJ 7ZX RQ YDULRXV DVSHFWV RI WKHUPRG\QDPLFV VWDWLVWLFDO PHFKDQLFV QXPHULFDO PHWKRGV DQG &KLQHVH FRRNLQJ 6mUHQ 7R[YDHUG 8QLYHUVLW\ RI &RSHQKDJHQ FRQWULEXWHG PXFK YDOXDEOH DGYLFH RQ WKH WKHRU\ DQG DVVRFLDWHG FDOFXODWLRQV IRU IOXLG LQWHUIDFHV 7KDQNV DUH DOVR GXH 'U 7R[YDHUG IRU SURYLGLQJ D FRS\ RI KLV FRPSXWHU SURJUDP IRU FDOFXODWLQJ WKH YDSRUOLTXLG LQWHUIDFLDO GHQVLW\ SURILOH IRU /HQQDUG-RQHV IOXLGV 3HWHU (JHOVWDII DOORZHG PH WR VSHQG VHYHUDO PRQWKV LQ WKH VWLPXODWLQJ DWPRVSKHUH RI WKH 3K\VLFV 'HSDUWPHQW DW WKH 8QLYHUVLW\ RI *XHOSK DP JUDWHIXO WR WKH IDFXOW\ DQG VWDII IRU WKHLU KRVSLn WDOLW\ DQG IRU WKH ODUJH DPRXQW RI 129$ FRPSXWHU WLPH PDGH DYDLODEOH WR PH DP HVSHFLDOO\ WKDQNIXO WR 'DQ /LWFKLQVN\ IRU XVHIXO DGYLFH RQ 129$ VRIWZDUH DQG WR 5RVV 0F3KHUVRQ IRU WLPHO\ KDUGZDUH VXSSRUW RQ WKH 129$ DP DOVR LQGHEWHG WR 6KLHQ6KLRQ :DQJ IRU VSHQGLQJ PDQ\ KRXUV LQ WHDFKLQJ PH WKH 0RQWH &DUOR PHWKRG 'LFN 'DOH DQG 5RQ )UDQNOLQ RI WKH (QJLQHHULQJ ,QIRUPDWLRQ 2IILFH 8QLYHUVLW\ RI )ORULGD JDYH WLPHO\ DQG HQWKXVLDVWLF SKRWRJUDSKLF WHFKn QLFDO DVVLVWDQFH LQ SURGXFLQJ WKH ILOPHG DQLPDWLRQ RI PROHFXODU G\QDPLFV VLPXODWLRQV /DUU\ 0L[RQ LQ WKH 1RUWKHDVW 5HJLRQDO 'DWD &HQWHU 8QLYHUn VLW\ RI )ORULGD SURYLGHG YDOXDEOH VRIWZDUH VXSSRUW LQ GHYHORSLQJ WKH ILOPHG DQLPDWLRQ WHFKQLTXH LLL

PAGE 4

, DP JUDWHIXO WR 'U : 'XIW\ IRU VHUYLQJ RQ WKH VXSHUYLVRU\ FRPPLWWHH ZRXOG DOVR OLNH WR UHPHPEHU 'U 7 0 5HHG ZKR ZDV DQ RULJLQDO PHPEHU RI WKH FRPPLWWHH DQG ZKR VWURQJO\ HQFRXUDJHG PH LQ WKH LQLWLDO SKDVHV RI WKH UHVHDUFK UHSRUWHG KHUH 36< &KHXQJ 7 .H\HV & *UD\ DQG 5 / +HQGHUVRQ : % 6WUHHWW DQG 6 7R[YDHUG NLQGO\ SURYLGHG PDQXVFULSWV RI WKHLU ZRUN SULRU WR SXEOLFDn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f DQG WKH 1DWLRQDO 6FLHQFH )RXQGDWLRQ IRU ILQDQFLDO VXSSRUW RI WKLV VWXG\ LY

PAGE 5

7$%/( 2) &217(176 3DJH $&.12:/('*(0(176 LL /,67 2) 7$%/(6 L[ /,67 2) ),*85(6 [Y .(< 72 6<0%2/6 [[LL $%675$&7 [[L[ &+$37(56 ,1752'8&7,21 7KHRU\ RI 6XUIDFH 3URSHUWLHV &RPSXWHU 6LPXODWLRQ 0HWKRGV 2XWOLQH RI 'LVVHUWDWLRQ 7+(25< 2) 685)$&( 7(16,21 *HQHUDO ([SUHVVLRQV IRU 6XUIDFH 7HQVLRQ RI 3RO\DWRPLF )OXLGV *HQHUDO )LUVW 2UGHU 3HUWXUEDWLRQ 7KHRU\ IRU 6XUIDFH 7HQVLRQ 3HUWXUEDWLRQ 7KHRU\ IRU 6XUIDFH 7HQVLRQ XVLQJ D 3RSOH 5HIHUHQFH )RZOHU 0RGHO ([SUHVVLRQV IRU 3HUWXUEDWLRQ 7HUPV <$f <%f <$f <% LQ 3r3OH ([3DQVLRQ 6XSHUILFLDO ([FHVV ,QWHUQDO (QHUJ\ IURP WKH 3DG 3HUWXUEDWLRQ 7KHRU\ IRU 6XUIDFH 7HQVLRQ 180(5,&$/ &$/&8/$7,216 2) 685)$&( 7(16,21 ) ) ) ) (YDOXDWLRQ RI <$
PAGE 6

7$%/( 2) &217(176 &RQWLQXHGf &+$37(56 3DJH &DOFXODWLRQ RI WKH 6XSHUILFLDO ([FHVV ,QWHUQDO (QHUJ\ IRU 0RGHO )OXLGV 6XUIDFH 7HQVLRQ &DOFXODWLRQV IRU 5HDO )OXLGV &RUUHODWLRQ RI 6XUIDFH 7HQVLRQ IRU 3XUH 3RO\n DWRPLF /LTXLGV 9$325/,48,' '(16,7<25,(17$7,21 352),/(6 )LUVW 2UGHU 3HUWXUEDWLRQ 7KHRU\ IRU S]8fAf &DOFXODWLRQV RI S]AFRAf IRU 2YHUODS DQG 'LVn SHUVLRQ 0217( &$5/2 6,08/$7,21 2) 02/(&8/$5 )/8,'6 21 $ 0,1,&20387(5 ,QWURGXFWLRQ 0RQWH &DUOR 0HWKRG IRU 1RQVSKHULFDO 0ROHFXOHV 'HVFULSWLRQ RI WKH 0LQLFRPSXWHU 6\VWHP 0RQWH &DUOR 3URJUDP IRU WKH 129$ &RPSDULVRQ RI 129$ 5HVXOWV ZLWK )XOO6L]H &RPSXWHU 5HVXOWV &RQFOXVLRQV n 02/(&8/$5 '<1$0,&6 0(7+2' )25 $;,$//< 6<00(75,& 02/(&8/(6 ,QWURGXFWLRQ ([SUHVVLRQV IRU WKH )RUFH DQG 7RUTXH IRU $[LDOO\ 6\PPHWULF 0ROHFXOHV 0HWKRG RI 6ROXWLRQ RI WKH (TXDWLRQV RI 0RWLRQ DQG WKH 0ROHFXODU '\QDPLFV $OJRULWKP (YDOXDWLRQ RI 3DLU &RUUHODWLRQ )XQFWLRQV (TXLOLEULXP 3URSHUWLHV IURP WKH JA A QUA

PAGE 7

7$%/( 2) &217(176 &RQWLQXHGf &+$37(56 3DJH 02/(&8/$5 '<1$0,&6 5(68/76 3RWHQWLDO 0RGHOV (TXLOLEULXP 3URSHUWLHV 6SKHULFDO +DUPRQLF &RHIILFLHQWV Jf f Uff $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ 6LWH6LWH 3DLU &RUUHODWLRQ )XQFWLRQV )LOPHG $QLPDWLRQ RI 0ROHFXODU 0RWLRQV &21&/86,216 7KHRU\ IRU 6XUIDFH 7HQVLRQ RI 3RO\DWRPLF )OXLGV 7KHRU\ IRU WKH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOH RI 3RO\DWRPLF )OXLGV &RPSXWHU 6LPXODWLRQ RI 3RO\DWRPLF )OXLGV $33(1',&(6 $ (;35(66,216 )25 7+( $1*/( $9(5$*(6 ,1 (48$7,216 f 72 f % &225',1$7( 75$16)250$7,21 $1' ,17(*5$7,21 29(5 (8/(5 $1*/(6 72 2%7$,1 (48$7,216 f $1' f %O &KRLFH RI (XOHU $QJOHV % (YDOXDWLRQ RI ,QWHJUDO & 02'(/6 )25 $1,627523,& 327(17,$/6 2) /,1($5 02/(&8/(6 (;35(66,216 )25 \A
PAGE 8

7$%/( 2) &217(176 &RQWLQXHGf $33(1',&(6 3DJH + 9$/8(6 )25 7+( Jf f U &2()),&,(176 ]P A 9$/8(6 )25 7+( ,17(*5$/6 Q 9$/8(6 2) 7+( 6,7(6,7( &255(/$7,21 )81&7,216 9$/8(6 2) 7+( ,17(*5$/ +AA /,7(5$785( &,7(' %,%/,2*5$3+< %,2*5$3+,&$/ 6.(7&+ YLLL

PAGE 9

3DJH /,67 2) 7$%/(6 ([DPSOHV RI 0DFURVFRSLF DQG 0LFURVFRSLF ,QWHUIDFLDO 3URSHUWLHV 5HODWHG E\ (TXDWLRQV RI WKH )RUP f 7HVW RI WKH *LEEV+HOPKROW] (TXDWLRQ LQ WKH )RZOHU 0RGHO 3HUWXUEDWLRQ 7KHRU\ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLGV 3RWHQWLDO 3DUDPHWHU 9DOXHV XVHG LQ &DOFXODWLQJ 6XUIDFH 7HQVLRQ 9DOXHV IRU WKH 3DUDPHWHUV DA DQG D LQ WKH 6XUIDFH 7HQVLRQ &RUUHODWLRQ RI (TXDWLRQ f (TXLOLEULXP 3URSHUWLHV LQ WKH )RUP RI (QVHPEOH $YHUDJHV $SSUR[LPDWH 1XPEHU RI 0RQWH &DUOR &RQILJXUDWLRQV *HQHUDWHG SHU +RXU RQ WKH 129$ &RPSDULVRQ RI 129$ DQG &'& 5HVXOWV IRU 3URSHUW\ 9DOXHV RI /HQQDUG-RQHV 4XDGUXSROH 0RGHO )OXLG N7H SD 4HDf ([SUHVVLRQV IRU X fJf! LQ 7HUPV RI Jf f UD -ADfA A ]P = IRU 9DULRXV 0RGHO 3RWHQWLDOV ([SUHVVLRQV O+mUf IRU WKH &RQILJXUDWLRQDO (QHUJ\ LQ 7HUPV IRU 9DULRXV 0RGHO 3RWHQWLDOV ([SUHVVLRQV IRU WKH 3UHVVXUH LQ 7HUPV RI JA A UAf 9DULRXV 0RGHO 3RWHQWLDOV RI IRU ([SUHVVLRQV IRU WKH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ LQ 7HUPV RI J f Uff IRU 9DULRXV 0RGHO 3RWHQWLDOV ([SUHVVLRQV IRU WKH )RZOHU 0RGHO 6XUIDFH ([FHVV ,QWHUQDO (QHUJ\ LQ 7HUPV RI Jf f U ff IRU 9DULRXV 0RGHO 3RWHQWLDOV ([SUHVVLRQV IRU WKH 0HDQ 6TXDUHG )RUFH LQ 7HUPV RI Jf f Uff IRU 9DULRXV 0RGHO 3RWHQWLDOV D -2,' -=

PAGE 10

7DEOH 3DJH ([SUHVVLRQV IRU WKH 0HDQ 6TXDUHG 7RUTXH LQ 7HUPV RI J e PAUA ArU ADUrRXV 0rGHO 3RWHQWLDOV ([SUHVVLRQV IRU WKH $QJXODU &RUUHODWLRQ )XQFWLRQV LQ 7HUPV RI JA A PAUA ARU ADUARXV 0RGHO 3RWHQWLDOV 3ULPDU\ 2ULHQWDWLRQV IRU 3DLUV RI /LQHDU 0ROHFXOHV 3URSHUW\ 9DOXHV RI D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG 2EWDLQHG LQ WKLV :RUN DQG &RPSDUHG ZLWK WKRVH JLYHQ E\ %HUQH DQG +DUS (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXn SROH )OXLG DW SD 4HDfr (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXn SROH )OXLG DW SR 4HDf (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXn SROH )OXLG DW SR 4HDf $QLVRWURSLF &RQWULEXWLRQV WR (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW SD HDf $QLVRWURSLF &RQWULEXWLRQV WR (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW SD 4LHFf $QLVRWURSLF &RQWULEXWLRQV WR (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW SD 4 HDf (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 2YHUODS )OXLG DW SD (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 2YHUODS )OXLG DW SD • (IIHFW RI 3RWHQWLDO 0RGHO DQG 6WDWH &RQGLWLRQ RQ WKH )LLnVW 3HDN +HLJKW RI WKH Jf f &RHIILFLHQWV O ]P &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEULXP 3URSHUWLHV ZLWK 9DOXHV 2EWDLQHG IURP eAeA ,QWHJUDOV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK SR N7H 4HDf@ [

PAGE 11

7DEOH 3DJH &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEn ULXP 3URSHUWLHV ZLWK 9DOXHV 2EWDLQHG IURP ,QWHJUDOV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK SD N7H 4HDfA &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEn ULXP 3URSHUWLHV ZLWK 9DOXHV 2EWDLQHG IURP OAP ,QWHJUDOV IRU /HQQDUG-RQHV SOXV 2YHUODS )OXLG ZLWK SR N7H &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEn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f DQG f \ ([SUHVVLRQV IRU \M IRU 9DULRXV $QLVRWURSLF 3RWHQWLDOV IRU /LQHDU 0ROHFXOHV ) ([SUHVVLRQV IRU \A IRU 0XOWLSROH 3RWHQWLDOV IRU /LQHDU 0ROHFXOHV ) ([SUHVVLRQV IRU \f% IRU 9DULRXV $QLVRWURSLF 3RWHQWLDOV IRU $[LDOO\ 6\PPHWULF 0ROHFXOHV ) ([SUHVVLRQV IRU \A% IRU 0XOWLSROH 3RWHQWLDOV IRU /LQHDU 0ROHFXOHV 7KH ,QWHJUDOV .AeeneQQnQf IRU 3XUH )OXLGV 7KH ,QWHJUDOV /AeQQnf IRU 3XUH )OXLGV [L

PAGE 12

7DEOH 3DJH ( 7KH &RQVWDQWV LQ (TXDWLRQ (O 7KH ,QWHJUDO A IRU 3XUH )OXLGV +, 9DOXHV RI JR22AUA a pAUA IrU )HQQDUGGRQHV SOXV 4XDGUXSROH )OXLG DW N7H S3G 4 HDf + 9DOXHV RI JAUA a iAUA IRU WKH )OXLG RI 7DEOH +, + 9DOXHV RI Af JAUA IRU WKH )OXLG RI 7DEOH +, + 9DOXHV RI JAUA f p4AUA IrU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK N7H SDG DQG 4HR G + 9DOXHV RI JAUA f JAUA IRU WKH )OXLG RI 7DEOH + + 9DOXHV RI JA&UAf f AA IRU WKH )OXLG RI 7DEOH + + 9DOXHV RI JAUA f JAUA IrU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK N7H SDG DQG 4HRf + 9DOXHV RI JAUA JAUA IrU WrLH )GXGG rI 7DEOH + + 9DOXHV RI Af AA IRU WKH )OXLG RI 7DEOH + + 9DOXHV RI JRRAUA JAUA IrU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWK N7H SDA DQG +OO 9DOXHV RI JAUA JAUA IrU WALH )GXGG rI 7DEOH + + 9DOXHV RI Af f JAUA IRU WKH )OXLG RI 7DEOH + + 9DOXHV RI J4AUA a JAUA IrU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWK N7H SDG DQG [LL

PAGE 13

7DEOH 3DJH + 9DOXHV RI JQUf 7DEOH + iAUA IRU WrLH ))XLG R) fff + 9DOXHV RI JU 7DEOH + iAA IRU WKH )OXLG RI 7KH ,QWHJUDOV -Ap 4XDGUXSROH )OXLG 4&HRf IRU D /HQQDUG-RQHV SOXV SDQ N7H 7KH ,QWHJUDOV -AA Q IRU WKH )OXLG RI 7DEOH Q f f 7KH ,QWHJUDOV Q -A2 IRU WKH )OXLG RI 7DEOH Q f f 7KH ,QWHJUDOV -A -A IRU WKH )OXLG RI 7DEOH Q f f 7KH ,QWHJUDOV -AA 4XDGUXSROH )OXLG 4&HDf IRU D /HQQDUG-RQHV SOXV SDQ N7H 7KH ,QWHJUDOV -Ar IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV -AA -AA IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV -Ap B IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLi SDAQ N7H 4&HRf f 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV -Ap B IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG SS N7H 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV -AA IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV -Ap IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG S N7H [LLL

PAGE 14

7DEOH 3DJH 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV -rr IRU WKH )OXLG RI 7DEOH Q Q 7KH ,QWHJUDOV -Ap B MA2 IRU )OXLG 4I 7DEOH Q Q 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK =R N7H SD 4&2f 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK LD N7H SD 4HRfO 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK LR N7H SD 4HDfO LR 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWK eR N7H SD 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWK LR N7H SR 7KH ,QWHJUDO IRU SXUH )OXLGV [LY

PAGE 15

/,67 2) ),*85(6 )LJXUH 3DJH 5HODWLRQ RI 7KHRU\ ([SHULPHQW DQG &RPSXWHU 6LPXODn WLRQ LQ WKH 6WXG\ RI /LTXLGV 9DULDWLRQ RI )OXLG 'HQVLW\ ZLWK 3RVLWLRQ WKURXJK D 3ODQDU 9DSRU/LTXLG ,QWHUIDFH 7ZR 3RVVLEOH 9DOXHV IRU WKH 0D[LPXP =M 9DOXH IRU D 3DLU RI 0ROHFXOHV LQ WKH )RZOHU 0RGHO ,QWHUIDFH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 'LSROH 0RGHO 3RWHQWLDO SDA N7H )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 'LSROH 0RGHO 3RWHQWLDO SDA N7H )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 4XDGUXSROH 0RGHO 3RWHQWLDO SDA N7H )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU )OXLGV RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 9DULRXV $QLVRWURSLF 3RWHQWLDOV SF" N7H &RUUHVSRQGLQJ 6WDWHV 3ORW IRU 6XUIDFH 7HQVLRQ RI 6LPSOH /LTXLGV 6XUIDFH 7HQVLRQ IRU &2 &RPSDULQJ 3HUWXUEDWLRQ 7KHRU\ &DOFXODWLRQV ZLWK ([SHULPHQWDO 9DOXHV 6XUIDFH 7HQVLRQV IRU &+ DQG +%U &RPSDULQJ 3HUWXUEDn WLRQ 7KHRU\ &DOFXODWLRQV ZLWK ([SHULPHQWDO 9DOXHV 7HVW RI 6XUIDFH 7HQVLRQ &RUUHODWLRQ IRU &2 7HVW RI 6XUIDFH 7HQVLRQ &RUUHODWLRQ IRU $FHWLF $FLG 7HVW RI 6XUIDFH 7HQVLRQ &RUUHODWLRQ IRU 0HWKDQRO [Y

PAGE 16

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

PAGE 17

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fO 4HDf 3DLU 3RWHQWLDO IRU /HQQDUG-RQHV SOXV 'LSROH 'LSROH 4XDGUXSROH DQG 4XDGUXSROH 0RGHO )OXLG DW 3ULPDU\ 3DLU 2ULHQWDWLRQV \HDfO 4HDf 3DLU 3RWHQWLDO IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS 0RGHO )OXLG DW 3ULPDU\ 3DLU 2ULHQWDWLRQV [YLL

PAGE 18

)LJXUH 3DJH 0HDQ6TXDUHG 'LVSODFHPHQW RI 0ROHFXODU &HQWHUV RI 0DVV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG )OXFWXDWLRQ LQ 7HPSHUDWXUH IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG )OXFWXDWLRQ LQ WKH 5DWLR RI 7UDQVODWLRQDO WR 5RWDWLRQDO .LQHWLF (QHUJ\ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG (IIHFW RI 4XDGUXSROH 0RPHQW RQ WKH &HQWHU&HQWHU 3DLU &RUUHODWLRQ )XQFWLRQ DW SD 6SKHULFDO +DUPRQLF &RHIILFLHQWV Jff IRU /HQQDUG-RQHV T \AP SOXV 4XDGUXSROH )OXLG DW SR N7H 4HDf 6SKHULFDO +DUPRQLF &RHIILFLHQWV JQ IRU WKH )OXLG RI ; R,7O )LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV J IRU WKH )OXLG RI P )LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV J f B IRU WKH )OXLG RI )LJXUH (IIHFW RI $QLVRWURSLF 2YHUODS 3DUDPHWHU RQ WKH &HQWHU &HQWHU 3DLU &RUUHODWLRQ )XQFWLRQ DW SD 6SKHULFDO +DUPRQLF &RHIILFLHQWV JQf IRU /HQQDUG-RQHV f SOXV $QLVRWURSLF 2YHUODS )OXLG DW SD N7H 6SKHULFDO +DUPRQLF &RHIILFLHQWV J IRU WKH )OXLG RI )LJXUH "7 6SKHULFDO +DUPRQLF &RHIILFLHQWV J IRU WKH )OXLG RI )LJXUH 7 6SKHULFDO +DUPRQLF &RHIILFLHQWV J IRU WKH )OXLG RI )LJXUH ,QWHJUDQGV WJ&fUf a JA AUAU DQG >JA A AUAUB IRU DQG 8Df UHVSHFWLYHO\ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG [YLLL

PAGE 19

MJX 3DJH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG RI )LJXUH IRU WKH 7HH 2ULHQWDn WLRQ A r S XQGHILQHGf $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG RI )LJXUH IRU WKH &URVV DQG 3DUDOOHO 2ULHQWDWLRQV A rf $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG RI )LJXUH IRU D 6NHZHG 2ULHQWDn WLRQ A # Mf rf $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG RI )LJXUH IRU WKH 7HH 2ULHQWDWLRQ A r k S XQGHILQHGf $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG RI )LJXUH IRU WKH (QGRQ 2ULHQWDWLRQ A # S f 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU M! ZLWK 4r 7r Sr 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU S ZLWK 4r 7r Sr 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU f ZLWK 4r 7r Sr 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU f ZLWK 4r 7r Sr 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU f ZLWK 4r 7r Sr 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU f ZLWK 4r 7r Sr &RPSDULVRQ RI 3HDN +HLJKWV LQ WKH $QJXODU 3DLU &RUUHODn WLRQ )XQFWLRQ ZLWK :HOO 'HSWKV LQ WKH 3DLU 3RWHQWLDO IRU WKH /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK 4&HDf ;,;

PAGE 20

)LJXUH 3DJH 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG IRU ; } Mf ZLWK 7r Sr 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG IRU S ZLWK 7r Sr R ,QWHJUDQG IRU ,QWHUQDO (QHUJ\ U XfJf IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLGV IRU WKH 7HH 2ULHQWDWLRQ ,QWHJUDQG IRU 3UHVVXUH UA Jf IRU /HQQDUG -RQHV SOXV 4XDGUXSROH )OXLGV IRU WKH 7HH 2ULHQWDWLRQ ,QWHJUDQG IRU ,QWHUQDO (QHUJ\ U XfJf IRU /HQQDUG -RQHV SOXV $QLVRWURSLF 2YHUODS )OXLGV IRU 3DUDOOHO 2ULHQWDWLRQ ,QWHJUDQG IRU 3UHVVXUH UA A Jf IRU /HQQDUG -RQHV SOXV $QLVRWURSLF 2YHUODS )OXLGV IRU 3DUDOOHO 2ULHQWDWLRQ 6LWH6LWH 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLGV 3RVVLEOH 6TXDUH 3DFNLQJ RI /HQQDUG-RQHV SOXV 4XDGUXn SROH 0ROHFXOHV IRU ,QWHUSUHWLQJ JDJUf 6LWH6LWH 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLGV %R[ 5HSUHVHQWLQJ WKH 0ROHFXODU '\QDPLFV 6\VWHP ZLWK WKH 9ROXPH (OHPHQW 6DPSOHG IRU WKH )LOPHG $QLPDWLRQ ,QGLFDWHG ,QLWLDO )&& /DWWLFH &RQILJXUDWLRQ RI /HQQDUG-RQHV 0ROHFXOHV LQ WKH 9ROXPH (OHPHQW 6DPSOHG LQ WKH )LOPHG $QLPDWLRQ )UDPH IURP WKH )LOPHG $QLPDWLRQ RI /HQQDUG-RQHV 0ROHn FXOHV &RUUHVSRQGLQJ WR WKH 6L[WK 7LPH6WHS LQ WKH 0ROHFXODU '\QDPLFV &DOFXODWLRQ )UDPH IURP WKH )LOPHG $QLPDWLRQ RI /HQQDUG-RQHV 0ROHn FXOHV &RUUHVSRQGLQJ WR WKH VW 7LPH6WHS LQ WKH 0ROHFXODU '\QDPLFV &DOFXODWLRQ [[

PAGE 21

3DJH )LJXUH % 5RWDWLRQV 'HILQLQJ WKH (XOHU $QJOHV [ fff % 5RWDWLRQV ,QWHUIDFH LQ WKH 7ULDQJOH LQ WKH WR 'HILQH 9DOXHV IRU ] PD[ )RZOHU 0RGHO [[L

PAGE 22

.(< 72 6<0%2/6 Fe $ F $ [ $]Lf]f %=O}=Of e L PQ er PQ )L )$)% )$f)% Q 5RPDQ 8SSHU &DVH +HOPKROW] IUHH HQHUJ\ &RQILJXUDWLRQDO +HOPKROW] IUHH HQHUJ\ 7KH LWK WHUP LQ WKH SHUWXUEDWLRQ H[SDQVLRQ IRU +HOPKROW] IUHH HQHUJ\ )XQFWLRQ GHILQHG LQ (TXDWLRQ f )XQFWLRQ GHILQHG LQ (TXDWLRQ f 5HVLGXDO FRQWULEXWLRQ WR FRQVWDQW YROXPH KHDW FDSDFLW\ &OHEVFK*RUGDQ FRHIILFLHQW 'LIIXVLRQ FRHIILFLHQW 5HSUHVHQWDWLRQ FRHIILFLHQW &RPSOH[ FRQMXJDWH RI UHSUHVHQWDWLRQ FRHIILFLHQW )RUFH RQ PROHFXOH )XQFWLRQ GHILQHG LQ (TXDWLRQ f )XQFWLRQV GHILQHG LQ (TXDWLRQ f DQG f UHVSHFWLYHO\ )XQFWLRQV GHILQHG LQ (TXDWLRQV f DQG f UHVSHFWLYHO\ $QJXODU FRUUHODWLRQ SDUDPHWHU ,QWHJUDO GHILQHG LQ (TXDWLRQ f [[LL

PAGE 23

, ,Uf AQQn = . .< / / /< 1 3 3 O 3s[f RLf 4 r = 7 0RPHQW RI LQHUWLD )XQFWLRQV GHILQHG LQ (TXDWLRQV f DQG f ,QWHJUDO GHILQHG LQ (TXDWLRQV f DQG %f ,QWHJUDO GHILQHG LQ (TXDWLRQ *Of ,QWHJUDO GHILQHG LQ (TXDWLRQ f ,QWHJUDO GHILQHG LQ (TXDWLRQ f &RHIILFLHQW RI HOOLSWLFLW\ IRU SODQH SRODUL]HG OLJKW ,QWHJUDO GHILQHG LQ (TXDWLRQ f ,QWHJUDO GHILQHG LQ (TXDWLRQ f )XQFWLRQV GHILQHG LQ (TXDWLRQV f DQG f UHVSHFWLYHO\ $QJXODU PRPHQWXP RSHUDWRU ,QWHJUDO GHILQHG LQ (TXDWLRQ f ,QWHJUDO GHILQHG LQ (TXDWLRQ f 1XPEHU RI PROHFXOHV $YRJDGURnV QXPEHU 3UHVVXUH W K 7KH L FRPSRQHQW RI WKH ORFDO SRODUL]DWLRQ YHFWRU /HJHQGUH SRO\QRPLDO RI RUGHU L 3UREDELOLW\ RI WKH LA VWDWH RFFXUULQJ GHILQHG LQ (TXDWLRQ f 4XDGUXSROH PRPHQW 5HGXFHG TXDGUXSROH PRPHQW 4HD f *HQHUDO PXOWLSROH PRPHQW 7HPSHUDWXUH &ULWLFDO WHPSHUDWXUH [[LLL

PAGE 24

WU r 7 5HGXFHG WHPSHUDWXUH 77A 5HGXFHG WHPSHUDWXUH N7H 8 V 6XSHUILFLDO H[FHVV LQWHUQDO HQHUJ\ 9 9ROXPH 9 F &ULWLFDO YROXPH
PAGE 25

JFUOf &HQWHUFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ JD%AUf 6LWHVLWH SDLU FRUUHODWLRQ IXQFWLRQ J]O]Uf ,QWHUIDFLDO SDLU FRUUHODWLRQ IXQFWLRQ JUZODff $QJXODU SDLU FRUUHODWLRQ IXQFWLRQ Jf JU-Wf JLLPUOf &RHIILFLHQWV LQ WKH H[SDQVLRQ RI Jf LQ VSKHULFDO KDUPRQLFV RI WKH PROHFXODU RULHQWDWLRQV IL f§ 8QLW YHFWRU DOLJQHG DORQJ WKH D[LV RI OLQHDU PROHFXOH L N %ROW]PDQQnV FRQVWDQW L 0ROHFXODU ERQG OHQJWK EHWZHHQ DWRPV P 0ROHFXODU PDVV Q ,QGH[ RI UHIUDFWLRQ Q &RQVWDQW YDOXHV JLYHQ LQ (TXDWLRQV f DQG f Q ([SRQHQW RI U LQ UHSXOVLYH SDUW RI 0LH SRWHQWLDO (TXDWLRQ f Q V 3RZHU RI U LQ YDULRXV PRGHO SRWHQWLDOV 31 1RUPDO FRPSRQHQW RI WKH VWUHVV WHQVRU SW 7DQJHQWLDO FRPSRQHQW RI WKH VWUHVV WHQVRU U f§L 9HFWRU ORFDWLRQ RI FHQWHU RI PDVV RI PROHFXOH L 9HFWRU VHSDUDWLRQ RI FHQWHUV RI PDVV RI PROHFXOHV DQG UO r U 0DJQLWXGH RI UA 5HGXFHG GLVWDQFH UAAR U P 9DOXH RI U ZKHUH SDLU SRWHQWLDO LV D PLQLPXP V L 6LQH RI DQJOH A W 7LPH XULf 6SKHULFDOO\ V\PPHWULF LQWHUPROHFXODU SDLU SRWHQWLDO ;;9

PAGE 26

XUfOXf 2ULHQWDWLRQ GHSHQGHQW LQWHUPROHFXODU SDLU SRWHQWLDO X f XUFff DR^]! ,VRWURSLF UHIHUHQFH FRQWULEXWLRQ WR Xf XDf rUf $QLVRWURSLF FRQWULEXWLRQ WR Xf 5HGXFHG SRWHQWLDO XUAfH X V 6XSHUILFLDO H[FHVV LQWHUQDO HQHUJ\ SHU XQLW RI VXUIDFH DUHD 9 0ROHFXODU WUDQVODWLRQDO YHORFLW\ ; [FRPSRQHQW RI UA \ n \FRPSRQHQW RI UA \ULf )XQFWLRQ GHILQHG LQ (TXDWLRQ f = ]FRRUGLQDWH ORFDWLRQ RI PROHFXOH = ]FRPSRQHQW RI UA ] PD[ r ]L 0D[LPXP YDOXH RI ] GHILQHG LQ (TXDWLRQV f DQG %f 5HGXFHG FRRUGLQDWH ]AR 5RPDQ 6FULSW ,QWHUIDFLDO DUHD 8 7RWDO LQWHUPROHFXODU SRWHQWLDO ( &RHIILFLHQWV LQ WKH VSKHULFDO KDUPRQLF H[SDQVLRQ RI WKH DQLVRWURSLF SDLU SRWHQWLDO *UHHN 8SSHU &DVH U L 6XUIDFH DGVRUSWLRQ RI FRPSRQHQW L $ 7ULSOHW RI LQGLFHV IRU H[DPSOH 9ROXPH LQ DQJOH VSDFH [[YL

PAGE 27

 $QJXODU YHORFLW\ 7KH OAN FRPSRQHQW RI 4B D D L D % < < < ) r < • LM H [f *UHHN /RZHU &DVH $GMXVWDEOH SDUDPHWHU LQ (TXDWLRQV f DQG f ,QWHULRU DQJOH LQ WKH WULDQJOH IRUPHG E\ UA} ULf Uf DW PROHFXOH L &RQVWDQWV LQ WKH SUHGLFWRUFRUUHFWRU DOJRULWKP YDOXHV JLYHQ LQ (TXDWLRQV f DQG f UHVSHFWLYHO\ $]LPXWKDO DQG SRODU DQJOHV UHVSHFWLYHO\ IRU PROHFXODU RULHQWDWLRQ LQ WKH VSDFH IL[HG IUDPH N7 $QJOH EHWZHHQ WKH D[HV RI D SDLU RI OLQHDU PROHFXOHV 6XUIDFH WHQVLRQ )RZOHU PRGHO VXUIDFH WHQVLRQ a 7KH LWK WHUP LQ WKH SHUWXUEDWLRQ H[SDQVLRQ IRU VXUIDFH WHQVLRQ 6XUIDFH WHQVLRQ UHGXFHG E\ SRWHQWLDO SDUDPHWHUV \2 H 6XUIDFH WHQVLRQ UHGXFHG E\ FULWLFDO FRQVWDQWV <9F1$fN7Ff 'LPHQVLRQOHVV DQLVRWURSLF RYHUODS SDUDPHWHU .URQHFNHU GHOWD ,QWHULQROHFXODU SRWHQWLDO HQHUJ\ SDUDPHWHU 8QLW VWHS IXQFWLRQ 3RODU DQJOH IRU PROHFXODU RULHQWDWLRQ LQ WKH LQWHU PROHFXODU IUDPH [[YLL

PAGE 28

$QJXODU GLVSODFHPHQW YHFWRU 3RODU DQJOH IRU UA LQ VSKHULFDO FRRUGLQDWHV 'LPHQVLRQOHVV DQLVRWURSLF SRODUL]DELOLW\ :DYHOHQJWK RI LQFLGHQW OLJKW LQ OLJKW VFDWWHULQJ H[SHULPHQWV 3HUWXUEDWLRQ SDUDPHWHU 'LSROH PRPHQW 5HGXFHG GLSROH PRPHQW \HR f 5DQGRP QXPEHUV )OXLG QXPEHU GHQVLW\ ,QWHUIDFLDO QXPEHU GHQVLW\ SURILOH ,QWHUIDFLDO QXPEHU GHQVLW\RULHQWDWLRQ SURILOH )LUVW RUGHU WHUP LQ SHUWXUEDWLRQ H[SDQVLRQ IRU S]A2-Af ,QWHUPROHFXODU SRWHQWLDO GLVWDQFH SDUDPHWHU 6WDQGDUG GHYLDWLRQ RI SURSHUW\ L 7RUTXH RQ PROHFXOH $]LPXWKDO DQJOH IRU PROHFXODU RULHQWDWLRQ LQ WKH LQWHUn PROHFXODU IUDPH $]LPXWKDO DQJOH IRU LQ VSKHULFDO FRRUGLQDWHV )XQFWLRQ GHILQHG LQ (TXDWLRQ f 6HW RI YDULDEOHV VSHFLI\LQJ WKH RULHQWDWLRQ RI PROHFXOH L [[YLLL

PAGE 29

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ff IRU WKH UHIHUHQFH IOXLG 7KHVH DUH UHPRYHG E\ XVLQJ WKH )RZOHU DSSUR[LPDn WLRQ IRU WKH LQWHUIDFH 7KH UHVXOWLQJ WKHRU\ KDV EHHQ WHVWHG DJDLQVW 0RQWH &DUOR UHVXOWV IRU WKH )RZOHU PRGHO VXUIDFH WHQVLRQ RI D IOXLG ZKRVH PROHFXOHV LQWHU£FW ZLWK D /HQQDUG-RQHV SOXV GLSROH SRWHQWLDO 7KH EHKDYLRU RI WKH WKHRU\ FRPSDUHG ZLWK VLPLODWLRQ SDUDOOHOV WKDW RI EXON IOXLG SURSHUWLHV QDPHO\ WKH VHFRQG RUGHU WKHRU\ DJUHHV ZLWK WKH 0RQWH &DUOR UHVXOWV IRU VPDOO YDOXHV RI WKH GLSROH PRPHQW XS WR \HD f )RU ODUJHU GLSROH VWUHQJWKV QHLWKHU WKH VHFRQG QRU WKLUG RUGHU WKHRULHV DJUHH ZLWK WKH FRPSXWHU VLPXODWLRQ UHVXOWV +RZn HYHU ZKHQ WKH WKLUG RUGHU H[SDQVLRQ LV UHFDVW LQ WKH IRUP RI D VLPSOH >@ 3DG DSSUR[LPDQW WKH WKHRU\ DJUHHV ZLWK WKH 0RQWH &DUOR UHVXOWV XS WR GLSROH PRPHQWV DV ODUJH DV XHFU f 7KLV 3DG WKHRU\ KDV EHHQ XVHG WR FDOFXODWH WKH VXUIDFH WHQVLRQ RI SXUH GLSRODU DQG [[ L[

PAGE 30

TXDGUXSRODU OLTXLGV 7KH WKHRU\ DJUHHV ZLWK H[SHULPHQWDO YDOXHV RI VXUIDFH WHQVLRQ LQ WKH QHLJKERUKRRG RI WKH WULSOH SRLQW KRZHYHU WKH )RZOHU PRGHO GRHV QRW JLYH WKH FRUUHFW WHPSHUDWXUH GHSHQGHQFH RI WKH VXUIDFH WHQVLRQ 7KH WKHRU\ KDV DOVR EHHQ XVHG DV D EDVLV IRU GHYHORSn LQJ XVHIXO FRUUHODWLRQV RI VXUIDFH WHQVLRQV RI SXUH SRO\DWRPLF OLTXLGV $ ILUVW RUGHU SHUWXUEDWLRQ WKHRU\ KDV EHHQ GHYHORSHG IRU WKH LQWHUIDFLDO GHQVLW\RULHQWDWLRQ SURILOH S]ARAf IRU SRO\DWRPLF IOXLGV 8SRQ LQWURGXFWLRQ RI D 3RSOH UHIHUHQFH WKH ILUVW RUGHU WHUP YDQLVKHV IRU PXOWLSRODU DQLVRWURSLHV EXW GRHV QRW YDQLVK IRU DQLVRWURSLF RYHUODS RU GLVSHUVLRQ SRWHQWLDO PRGHOV &DOFXODWLRQV RI S]ADMAf IRU D[LDOO\ V\PPHWULF PROHFXOHV LQWHUDFWLQJ ZLWK HDFK RI WKH ODWWHU SRWHQWLDOV KDYH EHHQ SHUIRUPHG XVLQJ D /HQQDUG-RQHV UHIHUHQFH IOXLG DQG WKH LQWHUn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f DW ILYH WR VHYHQ VSHFLILF RULHQWDWLRQV PD\ EH REWDLQHG DW D IUDFWLRQ RI WKH FRVW RI GRLQJ WKH FDOFXODWLRQV RQ D IXOO VL]H PDFKLQH 5HVXOWV IURP WKH PLQLFRPSXWHU FRPSDUH ZLWKLQ WKH VWDWLVWLFDO SUHFLVLRQ ZLWK UHVXOWV SUHYLRXVO\ REWDLQHG RQ &'& DQG ,%0 PDFKLQHV 7KH PHWKRG RI PROHFXODU G\QDPLFV KDV EHHQ XVHG WR VWXG\ V\VWHPV RI PROHFXOHV LQWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV TXDGUXSROH DQG [[[

PAGE 31

/HQQDUG-RQHV SOXV DQLVRWURSLF RYHUODS SRWHQWLDOV 7KH HTXLOLEULXP SURSHUWLHV GHWHUPLQHG LQFOXGH FRQILJXUDWLRQDO LQWHUQDO HQHUJ\ SUHVVXUH )RZOHU PRGHO VXUIDFH WHQVLRQ )RZOHU PRGHO VXUIDFH H[FHVV LQWHUQDO HQHUJ\ PHDQ VTXDUHG IRUFH DQG PHDQ VTXDUHG WRUTXH ,Q DGGLWLRQ WKH FRHIILFLHQWV JA A PAULA nnQ DQ H[3DQVLrQ IrU WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ J&UAAFLAf LQ WHUPV RI SURGXFWV RI VSKHULFDO KDUPRQLFV RI WKH PROHFXODU RULHQWDWLRQV DUH GHWHUPLQHG 6LWHVLWH SDLU FRUUHODWLRQ IXQFWLRQV J 'Uf DUH DOVR IRXQG 5HODWLRQV GS DUH GHYHORSHG EHWZHHQ WKH JJ f U ff FRHIILFLHQWV DQG WKH DERYH OLVWHG $ HTXLOLEULXP SURSHUWLHV IRU VHYHUDO DQLVRWURSLF SRWHQWLDO PRGHOV 6WXG\ LV PDGH RI RULHQWDWLRQDO VWUXFWXUH LQ TXDGUXSRODU DQG RYHUODS IOXLGV YLD JUA:A:A DV REWDLQHG IURP WKH UHFRPELQHG VSKHULFDO KDUPRQLF H[SDQVLRQ $ PHWKRG RI SURGXFLQJ ILOPHG DQLPDWLRQV RI PROHFXODU PRWLRQV IURP PROHFXODU G\QDPLFV GDWD LV GHVFULEHG [[[L

PAGE 32

&+$37(5 ,1752'8&7,21 7KH GHYHORSPHQW RI ULJRURXV PHWKRGV IRU SUHGLFWLRQ RI SURSHUWLHV RI OLTXLGV PXVW FRPH IURP DQ XQGHUVWDQGLQJ RI KRZ PROHFXOHV DUH GLVn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n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

PAGE 33

7+(25< (;3(5,0(17 &20387(5 6,08/$7,21 )LJXUH 5HODWLRQ RI 7KHRU\ ([SHULPHQW DQG &RPSXWHU 6LPXODWLRQ LQ WKH 6WXG\ RI /LTXLGV

PAGE 34

LQ )LJXUH 7KHVH SURSHUWLHV LQFOXGH GHQVLW\ UHIUDFWLYH LQGH[ GLHOHFWULF FRQVWDQW HWF 7KH RIW PHDVXUHG PDFURVFRSLFf SURSHUWLHV RI LQWHUIDFHV DUH LQ PDQ\ FDVHV UHODWHG WR LQWHJUDOV RYHU WKH PLFURVFRSLF SURSHUWLHV >@ ; 22 I 0]f@G] f ZKHUH ; UHSUHVHQWV D PDFURVFRSLF SURSHUW\ 0 LV D PLFURVFRSLF SURSHUW\ VXEVFULSW % LQGLFDWHV XVXDOO\ D EXON SKDVH YDOXH DQG N LV D SURn SRUWLRQDOLW\ FRQVWDQW ([DPSOHV RI VSHFLILF SURSHUWLHV KDYLQJ WKH JHQHUDO IRUP RI f DUH JLYHQ LQ 7DEOH (TXDWLRQ f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n WLRQ OLTXLG PHPEUDQH VHSDUDWLRQ WHFKQLTXHV DQG WHUWLDU\ RLO UHFRYHU\ ? ,QWHUIDFLDO WHQVLRQ LV DOVR LPSRUWDQW LQ XQGHUVWDQGLQJ YDULRXV ELRORJLFDO SURFHVVHV VXFK DV EORRG R[\JHQDWLRQ DQG H\H OXEULFDWLRQ 2I SDUWLFXODU

PAGE 35

] )LJXUH 9DULDWLRQ RI )OXLG 'HQVLW\ ZLWK 3RVLWLRQ WKURXJK D 3ODQDU 9DSRU/LTXLG ,QWHUIDFH

PAGE 36

7$%/( ([DPSOHV RI 0DFURVFRSLF DQG 0LFURVFRSLF ,QWHUIDFLDO 3URSHUWLHV 5HODWHG E\ (TXDWLRQV RI WKH )RUP f ; N >0 0]f@G] U L >3L]f Ss@G] >SL]f SA@G] f < >S1 f 37]f@G] f ; 3 ]f 3 ]f ] ; 3 rrf S rrf f§&2 ] ; G] f W 5HIHUHQFH >@ U L SL 3, SL < 1 5HIHUHQFHV >@n VXUIDFH DGVRUSWLRQ RI FRPSRQHQW L GHQVLW\ RI FRPSRQHQW L EXON YDSRU SKDVH GHQVLW\ EXON OLTXLG SKDVH GHQVLW\ VXUIDFH WHQVLRQ QRUPDO FRPSRQHQW RI VWUHVV WHQVRU WDQJHQWLDO FRPSRQHQW RI VWUHVV WHQVRU 37]f 3 L Q $ FRHIILFLHQW RI HOOLSWLFLW\ IRU OLJKW SODQH SRODUL]HG DW r WR WKH SODQH RI LQFLGHQFH ZKHQ LQFLGHQW DW %UHZVWHUnV DQJOH LWK FRPSRQHQW RI ORFDO SRODUL]DWLRQ YHFWRU EXON SKDVH LQGH[ RI UHIUDFWLRQ ZDYHOHQJWK RI LQFLGHQW EHDP

PAGE 37

LPSRUWDQFH LV WKH GHWHUPLQDWLRQ RI KRZ PROHFXODU FKDUDFWHULVWLFV RI WKH IOXLG FRQWULEXWH WR WKH LQWHUIDFLDO WHQVLRQ $Q LPSRUWDQW JRDO LQ GHVLJQ FRQVLGHUDWLRQV LV WKH LPSURYHG HIILFLHQF\ RI IOXLGIOXLG FRQWDFWLQJ RSHUDWLRQV E\ PRGLI\LQJ WKH V\VWHP HJ E\ LQWURGXFWLRQ RI DGGLWLYHVf LQ RUGHU WR ORZHU WKH LQWHUIDFLDO WHQVLRQ 7KLV LQ WXUQ OHDGV WR FRQVLGHUDWLRQ RI KRZ PROHFXOHV DEVRUE DQG RULHQW WKHPVHOYHV LQ WKH LQWHUIDFLDO UHJLRQ 7KHRU\ RI 6XUIDFH 3URSHUWLHV 5HDGLO\ XVHG PHWKRGV FXUUHQWO\ DYDLODEOH IRU SUHGLFWLQJ LQWHUn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n IDFLDO GHQVLW\ SURILOH S]Af DQG WKH LQWHUIDFLDO SDLU FRUUHODWLRQ IXQFWLRQ J]]Uf >@ < 7 G] Gf§ GXUf GU 3=f3=! 6]L9O! r f

PAGE 38

7KH .LUNZRRG%XII HTXDWLRQ LV YDOLG IRU VSKHULFDO PROHFXOHV DQG DVVXPHV WKH LQWHUPROHFXODU SRWHQWLDO WR EH D VXP RI SDLU SRWHQWLDOV 7KLV UHODWLRQ LV LQWUDFWDEOH DV LW VWDQGV EHFDXVH RI WKH XQNQRZQ IXQFWLRQ J&]A]AUAAf &RQVHTXHQWO\ WKH .LUNZRRG%XII UHODWLRQ KDV EHHQ VWXGLHG XVLQJ YDULRXV VLPSOLI\LQJ PRGHOV IRU WKH LQWHUIDFLDO UHJLRQ 7KH VLPSOHVW VXFK PRGHO LV GXH WR )RZOHU >@ DQG DVVXPHV DQ DEUXSW WUDQVLWLRQ IURP OLTXLG WR YDSRU SKDVH )XUWKHU WKH YDSRU SKDVH GHQVLW\ LV DVVXPHG WR EH QHJOLJLEOH FRPSDUHG WR WKH OLTXLG GHQVLW\ 7KXV WKH PRGHO PD\ EH H[SUHVVHG DV S]fS]f J]]ULA ]f]f S/J/Uf f ZKHUH LV WKH XQLW VWHS IXQFWLRQ [f LI [ A LI [ f VXEVFULSW / LQGLFDWHV D EXON OLTXLG SURSHUW\ DQG WKH QHJDWLYH ] GLUHFWLRQ LV LQWR WKH OLTXLG 7KH UHVXOWLQJ )RZOHU.LUNZRRG%XII H[SUHVVLRQ IRU VXUIDFH WHQVLRQ LV GXU` UO GU J/ U f $V FRXOG EH H[SHFWHG WKH )RZOHU.LUNZRRG%XII WKHRU\ ZRUNV ZHOO QHDU WKH IOXLG WULSOH SRLQW EXW JLYHV LQFUHDVLQJ HUURUV LQ VXUIDFH WHQVLRQ DV WKH WHPSHUDWXUH LV UDLVHG WRZDUGV WKH FULWLFDO SRLQW 5HFHQW

PAGE 39

HYLGHQFH LQGLFDWHV WKDW WKH JRRG DJUHHPHQW DW WKH WULSOH SRLQW LV GXH WR FDQFHOODWLRQ RI HUURUV >@ 7KH VHFRQG ULJRURXV UHODWLRQ IRU VXUIDFH WHQVLRQ LV D JHQHUDOL]HG YDQ GHU :DDOV HTXDWLRQ ZKLFK JLYHV WKH VXUIDFH WHQVLRQ LQ WHUPV RI WKH GHQVLW\ JUDGLHQW GS]AfG]A DQG WKH LQWHUIDFLDO GLUHFW FRUUHODWLRQ IXQFWLRQ F&]AAUAf >@ &2 RR RR f I G] G]f f§&2 &2 f§ G[ G\ GS]Af GS]f G] G] F]]Uf[A \Af f 7KLV UHODWLRQ LV PRUH JHQHUDO WKDQ WKH .LUNZRRG%XII H[SUHVVLRQ VLQFH QR DVVXPSWLRQ RI SDLUZLVH DGGLWLYH SRWHQWLDO LV PDGH LQ LWV GHULYDWLRQ ,W KDV WKH IXUWKHU DGYDQWDJH WKDW WKH GLUHFW FRUUHODWLRQ IXQFWLRQ FUAf LV JHQHUDOO\ RI VKRUWHU UDQJH WKDQ WKH SDLU FRUUHODWLRQ IXQFWLRQ JUAf 7KH JHQHUDOL]HG YDQ GHU :DDOV HTXDWLRQ f KDV QRW EHHQ DV WKRURXJKO\ VWXGLHG DV WKH ROGHU .LUNZRRG%XII IRUPXOD 7KH .LUNZRRG%XII HTXDWLRQ KDV UHFHQWO\ EHHQ JHQHUDOL]HG WR QRQVSKHULFDO PROHFXOHV >@ )URP ERWK D SUDFWLFDO VWDQGSRLQW DQG D GHVLUH WR JDLQ XQGHUVWDQGLQJ RI LQWHUIDFLDO SKHQRPHQD WKHUH LV VWURQJ QHHG IRU XVLQJ WKLV QHZ UHODWLRQ DV D EDVLV IRU GHYHORSLQJ SUHGLFWLYH PHWKRGV IRU VXUIDFH WHQVLRQ RI SRODU DQG TXDGUXSRODU V\VWHPV (YDOXDWLRQ RI WKH LQWHUIDFLDO GHQVLW\ SURILOH S]Af LV UHTXLUHG LQ RUGHU WR GHWHUPLQH WKH VXUIDFH WHQVLRQ IURP WKH ULJRURXV H[SUHVVLRQV f DQG f 6WXG\ RI WKH LQWHUIDFLDO GHQVLW\ SURILOH LV DOVR RI LQWHUHVW SHU VH 1HDUO\ DOO VWXGLHV WKXV IDU KDYH EHHQ IRU VSKHULFDO PROHFXOHV $ YDULHW\ RI DSSURDFKHV KDYH EHHQ XVHG Df YDQ GHU :DDOV WKHRU\ >@ Ef FRQVWDQW FKHPLFDO SRWHQWLDO WKURXJK

PAGE 40

WKH LQWHUIDFH >@ Ff ILUVW %RUQ*UHHQ@ Gf FRQVWDQW QRUPDO SUHVVXUH WKURXJK WKH LQWHUIDFH >@ Hf PLQLPL]Dn WLRQ RI V\VWHP IUHH HQHUJ\ REWDLQHG E\ SHUWXUEDWLRQ WKHRU\ >@ DQG If FRPSXWHU VLPXODWLRQ >@ 'HWHUPLQDWLRQ RI VXUIDFH WHQVLRQ IRU QRQVSKHULFDO PROHFXOHV ZLOO UHTXLUH NQRZOHGJH RI WKH LQWHUIDFLDO GHQVLW\RULHQWDWLRQ SURILOH S]A8fAf ZKHUH RRA LV D VHW RI (XOHU DQJOHV VSHFLI\LQJ WKH RULHQWDWLRQ RI PROHFXOH )XUWKHU WKHUH LV FRQVLGHUDEOH LQWHUHVW LQ GHWHUPLQLQJ KRZ PRGLILFDWLRQ RI PROHFXODU RULHQWDWLRQ DIIHFWV n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

PAGE 41

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n IDFLDO UHJLRQ

PAGE 42

3DUW ,, RI WKLV ZRUN GHVFULEHV FRPSXWHU VLPXODWLRQ VWXGLHV RI OLQHDU PROHFXOHV &KDSWHU UHSRUWV GHYHORSPHQW RI D PHWKRG IRU SHUIRUPLQJ 0RQWH &DUOR FDOFXODWLRQV IRU OLQHDU PROHFXOHV RQ D 129$ PLQLFRPSXWHU &KDSWHU GHVFULEHV WKH PROHFXODU G\QDPLFV PHWKRG IRU OLQHDU PROHFXOHV LQFOXGLQJ GHULYDWLRQ RI H[SUHVVLRQV IRU HIILFLHQW HYDOXDn WLRQ RI WKH IRUFH DQG WRUTXH H[HUWHG RQ D PROHFXOH GXH WR YDULRXV DQLVRWURSLF SRWHQWLDO LQWHUDFWLRQV WKH PHWKRG XVHG WR VROYH 1HZWRQn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n UHODWLRQ IXQFWLRQV $ PHWKRG IRU SURGXFLQJ ILOPHG DQLPDWLRQV RI PROHFXODU PRWLRQV IURP PROHFXODU G\QDPLFV FDOFXODWLRQV LV SUHVHQWHG &KDSWHU GUDZV FRQFOXVLRQV IURP WKLV VWXG\

PAGE 43

3$57 7+(25(7,&$/ 678'< 2) )/8,' ,17(5)$&(6

PAGE 44

&+$37(5 7+(25< 2) 685)$&( 7(16,21 *HQHUDO ([SUHVVLRQV IRU 6XUIDFH 7HQVLRQ RI 3RO\DWRPLF )OXLGV 7KHUH DUH WZR ULJRURXV H[SUHVVLRQV IRU WKH VXUIDFH WHQVLRQ RI SRO\DWRPLF IOXLGV 2QH LV WKH JHQHUDOL]HG YDQ GHU :DDOV HTXDWLRQ f 7KH VHFRQG LV D JHQHUDOL]DWLRQ RI WKH .LUNZRRG%XII H[SUHVVLRQ f ZKLFK KDV EHHQ SUHYLRXVO\ GHULYHG >@ ,Q WKLV VHFWLRQ WKH GHULYDn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f F 6LQFH RQO\ WKH FRQILJXUDWLRQDO SDUW RI WKH IUHH HQHUJ\ $ GHSHQGV RQ WKH LQWHUIDFLDO DUHD f PD\ EH ZULWWHQ DV < N7 / eQ = 197 f

PAGE 45

ZKHUH $ N7 eQ = f LV WKH VWDWLVWLFDO PHFKDQLFDO GHILQLWLRQ RI WKH FRQILJXUDWLRQDO +HOPKROW] IUHH HQHUJ\ N %ROW]PDQQnV FRQVWDQW DQG = LV WKH FRQn ILJXUDWLRQDO LQWHJUDO )RU QRQVSKHULFDO PROHFXOHV = U 1 1 U1Z1f GU GZ H f§ f ZKHUH N7 DQG 8 LV WKH IXOO LQWHUPROHFXODU SRWHQWLDO )RU QRQn VSKHULFDO PROHFXOHV 8 GHSHQGV RQ WKH RULHQWDWLRQV f RI WKH PROHFXOHV DV ZHOO DV WKH SRVLWLRQV RI WKHLU FHQWHUV RI PDVV BU 7KH RULHQWDWLRQV f DUH XVXDOO\ VSHFLILHG E\ D VHW RI (XOHU DQJOHV 2Mf[ EHWZHHQ D ERG\IL[HG UHIHUHQFH IUDPH RQ WKH PROHFXOH DQG D VSDFHIL[HG IUDPH ORFDWHG H[WHUQDO WR WKH V\VWHP 6XEVWLWXWLQJ f LQWR f < N7 B f f f I : ? 1 1 8U Z f GU GWR H 6 f 197 7KH GLIIHUHQWLDWLRQ LQ f FDQQRW EH GRQH LPPHGLDWHO\ VLQFH WKH 1 LQWHJUDWLRQ OLPLWV RQ WKH LQWHJUDOV RYHU UB GHSHQG RQ :H WKHUHIRUH IROORZ *UHHQ >@ DQG FKDQJH BU YDULDEOHV XVLQJ WKH WUDQVIRUPDWLRQ FO [ E [ 4O \ 6 \ 9 V f 3HUIRUPLQJ WKH GLIIHUHQWLDWLRQ LQ f DQG FKDQJLQJ EDFN WR WKH ROG YDULDEOHV JLYHV

PAGE 46

f f f I 00 1 1 J8U X f GU GLf H aOX 1 1 XU Z f f 197 f $VVXPLQJ WKH SRWHQWLDO WR EH SDLUZLVH DGGLWLYH f EHFRPHV 8 M L X U FR LRf f§LM L M LM f < GeOGeGMOGI ee-L-n! X f@ O 197 f ZKHUH Xf X&UAAMAA DQG WGH GHALQLWLRQ RI WKH DQJXODU SDLU GLVn WULEXWLRQ IXQFWLRQ KDV EHHQ XVHG I eeMLMf 11Of = Gen GL1Gfn ‘ GRU DQW 1 1Q SXU WR f H f§ f ,QWHJUDWLQJ f RYHU [A DQG WUDQVIRUPLQJ U? WR eA iGYHV G]O Gf§ GWDLGI A=&M-A 6 Xf 6 197 f 3 Xf DV FDQ EH HYDOXDWHG IRU QRQVSKHULFDO PROHFXOHV E\ FRQVLGHULQJ f Xf f Xf DV DV DU Xf B Xf >UL ] DOf§M f f +HQFH f EHFRPHV

PAGE 47

22 < G]O Ge" GFGFR I ]e "RMFf Xf Xf U aL ]f 22 'HILQLQJ WKH DQJOH DYHUDJH E\ f fmf! fO8 WW f f f fG-GR ZKHUH IL GZ f f (TXDWLRQ f FDQ EH ZULWWHQ < G] Gf§ I =--f AUO Xf U ] Xf @! ] f 9 f (TXDWLRQ f LV RQH IRUP RI WKH JHQHUDO .LUNZRRG%XII HTXDWLRQ $QRWKHU IRUP PD\ EH REWDLQHG E\ WUDQVIRUPLQJ f WR VSKHULFDO FRRUGLQDWHV 6LQFH >@ Xf ] FRV ;\ Xf U VLQ fA Xf UfA WKHQ f Xf U f f§A ] Xf U ] 3FRV HLfU U VLQ FRV 4 Xf f ZKHUH 3[f LV WKH VHFRQG RUGHU /HJHQGUH SRO\QRPLDO 3[f \ [ f 6XEVWLWXWLQJ f LQWR f JLYHV

PAGE 48

< <$ f IW G] GUL 3FRV f UIU0}f 9f IW % G] Gf§ V,Q FRV 6 e]OfOZf nf (TXDWLRQV f JLYH WKH JHQHUDO .LUNZRRG%XII HTXDWLRQ 7KH HTXDWLRQ DSSOLHV WR JHQHUDO VKDSHG PROHFXOHV WKH RQO\ DVVXPSWLRQ EHLQJ D SDLUZLVH DGGLWLYH SRWHQWLDO ,Q WKH FDVH RI VSKHULFDO PROHFXOHV WKH SRWHQWLDO Xf JRHV WR XUff VR WKDW WKH GHULYDWLYH LQ WKH @ I]UDff =Of ]f J/f f ZKHUH LV WKH XQLW VWHS IXQFWLRQ DV LQ f S LV WKH EXON OLTXLG M GHQVLW\ DQG J LV WKH EXON OLTXLG DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ /L ,QWURGXFLQJ WKH )RZOHU PRGHO f LQWR f DOORZV WKH LQWHJUDWLRQV RYHU ]A DQG FRA WR EH SHUIRUPHG JLYLQJ

PAGE 49

WWSW
PAGE 50

X$f XTf $X f f ZKHUH $ LV D SHUWXUEDWLRQ SDUDPHWHU VXFK WKDW ZKHQ $ f JLYHV WKH UHIHUHQFH SRWHQWLDO DQG ZKHQ $ f JLYHV WKH IXOO SRWHQWLDO ([SDQGLQJ WKH WZRSKDVH V\VWHP +HOPKROW] IUHH HQHUJ\ f LQ SRZHUV RI $ DQG VHWWLQJ $ JLYHV $ $ $ ‘ ‘ n R f ZKHUH $ Gf§OGf§ I R ]Of§WLfO-f X$f $ f $ DQG I4 LV WKH LQWHUIDFLDO DQJXODU SDLU GLVWULEXWLRQ IXQFWLRQ IRU WKH UHIHUHQFH V\VWHP 7KH FRUUHVSRQGLQJ H[SDQVLRQ IRU VXUIDFH WHQVLRQ LV REWDLQHG E\ DSSO\LQJ f WR f < <
PAGE 51

A G] GeO 6 )]f 6 197 f 6 6)A ]AU?Af6 PD\ EH HYDOXDWHG E\ FRQVLGHULQJ )L]LOL"f MUQ ) ]Uf ) ] U f ] A V BAe f f§.-s V f§LBB/B/eBBL f 6 6 f§ ]A 6 f§ )=Af f§ )]f = ] f 7KXV f EHFRPHV )O]f§! ] G] )]f )]f GUO = ] = ] )]UBf OY f§ f 7KH VHFRQG WZR WHUPV LQ f FDQ EH LQWHJUDWHG E\ SDUWV DQG VKRZQ WR FDQFHO OHDYLQJ < n Gf§ G]O = X$f $ $ IR]$9f ] fOX f (TXDWLRQ f LV YDOLG IRU VSKHULFDO RU QRQVSKHULFDO UHIHUHQFH IOXLGV WKH RQO\ DVVXPSWLRQ KDV EHHQ D SDLUZLVH DGGLWLYH SRWHQWLDO 1RWH WKDW LI WKH UHIHUHQFH IOXLG LV WDNHQ WR EH D 3RSOH UHIHUHQFH GHILQHG E\ f EHORZ WKHQ \A YDQLVKHV

PAGE 52

)LUVW 2UGHU 7HUP LQ WKH )RZOHU 0RGHO 7R PDNH f DPHQDEOH WR FDOFXODWLRQ WKH )RZOHU DSSUR[LPDWLRQ f PD\ EH LQWURGXFHG A G G]O = L ]Of ]f X$f $ Qr/AA!MDDff $ f 6XEVWLWXWLQJ f DQG FKDQJLQJ WKH RUGHU RI LQWHJUDWLRQ Gf§ GFRA GD! X$f $ f $ ZKHUH ] G]O = ca B=f f=f JR/f f 7KH LQWHJUDO PD\ EH HYDOXDWHG E\ SDUWV WR JLYH ] J 7 f ] R/ PD[ f ZKHUH PD[ UO FRV OI LI f DQG A LV WKH VSKHULFDO FRRUGLQDWH SRODU DQJOH IRU BUA! DV VKRZQ LQ )LJXUH (TXDWLRQ f EHFRPHV WKHUHIRUH
PAGE 53

] ] D p 7A E WW )LJXUH 7ZR 3RVVLEOH 9DOXHV IRU WKH 0D[LPXP ]A 9DOXH ]PD[f IRU D 3DLU RI 0ROHFXOHV LQ WKH )RZOHU 0RGHO ,QWHUIDFH

PAGE 54

5HLQWURGXFLQJ WKH DQJOH DYHUDJH f DQG QRWLQJ WKDW Jf LQ WKH EXON OLTXLG LV LQGHSHQGHQW RI f FDQ EH ZULWWHQ DV 3HUIRUPLQJ WKH LQWHJUDWLRQ RYHU 8fA JLYHV f (TXDWLRQ f LV WKH JHQHUDO UHVXOW IRU WKH ILUVW RUGHU SHUWXUEDWLRQ WHUP LQ WKH )RZOHU DSSUR[LPDWLRQ 3HUWXUEDWLRQ 7KHRU\ IRU 6XUIDFH 7HQVLRQ XVLQJ D 3RSOH 5HIHUHQFH :KHQ WKH JHQHUDO H[SDQVLRQ IRU VXUIDFH WHQVLRQ f LV LQn FUHDVHG WR KLJKHU RUGHU WKH VHFRQG RUGHU WHUP LV IRXQG WR LQFOXGH D WHUP FRQWDLQLQJ WKH UHIHUHQFH IRXUERG\ GLVWULEXWLRQ IXQFWLRQ +LJKHU RUGHU WHUPV LQ WKH H[SDQVLRQ FRQWDLQ HYHQ KLJKHU RUGHU PXOWLERG\ WHUPV 7KHVH FRPSOLFDWHG WHUPV FDQ EH PDGH WR YDQLVK XS WR DW OHDVW WKH VHFRQG RUGHU WHUP LQ WKH H[SDQVLRQ E\ XVLQJ WKH LVRWURSLF UHIHUHQFH SRWHQWLDO ILUVW VXJJHVWHG E\ 3RSOH >@ XULf Xf! f :LWK WKLV FKRLFH RI UHIHUHQFH f EHFRPHV X$f X U ff $X f R D f

PAGE 55

f DQG f WRJHWKHU JLYH WKH VLPSOLILFDWLRQ X$f $ $ X Lf X f! D 8OZ 7KHQ f DQG f KDYH f $[ < f DQG WKH VHFRQG DQG WKLUG RUGHU WHUPV $A$A\A\Af VLPSOLI\ 7KXV LQ WKH 3RSOH H[SDQVLRQ f DQG f EHFRPH WR WKLUG RUGHU $& $ $ f < @ AOGU XDf f ZKHUH JAf LV WKH ILUVW RUGHU WHUP LQ D SHUWXUEDWLRQ WKHRU\ H[SDQVLRQ RI WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ :KHQ WKH H[SDQVLRQ LV DERXW D 3RSOH UHIHUHQFH JAf LV JLYHQ E\ >@ if %XDf JRUf 3 GU >X f! X f! @ [ f§ D A D MOfA JRUUUf f

PAGE 56

ZKHUH iUAUOUA nrnV WrLH WULSOHW FRUUHODWLRQ IXQFWLRQ IRU WKH UHIHUHQFH :KHQ WKH DQLVRWURSLF SRWHQWLDO FRQWDLQV RQO\ VSKHULFDO KDUPRQLFV RI RUGHU LW A VXFK DV PXOWLSROHV WKHQ WKH DQJOH DYHUDJHV LQ WKH VHFRQG WHUP LQ f YDQLVK 6XFK SRWHQWLDO PRGHOV DV DQLVRWURSLF RYHUODS DQG GLVSHUVLRQ FRQWDLQ / VSKHULFDO KDUPRQLFV LQ ZKLFK FDVH WKH VHFRQG WHUP PXVW EH LQFOXGHG (TXDWLRQ f PD\ EH ZULWWHQ WKHUHIRUH DV $ $$ $% f ZKHUH 3% $ Gf§OGf§ JRUf 8Df?Df f %S U % GU GU GU J UULDUf4f X fX f X fX f! f§ f§ f§ tR D D D D ::8 f 6LQFH BUA DQG BUA DUH HDFK LQWHJUDWHG RYHU LQ f WKH LQGLFHV DUH GXPP\ LQGLFHV DQG f PD\ KHQFH EH ZULWWHQ DV 3%n % GU GU GUf J U fU UfBf X f X f! f f§ f§ f§ R D D )RU D IOXLG QRQXQLIRUP LQ WKH ]GLUHFWLRQ f DQG f JHQHUDOL]H WR $ £/OG 3R]Of3R]! XDf!fO0 f

PAGE 57

% GU GU GU S ]fS ]ffS ]ff J ]U fUAfX fX f! f§ f§ f§ R R R rR f§f§ D D AfA f 7KH FRUUHVSRQGLQJ WHUPV LQ WKH H[SDQVLRQ IRU VXUIDFH WHQVLRQ f PD\ EH IRXQG E\ DSSO\LQJ f WR f DQG f < $ % B DV DB DV GOG )$]ef 197 GUGUGU )A]Af f f 197 ZKHUH )$ 3R]Of3R]f JR]Of XDf!8O f )% ( 3R]f3R]f3R]f R]OeOf XD8QA&8f}A nf (TXDWLRQV f DQG f KDYH WKH VDPH IRUP DV f WKXV WKH\ PD\ EH HYDOXDWHG LQ D VLPLODU PDQQHU WR JLYH DQDORJRXV WR f U $ GU G] ] )$=(Of ] f < % Gf§Gf§ G] ] )%=A D] f 6XEVWLWXWLQJ f DQG f LQWR f DQG f UHVSHFWLYHO\ DQG XVLQJ WKH UHODWLRQV

PAGE 58

I R ]OLO! n S]LfS]A JRA=OOA f IR]Of 3R]f3R]f3R]f JR]Of f JLYHV $ GU f X f! f§ D Z WR G] ] IR]OLOf ] f % GUfGU X fX f! D D RL G]A ]A IR]OOOOOf ] f 7KH GHULYDWLRQ RI f DQG f RQO\ DVVXPHV D SDLUZLVH DGGLWLYH SRWHQWLDO DQG XVH RI D 3RSOH UHIHUHQFH IOXLG ,I WKH DQLVRn WURSLF SRWHQWLDO FRQWDLQV RQO\ e A VSKHULFDO KDUPRQLFV WKHQ YDQLVKHV EHFDXVH RI f 'HULYDWLRQ RI 7KLUG 2UGHU 7HUPV ,I RQO\ DQLVRWURSLF SRWHQWLDOV FRQWDLQLQJ e VSKHULFDO KDUPRQLFV DUH FRQVLGHUHG Jf VLPSOLILHV 8VLQJ DUJXPHQWV DQRORJRXV WR WKRVH IRU VLPSOLI\LQJ WKH $f' WHUP LQ f DQG JHQHUDOL]LQJ WR D IOXLG ZKLFK LV QRQXQLIRUP LQ WKH ]GLUHFWLRQ f EHFRPHV IRU e A KDUPRQLFVf

PAGE 59

$ $$ $% $$ % GULGU 3QAOARA"f JB]U 8 f! f§, f§ R R R f§ D :: $% Gf§OGf§Gf§ 3]OfSR]fSR]f JR]AAf X fX fX f! 0 D D D f f f $SSO\LQJ f WR f JLYHV < n <$ <% f ZKHUH <$ % U 6 f§ 6 ‘ GULGU )$=Oef f 197 Gf§OGf§Gf§ )%]Of§f§` f 197 DQG )$=OLOf 3R]f3Rf =OLO! XDf!:O f ) ] U U f S ] fS ] fS ] f J ] U fU fX fX fX f! % f§f§ R R R R f§f§ D D D RLAFRAWRA f $JDLQ f DQG f KDYH WKH VDPH IRUP DV f DQG FRQVHTXHQWO\ WKH\ PD\ EH HYDOXDWHG LQ WKH VDPH PDQQHU WR JLYH

PAGE 60

DQDORJRXV WR f WKH -DFRELDQ RI WKH WUDQVIRUPDWLRQ GMUAGMAGUB GUAGUAGUA LV XQLW\f <$ e GU X f! f§ D G] ] IR]OLOf f ] f % GU fGU X fX fX f! f§ f§ D D D f ,D: G]O = IR]Of ] f ZKHUH IA]AUAf DQG I4 ]MMUA-/Af DUH AYHQ E\ f DQG f UHVSHFn WLYHO\ 7KH GHULYDWLRQ RI f DQG f DVVXPHV Df SDLUZLVH DGGLWLYH SRWHQWLDO Ef XVH RI D 3RSOH UHIHUHQFH IOXLG DQG Ff WKH DQLVRWURSLF SRWHQWLDO FRQWDLQV RQO\ WHUPV ZLWK VSKHULFDO KDUPRQLFV RI RUGHU e A )RZOHU 0RGHO ([SUHVVLRQV IRU 3HUWXUEDWLRQ 7HUPV \A -QEBOB-/DMB/G LQ 3RSOH ([SDQVLRQ 7KH )RZOHU DSSUR[LPDWLRQ IRU WKH VSKHULFDOO\ V\PPHWULF 3RSOH UHIHUHQFH PD\ EH H[SUHVVHG DV IR]A` ]Lf]f S/R/Uf ff IRA=Of§f§A ]A]A]A 3/ JOUA f 3XWWLQJ f LQWR f f DQG f IRU <$! A DQG
PAGE 61

%S $ % %S rr GU U Xf! J U f D ZLf R/ GRR f ] f PD[ GU UO GU f U f X f X f! D D ::: GFR GFR f J UU BUfff ] rR/ PD[ f < ) A 3/ $ GU f U Xf! J Uf D fA R/ GFR f ] f PD[ < rr ) 36 % GU UO GU UIB X f X f X f! D D D ::: GZ GFR J U fU U f ] R/ PD[ f ,Q WKH FDVH RI WKH WZRERG\ WHUPV <$ DQA
PAGE 62

RI WKH WULDQJOH ZKRVH YHUWLFHV DUH WKH PROHFXOHV DQG 'HWDLOV RI WKH FRRUGLQDWH WUDQVIRUPDWLRQ DQG LQWHJUDWLRQ RYHU WKH (XOHU DQJOHV DUH UHVHUYHG IRU $SSHQGL[ % 7KH UHVXOWLQJ H[SUHVVLRQV DUH < ) r r!/ n % GU U GU UO U U GU U 6R/UUUf U U A [ X f X f! U f U UBBf D D ZL8fDf f f rr r 3/ n % GU U GU U U U GU U JR/UUU! UL UO ; X f X f X f! U B U Ufff f D D D ) ) ) ) &RPSXWDWLRQDOO\ FRQYHQLHQW IRUPV IRU \ \ \f' DQG @ KDYH REWDLQHG LPSURYHG UHVXOWV RYHU WKH SHUWXUEDWLRQ WKHRU\ E\ UHVXPPLQJ WKH VHULHV f LQ WKH IRUP RI D 3DG DSSUR[LPDQW f 7KH DQDORJRXV IRUP IRU WKH VXUIDFH WHQVLRQ H[SDQVLRQ f LV

PAGE 63

< < R < f &DOFXODWLRQV EDVHG RQ f LQ WKH )RZOHU PRGHO DUH JLYHQ LQ WKH QH[W FKDSWHU 6XSHUILFLDO ([FHVV ,QWHUQDO (QHUJ\ IURP WKH 3DG 3HUWXUEDWLRQ 7KHRU\ IRU 6XUIDFH 7HQVLRQ 7KH VXUIDFH H[FHVV LQWHUQDO HQHUJ\ LV UHODWHG WR WKH WHPSHUDWXUH GHSHQGHQFH RI WKH VXUIDFH WHQVLRQ E\ WKH FODVVLFDO *LEEV+HOPKROW] HTXDn WLRQ X V < 7 f ZKHUH XA LV WKH VXUIDFH H[FHVV LQWHUQDO HQHUJ\ 8V! SHU XQLW RI VXUIDFH DUHD f 7KH )RZOHU DSSUR[LPDWLRQ GRHV QRW SUHGLFW WKH FRUUHFW WHPSHUDWXUH GHSHQn GHQFH RI WKH VXUIDFH WHQVLRQ IRU IOXLGV RI VSKHULFDO PROHFXOHV )XUWKHU WKH )RZOHU PRGHO PD\ EH XVHG WR REWDLQ DQ HTXDWLRQ IRU XJ LQ WHUPV RI WKH SDLU GLVWULEXWLRQ IXQFWLRQ IRU WKH EXON OLTXLG >@ DQG WKLV VHFRQG HTXDWLRQ LV LQFRQVLVWHQW ZLWK f >@ )UHHPDQ DQG 0F'RQDOG VKRZ WKDW WKHVH WZR H[SUHVVLRQV JLYH TXLWH GLIIHUHQW UHVXOWV IRU XVH LQ WKH FDVH RI /HQQDUG-RQHV OLTXLGV >@ 7R GHWHUPLQH WKH YDOLGLW\ RI f IRU WKH 3DG SHUWXUEDWLRQ WKHUR\ IRU VXUIDFH WHQVLRQ SUHVHQWHG DERYH

PAGE 64

ZH FRPELQH f DQG f WR REWDLQ X X X f V VR VD ZKHUH X LV VR WKH LVRWURSLF UHIHUHQFH FRQWULEXWLRQ DQG X VD 7
PAGE 65

&+$37(5 180(5,&$/ &$/&8/$7,216 2) 685)$&( 7(16,21 7KH )RZOHU PRGHO SHUWXUEDWLRQ WKHRU\ GHYHORSHG LQ &KDSWHU KDV EHHQ XVHG WR FDOFXODWH VXUIDFH WHQVLRQV IRU SXUH SRO\DWRPLF IOXLGV 7KH LQWHUPROHFXODU SRWHQWLDO XVHG IRU WKH IOXLGV LV RI WKH IRUP XUDfO:f 8RUf 8DUOfO-f Bf ZKHUH XT LV WKH /HQQDUG-RQHV SRWHQWLDO 8RUL` H>DUf DUAf@ f DQG X LV WKH GLSROH TXDGUXSROH DQLVRWURSLF RYHUODS RU DQLVRWURSLF GLVSHUVLRQ SRWHQWLDO (TXDWLRQV IRU WKHVH SRWHQWLDOV DUH JLYHQ LQ $SSHQGL[ & ,Q WKHVH FDOFXODWLRQV WKH VXSHUSRVLWLRQ DSSUR[LPDWLRQ LV PDGH IRU J4ArUf JRUOUOUf 6RUOf JRUf JRUOf f DQG 9HUOHWnV PROHFXODU G\QDPLFV UHVXOWV DUH XVHG IRU J4Uf >@ ,Q 6HFWLRQ IRUPV DPHQDEOH WR FDOFXODWLRQ DUH SUHVHQWHG ,Q 6HFWLRQV DQG FDOFXODWLRQV ) ) ) ) IrU <$f <%f <$f DQG <%r

PAGE 66

RI VXUIDFH WHQVLRQ DQG VXUIDFH H[FHVV LQWHUQDO HQHUJ\ DUH SUHVHQWHG IRU YDULRXV PRGHO IOXLGV ZKLFK REH\ f FRPSDULVRQV ZLWK 0RQWH &DUOR FDOFXODWLRQV DUH PDGH IRU VXUIDFH WHQVLRQ RI SRODU OLTXLGV ,Q 6HFWLRQ UHVXOWV IRU UHDO IOXLGV DUH JLYHQ DQG FRPSDUHG ZLWK H[SHULPHQWDO PHDVXUHPHQWV ,Q 6HFWLRQ WKH SHUWXUEDWLRQ WKHRU\ LV XVHG DV D EDVLV IRU GHYHORSLQJ D FRUUHODWLRQ RI VXUIDFH WHQVLRQ IRU SRO\DWRPLF OLTXLGV ,UO (YDOXDWLRQ RI \A \A$ DQG 5HZULWLQJ f LQ GLPHQVLRQOHVV IRUP N FR )r B ) D 77 3/ <$ ‘ \$ H 7r )r B ) B f§WW 3/ U$ f <$ H Wr r rr GUr Ur J 7Ur fXrf! R/ D fAfA r GUf Uf J U ffX f! R/ )r B ) Rn N FR S7 ,7 / <% r <% H r r r GU U r r GU U r r U U r r GU U r r L U Un fOX f f JR/UOUUf U UO Uf8DfXDf!UXffff f r &2 RR D 77 9 r r GU U } -8 GU U e r r N I U r r GUU N N L Ua Un r r r r r r r r r ; JR/ UOU@ UA AUO U UAXa fA8 AAXa AA! DfO-: N ZKHUH S N7H r X D X H DQG U D UD f

PAGE 67

7KH DQJOH DYHUDJHV LQ f KDYH EHHQ HYDOXDWHG >@ DQG DUH WDEXODWHG LQ $SSHQGL[ $ 6XEVWLWXWLQJ WKRVH H[SUHVVLRQV LQWR f JLYHV \= , \e$VVff $ f $ VV \?O O \OOZVVnf $$n 66 % f < ,
PAGE 68

r e r e e e en e QQQ B } BB 0 BB 0 QO e r e A r r LO n f§ f Q-f QOQQ [ (VO,$QQffH f ,Q WKHVH HTXDWLRQV LV WKH .URQHFNHU GHOWD Q Q e en e LV D M V\PERO > e en e _P Pn P W2 LV D M V\PERO >@ DQG

PAGE 69

e M"n D L O O r e e LV D M V\PERO >@ 7KH (J DUH FRHIILFLHQWV LQ D U Y D VSKHULFDO KDUPRQLF H[SDQVLRQ RI WKH DQLVRWURSLF SRWHQWLDO X 7KH VXSHUVFULSW r RQ (J LQGLFDWHV D FRPSOH[ FRQMXJDWH 'HWDLOV RI WKH H[SDQVLRQ DQG HTXDWLRQV IRU (J IRU VSHFLILF LQWHUDFWLRQV DUH JLYHQ LQ $SSHQGL[ & e A ,Q f DQG f -QS 7 f LV WKH VLQJOH LQWHJUDO N N -QS 7 f r rQf r GUf UfY JULA f 9DOXHV RI WKH LQWHJUDOV DUH WDEXODWHG HOVHZKHUH DQG KDYH EHHQ ILWWHG Q N N WR DQ HPSLULFDO HTXDWLRQ LQ S DQG 7 >@ \ ,Q f / eAQJQ f LV WKH WULSOH LQWHJUDO /AeQQnf r rQOf GUO UO r rQnOf GU UO r r UOU U Un ; GU UAU UO UA JR/AUUOUA 3eAFRV fOA nf W K +HUH LV WKH e RUGHU /HJHQGUH SRO\QRPLDO DQG LV WKH LQWHULRU DQJOH \ DW PROHFXOH LQ WKH WULDQJOH IRUPHG E\ UAM UA! DQFr Uf 9DOXHV RI / DUH WDEXODWHG LQ $SSHQGL[ ( DQG KDYH EHHQ ILWWHG WR DQ HPSLULFDO HTXDWLRQ N N LQ S DQG 7 ,Q f eeneQJQVQAf LV WKH WULSOH LQWHJUDO

PAGE 70

.@ 7KH DUH VSKHULFDO KDUPRQLFV LQ WKH FRQYHQWLRQ RI 5RVH >@ ,Q f DQG f WKH FX DUH WKH LQWHULRU DQJOHV DW PROHFXOH L LQ WKH WULDQJOH IRUPHG E\ Uf UOf DQG Uf ([SUHVVLRQV IRU IRU PXOWLSROH LQWHUDFWLRQV DUH JLYHQ LQ $SSHQGL[ $ 1RWH WKDW \A$$ n $ VV n Vf JLYHQ E\ f YDQLVKHV LI eenef LV RGG RU LI WKH PROHFXOHV DUH OLQHDU DQG HLWKHU eAeeAf RU )r efenef LV RGG \27f $$nVV n f YDQLVKHV XQOHVV WKH DQLVRWURSLF SRWHQWLDO = = = =-G FRQWDLQV e VSKHULFDO KDUPRQLFV )r )r )r )r 6SHFLILF H[SUHVVLRQV IRU @

PAGE 71

A$DVnf I Q Q V V A Q Q V V $$$66nf “N7 f <)%0VVf 9 rr /9QV9! / eQ Q f V Vn $E$$nVVnf “N7 f D D <$$$n$VVfVf Q Q Q fO V V V Q Q Q n V V V $ $$n$VVnVf M $ 1N7 f <%$$f$VVnVf Sr7r .
PAGE 72

<9H WW3r>-8@ f ZKHUH WKH LQWHJUDOV DUH GHILQHG E\ f DQG DUH WDEXODWHG HOVHZKHUH >@ )LJXUHV DQG VKRZ WKH HIIHFW RI GLSRODU IRUFHV $ ( e f RQ WKH VXUIDFH WHQVLRQ DV SUHGLFWHG E\ WKH VHFRQG RUGHU WKHRU\ WKH WKLUG RUGHU WKHRU\ DQG WKH 3DG WKHRU\ f 7KH SRLQWV RQ WKHVH ILJXUHV DUH 0RQWH &DUOR FDOFXODWLRQV RI WKH )RZOHU PRGHO VXUIDFH WHQVLRQ IRU WKH /HQQDUG-RQHV SOXV GLSROH IOXLGV >@ )LJXUH VKRZV VLPLODU UHVXOWV IRU TXDGUXSROHV $ f 7KH VHFRQG RUGHU DQG WKLUG RUGHU WHUPV XVHG LQ WKHVH FDOFXODWLRQV ZHUH GHWHUPLQHG IURP WKH H[SUHVVLRQV JLYHQ LQ $SSHQGL[ 7KH UHVXOWV LQ )LJXUHV DQG DUH VLPLODU WR WKRVH IRXQG IURP WKH FRUUHVSRQGLQJ WKHRULHV IRU WKH +HOPKROW] IUHH HQHUJ\ >@ ,QFOXGLQJ WKH WKLUG RUGHU WHUP H[WHQGV WKH UDQJH RI DSSOLFDWLRQ RI WKH H[SDQVLRQ VRPHZKDW KRZHYHU WKH WKLUG RUGHU WHUP RYHUFRUUHFWV WKH VHFRQG RUGHU WKHRU\ IRU _Mr RU 4r 7KH 3DG WKHRU\ RQ WKH RWKHU KDQG LQWHUSRODWHV EHWZHHQ WKH VHFRQG DQG WKLUG RUGHU WKHRULHV ,Q WKH FDVH RI WKH SRODU IOXLGV WKH 3DG DJUHHV ZHOO ZLWK 0RQWH &DUOR UHVXOWV XS WR \r ,Q )LJXUH FRPSDULVRQ LV PDGH RI WKH HIIHFWV RI YDULRXV DQLVRn WURSLHV RQ WKH VXUIDFH WHQVLRQ 7KH GLSROH DQG TXDGUXSROH FXUYHV DUH WKH 3DG UHVXOWV IURP )LJXUHV DQG UHVSHFWLYHO\ $V LQ WKH FDVH RI EXON IOXLG WKHUPRG\QDPLF SURSHUWLHV IRU D JLYHQ YDOXH RI WKH PXOWLSROH VWUHQJWK \ RU 4 f WKH TXDGUXSROH SRWHQWLDO LV IRXQG WR KDYH D ODUJHU HIIHFW RQ VXUIDFH WHQVLRQ WKDQ WKH GLSROH SRWHQWLDO

PAGE 73

)LJXUH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 'LSROH 0RGHO 3RWHQWLDO SDA N7H

PAGE 74

8f )LJXUH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 'LSROH 0RGHO 3RWHQWLDO SDA N7H r

PAGE 75

4fFUf )LJXUH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 4XDGUXSROH 0RGHO 3RWHQWLDO SD N7H

PAGE 76

H )LJXUH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ IRU )OXLGV RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 9DULRXV $QLVRWURSLF 3RWHQWLDOV S N7H

PAGE 77

7KH DQLVRWURSLF RYHUODS DQG GLVSHUVLRQ UHVXOWV DUH IURP WKH VHFRQG RUGHU ) ) WKHRU\ XVLQJ H[SUHVVLRQV IRU
PAGE 79

8J4FUH WW3r >M -X@ f < ZKHUH WKH LQWHJUDOV DUH GHILQHG E\ f 7KH YDOXHV IRU X ZHUH Q VD GHWHUPLQHG LQ WKH VLPXODWLRQ E\ HYDOXDWLQJ WKH HQVHPEOH DYHUDJH JLYHQ LQ &KDSWHU ,Q YLHZ RI WKH GHPRQVWUDWHG LQDSSOLFDELOLW\ RI (TXDWLRQ f LQ WKH )RZOHU DSSUR[LPDWLRQ IRU /HQQDUG-RQHV IOXLGV >@ WKH DJUHHPHQW EHWZHHQ WKH WKHRU\ DQG FRPSXWHU VLPXODWLRQ VKRZQ LQ 7DEOH LV VXUSULVLQJ 6LQFH WKH LQFRQVLVWHQF\ LQ WKH )RZOHU H[SUHVVLRQV IRU \ DQG X LV QRW OLPLWHG WR VSKHULFDO SRWHQWLDOV >@ WKH UHVXOWV LQ 7DEOH V PXVW EH IRUWXLWRXV 7KH DJUHHPHQW PD\ LQ SDUW EH DWWULEXWHG WR WKH KLJK GHQVLW\ VWDWH FRQGLWLRQV FRQVLGHUHG ZKHUHLQ WKH )RZOHU DSSUR[LPDn WLRQ LV PRUH DFFXUDWH 0XFK RI WKH DJUHHPHQW KRZHYHU PXVW EH GXH WR FDQFHOODWLRQ RI HUURUV EHWZHHQ WKH \ DQG G\G7 WHUPV LQ f 6XUIDFH 7HQVLRQ &DOFXODWLRQV IRU 5HDO )OXLGV 7KH 3DG SHUWXUEDWLRQ WKHRU\ GHYHORSHG LQ &KDSWHU KDV EHHQ XVHG WR SUHGLFW SXUH OLTXLG VXUIDFH WHQVLRQV IRU &2A &A+A DQG +%U ,Q WKHVH FDOFXODWLRQV WKH UHIHUHQFH IOXLG ZDV WDNHQ WR EH D 0LH Qf IOXLG 7KH DQLVRWURSLHV FRQVLGHUHG ZHUH WKH PXOWLSROH LQWHUDFWLRQV XS WKURXJK WKH TXDGUXSROHTXDGUXSROH WHUP DV ZHOO DV DQLVRWURSLF GLVSHUVLRQ DQG RYHUODS FRQWULEXWLRQV 0LH Qf 5HIHUHQFH &RQWULEXWLRQ WR 6XUIDFH 7HQVLRQ 7KH 0LH Qf IOXLG ZDV WDNHQ DV WKH UHIHUHQFH LQ WKH SHUWXUEDWLRQ WKHRU\ FDOFXODWLRQV VLQFH 7ZX KDV GHWHUPLQHG YDOXHV IRU H 2 DQG Q E\

PAGE 80

ILWWLQJ SHUWXUEDWLRQ WKHRU\ FDOFXODWLRQV RI OLTXLG GHQVLWLHV DQG SUHVVXUHV WR H[SHULPHQWDO YDOXHV DORQJ WKH RUWKREDULF OLQH IRU WKH IOXLGV FRQVLGHUHG KHUH >@ ,W LV IHOW WKDW WKH WHVW RI WKH 3DG WKHRU\ IRU VXUIDFH WHQn VLRQ LV VWUHQJWKHQHG E\ XVLQJ WKHVH LQGHSHQGHQWO\ GHWHUPLQHG SRWHQWLDO SDUDPHWHUV 7KH 0LH Qf SRWHQWLDO XAQAUf LV JLYHQ E\ >@ XQ!fUf QH Qf Qf f 7R GHWHUPLQH WKH VXUIDFH WHQVLRQ IRU WKLV SRWHQWLDO WKH +HOPKROW] IUHH HQHUJ\ RI WKH QRQKRPRJHQHRXV WZR SKDVH Qf IOXLG LV H[SDQGHG WR ILUVW RUGHU LQ SRZHUV RI Q A DERXW WKH /HQQDUG-RQHV f IOXLG IUHH HQHUJ\ 7KH VXUIDFH WHQVLRQ LV REWDLQHG E\ DSSO\LQJ WKH WKHUPRG\QDPLF GHILQLWLRQ f 7KHQ WKH )RZOHU DSSUR[LPDWLRQ LV PDGH LQ RUGHU WR REWDLQ D IRUP DPHQDEOH WR FDOFXODWLRQ 7KH H[SDQVLRQ RI WKH Qf IUHH HQHUJ\ DERXW WKH f PD\ EH GRQH LQ WZR ZD\V ,Q RQH PHWKRG WKH YDOXHV RI e DQG D DUH WDNHQ WR EH WKH VDPH IRU WKH WZR IOXLGV 7KH H[SUHVVLRQ IRU  8m! -f` B B Q f ,Q f LV WKH VLQJOH LQWHJUDO KffS?Wrf H r rQnBf r f r GU U QU JY f U f f

PAGE 81

9DOXHV RI WKLV LQWHJUDO IRU Qn DUH WDEXODWHG LQ $SSHQGL[ DQG r r KDYH EHHQ ILWWHG WR DQ HPSLULFDO HTXDWLRQ LQ S DQG 7 ,Q WKH VHFRQG PHWKRG RI GRLQJ WKH H[SDQVLRQ WKH YDOXHV RI e DQG UA DUH WDNHQ WR EH WKH VDPH IRU WKH Qf DQG f IOXLGV +HUH UA LV WKH YDOXH RI U ZKHUH XUf e 7KH H[SUHVVLRQ IRU UHVXOWLQJ IURP WKLV VHFRQG H[SDQVLRQ LV )Qf \ f D R )f OeQf r B f f ‘ + f :KHQ WKH Qf IOXLG LV XVHG DV WKH UHIHUHQFH WKH VHFRQG DQG WKLUG WLRQ Q f RUGHU WHUPV DQG \A LQ WKH SHUWXUEDWLRQ WKHRU\ LQYROYH LQWHJUDOV RYHU WKH Qf SDLU FRUUHODWLRQ IXQFWLRQ JAffAUf 7KHVH Qf FRUUHODWLRQ IXQFWLRQV FDQ EH UHODWHG WR /HQQDUG-RQHV JAAfAAUf IXQFWLRQV IRU ZKLFK WKHUH DUH PROHFXODU G\QDPLFV UHVXOWV >@f LQ WKH IROORZLQJ ZD\ >@ DQG JQfUf V JfUHS HaX QfffUHS \+6GQf` JffUf JfffUHS H%X8fffUHS \+6Gf` f f ZKHUH WKH VXSHUVFULSWV UHS DQG +6 LQGLFDWH WKH UHSXOVLYH DQG KDUG VSKHUH SRWHQWLDO FRQWULEXWLRQV UHVSHFWLYHO\ 7KH IXQFWLRQ \ LV GHILQHG E\ V XUf V \ Uf H J Uf f

PAGE 82

,Q f DQG f WKH KDUG VSKHUH GLDPHWHU G LV WDNHQ WR EH >@ G > HA:MGU )XUWKHU DVVXPLQJ f \+6GQff r \+9ff f f f DQG f JLYH JQfUf JfUf H>XQf!UHO! X8fUm3M f ZKHUH XQ! fUFS XD!UHS YDQOVKHV eRU U U LQ 8VLQJ f ZLWK WKH 0LH Qf SRWHQWLDO LQ WKH LQWHJUDOV -A (TXDWLRQ f DQG .eeneQQnQf (TXDWLRQ f 7ZX KDV IRXQG WKH UHVXOWLQJ YDOXHV WR EH QHJOLJLEO\ GLIIHUHQW IURP WKRVH YDOXHV REWDLQHG IRU WKH /HQQDUG-RQHV f SRWHQWLDO >@ DW OHDVW IRU YDOXHV RI Q FORVH WR +HQFH LQ WKH FDOFXODWLRQV UHSRUWHG KHUH WKH /HQQDUG-RQHV f SDLU FRUUHODWLRQ IXQFWLRQ KDV EHHQ XVHG LQ HYDOXDWLQJ WKH WHUPV \A DQG LQ WKH VXUIDFH WHQVLRQ H[SDQVLRQ ,Q WKH FDOFXODWLRQV UHSRUWHG KHUH (TXDWLRQ f ZDV XVHG WR GHWHUPLQH WKH UHIHUHQFH IOXLG FRQWULEXWLRQ 9DOXHV IRU WKH /HQQDUG-RQHV WHUP \Af ZHUH REWDLQHG IURP D FRUUHVSRQGLQJ VWDWHV SORW RI VXUIDFH WHQVLRQ IRU VLPSOH IOXLGV )LJXUH )RU WKH WHPSHUDWXUH UDQJH  N7H WKH FXUYH LQ )LJXUH REH\V

PAGE 83

)LJXUH &RUUHVSRQGLQJ 6WDWHV 3ORW IRU 6XUIDFH 7HQVLRQ RI 6LPSOH /LTXLGV

PAGE 84

f r r @ DQG 7ZX >@ LQ VWXGLHV RI EXON IOXLG WKHUPRG\QDPLF SURSHUWLHV 7KH PXOWLSROH FRQWULEXn WLRQV WR VXUIDFH WHQVLRQ IRU WKLV PRGHO SRWHQWLDO DUH REWDLQHG E\ FRPELQLQJ f f f DQG f ZLWK f f f DQG f UHVSHFWLYHO\ <$ r <$f <$f <$f <$ f M8f \Mr f 7r ‘&t \A$ f \A f 7t 77t <% <%+f <%+f
PAGE 85

([SUHVVLRQV IRU WKH WHUPV RQ WKH ULJKW VLGH RI f f DQG f )r DUH JLYHQ LQ $SSHQGL[ 7KH WHUP LV ]HUR IRU PXOWLSROHV VLQFH RQO\ WHUPV ZLWK O  VSKHULFDO KDUPRQLFV RFFXU LQ WKH PXOWLSROH SRWHQWLDOV 7KH GLVSHUVLRQ DQG RYHUODS DQLVRWURSLHV LQ f KDYH EHHQ LQFOXGHG LQ RQO\ WKH VHFRQG RUGHU WHUP \A LQ FDOFXODWLQJ WKH VXUIDFH WHQVLRQ IURP WKH 3DG SHUWXUEDWLRQ WKHRU\ f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b IRU & DQG OHVV WKDQ b IRU &+ DQG +%U IRU WKH WHPSHUDWXUHV VKRZQ LQ WKH DFFRPSDQ\LQJ ILJXUHV 7KH FRQVLVWHQW GHYLDWLRQV EHWZHHQ WKHRU\ DQG H[SHULPHQW HVSHFLDOO\ IRU &+ DQG +%U VXJJHVW WKDW DGMXVWPHQW RI WKH SRWHQWLDO SDUDPHWHUV ZRXOG LPSURYH WKH DJUHHPHQW ,W LV QRW D YHU\ LQIRUPDWLYH WHVW RI WKH WKHRU\ KRZHYHU WR DGMXVW SRWHQWLDO

PAGE 86

FWf N7H )LJXUH 6XUIDFH 7HQVLRQ IRU &2 FRPSDULQJ 3HUWXUEDWLRQ 7KHRU\ &DOFXODWLRQV SRLQWVf ZLWK ([SHULPHQWDO 9DOXHV OLQHf

PAGE 87

\ Ra )LJXUH 6XUIDFH 7HQVLRQ IRU &+ DQG +%U FRPSDULQJ 3HUWXUEDWLRQ 7KHRU\ &DOFXODWLRQV SRLQWVf ZLWK ([SHULPHQWDO 9DOXHV OLQHVf

PAGE 88

7$%/( 3RWHQWLDO 3DUDPHWHU 9DOXHV 8VHG LQ &DOFXODWLQJ 6XUIDFH 7HQVLRQ )OXLG HN .f $f Q 0f HVX FPf 4f HVX FP f FR f§ >@ FK f§ >@ +%U >@ >@ f§ f§ $OO YDOXHV IRU H R Q DQG DUH WDNHQ IURP 7ZX >@

PAGE 89

SDUDPHWHUV XVLQJ VXUIDFH WHQVLRQ LQ RUGHU WR FDOFXODWH VXUIDFH WHQVLRQ 7KH GHYLDWLRQV EHWZHHQ WKHRU\ DQG H[SHULPHQW VKRZQ LQ )LJXUHV DQG GR QRW LQFUHDVH PXFK ZLWK WHPSHUDWXUH DQG LQ IDFW IRU &&A WKH GHYLDn WLRQV GHFUHDVH 7KLV LV LQ FRQWUDVW WR WKH UHVXOWV IRXQG IRU VLPSOH /HQQDUG-RQHV IOXLGV ZKHUHLQ VXUIDFH WHQVLRQ FDOFXODWHG LQ WKH )RZOHU PRGHO VKRZ UDSLGO\ LQFUHDVLQJ GLVDJUHHPHQW ZLWK H[SHULPHQW DV WKH WHPSHUDWXUH LV LQFUHDVHG >@ )XUWKHU WKH )RZOHU PRGHO YDOXHV RI VXUIDFH WHQVLRQ IRU /HQQDUG-RQHV IOXLGV DUH JHQHUDOO\ ODUJHU WKDQ WKH H[SHULPHQWDO YDOXHV DW WKH VDPH WHPSHUDWXUH ,W PD\ EH WKDW LQ DGGLn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n < <$ <% <$ <% f $ VLPSOH FRUUHODWLRQ IRU \ PD\ EH REWDLQHG E\ PDNLQJ WKH YDQ GHU :DDOV W\SH DVVXPSWLRQ WKDW WKH UHIHUHQFH IOXLG UDGLDO GLVWULEXWLRQ IXQFWLRQ LV D FRQVWDQW

PAGE 90

"R/U` & f 8VLQJ f LQ f f f DQG f WRJHWKHU ZLWK D VLPLODU DSSUR[LPDWLRQ IRU WKH WULSOHW FRUUHODWLRQ IXQFWLRQ J4UAUUAf f EHFRPHV LQ UHGXFHG IRUP r < r r r r D S DnS DnS DnS \ / / / fR r r r r f ZKHUH Dn DO f f§ D D" GU f U @IX f! D DfOW f &2 &2 r r $ $ GU U GU U U $ $ $ $ I U $$$ $ GU Uf X fX f! $ D D D ::: UOn [ U U U f N f D 7 $ $ $7 GU U X -f! D AA $ $ GU UO $ $ GU U Ur Ur $ $ GUU U U f $$$ $$$ ; X fX fX f! U U Ufaf D Dn DY DfOr-D! f (TXDWLRQ f FDQ EH ZULWWHQ LQ DQ HTXLYDOHQW IRUP XVLQJ WKH FULWLFDO FRQVWDQWV 7A 9 DQG SA DV UHGXFLQJ SDUDPHWHUV UDWKHU WKDQ WKH SRWHQWLDO SDUDPHWHUV H DQG 2

PAGE 91

\ 9e A $ 9 UR5 7f 7 5 5 5 5 ZKHUH @ +HUH KRZHYHU WKH SRWHQWLDO LV IRU SRO\DWRPLF IOXLGV DQG WKHUHIRUH FRQWDLQV SDUDPHWHUV LQ DGGLWLRQ WR WKH HQHUJ\ DQG GLVWDQFH SDUDPHWHUV e DQG 2 HJ WKH PXOWLSROH PRPHQWV \ 4 DQG DQLVRWURSLF SRODUL]DELOLW\ DQG RYHUODS SDUDPHWHUV )RU VXFK DQ LQWHUPROHFXODU SRWHQWLDO LI WKH XVXDO GHULYDWLRQ RI FRUUHVSRQGLQJ VWDWHV WKHRU\ LV IROORZHG >@ RQH ILQGV r F r r 7F aa FA\ 4 N f f ff r r r 3F SFD F\ 4 Nffff S D r & r r a A f§ \ 4 f f f ZKHUH FA FA DQG FA DUH FRQVWDQWV RQO\ LI WKH DQLVRWURSLF SRWHQWLDO SDUD r r PHWHUV \ 4 R fff DUH NHSW IL[HG +RZHYHU WKHVH FRQVWDQWV FDQ EH DEVRUEHG LQ WKH WHUPV DA D HWF DV KDV EHHQ GRQH LQ f 7KXV D_FFA HWF 7KH WUDQVIRUPDWLRQ IURP WKH SRWHQWLDO SDUDPHWHUV e DQG D DV UHGXFLQJ SDUDPHWHUV WR FULWLFDO FRQVWDQWV DV UHGXFLQJ SDUDPHWHUV PD\ WKHQ EH PDGH LQ WKH XVXDO ZD\ >@

PAGE 92

(TXDWLRQ f KDV EHHQ XVHG DV D EDVLV IRU FRUUHODWLQJ VXUIDFH WHQVLRQ E\ WUHDWLQJ WKH SDUDPHWHUV DA DV VHPLHPSLULFDO FRQVWDQWV 9DULRXV WUXQFDWHG IRUPV RI f LQFOXGLQJ LWV 3DG IRUP DQDORJRXV WR f KDYH EHHQ WHVWHG DJDLQVW H[SHULPHQWDO VXUIDFH WHQVLRQV IRU QXPHURXV SRO\n DWRPLF OLTXLGV 7KH IRUP JLYLQJ WKH EHVW FRPSDULVRQ ZLWK H[SHULPHQW ZDV IRXQG WR EH WKDW WHUPLQDWHG DIWHU WKH \ WHUP =% <5
PAGE 93

7$%/( 9DOXHV IRU WKH 3DUDPHWHUV DQG DA LQ WKH 6XUIDFH 7HQVLRQ &RUUHODWLRQ RI (TXDWLRQ f 6XEVWDQFH 5DQJH DAf D f 5HIHUHQFHV 7 U 3 < 3DUDIILQV (WKDQH 3URSDQH Q%XWDQH L%XWDQH Q3HQWDQH L3HQWDQH Q+H[DQH Q+HSWDQH Q2FWDQH L2FWDQH Q1RQDQH Q'HFDQH Q'RGHFDQH Q7ULGHFDQH Q7HWUDGHFDQH Q3HQWDGHFDQH Q+H[DGHFDQH Q+HSWDGHFDQH Q2FWDGHFDQH Q1RQDGHFDQH Q(LFRVDQH &\FORSDUDIILQV &\FORSHQWDQH 0HWK\OF\FOR SHQWDQH (WK\OF\FOR SHQWDQH &\FORKH[DQH 0HWK\OF\FOR KH[DQH W 5HGXFHG WHPSHUDWXUH UDQJH RYHU ZKLFK D DQG Df ZHUH ILWWHG 6RXUFHV RI H[SHULPHQWDO GDWD IRU OLTXLG GHQVLWLHV S DQG VXUIDFH WHQVLRQV \

PAGE 94

7$%/( &RQWLQXHGf 6XEVWDQFH 5DQJH D f D82 f 5HIHUHQFHV 7U 3 < 2OHILQV 3URS\OHQH %XWHQH %XWHQH +H[HQH 2FWHQH &\FORSHQWHQH $URPDWLFV %HQ]HQH 7ROXHQH (WK\OEHQ]HQH ,VRSURS\OEHQ]HQH $OFRKROV 0HWKDQRO Q3URSDQRO L3URSDQRO Q%XWDQRO 2UJDQLF +DOLGHV 0HWK\O &KORULGH (WK\O %URPLGH &DUERQ 7HWUD FKORULGH &KORUREHQ]HQH 2[LGHV &DUERQ PRQR[LGH &DUERQ GLR[LGH :DWHU 2WKHUV $FHWLF $FLG $FHWRQH $PPRQLD $QLOLQH &DUERQ GLVXOILGH &KORULQH 'LHWK\O HWKHU (WK\O DFHWDWH

PAGE 95

)LJXUH 7HVW RI 6XUIDFH 7HQVLRQ &RUUHODWLRQ OLQHf IRU &&!

PAGE 96

)LJXUH 7HVW RI 6XUIDFH 7HQVLRQ &RUUHODWLRQ OLQHf IRU $FHWLF $FLG

PAGE 97

)LJXUH 7HVW RI 6XUIDFH 7HQVLRQ &RUUHODWLRQ OLQHf IRU 0HWKDQRO

PAGE 98

RI SUHGLFWHG VXUIDFH WHQVLRQV ZLWK H[SHULPHQWDO YDOXHV IRU WKHVH DQG VHYHUDO RWKHU OLTXLGV DUH VKRZQ LQ )LJXUHV DQG 7KH OLQHDU UHODWLRQ VXJJHVWHG E\ f LV QRW REH\HG E\ H[SHULn PHQWDO GDWD IRU ORQJ FKDLQ K\GURFDUERQV KRZHYHU 7\SLFDO SORWV DUH VKRZQ LQ )LJXUH ,Q VSLWH RI WKH SRRU FRUUHODWLRQ IRU WKHVH VXEVWDQFHV WKH SUHGLFWHG VXUIDFH WHQVLRQV XVLQJ f ZHUH XVXDOO\ ZLWKLQ b RI WKH H[SHULPHQWDO YDOXHV DV VKRZQ LQ )LJXUH *HQHUDOO\ WKH FRUUHODWLRQ RI f UHSURGXFHG WKH H[SHULPHQWDO GDWD IRU VXEVWDQFHV WHVWHG KHUH ZLWKLQ b IRU YDOXHV RI 7 ,Q RUGHU IRU WKH FRUUHODWLRQ WR DSSO\ LQ WKH FULWLFDO UHJLRQ ZH PXVW KDYH &3 &3 f KHQFH IURP f f 7KHQ f \U ‘
PAGE 99

)LJXUH &RPSDULVRQ RI 6XUIDFH 7HQVLRQ &DOFXODWHG IURP WKH &RUUHODWLRQ OLQHVf ZLWK ([SHULPHQWDO 9DOXHV SRLQWVf IRU 6HYHUDO 3RO\DWRPLF /LTXLGV

PAGE 100

a ; 1r )LJXUH  &RPSDULVRQ RI 6XUIDFH 7HQVLRQ &DOFXODWHG IURP WKH &RUUHODWLRQ OLQHVf ZLWK ([SHULPHQWDO 9DOXHV SRLQWVf IRU 6HYHUDO 3RO\DWRPLF /LTXLGV

PAGE 101

)LJXUH 7HVW RI 6XUIDFH 7HQVLRQ &RUUHODWLRQ OLQHVf IRU Q+H[DQH DQG Q2FWDQH

PAGE 102

W7F )LJXUH &RPSDULVRQ RI 6XUIDFH 7HQVLRQV &DOFXODWHG IURP WKH &RUUHODWLRQ OLQHVf ZLWK ([SHULPHQWDO 9DOXHV SRLQWVf IRU 6HYHUDO +\GURFDUERQV

PAGE 103

&+$37(5 9$325/,48,' '(16,7<25,(17$7,21 352),/(6 &DOFXODWLRQ RI VXUIDFH WHQVLRQ IRU SRO\DWRPLF IOXLGV IURP WKH JHQHUDO .LUNZRRG%XII HTXDWLRQ f UHTXLUHV NQRZOHGJH RI WKH LQWHUIDFLDO SDLU GLVWULEXWLRQ IXQFWLRQ I ]AUAAMNA f 7KLV IXQFWLRQ PD\ EH ZULWWHQ LQ WHUPV RI WKH LQWHUIDFLDO VLQJOHW DQG SDLU FRUUHODWLRQ IXQFWLRQV S]AAf DQG J]Af§LM-O-A f UHV3HFWfYHr\fn I ]OWO:A SA]LZLA 3]8fA Jn=O&OWA f 7KXV f ZLWK f LQGLFDWH WKDW WKH VXUIDFH WHQVLRQ RI SRO\DWRPLF IOXLGV LV D IXQFWLRQ RI DPRQJ RWKHU WKLQJV WKH FRQFHQWUDn WLRQ DQG RULHQWDWLRQ RI PROHFXOHV LQ WKH LQWHUIDFLDO UHJLRQ $V LQGLFDWHG LQ &KDSWHU PXFK HIIRUW KDV EHHQ H[SHQGHG LQ GHWHUPLQLQJ WKH QDWXUH RI WKH LQWHUIDFLDO GHQVLW\ SURILOH S]Af IRU DWRPLF IOXLGV $OPRVW QRWKLQJ KDV EHHQ GRQH LQ GHWHUPLQLQJ WKH FRUUHVSRQGLQJ SURILOH S]ARMAf IRU PRUH FRPSOLFDWHG PROHFXOHV ,Q WKLV FKDSWHU D ILUVW RUGHU SHUWXUEDWLRQ WKHRU\ LV GHYHORSHG IRU S]A-Af DQG FDOFXODWLRQV DUH SUHVHQWHG IRU PRGHO IOXLGV LQWHUDFWLQJ ZLWK DQLVRWURSLF RYHUODS DQG GLVSHUVLRQ IRUFHV )LUVW 2UGHU 3HUWXUEDWLRQ 7KHRU\ IRU S]A4-Af 'HYHORSPHQW RI WKH *HQHUDO )LUVW 2UGHU 7HUP )RU D SXUH OLTXLG ZLWK 1 QRQVSKHULFDO PROHFXOHV LQWHUDFWLQJ ZLWK

PAGE 104

1 1 WRWDO SRWHQWLDO 8BU Z f DW WHPSHUDWXUH 7 WKH VLQJOHW GLVWULEXWLRQ IXQFWLRQ LV I UAXAf 1 = 1 1 1O 1O 8U : f GU GZ H f§ f = LV WKH FRQILJXUDWLRQDO LQWHJUDO RI f N7 DQG f ^!;A LV D VHW RI (XOHU DQJOHV JLYLQJ WKH PROHFXODU RULHQWDWLRQV 1RZ FRQVLGHU WKH SXUH OLTXLG WR EH LQ HTXLOLEULXP ZLWK LWV YDSRU DQG WR KDYH D SODQDU LQWHUIDFH EHWZHHQ WKH YDSRU DQG OLTXLG SKDVHV $VVXPH D VSDFHIL[HG FRRUGLQDWH V\VWHP RULHQWHG VXFK WKDW WKH [\ SODQH OLHV LQ WKH SODQH RI WKH LQWHUIDFH DQG WKH SRVLWLYH ]D[LV LV GLUHFWHG LQWR WKH YDSRU SKDVH ,I WKH IOXLG DW DQ\ KHLJKW ] LV FRQVLGHUHG WR EH XQLIRUP LQ WKH [ DQG \ GLUHFWLRQV WKHQ RQO\ WKH ]FRPSRQHQW RI f DSSOLHV S8MAf 1 1 1O 1O U : f GU GWR H f§ f f JLYHV WKH XQQRUPDOL]HG SUREDELOLW\ RI ILQGLQJ D PROHFXOH ZLWK RULHQWDWLRQ DW KHLJKW ]A LQ WKH IOXLG 7KH SHUWXUEDWLRQ WKHRU\ IRU S]AFRAf LV GHYHORSHG E\ ZULWLQJ WKH SRWHQWLDO DV D VXP RI LVRWURSLF UHIHUHQFH SOXV DQLVRWURSLF SHUWXUEDWLRQ WHUPV 8U1-1f 8 U1f ;8 $f1f f f§ R f§ D f§ ZKHUH ; LV D SHUWXUEDWLRQ SDUDPHWHU DV LQ f 1RZ H[SDQG S]A2-Af

PAGE 105

LQ D 7D\ORU VHULHV ZLWK UHVSHFW WR WKH SDUDPHWHU $ DERXW WKH UHIHUHQFH LH DERXW $ S&]Af S] f %S]MAf [ Ua S] LfA Q $ $ $ $ f f f f $ +HUH ZH FRQVLGHU RQO\ WKH ILUVW RUGHU WKHRU\ 3R]O` S A 3MA=M2Af f ZKHUH SA]Af $ S]WRf $ f $ 7DNLQJ WKH GHULYDWLYH RI f ZLWK UHVSHFW WR $ DQG DVVXPLQJ WKH SRWHQn WLDO WR EH SDLUZLVH DGGLWLYH DV LQ (TXDWLRQ f ZH ILQG 6SL]Af $ 4 N $ GL IR]OLOf XD9 B GU GU I4=OUUf AfA 7 3f]Lf = 2 GU GU I ] U f X f! f f§ f§ R f§ D AA ZKHUH WKH DQJOH DYHUDJH fff! LV GHILQHG E\ f ,I WKH UHIHUHQFH IOXLG LV QRZ FKRVHQ WR EH D 3RSOH UHIHUHQFH f WKHQ f FDXVHV WKH ODVW WZR WHUPV LQ f WR YDQLVK OHDYLQJ

PAGE 106

S]Xf G[ [ R 4 GU I ]QU f X f! f§ R f§ D rff f )RU DQLVRWURSLF SRWHQWLDOV KDYLQJ e RUGHU VSKHULFDO KDUPRQLFV X f! D X f 7KXV f YDQLVKHV IRU PXOWLSRODU LQWHUDFWLRQV DQG WKH GHQVLW\ RULHQWDWLRQ SURILOH LV WR ILUVW RUGHU MXVW WKH UHIHUHQFH SURILOH 3R]O` $QLVRWURSLF RYHUODS DQG GLVSHUVLRQ LQWHUDFWLRQV FRQWDLQ e VSKHULFDO KDUPRQLFV KRZHYHU VR f GRHV QRW YDQLVK IRU WKHVH SRWHQWLDOV &RPELQLQJ f ZLWK f DQG f DQG VHWWLQJ ; WR UHJDLQ WKH UHDO IOXLG S8M: f 3R]O` 4 GUB I f X f! f§ R D WR f (YDOXDWLRQ RI SA]A2OfAf IRU $QLVRWURSLF 2YHUODS DQG 'LVSHUVLRQ 7KH DQLVRWURSLF RYHUODS SRWHQWLDO LV DSSUR[LPDWHG E\ WKH ILUVW WZR WHUPV LQ D VSKHULFDO KDUPRQLF H[SDQVLRQ X RYHU X f X RYHU RYHU f f 8VLQJ WKH JHQHUDO VSKHULFDO KDUPRQLF H[SDQVLRQ IRU X JLYHQ LQ $SSHQGL[ & ZLWK FRHIILFLHQWV ( IURP 7DEOH &O f EHFRPHV RYHU

PAGE 107

U D I ? X D RYHU OrUA &PAPPf [ 'ORfO! '}\! @ f UHGXFHV WR X RYHU FR H 3 &26 Af 3F26 f f ZKHUH 3 LV WKH VHFRQG RUGHU /HJHQGUH SRO\QRPLDO 7R REWDLQ WKH H[SUHVVLRQ DQDORJRXV WR f IRU DQLVRWURSLF GLVn SHUVLRQ ZH SURFHHG LQ D VLPLODU PDQQHU 7KH PRGHO IRU WKH DQLVRWURSLF GLVSHUVLRQ SRWHQWLDO LV EDVHG RQ /RQGRQnV SRODUL]DELOLW\ DSSUR[LPDWLRQ DQG LV ZULWWHQ M f GLV X f GLV X f GLV fGLVf f

PAGE 108

7KH WHUPV RQ WKH ULJKW VLGH RI f DUH HYDOXDWHG IURP WKH H[SDQVLRQ IRU LQ $SSHQGL[ & DQG WKH H[SUHVVLRQV IRU LQ 7DEOH &O 7DNLQJ WKH DQJOH DYHUDJH RI f RYHU XA WKH VHFRQG WZR WHUPV YDQLVK GXH WR DUJXPHQWV DQDORJRXV WR f 7KHQ FDUU\LQJ RXW VWHSV DQDORJRXV WR WKRVH FLWHG DERYH IRU RYHUODS ZH REWDLQ X NJ GLV ZB UL 3F26 9 3F26 ` f &RPELQLQJ f RU f ZLWK f DQG f DQG QRWLQJ WKDW IRU OLQHDU PROHFXOHV WGA ^A` RQO\ 3O=OOf & 3AFRV A 3]Of rU G UO 3AFRV HLA 3Rn=nf JRA]OA f ZKHUH I R ]OeOf 3A]Lf S]A R ]LeL! f KDV EHHQ XVHG DQG IRU RYHUODS & 6HIW Q f ZKLOH IRU GLVSHUVLRQ & .( Q f 7R REWDLQ WKH LQWHUIDFLDO GHQVLW\ SURILOH S]Af (TXDWLRQ f LV LQWHJUDWHG RYHU WKH RULHQWDWLRQ ,QWHJUDWLRQ RI SA]AAf RYHU fA YDQLVKHV ZKHQ SA]AAf LV JLYHQ E\ f 7KXV IRU WKH ZHDNO\

PAGE 109

DQLVRWURSLF SRWHQWLDOV IRU ZKLFK WKH ILUVW RUGHU SHUWXUEDWLRQ WKHRU\ LV H[SHFWHG WR DSSO\ WKH LQWHUIDFLDO GHQVLW\ SURILOH S]Af LV MXVW WKH UHIHUHQFH IOXLG SURILOH S4]Af 7KXV WKHUH LV QR OD\HUHG VWUXFWXUH LQ WKH LQWHUIDFH RI IOXLGV ZKLFK LQWHUDFW ZLWK WKHVH ZHDNO\ DQLVRWURSLF SRWHQWLDOV 0RUHRYHU WKH *LEEV GLYLGLQJ VXUIDFH LV WKH VDPH WR ILUVW RUGHUf IRU WKH UHDO DQG UHIHUHQFH IOXLGV +RZHYHU LW LV FOHDU IURP f WKDW WKHVH IOXLGV ZLOO H[KLELW SUHIHUHQWLDO RULHQWDWLRQ RI WKH PROHFXOHV LQ WKH LQWHUIDFLDO UHJLRQ &DOFXODWLRQV RI S]AFDAf IRU 2YHUODS DQG 'LVSHUVLRQ &DOFXODWLRQDO )RUP IRU SA]A2-Af XVLQJ 7R[YDHUGfV 0RGHO IRU ,Q RUGHU WR DYRLG WKH LQWHJUDWLRQ RYHU A LQ f F\OLQGULFDO FRRUGLQDWHV DUH LQWURGXFHG JLYLQJ 3O=OOf & 3A&26 A S]Of G] 3R] GU U O Q 3f +@ LUOAn R]O]U! WW GM! f 3HUIRUPLQJ WKH -! LQWHJUDWLRQ DQG ZULWLQJ LQ GLPHQVLRQOHVV IRUP 3A=A 7& 3AFRV 9 3RA=A GUr UrA 3 n f } r r G] SR] ]O! BRR r N N N JR]O=Uf f 7R SURFHHG IXUWKHU DQ DSSUR[LPDWLRQ PXVW EH PDGH IRU WKH XQNQRZQ LQWHU N N N IDFLDO FRUUHODWLRQ IXQFWLRQ J ]]fUff :H FKRRVH WKH PRGHO XVHG E\ R

PAGE 110

N N 7R[YDHUG >@ LQ ZKLFK JA&]A]AU MAf LV DVVXPHG WR EH D GHQVLW\ ZHLJKWHG DYHUDJH RI WKH EXON IOXLG FRUUHODWLRQ IXQFWLRQV r r r R]O]U! r r r r $=O]f JR/U3/f %=O]f JR9U3Yf f ZKHUH r r $]]f N N N N DS ]f Df S ]f 3/ 39 3 f r t E=L]Lf 3W DSr]rf Df r r 3 ]ff 3W 3 f DQG D LV DQ DGMXVWDEOH SDUDPHWHU ,W KDV EHHQ SRLQWHG RXW WKDW WKH PRGHO f VDWLVILHV WKH V\PPHWU\ UHTXLUHPHQW RI 4]A]AUA AHnQ LQYDULDQW XQGHU D UHQXPEHULQJ RI PROHFXOHV DQG RQO\ ZKHQ D >@ 7R[YDHUG IRXQG D ZKHQ VROYLQJ WKH %RUQ*UHHQ@ ,Q WKDW FDOFXODWLRQ WKH DV\PSWRWLF OLPLW H[S Xf ZDV XVHG IRU WKH YDSRU SKDVH J\Uf ZKLOH WKH 3HUFXV@ / 2XU VROXWLRQ RI WKH %*< HTXDWLRQ IRU WKH /HQQDUG-RQHV S]Af XVLQJ 7R[YDHUGnV PHWKRG UHTXLUHG D 7KH UHVXOWLQJ SURILOH S]Af LV YHU\ QHDUO\ WKH VDPH DV 7R[YDHUGnV UHVXOWV DV VKRZQ LQ )LJXUH f )XUWKHU 7R[YDHUG KDV EHHQ DEOH WR UHSURGXFH WKH UHVXOWV IRU S]Af VKRZQ LQ )LJXUH E\ D PHWKRG ZKLFK GRHV QRW LQYROYH WKH PRGHO RI f >@

PAGE 111

=FU )LJXUH ,QWHUIDFLDO 'HQVLW\ 3URILOH IRU /HQQDUG-RQHV )OXLG

PAGE 112

7KXV ZH FRQFOXGH WKDW ZKLOH WKH PRGHO RI f LV RSHQ WR FULWLFLVP LW VHHPV WR SURYLGH WKH FRUUHFW UHVXOW &RPELQLQJ f ZLWK f JLYHV SA;f t N WW& 3 FRV f 3 ]Lf ] R $$$ $ G= 3R= =O! GUr Urrnf 3 fm, AU W$]O=` JR/Uf %]r]r! JR9Urf@ f 3A=3 O7& 3AFRV S 3RA=OA $$$ $ G] SR] ]O! $ $ $ $ >$]]f .O]f %]]f .Y]f@ f ZKHUH 9=! GUr rOQf N r n r n R/U! f .9]f a GUr rQf S n YU n $ rR9Uf f &DOFXODWLRQ 3URFHGXUH IRU S]ASf IRU $QLVRWURSLF 2YHUODS RU 'LVSHUVLRQ )RU D JLYHQ WHPSHUDWXUH 7 DQG OLTXLG DQG YDSRU EXON SKDVH fN N GHQVLWLHV S 7 DQG S FDOFXODWH WKH /HQQDUG-RQHV UHIHUHQFH IOXLG LQWHU R/ R9 N N IDFLDO GHQVLW\ SURILOH 34]Sf E\ VROYLQJ WKH %*< HTXDWLRQ

PAGE 113

Z r G] r r r r r r r r r r r AUOA ] Gf§ SR]O` 3R]O` R]O]UOf f f GU N N N 7KH PRGHO f LV XVHG IRU ]A]UOA DQFr WrLH YD3RU 3ADVH FRUUHODWLRQ IXQFWLRQ LV DSSUR[LPDWHG E\ H AXAUA 9HUOHWnV PROHFXODU G\QDPLFV UHVXOWV N DUH XVHG IRU J Uf >@ 'HWDLOV RI WKH VROXWLRQ PHWKRG DUH JLYHQ LQ >@ &DOFXODWH WKH LQWHJUDOV DQG .A f DQG f DV N IXQFWLRQV RI ]A OD WKHVH HTXDWLRQV Q LV JLYHQ E\ f DQG f IRU RYHUODS DQG GLVSHUVLRQ UHVSHFWLYHO\ )RU D JLYHQ YDOXH XVLQJ f IRU RYHUODS RU D JLYHQ N YDOXH XVLQJ f IRU GLVSHUVLRQ IL[ HLWKHU ]A RU A DQG VROYH r N f IRU DV D IXQFWLRQ RI A RU ]A UHVSHFWLYHO\ N N N N :LWK S DQG S GHWHUPLQHG REWDLQ S ]f IURP f R 5HVXOWV IRU S ]AAf IRU 'LVSHUVLRQ N N )LJXUH VKRZV WKH S ]AAf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

PAGE 114

)LJXUH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 6XUIDFH IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 'LVSHUVLRQ 0RGHO 3RWHQWLDO N7H

PAGE 115

)LJXUH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOHV IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 'LVSHUVLRQ 0RGHO 3RWHQWLDO N7H N

PAGE 116

)LJXUH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOHV IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV 'LVSHUVLRQ 0RGHO 3RWHQWLDO N7H

PAGE 117

2Q WKH YDSRU VLGH RI WKH LQWHUIDFH WKH FDOFXODWLRQV LQGLFDWH D VOLJKW SUHIHUHQFH IRU WKH RSSRVLWH RULHQWDWLRQV LH WKH PROHFXOHV WHQG WR VWDQG SHUSHQGLFXODU WR WKH SODQH RI WKH LQWHUIDFH $ SODXVLEOH H[SODQDWLRQ IRU WKLV FKDQJH LQ PRVW SUREDEOH RULHQWDn WLRQV IURP WKH OLTXLG WR WKH YDSRU SKDVHV FDQ EH REWDLQHG E\ FRQVLGHULQJ WKH GLIIHUHQFH EHWZHHQ WKH QRUPDO DQG WDQJHQWLDO FRPSRQHQWV RI WKH VWUHVV WHQVRU ^SA SA]f` 7R[YDHUG KDV FDOFXODWHG ^SA SA]f` DV D IXQFWLRQ RI ] E\ WZR GLIIHUHQW PHWKRGV >@ ,Q ERWK FDOFXODWLRQV KH ILQGV 3Q a 37]f` WR EH SRVLWLYH LQ WKH OLTXLG VLGH RI WKH LQWHUIDFH FRUn UHVSRQGLQJ WR D VXUIDFH WHQVLRQ LQ WKH OLTXLG 2Q WKH YDSRU VLGH KRZHYHU 7R[YDHUG ILQGV WKH GLIIHUHQFH ^SA SA]f`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r]AAf IRU 2YHUODS &DOFXODWLRQV KDYH EHHQ SHUIRUPHG IRU WKH /HQQDUG-RQHV SOXV RYHUODS SRWHQWLDO XVLQJ ERWK SRVLWLYH URGOLNH PROHFXOHVf DQG QHJDWLYH SODWHOLNH

PAGE 118

)LJXUH 'LIIHUHQFH LQ 1RUPDO DQG 7DQJHQWLDO &RPSRQHQWV RI 6WUHVV 7HQVRU IRU /HQQDUG-RQHV )OXLG )URP 7R[YDHUG >@f

PAGE 119

PROHFXOHVf YDOXHV IRU WKH RYHUODS SDUDPHWHU $V VKRZQ LQ )LJXUHV DQG WKH EHKDYLRU RI Sr]rAf IRU LV RSSRVLWH WR WKDW IRXQG IRU WKH GLVSHUVLRQ LQWHUDFWLRQ 7KH PROHFXOHV WHQG WR VWDQG SHUSHQGLFXODU WR WKH SODQH RI WKH LQWHUIDFH LQ WKH OLTXLG 2Q WKH YDSRU VLGH WKHUH LV D VOLJKW SUHIHUHQFH IRU WKH PROHFXOHV WR OLH LQ WKH SODQH RI WKH LQWHUIDFH 5HVXOWV IRU DUH VKRZQ LQ )LJXUHV DQG 7KHVH FDOn FXODWLRQV VKRZ WKH RSSRVLWH IHDWXUHV WR WKRVH IRU KRZHYHU IRU SODWHOLNH PROHFXOHV WKH V\PPHWU\ D[LV LV SHUSHQGLFXODU WR WKH SODQH RI WKH PROHFXOH 7KXV LQ WKH OLTXLG VLGH RI WKH LQWHUIDFH WKH SODWHOLNH PROHFXOHV SUHIHU DQ RULHQWDWLRQ ZLWK WKH V\PPHWU\ D[LV SDUDOOHO WR WKH LQWHUIDFLDO SODQH 7KH PROHFXOHV WKHPVHOYHV WHQG WR VWDQG SHUSHQGLFXODU WR WKH LQWHUIDFLDO SODQH $Q DQDORJRXV DUJXPHQW KROGV IRU PROHFXOHV LQ WKH YDSRU VLGH RI WKH LQWHUIDFH

PAGE 120

)LJXUH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOHV IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS 0RGHO 3RWHQWLDO N7H

PAGE 121

)LJXUH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOHV IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS 0RGHO 3RWHQWLDO N7H

PAGE 122

)LJXUH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOHV IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS 0RGHO 3RWHQWLDO N7H

PAGE 123

)LJXUH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOHV IRU D )OXLG RI $[LDOO\ 6\PPHWULF 0ROHFXOHV ,QWHUDFWLQJ ZLWK /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS 0RGHO 3RWHQWLDO N7H

PAGE 124

3$57 ,, &20387(5 6,08/$7,21 678',(6

PAGE 125

&+$37(5 0217( &$5/2 6,08/$7,21 2) 02/(&8/$5 )/8,'6 21 $ 0,1,&20387(5 ,QWURGXFWLRQ 7KH 0RQWH &DUOR PHWKRG XVHG LQ WKH VWXG\ RI IOXLGV LV D IRUP RI WKH VWDQGDUG 0RQWH &DUOR WHFKQLTXH IRU HYDOXDWLQJ PXOWLn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bf 6XFK FDOFXODWLRQV WKHUHIRUH UHTXLUH D VLJQLILFDQW DPRXQW RI FRPSXWHU WLPH 7KH DPRXQWV RI WLPH YDU\ JUHDWO\ IURP RQH PDFKLQH WR DQRWKHU RI FRXUVH )RU LOOXVWUDWLYH SXUSRVHV RQH 0RQWH &DUOR FDOFXODWLRQ IRU D VLPSOH /HQQDUG-RQHV IOXLG RI SDUWLFOHV UHTXLUHV D IHZ PLQXWHV RI &38 WLPH RQ D &'& D PDMRU SRUWLRQ RI DQ KRXU RQ DQ ,%0 DQG D IHZ KRXUV RQ DQ ,%0

PAGE 126

RU +RQH\ZHOO &DOFXODWLRQV IRU PRUH FRPSOLFDWHG IOXLGV QRQ VSKHULFDO PROHFXOHVf UHTXLUH VLJQLILFDQWO\ ORQJHU WLPHV XVXDOO\ E\ D IDFWRU RI ILYH RU VL[f 6XFK D UHTXLUHPHQW RI FRPSXWHU UHVRXUFHV KDV D FRQVWULFWLYH HIIHFW RQ XVH RI VLPXODWLRQ LQ IOXLGV UHVHDUFK $ UHODWLYHO\ VPDOO QXPEHU RI UHVHDUFKHUV KDYH DFFHVV WR PDFKLQHV IRU VXFK OHQJWK\ FDOFXODWLRQV 7KH YDOXH RI FRPSXWHU VLPXODWLRQ LQ WKH VWXG\ RI IOXLGV FDQQRW EH RYHUVWUHVVHG 6LQFH WKH SRVLWLRQ DQG LQ WKH FDVH RI QRQVSKHULFDO PROHFXOHV RULHQWDWLRQf RI HDFK PROHFXOH LV NQRZQ WKURXJKRXW WKH FDOFXODWLRQ FRPSXWHU VLPXODWLRQ SURYLGHV D GHJUHH RI GHWDLO ZKLFK LV DV \HW XQDWWDLQHG E\ HLWKHU WKHRU\ RU ODERUDWRU\ H[SHULPHQW 6XFK GHWDLO DOORZV WKH VWXG\ RI ORFDO VWUXFWXUH LQ IOXLGV DV ZHOO DV WKH HIIHFWV RI SRWHQWLDO PRGHOV DQLVRWURSLF SRWHQWLDO VWUHQJWKV PROHFXODU VKDSHV HWF RQ IOXLG SURSHUWLHV &RPSXWHU VLPXODWLRQ LV WKH RQO\ VRXUFH RI H[SHULPHQWDO GDWD IRU PRGHO IOXLGV ZKLFK DUH XVHG LQ GHYHORSLQJ WKHRULHV RI IOXLGV ,Q DGGLWLRQ FRPSXWHU VLPXODWLRQ SURYLGHV D YDOXDEOH PHWKRG IRU VWXG\LQJ LQWHUPROHFXODU SRWHQWLDO IXQFWLRQV DSSOLFDEOH WR UHDO IOXLGV ,Q OLJKW RI WKH YDOXH RI PDFKLQH VLPXODWLRQ DQG WKH UHVWULFWHG DYDLODELOLW\ RI ODUJH FRPSXWHUV LW ZDV IHOW ZRUWKZKLOH WR H[SORUH WKH SRVVLELOLW\ RI XVLQJ D PLQLFRPSXWHU IRU SHUIRUPLQJ 0RQWH &DUOR FDOFXODWLRQV 0DQ\ UHVHDUFKHUV LW LV IHOW KDYH DFFHVV WR PLQLn FRPSXWHUV DQG IXUWKHU WKHVH PDFKLQHV DUH RIWHQ RQO\ LQ XVH IRU VRPH IUDFWLRQ RI D KRXU GD\ 7KH PLQLFRPSXWHUV FXUUHQWO\ DYDLODEOH DUH FHUWDLQO\ VORZHU WKDQ WKH ELJ PDFKLQHV DQG VR WKH\ FRXOG QRW EH DSSOLHG WR PDQ\ VLPXODWLRQ SUREOHPV %XW WKRXJK WKH

PAGE 127

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f ^[` EHWZHHQ D ERG\IL[HG IUDPH DQG D UHIHUHQFH IUDPH IL[HG H[WHUQDO WR WKH V\VWHP 7KH IXOO SRWHQWLDO IRU D V\VWHP RI 1 PROHFXOHV LV WKHQ 8 8 $f1f f ZKLFK LV DVVXPHG WR EH D VXP RI SDLU LQWHUDFWLRQV 8U?f1f \ XU8fZf LM f

PAGE 128

+HUH ZH FRQVLGHU SDLU SRWHQWLDOV RI WKH IRUP XUMGMMf X77Uf X U WRff LO L M /L@ D f§LM L M f ,Q f LV WKH /HQQDUG-RQHV SDLU SRWHQWLDO RI (TXDWLRQ f DQG X LV DQ DQLVRWURSLF SRWHQWLDO HJ GLSROH TXDGUXSROH DQLVR FO WURSLF RYHUODS RU DQLVRWURSLF GLVSHUVLRQ ([SUHVVLRQV IRU WKHVH SRWHQWLDO PRGHOV DUH JLYHQ LQ $SSHQGL[ & 0RQWH &DUOR 3URFHGXUH 0RQWH &DUOR VLPXODWLRQV PD\ EH GRQH LQ DQ\ RI VHYHUDO HQVHPEOHV KHUH ZH XVH WKH FDQRQLFDO HQVHPEOH $ QXPEHU RI SDUWLFOHV 1 V\VWHP YROXPH 9 WHPSHUDWXUH 7 DQG IRUP IRU WKH LQWHUPROHFXODUASDLU SRWHQWLDO f DUH FKRVHQ ,QLWLDO SRVLWLRQV DQG RULHQWDWLRQV DUH DVVLJQHG WR HDFK RI WKH 1 SDUWLFOHV 7KH QDWXUH RI WKLV LQLWLDO FRQILJXUDWLRQ LV DUELWUDU\ LH LW PD\ EH UDQGRPO\ JHQHUDWHG SDWKRORJLFDO HJ )&& ODWWLFHf RU WKH ODVW FRQILJXUDWLRQ IURP D SUHYLRXV FDOFXODWLRQ 7KH SRVLWLRQ RI WKH Lrr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f WKH V\VWHP HQHUJ\ LV FDOFXODWHG

PAGE 129

' X \ \ X U MRMf f 2QH SDUWLFOH KDYLQJ SRVLWLRQ UBA DQG RULHQWDWLRQ fIA LV VHOHFWHG HLWKHU F\FOLFDOO\ RU DW UDQGRP DQG D QHZ SRVLWLRQ f f U? DQG RULHQWDWLRQ DUH SURSRVHG E\ UI! f§L f f f B f f ( GQ L L A f ZKHUH  LV D YHFWRU RI UDQGRP FRPSRQHQWV XQLIRUPO\ GLVWULEXWHG RQ f ZKLOH G DQG GA DUH PD[LPXP DOORZDEOH VWHSV OHQJWKV IRU WKH WUDQVODWLRQDO DQG URWDWLRQDO PRWLRQ RI WKH PROHFXOHV UHVSHFWLYHO\ 7KH HQHUJ\ RI WKH SURSRVHG QHZ FRQILJXUDWLRQ LV FDOFXODWHG f O LM f X U Z2f f L@ L@ f 7KH SURSRVHG QHZ FRQILJXUDWLRQ LV DFFHSWHG RU UHMHFWHG EDVHG RQ WKH UHODWLYH SUREDELOLWLHV RI RFFXUUHQFH RI WKH WZR FRQn ILJXUDWLRQV 7KH SUREDELOLWLHV DUH SURSRUWLRQDO WR WKH %ROW]PDQQ IDFWRU 3f H[S > 8fN7@ f 3f H[S > 8fN7@ f

PAGE 130

7KH FULWHULD IRU DFFHSWLQJ RU UHMHFWLQJ WKH SURSRVHG FRQILJXUDWLRQ LV A f Df ,I 3Z 3 DFFHSW WKH QHZ FRQILJXUDWLRQ Ef ,I 3A GR QRW UHMHFW WKH QHZ FRQILJXUDWLRQ RXW RI KDQG UDWKHU DFFHSW LW ZLWK SUREDELOLW\ SURSRUWLRQDO WR WKH %ROW]PDQQ IDFWRU 7KLV PD\ EH DFFRPSOLVKHG E\ JHQHUDWLQJ D UDQGRP QXPEHU e RQ WKH LQWHUYDO f DQG ,I SAfSAf DFFHSW WKH QHZ FRQILJXUDWLRQ ,I SAfnSnOf UHMHFW WKH QHZ FRQILJXUDWLRQ ,I WKH QHZ FRQILJXUDWLRQ LV UHMHFWHG WKH ROG FRQILJXUDWLRQ LV FRXQWHG DV WKH QHZ FRQILJXUDWLRQ DQG WKH SURSHUWLHV RI LQWHUHVW DUH GHWHUPLQHG IRU WKLV QHZ FRQILJXUDWLRQ 6WHSV DUH LWHUDWHG RYHU WKH OHQJWK RI WKH FDOFXODWLRQ $W WKH HQG RI WKH UXQ HQVHPEOH DYHUDJH SURSHUW\ YDOXHV DUH REWDLQHG E\ DYHUDJLQJ WKH SURSHUW\ YDOXHV REWDLQHG IRU HDFK FRQILJXUDWLRQ 5HODWLRQV IRU VHYHUDO HTXLOLEULXP SURSHUWLHV LQ WHUPV RI HQVHPEOH DYHUDJHV DUH JLYHQ LQ 7DEOH ,Q 7DEOH WKH DQJXODU FRUUHODWLRQ SDUDPHWHUV DUH PHDVXUHV RI WKH FRUUHODWLRQ RI RULHQWDWLRQ LQ WKH / IOXLG :KHQ IOXFWXDWLRQV LQ WKH FROOLVLRQLQGXFHG GLSROH PRPHQWV DUH QHJOHFWHG WKH SDUDPHWHUV DUH UHODWHG WR WKH GLHOHFWULF DQG .HUU / FRQVWDQWV )A DQG )Af IRU OLQHDU PROHFXOHV 7KH SDUDPHWHUV DOVR DULVH LQ WKHRULHV RI GHSRODUL]HG OLJKW VFDWWHULQJ >@ 7KHUH VHHPV WR EH QR VWDQGDUG QRWDWLRQ IRU WKHVH DQJXODU FRUUHODWLRQ SDUDPHWHUV FI &KHXQJ DQG 3RZOHV >@f

PAGE 131

7$%/( (TXLOLEULXP 3URSHUWLHV LQ WKH )RUP RI (QVHPEOH $YHUDJHV &RQILJXUDWLRQDO ,QWHUQDO (QHUJ\ P XU ZFRf! 1 1 K L L LM f 5HVLGXDO +HDW &DSDFLW\ K >WM! X!@ 17 f 3UHVVXUH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ SN7 XUZZf Y F LO L e O UB f LM 17 f§ LM U LM B f XU FRRff ) B3B \ U U \B n 1  LM U LL! f f ) f§ 4 Y 9 )RZOHU 0RGHO 6XUIDFH 8f WWW f f U XU f f f 7 6 1 +LM LM @ ([FHVV (QHUJ\ LM f $QJXODU &RUUHODWLRQ 3DUDPHWHUVf &7 O 37 FRV \ f! Mf}O f 0HDQ 6TXDUHG )RUFH )! O 9 XU Df f MAO ff! f 0HDQ 6TXDUHG 7RUTXH 7;! O9XULMfLfMf`n -AO f LV WKH DQJOH EHWZHHQ WKH D[HV RI PROHFXOHV DQG M 3A LV WKH RUGHU /HJHQGUH SRO\QRPLDO

PAGE 132

'XULQJ WKH ILUVW IHZ KXQGUHG FRQILJXUDWLRQV RI WKH FDOFXODWLRQ WKH V\VWHP LV UHOD[LQJ IURP WKH DUELWUDU\ VWDUWLQJ FRQILJXUDWLRQ WR DQ HTXLOLEULXP VWDWH ZKLFK LV UHIOHFWHG E\ WKH V\VWHP HQHUJ\ IOXFWXDWLQJ DERXW D PLQLPXP YDOXH 7KHVH LQLWLDO FRQILJXUDWLRQV DUH GLVFDUGHG LQ WKDW WKH\ DUH QRW XVHG LQ FDOFXODWLQJ WKH DYHUDJH YDOXHV IRU WKH SURSHUWLHV $W UHJXODU LQWHUYDOV WKURXJKRXW WKH FDOFXODWLRQ WKH VWHS OHQJWKV LQ f DQG f DUH DGMXVWHG WR PDLQWDLQ DFFHSWDQFH RI DERXW KDOI WKH SURSRVHG QHZ FRQILJXUDWLRQV ,Q RUGHU IRU WKH VPDOO QXPEHU RI SDUWLFOHV KHUH 1 f WR DGHTXDWHO\ PLPLF D EXON IOXLG V\VWHP VRFDOOHG SHULRGLF ERXQGDU\ FRQGLWLRQV DUH XVHG 0RUHRYHU WKH SRWHQWLDO LV VHW WR ]HUR EH\RQG VRPH FXWRII GLVWDQFH UA WDNHQ WR EH WKH UDGLXV RI WKH ODUJHVW VSKHUH ZKLFK FDQ EH LQVFULEHG LQ D FXEH RI YROXPH 9 0RUH GHWDLOHG GHVFULSWLRQV RI WKH PHWKRG FDQ EH IRXQG LQ >@r 'HVFULSWLRQ RI WKH 0LQLFRPSXWHU 6\VWHP 7KH FRPSXWHU XVHG LQ WKLV VWXG\ ZDV D 'DWD *HQHUDO 129$ ZLWK RSWLRQDO ELW ZRUGV RI FRUH VWRUDJH DQG QVHF PHPRU\ F\FOH WLPH 7KH &38 LQFOXGHG WKH IROORZLQJ RSWLRQV SRZHU PRQLWRU DXWR UHVWDUW DXWR SURJUDP ORDG UHDO WLPH FORFN KDUGZDUH PXOWLSO\GLYLGH DQG KLJK SHUIRUPDQFH KDUGZDUH IORDWLQJ SRLQW SURFHVVRU 7KH &38 VHUYLFHG WKH IROORZLQJ SHULSKHUDOV WZR PRYLQJ KHDG GLVF XQLWV ZLWK FRQWUROOHU IRU D WRWDO RI PHJDZRUGV RI VWRUDJH VHULDO PDWUL[ OLQH SULQWHU FSVf IDVW SDSHU WDSH UHDGHU DQG SXQFK DQG WHOHW\SH 7KH FRPSXWHU ZDV RSHUDWHG XQGHU WKH 'DWD *HQHUDO UHDO WLPH GLVF RSHUDWLQJ V\VWHP 5'26 UHYLVLRQ ZKLFK KDQGOHV WDVN VFKHGXOLQJ DQG V\VWHP PDLQWHQDQFH 7KH H[HFXWLYH UHPDLQHG FRUH UHVLGHQW DQG RFFXSLHG DERXW ZRUGV VR DERXW ZRUG ORFDWLRQV

PAGE 133

ZHUH DYDLODEOH IRU FRPSXWDWLRQ 7KH DYDLODEOH VRIWZDUH LQFOXGHG 'DWD *HQHUDOnV )RUWUDQ ,9 )RUWUDQ 9 %DVLF DQG $OJRO 7KH SURJUDP IRU WKLV ZRUN ZDV ZULWWHQ LQ GRXEOH SUHFLVLRQ DULWKPHWLF IRU WKH )RUWUDQ 9 FRPSXWHU 7KH 'DWD *HQHUDO )RUWUDQ 9 LV D FRGH RSWLPL]HU LQ WKDW UHGXQGDQW RSHUDWLRQV DUH HOLPLQDWHG DQG IORDWLQJ SRLQW RSHUDn WLRQV DUH RSWLPL]HG IRU HIIHFWLYH XVH RI WKH IORDWLQJ SRLQW KDUGZDUH 0RQWH &DUOR 3URJUDP IRU WKH 129$ &DOFXODWLRQ RI 6\VWHP (QHUJ\ 7KH PDMRU WLPHFRQVXPLQJ FDOFXODWLRQ LQ 0RQWH &DUOR VLPXODWLRQV LV HYDOXDWLRQ RI WKH V\VWHP HQHUJ\ 8 LQ f DQG f 7KH IXOO GRXEOH VXP LQ f DQG f GRHV QRW KDYH WR EH HYDOXDWHG KRZHYHU VLQFH WKH SDUWLFOHV DUH PRYHG VHTXHQWLDOO\ 7KXV ZKHQ SDUWLFOH L LV PRYHG IURP FRQILJXUDWLRQ WR FRQILJXUDWLRQ RQO\ 1Of SDLU HQHUJLHV FKDQJH 7KH QHZ V\VWHP HQHUJ\ FDQ EH IRXQG WKHUHIRUH IURP f 8f $8 f L ZKHUH $8 O XUAfRMIfD!f e X L+ -"L UA:Rf LM L f ,Q WKH XVXDO 0RQWH &DUOR FDOFXODWLRQ HDFK RI WKH SDLU HQHUJLHV XUf.-f IRU WKH VWDUWLQJ FRQILJXUDWLRQ LV FDOFXODWHG DQG VWRUHG LQ FRUH 7KH WRWDO V\VWHP HQHUJ\ IRU WKH VWDUWLQJ FRQILJXUDWLRQ LV REWDLQHG IURP f DQG VWRUHG DV 8 )RU HDFK VXEVHTXHQW FRQn ILJXUDWLRQ JHQHUDWHG WKH V\VWHP HQHUJ\ LV GHWHUPLQHG IURP f DQG f UDWKHU WKDQ f )RU WKLV FDOFXODWLRQ WKH SDLU HQHUJLHV

PAGE 134

IRU WKH ILUVW WHUP RQ WKH ULJKW VLGH RI f DUH FDOFXODWHG ZKLOH WKH SDLU HQHUJLHV IRU WKH VHFRQG WHUP DUH REWDLQHG IURP VWRUDJH :KHQHYHU D SURSRVHG FRQILJXUDWLRQ LV DFFHSWHG WKH SDLU HQHUJLHV f f ? f f XU FR XLf LQ VWRUDJH DUH XSGDWHG WR WKH QHZ YDOXHV XU f XLf LM L LM L M DQG WKH QHZ V\VWHP HQHUJ\ 8 LV VWRUHG DV 7KLV SURFHGXUH VLJQLILFDQWO\ GHFUHDVHV WKH DPRXQW RI FDOFXODWLRQ UHTXLUHG EXW SODFHV D KHDY\ GHPDQG RQ FRUH VWRUDJH (YHQ WKRXJK WKH PDWUL[ RI SDLU HQHUJLHV LV V\PPHWULF LH XUXLLf XUf LM L -L L f \ 11Of YDOXHV PXVW EH VWRUHG MXVW IRU WKH HQHUJ\ ,I RWKHU SURSHUWLHV DUH EHLQJ FDOFXODWHG VLPLODU VWRUDJH PXVW EH SURYLGHG IRU HDFK RI WKHP 7KLV ODUJH VWRUDJH UHTXLUHPHQW LV DYRLGHG LQ WKH SURJUDP IRU WKH 129$ E\ FDOFXODWLQJ ERWK VXPV LQ f IRU HDFK FRQILJXUDWLRQ JHQHUDWHG DQG WKHUHIRUH VWRULQJ QRQH RI WKH SDLU SURSHUWLHV ([HFXWLRQ WLPH LV LQFUHDVHG DFFRUGLQJO\ $OWHUQDWLYHO\ WKH SDLU HQHUJ\ PDWUL[ FRXOG EH VWRUHG RQ GLVF DQG WUDQVIHUUHG LQWR FRUH FROXPQ E\ FROXPQ DV LW LV QHHGHG :KHQ D PRYH LV DFFHSWHG KRZHYHU D PDMRU SRUWLRQ RI WKH PDWUL[ PXVW EH EURXJKW WKURXJK FRUH LQ RUGHU WR XSGDWH WKH FKDQJHG HOHPHQWV IURP XU IARLAADLf WR XU I "AXLI ARLf 4Q WKH 129$ V\VWHP IRU SDUWLFOHV ZH IRXQG WKLV WUDQVIHU WR EH VORZHU WKDQ UHFDOFXODWLQJ WKH QHHGHG PDWUL[ HOHPHQWV DV LQGLFDWHG DERYH +HQFH WKH SURJUDP GHVFULEHG KHUH GRHV QRW XVH FRUHGLVF GDWD WUDQVIHUV GXULQJ H[HFXWLRQ

PAGE 135

3URJUDP 6WUXFWXUH )LJXUH VKRZV D VLPSOLILHG VFKHPDWLF IORZ GLDJUDP RI WKH SURJUDP ZULWWHQ IRU WKH 129$ 7KH 0RQWH &DUOR FDOFXODWLRQ LV GRQH E\ WKH PDLQ SURJUDP ODEHOHG ZKLFK LV VXSSRUWHG E\ VXEURXWLQHV ,1,7,$/ DQG (1(5*< 6XEURXWLQH (1(5*< FRQVXPHV WKH PDMRU SRUWLRQ RI H[HFXWLRQ WLPH DV LW IRUPV WKH VXPV DQDORJRXV WR (TXDWLRQ f IRU DOO SURSHUWLHV RI LQWHUHVW 'LVF ILOH KROGV WKH VWDUWLQJ FRQILJXUDWLRQnV ORFDWLRQ YHFWRUV DQG GLUHFWLRQ FRVLQHV 'LVF ILOH KROGV LQWHUPHGLDWH FRQn ILJXUDWLRQ SDUWLFOH ORFDWLRQ YHFWRUV GLUHFWLRQ FRVLQHV DQG SURSHUW\ YDOXHV DOO RI ZKLFK DUH GXPSHG DW DERXW KDOIKRXU LQWHUYDOV WKURXJKRXW WKH FDOFXODWLRQ 7KLV SHULRGLF GDWD GXPS VHUYHV DV D EDFNXS DJDLQVW SRVVLEOH SRZHU IDLOXUH DQG DOORZV RWKHU V\VWHP XVHUV WR LQWHUUXSW SURJUDP H[HFXWLRQ 7KH HQWLUH SURJUDP IRU SDUWLFOHV UHVLGHV LQ FRUH WKURXJKRXW WKH FDOFXODWLRQ DQG UHTXLUHV DERXW ZRUGV RI 129$ PHPRU\ $ IXUWKHU VDYLQJ RI ZRUGV FRXOG EH DWWDLQHG E\ PDNLQJ WKH VXEURXWLQH ,1,7,$/ D VHSDUDWH PDLQ SURJUDP DV LW LV RQO\ XVHG DW WKH VWDUW RU UHVWDUWf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

PAGE 136

)LJXUH 6LPSOLILHG 6FKHPDWLF )ORZ 'LDJUDP RI )2575$1 0RQWH &DUOR 3URJUDP 'HYHORSHG IRU 129$

PAGE 137

7$%/( $SSUR[LPDWH 1XPEHU RI 0RQWH &DUOR &RQILJXUDWLRQV *HQHUDWHG SHU +RXU RQ WKH 129$

PAGE 138

QXPEHU RI FRQILJXUDWLRQV JHQHUDWHG SHU KRXU E\ WKH )RUWUDQ SURJUDP IRU DQG SDUWLFOHV XVLQJ WKH /HQQDUG-RQHV SRWHQWLDO f DQG WKH /HQQDUG-RQHV SOXV TXDGUXSROH SRWHQWLDO RI (TXDWLRQV f DQG &f 7DEOH LQGLFDWHV WKDW GRXEOLQJ WKH QXPEHU RI SDUWLFOHV URXJKO\ GRXEOHV H[HFXWLRQ WLPH ,Q DGGLWLRQ WR WKH UHVXOWV VKRZQ LQ WKH WDEOH WKH /HQQDUG-RQHV SOXV GLSROH PRGHO RI (TXDWLRQV f DQG &f H[HFXWHV DERXW b IDVWHU WKDQ WKH /HQQDUG-RQHV SOXV TXDGUXSROH PRGHO 7KH /HQQDUG-RQHV SOXV RYHUODS PRGHO (TXDWLRQV f DQG &f LQ WXUQ H[HFXWHV DERXW b IDVWHU WKDQ WKH /HQQDUG-RQHV SOXV GLSROH PRGHO 7KHVH SURJUDP H[HFXWLRQ VSHHGV RQ WKH 129$ PD\ EH FRPSDUHG ZLWK VSHHGV DWWDLQHG RQ ,%0 DQG &'& PDFKLQHV $ )RUWUDQ SURJUDP IRU VLPXODWLQJ WKH /HQQDUG-RQHV SOXV RYHUODS PRGHO IOXLG XVLQJ SDUWLFOHV DQG FDOFXODWLQJ RQO\ WKH V\VWHP LQWHUQDO HQHUJ\ DQG WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ DW VHYHQ RULHQWDWLRQV JHQHUDWHV DERXW f FRQILJXUDWLRQV SHU KRXU RQ DQ ,%0 DQG DERXW Af FRQn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

PAGE 139

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r 'HWHUPLQDWLRQ RI VXFK D KLVWRJUDP LV GLIILFXOW WR DFFRPSOLVK VLQFH Df ORQJ UXQV DUH UHTXLUHG LQ RUGHU WR REWDLQ VWDWLVWLFDOO\ UHOLDEOH VDPSOLQJ RI OHVV SUREDEOH FRQILJXUDWLRQV HVSHFLDOO\ LQ DQJOH VSDFHf DQG Ef VPDOO UDGLDO DQG DQJXODU LQFUHPHQWV DUH UHTXLUHG LQ RUGHU WR REWDLQ VXIILFLHQW GDWD RQ JUA:AA IrU DFFXUDJH LQWHJUDWLRQ 7KH YDOXHV IRU )A! DQG 7A! FDOFXODWHG RQ WKH 129$ ZHUH REWDLQHG E\ GLUHFW HYDOXDWLRQ RI WKH DYHUDJHV JLYHQ LQ 7DEOH ,W LV IHOW WKHUHIRUH WKDW WKH 129$ SURSHUW\ YDOXHV DUH PRUH UHOLDEOH WKDQ WKRVH REWDLQHG IURP WKH &'& PDFKLQH )XUWKHU FRPSDULVRQV RI 129$ UHVXOWV ZLWK &'& UHVXOWV DUH PDGH LQ )LJXUHV DQG )LJXUH VKRZV YDOXHV IRU WKH FHQWHUWRFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ J UAf IrU D /HQQDUG-RQHV SOXV TXDGUXSROH IOXLG 7KH &'& YDOXHV IRU JA ZHUH REWDLQHG IRU D V\VWHP RI OLQHDU PROHFXOHV IURP D 0RQWH &DUOR FKDLQ RI VRPH f FRQILJXUDWLRQV DIWHU WKH V\VWHP KDG UHDFKHG HTXLOLEULXP >@ 7KH PLQLFRPSXWHU UHVXOWV DUH IURP D UXQ RI VLPLODU OHQJWK RQ D V\VWHP RI PROHFXOHV

PAGE 140

7$%/( &RPSDULVRQ RI 129$ DQG &3& 5HVXOWV IRU 3URSHUW\ 9DOXHV RI /HQQDUG-RQHV 4XDGUXSROH 0RGHO )OXLG N7H SD 4HF9 3URSHUW\ 129$ &'& 81F s s I @ )!RH s s >@ 7!H s s >@

PAGE 141

)LJXUH &RPSDULVRQ RI &'& OLQHf DQG 129$ SRLQWVf 0RQWH &DUOR 5HVXOWV IRU WKH &HQWHU&HQWHU 3DLU &RUUHODWLRQ )XQFWLRQ IRU D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG

PAGE 142

,OO &RPSDULVRQ RI &'& OLQHf DQG 129$ SRLQWVf 5HVXOWV IRU WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU 0ROHFXODU 3DLUV LQ WKH 7HH 2ULHQWDWLRQ )LJXUH

PAGE 143

)LJXUH PDNHV D VLPLODU FRPSDULVRQ IRU WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JU ARU WAH /HQQDUG-RQHV SOXV TXDGUXSROH IOXLG 7KH YDOXHV RI JUAD-MA VKRZQ DUH IRU PROHFXODU SDLUV LQ WKH WHH RULHQWDWLRQ LH A WW XQGHILQHG ZKHUH A GA DQG !A DUH WKH RULHQWDWLRQ DQJOHV UHODWLYH WR WKH LQWHUPROHFXODU D[LV DV VKRZQ LQ )LJXUH &O 7KH &'& UHVXOWV DUH IURP D FKDLQ RI DERXW frf FRQILJXUDWLRQV IRU D V\VWHP RI SDUWLFOHV 7KH PLQLFRPSXWHU UHVXOWV DUH IURP WKH VDPH UXQ DV WKDW IRU )LJXUH 7R DFKLHYH FRQVLVWHQF\ LQ WKH FRPSDULVRQ WKH VDPH DQJXODU LQFUHPHQW RI s r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f FRQILJXUDWLRQV RI D 0RQWH &DUOR VHTXHQFH GHSHQGLQJ RQ WKH QXPEHU RI SDUWLFOHV SRWHQWLDO PRGHO HWF 7KXV RQH ZRXOG QRW FRQVLGHU XVLQJ SUHVHQW PLQLFRPSXWHUV IRU VSHFLDOL]HG VWXGLHV ZKLFK UHTXLUH VLJQLILFDQWO\ ORQJHU UXQV WKDQ FDOFXODWLRQ RI EXON IOXLG SURSHUWLHV HJ VWXG\ RI SKDVH WUDQVLWLRQV RU FULWLFDO SKHQRPHQD

PAGE 144

$ PDMRU LPSURYHPHQW LQ VSHHG RI SURJUDP H[HFXWLRQ FDQ EH DWWDLQHG E\ XVH RI IDVWHU KDUGZDUH WKDQ LV DYDLODEOH RQ WKH 129$ 7KXV )UHDVLHU HWB DO >@ KDYH UHFHQWO\ UHSRUWHG 0RQWH &DUOR UHVXOWV IRU D KDUG GXPEn EHOO IOXLG XVLQJ D 3'3 7KH FDOFXODWLRQV ZHUH GRQH IRU SDUWLFOHV DQG DERXW FRQILJXUDWLRQV ZHUH JHQHUDWHG IRU HDFK VWDWH FRQGLWLRQ VWXGLHG 7KH FRGH ZDV ZULWWHQ LQ )RUWUDQ H[FHSW WKH WLPH FRQVXPLQJ URXWLQHV IRU ZKLFK DVVHPEOHU ODQJXDJH ZDV XVHG

PAGE 145

&+$37(5 02/(&8/$5 '<1$0,&6 0(7+2' )25 $;,$//< 6<00(75,& 02/(&8/(6 ,QWURGXFWLRQ $ VHFRQG PHWKRG XVHG LQ WKH FRPSXWHU VLPXODWLRQ RI IOXLGV DV RSSRVHG WR WKH 0RQWH &DUOR WHFKQLTXH GHVFULEHG LQ WKH SUHYLRXV FKDSWHUf LV WKH PROHFXODU G\QDPLFV PHWKRG 7KLV VLPXODWLRQ PHWKRG LV LQ PDQ\ ZD\V VLPLODU WR WKH 0RQWH &DUOR WHFKQLTXH ERWK PHWKRGV DUH DSSOLHG WR VPDOO ILQLWH V\VWHPV ERWK XVH SHULRGLF ERXQGDU\ FRQGLWLRQV DQG ERWK XVXDOO\ DVVXPH D SDLUZLVH DGGLWLYH LQWHUPROHFXODU SRWHQWLDO 7KH PDMRU GLVWLQFWLRQ EHWZHHQ WKH WZR LV WKDW LQ PROHFXODU G\QDPLFV JHQHUDWLRQ RI VWDWHV DFFHVVLEOH WR WKH V\VWHP LV DFFRPSOLVKHG E\ VROYLQJ 1HZWRQnV HTXDWLRQV RI PRWLRQ IRU HDFK RI WKH SDUWLFOHV ZKHUHDV LQ WKH 0RQWH &DUOR PHWKRG DFFHVVLEOH VWDWHV DUH REWDLQHG E\ UDQGRP VDPSOLQJ RI FRQILJXUDWLRQDO SKDVH VSDFH 6HTXHQWLDO VROXWLRQ RI 1HZWRQnV HTXDWLRQV RI PRWLRQ IRU D V\VWHP RI SDUWLFOHV LPSOLHV GHYHORSPHQW RI WKH WLPH HYROXWLRQ RI WKH V\VWHP RQ WKH PROHFXODU OHYHO 7KLV WLPH HYROXWLRQ RI WKH V\VWHP SURYLGHV QHZ LQIRUPDWLRQ QRW DWWDLQDEOH LQ 0RQWH &DUOR +HQFH PROHFXODU G\QDPLFV PD\ EH XVHG WR VWXG\ YDULRXV WLPH GHSHQGHQW SURSHUWLHV RI WKH V\VWHP LQ DGGLWLRQ WR FRQILJXUDWLRQDO SURSHUWLHV WKRVH HJ OLVWHG LQ 7DEOH f 7KH IRUPHU LQFOXGH WUDQVSRUW FRHIILFLHQWV VXFK DV YLVFRVLW\ DQG GLIIXVLRQ FRHIILFLHQWV DV ZHOO DV D KRVW RI WLPH FRUUHODWLRQ IXQFWLRQV 7KH WLPH FRUUHODWLRQ IXQFWLRQV

PAGE 146

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f f f DQG E\ 1HZWRQnV VHFRQG ODZ )RU f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

PAGE 147

FRQYHQLHQW WR XVH LVREDULF RU JUDQG FDQRQLFDO HQVHPEOHV LQ VXFK FDVHV WKH 0RQWH &DUOR PHWKRG PD\ EH SUHIHUDEOHf &RPSDULVRQ RI FRQILJXUDWLRQDO SURSHUW\ YDOXHV IURP WKH PROHFXODU G\QDPLFV PLFURFDQRQLFDO HQVHPEOH ZLWK 0RQWH &DUOR UHVXOWV LQ WKH FDQRQLFDO HQVHPEOH DJUHH ZLWKLQ VWDWLVWLFDO IOXFWXDWLRQV >@ 7KLV ZRUN KDV LQYROYHG PROHFXODU G\QDPLFV VLPXODWLRQV RI IOXLGV RI D[LDOO\ V\PPHWULF OLQHDUf PROHFXOHV LQWHUDFWLQJ ZLWK SDLU SRWHQWLDOV RI WKH IRUP JLYHQ LQ f 7KH LPPHGLDWH JRDOV KDYH EHHQ Df FRPSDULVRQ RI VLPXODWLRQ UHVXOWV ZLWK SHUWXUEDWLRQ WKHRU\ SUHGLFWLRQV IRU HTXLOLEULXP SURSHUW\ YDOXHV DQG Ef VWXG\ RI WKH PROHFXODU VWUXFWXUH LQ WKH IOXLG YLD WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ 0RUH ORQJ UDQJH JRDOV LQFOXGH Ff VWXG\ RI WLPH GHSHQGHQW SURSHUWLHV DQG Gf FRPSDULVRQ RI UHVXOWV IRU YDULRXV PRGHO SRWHQWLDOV LQ DQ DWWHPSW WR JDLQ LQVLJKW LQWR WKH QDWXUH RI WKH LQWHUPROHFXODU SRWHQWLDO IRU UHDO IOXLGV ,QLWLDO VLPXODWLRQ VWXGLHV RI VWUXFWXUH LQ IOXLGV RI QRQVSKHULFDO PROHFXOHV ZHUH GRQH E\ GLUHFW HYDOXDWLRQ RI WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JU Xf RAf DV! HJ! UHSRUWHG LQ &KDSWHU f 7KLV GLUHFW HYDOXDWLRQ RI D PXOWLGLPHQVLRQDO IXQFWLRQ ZDV IRXQG WR KDYH VHYHUH OLPLWDWLRQV LQ WKDW VPDOO DQJXODU LQFUHPHQWV DUH UHTXLUHG IRU DFFXUDWH UHSUHVHQWDWLRQ RI JUT:O-MA f 7KHVH VPDOO LQFUHPHQWV LQ WXUQ GHPDQG ODUJH DPRXQWV RI FRPSXWHU PHPRU\ DQG ORQJ VLPXODWLRQ UXQV WR UHGXFH VWDWLVWLFDO HUURU HVSHFLDOO\ IRU OHVV IDYRUHG PROHFXODU SDLU RULHQWDn WLRQV 7KHVH SUREOHPV FDQ EH VLJQLILFDQWO\ UHGXFHG E\ H[SDQGLQJ JUA:A:A DQ WHUPV rI VSKHULFDO KDUPRQLFV DQG HYDOXDWLQJ WKH H[SDQVLRQ FRHIILFLHQWV JA A WrLH VLPXODWLrQ UDWKHU WKDQ JUAFRMAf > @ f ,Q DGGLWLRQ YDULRXV FRPELQDWLRQV RI WKHVH H[SDQVLRQ FRHIILFLHQWV DUH

PAGE 148

UHODWHG WR HTXLOLEULXP SURSHUWLHV WKXV DIIRUGLQJ D FRQVLVWHQF\ FKHFN RQ WKH SURSHUW\ YDOXHV GHWHUPLQHG LQ WKH FRXUVH RI WKH VLPXODWLRQ 7KLV FKDSWHU SUHVHQWV WKH PHWKRG RI SHUIRUPLQJ PROHFXODU G\QDPLFV VLPXODWLRQV RI IOXLGV FRQWDLQLQJ D[LDOO\ V\PPHWULF PROHFXOHV 6HFWLRQ GHYHORSV JHQHUDO H[SUHVVLRQV IRU REWDLQLQJ WKH IRUFH DQG WRUTXH IURP WKH LQWHUPROHFXODU SRWHQWLDO IRU VXFK PROHFXOHV 6HFWLRQ LQGLFDWHV WKH PHWKRG XVHG IRU VROYLQJ 1HZWRQnV HTXDWLRQV RI PRWLRQ DQG RXWOLQHV WKH PROHFXODU G\QDPLFV DOJRULWKP ,Q 6HFWLRQ WKH VSKHULFDO KDUPRQLF H[SDQVLRQ IRU WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ LV GHYHORSHG DQG WKH PHWKRG RI GHWHUPLQLQJ WKH FHQWHUFHQWHU DQG VLWHVLWH SDLU FRUn UHODWLRQ IXQFWLRQV LV GHVFULEHG 5HODWLRQV EHWZHHQ WKH H[SDQVLRQ FRHIILFLHQWV JA A PAUA DQA YDULrXV HTXLOLEULXP SURSHUWLHV DUH GHYHORSHG LQ 6HFWLRQ ([SUHVVLRQV IRU )RUFH DQG 7RUTXH IRU $[LDOO\ 6\PPHWULF 0ROHFXOHV $ QRQVSKHULFDO PROHFXOH H[KLELWV WUDQVODWLRQDO DQG URWDWLRQDO PRWLRQ GHVFULEHG E\ UHVSHFWLYHO\ A P a AL W A G4 c f f ,Q f ) LV WKH IRUFH H[HUWHG RQ PROHFXOH E\ DOO RWKHU PROHFXOHV LQ WKH V\VWHP P LV WKH PROHFXODU PDVV DQG BU LV D SRVLWLRQ YHFWRU ORFDWLQJ WKH FHQWHU RI PDVV RI PROHFXOH UHODWLYH WR VRPH DUELWUDU\

PAGE 149

VSDFH IL[HG D[LV ,Q f B7 LV WKH WRUTXH DSSOLHG WR PROHFXOH E\ DOO RWKHU PROHFXOHV LQ WKH V\VWHP LV WKH PROHFXODU PRPHQW RI LQHUWLD DQG LV WKH DQJXODU GLVSODFHPHQW RI WKH PROHFXODU D[LV GXH WR URWDWLRQ DERXW LWV FHQWHU RI PDVV ,Q D FRQVHUYDWLYH V\VWHP B) DQG DUH DOVR JLYHQ E\ UHVSHFWLYHO\ WKH QHJDWLYH UDGLDO DQG DQJXODU JUDGLHQWV RI WKH LQWHUPROHFXODU SRWHQWLDO WDNHQ WR EH D VXP RI SDLU FRQWULEXWLRQV )L D f§ RL f§ XU f WR @ ,, c/Q 8 f 7f 6X f§ XULLfL'f P f ,Q f ZA UHSUHVHQWV WKH DQJXODU JUDGLHQW LQ WHUPV RI (XOHU DQJOHV DQG RL LV WKH RULHQWDWLRQ RI PROHFXOH (YDOXDWLRQ RI WKH IRUFH DQG WRUTXH RQ HDFK PROHFXOH LQ WKH V\VWHP IURP f DQG f DW UHJXODU LQWHUYDOV WKURXJKRXW D PROHFXODU G\QDPLFV FDOFXODWLRQ LV D PDMRU WLPH FRQVXPLQJ SURFHGXUH ,W LV LPSRUWDQW WKHUHn IRUH WR GHYHORS HIILFLHQW PHWKRGV RI HYDOXDWLQJ f DQG f 6LJQLILFDQW LPSURYHPHQW LQ SURJUDP H[HFXWLRQ VSHHG PD\ EH UHDOL]HG E\ D SURSHU FKRLFH RI WKH PDQQHU LQ ZKLFK PROHFXODU RULHQWDWLRQV DUH VSHFLILHG 6SHFLILFDWLRQ RI 0ROHFXODU 3DLU 2ULHQWDWLRQ 7KH RULHQWDWLRQ RI DQ D[LDOO\ V\PPHWULF PROHFXOH LV FRPSOHWHO\ VSHFLILHG E\ WZR LQGHSHQGHQW YDULDEOHV $V VKRZQ LQ )LJXUH WKH RULHQWDWLRQ RN RI PROHFXOH L PD\ EH JLYHQ E\ WKH SRODU %Af DQG D]LPXWKDO DAf DQJOHV EHWZHHQ WKH PROHFXODU D[LV DQG D VSDFH IL[HG IUDPH [ \ ] $ VHFRQG PHWKRG RI JLYLQJ WKH RULHQWDWLRQ WG LV E\

PAGE 150

)LJXUH 0HWKRGV RI 6SHFLI\LQJ WKH 2ULHQWDWLRQ RI DQ $[LDOO\ 6\PPHWULF 0ROHFXOH

PAGE 151

D XQLW YHFWRU IL DOLJQHG DORQJ WKH PROHFXODU D[LV 7KH FRPSRQHQWV RI I/ DUH UHODWHG WR DQJOHV DABA E\ K VLQ ,; FRV D L L f K VLQ L\ VLQ D L L f K FRV D f 2EYLRXVO\ RQO\ WZR RI WKH FRPSRQHQWV RI -/ DUH LQGHSHQGHQW )RU LQWHUPROHFXODU SRWHQWLDOV ZKLFK DUH WDNHQ WR EH D VXP RI SDLU WHUPV WKH RULHQWDWLRQV RI ERWK PROHFXOHV DUH QHHGHG $ SRVVLEOH FKRLFH IRU VSHFLI\LQJ WKH RULHQWDWLRQV LV XVH RI WKH DQJOHV ^A` IRU HDFK PROHFXOH L VKRZQ LQ )LJXUH 8VH RI VXFK D V\VWHP LV FDO FXODWLRQDOO\ SURKLELWLYH KRZHYHU GXH WR Df WKH ODUJH QXPEHU RI WHUPV ZKLFK DULVH ZKHQ WKH SDLU SRWHQWLDO LV ZULWWHQ LQ WHUPV RI WKHVH DQJOHV Ef WKH QHFHVVLW\ IRU LQFOXGLQJ WKH RULHQWDWLRQ RI WKH LQWHU PROHFXODU D[LV UB LQ WKH SRWHQWLDO 6LPSOLILFDWLRQ RI WKH SDLU SRWHQWLDO UHVXOWV LI WKH UHIHUHQFH IUDPH LV FKRVHQ WR EH DOLJQHG UHODWLYH WR WKH SDLU RI PROHFXOHV 2QH VXFK LQWHUPROHFXODU UHIHUHQFH IUDPH ZKLFK LV RIWHQ XVHG KDV LWV ]D[LV DOLJQHG DORQJ WKH LQWHUPROHFXODU D[LV DV VKrZQ LQ )LJXUH 8VLQJ WKLV IUDPH WKH RULHQWDWLRQ GHSHQGHQFH RI BUA LQ WKH SRWHQWLDO YDQLVKHV )XUWKHU WKH GHSHQGHQFH RI WKH SRWHQWLDO RQ D]LPXWKDO DQJOHV !A DQG RFFXUV LQ WHUPV RI WKH GLIIHUHQFH SA ([SUHVVLRQV IRU VHYHUDO DQLVRWURSLF PRGHO SRWHQWLDOV LQ WKLV VHW RI YDULDEOHV DUH OLVWHG LQ $SSHQGL[ & 7KLV LQWHUPROHFXODU IUDPH DOVR KDV WKH DGYDQWDJH

PAGE 152

)LJXUH 2ULHQWDWLRQ $QJOHV IRU $[LDOO\ 6\PPHWULF 0ROHFXOHV LQ DQ $UELWUDU\ 6SDFH )L[HG )UDPH

PAGE 153

=6 )LJXUH 2ULHQWDWLRQ $QJOHV IRU $[LDOO\ 6\PPHWULF 0ROHFXOHV LQ WKH ,QWHUPROHFXODU )UDPH

PAGE 154

WKDW WKH SRODU DQJOHV A PD\ EH IRXQG IURP VLPSOH GRW SURGXFW UHODWLRQV EHWZHHQ D XQLW YHFWRU U DORQJ U DQG WKH XQLW YHFWRU IL DORQJ WKH LM LM L PROHFXODU D[LV FRV IL f U L O LM f 7KH DQJOH -!f LV QRW VR VWUDLJKWIRUZDUG WR GHWHUPLQH KRZHYHU )URP WKH ODZ RI FRVLQHV RI VSKHULFDO WULJRQRPHWU\ FRV ,f LM FRV < FRV L VLQ VLQ L FRV H M f ZKHUH \ LV WKH DQJOH EHWZHHQ WKH D[HV RI PROHFXOHV L DQG M JLYHQ E\ FRV < r ILM f 7KHUH UHPDLQ GLIILFXOWLHV LQ XVLQJ WKLV VHW RI LQWHUPROHFXODU DQJOHV ,Q SDUWLFXODU &KHXQJ > @ KDV SRLQWHG RXW WKDW XVH RI WKH DQJOHV A M -f UHTXLUHV HYDOXDWLRQ RI WLPH FRQVXPLQJ YHFWRU FURVV SURGXFWV WR GHWHUPLQH WKH GLUHFWLRQ RI WKH WRUTXH RQ D PROHFXOH )XUWKHU DV FDQ EH DSSUHFLDWHG IURP f WKHUH DUH FRPSXWDWLRQDO GLIILFXOWLHV LQ WKH QHLJKERUKRRG RI A (TXDWLRQ f LPSOLHV WKDW _! PD\ EH HOLPLQDWHG LQ IDYRU RI WKH DQJOH \ DQG WKDW WKHQ WKH FRVLQHV RI A DQG \ PD\ EH XVHG DV WKH LQGHSHQGHQW YDULDEOHV VSHFLI\LQJ WKH PROHFXODU SDLU RULHQWDWLRQ UDWKHU WKDQ WKH DQJOHV WKHPVHOYHV 8VH RI WKH VHW FRV A FRV A FRV \ UHPRYHV WKH DERYH PHQWLRQHG GLIILFXOWLHV DVVRFLDWHG ZLWK WKH LQWHUPROHFXODU UHIHUHQFH IUDPH $V LV VKRZQ EHORZ XVLQJ WKHVH DQJOHV DOORZV GHWHUPLQDWLRQ

PAGE 155

RI WKH GLUHFWLRQ RI WKH WRUTXH E\ HYDOXDWLQJ D VLQJOH YHFWRU FURVV SURGXFW 8VH RI f DQG f DOVR DYRLGV WKH SUREOHP DULVLQJ LQ f DURXQG A 7KXV LQ WKH PROHFXODU G\QDPLFV ZRUN UHSRUWHG KHUH ZH KDYH XVHG WKH SDLU SRWHQWLDO LQ WKH IRUP X LM f RU HTXLYDOHQWO\ X LM XUMILU IL f U IL rIL f LM fO OM fM LM f[ M f ([SUHVVLRQV IRU VHYHUDO DQLVRWURSLF SRWHQWLDO PRGHOV LQ WKH IRUP RI f DUH JLYHQ LQ $SSHQGL[ & *HQHUDO ([SUHVVLRQV IRU WKH )RUFH ,Q WKLV DQG WKH QH[W VXEVHFWLRQ GHULYDWLRQV DUH RXWOLQHG IRU WKH IRUFH DQG WRUTXHRQ D[LDOO\ V\PPHWULF PROHFXOHV ZKHQ WKH PROHFXODU SDLU RULHQWDWLRQV DUH JLYHQ E\ f DQG f DQG WKH SDLU SRWHQWLDO LV ZULWWHQ LQ WKH IRUP RI f RU f (TXDWLRQ f IRU WKH IRUFH RQ D PROHFXOH PD\ EH ZULWWHQ DV ) f§ B P XUAFRARBf f ZKHUH f KDV EHHQ LQWURGXFHG :ULWLQJ WKH JUDGLHQW RSHUDWRU [f§f§ LQ WHUPV RI WKH VSKHULFDO FRRUGLQDWH DQJOHV Df RI )LJXUH IRU WKH LQWHUPROHFXODU D[LV WKH FRPSRQHQWV RI DUH

PAGE 156

O[ LWL VLQ FRV D XLL UX D 6X f FRV S FRV D -M X VLQ D OM OM UA VLQ D f L\ VLQ VLQ D X f X OM FRV S VP D OM UOL XB FRV D OMB OM UA VLQ RW f )O] O FRV X D X /O VP  U U OM OM f 7R WUDQVIRUP f f DQG f WR WKH DQJOHV A k DQG < LQ WKH LQWHUPROHFXODU IUDPH ZH XVH WKH IROORZLQJ UHODWLRQV JLYHQ E\ WKH ODZ RI FRVLQHV RI VSKHULFDO WULJRQRPHWU\ FRV FRV FRV VLQ VLQ FRV D Df L L O L f FRV \ FRV L FRV VLQ VLQ FRV D Df L L f $SSO\LQJ WKH FKDLQ UXOH RI SDUWLDO GLIIHUHQWLDWLRQ WR f f DQG f JLYHV )O[ O MrO X f f OL FRV S FRV D VLQ VLQ D WWf§f§ U U OM OM FRV X /O FRV FRV X L /O FRV VLQ D U VLQ OM FRV X /O D FRV FRV X /O D FRV f

PAGE 157

, r VLQ % VLQ D X U OM OM FRV % VLQ D FRV X % ,, FRV FRV % FRV FRV D FRV A X FRV fLWn U VLQ % OM D FRV A D FRV f ) O] O r L f A8OM VLQ % FRV % f§ f§ U OM OM FRV A X OM L FRV 8OM % FRV A % FRV f FRV FRV (YDOXDWLQJ WKH GHULYDWLYHV f§ DQG IURP f DQG f XVLQJ WKH UHODWLRQV f f DQG f SHUIRUPLQJ VRPH DOJHEUDLF PDQLSXODWLRQ DQG XVLQJ WKH WULJRQRPHWULF UHODWLRQ VLQ [ FRV [\f FRV [ VLQ [\f VLQ \ f f f DQG f FDQ EH UHFDVW LQ WKH YHFWRU IRUP ,, MrO X U f§IU IL f U f OM U U /OM9OOM OM OM X ILM LL f§ FRV X ILM Ls fM FRV -f (TXDWLRQ f LV WKH JHQHUDO H[SUHVVLRQ IRU WKH IRUFH H[HUWHG RQ PROHFXOH E\ D V\VWHP RI D[LDOO\ V\PPHWULF PROHFXOHV ZKHQ WKH SDLU SRWHQWLDO LV ZULWWHQ LQ WKH IRUP RI f ,W LV UHDGLO\ DSSDUHQW

PAGE 158

WKDW IRU DQ LVRODWHG SDLU RI PROHFXOHV f JLYHV )A ([SUHVVLRQV IRU WKH GHULYDWLYHV LQ f IRU VHYHUDO DQLVRWURSLF SRWHQWLDOV DUH JLYHQ LQ $SSHQGL[ & *HQHUDO ([SUHVVLRQ IRU WKH 7RUTXH 7KH GHULYDWLRQ IRU WKH JHQHUDO H[SUHVVLRQ IRU WKH WRUTXH SURFHHGV LQ DQ DQDORJRXV IDVKLRQ WR WKDW IRU WKH IRUFH 7R REWDLQ f LQ WHUPV RI DQJOHV LQ WKH VSDFH IL[HG UHIHUHQFH IUDPH ZH XVH WKH IDFW WKDW WKH FRPSRQHQWV RI WKH DQJXODU JUDGLHQW LQ f FDQ EH VKRZQ WR EH SURSRUn WLRQDO WR FRPSRQHQWV RI WKH DQJXODU PRPHQWXP RSHUDWRU / > @ WKHQ LL n K O XULMfLfMf -AO ZKHUH LQ WKH SUHVHQW QRWDWLRQ f f 7KH FRPSRQHQWV RI LQ WKH VSKHULFDO FRRUGLQDWHV RI )LJXUH DUH JLYHQ E\ 5RVH > @ +HQFH ZH ILQG WKH FRPSRQHQWV RI B7A LQ DQJOHV UHODWLYH WR D VSDFH IL[HG IUDPH WR EH 7 O[ 7 L\ 7 O] O MO O MWIL LFRV FRV D X M NO OM VLQ % DA >FRV VLQ D X OM VLQ A DA GX VLQ 2W LM X FRV D f f f

PAGE 159

7R WUDQVIRUP f f DQG f WR IRUPV LQYROYLQJ 4M M!
PAGE 160

RWKHU 1Of PROHFXOHV LQ WKH V\VWHPV 7KLV UHVXOWV LQ D FRQVLGHUDEOH VDYLQJ RI H[HFXWLRQ WLPH )RU DQ LVRODWHG SDLU RI PROHFXOHV LW LV VWUDLJKWIRUZDUG WR VKRZ WKDW f DQG f VDWLVI\ f LH WKH DQJXODU PRPHQWXP RI WKH V\VWHP LV FRQVHUYHG ([SUHVVLRQV IRU WKH GHULYDWLYHV LQ f IRU YDULRXV DQLVRWURSLF PRGHO SRWHQWLDOV DUH JLYHQ LQ $SSHQGL[ & $Q LQGHSHQGHQW GHULYDWLRQ RI f DQG f KDV UHFHQWO\ EHHQ JLYHQ E\ &KHXQJ > @f 0HWKRG RI 6ROXWLRQ RI WKH (TXDWLRQV RI 0RWLRQ DQG WKH 0ROHFXODU '\QDPLFV $OJRULWKP 0HWKRG RI 6ROXWLRQ RI WKH (TXDWLRQV RI 0RWLRQ 6HYHUDO GLIIHUHQW PHWKRGV KDYH EHHQ XVHG WR VROYH WKH WUDQVODWLRQDO DQG URWDWLRQDO HTXDWLRQV RI PRWLRQ f DQG f IRU D[LDOO\ V\PPHWULF PROHFXOHV >@ 7KH PHWKRG XVHG KHUH LV WKDW GXH WR &KHXQJ DQG 3RZOHV > @ $ PDMRU DGYDQWDJH RI WKLV PHWKRG LV WKDW SUREOHPV DVVRFLDWHG ZLWK VROYLQJ WKH VHFRQG RUGHU GLIIHUHQWLDO HTXDWLRQ f IRU WKH PROHFXODU RULHQWDWLRQV LQ WKH LQWHUPROHFXODU IUDPH DUH DYRLGHG E\ VROYLQJ LQVWHDG WKH FRUUHVSRQGLQJ ILUVW RUGHU GLIIHUHQWLDO HTXDWLRQ IRU WKH DQJXODU YHORFLW\ f 7KH GLIILFXOWLHV ZKLFK DULVH LQ VROYLQJ f DUH Df WKH VHFRQG RUGHU HTXDWLRQV IRU WKH SRODU DQG D]LPXWKDO DQJOHV PXVW EH VROYHG VHSDUDWHO\

PAGE 161

WKXV PRUH FRPSXWDWLRQ ,V LQYROYHG Ef WKH VHFRQG RUGHU HTXDWLRQ IRU WKH D]LPXWKDO DQJOH FS FRQWDLQV D WHUP LQYROYLQJ VLQ 6XFK D WHUP LQWURGXFHV FRPSXWDWLRQDO GLIILFXOWLHV LQ WKH UHJLRQ DURXQG > @ 2QFH WKH DQJXODU YHORFLW\ LV REWDLQHG IURP f WKHUH UHPDLQV WKH GHWHUPLQDWLRQ RI WKH PROHFXODU RULHQWDWLRQ BIL 7KH RULHQWDWLRQ PD\ EH IRXQG E\ UHDOL]LQJ WKDW WKH DQJXODU YHORFLW\ DQG DQJXODU DFFHOHUDWLRQ DUH PXWXDOO\ SHUSHQGLFXODU WR RQH DQRWKHU DQG WR WKH PROHFXODU D[LV > @ 6LQFH WKH DQJXODU DFFHOHUDWLRQ LV SURSRUWLRQDO WR WKH WRUTXH REWDLQHG IURP f WKH XQLW YHFWRU _L PD\ EH IRXQG E\ IL f§L f§L f§L f Qf WK ZKHUH [ LQGLFDWHV WKH Q WLPH GHULYDWLYH RI [ 7KH PHWKRG XVHG IRU VROYLQJ WKH VHFRQG RUGHU WUDQVODWLRQDO HTXDWLRQV RI PRWLRQ f DQG WKH ILUVW RUGHU URWDWLRQDO HTXDWLRQV RI PRWLRQ f LV WKH SUHGLFWRUFRUUHFWRU DOJRULWKP RI *HDU > @f 7KH PHWKRG LQYROYHV WKUHH VWHSV SUHGLFWLRQ HYDOXDWLRQ DQG FRUUHFWLRQ ,Q WKH SUHGLFWRU VWHS WKH SRVLWLRQ BUA DQG RULHQWDWLRQ IL RI HDFK PROHFXOH DQG WKHLU ILUVW ILYH WLPH GHULYDWLYHV DW WLPH W $W DUH SUHGLFWHG IURP WKHLU YDOXHV DW WLPH W E\ D 7D\ORUnV H[SDQVLRQ 7KH DQJXODU YHORFLW\  RI HDFK PROHFXOH DQG LWV ILUVW IRXU WLPH GHULYDWLYHV DUH SUHGLFWHG LQ WKH VDPH PDQQHU U? W$Wf UWf UfWf$W UfWf$Wf fff UfWf$Wf U3fW$Wf UAfWf W U-fWf$W fff U>fWf$Wf U3fW$Wf UAfWf UAfWf$W fff U_fWf$Wf f U3f W$Wf U3f Wf

PAGE 162

IL3W$Wf IO Wf A;fWf$W fff QAf$Wf J3fW$Wf Q3fWf f IL3W$Wf ILWf ILDfWf$W f§L f§L f§L IL3fW$Wf ILIfWf ILMf$Wf f ,Q WKH HYDOXDWLRQ VWHS WKH IRUFH WUDQVODWLRQDO YHORFLW\f DQG WRUTXH DQJXODU YHORFLW\f DUH GHWHUPLQHG DW WKH SUHGLFWHG SRVLWLRQV 3 3 YB DQG RULHQWDWLRQV BIL IURP f DQG f S ,Q WKH FRUUHFWLRQ VWHS WKH SUHGLFWHG SRVLWLRQV UB DQG DQJXODU S YHORFLWLHV DQG WKHLU GHULYDWLYHV DUH FRUUHFWHG E\ UAQfW$Wf U3QfW$Wf D $U >$WfQQ@ f§L f§L Q f§L f LAQfW$Wf IW3QfW$Wf $ > $WfQQ @ f§O f§L Q f§L f ZKHUH WKH FRUUHFWLRQ WHUPV LQ f DQG f DUH SURSRUWLRQDO WR WKH GLIIHUHQFH EHWZHHQ WKH SUHGLFWHG DQG HYDOXDWHG DFFHOHUDWLRQV $U f§O >UI `W$Wf U3fW$Wf@$Wf f $ f§ >Q_fW$Wf IL3fW$Wf@$W f 7KH SDUDPHWHUV D DQG LQ f DQG f DUH FKRVHQ WR PDLQWDLQ Q Q

PAGE 163

VWDELOLW\ RI WKH VROXWLRQ DQG GHSHQG RQ WKH RUGHU RI WKH GLIIHUHQWLDO HTXDWLRQ DQG WKH GHJUHH RI H[SDQVLRQ LQ WKH 7D\ORU VHULHV SUHGLFWRU VWHS )RU WKH SUREOHP GHVFULEHG KHUH *HDU > @ JLYHV WKH YDOXHV Df n n n D f /! Lf L f f 7KLV SUHGLFWRUFRUUHFWRU PHWKRG LV QRW VHOIVWDUWLQJ FRQVHTXHQWO\ ZH XVH WKH IROORZLQJ VWDUWXS SURFHGXUH Df )RU WKH LQLWLDOO\ DVVLJQHG PROHFXODU SRVLWLRQV YHORFLWLHV RULHQWDWLRQV DQG DQJXODU YHORFLWLHV FDOFXODWH WKH WUDQVODWLRQDO DQG DQJXODU DFFHOHUDWLRQV IURP f f f DQG f DQG VHW WKH KLJKHU GHULYDWLYHV RI U DQG WR ]HUR f§L f§L Ef )URP WKH LQLWLDOO\ DVVLJQHG RULHQWDWLRQV HYDOXDWH WKH GHULYDWLYHV RI IL WR E\ UHSHDWHG GLIIHUHQWLDWLRQ RI f§Of§L ILIf IL f  f f§O f§L f§L 7KH FDOFXODWLRQ WKHQ SURFHHGV ZLWK WKH SUHGLFWRU VWHS DERYH 6HWWLQJ WKH KLJKHU GHULYDWLYHV RI UB f DQG ILAf WR ]HUR LQWURGXFHV D VOLJKW HUURU LQWR WKH ILUVW IHZ VROXWLRQV RI WKH HTXDWLRQV RI PRWLRQ +RZHYHU WKH DOJRULWKP FRUUHFWV LWVHOI WR WKH SURSHU VROXWLRQV DIWHU D IHZ WLPH VWHSV KDYH HYROYHG ,W VKRXOG EH HPSKDVL]HG WKDW QR FRPSDULVRQ RI YDULRXV PHWKRGV IRU VROYLQJ WKH HTXDWLRQV RI PRWLRQ KDYH EHHQ PDGH LQ WKLV VWXG\ 7KH PHWKRG

PAGE 164

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f 7KH EDVLF FHOO IRU WKH V\VWHP ZDV RI FXELF VKDSH DQG SHULRGLF ERXQGDU\ FRQGLWLRQV ZHUH XVHG WR QHJDWH VXUIDFH HIIHFWV ,Q RUGHU WR VROYH WKH URWDWLRQDO HTXDWLRQV RI PRWLRQ f WKH PROHFXOHV ZHUH DVVXPHG WR EH KRPRQXFOHDU GLDWRPLFV DQG WKH ERQG OHQJWK FRUUHVSRQGLQJ WR WKH QLWURJHQ PROHFXOH eR ZDV XVHG > @ 7KH FKRLFH RI ERQG OHQJWK FDQ KDYH QR HIIHFW RQ WKH HTXLOLEULXP SURSHUWLHV VLQFH SRWHQWLDOV RI WKH IRUP f DUH LQGHSHQGHQW RI ERQG OHQJWK ,Q WKHVH FDOFXODWLRQV WKH SRWHQWLDO ZDV FXW RII DW HLWKHU D RU D 7KH XQLW RI OHQJWK

PAGE 165

ZDV WDNHQ WR EH WKH PROHFXODU GLDPHWHU DQG WKH XQLW RI PDVV ZDV WDNHQ WR EH WKH PROHFXODU PDVV 7KH FDOFXODWLRQ SURFHGXUH LV DV IROORZV &KRRVH D SDUWLFXODU SRWHQWLDO PRGHO ZKRVH IRUP LV JLYHQ LQ r $SSHQGL[ & D IOXLG GHQVLW\ S SS DQG D WLPH VWHS $W DW ZKLFK WR GR N WKH VLPXODWLRQ &KRRVH DQ DSSUR[LPDWH WHPSHUDWXUH 7 N7H IRU WKH V\VWHP )RU WKH LQLWLDO SRVLWLRQV U?f RI WKH PROHFXODU FHQWHUV RI PDVV XVH HLWKHU DQ )&& ODWWLFH VWUXFWXUH RU WDNH WKH SRVLWLRQV IURP WKH HQG RI D SUHYLRXV FDOFXODWLRQ $VVLJQ LQLWLDO RULHQWDWLRQV LIX2f WUDQVODWLRQDO f DQG URWDWLRQDO ILAf YHORFLWLHV WR WKH PROHFXOHV IURP UDQGRP QXPEHUV XQLIRUPO\ GLVWULEXWHG RQ f 6FDOH WKH YHORFLWLHV VR WKHUH LV QR QHW WUDQVODWLRQDO RU URWDWLRQDO GULIW RQ WKH V\VWHP ^`Rf L fRf &' L O f Q Rf VLRf \ Q Rf f§L f§L 1 f§L f 6FDOH WKH YHORFLWLHV WR JLYH DSSUR[LPDWHO\ WKH FRUUHFW WHPSHU N DWXUH 7 UA`f N 7 1 O U f§ L L f QRf ILsf r 7 1 O  f§L f§ L f§ f f

PAGE 166

ZKHUH WKH PRPHQW RI LQHUWLD LV WDNHQ WR EH P P f %HJLQ WKH SUHGLFWRUFRUUHFWRU DOJRULWKP XVLQJ WKH VWDUWXS SURFHGXUH JLYHQ LQ 6HFWLRQ 3UHGLFW WKH PROHFXODU SRVLWLRQV DQJXODU YHORFLWLHV RULHQWDn WLRQV DQG WKHLU GHULYDWLYHV DW WLPH W$Wf E\ f f DQG f 8VLQJ WKH SUHGLFWHG SRVLWLRQV DQG RULHQWDWLRQV HYDOXDWH WKH IRUFH DQG WRUTXH RQ HDFK PROHFXOH XVLQJ f DQG f DQG KHQFH JHW WKH DFWXDO WUDQVODWLRQDO DQG DQJXODU DFFHOHUDWLRQV IURP f DQG f $OVR XVH WKH SUHGLFWHG SRVLWLRQV DQG RULHQWDWLRQV WR HYDOXDWH WKH HQVHPEOH DYHUDJHV IRU WKH HTXLOLEULXP SURSHUWLHV JLYHQ LQ 7DEOH 'HWHUPLQH WKH SDLU FRUUHODWLRQ IXQFWLRQV IURP WKH SUHGLFWHG UA DQG I/ E\ WKH PHWKRG JLYHQ LQ WKH QH[W VHFWLRQ &RUUHFW WKH SUHGLFWHG SRVLWLRQV DQG DQJXODU YHORFLWLHV XVLQJ f DQG f &DOFXODWH WKH PROHFXODU RULHQWDWLRQV DW W$Wf IURP f &KHFN WKDW HDFK SDUWLFOH LV LQ WKH EDVLF FHOO EDVHG RQ WKH FRUUHFWHG SRVLWLRQV 8VH WKH LPDJH KDYLQJ WKH PLQLPXP LPDJH GLVWDQFH IRU DQ\ SDUWLFOH QR ORQJHU LQ WKH FHOO > @ 'XULQJ WKH LQLWLDO WLPH VWHSV IRU ZKLFK WKH PHDQ VTXDUHG GLVSODFHPHQW RI WKH PROHFXOHV LV OHVV WKDQ VRPH H YDOXH WKH V\VWHP LV QRW FRQVLGHUHG WR EH DW HTXLOLEULXP DQG WKH YHORFLWLHV DUH UHVFDOHG DIWHU HDFK WLPH VWHS XVLQJ f DQG f :KHQ VWDUWLQJ IURP DQ )&& ODWWLFH VWUXFWXUH ( LV WDNHQ WR EH DERXW b RI WKH PROHFXODU UDGLXV ZKLFK LV D FRUUXSWLRQ RI /LQGHPDQQnV ODZ RI PHOWLQJ > @

PAGE 167

7KH EUHDNGRZQ RI WKH )&& ODWWLFH XVXDOO\ UHTXLUHV WLPH VWHSV &RQWULEXWLRQV WR SURSHUW\ DYHUDJHV DUH QRW FDOFXODWHG GXULQJ WKLV SRUWLRQ RI WKH FDOFXODWLRQ :KHQ WKH PHDQ VTXDUHG GLVSODFHPHQW H[FHHGV WKH GHVLUHG e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f 7KH DQJOH GHSHQGHQW GLVWULEXWLRQ IXQFWLRQV IRU PROHFXOHV LQWHUn DFWLQJ ZLWK SRWHQWLDOV RI WKH IRUP RI f DUH GHILQHG LQ D PDQQHU DQDORJRXV WR WKDW IRU GLVWULEXWLRQ IXQFWLRQV IRU VSKHULFDO PROHFXOHV 7KH JHQHULF GLVWULEXWLRQ IXQFWLRQ RI RUGHU K IU U Z Z f LV

PAGE 168

GHILQHG VXFK WKDW I U?fAf GUAGRA LV SURSRUWLRQDO WR WKH SUREDELOLW\ WKDW JLYHQ D PROHFXOH ZLWK SRVLWLRQ LQ GUA DQG RULHQWDWLRQ FRA LQ GFRA PROHFXOH LV DW UA LQ GUA ZLWK RULHQWDWLRQ FRA LQ GR! HWF XS WR PROHFXOH K LUUHVSHFWLYH RI WKH SRVLWLRQV DQG RULHQWDWLRQV RI WKH UHPDLQLQJ 1K PROHFXOHV >@ K K 1 I U f f 8U1f1f 1K 1K H f§ GU GRM 1 Kf J8U1&-1f 1 1 H f§ GU GLG f $Q DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JUA&Lfrf FDQ EH GHILQHG LQ WHUPV RI IBU?fnf DQDORJRXV WR WKH GHILQLWLRQ RI WKH UDGLDO GLVWULEXWLRQ IXQFWLRQ IRU VSKHULFDO PROHFXOHV IeKWRKf I&UMAf I eR!f f fI UAAf JUBKZKf f )RU DQ LVRWURSLF KRPRJHQHRXV IOXLG WKH YDOXH RI WKH VLQJOHW DQJXODU GLVWULEXWLRQ IXQFWLRQ I BUAAf LV LQGHSHQGHQW RI U DQG A KHQFH SXWWLQJ K LQ f JLYHV If§O8Of f ZKHUH LV WKH LQWHJUDO RYHU WKH DQJXODU YROXPH HOHPHQW GLR  WW IRU OLQHDU PROHFXOHV DQG  WW IRU QRQOLQHDU PROHFXOHV 7KXV IRU LVRWURSLF KRPRJHQHRXV IOXLGV f UHGXFHV WR U K K K YK K K I e &2 f S  f Je  f f

PAGE 169

7KH GLVWULEXWLRQ RI PROHFXODU FHQWHUV LQGHSHQGHQW RI PROHFXODU RULHQWDWLRQ LV REWDLQHG E\ LQWHJUDWLRQ RI IU?MAf RYHU RULHQWDWLRQ IUKf K K1 K I U WR fGWR f 8VLQJ f IUKf KA K K S Je X f! f f RU K1 A K KZ JFe f Je : f! K f 7KH SDLU FRUUHODWLRQ IXQFWLRQV DUH VWXGLHG LQ GHWDLO KHUH )URP f DQG f WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ LV JLYHQ E\ 11OfIL JnU-O-A 8U1f1f 1 1 H f§ GU GX 1 1 8U WR f 1 1 H f§ GU GWR f /LNHZLVH WKH FHQWHUWRFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ FDQ EH REWDLQHG IURP f JFUOf JUODfO:nf!WR f /HVV IRUPDOO\ JAUf PD\ EH GHILQHG DV WKH UDWLR RI WKH ORFDO QXPEHU GHQVLW\ RI PROHFXODU FHQWHUV RI PDVV DW GLVWDQFH U IURP WKH

PAGE 170

FHQWHU RI PDVV RI D JLYHQ PROHFXOH LQGHSHQGHQW RI WKH RULHQWDWLRQV RI WKH PROHFXOHV 3FUf WR WKH EXON IOXLG QXPEHU GHQVLW\ S 3 Uf FUf ff Bf ,W LV DOVR RI LQWHUHVW WR VWXG\ VRFDOOHG VLWHVLWH SDLU FRUn UHODWLRQ IXQFWLRQV J 'Uf ZKHUH D DUH VLWHV ORFDWHG RQ D PROHFXOH &WS 8VXDOO\ WKH PROHFXODU VLWHV RI LQWHUHVW DUH WKH DWRPLF FHQWHUV RQ D SRO\DWRPLF PROHFXOH 7KH JAUf LV WKHQ SURSRUWLRQDO WR WKH SUREn DELOLW\ RI ILQGLQJ WKH aVLWH RI VRPH PROHFXOH DW D GLVWDQFH U IURP WKH DVLWH RI VRPH GLIIHUHQW PROHFXOH > @ 7KH IXQFWLRQ DJUf PD\ EH REWDLQHG IURP WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ E\ :: n JLR f f ZKHUH U LV WKH YHFWRU IURP WKH FHQWHU RI PROHFXOH L WR VLWH D f§L /HVV IRUPDOO\ DJUDJf FDQ GHILQHG DV WKH UDWLR RI WKH ORFDO QXPEHU GHQVLW\ RI VLWHV DW GLVWDQFH UBA IURP WKH DVLWH RI D JLYHQ PROHFXOH LQGHSHQGHQW RI PROHFXODU RULHQWDWLRQV DQG H[FOXGLQJ WKH aVLWH RI WKH JLYHQ PROHFXOH WR WKH EXON IOXLG QXPEHU GHQVLW\ 3UD` DADA S f (YDOXDWLRQ RI &HQWHU&HQWHU DQG 6LWH6LWH 3DLU &RUUHODWLRQ )XQFWLRQV 7KH FHQWHUFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ JFUf FDQ EH GHWHUPLQHG LQ D PROHFXODU G\QDPLFV FDOFXODWLRQ E\ XVLQJ WKH GHILQLWLRQ f 7KH

PAGE 171

SURFHGXUH LV WR ILUVW GLYLGH WKH YROXPH RI WKH FXELF V\VWHP RI VLGH bR LQWR VSKHULFDO VKHOOV RI WKLFNQHVV $U ,Q WKLV ZRUN $U ZDV VHW HTXDO WR D 'XULQJ WKH HYDOXDWLRQ VWHS RI WKH PROHFXODU G\QDPLFV SURFHGXUH WKH GLVWDQFH UB EHWZHHQ FHQWHUV RI HDFK SDLU RI PROHFXOHV LV GHWHUPLQHG XVLQJ WKH SUHGLFWHG SRVLWLRQV )RU HDFK Uf WKH VSKHULFDO VKHOO LQ ZKLFK PROHFXOH M OLHV UHODWLYH WR PROHFXOH L LV GHWHUPLQHG DQG $YHUDJH QXPEHU RI PROHFXODU FHQWHUV LQ VSKHULFDO VKHOO KDYLQJ ERXQGDULHV f Q 7KH IDFWRU RI DULVHV LQ f WR LQFOXGH WKH FRQWULEXWLRQ RI WKH VKHOOV GXH WR U ZKLFK DUH QRW LQFOXGHG LQ ,Uf --L LM ,I WKH IOXLG ZHUH RI XQLIRUP GHQVLW\ S IRU DOO U UHJDUGOHVV RI WKH ORFDWLRQ RI DQ\ JLYHQ PROHFXOH WKH DYHUDJH QXPEHU RI PROHFXODU FHQWHUV LQ DQ\ VSKHULFDO VKHOO ZRXOG EH JLYHQ E\ 1S$9Uf ZKHUH $9Uf LV WKH YROXPH RI WKH VKHOO 7KHQ IURP f UfQ F f 1S $9Uf WW I ZKHUH $9Uf GM! GFRV f UA GU f f

PAGE 172

'HWHUPLQDWLRQ RI WKH VLWHVLWH SDLU FRUUHODWLRQ IXQFWLRQV J 4U 'f LV DFFRPSOLVKHG XVLQJ f LQ WKH VDPH PDQQHU DV IRU &WS 2WS JFUf GHVFULEHG DERYH 7KH UHVXOWLQJ UHODWLRQ LV VDUD% f ,D%UD%fQ 1S $9UD%f f 7R GHWHUPLQH WKH GLVWDQFHV U IRU D SDLU RI PROHFXOHV FRQVLGHU WKH &;S JHRPHWU\ IRU D SDLU RI GLDWRPLFV ZLWK VLWHV $ DQG % VHSDUDWHG E\ GLVWDQFH e RQ HDFK PROHFXOH DV VKRZQ LQ )LJXUH ,Q WKH PROHFXODU G\QDPLFV FDOFXODWLRQ WKH ORFDWLRQV RI WKH FHQWHUV RI PDVV RI HDFK PROHFXOH UHODWLYH WR VRPH VSDFH IL[HG IUDPH UA DUH NQRZQ 7KH RULHQWDWLRQV RI HDFK PROHFXOH DUH DOVR NQRZQ LQ WHUPV RI XQLW YHFWRUV IL DOLJQHG DORQJ HDFK PROHFXOH 7KXV ZLWK )LJXUH DQG VRPH VLPSOH YHFWRU DGGLWLRQ WKH GLVWDQFHV U Q DUH IRXQG WR EH U W[% f ZKHUH WKH ILUVW VLJQ RQ WKH ULJKW KDQG VLGH LV LI D $ LI D % DQG WKH VHFRQG VLJQ LV LI D A % LI D %! DQG UA LV JLYHQ E\ f 1RWH WKDW IRU KRPRQXFOHDU GLDWRPLFV JA JA GXH WR V\PPHWU\ 6SKHULFDO +DUPRQLF ([SDQVLRQ IRU JUAAf§f§ $V SRLQWHG RXW LQ 6HFWLRQ LW LV ERWK LQDFFXUDWH DQG LQHIILFLHQW WR DWWHPSW GLUHFW HYDOXDWLRQ RI WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JULM-L:A nrnUL FRP3XWHU VLPXODWLRQ VWXGLHV 7KH EHWWHU PHWKRG LV WR H[SDQG JWUAAWAf ff‘Q VSKHULFDO KDUPRQLFV DQG HYDOXDWH WKH H[SDQVLRQ

PAGE 174

FRHIILFLHQWV LQ WKH VLPXODWLRQ 7KH H[SDQVLRQ FRQVLGHUHG KHUH LV DQ LQILQLWH VHULHV LQ WHUPV RI SURGXFWV RI VSKHULFDO KDUPRQLFV RI WKH PROHFXODU RULHQWDWLRQV LQ WKH LQWHUPROHFXODU IUDPH RI )LJXUH U8ODfA WW O JeefPUf @ 7KH JS A UDAULA DUH WrLH H[3DQVrRQ FRHIILFLHQWV WR EH GHWHUPLQHG LQ WKH VLPXODWLRQ )RU PROHFXOHV ZLWK D SODQH RI V\PPHWU\ SHUSHQGLFXODU WR WKH PROHFXODU D[LV WKH J f Uff DUH ]HUR XQOHVV ERWK -/ DQG =f DUH ; [ P HYHQ KRPRQXFOHDU GLDWRPLFV HJf )RU KHWHURQXFOHDU GLDWRPLFV J ZLWK ERWK RGG DQG HYHQ YDOXHV FRQWULEXWH +A=AP IW IW 0XOWLSO\LQJ ERWK VLGHV RI f E\ @ GRf < RMf < RMf [P [ P += PP f 6R f EHFRPHV

PAGE 175

-OPUf WW GL GZ
PAGE 176

([SUHVVLRQV IRU DOO A PAUOA WHUPV KDYLQJ eA DQG e HYHQ XS WR ^eAeP` ^` DQG WKH DGGLWLRQDO WHUUDV ZLWK eA DQG HYHQ P WR ^eAeP` ^` DUH WDEXODWHG LQ $SSHQGL[ ) 6RPH DXWKRUV SUHIHU WR H[SDQG UA-OD!A rQ WHUPV rI SURGXFWV RI VSKHULFDO KDUPRQLFV RI WKH PROHFXODU RULHQWDWLRQV LQ WKH VSDFH IL[HG IUDPH RI )LJXUH UDWKHU WKDQ WKH LQWHUPROHFXODU IUDPH > @ 7KLV H[SDQVLRQ LV 6UO:O:f O JAALfULf &eeePPPf eAee PAPP [ < WR f < WR f ;AQLA \/Zf f +HUH Z ^RW` LV WKH RULHQWDWLRQ RI WKH LQWHUPROHFXODU D[LV DV LQ )LJXUH &MPAQAPf LV D &OHEVFK*RUGDQ FRHIILFLHQW LQ WKH FRQYHQWLRQ RI 5RVH > 7KH FRHIILFLHQWV RI WKH LQWHUPROHFXODU IUDPH H[SDQVLRQ f A PAUOAf &DQ GHWHUPLQHG IURP WKRVH RI WKH VSDFH IL[HG IUDPH H[SDQVLRQ f J eMe e Uf E\ JeePUOf 7fa ? e! JAUf & AAPA2f f &RQYHUVHO\ WKH JeAeeUAf PD\ EH IRXQG IURP WKH JA QUOA JeeeU@Bf r rL, 4 / te e P Uf FLneePO+rf [U./f f

PAGE 177

(TXLOLEULXP 3URSHUWLHV IURP WKH JA f XUOf§ 7KH FRHIILFLHQWV A PAULA 3URYLGH XVHIXO LQIRUPDWLRQ DERXW ORFDO VWUXFWXUH DQG PROHFXODU RULHQWDWLRQV LQ IOXLGV ZKHQ UHFRPELQHG LQ WKH VHULHV f WR REWDLQ WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JUZLAfr DGGLWLRQ LQWHJUDOV RYHU YDULRXV FRPELQDWLRQV RI WKH JI e PAUOA LYH WKH HTXLOLEULXP SURSHUWLHV RI WKH IOXLG ,Q WKH FRQWH[W RI PDFKLQH VLPXODWLRQ VWXGLHV VXFK UHODWLRQV SURYLGH XVHIXO FRQVLVWHQF\ FKHFNV EHWZHHQ WKH GLUHFW HYDOXDWLRQ RI WKH SURSHUWLHV YLD WKH HQVHPEOH DYHUDJHV RI 7DEOH DQG WKH GHWHUPLQDWLRQ RI WKH JR R Uff FRHIILFLHQWV ,Q WKLV VHFWLRQ WKH GHULYDWLRQ RI WKH UHODWLRQ EHWZHHQ WKH JA A PUA f DQG WKH FRQILJXUDWLRQDO HQHUJ\ 8 GXH WR D /HQQDUG-RQHV SOXV TXDGUXSRODU PRGHO SRWHQWLDO LV JLYHQ 'HULYDWLRQV RI WKH FRUUHVSRQGLQJ UHODWLRQV IRU RWKHU SRWHQWLDO PRGHOV DQG RWKHU SURSHUWLHV DUH DFFRPSOLVKHG LQ DQ DQDORJRXV PDQQHU WKHUHIRUH RQO\ WKH UHVXOWLQJ H[SUHVVLRQV DUH SUHVHQWHG LQ WDEXODU IRUP 7KH FRQILJXUDWLRQDO HQHUJ\ IRU SRO\DWRPLF IOXLGV LV JLYHQ E\ XfJf! GU fA f ZKHUH ;f DQG fff! LV GHILQHG E\ f 8VLQJ IRU Xf WKH PRGHO RI f JLYHV 81 8/8 D f 8/-Uf JUf G ZKHUH f

PAGE 178

DQG 8 e D XDfJf!:L: GU f (TXDWLRQ f KDV EHHQ REWDLQHG E\ XVLQJ f 6XEVWLWXWLQJ WKH /HQQDUG-RQHV SRWHQWLDO f LQWR f JLYHV 8/LUS H r r r r UOnf AU f U A U GU tefP r N 'HILQLQJ WKH LQWHJUDOV S 7 f E\ f Q [ rQf r PUOf U GU f f FDQ EH ZULWWHQ R r U 7 8/n F ,f 7R HYDOXDWH 8 IURP f ILUVW REWDLQ DQ H[SUHVVLRQ IRU X fJf! &RQVLGHU X f WR EH WKH TXDGUXSROHTXDGUXSROH F/ SRWHQWLDO 7KHQ IURP $SSHQGL[ & (TV &f DQG &f JLYH WKH H[SDQVLRQ IRU 8TTf LQ WHUPV RI VSKHULFDO KDUPRQLFV RI WKH PROHFXODU RULHQWDWLRQV LQ WKH VSDFH IL[HG IUDPH [ B UU 4B 844AA WWf U POP O &fPff
PAGE 179

9f WW f e &P P f < WR f < D! f f U PO B &RPELQLQJ f DQG f mXAG2G} f 4 I Je e PUr A F&PPf U nLnLQ PO ; A!f AP9 @ DQG ZLWK WKH IDFW WKDW VLQFH WKH Je UAf DUH UHDO eePUOf eePUf f f EHFRPHV 844ff!-W >JUf JUOf JUOf@ U f

PAGE 180

$Q DQDORJRXV SURFHGXUH JLYHV XAfJf!A A IRU RWKHU SRWHQWLDO PRGHOV 5HVXOWV DUH JLYHQ LQ 7DEOH &RPELQLQJ f DQG f DQG XVLQJ f WW r r" S 4 H >f 5HVXOWV IRU X IRU RWKHU SRWHQWLDO PRGHOV DUH JLYHQ LQ 7DEOH 6/ 7DEOHV DQG JLYH H[SUHVVLRQV UHODWLQJ WKH AAP r r LQWHJUDOV -Q S 7 f RYHU WKH FRHIILFLHQWV JA A PAUOA W WOLH 3UHVB VXUH )RZOHU PRGHO VXUIDFH WHQVLRQ )RZOHU PRGHO VXUIDFH H[FHVV LQWHUQDO HQHUJ\ PHDQ VTXDUHG IRUFH PHDQ VTXDUHG WRUTXH DQG DQJXODU FRUUHODWLRQ IXQFWLRQV UHVSHFWLYHO\ /M

PAGE 181

([SUHVVLRQV 7$%/( IRU X fJf! LQ 7HUPV D r fMf f IRU 9DULRXV 0RGHO 3RWHQWLDOV 8''ff!f FR >Uf OOOUf@ U f 844fJf!R FR 4 >JUf JUf JUf@ f U 8'4fJf!FR AU JUf A JUf JUf JUf@ f X fJf! f§ H RYHU 2OR A U LO R JRRUA f X fJf! f§ NJ GLV FRAFR B U JRRAUnf f e >JUf iAUA JAUA W '4RYHUGLV GLSROH TXDGUXSROH RYHUODS DQG GLVSHUVLRQ UHVSHFWLYHO\

PAGE 182

7$%/( ([SUHVVLRQV IRU WKH &RQILJXUDWLRQDO (QHUJ\ LQ 7HUPV RI A UDUAQf IRU 9DULRXV 0RGHO 3RWHQWLDOV 81 8 7 8 /D f R r U7 7 8/L7S H >-$ @ f P B WW r r U 7 8'' 3 e >@ f B WW r r U A 7 A 844 3 4 e >f f WW r r r I 5 W RW b f§ 3 9 4 e > @ f RYHU WW r S RH B f 8 GLV WW r W 3 e WW r f S e >8f

PAGE 183

7$%/( ([SUHVVLRQV IRU WKH 3UHVVXUH LQ 7HUPV RI JQ Q ULRf IrU 9DULRXV 0RGHO 3RWHQWLDOV [ f§A 3 SN7 f WWS U B /N7 f 3 D f§ 8 N7 D f ZKHUH QJ ef IRU GLVSHUVLRQ RYHUODS RU fPXOWLSROH UHVSHFWLYHO\ DQG H[SUHVVLRQV IRU 8 DUH DV JLYHQ LQ 7DEOH

PAGE 184

7$%/( ([SUHVVLRQV IRU WKH )RZOHU 0RGHO 6XUIDFH 7HQVLRQ LQ 7HUPV RI A QUO ARU ADUf486 0RGHO 3RWHQWLDOV ) ) ) < D H \/-D H \DR H ) Q r U7 27 -Q @ ) WW r r U7 7''r (nI X ) r r U W 7 7 <44r H A 4 >@ ) 77 r r r U SU U M 7 7 f 7 -f @ '4 ) < D H RYHU WW r f§ 3 ) WW r W r U 7 7 f f f f f f f

PAGE 185

7$%/( ([SUHVVLRQV IRU WKH )RZOHU 0RGHO 6XUIDFH ([FHVV ,QWHUQDO (QHUJ\ LQ 7HUPV RI JA A PAO=A )RU ADUfRXV 0RGHO 3RWHQWLDOV 8J&Ue 8J DH 8J DH /D W) B r Q W X D H 7LS >M @ E/f) WW r r 7 7OOO 8 D H ] S \ >''   WL) WW r r 7 7 8 D H f§ S 4 >f§ f§ @ 44 ) WW r r r U A + 7 7 7 8 D e S \ 4 > -4 6'4 WL) WW r[ 8V 2 f f§ 3 m -Q RYHU YG WW) WW r r U 8 &7e f§ S 73 >,, GLV f f f f f f f

PAGE 186

7$%/( ([SUHVVLRQV IRU WKH 0HDQ 6TXDUHG )RUFH LQ 7HUPV RI A PAUA ARU 9DUrRXV 0RGHO 3RWHQWLDOV r r RQQ QRQ )S!/R H 773 7 > -aX -J@ ) D H 08/7, r r I! R H 77f f§ S 7 MU RYHU f f f

PAGE 187

7$%/( IRU WKH 0HDQ 6TXDUHG 7RUTXH LQ OHUDV RI 9DULRXV 0RGHO 3RWHQWLDOV 7K4H f a 7f 844& f m:r 7r 9V f W "! H RYHU 7r 8 H f RYHU 7! ] G[V 7r 8 H f GLV f

PAGE 188

7$%/( ([SUHVVLRQV IRU WKH $QJXODU &RUUHODWLRQ )XQFWLRQV LQ 7HUPV RI JA A UA ARU 9DULRXV 0RGHO 3RWHQWLDOV ‘N I7S \ P //P / f /  & f P f HJ UUSr U 1 LUS 1 M-;@ f -T -T @ f

PAGE 189

&+$37(5 02/(&8/$5 '<1$0,&6 5(68/76 7KLV FKDSWHU SUHVHQWV WKH HTXLOLEULXP SURSHUWLHV REWDLQHG E\ WKH PROHFXODU G\QDPLFV PHWKRG GHVFULEHG LQ &KDSWHU ,Q 6HFWLRQ DQDO\VLV LV PDGH RI WKH SRWHQWLDO PRGHOV FRQVLGHUHG LQ WKLV VWXG\ 6HFWLRQ JLYHV YDOXHV IRU HTXLOLEULXP SURSHUWLHV REWDLQHG IURP PROHFXODU G\QDPLFV DQG FRPSDULVRQ ZLWK SUHGLFWLRQV IURP SHUWXUEDWLRQ WKHRU\ LV PDGH 9DOXHV IRU WKH VSKHULFDO KDUPRQLF FRHIILFLHQWV e PAUOA DUH 3UHVHQWHG LQ 6HFWLRQ $OVR YDOXHV IRU WKH LQWHJUDOV DUH JLYHQ DQG HTXLOLEULXP SURSHUWLHV REWDLQHG IURP WKH DUH FRPSDUHG ZLWK YDOXHV REWDLQHG E\ GLUHFW HYDOXDWLRQ LQ WKH FRXUVH RI WKH VLPXODWLRQ ,Q 6HFWLRQ WKH DQJXODU SDLU FRUn UHODWLRQ IXQFWLRQ JUA:A:A REWDLQHG E\ UHFRPELQLQJ WKH VSKHULFDO KDUPRQLF H[SDQVLRQ LV VWXGLHG 7KH VLWHVLWH SDLU FRUUHODWLRQ IXQFWLRQ LV GLVFXVVHG LQ 6HFWLRQ ,Q 6HFWLRQ D PHWKRG IRU SURGXFLQJ ILOPHG DQLPDWLRQV RI WKH PROHFXODU PRWLRQV IURP D PROHFXODU G\QDPLFV VLPXODWLRQ LV GHVFULEHG 3RWHQWLDO 0RGHOV 7KH PRGHO LQWHUPROHFXODU SDLU SRWHQWLDOV FRQVLGHUHG KHUH DUH RI WKH IRUP LQ f LH LVRWURSLF /HQQDUG-RQHV SOXV DQ DQLVRn WURSLF FRQWULEXWLRQ 7KH DQLVRWURSLF FRQWULEXWLRQV FRQVLGHUHG LQFOXGH GLSROH TXDGUXSROH DQLVRWURSLF RYHUODS DQG DQLVRWURSLF GLVSHUVLRQ 6SHFLILF H[SUHVVLRQV IRU WKHVH DQLVRWURSLF SRWHQWLDOV

PAGE 190

DUH JLYHQ LQ $SSHQGL[ & ,Q WKLV VHFWLRQ ZH H[SORUH WKH QDWXUH RI WKHVH SRWHQWLDO PRGHOV IRU YDULRXV PROHFXODU RULHQWDWLRQV DQG DV D IXQFWLRQ RI WKH VWUHQJWK FRQVWDQW \4 RU f DVVRFLDWHG ZLWK HDFK $ GJUHH RI DPELJXLW\ DULVHV LQ WU\LQJ WR UHVROYH D SK\VLFDO GHVFULSWLRQ RI WKH PROHFXOH DVVRFLDWHG ZLWK SRWHQWLDOV RI WKH IRUP LQ f 7KH SRWHQWLDO FRQWDLQV D VSKHULFDOO\ V\PPHWULF UHSXOVLYH FRUH GXH WR WKH /HQQDUG-RQHV FRQWULEXWLRQf SOXV D QRQVSKHULFDO D[LDOO\ V\PPHWULF DQLVRWURS\ ,Q WKH FDVH RI PXOWLSROHV WKH DQLVRWURS\ PD\ EH LQWHUSUHWHG DV DULVLQJ GXH WR SRLQW FKDUJHV LPEHGGHG LQ WKH FRUH 7KH UHVXOWLQJ IRUFH ILHOG LV WKHUHIRUH D[LDOO\ V\PPHWULF DERXW WKH PROHFXOH *HRPHWULFDOO\ WKH PROHFXOH PD\ EH LQWHUSUHWHG DV D[LDOO\ V\PPHWULF )RU GLSROH TXDGUXSROH GLVSHUVLRQ DQG RYHUODS ZLWK f DQLVRWURSLHV WKH PROHFXOHV DUH FRQVLGHUHG OLQHDU )RU WKH RYHUODS PRGHO ZLWK WKH PROHFXOH LV SODWHOLNH HJ EHQ]HQHf :KHQ WKH DQLVRWURS\ LV TXDGUXSRODU RU RYHUODS ZLWK WKH PROHFXOH FDQ EH FRQVLGHUHG WR EH SURODWH HOOLSVRLG HJ KRPRQXFOHDU GLDWRPLFf ,I WKH DQLVRWURS\ LV GLSRODU WKH PROHFXOH FDQ EH FRQVLGHUHG WR EH KHWHURQXFOHDU GLDWRPLF 7KHVH GLVWLQFWLRQV DUH LPSRUWDQW HJ LQ UHFRJQL]LQJ ZKLFK WHUPV FRQWULEXWH WR WKH VSKHULFDO KDUPRQLF H[SDQVLRQ IRU J ,Q VROYLQJ WKH HTXDWLRQV RI PRWLRQ LQ WKH PROHFXODU G\QDPLFV VLPXODWLRQ WKH PRPHQW RI LQHUWLD DSSURSULDWH WR WKH VKDSH RI WKH PROHFXOHV LV UHTXLUHG VHH 6HFWLRQ f )RU OLQHDU PROHFXOHV WKH PRPHQW RI LQHUWLD GHSHQGV RQ WKH OHQJWK RI WKH PROHFXODU D[LV FRQVLGHUHG WR EH WKH ERQG OHQJWK LQ GLDWRPLFVf DV LQ (TXDWLRQ f +RZHYHU WKH SRWHQWLDO PRGHO f LV LQGHSHQGHQW RI ERQG OHQJWK 7KXV HTXLOLEULXP SURSHUW\ YDOXHV REWDLQHG LQ WKH VLPXODWLRQ

PAGE 191

DUH LQGHSHQGHQW RI ERQG OHQJWK IRU WKH PRGHO RI f 7LPH GHSHQGHQW SURSHUWLHV RQ WKH RWKHU KDQG UHPDLQ D IXQFWLRQ RI ERQG OHQJWK WKURXJK WKH PRPHQW RI LQHUWLD DQG WKH URWDWLRQDO HTXDWLRQV RI PRWLRQ 7KH YDOXH FKRVHQ IRU WKH ERQG OHQJWK ZLOO KDYH DQ HIIHFW RQ WKH UDWH RI FRQn YHUJHQFH RI VWDWLF SURSHUWLHV WR WKHLU HTXLOLEULXP YDOXHV LH WKH QXPEHU RI WLPH VWHSV UHTXLUHG WR REWDLQ VWDWLVWLFDOO\ PHDQLQJIXO UHVXOWVf 7KXV IRU D JLYHQ YDOXH RI WKH WRUTXH RQ D OLQHDU PROHFXOH (TXDWLRQV f DQG f LQGLFDWH WKDW VPDOOHU ERQG OHQJWKV FRUUHVSRQG WR KLJKHU URWDWLRQDO YHORFLWLHV ZKLFK SURPRWH PRUH H[WHQVLYH VDPSOLQJ RI DQJXODU SKDVH VSDFH FRPSDUHG ZLWK WKDW UHVXOWLQJ IURP ORQJHU ERQG OHQJWKV /HQQDUG-RQHV 3OXV 'LSROH 3RWHQWLDO 7KLV SRWHQWLDO PRGHO LV REWDLQHG E\ FRPELQLQJ (TXDWLRQV f f DQG &f )LJXUH VKRZV YDOXHV RI WKH SDLU SRWHQWLDO IRU YDULRXV PROHFXODU SDLU RULHQWDWLRQV DV D IXQFWLRQ RI WKH VHSDUDWLRQ RI WKHLU FHQWHUV RI PDVV U 7KH VL[ SDLU RULHQWDWLRQV VKRZQ LQ )LJXUH DUH WKH SULPDU\ RULHQWDWLRQV LH IRU WKH PRGHOV FRQVLGHUHG LQ WKLV VWXG\ RQH RI WKHVH VL[ KDV DOZD\V EHHQ IRXQG WR EH WKH PRVW SUREDEOH RULHQWDWLRQ LQ WHUPV RI ERWK PLQLPXP HQHUJ\ DQG PD[LPXP YDOXH RI WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JUAAMAA f 'HILQLWLRQV RI WKHVH RULHQWDWLRQV LQ WHUPV RI WKH UHODWLYH DQJOHV LQ WKH LQWHU PROHFXODU IUDPH RI )LJXUH DUH JLYHL! LQ 7DEOH )LJXUH LQn GLFDWHV WKDW WKH SDLU RULHQWDWLRQ KDYLQJ WKH PLQLPXP FRQILJXUDWLRQDO HQHUJ\ LV WKH KHDGWRWDLO HQGRQ RULHQWDWLRQ ZKLOH WKDW KDYLQJ WKH ODUJHVW HQHUJ\ LV WKH KHDGWRKHDG RULHQWDWLRQ 1RWH WKDW WKH GLSROH

PAGE 192

)LJXUH 3DLU 3RWHQWLDO IRU /HQQDUG-RQHV SOXV 'LSROH 0RGHO )OXLG DW 3ULPDU\ 3DLU 2ULHQWDWLRQV

PAGE 193

7$%/( 3ULPDU\ 2ULHQWDWLRQV IRU 3DLUV RI /LQHDU 0ROHFXOHV r < KHDGWRWDLO HQGRQ ‘ KHDGWRKHDG HQGRQ W L SDUDOOHO W DQWLSDUDOOHO f r n WHH FURVV W LQGLFDWHV WKH DQJOH LV XQGHILQHG

PAGE 194

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n LFDQWO\ IURP WKH /HQQDUG-RQHV DWRP SOXV GLSROH PRGHO GHVFULEHG DERYH 7KH /HQQDUG-RQHV GLDWRPLF SRWHQWLDO LV JLYHQ E\ >@ XULR:Z, N O If If N N f ZKHUH WKH UA DUH WKH GLVWDQFHV EHWZHHQ DWRP FHQWHUV IRU D SDLU RI GLn DWRPLF PROHFXOHV DV VKRZQ LQ )LJXUH 7KH RULHQWDWLRQ IRU WKH /HQQDUG-RQHV /-f GLDWRPLF SOXV GLSROH SRWHQWLDO KDYLQJ WKH PLQLPXP HQHUJ\ LV WKH DQWLSDUDOOHO IRU \ DQG D PROHFXODU HORQJDWLRQ RI D 7KH GLIIHUHQFHV LQ WKH /GLDWRPLF SOXV GLSROH DQG /DWRP SOXV GLSROH SRWHQWLDOV DULVH IURP Df WKH FRPSHWLWLRQ LQ RULHQWDWLRQV EHWZHHQ WKH DQLVRWURSLF /GLDWRPLF DQG WKH GLSROH SRWHQWLDO ZKHUHDV WKH /DWRP SRWHQWLDO GRHV QRW FRQWULEXWH WR RULHQWDWLRQDO HIIHFWV Ef JHRPHWULF HIIHFWV ZKLFK RFFXU LQ WKH GLDWRPLF SRWHQWLDO GXH WR LWV GHSHQGHQFH RQ PROHFXODU HORQJDWLRQ ZKLFK GR QRW RFFXU LQ WKH /DWRP SRWHQWLDO

PAGE 195

/HQQDUG-RQHV 3OXV 4XDGUXSROH 3RWHQWLDO 7KLV SRWHQWLDO PRGHO LV REWDLQHG E\ FRPELQLQJ (TXDWLRQV f f DQG &f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f LV LQFUHDVHG LV VKRZQ LQ )LJXUH /HQQDUG-RQHV 3OXV 0XOWLSROH 3RWHQWLDO +HUH ZH FRQVLGHU WKH PRUH JHQHUDO PXOWLSROH PRGHO XUfOf! r /8'' 8 '4 X 44 f ZKHUH XAM LV WKH /HQQDUG-RQHV SRWHQWLDO RI f XA LV WKH GLSROHGLSROH LQWHUDFWLRQ RI &f XA LV WKH GLSROHTXDGUXSROH SRWHQWLDO RI &f DQG X LV WKH TXDGUXSROHTXDGUXSROH SRWHQWLDO RI &f )RU WKLV PRGHO WKH QDWXUH RI WKH SRWHQWLDO GHSHQGV RQ WKH UHODWLYH VWUHQJWKV RI WKH GLSROH DQG TXDGUXSROH PRPHQWV 7KXV LI WKH TXDGUXSROH PRPHQW LV VWURQJHU WKDQ r r WKH GLSROH DV VKRZQ LQ )LJXUH ZKHUHLQ 4 \ f WKH WHH RULHQWDWLRQ H[KLELWV WKH PLQLPXP SRWHQWLDO HQHUJ\ FXUYH IROORZHG E\ WKH FURVV DQG DQWLSDUDOOHO RULHQWDWLRQV 2Q WKH RWKHU KDQG LI WKH UHODWLYH VWUHQJWKV DUH UHYHUVHG DV LQ )LJXUH WKHQ WKH PLQLPXP

PAGE 196

)LJXUH 3DLU 3RWHQWLDO IRU /HQQDUG-RQHV SOXV 4XDGUXSROH 0RGHO )OXLG DW 3ULPDU\ 3DLU 2ULHQWDWLRQV .H\ DV LQ )LJXUH f

PAGE 197

)LJXUH 6XUIDFH RI WKH /HQQDUG-RQHV SOXV 4XDGUXSROH 3DLU 3RWHQWLDO IRU WKH 7HH 2ULHQWDWLRQ DV D )XQFWLRQ RI WKH 4XDGUXSROH 6WUHQJWK

PAGE 198

)LJXUH 3DLU 3RWHQWLDO IRU /HQQDUG-RQHV SOXV 'LSROH 'LSROH 4XDGUXSROH DQG 4XDGUXSROH 0RGHO )OXLG DW 3ULPDU\ 3DLU 2ULHQWDWLRQV \HDfO LR 4HFMfO .H\ DV LQ )LJXUH f

PAGE 199

)LJXUH 3DLU 3RWHQWLDO IRU /HQQDUG-RQHV SOXV 'LSROH 'LSROH 4XDGUXSROH DQG 4XDGUXSROH 0RGHO )OXLG DW 3ULPDU\ 3DLU 2ULHQWDWLRQV \HDfO 4HFMfO @B .H\ DV LQ )LJXUH f

PAGE 200

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et DQG $ tt LQ WKH VSKHULFDO KDUPRQLF H[SDQVLRQ RI WKH JHQHUDO DQLVRWURSLF SRWHQWLDO DV JLYHQ LQ $SSHQGL[ & )RU WKLV PRGHO WKH DSSURSULDWH YDOXHV DUH WKRVH RI WKH ILUVW FRQWULEXWLQJ WHUP DOORZHG E\ V\PPHWU\ 7KXV IRU D[LDOO\ V\PPHWULF PROHFXOHV >@ X X f X f RYHU RYHU RYHU f 8VLQJ (TXDWLRQ &f IRU HDFK RI WKH WHUPV RQ WKH ULJKW RI f ZLWK H[SUHVVLRQV IRU WKH H[SDQVLRQ FRHIILFLHQWV (Uf DQG (Uf IURP 7DEOH &O DQG ZLWK WKH KHOS RI WKH VSKHULFDO KDUPRQLF DGGLWLRQ WKHRUHP >@ (TXDWLRQ &f LV REWDLQHG IRU X )RU WKH WRWDO RYHU RYHUODS SRWHQWLDO WR EH SRVLWLYH WKH RYHUODS SDUDPHWHU PXVW OLH LQ WKH UDQJH e e )XUWKHU URGOLNH PROHFXOHV OLQHDUf KDYH ZKHUHDV SODWHOLNH PROHFXOHV KDYH )LJXUH VKRZV WKH /HQQDUG-RQHV SOXV DQLVRWURSLF RYHUODS PRGHO IRU ERWK D SRVLWLYH DQG QHJDWLYH )RU WKH RULHQWDWLRQV DQJOHV A DQG DUH ZLWK

PAGE 201

)LJXUH 3DLU 3RWHQWLDO IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS 0RGHO )OXLGV DW 3ULPDU\ 3DLU 2ULHQWDWLRQV

PAGE 202

UHIHUHQFH WR WKH V\PPHWU\ D[LV ZKLFK LV SHUSHQGLFXODU WR WKH SODQH RI WKH PROHFXOH 7KH SULPDU\ RULHQWDWLRQV RI 7DEOH UHIHU WR WKH V\Pn PHWU\ D[LV DQG QRW WKH PROHFXOHV WKHPVHOYHV ZKHQ 7KXV DQ HQGRQ RULHQWDWLRQ KDV WKH SODWHOLNH PROHFXOHV LQ SDUDOOHO ZKHQ 1RWH IURP (TXDWLRQ &f WKDW WKH RYHUODS SRWHQWLDO LV LQGHSHQGHQW RI WKH DQJOH S VHH )LJXUH f KHQFH WKH SDUDOOHO DQWLSDUDOOHO DQG FURVV RULHQWDWLRQV FDQQRW EH GLVWLQJXLVKHG $OVR WKH PRGHO FDQQRW GLVWLQJXLVK EHWZHHQ WKH WZR HQGRQ RULHQWDWLRQV GXH WR V\PPHWU\ RI WKH PROHFXOH DERXW D SODQH SHUSHQGLFXODU WR WKH PROHFXOH 7KXV WKHUH DUH RQO\ WKUHH SULPDU\ RULHQWDWLRQV IRU WKLV PRGHO )URP )LJXUH WKH PLQLPXP SRWHQWLDO HQHUJ\ FXUYH LV GXH WR WKH SDUDOOHO FURVVf RULHQWDWLRQ IRU 6 DQG LV GXH WR WKH HQGRQ PROHFXOHV SDUDOOHOf IRU (TXLOLEULXP 3URSHUWLHV 7LPH 'HYHORSPHQW RI WKH 6LPXODWLRQ 7KH GHYHORSPHQW RI D PROHFXODU G\QDPLFV FDOFXODWLRQ PD\ EH PRQLWRUHG E\ IROORZLQJ D QXPEHU RI V\VWHP YDULDEOHV WHPSHUDWXUH WRWDO HQHUJ\ WUDQVODWLRQDO DQG URWDWLRQDO NLQHWLF HQHUJ\ DQG PHDQ VTXDUHG PROHFXODU GLVSODFHPHQW 7KH PHDQ VTXDUHG GLVSODFHPHQW RI WKH PROHFXODU FHQWHU RI PDVV LV RI YDOXH VLQFH Df LW JLYHV DQ LQGLFDWLRQ RI ZKHWKHU WKH V\VWHP LV OLTXLG RU VROLG DQG Ef LW PD\ EH XVHG WR HVWLPDWH WKH PDVV GLIIXVLRQ FRHIILFLHQW 7KH VKRUW WLPH EHKDYLRU W fVf RI WKH PHDQ VTXDUHG GLVSODFHPHQW KDV EHHQ IRXQG WR EH WKH VDPH IRU ERWK WKH VROLG DQG OLTXLG VWDWHV LQ WKH FDVH RI WKH /HQQDUG-RQHV SRWHQWLDO PRGHO >@ )RU ORQJ WLPHV KRZHYHU WKH GLVSODFHPHQW EHFRPHV ERXQGHG

PAGE 203

IRU WKH VROLG EXW LQFUHDVHV PRQRWRQLFDOO\ ZLWK WLPH IRU WKH OLTXLG )LJXUH VKRZV D W\SLFDO SORW RI WKH PHDQ VTXDUHG GLVSODFHPHQW RI WKH FHQWHU RI PDVV ^$UAPWf` IRU D /HQQDUG-RQHV SOXV TXDGUXSROH OLTXLG )XUWKHU HYLGHQFH IRU VROLGLILFDWLRQ RI D OLTXLG V\VWHP PD\ EH REWDLQHG E\ GHWHUPLQLQJ HVVHQWLDOO\f WKH VWUXFWXUH IDFWRU >@ RU WKH FHQWHUV SDLU FRUUHODWLRQ IXQFWLRQ 7KH ODWWHU H[KLELWV EHKDYLRU VLPLODU WR WKH UDGLDO GLVWULEXWLRQ IXQFWLRQ IRU VSKHULFDO PROHFXOHV >@ ZKHQ VROLGLILFDn WLRQ RFFXUV 7KH PHDQ VTXDUHG GLVSODFHPHQW RI FHQWHUV RI PDVV LV UHODWHG WR WKH YHORFLW\ DXWRFRUUHODWLRQ IXQFWLRQ E\ W WWnfYWnfnYW f!GWn f DQG WKURXJK WKH *UHHQ.XER UHODWLRQ WR WKH GLIIXVLRQ FRHIILFLHQW >@ ^$U Wf`! 'W & f f§FP ^$U Wf` f§FP ZKHUH & LV D FRQVWDQW ZKLFK DOORZV IRU LQLWLDO FRKHUHQW PRWLRQ RI WKH PROHFXOHV 2WKHU V\VWHP SDUDPHWHUV ZKLFK PD\ EH IROORZHG DUH UHODWHG WR WKH V\VWHP HQHUJLHV 7KH WRWDO NLQHWLF HQHUJ\ PD\ EH PRQLWRUHG WKURXJK WKH NLQHWLF WHPSHUDWXUH ZKLFK IRU WUDQVODWLRQDO DQG URWDWLRQDO PRWLRQ LV JLYHQ E\ 1H 1 Q L N7H P f

PAGE 204

)LJXUH 0HDQ6TXDUHG 'LVSODFHPHQW RI 0ROHFXODU &HQWHUV RI 0DVV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG

PAGE 205

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f IRU PROHFXODU SDLUV VHSDUDWHG E\ GLVWDQFHV JUHDWHU WKDQ WKH SRWHQWLDO FXWRII UA %HUQH DQG +DUS XVHG UA 2 7KH YDOXH IRU WKH GLIIXVLRQ FRHIILFLHQW JLYHQ E\ %HUQH DQG +DUS ZDV REWDLQHG IURP WKH *UHHQ.XER IRUPXOD 22 YW frYWf!GW f

PAGE 206

)LJXUH )OXFWXDWLRQ LQ 7HPSHUDWXUH IRU /HQQDUG-RQHV SOXV 4XDGUXSROH 0RGHO )OXLG 6ROLG OLQH JLYHV LQVWDQn WDQHRXV YDOXHV RI N7H EURNHQ OLQH LV DYHUDJH YDOXH RI N7H SDA 4r

PAGE 207

)LJXUH )OXFWXDWLRQ LQ WKH 5DWLR RI 7UDQVODWLRQDO WR 5RWDWLRQDO .LQHWLF (QHUJ\ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG

PAGE 208

7$%/( 3URSHUW\ 9DOXHV RI D /HQQDUG-RQHV 3OXV 4XDGUXSROH )OXLG 2EWDLQHG LQ WKLV :RUN DQG &RPSDUHG ZLWK WKRVH JLYHQ E\ %HUQH DQG +DUS >@ 3URSHUW\ 7KLV :RUN %HUQH DQG +DUS 1R 3DUWLFOHV 1 /3DUDPHWHUV HN . D $ ƒ (ORQJDWLRQ OR 4 0RPHQW 4HD9 'HQVLW\ SD 7LPH 6WHS $W BfV BfV /HQJWK RI 5XQ 0 $W $W 7HPSHUDWXUH N7H s s 3RWHQWLDO (QHUJ\ WRWDO 81H s /$MQ" 0HDQ 6TXDUHG )RUFH ) A!D] s s 0HDQ 6TXDUHG 7RUTXH [_!H s s 'LII &RHIILFLHQW 'PHfD

PAGE 209

:KHUHDV LQ WKLV ZRUN ZDV HVWLPDWHG IURP WKH PHDQ VTXDUHG GLVSODFHn PHQW RI WKH PROHFXODU FHQWHUV RI PDVV E\ f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f RI WKH FDOFXODWLRQ KRZHYHU WKH PHDQ VTXDUHG GLVSODFHPHQW RFFDVLRQDOO\ UHPDLQHG XQFKDQJHG IRU SHULRGV RI WR WLPH VWHSV HYLGHQFLQJ WKH SUR[LPLW\ RI WKH SKDVH ERXQGDU\ ,Q 7DEOH DQG IRU UHVXOWV UHSRUWHG EHORZ WKH VWDQGDUG GHYLDWLRQ RI VRPH SURSHUW\ ) IURP LWV PHDQ YDOXH )! ZDV GHWHUPLQHG E\ D ) 0O O ) 0)!f L O f ZKLFK FDQ EH VKRZQ WR UHGXFH WR D) 0O 0 O )7 L O 0)! f ZKHUH 0 I!  ,I 0 L L r f

PAGE 210

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nrfAA $ PROHFXODU HORQJDWLRQ FRUUHVSRQGLQJ WR WKH QLWURJHQ PROHFXOH OR >@ DQG D WLPH VWHS RI DERXW Af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f DSSUR[LPDWHG E\ XQLW\ IRU UA U 7KXV WKH FRUUHFWLRQV ZHUH JLYHQ E\ 22 X r r r r X/MU f U GU f ; U F

PAGE 211

3 FRU SN7 22 -/ G8/-U ` r r GU U F r U r GU f ) < 7 r 3 &2 r G8/-U ` r r GU U F r U r GU f FRU H r @ +RZn HYHU WKH TXDGUXSRODU FRQWULEXWLRQ ZDV IRXQG WR EH QHJOLJLEOH FRPSDUHG WR WKH /HQQDUG-RQHV WHUPV VR QR TXDGUXSRODU ORQJ UDQJH FRUUHFWLRQ ZDV LQFOXGHG LQ WKH PROHFXODU G\QDPLFV SURSHUW\ YDOXHV $ ORQJ UDQJH FRUUHFWLRQ ZRXOG EH VLJQLILFDQW LQ WKH FDVH RI GLSROHVf nN 7KH IXOO HTXLOLEULXP SURSHUW\ YDOXHV REWDLQHG IRU 4 DQG DUH JLYHQ LQ 7DEOHV DQG UHVSHFWLYHO\ 7KH SURSHUWLHV ZHUH HYDOXDWHG IURP WKH HQVHPEOH DYHUDJHV LQ 7DEOH DQG VWDQGDUG GHYLDWLRQV ZHUH GHWHUPLQHG RYHU WKH HQWLUH WLPH VWHS FDOn FXODWLRQ XVLQJ (TXDWLRQ f $V LQGLFDWHG LQ WKH ILJXUHV WKH PHDQ VTXDUHG IRUFH DQG WRUTXH KDYH HDFK EHHQ HYDOXDWHG E\ WZR GLIIHUHQW PHWKRGV 7KH YDOXHV ODEHOHG ZLWK WKH VXEVFULSW / ) DQG f DUH ; -B; /L WKH OLWHUDO YDOXHV REWDLQHG E\ GLUHFW HYDOXDWLRQ RI (TXDWLRQV f DQG f IRU WKH IRUFH DQG WRUTXH UHVSHFWLYHO\ 7KH VHFRQG YDOXHV RI WKH IRUFH DQG WRUTXH ZHUH FDOFXODWHG IURP WKH H[SUHVVLRQV >@

PAGE 212

7$%/( (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW SJ 4HDf B 3URSHUW\ 0ROHFXODU '\QDPLFV 3HUWXUEDWLRQ 7KHRU\ N7H s LO1H s 3SN7 s ) < D H s ) XVD e s Ff1N s s )!DH s f IL9H s [!H s W!OH s 7 U 6\PEROV GHILQHG LQ 7DEOHV DQG

PAGE 213

7$%/( (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH V )OXLG DW SD 4HR f 3URSHUW\ 0ROHFXODU '\QDPLFV 3HUWXUEDWLRQ 7KHRU\ N7H 81H 3SN7 <)FH 8)DH 6 &51N 9 s s )!DH s )O!/De s [!H f A

PAGE 214

7$%/( (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH A )OXLG DW SJ 4 HD f 3URSHUW\ 0ROHFXODU '\QDPLFV 3HUWXUEDWLRQ 7KHRU\ N7H s 81H s 3SN7 s ) < R F s f) 8 c= V s &\1N s s )!DH s IOr F s n r f W!H s 7"!H s ; -/L

PAGE 215

!D H N7Hf n f D MrO NMf L!] N7Hf O O -/-/f X $f! M $ f ([SUHVVLRQV IRU X $f LQ f DUH JLYHQ LQ $SSHQGL[ & FO $OVR VKRZQ LQ 7DEOHV DQG DUH YDOXHV RI WKH SURSn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r r FDOFXODWHG DQG ILWWHG WR DQ HPSLULFDO HTXDWLRQ LQ S DQG 7 5HVXOWV DUH JLYHQ LQ $SSHQGL[ *f 7DEOHV DQG VKRZ JRRG DJUHHn PHQW EHWZHHQ WKH PROHFXODU G\QDPLFV UHVXOWV DQG SHUWXUEDWLRQ WKHRU\ SUHGLFWLRQV IRU WKRVH SURSHUWLHV ZKLFK DUH GLUHFWO\ UHODWHG WR WKH LQWHUPROHFXODU SRWHQWLDO LQWHUQDO HQHUJ\ VXUIDFH H[FHVV LQWHUQDO HQHUJ\ DQG PHDQ VTXDUHG WRUTXH 6LJQLILFDQW GHYLDWLRQV LQ WKH UHVXOWV IURP WKH WZR PHWKRGV DUH HYLGHQW KRZHYHU IRU WKRVH SURSHUWLHV ZKLFK DUH UHODWHG WR GHULYDWLYHV RI WKH SRWHQWLDO SUHVVXUH VXUIDFH WHQVLRQ DQG KHDW FDSDFLW\ HVSHFLDOO\ DW WKH KLJKHU TXDGUXSROH PRPHQWV

PAGE 216

7KH GLVSDUDWH YDOXHV IRU WKH KHDW FDSDFLW\ &A IRXQG E\ VLPXODn WLRQ DQG SHUWXUEDWLRQ WKHRU\ DUH QRW VXUSULVLQJ VLQFH LW LV ZHOO NQRZQ WKDW WKH UHOD[DWLRQ RI IOXFWXDWLRQV LQ WKH KHDW FDSDFLW\ UHTXLUH VLJQLILFDQWO\ ORQJHU WLPHV WKDQ FDQ JHQHUDOO\ EH DWWDLQHG LQ WKH VLPXODWLRQ RI HYHQ D VLPSOH /HQQDUG-RQHV V\VWHP >@ $V D FRQn VLVWHQF\ FKHFN RQ WKH VLPXODWLRQ YDOXH IRU &A /HERZLW] HW DO >@ KDYH VKRZQ WKDW WKH KHDW FDSDFLW\ PD\ EH REWDLQHG IURP WKH IOXFWXDWLRQ LQ WKH NLQHWLF HQHUJ\ LQ PROHFXODU G\QDPLFV VLQFH WKH WRWDO HQHUJ\ LV FRQVHUYHG LQ WKH PROHFXODU G\QDPLFV PLFURFDQRQLFDO HQVHPEOH +HQFH WKH KHDW FDSDFLW\ FDQ EH REWDLQHG IURP WKH IOXFWXDWLRQ LQ WKH WHPSHUn DWXUH GXH WR f &RPELQLQJ f f DQG f JLYHV &Y B 1 F f 1N Wr O 7! f ZKHUH LV WKH VWDQGDUG GHYLDWLRQ LQ WKH WHPSHUDWXUH 7KH YDOXHV IRU D7 DQG JLYHQ LQ 7DEOHV DQG VDWLVI\ f ZKHQ DOORZDQFH LV PDGH IRU WKH URXQGRII RI WKH YDOXHV JLYHQ LQ WKH WDEOHV 7KH UHODWLYHO\ ODUJH VWDQGDUG GHYLDWLRQV DQG GLVDJUHHPHQW EHWZHHQ WKH PROHFXODU G\QDPLFV DQG SHUWXUEDWLRQ WKHRU\ UHVXOWV IRU WKH SUHVVXUH DQG VXUIDFH WHQVLRQ DUH OHVV FOHDUO\ UHVROYHG %RWK WKHVH SURSHUWLHV DULVH IURP FDQFHOODWLRQ EHWZHHQ D QHJDWLYH FRQWULEXWLRQ IRU PROHFXODU SDLUV VHSDUDWHG E\ VPDOO U GLVWDQFHV DQG D SRVLWLYH FRQWULEXWLRQ RI WKH VDPH RUGHU RI PDJQLWXGH DW ODUJHU U GLVWDQFHV VHH 6HFWLRQ EHORZf +HQFH RQH PLJKW H[SHFW VRPH GLIILFXOWLHV LQ REWDLQLQJ DFFXUDWH FRPSXWHU UHVXOWV IRU WKHVH SURSHUWLHV 0F'RQDOG HWB DO LQ IDFW KDYH IRXQG LW

PAGE 217

QHFHVVDU\ WR SHUIRUP PLOOLRQ 0RQWH &DUOR FRQILJXUDWLRQV RQ D V\VWHP RI SDUWLFOHV LQWHUDFWLQJ ZLWK WKH /HQQDUG-RQHV SOXV GLSROH SRWHQWLDO LQ RUGHU WR REWDLQ UHOLDEOH YDOXHV IRU WKH )RZOHU PRGHO VXUIDFH WHQVLRQ >@ 2Q WKH RWKHU KDQG PLOOLRQ 0RQWH &DUOR FRQILJXUDWLRQV ZHUH VXIILFLHQW WR GHWHUPLQH WKH SUHVVXUH IRU WKH VDPH V\VWHP >@ ,Q VSLWH RI WKH ODUJH IOXFWXDWLRQV WKH YDOXHV IRU WKH VXUIDFH WHQVLRQ DQG SUHVVXUH LQ 7DEOHV DQG DUH FRQVLVWHQW ZLWK YDOXHV REWDLQHG E\ LQWHJUDWLQJ RYHU WKH DSSURn SULDWH VSKHULFDO KDUPRQLF FRHIILFLHQWV JA A PAULA VHH 6HFWLRQ EHORZf 7KH ODUJH XQFHUWDLQWLHV DVVRFLDWHG ZLWK WKH DQJXODU FRUUHODWLRQ IXQFWLRQV DQG FRQILUP WKH GLIILFXOWLHV H[SHULHQFHG E\ RWKHUV LQ WKH VWXG\ RI WKHVH SURSHUWLHV >@ &KHXQJ >@ KDV LQGLFDWHG WKDW IRU WKH /HQQDUG-RQHV GLDWRPLF SRWHQWLDO IOXLG FRQWULEXWLRQV WR VHHP WR RFFXU IRU PROHFXODU SDLUV ZLWK VHSDUDWLRQV LQ WKH UDQJH D e U 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s IRU FDUERQ PRQR[LGH DQG s IRU QLWURJHQ 7KH QHJDWLYH YDOXH

PAGE 218

IRU &2 LV DWWULEXWHG WR WKH HIIHFW RI WKH TXDGUXSROH PRPHQW 7KH SUHVHQW PROHFXODU G\QDPLFV DQG SHUWXUEDWLRQ UHVXOWV WHQG WR FRQILUP WKDW WKH TXDGUXSROH SRWHQWLDO OHDGV WR QHJDWLYH YDOXHV IRU *KRZHYHU WKH ODUJH XQFHUWDLQWLHV LQ ERWK WKH VLPXODWLRQ DQG H[SHULPHQWDO UHVXOWV SUHYHQW FRQILGHQFH DERXW HYHQ WKH VLJQ RI ,Q 7DEOHV DQG WKH SXUHO\ DQLVRWURSLF FRQWULEXWLRQV WR WKH HTXLOLEULXP SURSHUWLHV DUH JLYHQ 7KHVH DQLVRWURSLF FRQWULEXn WLRQV ZHUH REWDLQHG E\ VXEWUDFWLQJ WKH LVRWURSLF /HQQDUG-RQHV SDUW RI HDFK SURSHUW\ IURP WKH WRWDO SURSHUW\ YDOXH JLYHQ LQ 7DEOHV DQG 7KH LVRWURSLF FRQWULEXWLRQV WR WKH LQWHUQDO HQHUJ\ DQG SUHVVXUH ZHUH REWDLQHG IURP WKH 0RQWH &DUOR VWXG\ RI /HQQDUG-RQHV IOXLGV E\ 0F'RQDOG DQG 6LQJHU >@ 7KH LVRWURSLF FRQWULEXWLRQ WR WKH RWKHU SURSHUWLHV ZHUH REWDLQHG E\ HYDOXDWLQJ WKH DSSURSULDWH LQWHJUDOV DQDORJRXV WR (TXDWLRQV f WR ff LQ WKH UDGLDO GLVWULEXWLRQ IXQFWLRQ WKHRU\ IRU VLPSOH IOXLGV XVLQJ WKH PROHFXODU G\QDPLFV UHVXOWV RI 9HUOHW IRU JLUAf >@ 5HVXOWV IRU /HQQDUG-RQHV SOXV 2YHUODS )OXLG $ V\VWHP RI OLQHDU PROHFXOHV LQWHUDFWLQJ ZLWK WKH /HQQDUG -RQHV SOXV DQLVRWURSLF RYHUODS PRGHO RI 6HFWLRQ KDV EHHQ VWXGLHG IRU WZR YDOXHV RI WKH RYHUODS SDUDPHWHU DQG 7KH PROHn FXOH DQG V\VWHP SDUDPHWHUV XVHG ZHUH WKH VDPH DV WKRVH IRU WKH TXDGUX SRODU IOXLG VDYH WKDW WKH SRWHQWLDO ZDV VHW WR ]HUR IRU PROHFXODU SDLUV VHSDUDWHG E\ GLVWDQFHV JUHDWHU WKDQ UA R /RQJ UDQJH FRUUHFWLRQV IRU GLVWDQFHV JUHDWHU WKDQ UA ZHUH PDGH IRU WKH /HQQDUG-RQHV FRQWULEXWLRQ XVLQJ (TXDWLRQV f WR f 1R FRUUHFWLRQ ZDV LQFOXGHG IRU WKH ORQJ UDQJH DQLVRWURSLF RYHUODS FRQWULEXWLRQ WR SURSHUW\ YDOXHV

PAGE 219

7$%/( $QLVRWURSLF &RQWULEXWLRQV WR (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW SR 4HDf 3URSHUW\ 0ROHFXODU '\QDPLFV 3HUWXUEDWLRQ 7KHRU\ N7H 8 1H D 3 SN7 D )
PAGE 220

7$%/( $QLVRWURSLF &RQWULEXWLRQV WR (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW SD 4HRf 3URSHUW\ 0ROHFXODU '\QDPLFV 3HUWXUEDWLRQ 7KHRU\ N7H 8 1H D 3 SN7 D ) 9 ( D]H VD &5 1N YD

PAGE 221

7$%/( $QLVRWURSLF &RQWULEXWLRQV WR (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW SR 4HJf 3URSHUW\ 0ROHFXODU '\QDPLFV 3HUWXUEDWLRQ 7KHRU\ N7H 8 1H D 3 SN7 D ) 9 f 8) DH VD &5 1N YD

PAGE 222

)XOO HTXLOLEULXP SURSHUW\ YDOXHV DQG WKH FRUUHVSRQGLQJ DQLVRWURSLF FRQWULEXWLRQV DUH JLYHQ LQ 7DEOHV DQG $ SHUWXUEDWLRQ WKHRU\ LQFOXGLQJ WKUHH ERG\ WHUPV DSSOLFDEOH WR WKLV PRGHO SRWHQWLDO KDV RQO\ UHFHQWO\ EHHQ ZRUNHG RXW DQG WKHUHIRUH SHUWXUEDWLRQ WKHRU\ UHVXOWV DUH QRW \HW DYDLODEOH IRU FRPSDULVRQ 6SKHULFDO +DUPRQLF &RHIILFLHQWV = L e QUAnO a 7KH FRHIILFLHQWV Jf f U ff LQ WKH VSKHULFDO KDUPRQLF H[SDQVLRQ D  IRU WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ J&UAXQXA KDYH EHHQ GHWHUn PLQHG IURP WKH HQVHPEOH DYHUDJH RI f )RU WKH TXDGUXSROH DQG RYHUODS IOXLGV RQO\ WKH FRHIILFLHQWV ZLWK HYHQ YDOXHV IRU DQG A DUH QRQ]HUR GXH WR V\PPHWU\ RI WKH PROHFXOH $OO VXFK FRHIILFLHQWV XS WR Af} DQG WKH ie AUOA DQG JAUA FRHIILFLHQWV KDYH EHHQ HYDOXDWHG ([SOLFLW H[SUHVVLRQV IRU HDFK RI WKHVH DUH JLYHQ LQ $SSHQGL[ ) 6SKHULFDO +DUPRQLF &RHIILFLHQWV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG &HQWHUFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ )LJXUH VKRZV WKDW IRU WKH VWUHQJWKV RI WKH TXDGUXSROH PRPHQW FRQVLGHUHG LQ WKLV ZRUN WKH DGGLWLRQ RI WKH TXDGUXSROH SRWHQWLDO KDV OLWWOH HIIHFW RQ WKH FHQWHUFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ JAUAf} :LWKLQ WKH VWDWLVWLFDO SUHFLVLRQ RI WKH VLPXODWLRQ WKH FXUYHV LQ )LJXUH IRU WKH /HQQDUG-RQHV DQG /HQQDUG-RQHV SOXV TXDGUXSROH IOXLGV DUH WKH VDPH 7KH VOLJKW GLIIHUHQFH LQ WKH ILUVW SHDN KHLJKWV RI WKH WZR FXUYHV FRXOG EH GXH WR WKH WHPSHUDWXUH GLIIHUHQFH EHWZHHQ WKH

PAGE 223

7$%/( (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 2YHUODS )OXLG DW SJ 0ROHFXODU '\QDPLFV 3URSHUW\ )XOO 3URSHUW\ $QLVRWURSLF N7H s -1e s 3SN7 s \)DH s X)DH 6 s &p1N s 22 s )O!/rH s f [!H s A [ H On s

PAGE 224

7$%/( (TXLOLEULXP 3URSHUWLHV IRU /HQQDUG-RQHV SOXV 2YHUODS )OXLG DW SJ 0ROHFXODU '\QDPLFV 3URSHUW\ )XOO 3URSHUW\ $QLVRWURSLF N7H 81H 3SN7 \)DH 8)DH 6 F!N *L & I!OrH s W!H WL!Oe

PAGE 225

)LJXUH (IIHFW RI 4XDGUXSROH 0RPHQW RQ WKH &HQWHU&HQWHU 3DLU &RUUHODWLRQ )XQFWLRQ DW SF"

PAGE 226

WZR FDOFXODWLRQV 7KH SORW RI J UAf ARU WrLH XDGUXSROH IOXLG ZLWK 4HD f IDOOV HVVHQWLDOO\ RQ WKH /HQQDUG-RQHV FXUYH LQ )LJXUH ZLWK LWV ILUVW SHDN IDOOLQJ EHWZHHQ WKH WZR SHDNV VKRZQ LQ )LJXUH VLQFH WKH DYHUDJH WHPSHUDWXUH RI WKH 4HD f VLPXODWLRQ ZDV N7H EHWZHHQ DQG f 3DWH\ DQG 9DOOHDX > @ KDYH FRQGXFWHG 0RQWH &DUOR VWXGLHV RI KDUG VSKHUH SOXV TXDGUXSROH PRGHO IOXLGV 7KH\ ILQG GLVWRUWLRQ RI WKH FHQWHUWRFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ RFFXUULQJ DW KLJK YDOXHV RI WKH TXDGUXSROH PRPHQW LH DW 4N75 f ZKHUH 5 LV WKH KDUG VSKHUH GLDPHWHUf 7KLV GLVWRUWHG J UAf VHHPV WR EH HYLGHQFH IRU DQ )&&W\SH ORFDO VWUXFWXUH IRUPLQJ LQ WKH IOXLG ,Q WKH SUHVHQW ZRUN WKH KLJKHVW TXDGUXSROH PRPHQW VWXGLHG ZDV 4 HD f DQG DV VKRZQ LQ )LJXUH QR GLVWRUWLRQ RI JFUAf KDV EHHQ IRXQG 2WKHU VSKHULFDO KDUPRQLF FRHIILFLHQWV 5HVXOWV IRU WKH UHPDLQLQJ VSKHULFDO KDUPRQLF FRHIILFLHQWV DUH VKRZQ LQ )LJXUHV DQG IRU WKH TXDGUXSRODU IOXLG KDYLQJ 4HR f 7KH FRHIILFLHQWV IRU WKH IOXLGV ZLWK 4 HD f DQG DUH QRW VKRZQ IRU EUHYLW\ KRZHYHU WKHLU UHVXOWV DUH TXDOLWDWLYHO\ VLPLODU WR WKH FRHIILFLHQWV VKRZQ LQ WKH IROORZLQJ ILJXUHV ,Q JHQHUDO WKH PDJQLWXGHV RI WKH Jf FRHIILFLHQWV DUH IRXQG WR GHFUHDVH DV Df RU t LV LQFUHDVHG Ef P LV LQFUHDVHG DW IL[HG DQG Ff WKH IOXLG GHQVLW\ LV GHFUHDVHG RU Gf WKH VWUHQJWK RI WKH DQLVRWURSLF SRWHQWLDO LV GHFUHDVHG 7KHVH FRQFOXVLRQV DUH GHPRQVWUDWHG E\ WKH

PAGE 227

)LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV Jff IRU /HQQDUG -RQHV SOXV 4XDGUXSROH )OXLG DW SFI N7H 4HRfO LR

PAGE 228

)LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV Jf ;WQ RI )LJXUH IRU WKH )OXLG

PAGE 229

UFU )LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV RI )LJXUH IRU WKH )OXLG

PAGE 230

U FU )LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV J B IRU WKH )OXLG RI )LJXUH

PAGE 231

PDJQLWXGHV RI WKH ILUVW SHDN RI WKH A PAUA FRHIILFrHQWV JLYHQ LQ 7DEOH 7KH H[FHSWLRQ WR WKHVH JHQHUDO REVHUYDWLRQV LV WKH TXDGUX SRODU IOXLG ZKHUHLQ WKH PDJQLWXGH RI WKH t FRHIILRLHQW GRPLQDWHV WKH RWKHU FRHIILFLHQWV LQFOXGLQJ WKH if )URP 7DEOH WKH PDJQLWXGHV RI WKH ILUVW SHDN LQ WKH Jf IRU WKH TXDGUXSRODU IOXLGV 6," ? A 6," DW SD 4HD f DQG SD 4HR f n DUH URXJKO\ WKH VDPH LQGLFDWLQJ WKDW WKH HIIHFW RQ WKH Jf Q RI GH ]P FUHDVLQJ WKH TXDGUXSROH PRPHQW LV ODUJHO\ FDQFHOOHG RXW E\ WKH LQFUHDVH LQ IOXLG GHQVLW\ $QRWKHU HIIHFW RQ WKH JA A PAUA LQFUHDV!QJ WKH VWUHQJWK RI WKH DQLVRWURSLF SRWHQWLDO LV WR GHFUHDVH WKH GHJUHH RI UDQGRPO\ VFDWWHUHG YDOXHV RI WKH FRHIILFLHQWV EH\RQG WKH ILUVW SHDN 7KLV VFDWWHU LQ WKH JA ePAUA ADWD HVSHFLDOO\ DW UA YDOXHV EH\RQG WKH ILUVW SHDN RFFXUV LQ WKH FDVH RI ZHDNO\ TXDGUXSROH IOXLGV EHFDXVH RI WKH VKRUW UDQJH FRUUHODWLRQ RI RULHQWDWLRQ RI PROHFXODU SDLUV ,Q WKH FDVH RI VWURQJ TXDGUXSROH IOXLGV WKH VFDWWHU LV ODUJHO\ GXH WR VWURQJ FRUUHODWLRQ RI SDUWLFXODU RULHQWDWLRQV RI PROHFXODU SDLUV VR WKDW OHVV SUREDEOH SDLU RULHQWDWLRQV ZKLFK ZRXOG FRQWULEXWH WR FHUWDLQ JA A PAUA WHUPV RQO\ UDUHO\ RFFXU 7KH VFDWWHU KDV EHHQ PDGH V\VWHPDWLF E\ VPRRWKLQJ WKH JA A PAULAf XVDQi D VHYHQSRLQW WKLUG GHJUHH VPRRWKLQJ IRUPXOD >@ DSSOLHG WKUHH WLPHV WR WKH JA e QUOA YDAXHV EH\RQG WKH ILUVW SHDN 7KH FXUYHV LQ WKH DFFRPSDQ\LQJ ILJXUHV DUH WKH VPRRWKHG Jf U ff 7KH KLJK IUH ;\ TXHQF\ RVFLOODWLRQV VKRZQ LQ D IHZ RI WKH JA A PAUA UHIOHFW WKH VPRRWKLQJ RI QHDUO\ UDQGRPO\ VFDWWHUHG GDWD DQG VKRXOG QRW EH

PAGE 232

7$%/( (IIHFW RI 3RWHQWLDO 0RGHO DQG 6WDWH &RQGLWLRQ RQ WKH )LUVW 3HDN +HLJKW RI WKH Jf f &RHIILFLHQWV 4XDGUXSROHV 2YHUODS efP Df Ef Ff J 6 J 6 6 6 6 6 J J J J J J Df SR N7H 4HDf Ef SD N7H 4HD f Ff SD N7H

PAGE 233

LQWHUSUHWHG DV VLJQLILFDQW ORQJ UDQJH FKDUDFWHU IRU WKH FRHIILFLHQW 7KH RULJLQDO XQVPRRWKHG JA A PAULA GDWD IrU WAH V\VWHPV VWXGLHG DUH WDEXODWHG LQ $SSHQGL[ + 6SKHULFDO +DUPRQLF &RHIILFLHQWV IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG &HQWHUFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ )LJXUH VKRZV WKH FHQWHUFHQWHU SDLU FRUUHODWLRQ IXQFWLRQ J UAf REWDLQHG IURP VLPXODWLRQ RI WKH DQLVRWURSLF RYHUODS IOXLGV )RU WKH YDOXH RI WKH RYHUODS SDUDPHWHU WKH JA&UAf FXUYH IDOOV HVVHQWLDOO\ RQ WKH VLPSOH /HQQDUG-RQHV JUAf SORW RI )LJXUH H[FHSW WKDW WKH KHLJKW RI WKH ILUVW SHDN LV GHFUHDVHG VRPHZKDW 7KLV GLIIHUHQFH LQ SHDN KHLJKWV FRXOG EH GXH WR WKH VOLJKW GLIIHUHQFH LQ WHPSHUDWXUH EHWZHHQ WKH WZR FDOFXODWLRQV +RZHYHU ZKHQ WKHUH LV D GHILQLWH ORZHULQJ RI WKH ILUVW SHDN KHLJKW ZKLFK LV FRQVLVWHQW ZLWK LQWHUSUHWDWLRQ RI WKH UHSXOVLYH RYHUODS SRWHQWLDO DV WHQGLQJ WR UHVWULFW FORVH DSSURDFK RI PROHFXODU SDLUV 2WKHU VSKHULFDO KDUPRQLF FRHIILFLHQWV )LJXUHV DQG VKRZ WKH VPRRWKHG VSKHULFDO KDUPRQLF FRHIILFLHQWV IRU WKH DQLVRWURSLF RYHUODS IOXLG KDYLQJ 7KH J4 4 FRHIILFLHQWV IRU WKH IOXLG ZLWK DUH TXDOLWDWLYHO\ VLPLODU DQG WKH JHQHUDO VWDWHPHQWV FRQFHUQLQJ WKH IDFWRUV DIIHFWLQJ WKH PDJQLWXGHV RI WKH FRHIILFLHQWV PDGH LQ 6HFWLRQ DSSO\ WR WKH RYHUODS IOXLG DV VKRZQ LQ 7DEOH $OVR VHHQ LQ 7DEOH LV WKDW WKH TXDGUXSROH DQG DQLVRWURSLF RYHUODS IOXLGV VKRZ FRQVLGHUDEOH GLIIHUHQFHV LQ WKH QDWXUH RI WKHLU

PAGE 234

)LJXUH (IIHFW RI $QLVRWURSLF 2YHUODS 3DUDPHWHU RQ WKH &HQWHU&HQWHU 3DLU &RUUHODWLRQ )XQFWLRQ DW SR

PAGE 235

N7H .H\ DV LQ )LJXUH f

PAGE 236

LQ PDJQLWXGH WKDQ JAO IrU D+ U DUrG KDV EHHQ RPLWWHG IRU FODULW\f

PAGE 237

P )LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV J IRU WKH )OXLG RI )LJXUH .H\ DV LQ )LJXUH JA DQG JA DUH VPDOOHU LQ PDJQLWXGH WKDQ JA IrU U DQFr KDYH EHHQ RPLWWHG IRU FODULW\f

PAGE 238

)LJXUH 6SKHULFDO +DUPRQLF &RHIILFLHQWV J Q IRU WKH R;nn)OXLG RI )LJXUH .H\ DV LQ )LJXUH f

PAGE 239

V9P FRHIILFLHQWV 2I WKH FRHIILFLHQWV HYDOXDWHG LQ WKLV VWXG\ H[FOXGLQJ iAUOAf DARXW WZRWKLUGV RI WKHP H[KLELW VLJQ FKDQJHV LQ WKH ILUVW SHDN UHJLRQ ZKHQ FRPSDULQJ TXDGUXSROH ZLWK RYHUODS IOXLGV 7KH FRHIILFLHQW JRPU ZDV IRXQG WR EH QHJOLJLEOH IRU ERWK TXDGUXSROH DQG RYHUODS IOXLGVf +HQFH RQH ZRXOG H[SHFW VLJQLILFDQW GLIIHUHQFHV LQ ORFDO RULHQWDWLRQDO VWUXFWXUH EHWZHHQ TXDGUXSROH DQG RYHUODS IOXLGV 6XFK GLIIHUHQFHV LQ VWUXFWXUH KDYH EHHQ IRUHVKDGRZHG E\ VWXG\ RI WKH SDLU SRWHQWLDOV WKHPVHOYHV LQ 6HFWLRQ 'HWDLOHG VWXG\ RI WKH 6Q Q Wff FRHIILFLHQWV ZRXOG LQGLFDWH WKH FKDUDFWHU RI WKH RULHQ a A a A ,= WDWLRQDO VWUXFWXUH KRZHYHU LW LV PXFK PRUH VWUDLJKWIRUZDUG WR UHFRPELQH WKH FRHIILFLHQWV LQWR WKH VSKHULFDO KDUPRQLF H[SDQVLRQ DQG VWXG\ VWUXFWXUH YLD WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JUAANA r 7KLV DSSURDFK LV WDNHQ LQ 6HFWLRQ (TXLOLEULXP 3URSHUWLHV IURP WKH JA AUA 7KH JII S U ff FRHIILFLHQWV SURYLGH GHWDLOHG LQIRUPDWLRQ RQ A = ORFDO VWUXFWXUH WKURXJK J UAAAA f A SRWHQWLDOO\ PRUH YDOXH >@ LV WKH GHWHUPLQDWLRQ RI HTXLOLEULXP SURSHUWLHV IURP WKHVH FRHIILFLHQWV 7KH GHYHORSPHQW RI UHODWLRQV EHWZHHQ HTXLOLEULXP SURSHUWLHV DQG WKH LQWHJUDOV RYHU WKH JQ Q Uff KDV EHHQ GHVFULEHG LQ 6HFWLRQ Q -R P  7KH LQWHJUDOV f IRU WKH TXDGUXSROH DQG DQLVRWURSLF RYHUODS IOXLGV VWXGLHG KHUH KDYH EHHQ HYDOXDWHG WKURXJK Q DQG DUH WDEXODWHG LQ $SSHQGL[ 7KHVH LQWHJUDOV KDYH EHHQ XVHG ZLWK WKH UHODWLRQV LQ 6HFWLRQ WR FDOFXODWH WKRVH HTXLOLEULXP SURSHUWLHV ZKLFK KDYH DOVR EHHQ GHWHUPLQHG E\ GLUHFW HQVHPEOH DYHUDJLQJ LQ WKH PROHFXODU G\QDPLFV VLPXODWLRQ 7KXV WKH UHODWLRQV LQ 6HFWLRQ SURYLGH D FRQVLVWHQF\

PAGE 240

WHVW EHWZHHQ WKH HTXLOLEULXP SURSHUWLHV DQG WKH JQ f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f f Uf A P FRHIILFLHQWV 0RUH LPSRUWDQWO\ WKHVH UHVXOWV GHPRQVWUDWH WKH DELOLW\ RI WKH JA A PAUA FRHIILFLHQWV WR JLYH YDOXHV IRU HTXLOLEULXP SURSHUWLHV 7KH SURSHUW\ VKRZLQJ WKH ZRUVW DJUHHPHQW LQ 7DEOHV WR LV WKH DQJXODU FRUUHODWLRQ IXQFWLRQ 6RPH RI WKH GLIILFXOWLHV LQ HYDOXDWLQJ WKLV SURSHUW\ KDYH EHHQ VXJJHVWHG LQ 6HFWLRQ $UPHG QRZ ZLWK YDOXHV IRU WKH JA A PAUOA FRHIILFLHQWV IXUWKHU LQVLJKW LQWR PD\ EH DWWHPSWHG ,Q )LJXUH WKH LQWHJUDQG RI (TXDWLRQ f ZKLFK JLYHV IURP WKH DSSURSULDWH JA A AUAf FRHIILFLHQWV LV SORWWHG DV D IXQFWLRQ RI WKH PROHFXODU SDLU VHSDUDWLRQ UA! IrU D /HQQDUG-RQHV SOXV TXDGUXSROH IOXLG 7KLV ILJXUH LQGLFDWHV WZR SUREOHPV LQ GHWHUPLQLQJ Df WKH YDOXH RI LV WKH UHVXOW RI FDQFHOODWLRQ EHWZHHQ SRVLWLYH DQG QHJDWLYH FRQWULEXWLRQV RI QHDUO\ HTXDO PDJQLWXGH Ef WKH IXQFWLRQ

PAGE 241

7$%/( &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEULXP 3URSHUWLHV ZLWK 9DOXHV 2EWDLQHG AAP IURP ,QWHJUDOV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK S N7H 4eTf 3URSHUW\ 0ROHFXODU '\QDPLFV  e P Q XOM1H XTT1e 3/-SN7 9}N7 ) \OM H <44rH VOH XVTTre [!H )!DH

PAGE 242

7$%/( &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEULXP 3URSHUWLHV ZLWK 9DOXHV 2EWDLQHG e e IURP P ,QWHJUDOV IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK SD N7H 4HDf 3URSHUW\ 0ROHFXODU '\QDPLFV e e P Q 8/-1( 91e 3/MSN7 344SN7 ) $Mr H ) \TTr e 83 DH 6OM 83 DH 644 L!H )!DH

PAGE 243

7$%/( &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEULXP 3URSHUWLHV ZLWK 9DOXHV 2EWDLQHG eA P IURP ,QWHJUDOV IRU /HQQDUG-RQHV SOXV Q r a r 2YHUODS )OXLG ZLWK SD N7H 6 efP 3URSHUW\ 0ROHFXODU '\QDPLFV Q XOM1H 8 1H RYHU 3/-WN, 3 SN7 RYHU ) \OMr H \) RH RYHU f) 8f ] 6/f) 8 H fRYHU W!H

PAGE 244

7$%/( &RPSDULVRQ RI 0ROHFXODU '\QDPLFV 5HVXOWV IRU (TXLOLEULXP 3URSHUWLHV ZLWK 9DOXHV 2EWDLQHG IURP ,QWHJUDOV IRU /HQQDUG-RQHV SOXV 2YHUODS )OXLG ZLWK STar N7H teP 3URSHUW\ 0ROHFXODU '\QDPLFV Q XOM1H 8 1H RYHU 3/-W!N, 3 SN7 RYHU ) 9 H )
PAGE 245

)LJXUH ,QWHJUDQGV >J4Uf JUf JUf@U DQG >JRAUA AJA AUAUB IRU DQG 8 UHVSHFWLYHO\ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH 6 )OXLG SRN7H 4HF7f

PAGE 246

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f REWDLQHG IURP GLUHFW HQVHPEOH DYHUDJLQJ DQG IURP WKH P LQWHJUDOV PXVW EH GXH WR WKLV ORQJ UDQJH FRQWULEXWLRQ $ ORQJ Q UDQJH FRQWULEXWLRQ KDV EHHQ LQFOXGHG LQ WKH HYDOXDWLRQ RI WKH Q LQWHJUDOV ZKHUHDV QR VXFK FRUUHFWLRQ KDV EHHQ PDGH WR WKH HQVHPEOH DYHUDJHV IRU *A $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ &RQYHUJHQFH RI WKH 6SKHULFDO +DUPRQLF ([SDQVLRQ 7KH VPRRWKHG JQ f Uff FRHIILFLHQWV GHVFULEHG LQ 6HFWLRQ KDYH EHHQ UHFRPELQHG LQ WKH VSKHULFDO KDUPRQLF VHULHV f WR REWDLQ WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ J UAZAXAf +HUH ZH FRQVLGHU KRZ ZHOO WKH VHULHV UHIOHFWV WKH PXOWLGLPHQVLRQDO J UAZARAf E\ REWDLQLQJ YDOXHV IRU JUAFRAWAf XVrQJ GLIIHUHQW QXPEHUV RI JA A PAULA WHUPV LQ WKH H[SDQVLRQ $ WRWDO RI FRHIILFLHQWV LQFOXGLQJ 4RRAUA KDYH EHHQ H[SOLFLWO\ HYDOXDWHG LQ WKH PROHFXODU G\QDPLFV FDOFXODWLRQV IRU WKH TXDGUXSROH DQG RYHUODS IOXLGV VWXGLHG KHUH ([SUHVVLRQV IRU WKHVH FRHIILFLHQWV DUH JLYHQ LQ $SSHQGL[ ) DQG WKH UHVXOWLQJ YDOXHV

PAGE 247

DUH JLYHQ LQ $SSHQGL[ + ,Q DFWXDOLW\ WZLFH WKDW QXPEHU RI FRHIILFLHQWV KDYH EHHQ GHWHUPLQHG VLQFH GXH WR V\PPHWU\ RI WKH PROHFXOHV ePUOf e-PUf f +RZHYHU ZH ZLOO UHIHU WR D WRWDO VXP RYHU VD\ FRHIILFLHQWV LW EHLQJ XQGHUVWRRG WKDW ERWK Jf Q Uff DQG JQ Uff WHUPV KDYH EHHQ ; BA; A / M/ LQFOXGHG LQ IRUPLQJ f )RU HDFK RI WKH IOXLGV VWXGLHG WKH iJRRAUOA FRHIILFLHQWV ZHUH IRXQG WR EH QHJOLJLEOH DQG ZHUH QRW LQFOXGHG LQ IRUPLQJ J UAFRARAf ,Q DGGLWLRQ WR IRUPLQJ WKH VXP RYHU FRHIILFLHQWV VXPV IRU JUAWAZf RYHU WHQ DQG VL[ WHUPV KDYH EHHQ GHWHUPLQHG E\ LQFOXGLQJ UHVSHFWLYHO\ WKH WHUPV FRQWDLQLQJ JUAf JUA f JAUAf JUOff JUff JUOff Uff JUOff JUOff JUOf DQG JUOff UOfV JUOnff JUOff JfUff pAUAf 7KHVH P FRHIILFLHQWV DUH WKRVH ZKLFK DUH LQGHSHQGHQW RI WKH DQJOH FMf ZKHUH S LV WKH GLIIHUHQFH LQ D]LPXWKDO DQJOHV RI D SDLU RI PROHFXOHV LQ WKH LQWHUPROHFXODU IUDPH RI )LJXUH ,W LV RI LQWHUHVW WR VWXG\ WKH Mf GHSHQGHQFH RI JUA-rOM-A VrQFH WKH 3XUHO\ GLDWRPLF SRWHQWLDO PRGHOV H[KLELW ZHDN S GHSHQGHQFH RI WKHLU JUAZAWRf >@ /HQQDUG-RQHV SOXV TXDGUXSROH IOXLG 3ORWV RI WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ REWDLQHG IURP DQG WHUPV LQ WKH H[SDQVLRQ f DUH VKRZQ LQ )LJXUHV DQG IRU WKH WHH FURVV DQG SDUDOOHO SDLU RULHQWDWLRQV UHVSHFWLYHO\ 'XH WR D SURJUDPPLQJ HUURU WKH FRHIILFLHQW ZDV LQFRUUHFWO\ GHWHUPLQHG IRU WKH 4HR f

PAGE 248

J f )LJXUH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG -RQHV SOXV 4XDGUXSROH )OXLG RI )LJXUH IRU WKH 7HH 2ULHQWDWLRQ A r F_f XQGHILQHGf

PAGE 249

J f )LJXUH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG -RQHV SOXV 4XDGUXSROH )OXLG RI )LJXUH IRU WKH &URVV DQG 3DUDOOHO 2ULHQWDWLRQV A Arf

PAGE 250

IOXLG DQG KDV QRW EHHQ LQFOXGHG LQ WKH VXPPDWLRQ IRU Jn )RU WKH WHH RULHQWDWLRQ WKH DQJOH S LV XQGHILQHG VR RQO\ WHUPV ZLWK P FRQWULEXWH WR WKH VXP +HQFH ERWK WKH WHUP DQG WHUP VXPV JLYH WKH VDPH J&UAAFAf $V RQH ZRXAG H[SHFW FRQYHUJHQFH RI WKH H[SDQVLRQ LV JRRG IRU WKLV KLJKO\ SUREDEOH RULHQWDWLRQ WKHUH EHLQJ GLVWLQJXLVK DELOLW\ RI WKH DQG WHUP VXPV RQO\ DURXQG WKH ILUVW SHDN LQ JUfO8f )LJXUH VKRZV J&UAAAf ArU WALH FURVV DQWA SDUDOOHO RULHQWDn WLRQV REWDLQHG IURP D VXP RYHU VSKHULFDO KDUPRQLFV 7KH DQG WHUP VXPV HDFK JLYH WKH VDPH FXUYHV IRU WKHVH WZR SDLU RULHQWDWLRQV VLQFH RQO\ WKH DQJOH I! FKDQJHV LQ JRLQJ IURP D SDUDOOHO WR D FURVV RULHQWDWLRQ 7KLV ILJXUH LQGLFDWHV D QRQQHJOLJLEOH I! GHSHQGHQFH RI WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ (YLGHQFH IRU WKH Mf GHSHQGHQFH LV VWUHQJWKHQHG E\ FRQVLGHULQJ VNHZHG SDLU RULHQWDWLRQV LH RULHQWDWLRQV RWKHU WKDQ WKH SULPDU\ RQHV DV GHILQHG LQ 7DEOH 7KXV IRU WKH VNHZHG RULHQWDWLRQ RI )LJXUH WKHUH LV VXEVWDQWLDO YDULDWLRQ LQ WKH YDOXHV IRU JU RUf REWDLQHG IURP WKH DQG WHUP VXPV   7KH FRQYHUJHQFH RI WKH H[SDQVLRQ IRU WKH OHDVW SUREDEOH RULHQWDn WLRQV HJ WKH HQGRQ LV QRW YHU\ JRRG HYHQ ZKHQ WHUPV DUH LQFOXGHG LQ WKH VXP 7KXV HVSHFLDOO\ DW KLJK YDOXHV RI WKH TXDGUXSROH PRPHQW VSXULRXVO\ VWURQJ SHDNV RU XQSK\VLFDO QHJDWLYH YDOXHV PD\ EH REWDLQHG IRU JUAARU AHDVW SWREDEOH RULHQWDWLRQV 7R RYHUFRPH WKLV SUREOHP VLJQLILFDQWO\ ORQJHU PROHFXODU G\QDPLFV FDOFXODWLRQV DUH UHTXLUHG &RQYHUJHQFH RI WKH VSKHULFDO KDUPRQLF H[SDQVLRQ IRU WKH RWKHU TXDGUXSROH IOXLGV VWXGLHG 4HRf DQG f LV TXDOLWDWLYHO\

PAGE 251

)LJXUH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG -RQHV SOXV 4XDGUXSROH )OXLG RI )LJXUH IRU D 6NHZHG 2ULHQWDWLRQ A I! rf

PAGE 252

VLPLODU WR WKDW VKRZQ KHUH IRU WKH 4HD f IOXLG 7KH GHJUHH RI YDULDWLRQ EHWZHHQ WKH JLUAAMNA GHWHUPLQHG IURP WKH GLIIHUHQW QXPEHU RI WHUPV LV QRW DV VWURQJ IRU WKH ZHDNHU TXDGUXSROH VWUHQJWKV /HQQDUG-RQHV SOXV DQLVRWURSLF RYHUODS IOXLG &RQYHUJHQFH WHVWV IRU WKH VSKHULFDO KDUPRQLF H[SDQVLRQ RI J UMA-rA ARU WALH /HQQDUG-RQHV SOXV DQLVRWURSLF RYHUODS IOXLG KDYH EHHQ FRQGXFWHG LQ WKH VDPH PDQQHU DV GHVFULEHG DERYH IRU WKH TXDGUXSROH IOXLG 5HVXOWV DUH VKRZQ LQ )LJXUHV DQG IRU WKH RYHUODS IOXLG ZLWK IRU WKH SDUDOOHO DQG HQGRQ SDLU RULHQWDWLRQV UHVSHFWLYHO\ &DOFXODWLRQV RI J&UA:MAf IrU YDULRXV SDLU RULHQWDWLRQV VKRZV WKH PRVW SUREDEOH RULHQWDWLRQ WR EH WKH SDUDOOHO DV VXJJHVWHG IURP WKH SRWHQWLDO HQHUJ\ FXUYHV RI )LJXUH 7KXV WKH VHULHV FRQYHUJHV ZHOO IRU WKH PRVW SUREDEOH RULHQWDWLRQ EXW IRU OHVV SUREDEOH RULHQWDWLRQV WKH IXOO WHUPV DUH QHFHVVDU\ IRU DFFXUDWH UHVXOWV 7KH KLJK IUHTXHQF\ RVFLOODWLRQV DIWHU WKH ILUVW SHDN LQ WKH SORW RI JIrU WALH HQGRQ FRQILJXUDWLRQ )LJXUH UHIOHFW VWDWLVWLFDO IOXFWXDWLRQV LQ WKH VDPSOLQJ RI WKLV ORZ SUREDELOLW\ RULHQWDWLRQf 7KH EHKDYLRU RI WKH VHULHV IRU J&UARRAWAf IrU WKH RYHUODS IOXLG ZLWK LV TXDOLWDWLYHO\ VLPLODU WR WKDW VKRZQ IRU WKH IOXLG IDVWHU FRQYHUJHQFH RI WKH VHULHV LV IRXQG IRU WKH IOXLG 7KH TXHVWLRQ RI Mf GHSHQGHQFH RI WKH GRHV QRW DULVH KHUH VLQFH WKH RYHUODS SRWHQWLDO LV LQGHSHQGHQW RI WKH DQJOH F`! VHH (TXDWLRQ & $SSHQGL[ &f /RFDO 6WUXFWXUH IURP JUAAf§f§ 7KH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ JUMAOMAf LV SDUWLFXODU YDOXH LQ HOXFLGDWLQJ WKH QDWXUH RI PROHFXODU SDLU RULHQWDWLRQV LQ WKH

PAGE 253

Jf )LJXUH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG -RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG RI )LJXUH IRU WKH 7HH 2ULHQWDWLRQ r XQ GHILQHGf

PAGE 254

JGf )LJXUH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU WKH /HQQDUG -RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG RI )LJXUH IRU WKH (QGRQ 2ULHQWDWLRQ A a f

PAGE 255

IOXLG $ PDMRU GHWHUUHQW WR GHWDLOHG VWXG\ RI J&UAAWAf LV LWV PXOWLn GLPHQVLRQDOLW\ 7KXV IRU VLPSOH OLQHDU PROHFXOHV GHSHQGV RQ WKH GLVWDQFH RI VHSDUDWLRQ RI WKH FHQWHUV RI PDVV RI D PROHFXODU SDLU UO! DQG WKUHH DQJOHV A A DQG S VSHFLI\LQJ WKH UHODWLYH RULHQWDn WLRQ RI WKH SDLU $ ILYH GLPHQVLRQDO VSDFH LV UHTXLUHG WR SLFWRULDOO\ UHSUHVHQW WKH HQWLUH JUAfAff IXQFWLRQ IRU OLQHDU PROHFXOHV ,Q WKLV VHFWLRQ WKUHH GLPHQVLRQDO VXUIDFHV FXW IURP WKH JUA::A K\SHUVXUIDFH IRU WKH TXDGUXSROH DQG RYHUODS IOXLGV DUH SUHVHQWHG IRU VWXG\ 7KH REMHFWLYHV RI WKH VWXG\ DUH Df WR GLVFRYHU WKH UHODWLYH SUREDELOLW\ RI RFFXUUHQFH RI WKH SULPDU\ SDLU RULHQWDWLRQV DQG Ef WR GHWHUPLQH WKH HIIHFW RI WKH DQLVRWURSLF VWUHQJWK FRQVWDQW TXDGUXSROH PRPHQW 4 RU DQLVRWURSLF RYHUODS SDUDPHWHU f RQ WKH RFFXUUHQFH RI SDLU RULHQWDn WLRQV LQ WKH IOXLG 7KH WKUHH GLPHQVLRQDO ILJXUHV KDYH EHHQ JHQHUDWHG IURP WKH UHFRPELQHG VSKHULFDO KDUPRQLF H[SDQVLRQ f XVLQJ WKH VPRRWKHG JA \ PAUOA KDWD 7KH UHVXOWLQJ JUAfAZf FXUYHV KDYH QRW EHHQ VPRRWKHG VR WKDW D FRXSOH RI WKH VXUIDFHV VKRZQ FRQWDLQ VPDOO VWDWLVWLFDO IODZV ZKLFK DUH SRLQWHG RXW LQ WKH WH[W (DFK VXUn IDFH KDV EHHQ JHQHUDWHG E\ IL[LQJ WZR RI WKH UHODWLYH RULHQWDWLRQ DQJOHV A A RU M!f ZKLOH URWDWLQJ WKH WKLUG DQJOH WKURXJK WR X IRU DOO UA YDOXHV LQ WKH UDQJH  UA A )RU OLQHDU PROHFXOHV A DQG DUH UHVWULFWHG WR WKH UDQJH WWf DQG JUARAZf LV V\PPHWULF DERXW M! WW IRU WKH TXDGUXSROH DQG RYHUODS IOXLGV 2Q HDFK WKUHH GLPHQVLRQDO SORW D QXPHULFDO VFDOH LV VKRZQ ZKLFK DSSOLHV WR WKH ]D[LV LH WKH JUAA8fA8fAf D[LVf 7KH QXPEHUV RQ WKH OHIW VLGH RI WKH VFDOH UHIHU WR WKH SK\VLFDO KHLJKW RI WKH GUDZLQJ RQ WKH SDJH LQ LQFKHV

PAGE 256

7KH QXPEHUV RQ WKH ULJKW DUH WKH FRUUHVSRQGLQJ YDOXHV IRU J&UAARAf DW WKDW KHLJKW 7KH WRS QXPEHU RQ WKH ULJKW RI WKH VFDOH LV WKH PD[LPXP YDOXH RI JU ZKLFK RFFXUV RQ WKH SORW 7KH SORWV DUH SUHVHQWHG LQ SDLUV WKH VHFRQG SORW LQ HDFK SDLU LV IRU WKH VDPH RULHQWDWLRQV DV WKH ILUVW EXW DW D KLJKHU YDOXH RI WKH DQLVRWURSLF VWUHQJWK FRQVWDQW /HQQDUG-RQHV SOXV TXDGUXSROH IOXLG )LJXUHV DQG VKRZ WKH JLUAXWM8Af VXUIDFH JHQHUDWHG ZLWK A Mf DQG URWDWHG IURP WR 77 IRU WKH 4 HFU f DQG IOXLGV UHVSHFn WLYHO\ 7KH UHODWLYH SDLU RULHQWDWLRQV DUH WKXV URWDWHG IURP WKH WHH WKURXJK WKH SDUDOOHO DQG EDFN WR WKH WHH DV A FKDQJHV IURP WR WW )LJXUH VKRZV WKDW IRU WKH ZHDNO\ TXDGUXSROH IOXLG WKH WHH RULHQWDWLRQ LV DERXW WZLFH DV SUREDEOH DV WKH SDUDOOHO LQ WKH UHJLRQ RI UA DURXQG WKH ILUVW SHDN LQ UL Wr9rfA n :KHQ 4 HD f LV LQn FUHDVHG WR WKHUH LV D GUDPDWLF VKLIW LQ SUREDELOLW\ IDYRULQJ WKH WHH RYHU WKH SDUDOOHO DV VKRZQ LQ )LJXUH 7KH KHLJKW RI WKH ILUVW SHDN LQ J U -r A ARU WrLH WHH RULHQWDWLRQ IrU 4 *2 f LV PRUH WKDQ DQ RUGHU RI PDJQLWXGH ODUJHU WKDQ WKDW IRU WKH SDUDOOHO RULHQWDWLRQ DQG LV DERXW WKUHH WLPHV ODUJHU WKDQ IRU WKH VDPH WHH RULHQWDWLRQ ZLWK 4 HD f )LJXUHV DQG VKRZ WKH JVXUIDFH JHQHUDWHG ZLWK A k r DQG M! URWDWHG IURP WR WW IRU WKH 4HD f DQG IOXLGV UHVSHFWLYHO\ 7KH UHODWLYH SDLU RULHQWDWLRQV DUH WKXV URWDWHG IURP WKH SDUDOOHO WKURXJK WKH FURVV DQG EDFN WR WKH SDUDOOHO )RU WKH ZHDNO\ TXDGUXSROH IOXLG )LJXUH LQGLFDWHV D ZHDN EXW GLVFHUQ DEOH I! GHSHQGHQFH LQ J&UA-MAf LQ WKH ILUVW SHDN UHJLRQ DQG YHU\ VOLJKW

PAGE 257

" VR f§} UL U L U \RR n 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU A S ZLWK 4r 7r DQG Sr

PAGE 258

c c )LJXUH 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU -f ZLWK 4r 7r DQG Sr

PAGE 259

VR Lm 286 )LJXUH 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU f ZLWK 4r 7 DQG Sr

PAGE 260

)LJXUH

PAGE 261

S GHSHQGHQFH LQ WKH VHFRQG SHDN UHJLRQ :KHQ 4 HR f LV LQFUHDVHG )LJXUH VKRZV WKDW WKH Mf GHSHQGHQFH LQ WKH ILUVW SHDN UHJLRQ RI JUO-O-f VWUHQWALHQHA IDYRU RI WKH FURVV RULHQWDWLRQ 7KH S GHSHQGHQFH LQ WKH VHFRQG SHDN UHJLRQ UHPDLQV ZHDN 1RWH WKDW RQ LQn FUHDVLQJ WKH TXDGUXSROH PRPHQW WKH SUREDELOLWLHV IRU ILQGLQJ PROHFXOHV LQ WKH ILUVW SHDN UHJLRQ DUH UHGXFHG FRPSDUHG WR WKRVH LQ WKH VHFRQG SHDN UHJLRQ +HQFH WKH SUREDELOLW\ RI ILQGLQJ D PROHFXODU SDLU LQ WKH ILUVW SHDN UHJLRQ ZLWK A A LUUHVSHFWLYH RI WKH YDOXH RI S LV UHGXFHG ZKHQ WKH VWUHQJWK RI WKH TXDGUXSROH PRPHQW LV LQFUHDVHG 7KH EOLS RQ WKH VPDOO UA VLGH RI WKH ILUVW SHDN LQ )LJXUH LV D VWDWLVWLFDO IOXFWXDWLRQ DQG LV QRW LQGLFDWLYH RI IOXLG VWUXFWXUHf )LJXUHV DQG VKRZ WKH J&UAXAWA VXUIDFH JHQHUDWHG IRU D VHW RI VNHZHG SDLU RULHQWDWLRQV VSHFLILFDOO\ p DQA S URWDWHG IURP WR ,7 IRU WKH 4e f DQG IOXLGV UHVSHFn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f IRU WKHVH RULHQWDWLRQV )LJXUHV DQG f ZLWK WKH ORFDWLRQ RI WKH PLQLPD LQ WKH FRUUHVSRQGLQJ XLUAAMAf )LJXUH f DV VKRZQ LQ

PAGE 262

)LJXUH 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG IRU f ZLWK 4r 7r DQG Sr

PAGE 264

)LJXUH IRU WKH 4HD f IOXLG 7KHVH UHVXOWV DOVR FRQn ILUP HDUOLHU 0RQWH &DUOR VWXGLHV RI WKH /HQQDUG-RQHV SOXV TXDGUXSROH IOXLG >@ /HQQDUG-RQHV SOXV DQLVRWURSLF RYHUODS IOXLG )LJXUHV DQG VKRZ WKH J&UAAMAf VXUIDFH JHQHUDWHG ZLWK A I! DQG URWDWHG IURP WR 7 IRU WKH DQG IOXLGV UHVSHFWLYHO\ 7KH UHODWLYH SDLU RULHQWDWLRQV DUH WKXV URWDWHG IURP WKH WHH WKURXJK WKH SDUDOOHO DQG EDFN WR WKH WHH DV FKDQJHV IURP WR 77 )LJXUH VKRZV ZHDN GHSHQGHQFH DW WKHVH UHODWLYH SDLU RULHQWDWLRQV IRU WKH VPDOO YDOXH IOXLG 7KH SUHIHUHQFH RI WKH SDUDOOHO RULHQWDWLRQ RYHU WKH WHH LV VOLJKW 1RWH WKH KLQW LQ )LJXUH RI WKH HIIHFW RI LQn FUHDVLQJ QDPHO\ WKH SUHGRPLQDQFH RI WKH SDUDOOHO RYHU WKH WHH DQG WKH VOLJKW VKLIW RI WKH ILUVW SHDN WR VPDOOHU UA YDOXHV DV VKRZQ LQ )LJXUH 7KH RUGHU RI OLNHO\ RFFXUUHQFH RI WKH SULPDU\ SDLU RULHQWDWLRQV IRU WKH /HQQDUG-RQHV SOXV DQLVRWURSLF RYHUODS IOXLG ZLWK LV IRXQG WR EH SDUDOOHO WHH DQG HQGRQ 7KLV LV LQ DJUHHPHQW ZLWK WKH UHODWLYH GHSWKV RI WKH SRWHQWLDO ZHOOV IRU RYHUODS IOXLGV ZLWK VKRZQ LQ )LJXUH )RU WKH IOXLG WKH HQGRQ FRQILJXUDWLRQ LV IRXQG WR EH VOLJKWO\ PRUH SUREDEOH WKDQ WKH WHH 7KLV GRHV QRW FRUUHODWH ZLWK WKH FRUUHVSRQGLQJ SRWHQWLDO ZHOO GHSWKV RI )LJXUH ,W LV IHOW WKDW WKLV VPDOOf GLVFUHSDQF\ LV GXH WR SRRU FRQYHUJHQFH RI WKH VSKHULFDO KDUPRQLF H[SDQVLRQ IRU J FDXVHG E\ LQIUHTXHQW VDPSOLQJ RI DOPRVW HTXDOO\ ORZ SUREDELOLW\ SDLU RULHQWDWLRQV LQ WKH FRXUVH RI WKH PROHFXODU G\QDPLFV FDOFXODWLRQ $ ORQJHU PROHFXODU G\QDPLFV FDOFXODWLRQ ZRXOG VKRZ ZKHWKHU WKLV LV WKH SUREOHP

PAGE 265

J f M X f )LJXUH &RPSDULVRQ RI 3HDN +HLJKWV LQ WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ ZLWK :HOO 'HSWKV LQ WKH 3DLU 3RWHQWLDO IRU WKH /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK 4HRfO S

PAGE 266

B = O2& M )LJXUH 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG IRU ZLWK 7r DQG Sr

PAGE 267

)LJXUH 6XUIDFH RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG IRU -! ZLWK 7r DQG Sr

PAGE 268

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eO 1H WW f f GDf GRMf A A A UO XUOD-Off JUDfO:f GUO f SN7 OIL f n GW GRRf 22 N r XUZZf r r UO r JUO8O-fGUO f U ) r U U < D 3 H IW GLR GFf 2 N r XULZLD-f r r U f r JUDfLDff GU f U ,Q DGGLWLRQ WKH PHDQ VTXDUHG WRUTXH LV GLUHFWO\ SURSRUWLRQDO WR f E\ WKH UHODWLRQV LQ 7DEOH 8VLQJ WKH UHFRPELQHG VSKHULFDO KDUPRQLF H[SDQVLRQ IRU JUAWRARAf N WKH LQWHJUDQGV XQGHU WKH UA LQWHJUDOV LQ f f DQG f KDYH EHHQ FDOFXODWHG IRU WKH SULPDU\ SDLU RULHQWDWLRQV IRU ERWK WKH TXDGUXSROH DQG RYHUODS IOXLGV 6XFK FDOFXODWLRQV SURYLGH LQVLJKW LQWR WKH HIIHFW RI RULHQWDWLRQ RQ SURSHUW\ YDOXHV :KHQ FRXSOHG ZLWK VLPLODU FDOFXODn WLRQV IRU RWKHU PRGHO SRWHQWLDOV WKH\ SURYLGH D WRRO IRU GLVWLQJXLVKLQJ

PAGE 269

DPRQJ WKH YDULRXV PRGHOV DQG FRXOG SRWHQWLDOO\ VXJJHVW H[SHULPHQWDO ZRUN ZKLFK ZRXOG LOOXPLQDWH WKH QDWXUH RI WKH SRWHQWLDO LQ UHDO IOXLGV )RU ERWK WKH TXDGUXSROH DQG RYHUODS IOXLGV WKH LQWHJUDQGV LQ f WR f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fU Qf KDV EHHQ D D GHWHUPLQHG LQ WKH PROHFXODU G\QDPLFV FDOFXODWLRQ E\ WKH PHWKRG RXWn OLQHG LQ 6HFWLRQ 7KH OLQHDU PROHFXOHV VWXGLHG KHUH ZHUH FRQVLGHUHG WR EH KRPRQXFOHDU GLDWRPLFV IRU ZKLFK WKH VLWHV RI LQWHUHVW DUH WKH DWRPLF QXFOHL 7KH VLWHV ZHUH WDNHQ WR EH KDOI WKH DWRP ERQG OHQJWK = IURP WKH PROHFXODU FHQWHU RI PDVV 7KH ERQG OHQJWK IRU WKH QLWURJHQ PROHFXOH ZDV XVHG LQ WKH FDOFXODWLRQ LH =R >@

PAGE 270

)LJXUH ,QWHJUDQG IRU ,QWHUQDO (QHUJ\ UAXfJf IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLGV IRU WKH 7HH 2ULHQWDn WLRQ

PAGE 271

U GXfGU Jf )LJXUH ,QWHJUDQG IRU 3UHVVXUH U GXfGU Jf IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLGV IRU WKH 7HH 2ULHQWDWLRQ

PAGE 272

Xf Jf )LJXUH ,QWHJUDQG IRU ,QWHUQDO (QHUJ\ U XfJf IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLGV IRU 3DUDOOHO 2ULHQWDWLRQ

PAGE 273

/HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLGV IRU 3DUDOOHO 2ULHQWDWLRQ

PAGE 274

7$%/( 5DQJH RI 9DOXHV IRU 2ULHQWDWLRQDO &RQWULEXWLRQV WR 3URSHUW\ ,QWHJUDQGV IRU 4XDGUXSROH DQG 2YHUODS )OXLGV 3URSHUW\ 81H 3SN7 )OXLG 6WUHQJWK &RQVWDQW 3DLU 2ULHQWDWLRQ 5DQJH RI 9DOXHV IRU ,QWHJUDQGV /44 r 4 WHH WR FURVV WR SDUDOOHO WR 2L HQGRQ WR /2YHU WHH WR SDUDOOHO WR HQGRQ WR /44 r 4 WHH WR FURVV WR SDUDOOHO WR HQGRQ WR /2YHU WHH WR SDUDOOHO WR HQGRQ WR

PAGE 275

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iDJUDJf EURDGHQV LQ WKH ODUJH UA GLUHFWLRQ >@ $Q H[SODQDWLRQ IRU WKH QDWXUH RI WKLV JA FXUYH DW KLJK TXDGUXn SROH VWUHQJWK FDQ EH VXJJHVWHG EDVHG RQ PROHFXODU VWUXFWXUH &RQVLGHU WKH /HQQDUG-RQHV SOXV TXDGUXSROH PROHFXOH WR EH D UHSXOVLYH VSKHULFDO VKHOO ZLWK HPEHGGHG SRLQW FKDUJHV VXFK WKDW DQ DWWUDFWLYH D[LDOO\ V\PPHWULF TXDGUXSROH ILHOG H[LVWV DERXW WKH VSKHULFDO VKHOO /RFDWHG RQ WKH OLQHDU D[LV RI WKH TXDGUXSROH ILHOG DUH VLWHV $ DQG % FHQWHUHG DERXW WKH FHQWHU RI WKH VSKHUH DQG VHSDUDWHG IURP RQH DQRWKHU E\ DERXW D WKLUG RI D PROHFXODU GLDPHWHU e Df )URP WKH DQDO\VLV RI WKH SDLU SRWHQWLDO LQ 6HFWLRQ DQG WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ LQ 6HFWLRQ SDLUV RI WKHVH TXDGUXSRODU PROHFXOHV WHQG WR DOLJQ LQ D WHH RULHQWDWLRQ /HW XV DVVXPH WKDW WKH TXDGUXSROH VWUHQJWK LV VXIILFLHQW VR WKDW QRW RQO\ SDLUV RI PROHFXOHV WHQG WR FRQWULEXWH WR VWUXFWXUH EXW WKDW FOXVWHUV RI DW OHDVW IRXU PROHFXOHV IRUP ,Q VXFK D FOXVWHU HDFK SDLU RI PROHFXOHV LV LQ D UHODWLYH WHH RULHQWDWLRQ

PAGE 276

)LJXUH 6LWH6LWH 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG -RQHV SOXV 4XDGUXSROH )OXLGV

PAGE 277

DQG LV QHDU WKH FORVHVW SRLQW RI DSSURDFK OLPLWHG E\ WKH UHSXOVLYH VKHOOf DV LQ )LJXUH :KHQ WKH FOXVWHUV FRQWDLQ IRXU PROHFXOHV HDFK PROHFXOH KDV VLWHVLWH GLVWDQFHV U B EHWZHHQ LWV WZR VLWHV D6 DQG WKH VL[ RWKHU VLWHV LQ WKH FOXVWHU :KHQ WKH GLVWDQFH RI VHSDUDWLRQ RI VLWHV RQ D PROHFXOH LV DERXW D WKLUG RI WKH PROHFXODU GLDPHWHU DQG WKH VLWHV DUH LQGLVWLQJXLVKDEOH WZR JURXSV RI WKHVH VLWHVLWH GLVWDQFHV DUH HTXDO DV LQGLFDWHG LQ )LJXUH UO$$ UO$% UO$% UO%% UO%$ f UO$$ UO%$ UO%$ UO%% f 7KH ILUVW FRQQHFWLRQ EHWZHHQ WKH SURSRVHG FOXVWHU VWUXFWXUH LQ )LJXUH DQG WKH J LQ )LJXUH LV WKDW WKH QXPEHU RI HTXDO GLVWDQFHV LQ RWS f DQG f DUH LQ WKH UDWLR ZKLOH WKH UDWLR RI WKH PDLQ SHDN KHLJKW WR WKH KHLJKW RI WKH VKRXOGHU LQ )LJXUH LV IURP $SSHQGL[ -f )XUWKHU WKHUH LV VRPH FRUUHODWLRQ EHWZHHQ WKH GLVWDQFHV LQ f DQG f DQG WKH UA YDOXHV DW ZKLFK WKH VKRXOGHU DQG ILUVW PDLQ SHDN RFFXU 7KH GLVWDQFHV Aaf DUH IRXQG WR EH a D ZKLOH IURP $SSHQGL[ -f WKH ILUVW SHDN PD[LPXP RFFXUV DW U f D 7KH GLVWDQFHV U$2RI f DUH a D D% $$ ZKLOH WKH VKRXOGHU RFFXUV DW a D 7KH U ORFDWLRQV RI WKH VKRXOGHU D% DQG ILUVW SHDN ZRXOG EH H[SHFWHG WR EH DW VOLJKWO\ ORQJHU GLVWDQFHV WKDQ WKRVH LQ f DQG f VLQFH WKH PLQLPXP HQHUJ\ RFFXUV MXVW EH\RQG UA D IRU WKH PRVW SUREDEOH RULHQWDWLRQ 7KHUHIRUH D FOXVWHU RI PROHFXODU SDLUV H[KLELWLQJ VTXDUH SDFNLQJ ZLWK LQWHUORFNLQJ

PAGE 278

)LJXUH 3RVVLEOH 6TXDUH 3DFNLQJ RI /HQQDUG-RQHV SOXV 4XDGUXSROH 0ROHFXOHV IRU ,QWHUSUHWLQJ J fUf &LS

PAGE 279

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f IRU YDULRXV PROHFXODU HORQJDWLRQV $V WKH 4WS HORQJDWLRQ LV LQFUHDVHG WKHVH VWUXFWXUDO DUJXPHQWV VXJJHVW WKH PDLQ SHDN DQG VKRXOGHU ZLOO EHFRPH VHSDUDWHG IURP RQH DQRWKHU VLQFH WKH GLIIHUHQFH LQ YDOXHV RI WKH GLVWDQFHV RI f DQG f ZRXOG LQFUHDVH 7KHUH ZRXOG DOVR EH VOLJKW FKDQJHV LQ WKH UHODWLYH KHLJKWV RI WKH SHDN DQG VKRXOGHU ZLWK HORQJDWLRQ DV YDULRXV WHUPV ZRXOG VDWLVI\ f /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG 7KH JAJUf IRU WKH /HQQDUG-RQHV SOXV RYHUODS IOXLGV DUH VKRZQ LQ )LJXUH $V LQ WKH FDVH RI WKH TXDGUXSROH WKH ILUVW SHDN LQ J IRU RYHUODS LV VKRUWHQHG DQG EURDGHQHG FRPSDUHG ZLWK WKH FHQWHU 2&S FHQWHU SDLU FRUUHODWLRQ IXQFWLRQ RI )LJXUH 1R VSHFLDO IHDWXUHV DUH IRXQG RQ WKH RYHUODS JA FXUYHV WKRXJK WKHUH LV D KLQW RI D VKRXOGHU IRUPLQJ QHDU WKH KLJK UA VLGH RI WKH ILUVW SHDN IRU WKH IOXLG

PAGE 280

)LJXUH 6LWH6LWH 3DLU &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG -RQHV SOXV $QLVRWURSLF 2YHUODS )OXLGV

PAGE 281

9DOXHV IRU JDJUf IRU ERWK WKH TXDGUXSROH DQG RYHUODS IOXLGV DW WKH FRQGLWLRQV VWXGLHG LQ WKLV ZRUN DUH WDEXODWHG LQ $SSHQGL[ )LOPHG $QLPDWLRQ RI 0ROHFXODU 0RWLRQV ,QWURGXFWLRQ $ SULQFLSDO DGYDQWDJH RI SHUIRUPLQJ FRPSXWHU VLPXODWLRQ RI IOXLGV E\ WKH PROHFXODU G\QDPLFV PHWKRG UDWKHU WKDQ E\ 0RQWH &DUOR LV WKDW LQ WKH FRXUVH RI WKH VLPXODWLRQ WKH FODVVLFDO WLPH GHYHORSn PHQW RI WKH V\VWHP LV HYROYHG %\ VDYLQJ XVXDOO\ RQ PDJQHWLF WDSHf WKH SRVLWLRQV DQG RULHQWDWLRQV RI WKH SDUWLFOHV DW GLVFUHWH WLPH VWHSV IRU ZKLFK WKH HTXDWLRQV RI PRWLRQ DUH VROYHG RQH REWDLQV D SHUPDQHQW UHFRUG RI WKH HYROXWLRQ RI WKH V\VWHP 7KLV UHFRUG GDWDf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

PAGE 282

ZRXOG EH D PHDQV RI FRPSDULQJ URWDWLRQDO DQG WUDQVODWLRQDO PRWLRQ DQG ZRXOG LQGLFDWH WKH GHJUHH RI KLQGHUHG URWDWLRQ H[SHULHQFHG E\ WKH PROHFXOHV &HUWDLQO\ PXFK HIIHFWLYH VWXG\ RI WKHVH DUHDV FDQ EH GRQH LQ D PRUH TXDQWLWDWLYH PDQQHU WKURXJK WKHRU\ H[SHULPHQW DQG VLPXODWLRQ 7KH PRYLH KRZHYHU SURYLGHV D PHGLXP IRU FRPPXQLFDWLQJ GHWDLOHG DVSHFWV RI D V\VWHP HVSHFLDOO\ WR WKRVH DXGLHQFHV ZKR KDYH DQ LQWHUHVW LQ WKH SUREOHPV EXW ZKR KDYH OLWWOH WLPH RU LQFOLQDn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

PAGE 283

)OXLG 6\VWHP )LOPHG 'DWD IRU D V\VWHP RI /HQQDUG-RQHV PROHFXOHV ZHUH JHQHUDWHG E\ WKH PROHFXODU G\QDPLFV PHWKRG GHVFULEHG LQ &KDSWHU 7KH VWDWH FRQGLWLRQ FKRVHQ ZDV N7H DQG SD 7KLV VWDWH FRQGLWLRQ KDV EHHQ VWXGLHG SUHYLRXVO\ E\ 9HUOHW DQG WKH YDOXHV IRU WKH LQWHUQDO HQHUJ\ SUHVVXUH DQG UDGLDO GLVWULEXWLRQ IXQFWLRQ IRXQG KHUH DJUHHG ZLWK WKRVH JLYHQ E\ 9HUOHW >@ 7KH FDOFXODWLRQ ZDV VWDUWHG IURP DQ )&& ODWWLFH VWUXFWXUH DQG D WRWDO RI WLPH VWHSV ZHUH JHQHUDWHG HDFK RI f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f 7KH YROXPH HOHPHQW LV DQ RSHQ V\VWHP LQ WKH s ;n GLUHFWLRQ 0ROHFXOHV ZKLFK HQWHU DQG OHDYH WKH YROXPH HOHPHQW EHFRPH UHFRUGHG DV DSSHDULQJ DQG GLVDSSHDULQJ RQ WKH ILOP

PAGE 284

)LJXUH %R[ 5HSUHVHQWLQJ WKH 0ROHFXODU '\QDPLFV 6\VWHP ZLWK WKH 9ROXPH (OHPHQW 6DPSOHG IRU WKH )LOPHG $QLPDWLRQ ,QGLFDWHG $UURZ LQGLFDWHV WKH OLQH RI VLJKW LQ WKH PRYLHf

PAGE 285

f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f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

PAGE 286

HDFK WLPH VWHS ZHUH GLVSOD\HG DQG SKRWRJUDSKHG RQH DW D WLPH 7R FUHDWH D VORZO\ HYROYLQJ \HW VPRRWK DQLPDWLRQ IRXU PRYLH IUDPHV ZHUH SKRWRJUDSKHG RI HDFK WLPH VWHS GUDZLQJ :KHQ SURMHFWHG DW D VSHHG RI IUDPHV SHU VHFRQG WKLV FRUUHVSRQGV WR VL[ WLPH VWHSV SHU VHFRQG RI V\VWHP HYROXWLRQ +HQFH WLPH VWHSV JDYH DERXW PLQXWHV RI ILOP WLPH )LJXUHV DQG VKRZ WKH IUDPHV IRU WKH ILUVW V W VL[WK DQG WLPH VWHSV ,Q WKHVH ILJXUHV DV LQ WKH PRYLHf WKH OHQJWKV RI ERWK VLGHV RI WKH ER[ KDYH EHHQ OHQJWKHQHG E\ RQH PROHFXODU GLDPHWHU ODf VR WKDW DQ\ FLUFOH QHDU WKH HGJH RI WKH ER[ ZLOO DSSHDU WR EH ZKROO\ ZLWKLQ WKH ER[ )LJXUH VKRZV WKH )&& VWDUWLQJ FRQILJXUDWLRQ IRU WKH YROXPH HOHPHQW )LJXUH VKRZV WKH EHJLQQLQJ RI EUHDNGRZQ RI WKH )&& VWUXFWXUH GXH DW WKLV SRLQW ODUJHO\ WR WKH HIIHFW RI SHULRGLF ERXQGDU\ FRQGLWLRQV 1RWH WKDW VHYHUDO SDLUV RI PROHFXOHV DUH FROOLGLQJ DIWHU RQO\ VL[ WLPH VWHSV )LJXUH VKRZV WKH )&& ODWWLFH WR EH GLVVROYHG DIWHU WLPH VWHSV DQG LOOXVWUDWHV WKH PLVOHDGLQJ RYHUODS RI PROHFXOHV GLVFXVVHG LQ 6HFWLRQ

PAGE 287

)LJXUH ,QLWLDO )&& /DWWLFH &RQILJXUDWLRQ RI /HQQDUG-RQHV 0ROHFXOHV LQ WKH 9ROXPH (OHPHQW 6DPSOHG LQ WKH )LOPHG $QLPDWLRQ

PAGE 288

)LJXUH )UDPH IURP WKH )LOPHG $QLPDWLRQ RI /HQQDUG-RQHV 0ROHFXOHV &RUUHVSRQGLQJ WR WKH 6L[WK 7LPH6WHS LQ WKH 0ROHFXODU '\QDPLFV &DOFXODWLRQ

PAGE 289

)LJXUH )UDPH IURP WKH )LOPHG $QLPDWLRQ RI /HQQDUG-RQHV 0ROHFXOHV &RUUHVSRQGLQJ WR WKH VW 7LPH6WHS LQ WKH 0ROHFXODU '\QDPLFV &DOFXODWLRQ

PAGE 290

&+$37(5 &21&/86,216 7KHRU\ IRU 6XUIDFH 7HQVLRQ RI 3RO\DWRPLF )OXLGV $ JHQHUDO ILUVW RUGHU SHUWXUEDWLRQ WKHRU\ IRU WKH VXUIDFH WHQVLRQ RI SRO\DWRPLF IOXLGV KDV EHHQ GHYHORSHG 8SRQ LQWURGXFWLRQ RI D 3RSOH UHIHUHQFH WKH GLIILFXOW PXOWLERG\ HIIHFWV LQ WKH KLJKHU RUGHU WHUPV DV ZHOO DV WKH ILUVW RUGHU WHUP YDQLVK DOORZLQJ H[WHQVLRQ RI WKH WKHRU\ WR WKLUG RUGHU 7KH QRQYDQLVKLQJ WHUPV LQ WKLV WKHRU\ LQYROYH WKH XQNQRZQ LQWHUIDFLDO SDLU DQG WULSOHW FRUUHODWLRQ IXQFWLRQV J ]Uf DQG J ]UfUf ,Q RUGHU WR rR f§ R f§f§ SHUIRUP FDOFXODWLRQV WKH )RZOHU PRGHO RI WKH LQWHUIDFH LV LQWURGXFHG 7KH WKHRU\ KDV EHHQ WHVWHG DJDLQVW 0RQWH &DUOR FDOFXODWLRQV RI WKH )RZOHU PRGHO VXUIDFH WHQVLRQ IRU D 6WRFNPD\HU IOXLG 7KH EHKDYLRU RI WKH VHFRQG DQG WKLUG RUGHU SHUWXUEDWLRQ WKHRULHV IRU VXUIDFH WHQVLRQ LV PXFK WKH VDPH DV WKDW IRXQG IRU EXON IOXLG WKHUPRG\QDPLF SURSHUWLHV 7KH VHFRQG RUGHU WKHRU\ LV IRXQG WR ZRUN ZHOO IRU ZHDN GLSROH VWUHQJWKV \&2 f ef EXW ERWK WKH VHFRQG DQG WKLUG RUGHU WKHRULHV JLYH SRRU UHVXOWV DW KLJK YDOXHV RI WKH GLSROH PRPHQW FRPSDUHG WR WKH 0RQWH &DUOR ZRUN :KHQ WKH WKLUG RUGHU WKHRU\ LV UHFDVW LQ WKH IRUP RI D >@ 3DG DSSUR[LPDQW KRZHYHU WKH WKHRU\ DJUHHV ZLWK WKH 0RQWH &DUOR UHVXOWV XS WR \HD f $V LV WKH FDVH IRU EXON IOXLG WKHUPRG\QDPLF SURSHUWLHV WKH UHDVRQ IRU WKH VXFFHVV RI WKH 3DGH LV REVFXUH LW JLYHV FRUUHFW UHVXOWV LQ WKH OLPLWV RI KLJK DQG ORZ DQLVRWURSLF VWUHQJWK DQG DSSDUHQWO\ LQWHUSRODWHV FRUUHFWO\ EHWZHHQ WKHVH OLPLWV

PAGE 291

7KLV 3DG WKHRU\ KDV EHHQ XVHG WR SUHGLFW VXUIDFH WHQVLRQV IRU SXUH SRO\DWRPLF IOXLGV $ VLPSOH WR XVH FRUUHODWLRQ IRU SXUH IOXLG VXUIDFH WHQVLRQV KDV EHHQ GHYHORSHG EDVHG RQ WKH SHUWXUEDWLRQ WKHRU\ )XWXUH ZRUN WR EH GRQH RQ WKH SXUH IOXLG WKHRU\ LV WKH UHSODFHPHQW RI WKH )RZOHU PRGHO ZLWK PRUH UHDOLVWLF H[SUHVVLRQV IRU WKH UHIHUHQFH IOXLG GLVWULEXWLRQ IXQFWLRQV $Q REYLRXV UHDOLVWLF FKRLFH ZRXOG EH WKH XVH RI LQWHUIDFLDO GLVWULEXWLRQ IXQFWLRQV REWDLQHG IURP FRPSXWHU VLPXODWLRQ VWXGLHV 7KH VLQJOHW GLVWULEXWLRQ IXQFWLRQ 3T]Af KDV EHHQ REWDLQHG IRU D /HQQDUG-RQHV IOXLG >@ WKH UHIHUHQFH IOXLG XVHG LQ WKLV ZRUN +RZHYHU WKH LQWHUIDFLDO SDLU FRUUHODWLRQ IXQFWLRQ J4]SUHVHQWV PRUH GLIILFXOWLHV GXH WR LWV KLJKHU GLPHQVLRQDOLW\ DQG WKH VORZ UHOD[DWLRQ WLPHV LQKHUHQW LQ LQWHUIDFLDO SKHQRPHQD 7KH WKHRULHV IRU SXUH IOXLG VXUIDFH WHQVLRQ DUH DW D VWDJH ZKHUH H[SUHVn VLRQV IRU DQ LQWHUIDFLDO SDLU FRUUHODWLRQ IXQFWLRQ JA]AUAf RU FA]AUAf ZRXOG EH RI JUHDW YDOXH 7R H[WHQG WKLV SXUH IOXLG WKHRU\ WR PL[WXUHV RQH LV IDFHG LPPHGLDWHO\ ZLWK WKH SUREOHP RI REWDLQLQJ WKH LQWHUIDFLDO GHQVLW\ SURILOHV IRU HDFK FRPSRQHQW LQ WKH V\VWHP )RU PXOWLFRPSRQHQW V\VWHPV WKLV LV D IRUPLGDEOH SUREOHP UHTXLULQJ WKH VLPXOWDQHRXV VROXWLRQ RI LQWHJUDO RU HYHQ LQWHJURGLIIHUHQWLDO HTXDWLRQV (YHQ LQ WKH XVXDOO\f VLPSOH FRQIRUPDO VROXWLRQ WKHRU\ DSSURDFK WKH DYHUDJH FRPSRVLWLRQ RI WKH LQWHUIDFLDO UHJLRQ LV UHTXLUHG 2Q WKH H[SHULPHQWDO VLGH RI WKH PL[WXUH SUREOHP WKHUH LV D SDXFLW\ RI H[SHULPHQWDO VXUIDFH WHQVLRQ GDWD IRU HYHQ ELQDU\ PL[WXUHV RYHU WKH ZKROH FRPSRVLWLRQ UDQJH DQG PXFK RI WKH SKDVH GLDJUDP

PAGE 292

7KHRU\ IRU WKH ,QWHUIDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOH RI 3RO\DWRPLF )OXLGV $ ILUVW RUGHU SHUWXUEDWLRQ WKHRU\ KDV EHHQ GHYHORSHG IRU WKH LQWHUIDFLDO GHQVLW\RULHQWDWLRQ SURILOH S]ARAf IRU SRO\DWRPLF IOXLGV 8SRQ LQWURGXFWLRQ RI D 3RSOH UHIHUHQFH WKH ILUVW RUGHU WHUP YDQLVKHV IRU PXOWLSRODU DQLVRWURSLHV EXW GRHV QRW YDQLVK IRU DQLVRn WURSLF RYHUODS RU GLVSHUVLRQ SRWHQWLDOV &DOFXODWLRQV RI S]ADAf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f IRU WKHVH VLPSOH DQLVRWURSLF IOXLGV LQ RUGHU WR WHVW WKH WKHRU\ f 7R DWWHPSW SUHGLFWLRQ RI S]Zf IRU PXOWLSRODU IOXLGV WKH WKHRU\ PXVW EH H[WHQGHG WR KLJKHU RUGHU 7KHUH DUH GLIILFXOWLHV LQ H[WHQGLQJ WKH H[SDQVLRQ KRZHYHU 7R REWDLQ WKH LQWHUIDFLDO GHQVLW\ SURILOH S]Af IURP WKH GHQVLW\RULHQWDWLRQ SURILOH S&]AZAf} RQH VLPSO\ LQWHJUDWHV S]AFRAf RYHU WKH RULHQWDWLRQ :KHQ WKLV LV SHUIRUPHG RQ WKH ILUVW RUGHU WKHRU\ RI &KDSWHU WKH UHVXOW LV PHUHO\ WKH UHIHUHQFH IOXLG SURILOH S4]Af ,W DSSHDUV KRZHYHU WKDW WKH GHQVLW\ SURILOH REWDLQHG IURP WKH KLJKHU RUGHU WKHRU\ ZLOO QRW EH WKH UHIHUHQFH SURILOH LH WKH SURILOH PD\ EH GLVSODFHG IURP WKH UHIHUHQFH SURILOH 34]Af

PAGE 293

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f WKH VORZ H[HFXWLRQ WLPH FRPSDUHG WR ODUJH PDFKLQHV KHQFH Ef WKH QHHG IRU D PLQLFRPSXWHU GHGLFDWHG WR VLPXODWLRQ ZRUN Ff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

PAGE 294

%HFDXVH RI WKH ODUJH DPRXQWV RI FRPSXWLQJ WLPH UHTXLUHG LQ PROHFXODU G\QDPLFV FDOFXODWLRQV DQ\ HIIRUW ZKLFK FDQ HYHQ IUDFn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n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

PAGE 295

H[SHULPHQW RU VLPXODWLRQ 7KH HYLGHQFH KHUH LQGLFDWHV WKDW DFFXUDF\ LQ VLPXODWLRQ YDOXHV IRU ZLOO DOVR UHTXLUH ORQJHU FDOFXODWLRQV ZLWK PRUH SDUWLFOHV ,Q DGGLWLRQ WR WKHVH SURSHUWLHV WKH FRHIILFLHQWV Jf U ff LQ DQ H[SDQVLRQ IRU WKH DQJXODU SDLU FRUUHODWLRQ IXQFWLRQ LQ WHUPV RI SURGXFWV RI VSKHULFDO KDUPRQLFV RI WKH PROHFXODU RULHQWDn WLRQV LQ WKH LQWHUPROHFXODU IUDPH KDYH EHHQ GHWHUPLQHG 7KHVH AAP r r J S U f KDYH Df EHHQ UHODWHG WKURXJK LQWHJUDOV S 7 f ]P A WR WKH HTXLOLEULXP SURSHUWLHV GLVFXVVHG DERYH IRU VHYHUDO DQLVRWURSLF SRWHQWLDOV DQG Ef EHHQ UHFRPELQHG LQ WKH VSKHULFDO KDUPRQLF H[SDQVLRQ WR REWDLQ J UAXARA f ZKLFK LQ WXUQ ZDV XVHG WR VWXG\ ORFDO IOXLG VWUXFWXUH 7KH JA A PAULA FRHIILFLHQWV SURYLGH D QHZ DYHQXH IRU H[SORUDWLRQ RI HTXLOLEULXP SURSHUWLHV DQG KHQFH DUH RI FRQVLGHUDEOH YDOXH HYHQ LI WKH VSKHULFDO KDUPRQLF H[SDQVLRQ IRU J&UAAWAf GLG QRW FRQYHUJH >@ 7KH VHULHV IRU J UAAXAf rQ ADFW! VHHPV WR FRQYHUJH ZHOO IRU WKH PRVW SUREDEOH SDLU RULHQWDWLRQV WKRXJK WKH FRQYHUJHQFH LV TXHVWLRQDEOH IRU WKH OHDVW SUREDEOH RULHQWDWLRQV 6WXG\ RI 6AUDfO:A VARZV Wn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

PAGE 296

WHQG WR H[KLELW VTXDUH SDFNLQJ UHLQIRUFHG E\ WKH WHH RULHQWDWLRQ EHWZHHQ DGMDFHQW PROHFXODU SDLUV 7KH DYDLODELOLW\ RI YDOXHV IRU WKH JJ U f FRHIILFLHQWV n P ZLOO XQGRXEWHGO\ VSDZQ QHZ WKHRUHWLFDO HIIRUWV LQ OLTXLG SK\VLFV DV LQGHHG LW DOUHDG\ KDV >@ ,W LV DOVR OLNHO\ WKDW RWKHU H[SDQVLRQV DSSURSULDWH IRU PRUH VWURQJO\ DQLVRWURSLF PROHFXOHV WKDQ WKRVH FRQVLGHUHG KHUH ZLOO EH VWXGLHG H[DPSOHV DUH H[SDQVLRQV LQ HOOLSVRLGDO KDUPRQLFV >@ DQG H[SDQVLRQV RI WKH VLWHVLWH SDLU FRUUHODWLRQ IXQFWLRQ )LQDOO\ WKH DQDO\VLV RI WKH PROHFXODU G\QDPLFV FDOFXODWLRQV SUHVHQWHG KHUH LV RQO\ KDOI WKH VWRU\ 7KH DSSURSULDWH GDWD DUH VWRUHG RQ PDJQHWLF WDSH IRU IXWXUH DQDO\VLV RI WLPH GHSHQGHQW SURSHUWLHV LQFOXGLQJ D KRVW RI WLPH FRUUHODWLRQ IXQFWLRQV WKH YDQ +RYH GLVWULEXWLRQ IXQFWLRQ DQG WKH G\QDPLF VWUXFWXUH IDFWRU 6WXG\ RI WKHVH WLPH UHODWHG SURSHUWLHV PD\ SURYH WR EH RI PRUH YDOXH WKDQ WKH HTXLOLEULXP SURSHUWLHV VLQFH PDQ\ RI WKHVH DUH PRUH VHQVLWLYH WR WKH LQWHUDFWLRQ SRWHQWLDO WKDQ DQ\ HTXLOLEULXP SURSHUW\ 7KH SRWHQWLDO WKHQ EULQJV XV EDFN WR VTXDUH RQH IRU LW LV WKH REMHFWLYH RI VWDWLVWLFDO PHFKDQLFV WR FKDUDFWHUL]H IOXLGV LQ WHUPV RI FRQWULEXWLRQV IURP WKH GLVWULEXWLRQ DQG LQWHUDFWLRQ RI WKH PROHFXOHV LQ WKH V\VWHP 7KH VSKHULFDO KDUPRQLF H[SDQVLRQ LQ WKH SDLU WKHRU\ RI IOXLGV DW OHDVW JLYHV D ILUP KDQGKROG RQ WKH SDLU GLVWULEXWLRQ IXQFWLRQ IRU SRO\DWRPLF IOXLGV 7KH LQWHUDFWLRQ SRWHQWLDO KRZHYHU UHPDLQV HOXVLYH ,W ZLOO UHTXLUH D JHQWOH PDUU\LQJ RI WKHRU\ H[SHULPHQW DQG VLPXODWLRQ )LJXUH f WR UHYHDO DQG TXDQWLI\ WKH SRWHQWLDO IURP D VWDWLVWLFDO SK\VLFV DSSURDFK

PAGE 297

$33(1',&(6

PAGE 298

$33(1',; $ (;35(66,216 )25 7+( $1*/( $9(5$*(6 ,1 (48$7,216 f 72 f 7KH DQJOH DYHUDJHV X Af! X Af! X fX f! D fA D fA DY DY ::: N N N X fX fX f! ZKLFK RFFXU LQ (TXDWLRQV f f PD\ EH DY D D M-LrHYDOXDWHG E\ VXEVWLWXWLQJ IRU X LWV VSKHULFDO KDUPRQLF H[SDQVLRQ &f DQG SHUIRUPLQJ WKH DQJXODU LQWHJUDWLRQV RYHU VSKHULFDO KDUPRQLFV 7KH SURSHUWLHV RI VSKHULFDO KDUPRQLFV >@ VLPSOLI\ WKH SURFHGXUH 'HWDLOV FDQ EH IRXQG LQ UHIHUHQFH >@ 7KH UHVXOWLQJ H[SUHVVLRQV DUH r X f! D fA Y eOf U 9 U D ? r WW ? eOf efOf  (V$QOQf U $ QSQ VV Q r V ; (r$QLQf Ur 6 $Of r X f! D XfZ $$n$ U\\ e eOf enOf eOf ] $ $ W $ e en e [ ‘ e 6n 6 ? e en r ] e en e QOQQO VVnV B ,,, QQM e r eLM e ] n ] N AL f§ B L rBQVQVQVf ; (V$QQf(V$nQ_Qff(V,$QAf UA H$f

PAGE 299

XDfXDf!W D X WW L ene e•e en•e $O2 b (  9$ QA VV Q [ f (VeQf U (J eAeAf N QR I ; U 3e AFRV DLAe $f XrfXrfXrf! D XOf8 $$n$ eene af rAe RnAR eAene >eOfenOf@ efefeAf Rn e en e e eMB AeeneDODDf QOQQ VVnV QQfQn f (J $ QQf (J$QQS r QV r QVn r QV ; (VA$ ff§f§A U U U $f ,Q WKHVH HTXDWLRQV WKH QRWDWLRQ LV WKH VDPH DV WKDW IRU f WR f DQG LV GHVFULEHG XQGHU f DQG f ,Q (TXDWLRQ $f WKH IXQFWLRQ LV GHILQHG E\ WKH H[SDQVLRQ LQ (TXDWLRQ f ([SUHVVLRQV IRU WKH KDYH EHHQ HYDOXDWHG IRU PXOWLSROHV DQG DUH JLYHQ E\ >@ } B WW FRV FRV F FRV DAf $f > FRV FWM FRV FRV DA D FRV DAf@ $f

PAGE 300

WW WW >FRV FRV DAf FRV DA DAf FRV DAf FRV OLDA DAf FRV DA@ $f WW > FRV FRV DA FRV FRV FRV FRV ^FRV DA mf FRV f DAf FRV DA DAf`@ $f

PAGE 301

$33(1',; % &225',1$7( 75$16)250$7,21 $1' ,17(*5$7,21 29(5 (8/(5 $1*/(6 72 2%7$,1 (48$7,216 f $1' f ) ) 7KH
PAGE 302

)LJXUH %O 5RWDWLRQV 'HILQLQJ WKH (XOHU $QJOHV ^[

PAGE 303

7KH (XOHU DQJOHV ^-!;A WKXV VSHFLI\ WKH RULHQWDWLRQ RI WKH WKUHH ERG\ V\VWHP LQ WKH IOXLG $ IRXUWK DQJOH LV UHTXLUHG ZKLFK ZLWK WKH OHQJWKV UO! UA DQG A VSHFLILHV WKH VKDSH RI WKH WULDQJOH LWVHOI :H FKRRVH WKH LQWHULRU DQJOH DW PROHFXOH WR EH WKLV IRXUWK DQJOH +HQFH ZH FRQVLGHU WKH WUDQVIRUPDWLRQ GXfA GfA G-!A G FRV A A G FRV A GI! G FRV G\ G FRV %Of 7KH -DFRELDQ LQ %Of FDQ EH VKRZQ WR EH XQLW\ (TXDWLRQV f DQG f WKXV EHFRPH % 3/ GU U GUO U G FRV D X fX f! D D M-MM[ J U U U f ; JR/Yn ] %f \I AS n% 3/ GUnU GUO UO G FRV D X fX fX f! D Dn DY ML&JR/UOUUf ,] %f ZKHUH ] LU GI! G FRV WW G; ] PD[ %f DQG ] LV WKH YDOXH RI ] ZKHQ RQH RI WKH PROHFXOHV RU ILUVW PD[ f f FXWV WKH ] VXUIDFH ZKHQ WKH ULJLG WULDQJOH DSSURDFKHV WKH VXUIDFH IURP WKH OLTXLG VLGH

PAGE 304

% (YDOXDWLRQ RI ,QWHJUDO &RQVLGHU WKH WULDQJOH RI PROHFXOHV WR EH ORFDWHG RQ WKH OLTXLG VLGH RI WKH YDSRUOLTXLG LQWHUIDFLDO UHJLRQ RI D )RZOHU PRGHO WZR SKDVH V\VWHP 7KH VSDFHIL[HG IUDPH LV RULHQWHG VXFK WKDW WKH [ \ SODQH OLHV X LQ WKH LQWHUIDFLDO SODQH DQG WKH SRVLWLYH ] D[LV LV GLUHFWHG LQWR WKH YDSRU SKDVH VHH )LJXUH %f 9DOXHV IRU ] LQ %f DUH JLYHQ E\ WKH PD[ PDQQHU LQ ZKLFK WULDQJOH ILUVW FXWV WKH ] SODQH DV WKH WULDQJOH LV URWDWHG LQ WXUQ WKURXJK HDFK RI WKH (XOHU DQJOHV )LJXUH % VKRZ LPPHGLDWHO\ WKDW ] LV LQGHSHQGHQW RI -f VLQFH URWDWLRQ RI WKH WULDQJOH DERXW ]A GRHV QRW FKDQJH WKH ]SRVLWLRQ RI DQ\ SDUW RI WKH WULDQJOH +HQFH %f UHGXFHV WR ] WW G FRV f WW G; ] PD[ %f 5RWDWLRQ RI WKH WULDQJOH WKURXJK ; JLYHV WKUHH SRVVLEOH YDOXHV IRU ] GHSHQGLQJ RQ ZKLFK PROHFXOH ILUVW FXWV WKH ] SODQH PD[ ZKHQ PROHFXOH FXWV ] ILUVW ] ‘ PD[ U FRV H ZKHQ PROHFXOH FXWV ] ILUVW %f ‘ BU FRV HL ZKHQ PROHFXOH FXWV ] I LUVW 7KXV %f EHFRPHV WW ] G FRV G; aUL FRV LA G; U[ FRV f r %f ZKHUH [A! ;f DQG ; DUH WOLH YDOXHV RI [ ZKHQ PROHFXOH RU ILUVW FXWV WKH ] SODQH UHVSHFWLYHO\

PAGE 305

)LJXUH % 5RWDWLRQV RI WKH 7ULDQJOH LQ WKH )RZOHU 0RGHO ,QWHUIDFH WR 'HILQH 9DOXHV IRU ] PD[

PAGE 306

7KH VSKHULFDO FRRUGLQDWH DQJOHV A DQG A DUH UHODWHG WR WKH (XOHU DQJOHV E\ WKH ODZ RI FRVLQHV RI VSKHULFDO WULJRQRPHWU\ FRV A VLQ FRV \ 2LAf %f FRV A VLQ FRV \ %f (YDOXDWLQJ WKH LQWHJUDWLRQ OLPLWV \A \e DQG \A LQ %f IRU WW JLYHV ; %,2f [ 7 DO WW DL LI DO Le fO %f ; FRV UO &6 DO U UA VLQ %f 7KH QHJDWLYH VLJQ LQ %f DULVHV EHFDXVH \A PXVW OLH EHWZHHQ WW DQG WW ZKHQ UA FRV UA DQG EHWZHHQ WW DQG WW ZKHQ UA FRV RWA UA 8VLQJ %f WR %f LQ %f JLYHV G FRV VLQ >U VLQ ; Df a U VLQ \A Ua UA %f WW ] ,QWHJUDWLQJ RYHU JLYHV

PAGE 307

=] WW WU VLQ [ D[f U VLQ [ a r a U@ %f 8VLQJ %f DQG WKH ODZ RI FRVLQHV RI SODQH WULJRQRPHWU\ %f UHGXFHV WR ,] UO U Uf %f &RPELQLQJ %f ZLWK %f DQG %f JLYHV 22 < % % 3/ GUO UL GUO U G FRV D X fX f! D D ZL Wf ; JR/UOUOUfUO UO Uf %f % % 3/ GU UO GU U ; JR/UUUfUO U U` G FRV D X fX fX f! D D D DfL88 %f 7KH LQWHJUDWLRQ RYHU FRV PD\ EH WUDQVIRUPHG WR DQ LQWHJUDWLRQ RYHU Un 7KH ODZ RI FRVLQHV JLYHV U U G FRV GU U U UU %f 6XEVWLWXWLQJ %f LQWR %f DQG %f JLYHV f DQG f UHVSHFWLYHO\

PAGE 308

$33(1',; & 02'(/6 )25 $1,627523,& 327(17,$/6 2) /,1($5 02/(&8/(6 7KH DQLVRWURSLF SRWHQWLDO X FDQ EH H[SDQGHG LQ WHUPV RI SURGXFWV FO RI VSKHULFDO KDUPRPLFV RI WKH PROHFXODU RULHQWDWLRQV 7KH H[SDQVLRQ LV D VXP RYHU DOO WHUPV LQ =A =A DQG = H[FHSW =A =A = DQG KDV WKH IRUP >@ X D ULP! ^ XD$f &Of XD$f A ($QLQUf &&$MPAPf fA QOQ PAPP = A U? A A &f ZKHUH $ =A=A= Xf RULHQWDWLRQ RI YHFWRU BUA DORQJ WKH OLQH RI FHQWHUV IURP PROHFXOH WR PROHFXOH &$PAPPf D &OHEVFK*RUGDQ FRHIILFLHQW = WRf D UHSUHVHQWDWLRQ FRHIILFLHQW DQG @ 7KH VXSHUVFULSW r LQGLFDWHV D FRPSOH[ FRQMXJDWH 7KH FRHIILFLHQW ( LV D VWUHQJWK FRQVWDQW WDNHQ WR EH D VXP RYHU YDULRXV LQWHUDFWLRQV (A ZKHUH V UHSUHVHQWV PXOWLSROH RYHUODS RU GLVn SHUVLRQ 7KH U GHSHQGHQFH RI VXFK LQWHUDFWLRQV LV XVXDOO\ DVVXPHG WR EH RI D SRZHUODZ IRUP

PAGE 309

($Q/QUf e (A$MQAf V &f ZKHUH QJ ef RU IRU PXOWLSROH RYHUODS RU GLVSHUVLRQ UHVSHFWLYHO\ )RU D[LDOO\ V\PPHWULF PROHFXOHV (V$QAQf YDQLVKHV H[FHSW IRU QA Q ([SUHVVLRQV IRU (J IRU SDUWLFXODU SRWHQWLDO PRGHOV DUH JLYHQ LQ 7DEOH &O e 6LQFH Q Qf IRU D[LDOO\ V\PPHWULF PROHFXOHV LQ &f PQ VLPSOLILHV WR 'P2r;f WW eO &f DQG &f UHGXFHV WR D VXP RYHU WKUHH RUGLQDU\ VSKHULFDO KDUPRQLFV XD$f WW > eAOf eOf @ PLP @ HTXDWLRQ &f JLYHV H[SUHVVLRQV ZKLFK DUH DPHQDEOH WR FDOFXODWLRQ ([SUHVVLRQV IRU YDULRXV DQLVRWURSLF PRGHO SRWHQWLDOV LQ WKH LQWHUPROHFXODU IUDPH RI )LJXUH DUH JLYHQ LQ 7DEOH & ([SUHVVLRQV IRU WKH VDPH PRGHOV EXW XVLQJ WKH DQJOH \ RI (TXDWLRQ f UDWKHU WKDQ -fA IURP f DUH WDEXODWHG LQ 7DEOH &

PAGE 310

7$%/( &O ([SUHVVLRQV IRU WKH ([SDQVLRQ &RHIILFLHQWV ( IRU 9DULRXV ,QWHUDFWLRQ 3RWHQWLDOV IRU /LQHDU 0ROHFXOHV >@ *HQHUDO 0XOWLSROH 'LSROH'LSROH 4XDGUXSROH4XDGUXSOH 'LSROH4XDGUXSROH 'LVSHUVLRQ'LVSHUVLRQ 2YHUODS2YHUODS f (PXOWOnLUf mOf WWef f f 4/ 4e r ] OO (GLSROHUf O7f AU (TXDGUf I WWf 4U ('4Uf UUf \4U ( Uf WWf HLDUf DLV ( Uf GLV (GLVUf WWf HDUf ( Uf WWf HDUf RYHU ( Uf RYHU &f &f &f &f &,2f &OOf &f &f &f \ 4 DQG DUH WKH GLSROH PRPHQW TXDGUXSROH PRPHQW DQG GLPHQVLRQOHVV DQLVRWURSLF RYHUODS SDUDPHWHU UHVSHFWLYHO\ GLPHQVLRQOHVV DQG 4 6/U DQLVRWURSLF SRODUL]DELOLW\ DUH JHQHUDO PXOWLSROH PRPHQWV

PAGE 311

7$%/( & ([SUHVVLRQV IRU $QLVRWURSLF 3RWHQWLDO 0RGHOV LQ WKH ,QWHUPROHFXODU )UDPH RI )LJXUH 8''-f n >L6M& FL&M! LM 4 U XTTf r A FL FM LM Vr \4 U 8'4-f f 77 >FLFM ;f LM X LMf H RYHU U Y L>FI L 8GLVLMf e. U Y LMn >F L e U Y LMn IO F"F" ^VVF FF`@ L L L FF" f VVFF Ff@ M L  FM f @ F" @ BU F F FF WVVF L L &f &f &f &f &f FF`@ L GLSROH 4 TXDGUXSROH RYHU RYHUODS GLV GLVSHUVLRQ FRV A VA VLQ BA F FRV \ 4 DQG DUH WKH GLSROH PRPHQW TXDGUXSROH PRPHQW RYHUODS SDUDPHWHU DQG DQLVRWURSLF SRODUL]DELOLW\ UHVSHFWLYHO\ (T f LQ UHI >@ LV WRR VPDOO E\ D IDFWRU RI A (T f LQ UHI >@ LV WRR VPDOO E\ D IDFWRU RI FI (T f LQ UHI >@f

PAGE 312

7$%/( & ([SUHVVLRQV IRU $QLVRWURSLF 3RWHQWLDO 0RGHOV LQ WKH ,QWHUPROHFXODU )UDPH XVLQJ \ UDWKHU WKDQ S 9 >F F X L U LM FLf H > U A LU >F7 FI L n Ne U LM >FI F I D @ UL F X''OM! 44 '4 RYHU GLV N H U Y L&f ^F\f FL&-@ &f F\ff@ &f @ &f @ &f F" F"F" ^F\f FF`@ L L GLSROH 4 TXDGUXSROH RYHU RYHUODS GLV GLVSHUVLRQ F FRV F\f FRV
PAGE 313

7$%/( & 'HULYDWLYHV RI 9DULRXV $QLVRWURSLF 3RWHQWLDOV IRU (YDOXDWLQJ WKH )RUFH DQG 7RUTXH IURP (TV f DQG f 3DLU ,QWHUDFWLRQ U X F F \f '' 44 X '' X 42 U LM \ F f B4B U f§ >F FsF U LM FAF\f U LM A >F \f F F @ U '4 XG4 U LM \4 U7 LM >F\f F FF @ M >F B F @ / L LU LRYHU X RYHU U LH U Y F GLV X GLV U Ne n‘ULM F e. U A D AULMn >F f§F F F &M F\f @

PAGE 314

$33(1',; (;35(66,216 )25 \A
PAGE 315

7$%/( ', ([SUHVVLRQV IRU \ IRU 9DULRXV $QLVRWURSLF 3RWHQWLDOV IRU /LQHDU 0ROHFXOHV 'LSROH'LSROH 4XDGUXSROH4XDGUXSROH 'LSROH4XDGUXSROH 2YHUODS2YHUODS \ $f 3L 'LVSHUVLRQ'LVSHUVLRQ \$f \$f 'LVSHUVLRQ4XDGUXSROH 'LVSHUVLRQ2YHUODS \ f r 3W 7 3/ r r \ 7 'Of r )r \DAf WW 3/ r n LR 7r T 'f r )r A$f 77 3/ r r \ 7 r 4 'f r \)!f WW 3/ r 7 'f r f WW 3/ f f§ r 7 L @ 'f A$f r WW 3/ 7r r 4 'f r \Drf WW 3/ N -A 'f

PAGE 316

7$%/( ) ([SUHVVLRQV IRU \A$ IRU 0XOWLSROH 3RWHQWLDOV IRU /LQHDU 0ROHFXOHV )r
PAGE 317

7$%/( ) ([SUHVVLRQV IRU \Rf IRU 9DULRXV $QLVRWURSLF  $ 3RWHQWLDOV IRU $[LDOO\ 6\PPHWULF 0ROHFXOHV 2YHUODS2YHUODS \Ef 'LVSHUVLRQ'LVSHUVLRQ \Ef 2YHUODS'LVSHUVLRQ :f r WW 3/ U f§ f§ r 7 r WW 3/ f§ f§ 7 r LW 3/ 7r /
PAGE 318

7$%/( ([SUHVVLRQV IRU \ ) % IRU 0XOWLSROH 3RWHQWLDOV IRU /LQHDU 0ROHFXOHV A% += L f WW WWn_ S A \ .
PAGE 319

$33(1',; ( 7+( ,17(*5$/6 .<0n$QQnQf $1' /@ 7R IDFLOLWDWH LQWHUSRODWLRQ < < EHWZHHQ WKHVH VWDWH FRQGLWLRQV WKHVH UHVXOWV IRU DQG / KDYH EHHQ ILWWHG WR DQ HPSLULFDO HTXDWLRQ RI WKH IRUP eQ_. @ DQG E\ )O\W]DQL6WHSKDQRSRXORV ALW DO > @ IRU DQG .eeneQQnQf DQG YDOXHV RI WKH FRQVWDQWV IRU WKHVH KDYH EHHQ WDEXODWHG 7KH FRQVWDQWV LQ (Of KDYH EHHQ GHWHUPLQHG E\ D OHDVW VTXDUHV ILW DQG DUH JLYHQ LQ 7DEOH ( 7KH LQWHJUDOV .< DQG /< IRU S 7 DQG r r S 7 GR QRW IDOO RQ DQ\ VPRRWK FXUYH WKURXJK WKH RWKHUV DQG DUH WKHUHIRUH RPLWWHG IURP 7DEOHV (O DQG ( DQG LQ ILWWLQJ WKH FRQVWDQWV LQ 7DEOH (f 7KH PD[LPXP GHYLDWLRQ IRU WKH SUHGLFWLRQV RI (TXDWLRQ (Of LV OHVV WKDQ b RI WKH YDOXHV LQ 7DEOH (O IRU WKH < < LQWHJUDOV DQG OHVV WKDQ b RI WKH YDOXHV LQ 7DEOH ( IRU WKH / LQWHJUDOV

PAGE 320

7$%/( (O r S 7KH ,QWHJUDOV .AeeneQQnQf IRU 3XUH )OXLGVA r 7 N< < \ N< <,Q WKH WDEOH VXEVFULSWV RQ DUH 6//n DQG VXSHUVFULSWV DUH QQnQ 7KXV N N
PAGE 321

7$%/( ( 7KH ,QWHJUDOV /
PAGE 322

7$%/( ( 7KH &RQVWDQWV LQ (TXDWLRQ (O ,QWHJUDO $ Q % & Q Q Q ( Q ) Q MU \f Y\f U L
PAGE 323

$33(1',; ) ([SUHVVLRQV IRU WKH 6SKHULFDO +DUPRQLF &RHIILFLHQWV Jt e PUf LQ (TXDWLRQ f JAUnf A JA & JAUnf Z OZ nUnf A& & JAUf JAUA6OFO6F&! JA nUAVOVAF nf! JUf UfF $ FB! JU n JUf BZ Uf FO & A& JUf U L f f§ Uf6O&OLFO -` 6&&! JUnf JUf A Z 1 Z A UfO & 6 & f ;f! BZ B Uf & & r & ` JUf AUnf 6OFOFO 6& F & JUf JUf OZB AVO A& 6 F A F A f ,a JUnfVOFO 6& AF FA! JUf 0 F JUf6O6 nF F A F FRV V VLQ F FRV L L Lf

PAGE 324

JUf JUf JUf JUf JUf A JfUfFM FM JUfFM FM f Fr Mf! f§A JUfFM FM FM fFM \ FM _Mf! A JUfFM FM FfA A O26Fr J4UfFFFFA f!

PAGE 325

7+( ,17(*5$/ $33(1',; 86(' 72 &$/&8/$7( 7+( $1*8/$5 &255(/$7,21 3$5$0(7(5 )25 48$'5832/(6 7DEOH JLYHV YDOXHV IRU WKH LQWHJUDO ZKLFK DULVHV LQ WKH VHFRQG RUGHU SHUWXUEDWLRQ WKHRU\ IRU WKH DQJXODU FRUUHODWLRQ SDUDPHWHU ‘N N *f IRU TXDGUXSRODU IOXLGV > @f 7KH LQWHJUDOV fS 7 f DUH GHILQHG ] QQ ; E\ N ,RS 7 f QQ N N GUO U GU rQf r r r r r U U f r rQnOf r UO Un ; JRUUUf 3eFRV D` *Of ZKHUH 3A LV WKH RUGHU /HJHQGUH SRO\QRPLDO DQG LV WKH LQWHULRU DQJOH DW PROHFXOH LQ WKH WULDQJOH IRUPHG E\ PROHFXOHV DQG ,Q FDOFXODWLQJ IURP *Of WKH VXSHUSRVLWLRQ DSSUR[LPDWLRQ LV HPSOR\HG IRU WKH WULSOHW FRUUHODWLRQ IXQFWLRQ DQG 9HUOHWnV PROHFXODU G\QDPLFV UHVXOWV DUH XVHG IRU WKH SDLU FRUUHODWLRQ IXQFWLRQV J4U f W @ 7R IDFLOLWDWH LQWHUSRODWLRQ EHWZHHQ WKH VWDWH FRQGLWLRQV LQ 7DEOH *O WKH UHVXOWV IRU KDYH EHHQ ILWWHG WR DQ HPSLULFDO HTXDWLRQ RI WKH IRUP eQ _O $ Sr eQ 7r % Sr *f N N N &S eQ 7 'S N ( eQ 7 )

PAGE 326

_O UHSUHVHQWV WKH PDJQLWXGH RI ,A VLQFH A LV QHJDWLYH 7KH FRQVWDQWV LQ *f KDYH EHHQ GHWHUPLQHG E\ D OHDVW VTXDUHV ILW DQG DUH $%&'() *f 7KH PD[LPXP GHYLDWLRQ IRU WKH SUHGLFWLRQV RI (TXDWLRQV *f DQG *f LV fN OHVV WKDQ b RI WKH YDOXHV LQ 7DEOH *O 7KH LQWHJUDO IRU S r r r 7 DQG S WKURXJK WKH RWKHUV 7 DQG DUH WKHUHIRUH GR QRW IDOO RQ DQ\ RPLWWHG IURP 7DEOH VPRRWK FXUYH *O DQG LQ ILWWLQJ WKH FRQVWDQWV LQ *ff

PAGE 327

7$%/( 7KH ,QWHJUDO A IRU 3XUH )OXLGV r S r 7 r

PAGE 328

$33(1',; + 9$/8(6 )25 7+( Je  PUf &2()),&,(176

PAGE 329

7$%/( +, 9DOXHV RI J44Uf RUf IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG DW .7H S2A 4FDf 56,*0$ *f *f *f *^f *f *&f f & m O 2& & & & & & &22 & &2 & &O &O & & &22 &2 & & &O &2 & 2& & &2 &2 &2 &2 & & &2 &O &22 &2, & & &2

PAGE 330

7$%/( +, &RQWLQXHGf 6,*0$ *f *f *f *f *"f &2f & 2& O &2 & & & &2 &2, & & &2 & & &22 & &22 & &

PAGE 331

9DOXHV RI 5UOf 6Uf IRU WKH )OXLG RI 7DEOH +, 0 7 222L2L22f§LLrf§O22222222222222222R2222222222222L222222 222222222222222222222222222222222222222222222222222222222 H! RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR ,, , , , , , , , , , , , , ,,, , `‘ 2LUL2/QQLQII2WnY nLU?OYLWf.f23?2YI2LA$MUfL!1,f-2Y8LQRF?-Y L1U2?OnO RRR1-QLQWWI?ORRRRRBRrLRRRRRRRRRRRRRRRRRRRRRRRR f§ RRRRRrR 222222222222222222222222222222222222222222222222222222222 RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR ,, , , , , WLOO R `f RAWRRLPQ?L0QAIWMLLRRRRLY1RF1RLW1UnL$LQPfLQLDQWRZmRUL?M0n0W FR FR Q WR ?OnLfIIfW0?,2D_+2O0?_L02222222222222222+22+UI2222222 RRRRR 222222222222222222222222222222222222222222222222222222222 222222222222222222222222222222222222222222222222222R22222 ,,, , W , RR RR mU2&21LUf12OIfUHRn
PAGE 332

U?M LY IY! UY UR UR UY L? 7$%/( + &RQWLQXHGf N6,*0$  r & f  f f f ER L &  & *& & f & 24FO RRRL f 2& & f & & X f *f & & 222G 22R &

PAGE 333

R 0f Y2 LURLVnLUfLQAUYMRnQLWfnY2DRnDn LLnVGn2F?Mf&0RLQ121n2RF0n;fFRWI0FRLQ&7!Rn02W 2nn U0nr+2222 aLL 222222222222222222222222222222222222222222222222222222222 R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR , , ,, OLOL , , OLOL ,, 7 f+ D Uf§W IH f OO R R V R 2L02Dnnnn1fA121LnnQ;f/U!&?-A1UL-Ln02LI?,2I?-0fYW?-&?-m2LIfV`2Ynnf0FU2 UM2nnr+! RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR , , , , OLOL , OLOLO ,,, R Y2 2&02620nn21A&06&2&0Qn&22nFW2D nnQr&nY&7n6nn02!rW7Y27}n2UfnnYn!?-n$-6 FL f§ LaLa 222222222+22222222222 f§ 2L22222222 222222222222222222222222222222222222222222222222222222222 R 222222222222222222222222222222222222222222222222222222222 ,,, , , , , , , ,,, ,,, ,, ,,, ,,, UV R &2 &2 & Z &4 + &6 Uf§, r 97 f YS &VO U+ U VU R &2 FX =U+ R R Y2 .' 1&2Y'Y2Wn?MQR&AUAO2r1A-R12-&22U2I?LAURLQnWU21&A12n1-LUfA2&2}2&?Mf2&YMfr&?-&?-LUY-n\fFn?-1 RRRmRRRmQ+}RRRRRRrARRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR 222222222224222222222222222224222222222222242222222222222 222222222222222222222222222222222222222222222222222222222 , I W ,,, , , , , , , , , , , r [W R0n;fDnL!n!nr0n7nW0LfLUfRRLUfFR1YRRF2FR f§ mn0RLQRL2nRrnMInRUFM?L2YYUfRRLQFW20 rm f§ MmRLf 222222222222222222222222222222222222222222222222222222222 222222222222222222222222222222222222222222222222222222222 ,, , , , , ,,, ,, ,, , , , , , , , R 6n ‘V R Q LLf Q n R R Q WR U R UM LQ R K FR K R R A fL R Rn K !R Q Q WXQ LQ F R LIf K L Q R V R R R V cR V 2 2222n2&02 &?M f§ O22L22 222222 222222222222222222222222222222222222222222222222222222222 222222222222222222222222222222222222222222222222222222222 ,, , ,, ,W ,,, ,,, ‘ f§ LQ V ,; R LQ R R LQ R P R LQ R P 2 P R LQ 2 P R Q R P R LQ R LQ R P R F?O LQ UA R &0 P 1 R &0 LQ 1 2 0 LQ 1 R 0 Q Q R U?LQ 1 R $P 1 A Q ;f 2n 2n 2n 2n R R R 2 f§m f§L f§m $0 &0 QM U2 Q R V ‘V R[LRARLQRLLfR.LRLUfRLQRrQRLQRLIfRQfRDLRLQRPR RF?MRVRn0WQ120LIf1RF?ML212nMLQAR?LnQ1Rn?MLQ1R LUfLQPLQFRRR!2111AnQDfnWfFQRnF?nL7nRRRRnnF?M 2R2R2222rrA f§ f§ f§r f§ f§ f§K f§ f§f§ f§ f§ f§ f§ &?-n9n?OUn2I?-?-&-&?0

PAGE 334

2M LY U LY WY UY UY UY UY UY UY WY LY UY WY UY UY WY WY WY WY WY WY UY UY UY UY WY WY UY WY WY -f 1 2n nLfn2Y2!RDDDMDVM01L12nnnLXLXLXLA!n!AX8L88LYWY. rm R !L WY R YM RL WY R V X LY R F LY R QL XL LY R VL XL UY R J XL UY R VL XL WY R 28O2L28-O28f2XL28L28,2WI28O288,28XL2-OO LOO ,,, 22222222222222222222222222222222 F 22222222&22222222222222&22222222 2222222222222&222222&2 RRRRRRRF X YL!XWYUYLRLY-8!IY}XLXLRRRnUYRWYnf8nL2 UY La R m UR UY X R f ,, OLW FFRRFRRRRRFRRRRRRRRRRRRRRRRRFFRF &L rrrr RRRRRRRRRRRRRRRRRRRRRRFRFRRRRRRR ‘7n RRRRRRRFRRRRRRRR!RRRRRR!RRRRRRRr 8r&nV_&,f8O9W98-!fA!nnAI9!r!222-I9nY-An8-8nf!88O ,,, ,,, ,, L 22222222222222222222222222222222 FQ 2n RRRFIFRRRRtRRRRRRRRRRRRRRRRRRFRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRFRRF R Z ,, 2222222222&22222&222222222222222 F 2n UY 22222222222222222222*22222222222 2222222&2222&2*222222222*22&2*22 R RM WQ R r 2nXRXWYUYR!!fX XSUYUYRRRn ARLWYUYLYURDSWY 9r ,,, ,,, ,,, ,,, LOO 2*22222222222222&222222222222222 &O ‘r R &222222&2222222222222&2222222222 22222222222222222222222222222222 R fI!!X-XUY2LRX/Mr!!Xc3nUYrAnRFMXLRLYrLY}RMSUYXLVDf ,, ,,, 22222222222222222222222222222222 F R F RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR 2*2 2 22 2 2 2 2 2 2 RR r R 2 2 2 2 R R R RRRRRR D YJFMR KXL!UYr!LYAr!YXR*FQXXnR6* XRRA2n&GXL 7$%/( + &RQWLQXHGf

PAGE 335

7$%/( + 9DOXHV RI J JQQ7BLRf IRU /HQQDUG -RQHV 89-8 88 SOXV 4XDGUXSROH )OXLG ZLWK N7H SR DQG 4HD f 56,*0$ *^ f f f *f &2f & & & & & & &O & RRH & & O & O f O & & &2 &2 &2 &2 &O &O & &

PAGE 336

7$%/( + &RQWLQXHGf 36,*9$ *f 2f f *f *&f *&f & &22 & &O & & &2 & & &2 &2 & & & & & &O &

PAGE 337

AAARFRR2YX?RYIfDf2DRYLnYL!fmMR2nn88MLDL/QSA!W!L8XRMLYLURURU?mn!!RRRRYRYfY/fn&&'D-D RVMXLLYRYLL7 L?MR!QLL9nRVLFQURR1XnU?fRAXLURR1LFQU?RVXL0R!MLLYRJXnUARA8fUYL2n-8nnUnM2YQ-fUYMRAWFQU?R RMLRLURRLRXLRDcRQRXLRXLRLURRLRXLRRL2LR7n28LRnMLFLQRLRWQRLQRXLRXLRXLRWURXLRXLRMLRUR 6 L & OLOLO L , , O , , , 2222222222222222222222222222222222222222222&2222222222222 RRRRRRRRRRRRRRJRJRTRRRRRRRRRRRRRRRRRRRRRRR f§ fIYUXURnRRRRFRRF RRRRR!RRR+Lf2L+23RRLRLRRRFLRRRRFRFRR+L?LnLFWnnnLYLLUnL!RR !rFAUR[&f&.U?0nArRI!SLY!RXR!XrLXDRRnFL!M!S!XLYLSUYYRALYFQXDfD81nL&nAXLD!LR R ‘S ,9 R , , , , , , 22222&22&22222&n222222222222222&2222222222222222222&22&22& 2222222222222222222222&22222222222&2222222222222222222222 RFRRRFRRRFRRRRRn!fnnf2RRRAr2nRRRRRRRRRRrRRRrrUYfADALnA&-fF}LrRR YA!SI!XLWUDrAn08frYLRWY!DLR!I8L!MnRFURmMU?nRrYFDnSn!RIYLYR-RF[nL9f&'L?!-MMnI!0RUF!M2 F S UR , ,, , , , , , L , , , , RRRRRRRRRRRRRRRRRRRRRRRFRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRWRR 222L222222222222222222222222222222222222&,9U9Lr,9n,9!f!!f§RR 2'8LArf2QRXLYF'ALXLAXLDn82I!0UYLnnAXrMUYXWQ!L!2M8.-XLSUR2MA0F!MAFfRXL!D8XL2nJAMA &L t UY! t Kr F IO! HQ R }K 24 R 1! "a ,f§r OV! + O + f FU 4 : 8Kr U f f! UR P 2 21 UH VH IW 9Q WY! ,, , , , , , , RRRRRRRRRRRFRFRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRFRRRRRRRRRRFRRRRJRRRRRRRRnnP&M2MO URRRRRRRR 2222222222nn++R2222&222L22222222nn2n2WR0,;W8OOXD8LO0O.On2 fA!F!0DDn}&%AAURrXnDXX}LRXXLnDL!W%2 n-VXUXYRrDRXL!IYnXL2nD!YR2nA!!RALML*2nUY F S S R OLOL , L , , , , , 22222222&222222222222222222222222&*22*2222222222222222222 2222222222222222222222222222222*22222222222*!r,9I9_?m 22222A*.*2222222!2222n222222222n2r!222!}UYAnJYLr68_O-23n22 *0WYfN2W[MM8LR!U?`nf8LDfURRFUUX!fFUn!nn!XLRcRMUYfL?fAArUYRRRAYRAQ?n&'ARWR!L)!0XLXLDYFI!nn F LQ R ,, ‘\ ,, rUL ,f§ F 0 FY R LQ L , , , ,,, OLOL 2222222222222222222&22222222&22222*22222*2222222222222222 "L 22222222222222222222222222&2222&2222222222222n2n222222222 !RR2R2R2222222222222222222++*2&2*2L2+L0AX8+,}OLUXL28O02* f! ,9

PAGE 338

8 ,9f ,9 I? 1 ,9 ,9 ,9 W? ,9f ,9 0 ,9 9 ,9 ,9 UR UY UR UY 9 ,9 UR UR ,9 IY ,9 UY UY UY IY ,9 r r f } f r f f f f f f f 9 f f f f } 2n8nYe?&&QDF&&;nL!-aA 2 2n 2n 8L 8  8f S X! 8M 8L 2R U UR UR R [L Q LY R M X! UY R YL FQ UY! R UY R 2n UY R FQ 0 R fYO 8O UR R Q8O UR 28,2:28,22,22,22 F FQ R FQ R FQ F 8 R FQ F FQ R FQ R FQ R FQ R FQ ,, OLOLO ,,, ,,, , , ,,, 222222222222&2222222222222222222 22222222222222222222222222222222 222&-2222222222&&222222222222222& FQS!UYIYFQFQ!LFQSSUYXD!3LYUYXRLYSLY!1L3n3UY!XLDn ;, ? 8 f! F &O S I9 R , , OLOL OLOL , , RFRRRRRRDRRRRRRRRRRRFRRRDRRRRRRR RRRRRRRFRRRRRRRRRRRRRRRRRRRRRRRR 22&222&2222222222222222222222222 XL8nRLUX8LUYRX!XL38-8UXLYUYXrUYf!3R}UYnX-LYFQXRUXS H S 0 , , ,,, ,,, RRRRRRFRRRRRRRRRRFRRRRRRRRRFRRRR F 22222222222222*2222222222&222222 2222222*2 RRRRFFRRRRRRRRRRRRRRRR RLURUZLXMUY!nn2L!LXLRURUYL1fL?MUYfLYnrnXRMnf S .f WY ,,, , , , ,,, , , , , 2222222222222*2&2222222222222222 *2222222222222222*22&2222222222& RRRRJR!!RRRRRRFRRJFRRRRFRRRRR!RR &'!8Y&&Q8nn8L&,!!S8-&Q8Lnn-83I938f,93rn-!!Dn?3an-3 F S S R , , ,,, , , , L , ,,, *D222222222222&22222222222222222 &O 2222222222&22222222222222*222222 222222222**22222*22222222222,*22 WYIXUYXMnRrn8LIYR!8L2LYIYFUR!FQr}FQWWm2nrrUYL8-8 S S , , , , ,,, ,, RRRRRRRRRDRRRRRRRRFRRRRRRRRRRRRR Q 222&22222&&222222222222222222*22 **222222A22*222222222222222 2222 88+nL8On0n2208308O8O2n8:0n-3332L31 ff§ A &' ff§ S S 9 7$%/( + &RQWLQXHGf

PAGE 339

UYLYUFURUAUX00L9n f§ f§ f§ f!fffff§f!ffnf!!f‘ A!!!!RRRR!L!2nne}DDF[!WFAMAMAMnDnn8f8LX!FLWA!nm8XXcUYL?MIYUYf!! f§ f§ RRRRY!nefYFnFL!RFFLLF RYMXLL?MR18LLY2nMXLLYRnmXLU?Rn1LMLUR2nMFPYRAMLU?fn!MZR!fA 2nVLLUYRnYLXL.RVRLL12nYLLL[L2nYM2O:F R XL R XL R XL R WQ R LQ R HQ R ! R DL R R R R R R R X! R XL R WL R U R WQ R R HQ R HQ R XL R ML R XL R U R XL R FQ R P R X R ;n 1 rf§} Q L ,, ,,, OLOW WLOO ,,, RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR kRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR 2222222822/DWO2222 !&'2ARFU!!Ln&n8LfLnLY1RFUAn8LYRUYnR!RL2nRRL8YLRR UY L f§ R !f§VM FS P FG R R UY LY X YL UR I?! R F WR WR 2,,, , , , , , , ,,, , ,, 222222222222&222222*!222222&22222222222222222222222222222 R 222222222222222222*2222222222222222222222222222222222&222 f§f§f§ RRRRRR 2 f§ 2 f§ 2O922222 LrL1UYDn22RXLUnRRMLYXLn2RrRWRXLWRRALRL-MU?WR f§ UY f§ f§ XLLYp f§ XL p D WR p L?f0I!XLUYDDFLn&7! R f§m WR WR WR ,, OLOL ,, , , , , , , , , , RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRFRFRRRRRRRR2RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRR f§ RRRRRRRRRRRRR f§ f§ f§ R f§ RRRRRRRRRRRRRRRRRRRRIFZWRYLWR-LXLLYRRR 6WRW8 f§ 8WR8OG2n1-8n02+2nMUn-21L2n& f§ AWR228LU?!On2:m_nn2WRWRK-WR-I82nWRWR280!L8O--D!WRn:8O2 F 2n E R , , , , , , OLOL , , , , FRRRRRRRRRRRRRRRRRRRRFRRRRRRRRRRRRRRFRRRRRRRRFRRRRRRRRRRR 222222 RRRRRRRRRRRRRRRRRR f§ RRR f§ RRRRRRRR f§ R f§ RRRRRRR f§ RUYRRXLALFXLRR ARM2nRUY[UR!UA!RMn&n!RML?FUn!!RDRXL!RM!!U?M8MRRRLXL2nURLY2nn!URXL&/LFUn21n[1L3n &O 2n 0 R , , , , ,,, OLOL , , , , , , , RRRRRRRRRRRRRDRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRFRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRR f§ RR f§ f§ URL?L?MXL12naLnMWR f§ RR 8 0 Q_ : 2 8 f§ WR2Y,WRpkIn-A,!n&8!n8n f§ f§ f§ WR2WRWLWR8O2 f§ 8WR8&'228&22nnnWUL &n R K F f R +L 24 6 IW Kf§ WVO + WR FU WR +! 21 WR RQ + &2 WR P FF 2n 2R , ,, ,, OLOL , , ,, ,,, , , , , , , 222222222222222222222222&22222&22*RRRRRRRRF&2222222222222 2222222222222222222222222222222222222222222&22&2&22222222 RRR f§ f§RRRRF f§ RRR f§ RRRR f§ RR f§ RRRRRRRFFRR f§ RRRRURRR f§ WRf§f§WRWRXLUR f§ RR WRWRWRURRLRWRLYRn2n-XVWRRXLRWnMRRnnLIYRWR2-WRnWR1LRLRR*Wn-8LnRpp&M2RURUY XLWRUR f§ DRRR FQ FU R

PAGE 340

7$%/( + &RQWLQXHGf 56,*0$ r } r r r R & r & r } f r f & G *^ f & f *f & f§ f 22R *f & *f D

PAGE 341

7$%/( + 9DOXHV RI 6TTT&ARf a ATT&Af IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK N7H S DQG 4e4f 56,*0$ *222f *f *f *f *f *&f f & & & &2 & & &O &O & & & & & & & & & & & & &O & & & &2 &2 & & & & & &O & &

PAGE 342

7$%/( + &RQWLQXHGf 56,*0$ f I & & f f *f &2 f F 2 & &2 &O & &O & &2 &O & & &2 &2 &2 &2 &2, &22 R FRH & & 7 'XH WR D SURJUDPPLQJ HUURU WKH YDOXHV RI i444Uf LQ WKLV WDEOH IRU U D DUH LQ HUURU

PAGE 343

URLYI?fLYf YIWURLYUY f!f§!f§}!RF&RFRRR IMDD.LRFRRAL&nRLWWDDVVVV2n& 2n R P LQ XL R S S SSn82-8M8MWYLYLYUYnn!RRRRLe02n2YRF'D!DD R YM Xc LY R A LQ LY R QM RQ LY R YM LY R Rn LY R R UR R M LQ L? R fn‘M LU LY R fYL RQ LY R a!L P LY R bM LQ Q R nfM LQ LY R r!M LQ LY R A X} Q R R Q R LQ R LQ R P R X RXLRXLRXLRLLRRLRFfLRDWRXLRXLRXLRXLRXLRFQRLQRXLRXLRDLRXnRXLRDLRXLRGLRXLR ;f V f§ R ‘ L L L L L L L L L L L L L L L O L L L L L L L L L L L 222222&22222222222222222222222222222222222222222222222222 R 222222222222222222222222222222222222222222 UR ,9 rf!f fRRRRFRRRRRRRRRRRRRRRFLRRRRRRRRRRRfQUXLWI8.03LXXL2nWXRIILLY2n.RR QLQnMXnMSRUYFFDRnSUYUXLQLQLYLYXMLFQRYLMDXMLQnRXLRRXnLYYRF[SR fMWYDLQnLYLSXWYYIMUYLU SXXR S ,9n R , , , , , , ,, ,, 2222222222222222222222222&222222222222222&222222222222222 22222&22222222&222222222222&22222*222222222222&2222222222 R rRFRnUYLUYnYnYefYOS L?M1L*2nUY!UY!LQLY!UR!LQXn3XLQRLQ&Ln28LRSYLSLQLRSRLY!FnRLYLQUYn!RY2nRD2nYLAM!S[RXn1LLY R S ,9 , , , , , , , , ,, ,,, ,, RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRFRRRRRRRRRRRRRR&RRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRFRRRRRRRR}RRRRRRRRRRRRRRRRRR!RRRRRRRR!RR!RM8M!RRR nYRS!RSLYnXRLYIYRX!LSn&'*nFQnRLMMnfnrnAQRDfa-2nn&0Y&7nVLYLIf&QRM&0LLnfnSnAnVLDLRMRMW!WM!XLSRR &' S WY ,9 ,, ,, ,, ,, ,,, 2&2222222222&222222*222222&22*22*22222222222222222222*222 2222222222222222&22222222222222222222222222DD,98MS 3 Or! RRRRRnRRRnRRRRnRA R R RRrRnR!LY!frUYrRnRR!LYLYrRMRMF&(LYD!!RA!LI!!nXXRR XRXIY!MI9n!Mn!M!nLf§MLY!2LrRYLXLRRMSVRVRLLY3n&n!DL!LYRmMUY2n2L3WQDS!FRR'&-LY8nfURRL}fA3R R S S R , , , , , RRRRRRRRRRRRFRRRRRRRRRRRRRRRRRRRRRRRRRRRRFRRRRRRRRRRRRRRR 2222222222222222222222222222222222222222222nnnO9,98-8UX222222 RRRRRRRRRRRRRRRRRRRr}RRRRR!mf2RRRRRRLYrm f§ LYRnDM!LYYLXYULYR2'LYF f-LQLQSL0YR!LRRLQXL31LRX-n83n3-LQLQD!XXcLQDUYr38Mn2nL!Lna!LRRY f§ RSLYDfSLQLQRLY &' S S ,, , , , , RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRFRRRRRRRRRRRRRRRRRRRRRRRRRR!n!RRFRRRR RRRRRRRRRRRRRR}SRRRRRRRRnnRRRrRRnr!AR!f!UYnRRLFQFFRRXX1L2L2RR 3&09!A88OLQW9O9&086U?8L3&0QrrRUXn,9!Y23!S3Y&LUL8LO9!On23r12!OnX&Un22-1!&'32n287r &' S S ,9 9DOXHV RI 4UAAU[f IRU WKH )OXLG RI 7DEOH +

PAGE 344

7$%/( + &RQWLQXHGf 56,*0$  f E  & & 6& eL&  f & & f & 222L f & & 2& f§ f M f 2 & & f & *f &

PAGE 345

9DOXHV RI JAR&ARf J$2U! IRU WKH )XLG RI 7DEOH + R W R R '1:nQLILn0LLVM+ P Y LR R P XL P WW R R LQ n7 \ m ?M B U V R R R P R f§f mQ RM f§f Uf§ R U?L LQ R UR WR LQ W f§ U! R Fr UYL Q R f§f Y! RQ W RM f§‘ R R R R R R R R R R R R R R R R FL R f§ R R R R R R R R R R R R R R R R R R R R R 2 2 2 2 2 2 2 2 f§ 2 2 2 2 2 &f 2 2 2 2 2 &f 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 n n 2 2 2 2 2 2 &O R R R R R R R R R LR R R R R R R R R R U R R R R R R R R R R R R R R R R R R R R ‘! R R R Hn R R R R R R R R R R R R R R R R R R R R R R W , , , , , , , OLOL ,, ,,, ,, ,, , , , , F R ?M UR LQ LG Uf§L UYL R W Q Q OW! R YR R LUW R P R Q R R W + R D I P n?M FU} M RM L?M M P D Q P W Q RM R R R R! R R DM F R RWnUnInLQFX Df FPn8RRRRRRRRRL f§ &0 2R2222 f§ m2222222222222222222222222222222222222222222222 , 2 2 2 2 2 2 2 2 2 2 2 2 R 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 R R R R R R R R /2 L , , , , , , , , ,, , L , ,,, R R YOf n UR RQL Q DLI?MU R R! L LQ LR m LQUR ‘n!W UR WR P N R XL D L W?L YR 0 YD R I P FrLQ X2 FX fef YDR LQ R Q FUYR L aLR 2 2 2 &0 O??O UR O?O U9f < &n 2 2 2 2 2 2 2 UL 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 R 2 2 2 2 U Q R R R R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR2R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRF!RRRRRRRRR , , , , , , , , , ,,, ,, , , , <  LR ?M ?M aR M Q P Vf UW! R R R ?L P FW UXQ Q M R rU} LQ?M P R LQ LQ QM W nM Q V M LQL'21QR UL FP U L22r RIn 2&n2UW2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R FL R R R f L R ‘! 2 R L R R R R R R R ‘! R R R R R R R ‘ R R R R F A U} R R U , ,, , , , , , , , , , , , , ,, Q R m R Q R Q UR Q R Q UR Q R Q R Q WR R R R L UX LQ aL R L FYL Q QL P f§ R R FP a P Q B WQ U F WR L U P LQ ?L f‘ P f§ f§ RRRRAXRLQLQQQQUR2nrF?LrRRRRRn+RRRR&nRRRRRRRnRRRRRRRRRRRRRRRRRRRRm Bf 2 2 2 2 R R 2 2 2 2 R 2 R 2 2 2 2 2 2 2 2 R R 2 &R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R ‘ , , , , , , 9 R WQ R LR R P F! P R LQ R LQ R LQ R P R LQ R LQ R LQ R LQ R Q R XQ R LQ R LQ R LQ R P R LU! R P R P R Q R LQ R LQ FL P R LQ R LQ R LQ R Rn0LQ1RZLIL1Q1LLVR?MQVR1LQVR0LQ1RnLLLIL12n0LUL1R0LUf12nnLnQVR0 PfK RWnLLQ1RFQLRVR0LUL1R LW8.2LQL0LLRWLnQRRRL QM RM RM RM WR R UR WR FW LQ LQ XO LQ mR 2 LKR V V Q V P ‘f L LL LQ 7n2n2nRRRRf§‘ f§ m R Y ,7 2 2 R R &2 2 R R I?O 0 0 : 0 ,0 f0 0 0

PAGE 346

X FP U?f UM &0 ?L 0 ? Df FY U?L 9 P U?M FP FP ?L IYO U?f U?M U?M Z n2 7$%/( + &RQWLQXHGf ,*0$ f f f f f  & U? f R & f X & &2R & & & & & R f *2 f && f & 2& & R R R D f§ A G & & & RFF & & 22nM B n$ & & & & & & 2&2 & R F2 & F 22& 22& 2& Q A & &2, f & & & f

PAGE 347

7$%/( + 9DOXHV RI JS44Aa IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWK N7H SHUr DQG 6 R 0 $ & f f * F F f f f f F F & 4 M nf & f F U & 4 f &2, U? f & F f f  f ff & & E & & & & V U & & fA & n& F & & R & f L A R fR & & E & f & & M G & R F f & E L  f r  f G R M & f 4R 9 F  & S & && F R  2n f 2 A BB U M nV F V F & f G & & & F L A' fY! f & e & & Yr f F & f V F & R & & F & & f & & f 9  & f  & & & E f F M F &2M & e & f L f & O & & & & & f§ n? & & F f R f & & f§ f & f & & & F O R & & E & 8 O f B Q m LB f§ 4 & & & 2 IM f R M ‘ f f§ f  & & & * FO f & r & r F F U f ‘ YM f  R & F G f F & B U UV r B f ?M r B U L Q L Z f n & F & X & ‘ & & F f M && & & L RRFH U! rW f r  & & & U Q A -

PAGE 348

7$%/( + &RQWLQXHGf 6,*0$ *222f f *f *f *f *&f O & & r D & & & & && & & && & & 22& & &22 & & &2 &22 & &2 & &2 & & & &&22 &2, &2, & & & &22 & &&

PAGE 349

7$%/( +OO 9DOXHV RI J4Uf a J RLO [ f IRU WKH )OXLG RI 7DEOH + 6,*0$ *f *f *f *&f *! *f & O && O & A FRR & 2& & RRF W RFR & 2& &O 2 & 2 && & & &&22 &2, & & &2 & & &O & & & &22 & & & & 2 & 2 & & 2 & & & & &2 & &2, &O & &2 &2 &O & & &

PAGE 350

&0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &9 Q 7$%/( +OO &RQWLQXHGf 6,*0$ 6&2 f &2 &2 2 & & f & & & & *f & *f & & & Wf R R R H & & O *f &22 &2 2 && 2& & & &22 & &2 2 &2, & &2 &2 & & 2 & &2 & & & 2 & & & & &4

PAGE 351

SRUXUnRURUXUYMURZZf§f§ f§ f§ f§ f§ f§ RRRRRRRR UR f§ ARRRR?2Y&Y'YQL%4F[f&'1!LVLAM2n&7nnn8LXLXLDL3nRFVL!M2-L!LXURURURUR!!ARRR2YY2Y2YRDf&'&'& R!LRLL?fRALURR!LXLARAXLZRYLLnRRn!LFUZRMXnnUXR!L\LL?fVLXLURR1LURR!LDL0RmMMfURRALL021QURR \L-O8O2&52O8Ofn88,8OR8,888OR8L8O8O8,4828O28O4n8O28P 2 1 LQ R , , , , OLOL OLOL ,,, , , , , 22222222222222222222222222&222222222222222222222222222222 222222222222222222222222222222222222222222222222222222222 2222222R22222222222222R222222222+22U f§ 2Q2 f§ R R f§ 0R222 R n‘R7RRURXLXLrfnXLLRDfURFnLrL!MUYfA-LAMMS!LcrLRV!MLnURAWL2nF/QAR-1LMLLnRW[fAnn'!MY2RRn8nRXLRA nMMnP FrM R Q O! MM ,,, OLOL ‘ , , , , , 222222222222222222222222222222222222222222222222222222222 222222222222222222222222222222222222222222222222222222222 2222222222222222222 !! f§ ,1-222222 RJRRML&7nRURI4FAURXL2LRnXMURF-n RrVRMRRR! f§ f§ RR a R H f§ URXL0RR HQ F fI! , , , , , , , , , , , , , , , , , 222222222222222222222222222222222222222222222222222222222 222222222222222222222222222222222222222222222222222222222 22222222222222222 f§ n2222222222222Lrf§ f§ 2222+28U2222Xn22222 rMMY2RGrf§r2/QXL!212nnUYfRDR}RRL?fL?fX3nnf2!MMMMW?MURMMFnAfU?M2Y20FUFUn8LU?ML?M!ffnSM3nnL?f!frAf§RR HQ 2n R R OLOL OLOLO OLOL ,,, ,, , , , , , , , 222222222222222222222222222222222222222222222222422222222 222222222222222222222222222222222222222222222222222222222 }AA f§ r f§ XAURUR8RFMMA2Y2&7nRRMASURRUR[QAXRDnA1XMURURA8YQURURXLARURDLDLn3A f§ R 2n U?M R ,,, ,, , , , OLOL ,, , , 222222222222222222222222222222222222222222222222222222222 RRRRRRRRRRRRRRRRRRRRRRRRRRRRDRRRRRDRRRRRRRRRRRRRRRRRRRRRR 222222222222222222222222222222222 AAArA} R R FQASn&nAY2MMRRnA1AURRnRF-LAfSf2LA2!MM!MAXArF!MFQVAXL23n2&'2LARUnR!rSf4'&$MAIXXL}! RR &f R R ,f§ F f R A 6 7 + R UR Q n!f§ + + 26 /2 WX U UR FU Z R Lf§r F f 21 VH F f§r 1 K LW 2 +f§ UR , , , ,, OLOLO ,, , ,, OLOLO OLOL 222222222222222222222222222222222222222222222222222222222 F RU!fU! RRRR!!RR}RR+RRRRRRRRRRRRRRRRRRRRRRRRRRRURnrXM!nRARMAf f§ R R R R R Un-2,?I?O?f-18O+nO?f2,n2nnA1&%8O3n23fn!On&n!O!OISnUM--3nn&'M-6W!L62'n2O?f628O!IfM-!n2!2 Q 2n Qr R

PAGE 352

7$%/( + &RQWLQXHGf 6O*A$ f & & QO f O 2&2 *f f O & f & & 22& & & f F &22 &2 & 2 & & & &2 & &2 & &2 & & & & &O 2 & &22 2 & && & FRR &2 & & & & &O

PAGE 353

7$%/( + 9DOXHV RI JA SOXV $QLVRWU JfQU IRU /HQQDUG -RQHV ,88 = 88 RSLF 2YHUODS )OXLG ZLWK N7H SDA DQG 56,*0$ f f R f *f f & f & & R &r & f f H r & R } • %H 22& f f f H  R f & r f R Y L f r & & & RFRR R &O O & & & &&2E & G2 r } & & R* f§ & & & G f & F/ f 2R R

PAGE 354

7$%/( + &RQWLQXHGf 56,*0$ e m G L G r f & r r r r f f & 2R2 *f & *f *f G f & &2, 2G f O

PAGE 355

I?r W f§ FYMURU?MDnAWLQRRR1?-LRI?MnQAI?M f§ UQWUfURURWRFRrAAU?M121URIRUAUYMR1AMUAY2YQAARLQURn0YRrWRDnAL}f R ffI?O&Y-UArnrf§n2U22222222222A22222222222222222222222 222222222222222222222222222222222222222222222222222222222 a! , , , , , , , , , , , 2Q@ P 7 f+ ,f§, S+ 9 VL ‘8 R 90 &0 Ur + P X Uf§ f§ nZn (& F VL Z }f§ ;c &4 G L + + 0 &1 2 ,f§L &0 U R &2 ,f§, Q! f§ RF?MLRM?MFRD'2nUQ1YR`n&0nQLUfUMURW?MLW!XQmXYQDRWL1MYLnLQWnnn1?QUJnefnLM12nnF?LRA-\L?M0WQ f§ RRRRRRRRrL2RrmKRRRrRRRRRRRRRRRRRRmmRRRRRRRRRRRRRRRRRRR }‘ 222222222222222222222222222222222222222222222222222222222 A Z 222222222222222222222222222222222222222222222222222222222 ,, , , , , , , , } ,, ,, ,, R U UfR12nRnQ1L[!RLnYMLQLYR1r"RLnLRR0FRRLn?Mn2WQDfR/QRn+LL&0Uf1LUfnDnLnn?MLUfAfARRIR 22!'2Q_2222L?_Uf?,2+UW0R2UW22+&L222222222222n++222222222222 nnm2222222222R2DR222222222222222222222222222224222222 222222222222222222222222222222222222222222222222222222222 ,, ,, , , , , , , ,,, ,,, 0 0 f/2UMQDnRRWQF0&0F02112FAQY2&0-1&0LQ[UVP1[LQn2FRY21RM32 VQQQRRKf§ R UR F?M R 22200 f§ 22L222L222L22222222222222222222222222 RRRRRRRRRRRR 222422222222222222222222422222222222222222222422222224422 R 222222222222222222222222222222222222222222222222222222222 , , , , , , , ,, , , , 0 2&?_nf22nIn?-n!rWUOP1&?-UW1-FU\nn nnU\f0nfm6F?_&?-2n2-r211 n A 2 U2 U2 2 &?RRRRRfrPRaRRRrrRRRRRRRRRRRRRR RRRRRRRRRRRR RRRRRRRR 222222222222222222222222222222222222222222222222222222222 222222222222222222222222222222222222222222222222222222222 , , , , , , , , , , , , , , , ,, ,, R 2YO RAWQDnAQRn! f§ f§f§ffRRRRRRRRr!rRn2R!RRRRRRRRRR f§ 222L&01OA-f§&2222222222222222222222222222222222222222222222222 222222222222222222222222222222222222222222222222222222222 , , , , , , , , , ,,, , , , , , , , , , V -f 1 W RWILRLfRLURLRRLRRXLRLRRLQRLQXQRLfRLQLQRLfRLRRLQRLIfRARLLDLfRLQRLQRLfRLRRQRQRLLfRLRR 20,120,262 ?-LQ1R?-nUf121-LQUA20L212A-LQ121'n?MQ12UQLY20 '62?_L62?OLU!12n?RVR I2p&2,fI2!L0A0,?O722U2nWQLQL2Lfnf26660W+f2'0fnnnn>?O 2 R 2 2 2 2 2 2 ‘f§ r1r ‘f§r f§ f§ ‘f§ f§ ‘f§r f§ f§f nf§ f§ f§ f§ rf§ f§ f§ ‘f§L f§r r f§ f§ f§r ff§f ff§ ff§r ‘f§r f§ f§ f§ f§f f§ ‘f§m f§r ?O n$?1101-&YO2-

PAGE 356

LY UR LY WY WY LY Un 7$%/( + &RQWLQXHGf 56,*0$ f *^ *f & V f ZL } & RRRR & O r f R & } R R & FO } & } O *f &22 222M & & & f§ & f & & *f

PAGE 357

Yf WYRRRRRRRR UY!!r!RRFRRnefYR! DF[&'&/1A-AAMnnnn&U8Ln8fILInI!XLr-8L8LUYL?MUYY!nfrnRRFRYL!nLMnInnWDLDn&XD R V LY R YL IY R LL LY R X! LY R Q R LY R R R UY R R XL UY R QL R WY R Q R LY R M XL LY R VM R UR R !M LY R Q WY R M LY R RXnDRRLRDLRRLQRRLRXLRXLRXLRRLRXnLRXLRXLRARXLRURRLRXnRWARFQRQRXLRFURXLRFQRX!RFQR 7 ? : Wf§L 2 n , , L , , OLOL , , , , , , , ,, O 222222222222222222222222222222222222222222222222222222222 22222222222222222222&222222222222222222222222222222222222 RrRRRHRRFRRRRFRRR!fnf2F!RRR!RRRRRRR!RFrDRRRW!LaRR!RR!a2*22 228228,InRO28221fDR822O,Lnn.22nUY,9O8In62LXW6n!*L_;,n,2n212Lf F S X L , , , , , , , , L , ,,, 22222&&2222222222222* 22&&222222222222&222& 2&22&222222&222 22&2222222*&2222*22222222222222222222222222*2222222222222 rn22R*222rr!!m22&2222222&Kn !f!n!!r&!LLa!!!!a ‘!LRLYR-FQmnR}R!nR!X8fn!2MRLYR0'2-LYYRRnDLUY! UYRPXUYIYRRLYRRnDnRnRRRRMRFUnXnMDRM &O ‘S S S , , , , ,, OLOL , , , , , , , , , , RRRRRRRRRRRRRRRRFRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRFRRRRRRRRRRRJRRRFRRRRRRFRtRRRRRRRRRRRRRRRR}!!RRJR RR}RRR!RRRRRRRR}R!L!RRRRRRRFRRRRRRRRRYUYIYLYUYLYr!fLYRM!UY8n-LYXL3Y22R FAIYR2n2MFQfDf8O‘3‘R‘SnFQRnRFADfWFAOALAnfn2-OYfXRn!f}FAOY‘3‘!n&2RfS‘nA-nAOX Y2Y'2-8LAYRDXALY &L r‘} 2n R R , , , , , , , , OLOL , , 22222&22222222222&22&222222*2222222222222*222222222&22222 RRRFRRRRRRRRRRRRRRRRRFRRRRRRRRRRRRRRRRRW'RRRRRRRRRRRrRRRR 222&22R222222222n22222!2!a!*R2222222!nL22m2222!n223D-8nM2 IY!LYX2n1n3&n!XLR32L2RLYnR LYLYIYRLY!fIR!Z-M2n9RR?FNcARLLY!1L?fYRDSDAn-FU!LUYnLR!R!S LY F 2n UY R , , ,,, , , , , , , , ,, ,, , , , 2222222222222*22222222222222222&2222222222222222222222222 2*2222222222222222222222222222222222222222222222222222222 222*222*222222**,222**2&22*22*222222 f§ 22222nn2n2-2nL22nUYR f3nJ8On-W9A2'Y2O9fA+nn-22nAI928Anr28n!,3nn-Inn-2,9 ;2-2W9n2M22&2OA23n3A-2n,9 F ‘}r H e F 3 ‘2 7 ,f§ 1 .72 21 7 KR + W Z FF 2Q 22n ,,, , , ,, ,,, , , , , , , , 2222222222222222222222222222R22&2222222222222222222222222 &O 222222222222222222222222222**2222222222222222222222222222 222FL+222R222222+222222R22222222L2Wn2S+02::2LrAn!Ln2:8f_L+2 XLUYUYrUYXLX RRXnLRMWYI!I!RFSRXL!nI!n!LUY*n!R!f}RRn8c3n2'2cFXL)030*RMR3n68LL!RXRXfn2IY 2 2n R

PAGE 358

WY WY WY WY LY WY UY WY WY WY 7$%/( + &RQWLQXHGf N6,*0$ } L & -2& r M & } R Fr r E22 N r f N *f E f§ *f & & & & R O & & f RRF 22&2L *f n & & *f f§ & W'

PAGE 359

$33(1',; 6 P 9$/8(6 )25 7+( Q ,17(*5$/6

PAGE 360

7$%/( ,, 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV ,, a SOXV 4XDGUXSROH )OXLG SD N7H 4HD9 1 -1f -1f -1f -1f -1f f f f f f f f eeaP nM ( -Qe efPf Q 

PAGE 361

2 0 &1 2 0 2R Lf§, R : U2 UQ FW &2 34 + F R + R U R QL 9M 0 f f+ R &2 R QM UR ‘' R UQ 1 f§} LQ R R UR R LQ &0 7Y R 1 R &0 2 /2 R &0 2n 1 Q LQ Q FP 2m: R R R ?2UR LQ R QM 1 nR &0 ff§m R R R R R R R R R R R R R R R R R R R R R 2 R R R R R R D R R R R R R R R R R R R R R R R R 2 R R R R R R R R R R R R R R R R R R R R R R R R a! R R R R ? R } R R R R R R R R R R R R R R R R R R R R Q FR W P n2 R 2n Q F?M LQ UQ R f Ur R LQ R ‘e! W UR P W WR Q 2LQLQ\n_QF?fUKnALQF?LRRFnn&!nnnnn L WQ P R R cR Q Q Q !e! FR Y2 FR !R FR FR R P P LPR LQ LQ FYMmBRRRRRRRRRRRRRRRRRRRRR 2222222222222222222222222 RRRRRRRRRRRRRRRRRRRRRRRRR FQGRLn1QM?URQ!LLQRF2Q?LUQLUfFRF01QMQn LnQDn1LQLnL0an2RF!nnAfnQnn7LnQFR7fFRRn&nRR LQPUQQIQQnQQQL0F?MQMI?LQMQMR?MF0F?-n?MF?MnnQLQ 2222222222222222222222222 2222222222222222222222222 2222222222222222222222222 LL&0F?LRDDn1n2RVFn1DnF?LLQRLQr1nQ F'QLQQRnWmDnF'nLf2LQ!QWFWWWLQLQRn21U!FQ R1LQAQQLUfQMQMn?MAML00RMQMQM?MU?MU?M$M?MF?MUJM$M O222222222222222222222222 2222222222222222222222222 2222222222222222222222222 1 FP 2n UM? 2n LQ + Q ?1 2n LWr R 2n R R Q m+ R R ? n IU Q! R 1 fR R R &0 Q 6f 1 2n UQ R &0 &0 ?2&0 U?2UYM P &9QM 2FP FP 2R P UR P r R R R R R R R R R R R R R R R R R R R R R R R R R f A R R R R R R R R R R R R R R R R R R R R R R R R R 2222222222222222222222222 LF?LUQWLQQ1FRR!R F?MUQLQYR1FRRnQMUQA

PAGE 362

1 2 2 X 7$%/( 7KH ,QWHJUDOV 7 F X IRU WKH )OXLG ,, ,, RI 7DEOH L 1O f 1 f -1f -1 f -1 f & T & R & R& & 2& & & R & e & & & A &

PAGE 363

7$%/( 7KH B W ,QWHJUDOV 7 S 8 IRU WKH )OXLG /, ,, RI 7DEOH 1 -Q& -1f -1f L r & r r O & f§ *& f§ r

PAGE 364

7$%/( P 7 W B 7 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG SDA N7H m9 1 M Q RRR f -1 f 1 f -1  f -1f f f f f f f f f f f f f f f f f f &R R D M & f§ R R R f§ & f§ G& &O & R f§ &  f§ A M & WO9 ( -QPf Q

PAGE 365

1 2 2 7$%/( 7 B W W F M 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH MQ! MQf MQ & f& G & & & & 2 2& 22 R 2& -1 2& 2& & &22& -1f & &

PAGE 366

1 2 2 R 7$%/( 7KH ,QWHJUDOV RI 7DEOH IRU WKH )OXLG M Q L f r RR B! f fD & R R 1 f & R O -1-f &r & & -1f & & & R & 1 & f f§

PAGE 367

7$%/( B W W 7KH ,QWHJUDOV RI 7DEOH IRU WKH )OXLG 1 -1  f -1f -1f f§ R f§£ & && f§ f G

PAGE 368

1 2 ,6 7$%/( A 7 7 7 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG SJ N7R 4HRf -O1L2&2f -,1f -1f -1 -1f f f f f f f f f R R f & & & f & & f§ f§ f§ f§ a W R f§ & f§ & & & & & &

PAGE 369

1 2 & 7$%/( 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH -Q G M f V -1f & fA & & -1  f & -1 f f R & & -1&f & & &

PAGE 370

1 2 2 7$%/( ,OO 7KH ,QWHJUDOV U Y IRU WKH )OXLG RI 7DEOH 1 O f -1f -1Lf MQ f n -&1f &R & & &2 & f R & & œ R &  R &2 &2 & f R f • &

PAGE 371

7$%/( 7 W U 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH 1 -1f -1-f & f§  & A f f§ f§

PAGE 372

1 2 7$%/( P 7 W 7 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG SF7 N7H -1242-1-f -1f -1f -1f f f f f & RR & A M f§ & &6 2 2LO && & & &

PAGE 373

7$%/( A B 7KH ,QWHJUDOV 7 & W8 IRU WKH )OXLG RI 7DEOH Q 1 MQ f -1 f -1f -1f -1 f & f§& & G O f R ,G & 2& D & & & f§  & L  & f 2MM O & & & 2222OR  & L R f§ & &

PAGE 374

1 2 D 6 DR DL DD  7$%/( 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH 1L f & R RRRDD R RRRRDR -1RRRLD R RRLDL RRRRD & RRRRD L & R FRRDD -1f f§ R & f§ & f§ &• &22 &G -1  f 22& & RRFRD L R F R R D & & *2 2& -1cf & &

PAGE 375

7$%/( 7KH ,QWHJUDOV r IRU WKH )OXLG /, RI 7DEOH 1 -1f -1f -1L f & R & L L D r f§ & L D & r

PAGE 376

7$%/( 7KH ,QWHJUDOV IRU D /HQQDUG-RQHV UW Q A SOXV $QLVRWURSLF 2YHUODS )OXLG SD N7H f 1 M ^ Q R R R f -OQ f -1f -1f -1 f f f f f f f f f f f } & f r f H & & m f§ • & r & &O & 2f

PAGE 377

1 2 7$%/( 7KH ,QWHJUDOV Q Q RI 7DEOH IRU WKH )OXLG -Q f O G -1  f & & fGR f§ -1 G & *& -1L & & & *2 & -1 -f & & & & /

PAGE 378

7$%/( B 7KH ,QWHJUDOV IRU WKH )OXLG RI 7DEOH 1 1  1 1 L -1-f O 1 f & & A 8 & & r R L D f& & &E & & R & & && & & f

PAGE 379

7$%/( 7KH W 7 ,QWHJUDOV B IRU WKH )OXLG /, RI 7DEOH 1 -1-f M Q f -1 f D & f r fMG f§ & f§ r f§ r f§ O

PAGE 380

L $33(1',; 9$/8(6 2) 7+( 6,7(6,7( &255(/$7,21 )81&7,216

PAGE 381

7$%/( 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK tR N7H SD 4HRf 56,*0$ 6,7( 56,*0$ 6,7( HFR f f F f f f f & f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f O f f f f f f f f f f f f f f  f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f F f f f f f f f f f f f R f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f

PAGE 382

7$%/( 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG -RQHV SOXV 4XDGUXSROH )OXLG ZLWK +R N7H SD HRf 5 *0$ 6,7( 56,*0$ 6,7( f R R f  ]f f & % f  f G2 R O D f f R R &

PAGE 383

7$%/( 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV 4XDGUXSROH )OXLG ZLWK =R N7H SHU 4HJ-f 56 *0$ XO \ 5 6 0 $ 7 V W R & & f R f & 2% & FL f L & R R]! L e OR2 f A  H &  R F R e f & A f R R } f & f % O R D

PAGE 384

7$%/( -$ 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWAK +] N7H SJ 56,*0$ 6, 7( O f A f & A 6,*0$ 6,7( &

PAGE 385

7$%/( 6LWH6LWH &RUUHODWLRQ )XQFWLRQ IRU /HQQDUG-RQHV SOXV $QLVRWURSLF 2YHUODS )OXLG ZLWK /R N7H SD 56,*0$ 7 e 5,*0$ 6,7( f f f f -2 f f f f f f f  f f f f F f f f f f f f f f f f f f f O f f R f f f f f f f f f f f f f f f f f f f f f f 2GG f f f f f f f f f f / f f f f f f f f f G f f f f f f f f f f f f f f f f f &22 f f f f f eB f f

PAGE 386

$33(1',; 9$/8(6 )25 7+( ,17(*5$/ +Af 7DEOH JLYHV YDOXHV IRU WKH VLQJOH LQWHJUDO GHILQHG LQ (TXDWLRQ f 9DOXHV IRU WKH /HQQDUG-RQHV f UDGLDO GLVn WULEXWLRQ IXQFWLRQ ZHUH WDNHQ IURP WKH PROHFXODU G\QDPLFV VWXG\ E\ 9HUOHW > @ 7R IDFLOLWDWH LQWHUSRODWLRQ EHWZHHQ WKHVH YDOXHV IRU WKH VWDWH FRQGLWLRQV XVHG E\ 9HUOHW WKH UHVXOWV LQ 7DEOH KDYH EHHQ ILWWHG WR WKH IROORZLQJ HPSLULFDO HTXDWLRQ Q+S7f $S7 %S&S7 'S (7 ) .Of Q Q Q Q Q Q Q 1RWH WKDW (TXDWLRQ .Of LV GLVWLQFW IURP (TXDWLRQV (Of DQG *f 7KH FRQVWDQWV LQ .Of KDYH EHHQ GHWHUPLQHG E\ D OHDVW VTXDUHV ILW DQG DUH $%f&f'f(f) B! .f 9DOXHV IRU IRU S 7 DQG S 7 GR QRW IDOO RQ DQ\ VPRRWK FXUYH WKURXJK WKH RWKHUV DQG DUH WKHUHIRUH RPLWWHG IURP 7DEOH .O DQG LQ ILWWLQJ WKH FRQVWDQWV LQ (TXDWLRQ .f 7KH PD[LPXP GHYLDWLRQ IRU WKH SUHGLFWLRQV RI (TXDWLRQV .Of DQG .f LV OHVV WKDQ b RI WKH YDOXHV LQ 7DEOH .O

PAGE 387

7KH ,QWHJUDO + 7$%/( f e2-B 3XUH )OXLGV r S r 7 Kff +

PAGE 388

/,7(5$785( &,7(' >@ $ 5XVDQRY 5HFHQW ,QYHVWLJDWLRQV RQ WKH 7KLFNQHVV RI 6XUIDFH /D\HUV LQ 3URJU 6XUIDFH 0HPEUDQH 6FL ) 'DQLHOOL 0 5RVHQEHUJ DQG $ &DGHQKHDG HGVf &KDS $FDGHPLF 3UHVV 1HZ @ .LUNZRRG DQG ) 3 %XII 6WDWLVWLFDO 0HFKDQLFDO 7KHRU\ RI 6XUIDFH 7HQVLRQ &KHP 3K\V f >@ $ 5XVDQRY DQG 9 3VKHQLWV\Q (OOLSVRPHWU\ DQG 7KLFNQHVV RI 6XUIDFH /D\HUV 'RNO $NDG 1DXN 6665 f (QJOLVK WUDQVODWLRQ LQ 'RNO 3K\V &KHP 3URF $FDG 6FL 8665 f` >@ 9 6LYXNKLQ (OOLSWLFDO 3RODUL]DWLRQ LQ WKH 5HIOHFWLRQ RI /LJKW IURP /LTXLGV =KXU (NVSWO 7HRUHW )L] f >@ 5 & 5HLG DQG 7 6KHUZRRG 7KH 3URSHUWLHV RI *DVHV DQG /LTXLGV QG HG &KDS 0F*UDZ+LOO 1HZ @ ( *XEELQV DQG 0 +DLOH 0ROHFXODU 7KHRULHV RI ,QWHUIDFLDO 7HQVLRQ LQ ,PSURYHG 2LO 5HFRYHU\ E\ 6XUIDFWDQW DQG 3RO\PHU )ORRGLQJ 6KDK DQG 5 6 6FKHFKWHU HGVf $FDGHPLF 3UHVV 1HZ @ ) 3 %XII 6RPH &RQVLGHUDWLRQV RI 6XUIDFH 7HQVLRQ = (OHF WURFKHP B f >@ 5 + )RZOHU $ 7HQWDWLYH 6WDWLVWLFDO 7KHRU\ RI 0DFOHRGnV (TXDWLRQ IRU 6XUIDFH 7HQVLRQ DQG WKH 3DUDFKRU 3URF 5R\ 6RF $ f >@ 5 $ /RYHWW 3 : 'H+DYHQ 9LHFHOL -U DQG ) 3 %XII *HQHUDOL]HG YDQ GHU :DDOV 7KHRULHV IRU 6XUIDFH 7HQVLRQ DQG ,QWHUIDFLDO :LGWK &KHP 3K\V B f >@ & *UD\ DQG ( *XEELQV 7KHRU\ RI 6XUIDFH 7HQVLRQ IRU 0ROHFXODU )OXLGV 0ROHF 3K\V B f >@ 9 %RQJLRUQR DQG + 7 'DYLV 0RGLILHG YDQ GHU :DDOV 7KHRU\ RI )OXLG ,QWHUIDFHV 3K\V 5HY $ f

PAGE 389

>@ 7 / +LOO 6WDWLVWLFDO 7KHUPRG\QDPLFV RI WKH 7UDQVLWLRQ 5HJLRQ %HWZHHQ 7ZR 3KDVHV ,, 2QH &RPSRQHQW 6\VWHPV ZLWK D 3ODQH ,QWHUIDFH &KHP 3K\V =2 f >@ : 3OHVQHU DQG 3ODW] 6WDWLVWLFDO 0HFKDQLFDO &DOFXODWLRQ RI 6XUIDFH 3URSHUWLHV RI 6LPSOH /LTXLGV DQG /LTXLG 0L[WXUHV 3XUH /LTXLGV &KHP 3K\V M f >@ 6 7R[YDHUG 6WDWLVWLFDO 0HFKDQLFDO DQG 4XDVLWKHUPRG\QDPLF &DOFXODWLRQV RI 6XUIDFH 'HQVLWLHV DQG 6XUIDFH 7HQVLRQ 0ROHF 3K\V A f >@ 3UHVVLQJ DQG ( 0D\HU 6XUIDFH 7HQVLRQ DQG ,QWHUIDFLDO 'HQVLW\ 3URILOH RI )OXLGV QHDU WKH &ULWLFDO 3RLQW &KHP 3K\V f >@ 6 7R[YDHUG +\GURVWDWLF (TXLOLEULXP LQ )OXLG ,QWHUIDFHV &KHP 3K\V B f >@ 6 7R[YDHUG 3HUWXUEDWLRQ 7KHRU\ IRU 1RQXQLIRUP )OXLGV 6XUIDFH 7HQVLRQ &KHP 3K\V B f >@ ) ) $EUDKDP $ 7KHRU\ IRU WKH 7KHUPRG\QDPLFV DQG 6WUXFWXUH RI 1RQXQLIRUP 6\VWHPV ZLWK $SSOLFDWLRQV WR WKH /LTXLG9DSRU ,QWHUIDFH DQG 6SLQRGDO 'HFRPSRVLWLRQ &KHP 3K\V f >@ /HH $ %DUNHU DQG 0 3RXQG 6XUIDFH 6WUXFWXUH DQG 6XUIDFH 7HQVLRQ 3HUWXUEDWLRQ 7KHRU\ DQG 0RQWH &DUOR &DOFXODWLRQV &KHP 3K\V A f >@ ) ) $EUDKDP ( 6FKUHLEHU DQG $ %DUNHU 2Q WKH 6WUXFWXUH RI D )UHH 6XUIDFH RI D /HQQDUG-RQHV )OXLG $ 0RQWH &DUOR &DOFXODWLRQ &KHP 3K\V A f >@ $ &KDSHOD 6DYLOOH DQG 6 5RZOLQVRQ &RPSXWHU 6LPXODn WLRQ RI WKH *DV/LTXLG 6XUIDFH )DUDGD\ 'LVF &KHP 6RF f >@ + 7 'DYLV 6WDWLVWLFDO 0HFKDQLFV RI ,QWHUIDFLDO 3URSHUWLHV RI 3RO\DWRPLF )OXLGV 6XUIDFH 7HQVLRQ &KHP 3K\V f >@ : *LEEV (TXLOLEULXP RI +HWHURJHQHRXV 6XEVWDQFHV 7KH 6FLHQWLILF 3DSHUV RI : *LEEV 9RO 'RYHU 3XEOLFDn WLRQV 1HZ @ + 6 *UHHQ $ *HQHUDO .LQHWLF 7KHRU\ RI /LTXLGV (TXLOLEULXP 3URSHUWLHV 3URF 5R\ 6RF $ f

PAGE 390

>@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ 5 % %LUG : ( 6WHZDUW DQG ( 1 /LJKWIRRW 7UDQVSRUW 3KHQRPHQD $SSHQGL[ $ :LOH\ DQG 6RQV 1HZ
PAGE 391

>@ 3 $ (JHOVWDII & *UD\ DQG ( *XEELQV (TXLOLEULXP 3URSHUWLHV RI 0ROHFXODU )OXLGV LQ 0ROHFXODU 6WUXFWXUH DQG 3URSHUWLHV ,QWHUQDWLRQDO 5HYLHZ RI 6FLHQFH 3K\VLFDO &KHPLVWU\ 6HULHV 9RO $ %XFNLQJKDP HGf %XWWHUZRUWKV /RQGRQ >@ 7 0 5HHG DQG ( *XEELQV $SSOLHG 6WDWLVWLFDO 0HFKDQLFV 0F*UDZ+LOO 1HZ @ :HHNV &KDQGOHU DQG + & $QGHUVRQ 5ROH RI 5HSXOVLYH )RUFHV LQ 'HWHUPLQLQJ WKH (TXLOLEULXP 6WUXFWXUH RI 6LPSOH )OXLGV &KHP 3K\V MYL f >@ ( *XEELQV : 5 6PLWK 0 7KDP DQG ( : 7LHSHO 3HUWXUEDWLRQ 7KHRU\ IRU WKH 5DGLDO 'LVWULEXWLRQ )XQFn WLRQ 0ROHF 3K\V A f >@ 6WDQVILHOG 7KH 6XUIDFH 7HQVLRQV RI /LTXLG $UJRQ DQG 1LWURJHQ 3URF 3K\V 6RF /RQGRQf A f >@ 6 )XNV DQG $ %HOOHPDQV 7KH 6XUIDFH 7HQVLRQ RI .U\SWRQ 0HWKDQH DQG 7KHLU 0L[WXUHV 3K\VLFD > f >@ ) 3 %XII DQG 5 $ /RYHWW 7KH 6XUIDFH 7HQVLRQ RI 6LPSOH )OXLGV LQ 6LPSOH 'HQVH )OXLGV + / )ULVFK DQG = : 6DOVEXUJ HGVf $FDGHPLF 3UHVV 1HZ @ ) % 6SURZ DQG 0 3UDXVQLW] 6XUIDFH 7HQVLRQV RI 6LPSOH /LTXLGV 7UDQV )DUDGD\ 6RF f >@ -DVSHU 7KH 6XUIDFH 7HQVLRQ RI 3XUH /LTXLG &RPSRXQGV 3K\V &KHP 5HI 'DWD A/ f >@ 1 % 9DUJDIWLN 7DEOHV RQ WKH 7KHUPRSK\VLFDO 3URSHUWLHV RI /LTXLGV DQG *DVHV QG HG +DOVWHG 3UHVV 'LY :LOH\ DQG 6RQV 1HZ @ 7 3HDUVRQ DQG 3 / 5RELQVRQ 3DUDFKRU RI +\GURJHQ %URPLGH &KHP 6RF /RQGRQf >@ ( 6WRJU\Q DQG $ 3 6WRJU\Q 0ROHFXODU 0XOWLSROH 0RPHQWV 0ROHF 3K\V B/ f >@ 7 + 6SXUOLQJ DQG ( $ 0DVRQ 'HWHUPLQDWLRQ RI 0ROHFXODU 4XDGUXSROH 0RPHQWV IURP 9LVFRVLWLHV DQG 6HFRQG 9LULDO &RHIILFLHQWV &KHP 3K\V f >@ & ) 6SHQFHU DQG 5 3 'DQQHU ,PSURYHG (TXDWLRQ IRU 3UHGLFWLRQ RI 6DWXUDWHG /LTXLG 'HQVLW\ &KHP (QJ 'DWD BB f

PAGE 392

>@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ 6 .HOO 'HQVLW\ 7KHUPDO ([SDQVLYLW\ DQG &RPSUHVVLELOLW\ RI /LTXLG :DWHU IURP r WR r& &RUUHODWLRQV DQG 7DEOHV IRU $WPRVSKHULF 3UHVVXUH DQG 6DWXUDWLRQ 5HYLHZHG DQG ([SUHVVHG RQ 7HPSHUDWXUH 6FDOH &KHP (QJ 'DWD 4 f % %RUVWQLN DQG $ $]PDQ 7KH 1XPHULFDO 6ROXWLRQ RI WKH %*<% (TXDWLRQ IRU WKH /LTXLG9DSRU ,QWHUIDFH 0ROHF 3K\V f 6 7R[YDHUG 6WDWLVWLFDO 0HFKDQLFV RI 6XUIDFHV LQ 6WDWLVWLFDO 0HFKDQLFV 9RO 6LQJHU HGf &KDS 7KH &KHPLFDO 6RFLHW\ /RQGRQ : : :RRG 0RQWH &DUOR 6WXGLHV RI 6LPSOH /LTXLG 0RGHOV LQ 3K\VLFV RI 6LPSOH /LTXLGV +19 7HPSHUOH\ 6 5RZOLQVRQ DQG 6 5XVKEURRNH HGVf &KDS 1RUWK+ROODQG 3XEOLVKLQJ &R $PVWHUGDP 5 0F'RQDOG DQG 6LQJHU 7KH 6WXG\ RI 6LPSOH /LTXLGV E\ &RPSXWHU 6LPXODWLRQ 4 5HY &KHP 6RF $ f $ %XFNLQJKDP $QJXODU &RUUHODWLRQV LQ /LTXLGV 'LVF )DUDGD\ 6RFLHW\ f 36< &KHXQJ DQG 3RZOHV 7KH 3URSHUWLHV RI /LTXLG 1LWURJHQ ,9 $ &RPSXWHU 6LPXODWLRQ 0ROHF 3K\V B f 6 6 :DQJ & *UD\ 3 $ (JHOVWDII DQG ( *XEELQV 0RQWH &DUOR 6WXG\ RI WKH 3DLU &RUUHODWLRQ )XQFWLRQ IRU D /LTXLG ZLWK 1RQ&HQWUDO )RUFHV &KHP 3K\V /HWWHUV f & *UD\ 6 6 :DQJ DQG ( *XEELQV 0RQWH &DUOR &DOFXODWLRQV RI WKH 0HDQ 6TXDUHG )RUFH LQ 0ROHFXODU /LTXLGV &KHP 3K\V /HWWHUV A f & + 7ZX ( *XEELQV DQG & *UD\ 7KH 0HDQ 6TXDUHG 7RUTXH LQ 3XUH DQG 0L[HG 'HQVH )OXLGV 0ROHF 3K\V B f 6 6 :DQJ 3 $ (JHOVWDII & *UD\ DQG ( *XEELQV 0RQWH &DUOR 6WXG\ RI WKH $QJXODU 3DLU &RUUHODWLRQ )XQFWLRQ LQ D /LTXLG ZLWK 4XDGUXSRODU )RUFHV &KHP 3K\V /HWWHUV $B f % & )UHDVLHU -ROO\ DQG 5 %HDUPDQ +DUG 'XPEHOOV 0RQWH &DUOR 3UHVVXUHV DQG 9LULDO &RHIILFLHQWV 0ROHF 3K\V -/ f % %HUQH DQG +DUS 2Q WKH &DOFXODWLRQ RI 7LPH &RUUHODWLRQ )XQFWLRQV $GY &KHP 3K\V f % %HUQH DQG 5 3FRUD '\QDPLF /LJKW 6FDWWHULQJ :LOH\ DQG 6RQV 1HZ
PAGE 393

>@ : +RRYHU DQG : 7 $VKXUVW 1RQHTXLOLEULXP 0ROHFXODU '\QDPLFV LQ 7KHRUHWLFDO &KHPLVWU\ 9RO + (\ULQJ DQG +HQGHUVRQ HGVf $FDGHPLF 3UHVV 1HZ @ : % 6WUHHWW DQG 7LOGHVOH\ &RPSXWHU 6LPXODWLRQV RI 3RO\n DWRPLF 0ROHFXOHV 0RQWH &DUOR 6WXGLHV RI +DUG 'LDWRPLFV 3URF 5R\ 6RF $ f >@ 36< &KHXQJ 2Q WKH (IILFLHQW (YDOXDWLRQ RI 7RUTXHV DQG )RUFHV IRU $QLVRWURSLF 3RWHQWLDOV LQ &RPSXWHU 6LPXODWLRQ RI /LTXLGV &RPSRVHG RI /LQHDU 0ROHFXOHV &KHP 3K\V /HWWHUV f f >@ %DURMDV /HYHVTXH DQG % 4XHQWUHF 6LPXODWLRQ RI 'LDWRPLF +RPRQXFOHDU /LTXLGV 3K\V 5HY $ f >@ & : *HDU &RPSXWDWLRQDO 0HWKRGV LQ 2UGLQDU\ 'LIIHUHQWLDO (TXDWLRQV 3UHQWLFH+DOO (QJOHZRRG &OLIIV 1HZ -HUVH\ >@ %HHPDQ 6RPH 0XOWLVWHS 0HWKRGV IRU 8VH LQ 0ROHFXODU '\QDPLFV &DOFXODWLRQV &RPSXWDWLRQDO 3K\V 4 f >@ & + %HQQHWW 0DVV 7HQVRU 0ROHFXODU '\QDPLFV &RPSXWDWLRQDO 3K\V f >@ % $OGHU DQG : +RRYHU 1XPHULFDO 6WDWLVWLFDO 0HFKDQLFV LQ 3K\VLFV RI 6LPSOH /LTXLGV +19 7HPSHUOH\ 6 5RZOLQVRQ DQG 6 5XVKEURRNH HGVf &KDS 1RUWK+ROODQG 3XEOLVKLQJ &R $PVWHUGDP >@ ( *XEELQV & *UD\ 3 $ (JHOVWDII DQG 0 6 $QDQWK $QJXODU &RUUHODWLRQ (IIHFWV LQ 1HXWUDQ 'LIIUDFWLRQ IURP 0ROHFXODU )OXLGV 0ROHF 3K\V B f >@ & *UD\ DQG 5 / +HQGHUVRQ 3HUWXUEDWLRQ 7KHRU\ RI WKH 3DLU &RUUHODWLRQ )XQFWLRQ LQ 0ROHFXODU )OXLGV 0ROHF 3K\V LQ SUHVVf f >@ + 5DYHFKH 5 0RXQWDLQ DQG : % 6WUHHWW )UHH]LQJ DQG 0HOWLQJ 3URSHUWLHV RI WKH /HQQDUG-RQHV 6\VWHP &KHP 3K\V f >@ & + 7ZX ( *XEELQV DQG & *UD\ 0HDQ 6TXDUHG 7RUTXH LQ 'HQVH )OXLGV 6WURQJ 3RODU DQG 4XDGUXSRODU )RUFHV 0ROHF 3K\V f >@ & *UD\ DQG ‹ *XEELQV &DOFXODWLRQ RI WKH 'LHOHFWULF DQG .HUU &RQVWDQW $QJXODU &RUUHODWLRQ 3DUDPHWHUV 0ROHF 3K\V f >@ / /HERZLW] 3HUFXV DQG / 9HUOHW (QVHPEOH 'HSHQGHQFH RI )OXFWXDWLRQV ZLWK $SSOLFDWLRQ WR 0DFKLQH &RPSXWDWLRQV 3K\V 5HY f

PAGE 394

>@ 5 0F'RQDOG $SSOLFDWLRQ RI 7KHUPRG\QDPLF 3HUWXUEDWLRQ 7KHRU\ WR 3RODU DQG 3RODUL]DEOH )OXLGV 3K\V & B f >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ >@ 36< &KHXQJ 7KH 'LDWRPLF n$WRP$WRPn 3RWHQWLDO DQG WKH 6WDWLF 3URSHUWLHV RI 2[\JHQ DQG 1LWURJHQ 0ROHF 3K\V LQ SUHVVf f %UXLQLQJ DQG -+5 &ODUNH 0ROHFXODU 2ULHQWDWLRQ &RUUHODWLRQV DQG 5HRULHQWDWLRQDO 0RWLRQV LQ /LTXLGV &DUERQ 0RQR[LGH 1LWURJHQ DQG 2[\JHQ DW $ 5DPDQ DQG 5D\OHLJK /LJKW6FDWWHULQJ 6WXG\ 0ROHF 3K\V f 5 0F'RQDOG DQG 6LQJHU $Q (TXDWLRQ RI 6WDWH IRU 6LPSOH /LTXLGV 0ROHF 3K\V M f (UUDWXP 0ROHF 3K\V f 1 3DWH\ DQG 3 9DOOHDX )OXLGV RI 6SKHUHV &RQWDLQLQJ 4XDGUXSROHV DQG 'LSROHV $ 6WXG\ 8VLQJ 3HUWXUEDWLRQ 7KHRU\ DQG 0RQWH &DUOR &RPSXWDWLRQV &KHP 3K\V M! f ) % +LOGHEUDQG ,QWURGXFWLRQ WR 1XPHULFDO $QDO\VLV 0F*UDZ+LOO 1HZ
PAGE 395

>@ 6 $ 5LFH DQG $ 5 $OOQDWW 2Q WKH .LQHWLF 7KHRU\ RI 'HQVH )OXLGV 9, 6LQJOHW 'LVWULEXWLRQ )XQFWLRQ IRU 5LJLG 6SKHUHV ZLWK DQ $WWUDFWLYH 3RWHQWLDO &KHP 3K\V f >@ + +LOGHEUDQG 0 3UDXVQLW] DQG 5 / 6FRWW 5HJXODU DQG 5HODWHG 6ROXWLRQV 9DQ 1RVWUDQG 5HLQKROG &R 1HZ @ / 9HUOHW &RPSXWHU ([SHULPHQWV RQ &ODVVLFDO )OXLGV 7KHUPRn G\QDPLFDO 3URSHUWLHV RI /HQQDUG-RQHV 0ROHFXOHV 3K\V 5HY f >@ $ / 5RELQVRQ &RPSXWDWLRQDO &KHPLVWU\ *HWWLQJ 0RUH IURP D 0LQLFRPSXWHU 6FLHQFH f >@ $ $KPDG DQG / &RKHQ $ 1XPHULFDO ,QWHJUDWLRQ 6FKHPH IRU WKH 1%RG\ *UDYLWDWLRQDO 3UREOHP &RPSXWDWLRQDO 3K\V f >@ : 3HUUDP DQG 3 6WLOHV 2Q WKH $SSOLFDWLRQ RI (OOLSVRLGDO +DUPRQLFV WR 3RWHQWLDO 3UREOHPV LQ 0ROHFXODU (OHFWURVWDWLFV DQG 0DJQHWRVWDWLFV 3URF 5R\ 6RF $ f >@ +LUVFKIHOGHU & ) &XUWLVV DQG 5 % %LUG 0ROHFXODU 7KHRU\ RI *DVHV DQG /LTXLGV :LOH\ DQG 6RQV 1HZ
PAGE 396

%,%/,2*5$3+< 7KLV ELEOLRJUDSK\ UHIHUV WR PDLQO\ WKHRUHWLFDO DQG FRPSXWHU VLPXODWLRQ ZRUN RQ IOXLG LQWHUIDFHV SXEOLVKHG VLQFH WKH FRPSUHKHQVLYH UHYLHZ E\ 2QR DQG .RQGR LQ 5HYLHZV 5 & %URZQ 7KH 6XUIDFH 7HQVLRQ RI /LTXLGV &RQWHPSRUDU\ 3K\V f 5 & %URZQ DQG 1 + 0DUFK 6WUXFWXUH DQG ([FLWDWLRQV LQ /LTXLG DQG 6ROLG 6XUIDFHV 3K\V 5HSWV 3K\V /HWWHUV &f B f ) 3 %XII 7KH 7KHRU\ RI &DSLOODULW\ +DQGEXFK GHU 3K\V f ) 3 %XII DQG 5 $ /RYHWW 7KH 6XUIDFH 7HQVLRQ RI 6LPSOH )OXLGV LQ 6LPSOH 'HQVH )OXLGV + / )ULVFK DQG = : 6DOVEXUJ HGVf $FDGHPLF 3UHVV 1HZ
PAGE 397

$ 6DQIHOG 7KHUPRG\QDPLFV RI 6XUIDFHV LQ 3K\VLFDO &KHPLVWU\ DQ $GYDQFHG 7UHDWLVH 9RO + (\ULQJ +HQGHUVRQ DQG : -RVW HGVf $FDGHPLF 3UHVV 1HZ
PAGE 398

DQG 6 5 5H]QHN $ 6LPSOH 7KHRU\ IRU WKH 'HQVLWLHV RI &RH[LVWHQW /LTXLG DQG 9DSRU WKURXJK WKH 7UDQVLWLRQ 5HJLRQ 3K\V $ A f % %RUVWQLN DQG $ $]PDQ 7KH 1XPHULFDO 6ROXWLRQ RI WKH %*<% (TXDWLRQ IRU WKH /LTXLG9DSRU ,QWHUIDFH 0ROHF 3K\V f 1XPHULFDO 6ROXWLRQ RI WKH %*<% (TXDWLRQ IRU /LTXLG ,QWHUDFWLQJ ZLWK D 5LJLG :DOO 0ROHF 3K\V B f 0ROHFXODU '\QDPLFV 6LPXODWLRQV RI WKH /LTXLG6ROLG 7UDQVLWLRQ 7KH '\QDPLFDO 3URSHUWLHV RI WKH /LTXLG6ROLG ,QWHUIDFH &KHP 3K\V /HWWHUV f : %URXZHU DQG 5 3DWKULD 6XUIDFH 7HQVLRQ DW WKH +H +H ,QWHUIDFH 3K\V 5HY f ) 3 %XII 6WDWLVWLFDO 0HFKDQLFDO 9HULILFDWLRQ RI WKH *LEEV $GVRUSWLRQ (TXDWLRQ $GY &KHP 6HU B f ) 3 %XII DQG 1 6 *RHO 7KHRU\ RI 6XUIDFH 7HQVLRQ RI $TXHRXV 6ROXWLRQV RI $PLQR $FLGV &KHP 3K\V B f (OHFWURVWDWLFV RI 'LIIXVH $QLVRWURSLF ,QWHUIDFHV 3ODQDU /D\HU 0RGHO &KHP 3K\V f (OHFWURVWDWLFV RI 'LIIXVH $QLVRWURSLF ,QWHUIDFHV ,, (IIHFWV RI /RQJ5DQJH 'LIIXVHQHVV &KHP 3K\V f 2Q 6RPH 5HODWLRQV EHWZHHQ WKH 6ROXWLRQV DQG WKH 3DUDn PHWHUV RI 6HFRQG2UGHU /LQHDU 'LIIHUHQWLDO (TXDWLRQV 0DWK 3K\V f (OHFWURVWDWLFV RI 'LIIXVH $QLVRWURSLF ,QWHUIDFHV ,,, 3RLQW &KDUJH DQG 'LSROH ,PDJH 3RWHQWLDOV IRU $LU:DWHU DQG 0HWDO:DWHU ,QWHUIDFHV &KHP 3K\V -W£ f ) 3 %XII 5 /RYHWW DQG ) + 6WLOOLQJHU -U ,QWHUIDFLDO 'HQVLW\ 3URILOH IRU )OXLGV LQ WKH &ULWLFDO 5HJLRQ 3K\V 5HY /HWWHUV OMM f ) 3 %XII DQG ) + 6WLOOLQJHU -U 6WDWLVWLFDO 0HFKDQLFDO 7KHRU\ RI 'RXEOH/D\HU 6WUXFWXUH DQG 3URSHUWLHV &KHP 3K\V f & & &KDQJ DQG 0 &RKHQ 6WUXFWXUH RI WKH 6XUIDFH RI /LTXLG +H DW =HUR 7HPSHUDWXUH 3K\V 5HY $ B f

PAGE 399

& 7 &KHQ DQG 5 3UHVHQW ,VRWRSLF'LVWLOODWLRQ /DSODFLDQ DQG WKH 6XUIDFH (QHUJ\ DQG 7HQVLRQ RI /LTXLG $UJRQ DW r. &KHP 3K\V f $ &KDSHOD 6DYLOOH DQG 6 5RZOLQVRQ &RPSXWHU 6LPXODWLRQ RI WKH *DV/LTXLG 6XUIDFH )DUDGD\ 'LVF &KHP 6RF f ( & &KHQ &KDQJHV LQ 6XUIDFH 7HQVLRQ RI 6RPH +\GURFDUERQ 0L[WXUHV &DQ &KHP (QJ A f 0 : &ROH :LGWK RI WKH 6XUIDFH /D\HU RI /LTXLG +H? 3K\V 5HY $ f & $ &UR[WRQ DQG 5 3 )HUULHU 6WDWLVWLFDO 0HFKDQLFDO &DOFXODn WLRQ RI WKH 6XUIDFH 3URSHUWLHV RI 6LPSOH /LTXLGV 7KH 'LVWULEXWLRQ )XQFWLRQ JA]f 3K\V & B f 6WDWLVWLFDO 0HFKDQLFDO &DOFXODWLRQ RI 6XUIDFH 3URSHUWLHV RI 6LPSOH /LTXLGV ,, 7KH 'LVWULEXWLRQ )XQFWLRQ S ]Uff 3K\V & f N n 6WDWLVWLFDO 0HFKDQLFDO &DOFXODWLRQ RI 6XUIDFH 3URSHUWLHV RI 6LPSOH /LTXLGV ,,, 6XUIDFH 7HQVLRQ 3K\V & A f &ODVVLFDO 6LQJOHW 'LVWULEXWLRQ )XQFWLRQ IRU /LTXLG9DSRU 'HQVLW\ 7UDQVLWLRQ 3KLO 0DJ AB f 6WDWLVWLFDO 0HFKDQLFDO &DOFXODWLRQ RI 6XUIDFH 3URSHUWLHV RI 6LPSOH /LTXLGV ,9 0ROHFXODU '\QDPLFV 3K\V & A f & $ &UR[WRQ DQG 7 5 2VERUQ 7KH 'HYHORSPHQW RI 7RUTXH )LHOGV DW WKH 6XUIDFH RI 0ROHFXODU )OXLGV 3K\V /HWWHUV $ f 'DVK 6WDWLVWLFDO 7KHUPRG\QDPLFV RI 3K\VLVRUSWLRQ 3URJU 6XUIDFH 6FL f + 7 'DYLV 6WDWLVWLFDO 0HFKDQLFV RI ,QWHUIDFLDO 3URSHUWLHV RI 3RO\DWRPLF )OXLGV 6XUIDFH 7HQVLRQ &KHP 3K\V f DQG 9 %RQJLRUQR 0RGLILHG YDQ GHU :DDOV 7KHRU\ RI )OXLG ,QWHUIDFHV 3K\V 5HY $ / f & $ (FNHUW DQG 0 3UDXVQLW] 6WDWLVWLFDO 6XUIDFH 7KHUPRn G\QDPLFV RI 6LPSOH /LTXLG 0L[WXUHV $ &K ( B f

PAGE 400

' (GZDUGV 5 (FNDUGW DQG ) 0 *DVSDULQL 6XUIDFH ([FLWDWLRQ DQG 6XUIDFH 7HQVLRQ RI 6XSHUIOXLG +HA 3K\V 5HY $ B f 3 $ (JHOVWDII DQG % :LGRP /LTXLG 6XUIDFH 7HQVLRQ 1HDU WKH 7ULSOH 3RLQW &KHP 3K\V f 5 (YDQV $ 3VHXGR$WRP 7KHRU\ IRU WKH 6XUIDFH 7HQVLRQ RI /LTXLG 0HWDOV 3K\V & B f % 8 )HOGHUKRI '\QDPLFV RI WKH 'LIIXVH *DV/LTXLG ,QWHUIDFH QHDU WKH &ULWLFDO 3RLQW 3K\VLFD A f ,= )LVKHU 6XUIDFH 3KHQRPHQD LQ /LTXLGV LQ 6WDWLVWLFDO 7KHRU\ RI /LTXLGV &KDS 8QLYHUVLW\ RI &KLFDJR 3UHVV &KLFDJR 6 )LVN DQG % :LGRP 6WUXFWXUH DQG )UHH (QHUJ\ RI WKH ,QWHUIDFH EHWZHHQ )OXLG 3KDVHV LQ (TXLOLEULXP QHDU WKH &ULWLFDO 3RLQW -&KHP 3K\V f ' )LWWV 7KHRUHWLFDO 'HWHUPLQDWLRQ RI WKH 6XUIDFH 7KLFNQHVV RI /LTXLG +HA DQG +HA 3K\VLFD f 3 )OHPLQJ $-0
PAGE 401

0 +DLOH ( *XEELQV DQG & *UD\ 9DSRU/LTXLG ,QWHUn IDFLDO 'HQVLW\2ULHQWDWLRQ 3URILOHV IRU )OXLGV ZLWK $QLVRn WURSLF 3RWHQWLDOV &KHP 3K\V f 7KHRU\ RI 6XUIDFH 7HQVLRQ IRU 0ROHFXODU /LTXLGV ,, 3HUWXUEDWLRQ 7KHRU\ &DOFXODWLRQV &KHP 3K\V Mf f ( : +DUW 7KHUPRG\QDPLF )XQFWLRQV IRU 1RQXQLIRUP 6\VWHPV &KHP 3K\V f +HQGHUVRQ ) ) $EUDKDP DQG $ %DUNHU 7KH 2UQVWHLQ =HUQLNH (TXDWLRQ IRU D )OXLG LQ &RQWDFW ZLWK D 6XUIDFH 0ROHF 3K\V f 6 +XDQJ DQG : : :HEE 'LIIXVH ,QWHUIDFH LQ D &ULWLFDO )OXLG 0L[WXUH &KHP 3K\V f & -RXDQLQ 6XUIDFH 3URSHUWLHV RI 6LPSOH /LTXLGV & 5 $FDG 6FL 3DULV 6HU $% f ) 0 .XQL DQG $ 5XVDQRY 'LVWULEXWLRQ )XQFWLRQV LQ 6XUIDFH /D\HUV ,9, 5XVV 3K\V &KHP A f 'LVWULEXWLRQ )XQFWLRQV LQ 5XVV 3K\V &KHP B 6XUIDFH /D\HUV f 9,,9,,, 'LVWULEXWLRQ )XQFWLRQV LQ 5XVV 3K\V &KHP 6XUIDFH /D\HUV ,;;,, f 'LVWULEXWLRQ )XQFWLRQV LQ 5XVV 3K\V &KHP A 6XUIDFH /D\HUV f ;,,,;,9 / /HERZLW] DQG 3HUFXV /RQJ5DQJH &RUUHODWLRQV LQ D &ORVHG 6\VWHP ZLWK $SSOLFDWLRQV WR 1RQXQLIRUP )OXLGV 3K\V 5HY f 6WDWLVWLFDO 7KHUPRG\QDPLFV RI 1RQXQLIRUP )OXLGV 0DWK 3K\V f /HH $ %DUNHU DQG 0 3RXQG 6XUIDFH 6WUXFWXUH DQG 6XUIDFH 7HQVLRQ 3HUWXUEDWLRQ 7KHRU\ DQG 0RQWH &DUOR &DOn FXODWLRQ &KHP 3K\V A f 6 /XL 3KDVH 6HSDUDWLRQ RI /HQQDUG-RQHV 6\VWHPV $ )LOP LQ (TXLOLEULXP ZLWK 9DSRU &KHP 3K\V f 5 $ /RYHWW 6WDWLVWLFDO 0HFKDQLFDO 7KHRULHV RI )OXLG ,QWHUIDFHV 3K' 'LVVHUWDWLRQ 8QLYHUVLW\ RI 5RFKHVWHU 'LVV $EVWU % f

PAGE 402

5 /RYHWW 3 : 'H +DYHQ 9LHFHOL -U DQG ) 3 %XII *HQHUDOL]HG YDQ GHU :DDOV 7KHRULHV IRU 6XUIDFH 7HQVLRQ DQG ,QWHUIDFLDO :LGWK &KHP 3K\V B f 5 /RYHWW & < 0RX DQG ) 3 %XII 7KH 6WUXFWXUH RI WKH /LTXLG9DSRU ,QWHUIDFH &KHP 3K\V f 0 0DQGHOO 7KH $GVRUSWLRQ RI +DUG 6SKHUHV 1HDU D &XUYHG 6XUIDFH &KHP 3K\V A f $ 0DQQ -U /DWHUDO 7UDQVSRUW 3URSHUWLHV RI ,QWHUIDFLDO 5HJLRQV WK 1DWLRQDO $,&K( 0HHWLQJ %RVWRQ 6 : 0D\HU &DOFXODWLRQ RI 0HWDO 6XUIDFH 7HQVLRQV ,RQLFVDOW DQG 0RQDWRPLF 0RGHOV IRU /LTXLG 0HWDOV &KHP 3K\V f 'HSHQGHQFH RI 6XUIDFH 7HQVLRQ RQ 7HPSHUDWXUH &KHP 3K\V f $ 0ROHFXODU 3DUDPHWHU 5HODWLRQVKLS %HWZHHQ 6XUIDFH 7HQVLRQ DQG /LTXLG &RPSUHVVLELOLW\ 3K\V &KHP A f ,QWHUUHODWLRQVKLSV $PRQJ 7KHUPDO ([SDQVLYLWLHV 6XUIDFH 7HQVLRQV DQG &RPSUHVVLELOLWLHV IRU 0ROWHQ 6DOWV &KHP 3K\V f 5 0F'RQDOG ( *XEELQV DQG 0 +DLOH 6XUIDFH 7HQVLRQ RI 3RODU /LTXLGV VXEPLWWHG WR &KHP 3K\V f & 0HOURVH 7KHUPRG\QDPLF $VSHFWV RI &DSLOODULW\ ,QG (QJ &KHP f f 7KHUPRG\QDPLFV RI 6XUIDFH 3KHQRPHQD 3XUH $SSO &KHP f 0L\D]DNL $ %DUNHU DQG 0 3RXQG $ 1HZ 0RQWH &DUOR 0HWKRG IRU &DOFXODWLQJ 6XUIDFH 7HQVLRQ &KHP 3K\V f : 0RUULV -U 7KH 7ZR3KDVH )OXLG ,QWHUIDFH DW (TXLOLEULXP $ &RQWLQXXP 0RGHO &KHP 3K\V B f 0 0XOKROODQG 6HOI'LIIXVLRQ 7KURXJK D 9DSRU/LTXLG ,QWHUIDFH &KHP 3K\V !B f 0 1D]DULDQ 6WDWLVWLFDO 0HFKDQLFDO &DOFXODWLRQ RI WKH 'HQVLW\ 9DULDWLRQ WKURXJK D /LTXLG9DSRU ,QWHUIDFH &KHP 3K\V f

PAGE 403

$&/ 2SLW] 0ROHFXODU '\QDPLFV ,QYHVWLJDWLRQ RI D )UHH 6XUIDFH RI /LTXLG $UJRQ 3K\V /HWWHUV $ f ( 2URZDQ 6XUIDFH (QHUJ\ DQG 6XUIDFH 7HQVLRQ LQ 6ROLGV DQG /LTXLGV 3URF 5R\ 6RF $ f + $ 3DSD]LDQ &RUUHODWLRQ RI 6XUIDFH 7HQVLRQ EHWZHHQ 9DULRXV /LTXLGV $P &KHP 6RF A f 1 3DUVRQDJH / $ 5RZOH\ DQG 1LFKROVRQ 0RQWH &DUOR *UDQG &DQRQLFDO (QVHPEOH &DOFXODWLRQ LQ 9DSRU/LTXLG 7UDQVLWLRQ 5HJLRQ IRU $UJRQ &RPSXWDWLRQDO 3K\VLFV
PAGE 404

3UHVVLQJ DQG ( 0D\HU 6XUIDFH 7HQVLRQ DQG ,QWHUIDFLDO 'HQVLW\ 3URILOH 1HDU WKH &ULWLFDO 3RLQW &KHP 3K\V f 3UHVVLQJ 1HZ (TXDWLRQ IRU 6XUIDFH 7HQVLRQ 1HDU WKH &ULWLFDO 3RLQW &KHP 3K\V B f 9 3VKHQLWV\Q DQG $ 5XVDQRY 2SWLFDO 0HWKRGV IRU 6WXG\LQJ WKH 6XUIDFH /D\HUV RI /LTXLG 6ROXWLRQV 9RS 7HUPRGLQ *HWHURJHXQ\NK 6LVW 7HRU 3RYHUNK
PAGE 405

$ 5XVDQRY DQG 9 3VKHQLWV\Q (OOLSVRPHWU\ DQG 7KLFNQHVV RI 6XUIDFH /D\HUV 'RNO 3K\V &KHP 3URF $FDG 6FL 8665 3K\V &KHP 6HF f $ 5XVDQRY &DOFXODWLRQ RI WKH &RPSRVLWLRQ RI 6XUIDFH /D\HUV DW WKH /LTXLG9DSRU ,QWHUIDFH LQ %LQDU\ DQG 7HUQDU\ 6\VWHPV 3RYHUNK
PAGE 406

6 7R[YDHUG 6XUIDFH 6WUXFWXUH RI D 6TXDUH:HOO )OXLG &KHP 3K\V f 6WDWLVWLFDO 0HFKDQLFDO DQG 4XDVLWKHUPRG\QDPLF &DOFXODn WLRQV RI 6XUIDFH 'HQVLWLHV DQG 6XUIDFH 7HQVLRQ 0ROHF 3K\V f 0ROHFXODU '\QDPLFV &DOFXODWLRQV RI WKH /LTXLG*DV ,QWHUn IDFH IRU D 7ZR'LPHQVLRQDO )OXLG &KHP 3K\V B f +\GURVWDWLF (TXLOLEULXP LQ )OXLG ,QWHUIDFHV &KHP 3K\V f 7ULH]HQEHUJ DQG 5 : =ZDQ]LJ )OXFWXDWLRQ 7KHRU\ RI 6XUIDFH 7HQVLRQ 3K\V 5HY /HWWHUV A f : :HOVFK 7KHRUHWLFDO 6WXGLHV LQ 6XUIDFH 3KHQRPHQD 3K' 'LVVHUWDWLRQ 8QLYHUVLW\ RI 3HQQV\OYDQLD 'LVV $EVWU % f % :LGRP 6XUIDFH 7HQVLRQ DQG 0ROHFXODU &RUUHODWLRQV QHDU WKH &ULWLFDO 3RLQW &KHP 3K\V B f ,QWHUIDFLDO 7HQVLRQV RI 7KUHH )OXLG 3KDVHV LQ (TXLOLEULXP &KHP 3K\V f : :RRGEXU\ -U 6XUIDFH 7HQVLRQ &DOFXODWLRQV IRU WKH +DUG 6TXDUH /DWWLFH *DV &KHP 3K\V A f 2Q WKH 6XUIDFH 7HQVLRQ 7KHRU\ RI YDQ GHU :DDOV DQG RI &DKQ DQG +LOOLDUG &KHP 3K\V 4 f ( 6 :X DQG : : :HEE &ULWLFDO /LTXLG9DSRU ,QWHUIDFH LQ 6)A 7KLFNQHVV RI WKH 'LIIXVH 7UDQVLWLRQ /D\HU 3K\V 5HY $ f &ULWLFDO /LTXLG9DSRU ,QWHUIDFH LQ 6)A ,, 7KHUPDO ([FLWDWLRQV 6XUIDFH 7HQVLRQ DQG 9LVFRVLW\ 3K\V 5HY $ f $ 6 9DUJDV 2Q WKH 0ROHFXODU 7KHRU\ RI 'HQVH )OXLGV DQG )OXLG ,QWHUIDFHV 3K' 'LVVHUWDWLRQ 8QLYHUVLW\ RI 0LQQHVRWD $-0
PAGE 407

%,2*5$3+,&$/ 6.(7&+ -$0(6 0,7&+(// +$,/( ZDV ERUQ RQ 'HFHPEHU LQ $WODQWD *HRUJLD DQG ILYH \HDUV ODWHU VXIIHUHG KLV ILUVW H[SRVXUH WR DFDGHPLD DW WKH KDQGV RI WKH GLVFLSOLQHG 6LVWHUV RI 6W 0DU\nV LQ +XQWVYLOOH $ODEDPD +H DWWHQGHG 9DQGHUELOW 8QLYHUVLW\ RQ SDUWLDO DFDGHPLF VFKRODUVKLS DQG HDUQHG D %DFKHORUnV 'HJUHH LQ &KHPLFDO (QJLQHHULQJ LQ -XQH 'XULQJ WKH VXPPHU RI KH ZDV HPSOR\HG DV D 3URFHVV 'HVLJQ (QJLQHHU DW WKH 3OXWRQLXP 'HYHORSPHQW /DERUDWRU\ %DEFRFN DQG :LOFR[ &RUSRUDWLRQ /\QFKEXUJ 9LUJLQLD ,Q 6HSWHPEHU KH HQWHUHG WKH 1DYDO 2IILFHUV &DQGLGDWH 6FKRRO 1HZSRUW 5KRGH ,VODQG DQG ZDV FRPPLVVLRQHG DQ (QVLJQ LQ WKH 86 1DYDO 5HVHUYH RQ 9DOHQWLQHnV 'D\ +H VHUYHG WRXUV DERDUG WKH 866 6+$1*5,/$ &96f RQ
PAGE 408

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ tO0cA .HLWK ( *XEELQV &KDLUPDQ 3URIHVVRU RI &KHPLFDO (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ A!L -RKQ3 2n&RQQHOO 3URIHVVRU RI &KHPLFDO (QJLQHHULQJ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ -DMWLLn6n : 'XIW\ $VVRFLDWH 3URIHVVRU RI 3K\VLFV 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH &ROOHJH RI (QJLQHHULQJ DQG WR WKH *UDGXDWH &RXQFLO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 'HFHPEHU 'HDQ *UDGXDWH 6FKRRO

PAGE 409

81,9(56,7< 2) )/25,'$


UNIVERSITY OF FLORIDA
3 1262 08554 5878


SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS
By
JAMES MITCHELL HAILE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976

ACKNOWLEDGEMENTS
It is a pleasure to express my gratitude to those who have freely
contributed to this work through their instruction, guidance, and advice.
Keith Gubbins initiated the research reported herein and enthu¬
siastically stimulated and supported its development, as well as my own
professional growth. Dr. Gubbins consistently provided an extraordinary
environment for learning, provided a strong example of the scientific
method, and exposed me to numerous knowledgeable scientists and engineers
on both sides of the Atlantic. In addition, he expended considerable
effort in obtaining the financial support, travel and computer funds
which made this work possible.
Bill Streett generously allowed me to spend several months in
his laboratory at the U.S. Military Academy and patiently taught me
molecular dynamics. He obtained the copious amounts of computer time
used in the molecular dynamics work reported here and kept the program
running in my absence. Many of the ideas for presenting the molecular
dynamics results came to light in discussions with Colonel Streett.
Further, I am grateful to Colonel and Mrs. Streett for the hospitality
extended to me during my visits to West Point.
John O'Connell, University of Florida, continually inspired me
through open-ended questioning concerning classical and statistical
thermodynamics, science, engineering, and, most importantly, the
character of life.
ii

Chris Gray, University of Guelph, instructed me in spherical
trigonometry, spherical harmonic expansions, Racah algebra, etc.,
thereby developing in me a healthy respect for the physicist's view
of applied science.
I have benefited greatly from countless discussions with my
colleague Chorng-Horng Twu on various aspects of thermodynamics,
statistical mechanics, numerical methods, and Chinese cooking.
S«5ren Toxvaerd, University of Copenhagen, contributed much
valuable advice on the theory and associated calculations for fluid
interfaces. Thanks are also due Dr. Toxvaerd for providing a copy
of his computer program for calculating the vapor-liquid interfacial
density profile for Lennard-Jones fluids.
Peter Egelstaff allowed me to spend several months in the
stimulating atmosphere of the Physics Department at the University
of Guelph. I am grateful to the faculty and staff for their hospi¬
tality and for the large amount of NOVA 2 computer time made available
to me. I am especially thankful to Dan Litchinsky for useful advice
on NOVA 2 software and to Ross McPherson for timely hardware support
on the NOVA. I am also indebted to Shien-Shion Wang for spending many
hours in teaching me the Monte Carlo method.
Dick Dale and Ron Franklin of the Engineering Information Office,
University of Florida, gave timely and enthusiastic photographic tech¬
nical assistance in producing the filmed animation of molecular dynamics
simulations. Larry Mixon in the Northeast Regional Data Center, Univer¬
sity of Florida, provided valuable software support in developing the
filmed animation technique.
iii

I am grateful to Dr. J. W. Dufty for serving on the supervisory
committee. I would also like to remember Dr. T. M. Reed who was an
original member of the committee and who strongly encouraged me in the
initial phases of the research reported here.
P.S.Y. Cheung, T. Keyes, C. G. Gray and R. L. Henderson, W. B. Streett,
and S. Toxvaerd kindly provided manuscripts of their work prior to publica¬
tion.
Mrs. J. Ojeda, University of Florida, performed the remarkably
excellent typing of the manuscript.
Finally, I am grateful to Tricia who, in addition to all the
\
usual annoyances with which wives of Ph.D. students are plagued,
quietly endured our being separated for the greater part of the last
year and a half of this work.
The three dimensional drawings presented in Chapter 7 were done
on a Gould 5100 electrostatic plotter driven by the IBM 370/165 at the
Northeast Regional Data Center, University of Florida. The associated
k
software was the SYMVU Computer Graphics Program, Version 1.0, of the
Laboratory for Computer Graphics and Spatial Analysis, Harvard University.
I thank the Petroleum Research Fund (administered by the American
Chemical Society) and the National Science Foundation for financial
support of this study.
iv

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF TABLES ix
LIST OF FIGURES xv
KEY TO SYMBOLS xxii
ABSTRACT xxix
CHAPTERS:
1 INTRODUCTION 1
1.1 Theory of Surface Properties 6
1.2 Computer Simulation Methods 9
1.3 Outline of Dissertation 10
2 THEORY OF SURFACE TENSION 13
2.1 General Expressions for Surface Tension of
Polyatomic Fluids 13
2.2 General First Order Perturbation Theory for
Surface Tension 18
2.3 Perturbation Theory for Surface Tension using
a Pople Reference 23
2.4 Fowler Model Expressions for Perturbation Terms
Y2A’ Y2B’ Y3A’ Y3B in P°Ple ExPansion 29
2.5 Superficial Excess Internal Energy from the
Padé Perturbation Theory for Surface Tension 32
3 NUMERICAL CALCULATIONS OF SURFACE TENSION 34
F F F F
3.1 Evaluation of Y2A, Y3g> Y3A> and y3b 35
3.2 Surface Tension Calculations for Model Fluids.... 40
v

TABLE OF CONTENTS (Continued)
CHAPTERS: Page
3.3 Calculation of the Superficial Excess Internal
Energy for Model Fluids 46
3.4 Surface Tension Calculations for Real Fluids 48
3.5 Correlation of Surface Tension for Pure Poly¬
atomic Liquids.... 58
4 VAPOR-LIQUID DENSITY-ORIENTATION PROFILES 72
4.1 First Order Perturbation Theory for p(z.U)^) 72
4.2 Calculations of p(z^co^) for Overlap and Dis¬
persion 78
5
6
MONTE CARLO SIMULATION OF MOLECULAR FLUIDS ON. A
MINICOMPUTER 94
5.1 Introduction 94
5.2 Monte Carlo Method for Nonspherical Molecules 96
5.3 Description of the Minicomputer System 101
5.4 Monte Carlo Program for the NOVA 102
5.5 Comparison of NOVA Results with Full-Size
Computer Results 107
5.6 Conclusions '. 112
MOLECULAR DYNAMICS METHOD FOR AXIALLY SYMMETRIC
MOLECULES 114
6.1Introduction 114
6.2Expressions for the Force and Torque for Axially
Symmetric Molecules 117
6.3 Method of Solution of the Equations of Motion
and the Molecular Dynamics Algorithm 129
6.4 Evaluation of Pair Correlation Functions 136
6.5Equilibrium Properties from the g^ ^ n/r12^
146

TABLE OF CONTENTS (Continued)
CHAPTERS: Page
7 MOLECULAR DYNAMICS RESULTS 158
7.1 Potential Models 158
7.2 Equilibrium Properties 171
7.3 Spherical Harmonic Coefficients, g„ „ (r,„) 191
7.4 Angular Pair Correlation Function 215
7.5 Site-Site Pair Correlation Functions 238
7.6 Filmed Animation of Molecular Motions 250
8 CONCLUSIONS 259
8.1 Theory for Surface Tension of Polyatomic Fluids... 259
8.2 Theory for the Interfacial Density-Orientation
Profile of Polyatomic Fluids 261
8.3 Computer Simulation of Polyatomic Fluids 262
APPENDICES:
A EXPRESSIONS FOR THE ANGLE AVERAGES IN EQUATIONS (3-4)
TO (3-7) 267
B COORDINATE TRANSFORMATION AND INTEGRATION OVER EULER
ANGLES TO OBTAIN EQUATIONS (2-89) AND (2-90) 270
B.l Choice of Euler Angles 270
B.2 Evaluation of Integral I 273
C MODELS FOR ANISOTROPIC POTENTIALS OF LINEAR MOLECULES... 277
D EXPRESSIONS FOR y^, y^g. AND y^g FOR VARIOUS
ANISOTROPIC POTENTIALS FOR AXIALLY SYMMETRIC MOLECULES.. 283
E THE INTEGRALS KY(U';nn'n") AND LY(£;nn') 288
F EXPRESSIONS FOR THE SPHERICAL HARMONIC COEFFICIENTS
g£ £ m(r12> IN EQUATION (6-77) 292
G THE INTEGRAL I USED TO CALCULATE THE ANGULAR COR¬
RELATION PARAMETER G2 FOR QUADRUPOLES 294
vii

TABLE OF CONTENTS (Continued)
APPENDICES: Page
H VALUES FOR THE g„ „ (r. J COEFFICIENTS 297
I VALUES FOR THE J INTEGRALS 328
n
J VALUES OF THE SITE-SITE CORRELATION FUNCTIONS 349
K VALUES OF THE INTEGRAL H^2,6^ 335
LITERATURE CITED 357
BIBLIOGRAPHY 365
BIOGRAPHICAL SKETCH 376
viii

2
3
4
5
6
7
8
9
10
11
12
13
Page
5
47
57
62
100
106
109
150
151
152
153
154
155
LIST OF TABLES
Examples of Macroscopic and Microscopic Interfacial
Properties Related by Equations of the Form (1-1)
Test of the Gibbs-Helmholtz Equation in the Fowler
Model Perturbation Theory for Lennard-Jones plus
Quadrupole Fluids
Potential Parameter Values used in Calculating Surface
Tension.
Values for the Parameters a^ and a2 in the Surface
Tension Correlation of Equation (3-61)
Equilibrium Properties in the Form of Ensemble Averages...
Approximate Number of Monte Carlo Configurations
Generated per Hour on the NOVA 2
Comparison of NOVA and CDC Results for Property Values
of Lennard-Jones + Quadrupole Model Fluid. kT/e = 0.719,
pa3 = 0.80, Q/(ea5)!/2 = 1
Expressions for in Terms of gn „ (r,J
a ^1 zm 1Z
for Various Model Potentials
Expressions
8H1H2n,(r12)
for the Configurational Energy in Terms
for Various Model Potentials
Expressions for the Pressure in Terms of g^ ^ (r-^)
Various Model Potentials 1.2
of
for
Expressions for the Fowler Model Surface Tension in
Terms of g0 „ (r,„) for Various Model Potentials
Expressions for the Fowler Model Surface Excess Internal
Energy in Terms of g„ „ (r „) for Various Model
Potentials 1.2?
Expressions for the Mean Squared Force in Terms of
gn n (r10) for Various Model Potentials
a/. JO-ID J.Z

Table
Page
14
15
16
17
18
19
20
21
22
23
24
25
26
27
Expressions for the Mean Squared Torque in Terms of
g¿ £ m^r12^ ^°r ^ar*ous M°del Potentials 156
Expressions for the Angular Correlation Functions in
Terms of g^ ^ m^r12^ ^or ^ar^ous Model Potentials 157
Primary Orientations for Pairs of Linear Molecules 162
Property Values of a Lennard-Jones plus Quadrupole
Fluid Obtained in this Work and Compared with those
given by Berne and Harp 177
Equilibrium Properties for Lennard-Jones plus Quadru¬
pole Fluid at pa3 = 0.85, Q/(ea3)-*-/2 = 2/2 181
Equilibrium Properties for Lennard-Jones plus Quadru¬
pole Fluid at po3 = .931, Q/(ea5)1/2 = 0.707 182
Equilibrium Properties for Lennard-Jones plus Quadru¬
pole Fluid at po3 = 0.85, Q/(ea5)1/2 = 1.0 183
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at po3 = 0.85,
0 /(ea5)1/2 = 1/2 188
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = .931,
Q/(ea5)1/2 = 0.707 189
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at pa3 = 0.85,
Q/( ea5)1/2 = 1.0 190
Equilibrium Properties for Lennard-Jones plus Overlap
Fluid at pa3 = 0.85, ó = 0.10 192
Equilibrium Properties for Lennard-Jones plus Overlap
Fluid at pa3 = 0.85, Ó = 0.30 193
Effect of Potential Model and State Condition on the
Fii'st Peak Height of the g„ „ Coefficients 201
l zm
Comparison of Molecular Dynamics Results for Equilibrium
Properties with Values Obtained from
£^£^1
J Integrals for Lennard-Jones plus Quadrupole Fluid
with po3 = 0.85, kT/e = 1.277 , Q/(ea5)5/2 = 0.5 210
x

Table
Page
28 Comparison of Molecular Dynamics Results for Equilib¬
rium Properties with Values Obtained from
J Integrals for Lennard-Jones plus Quadrupole
Fluid with pa3 = 0.85, kT/e = 1.294, Q/(ea5)1^ = 1.0... 211
29 Comparison of Molecular Dynamics Results for Equilib¬
rium Properties with Values Obtained from
J Integrals for Lennard-Jones plus Overlap Fluid
with po3 = 0.85, kT/e = 1.291, 6 = 0.10 212
30 Comparison of Molecular Dynamics Results for Equilib¬
rium Properties with Values Obtained from
J Integrals for Lennard-Jones plus Overlap Fluid
with po3 = 0.85, kT/e = 1.287, 6 = 0.30 213
31 Range of Values for Orientational Contributions to
Property Integrands for Quadrupole and Overlap Fluids... 243
Cl Expressions for the Expansion Coefficients E for
Various Interaction Potentials for Linear Molecules 279
C2
C3
C4
D1
D2
D3
D4
El
E2
Expressions for Anisotropic Potential Models in the
Intermolecular Frame of Figure 32 280
Expressions for Anisotropic Potential Models in the
Intermolecular Frame, using y rather than <(>
Derivatives of Various Anisotropic Potentials for
Evaluating the Force and Torque from Equations (6-25)
and (6-34)
281
282
IT
Expressions for yj: for Various Anisotropic Potentials
for Linear Molecules 284
F
Expressions for y^ for Multipole Potentials for
Linear Molecules 285
F
Expressions for y„B for Various Anisotropic Potentials
for Axially Symmetric Molecules 286
F
Expressions for y^B for Multipole Potentials for
Linear Molecules 287
The Integrals K^(££'£";nn'n") for Pure Fluids 289
The Integrals L^(£;nn') for Pure Fluids 290
xi

Table
Page
E3 The Constants in Equation El 291
G1 The Integral I ^ for Pure Fluids 296
HI Values of goOO^r12^ “ §400^r12^ For Lennard-Jones plus
Quadrupole Fluid at kT/e = 1.277, pad = 0.85,
Q/( ea5)1/2 = 0.5 298
H2 Values of g420^r12^ “ §442^r12^ for the Fluid of
Table HI 300
H3 Values of 8443(1^) - g660^r12^ for the Fluid of
Table HI 302
H4 Values of g000^r12^ “ ®4Q0^r12^ f°r Fennard-dones plus
Quadrupole Fluid with kT/e = 0.765, pad = 0.931, and
Q/(eo5 i/2 = 0.707 304
H5 Values of g420^r12^ “ g442^r12^ for the Fluid of
Table H4 306
H6 Values of g^Cr^) “ 8660^12^ for the Fluid of
Table H4 308
H7 Values of g000^r12^ “ g400^r12^ f°r Lennard-Jones plus
Quadrupole Fluid with kT/e = 1.294, pa^ = 0.85, and
Q/(eo5)1/2 = 1.0 310
H8 Values of g420^r12^ " g442^r12^ f°r t*ie Fdudd °f
Table H7 312
H9 Values of 8443(^2) - 8640^12^ for the Fluid of
Table H7 314
H10 Values of goo0^r12^ ~ g400^r12^ f°r Lennard-Jones plus
Anisotropic Overlap Fluid with kT/e = 1.291, pa^ = 0.85,
and ó = 0.10 316
Hll Values of g420^r12^ ” g442^r12^ f°r t*ie Fdudd °f
Table H10 318
H12 Values of 8443(^2) “ g660^r12^ for the Fluid of
Table H10 320
H13 Values of gQ00^r12^ ~ g400^r12^ f°r Lennard-Jones plus
Anisotropic Overlap Fluid with kT/e = 1.287, pad = 0.85,
and 6 = 0.30 322
xii

Table
Page
H14
Values of g49n(r19)
Table H13..7ÍV...7.
- §442^r12^ ^°r t*ie FFuid oF
••• 324
H15
Values of g,,.(r J
Table H13..777...7.
- §660^12^ for the Fluid of
11
The Integrals J^^
Quadrupole Fluid.
Q/Cea5)1/2 - 0.5...
222
- J for a Lennard-Jones plus
pa3n= 0.85, kT/e = 1.277,
12
The Integrals J ^
440
- J for the Fluid of Table 11...
n
• • 330
13
441
The Integrals J
n
- J^O for the Fluid of Table 11...
n
• • 331
14
620
The Integrals J^
- J^ 0 for the Fluid of Table 11...
n
• • 332
15
The Integrals J^^
Quadrupole Fluid.
QCea5)1/2 - 0.707..
222
- J for a Lennard-Jones plus
pa3n= 0.931, kT/e = 0.765,
16 The Integrals for the Fluid of Table 15 334
n n
17 The Integrals J^^ - J^^ for the Fluid of Table 15 335
n n
18 The Integrals J^® _ for the Fluid of Table 15 336
n n
000 222
19 The Integrals J - J for a Lennard-Jones plus
Quadrupole Flui§. pa^n= 0.85, kT/e = 1.294,
Q/Ceo5)1/2 = 1.0 337
‘ 400 440
110 The Integrals J - J for the Fluid of Table 19 338
n n
111 The Integrals J^^ - J*^*3 for the Fluid of Table 19 339
n n
112 The Integrals J^® _ for the Fluid of Table 19 340
n n
000 222
113 The Integrals J - J for a Lennard-Jones plus
Anisotropic Overlap Fluid. pa3 = 0.85, kT/e = 1.291,
6 = 0.10 341
400 440
114 The Integrals J - J for the Fluid of Table 113.... 342
n n
115 The Integrals J^^ - J^® for the Fluid of Table 113.... 343
n n
116 The Integrals J^® - for the Fluid of Table 113.... 344
0 n n
000 222
117 The Integrals J - J for a Lennard-Jones plus
Anisotropic Overlap Fluid. pa3 = 0.85, kT/e = 1.287,
ó = 0.30 345
xiii

Table
Page
400 440
118 The Integrals J - J for the Fluid of Table 117.... 346
n n
119 The Integrals J441 - J6°° for the Fluid of Table 117.... 347
n n
120 The Integrals J620 - J660 for the Fluid of Table 117.... 348
n n
J1 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with £/a = 0.3292, kT/e = 1.277,
pa3 = 0.85, Q/(CO5)1/2 = 0.5 350
J2 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with &/a = 0.2955, kT/e = 0.765,
pa3 = 0.931, Q/(eo5)l/2 = 0.707 351
J3 Site-Site Correlation Function for Lennard-Jones plus
Quadrupole Fluid with £/0 = 0.3292, kT/e = 1.294,
pa3 = 0.85, Q/(ea3)l/2 = i.o 352
J4 Site-Site Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid with £/o = 0.3292, kT/e = 1.287,
pa3 = 0.85, 6 = 0.10 353
J5 Site-Site Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid with ¿/a = 0.3292, kT/e = 1.287,
pa3 = 0.85, ó = 0.30 354
K1 The Integral for pUre Fluids 356
xiv

LIST OF FIGURES
Figure Page
1 Relation of Theory, Experiment, and Computer Simula¬
tion in the Study of Liquids 2
2 Variation of Fluid Density with Position through a
Planar Vapor-Liquid Interface 4
3 Two Possible Values for the Maximum Zj Value for a Pair
of Molecules in the Fowler Model Interface 22
4 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.85,
kT/e = 1.273 42
5 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. po^ = 0.45,
kT/e = 2.934 43
6 Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Quadrupole Model Potential. p0^ = 0.85,
kT/e = 1.273 44
7 Fowler Model Surface Tension for Fluids of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Various Anisotropic Potentials. pc?3 = 0.85,
kT/e = 1.273 45
8 Corresponding States Plot for Surface Tension of
Simple Liquids 52
9 Surface Tension for CO2 Comparing Perturbation Theory
Calculations with Experimental Values 55
10 Surface Tensions for C2H2 and HBr Comparing Perturba¬
tion Theory Calculations with Experimental Values 56
11 Test of Surface Tension Correlation for CO2 64
12 Test of Surface Tension Correlation for Acetic Acid 65
13 Test of Surface Tension Correlation for Methanol 66
xv

Figure Page
14 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Polyatomic Liquids 68
15 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Polyatomic Liquids 69
16 Test of Surface Tension Correlation for n-Hexane and
n-Octane 70
17 Comparison of Surface Tensions Calculated from the
Correlation with Experimental Values for Several
Hydrocarbons 71
18 Interfacial Density Profile for Lennard-Jones Fluid 80
19 Interfacial Density-Orientation Surface for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 83
20 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 84
21 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Dispersion Model Potential. kT/e = 0.85,
K = 0.25 85
22 Difference in Normal and Tangential Components of Stress
Tensor for Lennard-Jones Fluid 87
23 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, ó = 0.10 89
24 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = 0.10 90
25 Interfacial Density-Orientation Profiles for a Fluid of
Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = -0.10 91
xv i

Figure Page
26 Interfacial Density-Orientation Profiles for a Fluid
of Axially Symmetric Molecules Interacting with Lennard-
Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = -0.10 92
27 Simplified Schematic Flow Diagram of Fortran Monte
Carlo Program Developed for NOVA 2 2.05
28 Comparison of CDC and NOVA 2 Monte Carlo Results for
the Center-Center Pair Correlation Function for a
Lennard-Jones plus Quadrupole Fluid 210
29 Comparison of CDC and NOVA 2 Monte Carlo Results for
the Angular Pair Correlation Function for a Lennard-
Jones plus Quadrupole Fluid for Molecular Pairs in
the Tee Orientation Ill
30 Methods of Specifying the Orientation of an Axially
Symmetric Molecule 119
31 Orientation Angles for Axially Symmetric Molecules in
an Arbitrary Space Fixed Frame 121
32 Orientation Angles for Axially Symmetric Molecules in
the Intermolecular Frame 122
33 Geometry of a Pair of Diatomic Molecules 142
34 Pair Potential for Lennard-Jones plus Dipole Model
Fluid at Primary Pair Orientations 161
35 Pair Potential for Lennard-Jones plus Quadrupole Model
Fluid at Primary Pair Orientations 165
36 Surface of the Lennard-Jones plus Quadrupole Pair
Potential for the Tee Orientation as a Function of
the Quadrupole Strength 1 166
37 Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. p/(ecr3)l/2 = 1.0, Q/(ea5)l/2 = 2.75 167
38 Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. y/(ea3)^/2 = 1.75, Q/(ea^)!/2 = l.o 168
39 Pair Potential for Lennard-Jones plus Anisotropic
Overlap Model Fluid at Primary Pair Orientations 270
xvii

Figure
Page
40
41
42
43
44
45
46
47
48
49
50
51
52
53
Mean-Squared Displacement of Molecular Centers of
Mass for Lennard-Jones plus Quadrupole Fluid
Fluctuation in Temperature for Lennard-Jones plus
Quadrupole Fluid
Fluctuation in the Ratio of Translational to Rotational
Kinetic Energy for Lennard-Jones plus Quadrupole Fluid 176
Effect of Quadrupole Moment on the Center-Center Pair
Correlation Function at pa3 = 0.85 194
Spherical Harmonic Coefficients g„„ for Lennard-Jones
plus Quadrupole Fluid at po = 0.85, kT/e = 1.294,
Q/(ea5)1/2 = 1.0 196
Spherical Harmonic Coefficients g/n for the Fluid of
. . . 4 X/ oITl
Figure 44 4 197
Spherical Harmonic Coefficients g,, for the Fluid of
Figure 44 T 198
Spherical Harmonic Coefficients g „ _ for the Fluid of
Figure 44 2 199
Effect of Anisotropic Overlap Parameter on the Center-
Center Pair Correlation Function at pa3 = 0.85 203
Spherical Harmonic Coefficients gn„ for Lennard-Jones
plus Anisotropic Overlap Fluid at pa3 = 0.85, kT/e = 1.287,
6 = 0.30 204
Spherical Harmonic Coefficients g. . for the Fluid of
Figure 49 ?T 205
Spherical Harmonic Coefficients g,, for the Fluid of
Figure 49 T 206
Spherical Harmonic Coefficients g,5 for the Fluid of
Figure 49 .2 207
2
Integrands [g22o^r^ ~ 2g221('r^ + 28222^r^r and
[g220^ + 4/38221^ + 1/38222^r^r_3 for G2 and Ua’
respectively, for Lennard-Jones plus Quadrupole Fluid 214
xviii

jgu
54
55
56
57
58
59
60
61
62
63
64
65
Page
217
218
220
222
223
226
227
228
229
231
232
234
Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for the Tee Orienta¬
tion (0^ = 90°, 02 = 0,

Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for the Cross and
Parallel Orientations (0^ = = 90°)
Angular Pair Correlation Function for the Lennard-Jones
plus Quadrupole Fluid of Figure 44 for a Skewed Orienta¬
tion (0^ = @2 = Angular Pair Correlation Function for the Lennard-Jones
plus Anisotropic Overlap Fluid of Figure 49 for the Tee
Orientation (0^ = 90°, ©2 = 0,

Angular Pair Correlation Function for the Lennard-Jones
plus Anisotropic Overlap Fluid of Figure 49 for the Endon
Orientation (0^ = @2 =

Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0 = 90, = 0,
with Q* = 0.5, T* = 1.277, p* = .85 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0 = 90,

with Q* = 1.0, T* = 1.294, p* = .85 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones .plus Quadrupole Fluid for 0.. = 90, 0„ = 90,
with Q* = 0.5, T* = 1.277, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 01 = 90, 0„ = 90,
with Q* = 1.0, T* = 1.294, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0, = 45, 0„ = 45,
with Q* = 0.5, T* = 1.277, p* = .85 7 7
Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Quadrupole Fluid for 0, =45, 0„ = 45,
with Q* = 1.0, T* = 1.294, p* = .85 7 7
Comparison of Peak Heights in the Angular Pair Correla¬
tion Function with Well Depths in the Pair Potential
for the Lennard-Jones plus Quadrupole Fluid with
Q/Cea5)1/2 = 1.0
XIX

Figure Page
66 Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Anisotropic Overlap Fluid for
0X = 90, 4> = 0, with 6 = 0.10, T* = 1.291, p* = 0.85 235
67 Surface of the Angular Pair Correlation Function for
Lennard-Jones plus Anisotropic Overlap Fluid for
= 90, = 0, with 6 = 0.30, T* = 1.287, p* = 0.85 236
2
68 Integrand for Internal Energy, r u(12)g(12), for
Lennard-Jones plus Quadrupole Fluids for the Tee
Orientation 239
69 Integrand for Pressure, r^ g(12), for Lennard-
Jones plus Quadrupole Fluids for the Tee Orientation 240
2
70 Integrand for Internal Energy, r u(12)g(12), for Lennard-
Jones plus Anisotropic Overlap Fluids for Parallel
Orientation 241
71Integrand for Pressure, r^ ^ g(12), for Lennard-
Jones plus Anisotropic Overlap Fluids for Parallel
Orientation 242
72 Site-Site Pair Correlation Function for Lennard-Jones
plus Quadrupole Fluids 245
73 Possible Square Packing of Lennard-Jones plus Quadru¬
pole Molecules for Interpreting gag(r) 247
74 Site-Site Pair Correlation Function for Lennard-Jones
plus Anisotropic Overlap Fluids 249
75Box Representing the Molecular Dynamics System with the
Volume Element Sampled for the Filmed Animation Indicated. 253
76 Initial FCC Lattice Configuration of Lennard-Jones
Molecules in the Volume Element Sampled in the Filmed
Animation 256
77 Frame from the Filmed Animation of Lennard-Jones Mole¬
cules Corresponding to the Sixth Time-Step in the
Molecular Dynamics Calculation 257
78 Frame from the Filmed Animation of Lennard-Jones Mole¬
cules Corresponding to the 101st Time-Step in the
Molecular Dynamics Calculation 258
xx

Page
Figure
B1
Rotations
Defining the Euler Angles
40x1
••• 271
B2
Rotations
Interface
in the Triangle 123 in the
to Define Values for z
max
Fowler Model
... 274
xxi

KEY TO SYMBOLS
c(£1
A
. c
A.
x
A(zi’z12)
B(zi,z12)
V
D
£
i
mn
£*
mn
Fi
F2A,F2B
F3A’F3B
n
Roman Upper Case
Helmholtz free energy
Configurational Helmholtz free energy
The 1th term in the perturbation expansion for
Helmholtz free energy
Function defined in Equation (4-26)
Function defined in Equation (4-27)
Residual contribution to constant volume heat capacity
Clebsch-Gordan coefficient
Diffusion coefficient
Representation coefficient
Complex conjugate of representation coefficient
Force on molecule 1
Function defined in Equation (2-32)
Functions defined in Equation (2-62) and (2-63),
respectively
Functions defined in Equations (2-77) and (2-78),
respectively
Angular correlation parameter
Integral defined in Equation (3-34)
xxii

I =
I.(r)
^nn' Z
K =
K
KY
L =
L =
LY =
N =
=
P =
P.
l
P^x)
o(i)
Q
*
T
Moment of inertia
Functions defined in Equations (6-65) and (6-69)
Integral defined in Equations (2-40) and (B7)
Integral defined in Equation (Gl)
Integral defined in Equation (3-16)
Integral defined in Equation (6-89)
Coefficient of ellipticity for plane polarized light
Integral defined in Equation (3-25)
Integral defined in Equation (3-18)
Functions defined in Equations (4-30) and (4-31),
respectively
Angular momentum operator
Integral defined in Equation (3-24)
Integral defined in Equation (3-17)
Number of molecules
Avogadro's number
Pressure
t h
The i component of the local polarization vector
Legendre polynomial of order i
Probability of the i^ state occurring, defined in
Equation (5-8)
Quadrupole moment
5 1/2
Reduced quadrupole moment = Q/(ea )
General multipole moment
Temperature
Critical temperature
xxiii

tr
*
T
= Reduced temperature, T/T^
= Reduced temperature, kT/e
U
s
= Superficial excess internal energy
V
= Volume
V
c
= Critical volume
Y£m
*
Y*m :
= Spherical harmonic
= Complex conjugate of
Z
= Configurational integral
Roman Lower Case
a. =
i
= Semiempirical parameters defined in Equations (3-50)
to (3-53)
c =
= Constant defined in Equations (4-21) and (4-22)
c =
= Cosine of angle d). .
ij
c. =
1
= Cosine of angle 0^
c(y) =
= Cosine of angle y
c(zlz2r12) =
= Interfacial direct correlation function
d =
= Hard sphere diameter defined in Equation (3-39)
d,dft
5 Maximum allowable step lengths for translational and
rotational motion, respectively, of molecules in one
Monte Carlo generated configuration
t(zl-El 2> '
Interfacial pair distribution function for spherical
molecules
£(zli12£13) '
Interfacial triplet distribution function for spherical
molecules
g(r12) =
Radial distribution function
xxiv

gc(rl2)
= Center-center pair correlation function
gaB^r12)
= Site-site pair correlation function
g(zlz2r12)
= Interfacial pair correlation function
g(r12a)lW2)
= Angular pair correlation function
g(12)
= g(r12UJ1t02)
gi.1i,2m(rl2)
= Coefficients in the expansion of g(12) in spherical
harmonics of the molecular orientations
fi. =
—1
= Unit vector aligned along the axis of linear molecule i
k =
= Boltzmann's constant
i =
= Molecular bond length between atoms
m :
= Molecular mass
n :
= Index of refraction
n =
= Constant, values given in Equations (4-21) and (4-22)
n =
= Exponent of r in repulsive part of Mie potential,
Equation (3-32)
n =
s
= Power of r in various model potentials
PN
= Normal component of the stress tensor
pt =
= Tangential component of the stress tensor
r. =
—i
= Vector location of center of mass of molecule i
-12
= Vector separation of centers of mass of molecules 1 and 2
rl2
*
r12
: Magnitude of r^2
Reduced distance, r^/a
r =
m
Value of r where pair potential is a minimum
s. =
i
Sine of angle 0^
t =
Time
u(ri2) =
Spherically symmetric intermolecular pair potential
XXV

u(r12“lu2)
= Orientation dependent intermolecular pair potential
u (12)
= u (r12ü)1a)2)
= Isotropic, reference contribution to u(12)
ua(12)
"*(rl2)
= Anisotropic contribution to u(12)
= Reduced potential, uCr^)/^
u
s
= Superficial excess internal energy per unit of surface area
V
= Molecular translational velocity
X12
= x-component of r^
y12 '
= y-component of r^
y(ri2) ;
= Function defined in Equation (3-38)
Z1 :
= z-coordinate location of molecule 1
Z12 :
= z-component of r^
z =
max
*
zi =
= Maximum value of z-, defined in Equations (2-42) and (B6)
= Reduced coordinate, z^/o
5 =
Roman Script
= Interfacial area
U =
= Total intermolecular potential
E =
= Coefficients in the spherical harmonic expansion of the
anisotropic pair potential
Greek Upper Case
r. =
i
: Surface adsorption of component i
A =
Triplet of indices, for example
52 =
Volume in angle space
xxvi

Í2 = Angular velocity
= The l^k component of Q_
a
a.
i
a. ,B.
3
Y
Y
Y
F
*
Y
6
Ó. .
ij
e
0(x)
0.
Greek Lover Case
= Adjustable parameter in Equations (4-26) and (4-27)
= Interior angle in the triangle formed by r^2» ri3’ r23’
at molecule i
= Constants in the predictor-corrector algorithm; values
given in Equations (6-45) and (6-46), respectively
= Azimuthal and polar angles, respectively, for molecular
orientation in the space fixed frame
= 1/kT
= Angle between the axes of a pair of linear molecules
= Surface tension
= Fowler model surface tension
=â–  The i*1^ term in the perturbation expansion for surface
tension
2
= Surface tension reduced by potential parameters, yO /e
= Surface tension reduced by critical constants,
Y(Vc/NA)2/3(kTc)"1
= Dimensionless anisotropic overlap parameter
= Kronecker delta
= Interinolecular potential energy parameter
= Unit step function
- Polar angle for molecular orientation in the inter-
molecular frame
xxv ii

Angular displacement vector
Polar angle for r^ in spherical coordinates
Dimensionless anisotropic polarizability
Wavelength of incident light in light scattering
experiments
Perturbation parameter
Dipole moment
3 1/2
Reduced dipole moment, y/(eo )
Random numbers
Fluid number density
Interfacial number density profile
Interfacial number density-orientation profile
First order term in perturbation expansion for p(z^OJ^)
Intermolecular potential distance parameter
Standard deviation of property i
Torque on molecule 1
Azimuthal angle for molecular orientation in the inter¬
molecular frame
Azimuthal angle for in spherical coordinates
Function defined in Equation (3-19)
Set of variables specifying the orientation of molecule i
xxviii

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
SURFACE TENSION AND COMPUTER SIMULATION
OF POLYATOMIC FLUIDS
By
James Mitchell Haile
December, 1976
Chairman: Keith E. Gubbins
Major Department: Chemical Engineering
A third order thermodynamic perturbation theory for the surface
tension of polyatomic liquids has been developed using a Pople reference
fluid. The expansion includes terms containing the unknown pair and
triplet interfacial correlation functions, gQ(zand ^Zl—12—23^ ’
for the reference fluid. These are removed by using the Fowler approxima¬
tion for the interface. The resulting theory has been tested against
Monte Carlo results for the Fowler model surface tension of a fluid
whose molecules interáct with a Lennard-Jones plus dipole potential.
The behavior of the theory compared with similation parallels that
of bulk fluid properties; namely, the second order theory agrees with
the Monte Carlo results for small values of the dipole moment up to
3 1/2
y/(ea ) - 0.6. For larger dipole strengths neither the second nor
third order theories agree with the computer simulation results. How¬
ever, when the third order expansion is recast in the form of a simple
[1,2] Padé approximant, the theory agrees with the Monte Carlo results
up to dipole moments as large as u/(ecr ) = 1.75. This Padé theory
has been used to calculate the surface tension of pure dipolar and
xx ix

quadrupolar liquids. The theory agrees with experimental values of
surface tension in the neighborhood of the triple point; however, the
Fowler model does not give the correct temperature dependence of the
surface tension. The theory has also been used as a basis for develop¬
ing useful correlations of surface tensions of pure polyatomic liquids.
A first order perturbation theory has been developed for the
interfacial density-orientation profile p(z^üo^) for polyatomic fluids.
Upon introduction of a Pople reference, the first order term
vanishes for multipolar anisotropies, but does not vanish for anisotropic
overlap or dispersion potential models. Calculations of p(z^O)^) for
axially symmetric molecules interacting with each of the latter potentials
have been performed, using a Lennard-Jones reference fluid and the inter¬
facial pair correlation function model used by Toxvaerd. The calculations
indicate that the axially symmetric molecules have preferred orientations
in the interfacial region.
A method has been devised for using a NOVA 2 minicomputer with
32K words of core and external disc storage to perform Monte Carlo
simulations of 128 nonspherical molecules. The simulations generally
require several days of continuous calculation; however, several
equilibrium property values and values for the angular pair correlation
function gCr^co^t^) at five to seven specific orientations may be obtained
at a fraction of the cost of doing the calculations on a full size machine.
Results from the minicomputer compare within the statistical precision
with results previously obtained on CDC and IBM machines.
The method of molecular dynamics has been used to study systems
of 256 molecules interacting with Lennard-Jones plus quadrupole and
xxx

Lennard-Jones plus anisotropic overlap potentials. The equilibrium
properties determined include configurational internal energy,
pressure, Fowler model surface tension, Fowler model surface excess
internal energy, mean squared force, and mean squared torque. In
addition, the coefficients g^ ^ m^ri2^ :*'n an exPansi°n f°r the
angular pair correlation function gCr^^ci^) in terms of products
of spherical harmonics of the molecular orientations are determined.
Site-site pair correlation functions g D(r) are also found. Relations
otp
are developed between the g„ „ (r „) coefficients and the above listed
A/
equilibrium properties for several anisotropic potential models. Study
is made of orientational structure in quadrupolar and overlap fluids
via g(r^2W-^W2^ as obtained from the recombined spherical harmonic
expansion. A method of producing filmed animations of molecular
motions from molecular dynamics data is described.
xxxi

CHAPTER 1
INTRODUCTION
The development of rigorous methods for prediction of properties
of liquids must come from an understanding of how molecules are dis¬
tributed and interact with one another. Such fundamental understanding
of liquids may be pursued from three directions: theory, experiment,
and computer simulation. Molecular theory is the domain of statistical
mechanics; molecular level experimentation usually involves light,
x-ray, or neutron scattering, while computer simulation embodies the
Monte Carlo and molecular dynamics techniques. Computer simulation
adds a powerful new dimension to the study of matter which in no way
supplants the other two methods of study. As indicated in Figure 1,
simulation complements both theory and experiment. Thus, comparison
of simulation with laboratory experiment provides information about
the molecular interactions in the liquid. On the other hand, com¬
parison of simulation with theory provides a stringent test of the
theory.
In spite of current gains being made in the study of liquids,
fundamental understanding of fluid-fluid interfacial phenomena has
been slow to develop. The difficulty with study of interfacial
properties arises from the inhomogeneity, nonuniformity and anisotropy
of the interfacial region between homogeneous fluid phases. In general,
microscopic fluid properties vary through the interface from their
values in one bulk phase to their values in the other bulk phase, as
1

THEORY
EXPERIMENT
COMPUTER
SIMULATION
Figure 1. Relation of Theory, Experiment, and Computer
Simulation in the Study of Liquids

3
in Figure 2. These properties include density, refractive index,
dielectric constant, etc. The oft measured (macroscopic) properties
of interfaces are, in many cases, related to integrals over the
microscopic properties [1]:
X
OO
f
M(z)]dz
(1-1)
where X represents a macroscopic property, M is a microscopic property,
subscript B indicates, usually, a bulk phase value and k is a pro¬
portionality constant. Examples of specific properties having the
general form of (1-1) are given in Table 1. Equation (1-1) indicates
that prediction of macroscopic interfacial properties and comparison
with experiment is not a completely satisfactory test of theory. Of
more value is understanding of the microscopic properties and their
relations to macroscopic properties of interest. In the case of
theoretical study, such an approach must lie in statistical mechanics.
The prediction of macroscopic property values in a variety of
situations, e.g., over a large portion of the phase diagram and in
multicomponent systems, remains a vital engineering concern. In the
case of interfacial problems, the macroscopic property of interest is
the interfacial tension. The ability to predict interfacial tensions
is required in the equipment and process design and operation of numerous
fluid-fluid contacting operations, such as distillation, solvent extrac¬
tion, liquid membrane separation techniques, and tertiary oil recovery.
\
Interfacial tension is also important in understanding various biological
processes such as blood oxygenation and eye lubrication. Of particular

4
z
Figure 2.
Variation of Fluid Density with Position through
a Planar Vapor-Liquid Interface

5
TABLE 1
Examples of Macroscopic and Microscopic
Interfacial Properties Related by Equations
of the Form (1-1)
X = k
[M - M(z)]dz
D
r.
i
[Pi(z) - p±]dz +
[pi(z) - P^]dz
Y =
[pN “ PT(z)]dz
K =
X + D
00
P (z)
P (z)
z
X
P (°°)
p (°°)
—CO
z
X
dz
(1-2)
(1-3)
(1-4)
t
Reference [2]
r.
i
pi
PI
pi
Y
N
References [3,4]'
surface adsorption of component i
density of component i
bulk vapor phase density
bulk liquid phase density
surface tension
normal component of stress tensor
tangential component of stress tensor
PT(z)
K
P.
i
n
A
= coefficient of ellipticity for light plane polarized at
45° to the plane of incidence, when incident at Brewster's
angle
= ith component of local polarization vector
= bulk phase index of refraction
= wavelength of incident beam

6
importance is the determination of how molecular characteristics of
the fluid contribute to the interfacial tension. An important goal
in design considerations is the improved efficiency of fluid-fluid
contacting operations by modifying the system (e.g., by introduction
of additives) in order to lower the interfacial tension. This, in
turn, leads to consideration of how molecules absorb and orient
themselves in the interfacial region.
1.1 Theory of Surface Properties
Readily used methods currently available for predicting inter¬
facial tension have been reviewed [5,6]. These methods include
corresponding states approaches, techniques based on regular solution
theory and scaled particle theory, and more empirical methods. These
methods are generally applicable to spherical and nearly spherical
molecules, nonpolar polyatomics and their mixtures. These methods
fail to accurately predict interfacial tensions for strongly polar,
quadrupolar or associating liquids.
Almost all rigorous theoretical work on interfacial phenomena
has been for fluids with spherically symmetric molecular interactions.
Two rigorous relations have been developed for surface tension. The
Kirkwood-Buff equation relates the surface tension y to the inter¬
facial density profile p(z^) and the interfacial pair correlation
function g(z1z2r12^ ^2’7^
Y = T
dz,
d—12
du(r12)
dr
12
P S
*12-
J12
12
(1-5)

7
The Kirkwood-Buff equation is valid for spherical molecules and
assumes the intermolecular potential to be a sum of pair potentials.
This relation is intractable as it stands because of the unknown
function Consequently, the Kirkwood-Buff relation has
been studied using various simplifying models for the interfacial
region. The simplest such model is due to Fowler [8] and assumes
an abrupt transition from liquid to vapor phase. Further, the vapor
phase density is assumed to be negligible compared to the liquid
density. Thus, the model may be expressed as:
p(z1)p(z2) g(z1z2ri2^
0(-z1)0(-z2) pLgL(r12)
(1-6)
where 0 is the unit step function,
0(x)
1 if x ^ 0
0 if x < 0
(1-7)
subscript L indicates a bulk liquid property and the negative z
direction is into the liquid. The resulting Fowler-Kirkwood-Buff
expression for surface tension is:
0
4 du(r12} , ,
rl2 dr12 gL r12
(1-8)
As could be expected, the Fowler-Kirkwood-Buff theory works well near
the fluid triple point but gives increasing errors in surface tension
as the temperature is raised towards the critical point. Recent

8
evidence indicates that the good agreement at the triple point is
due to cancellation of errors [6].
The second rigorous relation for surface tension is a
generalized van der Waals equation which gives the surface tension
in terms of the density gradient dp(z^)/dz^ and the interfacial
direct correlation function cCz^^r^) [9]:
CO oo oo
•
f
dz..
dz„
1
2
—oo
-OO _
dx
12
dy
dp(z^) dp(z2)
12 dz.
dz,
c(z1z2r12)(x^2+ y^2) (1-9)
This relation is more general than the Kirkwood-Buff expression since
no assumption of pairwise additive potential is made in its derivation.
It has the further advantage that the direct correlation function c(r^2)
is generally of shorter range than the pair correlation function g(r^2).
The generalized van der Waals equation (1-9) has not been as thoroughly
studied as the older Kirkwood-Buff formula.
The Kirkwood-Buff equation has recently been generalized to
nonspherical molecules [10]. From both a practical standpoint and a
desire to gain understanding of interfacial phenomena, there is strong
need for using this new relation as a basis for developing predictive
methods for surface tension of polar and quadrupolar systems.
Evaluation of the interfacial density profile p(z^) is required
in order to determine the surface tension from the rigorous expressions
(1-5) and (1-9). Study of the interfacial density profile is also
of interest per se. Nearly all studies thus far have been for
spherical molecules. A variety of approaches have been used: a)
van der Waals theory [11], b) constant chemical potential through

9
the interface [12,13], c) first Born-Green-Yvon equation [14], d)
constant normal pressure through the interface [15,16], e) minimiza¬
tion of system free energy obtained by perturbation theory [17,18],
and f) computer simulation [19,20,21]. Determination of surface
tension for nonspherical molecules will require knowledge of the
interfacial density-orientation profile p(z^oo^), where is a set
of Euler angles specifying the orientation of molecule 1. Further,
there is considerable interest in determining how modification of
molecular orientation affects 'the surface tension.
1.2 Computer Simulation Methods
In the computer simulation approach to the study of liquids
either the Monte Carlo or molecular dynamics technique may be used.
These methods provide detailed information on equilibrium and time
dependent properties and on liquid structure for molecules interacting
with known force laws. In addition to use of simulation results as
a standard for evaluating theories of liquids, there is now serious
interest in using simulation as a vehicle for evolving and evaluating
realistic potential models for liquids. Much of the simulation work
has been performed for simple, spherical molecules. Only in the past
few years has simulation of nonspherical molecules been undertaken.
There is need of exploiting these simulation methods as fully as
possible since they provide a wealth of detailed information for
study.
Much of the effort in the history of computer simulation has
been expended in developing the methods, demonstrating their usefulness
and potential applicability and, in general, making simulation a

10
respectable research tool. There is currently a need to explore
methods for improving the efficiency of these calculations with a
view towards conserving computer resources and making the simulation
techniques accessible to more users. One possible approach is the
use of a minicomputer for performing simulations of liquids.
1.3 Outline of Dissertation
This work is divided into two parts. Part I includes
Chapters 2-4 and is concerned with the theory of vapor-liquid
interfaces for nonspherical molecules. In Chapter 2 a statistical
mechanical perturbation theory is developed for the surface tension
of polyatomic fluids. The general first order perturbation term is
obtained and the second and third order terms are found when a Pople
reference is used. Specific relations for the perturbation terms
are given for several anisotropic potential models: dipole, quadrupole,
overlap and dispersion. The corresponding perturbation terms are also
derived when the Fowler" approximation for the interface is introduced.
In Chapter 3 numerical calculations are reported for surface
tension using the perturbation theory in the Fowler model. The results
are compared with experiment and computer simulation. Semiempirical
methods based on perturbation theory are explored for correlating
surface tensions of polyatomic and polar substances.
Chapter 4 develops a perturbation theory for determining the
vapor-liquid interfacial density-orientation profile of fluids with
anisotropic potentials. Calculations are presented for overlap and
dispersion model interactions. The calculations predict that these
nonspherical molecules exhibit preferred orientations in the inter¬
facial region.

11
Part II of this work describes computer simulation studies
of linear molecules. Chapter 5 reports development of a method for
performing Monte Carlo calculations for linear molecules on a NOVA 2
minicomputer.
Chapter 6 describes the molecular dynamics method for linear
molecules, including: derivation of expressions for efficient evalua¬
tion of the force and torque exerted on a molecule due to various
anisotropic potential interactions, the method used to solve Newton's
equations of motion and the molecular dynamics algorithm for linear
molecules, evaluation of the coefficients in a spherical harmonic
expansion of the angular pair correlation function, and development
of relations between these coefficients and various fluid equilibrium
properties.
Chapter 7 reports the results of the molecular dynamics
calculations for equilibrium property values obtained for Lennard-
Jones plus quadrupole and Lennard-Jones plus anisotropic overlap
fluids. Comparisons of the equilibrium property values are made
with perturbation theory predictions in the case of the quadrupole
fluid calculations. Study is made of the local structure in these
fluids using the molecular dynamics determined angular pair cor¬
relation functions. A method for producing filmed animations of
molecular motions from molecular dynamics calculations is presented.
Chapter 8 draws conclusions from this study.

PART I
THEORETICAL STUDY OF FLUID INTERFACES

CHAPTER 2
THEORY OF SURFACE TENSION
2.1 General Expressions for Surface Tension of Polyatomic Fluids
There are two rigorous expressions for the surface tension of
polyatomic fluids. One is the generalized van der Waals equation (1-9).
The second is a generalization of the Kirkwood-Buff expression (1-5)
which has been previously derived [10,22]. In this section the deriva¬
tion for the general Kirkwood-Buff equation is summarized and its
simplified form obtained when the Fowler approximation is made.
2.1.1 Generalized Kirkwood-Buff Formula
Consider a planar interface between vapor and liquid phases with
a coordinate system oriented such that the xy plane is in the plane of
the interface and the +z direction points into the vapor. The two phase
system has N molecules .in volume V at temperature T and interfacial area
S. Thermodynamically, the surface tension y is related to the Helmholtz
free energy A of the two phase system by [23]:
Y =
3A
9S
NVT
(2-1)
c
Since only the configurational part of the free energy A depends on
the interfacial area, (2-1) may be written as:
Y = - kT
L £n Z
NVT
(2-2)

14
where
A = - kT £n Z
(2-3)
is the statistical mechanical definition of the configurational
Helmholtz free energy, k = Boltzmann's constant, and Z is the con¬
figurational integral. For nonspherical molecules:
Z =
r N N -3U(rNwN)
dr dw e —
(2-4)
where 3 = 1/kT and U is the full intermolecular potential. For non¬
spherical molecules U depends on the orientations ü) of the molecules,
as well as the positions of their centers of mass _r. The orientations
0) are usually specified by a set of Euler angles OJ = body-fixed reference frame on the molecule and a space-fixed frame
located external to the system. Substituting (2-4) into (2-2):
Y = -
kT
3_
• • •
f M W \
, N , N -3U(r w )
dr dto e -
3 S
\
.
(2-5)
NVT
The differentiation in (2-5) cannot be done immediately since the
N
integration limits on the integrals over r_ depend on 5. We, therefore,
follow Green [24] and change _r variables using the transformation:
cl/2 ,
x = b x
q1/2 x ,
y = S y
V ,
Z = I 2
(2-6)
Performing the differentiation in (2-5) and changing back to the old
variables gives:

15
• • •
f MM
, N N -gU(r a) )
dr di) e -
(~lu N N.
8u(r w )
•*
.35
NVT
(2-7)
Assuming the potential to be pairwise additive,
(2-7) becomes
U = j i u (r . .co .to.)
—ij i j
i (2-8)
Y
d£ld£2díjüidüJ2
f (£1£2ÜJiÜJ2'>
8u (12)]
l 95 J
NVT
(2-9)
where u(12) = u Cr^to^o^) and tde definition of the angular pair dis¬
tribution function has been used:
f (£1£2üjiüj2)
N(N-l)
Z
d£3'
■dV“Y
â– dor
a„, N N.
-pu(r to )
e —
(2-10)
Integrating (2-9) over x^, and transforming r_2 to £^2 8dves
dzl d—12 dü)idüJ2
f ^Z1-12CÜ1Ü32^ S
9u(12)
9S
\ y
NVT
(2-11)
P 3u(12)
as
can be evaluated for nonspherical molecules by considering:
„ 3u(12) „ 9-12 3u(12)
as 6 as ar12
_ 1 L 3u(12) _ 3u(12)~
' 2 [r12 3r12 JZ12 az12 _
(2-12)
(2-13)
Hence, (2-11) becomes:

16
Y = T
dz.
dr_12 dcü1dco2 f (ziL12U)iU)2^
3u(12)
12 3r12
- 3z
8u(12)
12 3z
12
Defining the angle average by:
(2-14)
<• • •>
“lU2 ft2
(• • • )düJ1doü2
(2-15)
where
fi =
dw
(2-16)
Equation (2-14) can be written:
Si
Y = 4-
dz,
, sr 3u (12) „ 3u (12),
-12 '“1-12 12'1 12 8r
12
12 8z12 üj1ü)2
(2-17)
Equation (2-17) is one form of the general Kirkwood-Buff equation.
Another form may be obtained by transforming (2-17) to spherical
coordinates. Since [25]
then
3u(12)
8z
12
= cos 0
X12,y12
12
3u(12)
3r
12
sin 0
12
012 ,(^12
12
3u(12)
80
12 r12^12
(2-18)
8u(12) „ 8u(12)
r, „ —^ - 3z
12 8r
12
'12 8z
12
2P2(cos 612)r12 8r12)
+ 3 sin 0^2 cos 0^2 '0gQ^2— (2-19)
where P2(x) is the second order Legendre polynomial, P2(x) = — (3x - 1)
Substituting (2-19) into (2-17) gives:

17
Y = YA +
(2-20)
ft
2
dz,
dri2 P2(cos 012) r12
V„ -
3ft
B 4
dz.
d—12 sIn 012 cos S12 <£(zl¿12“lw2) <2-22)
Equations (2-20,21,22) give the general Kirkwood-Buff equation. The
equation applies to general shaped molecules, the only assumption being
a pairwise additive potential.
In the case of spherical molecules, the potential u(12) goes to
u(r,„) so that the derivative in the Y„ term vanishes and Y, reduces to
12 B A
the Kirkwood-Buff equation (1-5).
2.1.2 Fowler-Kirkwood-Buff Equation for Nonspherical Molecules
The general expression for the surface tension (2-20) is in¬
tractable as it stands due to the unknown distribution function
f (z^r^co (02)' Vari°us simplifying assumptions can be made for f to
enable calculations to be performed. The simplest assumption is that
due to Fowler [8]:
2
f(z1r12(a10)2) = 0(-Zl) 0(-z2) gL(12) (2-23)
where 0 is the unit step function as in (1-7), p is the bulk liquid
density, and g is the bulk liquid angular pair correlation function.
Li
Introducing the Fowler model (2-23) into (2-21) allows the integrations
over z^ and co^2 to be performed giving:

18
ttpt
Ya =
dr
12 12
\‘12>
3u(12).
9r. „ oo,co„
12 12
(2-24)
where the superscript F indicates Fowler model. Similarly, (2-22)
reduces to:
YB 32
3tt2 2
PL
3 . 9u(12)
drl2 ri2 gL(12) 90, „ u,(i)
(2-25)
0
12
12
Further, the y term can be shown to vanish by symmetry arguments. One
D
such argument is that 9u(12)/90^2 is proportional to the 0^ component
of the force on molecule 2 due to molecule 1. This should vanish by
symmetry when averaged over U)^ and • Thus, the Fowler-Kirkwood-Buff
equation for nonspherical molecules is:
TTp
Y =
, 4 „ 9u(12)^
dr, „ r. „
12 12
0
3r12 “l“2
(2-26)
For the special case of spherical molecules, (2-26) reduces to
the usual FKB expression (1-8).
2.2 General First Order Perturbation Theory
for Surface Tension
2.2.1 Rigorous First Order Term
The difficulty with using the general Kirkwood-Buff expression
(2-20) lies with the, in general, unknown interfacial distribution
function f (z^_r^2u)^lJJ2^ ‘ This problem may be avoided by use of per¬
turbation theory, if the interfacial pair distribution function is
available (or may be approximated) for some reference substance.
Perturbation theory for fluid properties may be developed by con¬
sidering the anisotropic pair potential u(12) to be the sum of a
(known) reference term uq and a perturbing term u^:

19
u(12;A) = uq(12) + Au (12) (2-27)
where A is a perturbation parameter such that when A = 0, (2-27) gives
the reference potential and when A = 1, (2-27) gives the full potential.
Expanding the two-phase system Helmholtz free energy (2-3) in powers of
A and setting A = 1 gives:
A = A + A. + â–  â–  â– 
o 1
(2-28)
where
A1 2
d£xdr2
3u(12;A)
9A
(2-29)
A=0
and fQ is the interfacial angular pair distribution function for the
reference system.
The corresponding expansion for surface tension is obtained by
applying (2-1) to (2-28):
Y - Y0 + Y! +
where = ~2
9_
9S
dildÍ2 Fl(zl£l2)
NVT
and V2!^) E 9u(12;A)
9A
A=0 0)^2
(2-30)
(2-31)
(2-32)
Applying Green's method as in section (2.1), integrating over x^ and
y^, and transforming _r^ to _r 2» (2-31) becomes:

20
^1 2
dz.
d£l2 S
3F1(z1Í12)
3S
NVT
(2-33)
S SF^ (z^r\^2)/3S may be evaluated by considering:
9F U r ) 3r SF^r^) ^
•S —— = S —¡ra— * ^ 1 S ^ TTñ— (2-34)
9 S
3 S
3—12
3z^
9S
—12 9F1(Z1^12)
9—12
3 9F1(z1-12)
2 Z12 9z12
(2-35)
Thus, (2-33) becomes:
3Fl(z!—12>
3z,
dz.
9F1(z1-12) , 3 9F1(z1-12)
drl2 Z1 3z, + 2 Z12 3z12
3F,(z.r,„)
-12
11—12
9—12
(2-36)
The second two terms in (2-36) can be integrated by parts and shown
to cancel, leaving:
Y1 = '
d—12
dzl Zl<
3u(12;A)
3A
A=0
9fo(zA2V2)
3z,
“lu2
(2-37)
Equation (2-37) is valid for spherical or nonspherical reference fluids
the only assumption has been a pairwise additive potential. Note that
if the reference fluid is taken to be a Pople reference, defined by
(2-48) below, then y^ vanishes.

21
2.2.2 First Order Term in the Fowler Model
To make (2-37) amenable to calculation, the Fowler approximation
(2-23) may be introduced:
^1 2
2fl
dr
12
dzl Z1 3i7 0(-zl} 0(-z2}<
9u(12;A)
9A
n8°L^12^>(ja1a)„
A=0 12
(2-38)
Substituting (2-15) and changing the order of integration:
2Í2
d¿12
do)^ da>2
9u(12;A)
9A
(2-39)
A=0
where I =
z
dzl Z1 9¡~ 9('Z1) 9(-z2) 8oL(12)
(2-40)
The integral I may be evaluated by parts to give:
I
z
g T (12) z
oL max
(2-41)
where
max
-rl2 cos 012
lf 012 > 2
if 012 < I
(2-42)
and 0^2 is the spherical coordinate polar angle for _r 2> as shown in
Figure 3. Equation (2-39) becomes, therefore:
2
Yl = +
2u
dr
12
düJ^d(jJ2
9u(12;A)
9A
A=0
g T (12) z
oL max
(2-43)

22
z
z
a. ®12 < 1T^2
b. 012 > tt/2
Figure 3. Two Possible Values for the Maximum z, Value (z )
_ 1 max
for a Pair of Molecules in the Fowler Model Interface

23
Reintroducing the angle average (2-15) and noting that g(12) in the
bulk liquid is independent of (2-43) can be written as:
Performing the integration over gives:
(2-45)
Equation (2-45) is the general result for the first order perturbation
term in the Fowler approximation.
2.3 Perturbation Theory for Surface Tension
using a Pople Reference
When the general expansion for surface tension (2-30) is in¬
creased to higher order, the second order term is found to include a
term containing the reference four-body distribution function. Higher
order terms in the expansion contain even higher order multibody terms.
These complicated terms can be made to vanish up to at least the second
order term in the expansion by using the isotropic reference potential
first suggested by Pople [26]:
u0(ri2) =
(2-46)
With this choice of reference, (2-27) becomes:
u(12;A) = u (r „) + Au (12)
o 1/ a
(2-47)

24
(2-46) and (2-47) together give the simplification:
9u(12;A)
9A
>
A=0 u
i“2
= = 0
a 0)^2
Then (2-29) and (2-37) have:
(2-48)
Ax = Y1 = 0 (2-49)
and the second and third order terms (A^A^.y^y^) simplify. Thus, in
the Pople expansion, (2-28) and (2-30) become, to third order:
AC = A° + A2 + A3 (2-50)
Y = Yo + Y2 + Y3 (2-51)
2.3.1 Derivation of Second Order Terms, y and y
Z.i\ Z D
The second order term in (2-50) for the bulk phase liquid is [27]:
dLlá-2 alla.2
(2-52)
where g^(12) is the first order term in a perturbation theory expansion
of the angular pair correlation function. When the expansion is about
a Pople reference, g^(12) is given by [28]:
§1(12) = - Bua(12) go(r12) - 6P
dr [ + # ] x
—3 a 03^ a (jl)^
go(r12r13r23)
(2-53)

25
where §0(r^2rl3r23^ '*'s t*ie triplet correlation function for the reference.
When the anisotropic potential contains only spherical harmonics of order
it ^ 0, such as multipoles, then the angle averages in the second term in
(2-53) vanish. Such potential models as anisotropic overlap and dispersion
contain ÍL = 0 spherical harmonics, in which case the second term must be
included.
Equation (2-52) may be written, therefore, as
A2 A2A + A2B
(2-54)
where
P2B
2 A
d—ld—2 go(rl2) U)10)2
(2-55)
Bp
3 r
2B
dr dr dr, g (r10r1Qr„„)
—1 —2 —3 &o 12 13 23 a a a a CJlü)2tú3
(2-56)
Since _r^, and _r^ are each integrated over in (2-56), the indices are
dummy indices and (2-56) may, hence, be written as:
PB'
2B
dr dr dr„ g (r „r r„_) (2-57)
—1 —2 —3 o 12 13 23 a a
For a fluid nonuniform in the z-direction, (2-55) and (2-57)
generalize to:
2A 4
d—l^—2 Po(zl)Po(z2> 8o(zlil2> Vz
(2-58)

26
2B 2
dr dr dr p (z,)p (z„)p (z,) g (znr „r „)
—1 —2 —3 o 1 o 2 o 3 °o 1—12—13 a a
(2-59)
The corresponding terms in the expansion for surface tension (2-51)
may be found by applying (2-1) to (2-58) and (2-59):
Y
2A 4
2B 2
3_
9S
3_
3S
^2 F2A(ziri2>
- NVT
d^dr^ F2B(Z;Lr12)
(2-60)
(2-61)
NVT
where
F2A - P0(zl)P0(z2) go(zl¿12) Ul(02
(2-62)
F2B E Po(z1)Po(z2)Po(z3) go3 <2'63)
Equations (2-60) and (2-61) have the same form as (2-31); thus they may
be evaluated in a similar manner to give, analogous to (2-37):
f -1
2A 4
dr
12
dz, z
3F2A(21-El2)
1 1 8z,
(2-64)
Y
2B 2
d—12d—13
dz, z
9F2B(Z1-12^
1 1 3z,
(2-65)
Substituting (2-62) and (2-63) into (2-64) and (2-65), respectively,
and using the relations:

27
f
o
(Z1—12} = po(zl)po(z2) go(zl^l2)
(2-66)
f
o
(zl-12-13) Po(z1)Po(z2)Po(z3) go(zl-12-13)
(2-67)
gives,
2A
3
4
dr „
—12 a 0)^2
dz, z
3fo(zlil2)
1 1 3z,
(2-68)
2B
dr,„dr10
-12 -13 a a U)iü)2tú3
dz^ z^
3fo(zlll2ll3)
3z,
(2-69)
The derivation of (2-68) and (2-69) only assumes a pairwise
additive potential and use of a Pople reference fluid. If the aniso¬
tropic potential contains only £ ¿ 0 spherical harmonics, then 2$
vanishes because of (2-53).
2.3.2 Derivation of Third Order Terms, y_. and Yon
j_i-3 A L3 B
The third order term in (2-50) for the bulk phase liquid is:
A
3
d£]dJr2
g9 (12)>
0)^2
(2-70)
where is the second order term in the perturbation theory expansion
for g(12) [28J. If only anisotropic potentials containing £ / 0 spherical
harmonics are considered, g2(12) simplifies. Using arguments anologous
to those for simplifying the A„D term in (2-56), and generalizing to a
ZD
fluid which is nonuniform in the z-direction, (2-70) becomes (for £ ^ 0
harmonics):

28
A3 A3A + A3B
A3A
6
12 J
dridr9 g (Z-T ,)
—I —2 o 1 o 2 o 1—12 a W1W2
A3B 6
d—!d—2d—3 P0(zl)po(z2)po(z3) ^zl—12—13}
M
a a a
(2-71)
(2-72)
(2-73)
Applying (2-1) to (2-71) gives:
Y3 - ^3A + Y3B
(2-74)
where
Y3A 12
3B
£
6
3
as
—
'
a
as
-
dridr2 F3A(Zl£12)
(2-75)
NVT
d—!d—2d—3 F3B(zl—12—13}
(2-76)
NVT
and
F3A Wl0,2
(2-77)
F (z r r ) = p (z )p (z )p (z ) g (z r „r )
3B 1—12—11 o 1 o 2 o 1 o 1—12—11 a a a oi-^co^to^
(2-78)
Again, (2-75) and (2-76) have the same form as (2-31) and,
consequently, they may be evaluated in the same manner to give

29
analogous to (2-37) (the Jacobian of the transformation djr^ch:^dr_3 = J
drr^dr^dj:^ is unity):
Y3A 10
12
dr19
—12 a a)
dz. z
^o(zl—12>
1 ‘1 3z,
(2-79)
3B
dr 9dr19
—12 —13 a a a “ 2W3
dzl Z1
9fo(zl-12-13)
9z,
(2-80)
where f^z^r^) and fQ (zjjr^jr^) are 8*-ven by (2-66) and (2-67), respec¬
tively.
The derivation of (2-79) and (2-80) assumes: a) pairwise additive
potential, b) use of a Pople reference fluid, and c) the anisotropic
potential contains only terms with spherical harmonics of order £ ^ 0.
2.4 Fowler Model Expressions for Perturbation
Terms_Y2A_s_^2c> Y3A» Y3D in Pople Expansion
The Fowler approximation for the spherically symmetric Pople
reference may be expressed as:
fo(zl—12} = 0(-z1)0(-z2) pL.8oL(r12) (2“81)
fo^Zl—12—13^ = e(-z1)0(-z2)0(-z3) pl gol/r12^ (2-82)
Putting (2-81) into (2-68), (2-69), and (2-79) for y2 , y^R and y^,
respectively, and using (2-82) in (2-80) for y9 , all give terms analo-
gous to (2-39) for y^ in the Fowler model. In each case, the integration
over z^ may be done by parts giving, analogous to (2-44):

30
Bp
2 00
2A
2B
Bp
0
3 °°
dr r2 g (r )
12 12 a wiw2 oL 12
doo.. „ z (2-83)
12 max
dr12 rl2
dr „ r.. „
13 13 a a W1W2W3
dco, 0 dco1 „ g (r10r,.r„) z
12 13 °oL 12 13 23 max
(2-84)
Y
D2 2 00
F ^ PL
3A ” 12
0
dr „ r2 g (r10)
12 12 a 0)^2 6oL 12
doj „ z (2-85)
12 max
Y
2 3 °°
F 6 PL
3B 6
dr12 rl2
dr rf„
13 13 a a a W1W2W3
dw dto g (r „r r ) z
12 13 oL 12 13 23 max
(2-86)
In the case of the two-body terms, Y2A anc* Y^> t^ie va^ues f°r
Zmax are 8aven by (2-42); see Figure 3. Hence, the integration over
co
12 can be performed giving:
Y
2 00
,F *BpL t
2A 4
0
dr 0 r2 g (r )
12 12 &oL 12 a co.^
(2-87)
Y
„2 2 00
F 716 PL r
3A 12
dr r2 g (r )
12 12 oL 12 a ü)it°2
(2-88)
The integrations over un „ and co, _ in the three-body terms, Yo,,
lz ±J Zd
and Ytt>> cannot be performed immediately since z and g T (r, „,r,,,r„~)
Jij ITlclX • OL 1Z 1 j Z j
depend on these angles. A portion of this angle dependence may be
integrated out, however, if the angles ^2^12^13^13 are transformed to
a set of angles which include Euler angles specifying the orientation

31
of the triangle whose vertices are the molecules 1, 2 and 3. Details
of the coordinate transformation and integration over the Euler angles
are reserved for Appendix B. The resulting expressions are:
Y
2 3 00
F * *>L '
2B 2
dr12 r12
dr13 rl3
r12+ r13
dr23 r23 SoL(r12r13r23)
r12" r13 ^
x (r „ + r , + r__)
a a üji0J2ÜJ3 12 13 23
(2-89)
2„2 3 °°
* 6 PL '
3B
dr12 r12
dr13 r13
r12+ r13
dr23 r23 goL
ri2- r13l
X (r _ + r + r„„) (2-90)
a a a 12 13 23
F F F F
Computationally convenient forms for , Yo«> YtD> anc^ You are
zA 3A zB 3B
derived from (2-87), (2-88), (2-89), and (2-90), respectively, in the
F F
next chapter. Equations for each of these perturbation terms YnA> You’
2A 2B
F F
Y3A’ Y^g f°r specific anisotropic potentials are tabulated in Appendix D.
In the case of bulk fluid properties, Stell et al. [29] have
obtained improved results over the perturbation theory by resumming
the series (2-50) in the form of a Padé approximant:
(2-91)
The analogous form for the surface tension expansion (2-51) is:

32
Y = Y
o
+
1
Y
2
(2-92)
Calculations based on (2-92) in the Fowler model are given in the next
chapter.
2.5 Superficial Excess Internal Energy from the
Padé Perturbation Theory for Surface Tension
The surface excess internal energy is related to the temperature
dependence of the surface tension by the classical Gibbs-Helmholtz equa¬
tion:
u
s
= Y - T
(2-93)
where u^ is the surface excess internal energy, U , per unit of surface
area 5:
(2-94)
The Fowler approximation does not predict the correct temperature depen¬
dence of the surface tension for fluids of spherical molecules. Further,
the Fowler model may be used to obtain an equation for ug in terms of
the pair distribution function for the bulk liquid [2], and this second
equation is inconsistent with (2-93) [30,31]; Freeman and McDonald show
that these two expressions give quite different results for use in the
case of Lennard-Jones liquids [31]. To determine the validity of (2-93)
for the Padé perturbation theroy for surface tension presented above,

33
we combine (2-92) and (2-93) to obtain:
u = u + u (2-93)
s so sa
where u is
so
the isotropic reference contribution,
and
u
sa
T
(2-96)
1

CHAPTER 3
NUMERICAL CALCULATIONS OF SURFACE TENSION
The Fowler model perturbation theory developed in Chapter 2
has been used to calculate surface tensions for pure polyatomic
fluids. The intermolecular potential used for the fluids is of
the form:
U(r12a)lü)2) = Uo(r12) + Ua(rl2ü)lÜJ2) (3_1)
where uq is the Lennard-Jones potential,
Uo(r12} = ^[(a/r12)12 - (a/r12)6] (3-2)
and u is the dipole, quadrupole, anisotropic overlap or anisotropic
dispersion potential. Equations for these potentials are given in
Appendix C. In these calculations the superposition approximation
is made for ^(r^):
go(rl2rl3r23) So(rl2) go(r23) go(rl3)
(3-3)
and Verlet's molecular dynamics results are used for gQ(r) [32].
In Section 3.1 forms amenable to calculation are presented
In Sections 3.2 and 3.3 calculations
, F F F J F
f°r Y2A’ Y2B’ Y3A’ and Y3B*
34

35
of surface tension and surface excess internal energy are presented
for various model fluids which obey (3-1); comparisons with Monte Carlo
calculations are made for surface tension of polar liquids. In Section
3.4 results for real fluids are given and compared with experimental
measurements. In Section 3.5 the perturbation theory is used as a
basis for developing a correlation of surface tension for polyatomic
liquids.
IrA . Evaluation of y^, y^A and
Rewriting (2-87,88,89,90) in dimensionless form:
k ? co
F* _ F a TT PL
Y2A â–  y2A e 4 T*
F* _ F a
Y3A " Y3A e
-TT PL
0
*2 °°
dr* r*^ g T(r* )
12 12 oL 12 a 0)^0)^
12 t*2
* *3 * *3
dr.„ r.„ g . (r „)
12 12 6oL 12
F* _ F
k 7 co
2 pT
IT L
Y2B ~ Y2B e 2 *
* *
dr12 r12
* *
drl3 rl3
'0
0
* *
rl2+ rl3
. * *
dr23 r23
* * i
r12" r13'
“l“2
(3-4)
(3-5)
x 8„t (rtort-rro^) O?o+ r*i+ r*T)(j0 u w <3"6)
, ,F* _ F a2 -TT2 PL
Y3B ' Y3B e " 6 *2
oLv 12 13 23' v 12 13 23'
*3 oo
12 3
* *
dr12 r12
k k
dr13 r13
0
0
* *
r12+ r13
* *
Z"23/23
I r12 r13'
* * * * * * * * *
X goL rl2r] 3r23^ ^rl2+ r13+ r23^
t°iaJ2t°3
k
where p
kT/e,
A
U
a
u /e, and r
a
r/a.
(3-7)

36
The angle averages in (3-4,5,6,7) have been evaluated [33] and
are tabulated in Appendix A. Substituting those expressions into
(3-4,5,6,7) gives:
- I I, y">ss,)
2A
(3-8)
A
ss
I l yllw-.ss')
AA'
SS
, 2B
(3-9)
Y31= I I y^(AA'A";ss’s”)
J AA'A" ss’s" J
(3-10)
p*
r3B - l l Y^(AA'A";ss's")
J AA’A" ss’s" J
(3-11)
In each of these equations A = in (3-10) A' = y A" =
while in (3-9) and (3-11) A' = A" = In these equations,
F* . n l. u*- t- J.J T v
Y2a(A;ss ) - * (2£1+1)(2£0+l) Jn +n ,-l(P ,T ) ^
*2
'L • (2£ + 1)
* *
16T '‘~l-~/ ' s s'
x Eg(A;n1n2) Eg,(A;n1n2)/e2
nln2
(3-12)
*3
^2B(AA' ’ss') g T* ni
x l (-) Es(£10£1;n10) Es, (^O^jn^/e (3-13)

37
*2
Yo*(AA,A";ss,s") = - ,/9 -r7 (2£+l)(2£'+l)(2£"+l)
J 96tt 7 T
x Jn . . i(P*>T*)
n +n ,+n
s s s
£" v a
0 0 0
£.[ £]
*2 £2
£" £' £
n1n2n1
_ » __ M __ M
2nl 2
(0"
*1
£i
*1
'£2
*2
*2
iül
-1'
—2
-2
-2-
(3-14)
Y~*(AA'A";ss's")
Jd
Es(A;nin2) E^iA'jnjn^) /e~
2 p*3 £_+£'+£"
L /• \ Z o r
6 x*2 ^ \£' °£2£^
x ^ o ' o"
f(2£+l)(2£'+l)]
1/2
£’£^ (2£1+1)(2£2+1)(2£^+1)
x
£ £' £"
0 ' 0 £
j¿3 x,2
Kr(££'£";n n'n")
s s s
n1 +n„+nl
x I (-) Es(A;nin2) Es,(A,;n[nJ)
nln2n3
x Es„(A";n2np/e-
(3-15)
In these equations, ó is the Kronecker delta, n = -n,
£" £' £
0 0 0
is a 3j symbol,
[ £ £' £
|m m' m
n)
is a 6j symbol [34,35], and

38
p," o » o
1 1 1
*2 £2 *2
is a 9j symbol [36]. The Eg are coefficients in a
A" A' £
spherical harmonic expansion of the anisotropic potential u . The
superscript * on Eg indicates a complex conjugate. Details of the
expansion and equations for Eg for specific interactions are given in
Appendix C.
£ ^
In (3-12) and (3-14) Jn(P ,T ) is the single integral:
. -k k
Jn(p ,T )
* *-(n-2) , * ,
dr12 r12 g(r12)
(3-16)
Values of the J integrals are tabulated elsewhere and have been fitted
n
k k
to an empirical equation in p and T [37].
Y
In (3-13) L (£^;nsngt) is the triple integral:
L^(£;nn') =
* *-(n-l)
drl2 rl2
* *-(n'-l)
dr13 rl3
* *
rl2+r13
r12 r13'
X dr23 r23^r12+ rl3+ r23^ goL^r12rl3r23^ P£^cos “l^ 17')
t h
Here is the £ order Legendre polynomial and is the interior angle
y
at molecule 1 in the triangle formed by r^j r^3> anc* r23‘ Values of L
are tabulated in Appendix E and have been fitted to an empirical equation
* *
in p and T .
In (3-15) K1 (££'£";ngns,n^,,) is the triple integral:

39
KY(££,£,,;nn'n")
dr* r*-(n-X)
12 12
* *-(n'-l)
drl3 ri3
* * * *
x dr r ' ^ a (r r r )
23 23 ^ol/ 12 13 23'
* *
ri2+ rl3
* * ,
r12~ r13'
(ri2+ r13+ r23) (a10t20t3)
(3-18)
The function is given by a spherical harmonic expansion:
^££'5,"^aia2a3^ = ^ C(££'£";mm'm") Y£m(“12)
mm'm"
* Y£'m'(W13) Y£"m"(W23)
(3-19)
where C(££'£";mm'm") is a Clebsch-Gordan coefficient [33]. The are
spherical harmonics in the convention of Rose [38]. In (3-18) and (3-19)
the cu are the interior angles at molecule i in the triangle formed by
r12’ rl3’ and r23‘ Expressions for for multipole interactions
are given in Appendix A.
Note that Y3^(AA'A";ss's") given by (3-14) vanishes if (£+£'+£")
is odd or if the molecules are linear and either (£^+£j+£!^) or
F*
(£„+£'+£") is odd. yOT) (AA';ss ' ) vanishes unless the anisotropic potential
Z Z Z ZJd
contains £=0 spherical harmonics.
F* F* F* F*
Specific expressions for , y„„, Y~Á, and y.„ have been evaluated
2A 2B 3A 3B
from (3-12), (3-13), (3-14), and (3-15), respectively, for various aniso¬
tropic potentials. The results are tabulated in Appendix D.
F
The contributions to y given in (3-12,13,14,15) are simply
related to the corresponding terms in the free energy expansion [33]:

40
Y^(A;ss') - - ¿f
n +n .-1
s s
^ n +n ,
s s
A2A(-A;SS'-)
ÑkT
(3-20)
YF2B(M,;ss,) - - V
** L' (^i ;nsns *)
L (£..;n n ,)
1 s s' J
A2b(AA';ss')
ÑkT
(3-21)
a a
Y3A(AA'A";ss’s") = -
n +n ,+n „-l
s s s
n +n ,+n ,, '
s s s
A (AA'A";ss's")
-j A
NkT
(3-22)
Y3B(AA’A";ss's") = -
p*T* KY(££*r’;nsns,nsll) A3fi (AA’ A"; ss ' s")
8 K (££'£";n n ,n ,,)
NkT
(3-23)
The L and K integrals in (3-21) and (3-23) are defined by:
L(£;nn') =
H * *-(n-l)
d 12 12
* *-(n'-l)
d 13 13
* *
ri2+ rl3
•k k
dr23.r23
* A .
rl2_ r13'
a a a
X go(r12r13r23) VC0S V
(3-24)
K(££'£";nn'n")
dr* r*-(n-1)
12 12
, * *-(n'-l)
d 13 13
A A
r + r
12 13
r12 r13'
x dr23 r23 go^rl2r13r23^ ^££'£"^aia2a3^ ^ 25^
3.2 Surface Tension Calculations for Model Fluids
In this section model fluid calculations using the Padé per¬
turbation theory for molecules obeying potentials of the form (3-1)
are presented. For these calculations, the reference fluid surface
F
tension YQ was obtained from the Fowler model expression for a
Lennard-Jones fluid:

41
YV/e = 3ttP*2[J5 - 2Jn] (3-26)
where the integrals are defined by (3-16) and are tabulated elsewhere
[37]. Figures 4 and 5 show the effect of dipolar forces (A E £ = 112)
on the surface tension as predicted by the second order theory, the third
order theory, and the Pad! theory (2-92). The points on these figures
are Monte Carlo calculations of the Fowler model surface tension for
the Lennard-Jones plus dipole fluids [39]. Figure 6 shows similar
results for quadrupoles (A = 224). The second order and third order
terms used in these calculations were determined from the expressions
given in Appendix D.
The results in Figures 4, 5, and 6 are similar to those found
from the corresponding theories for the Helmholtz free energy [29,40].
Including the third order term extends the range of application of the
expansion somewhat; however, the third order term overcorrects the
second order theory for |j* or Q* > 1. The Pad! theory, on the other
hand, interpolates between the second and third order theories. In
the case of the polar fluids, the Padé agrees well with Monte Carlo
results up to y* - 1.75.
In Figure 7 comparison is made of the effects of various aniso¬
tropies on the surface tension. The dipole and quadrupole curves are
the Pad! results from Figures 4 and 6, respectively. As in the case
of bulk fluid thermodynamic properties, for a given value of the
"k k
multipole strength (y or Q ), the quadrupole potential is found to
have a larger effect on surface tension than the dipole potential.

hi
Figure h. Fowler Model Surface Tension for a Fluid of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Dipole Model Potential. pa^ = 0.85, kT/e = 1.273

U)
Figure 5. Fowler Model Surface Tension for a Fluid of Axially Symmetric Molecules Interacting
with Lennard-Jones plus Dipole Model Potential. = 0.45, kT/£ 88 2.934

44
Q/(6cr5)1/2
Figure 6. Fowler Model Surface Tension for a Fluid of
Axially Symmetric Molecules Interacting with
Lennard-Jones plus Quadrupole Model Potential.
po3 = 0.85, kT/e = 1.273

o-2/e
45
Figure 7. Fowler Model Surface Tension for Fluids of Axially
Symmetric Molecules Interacting with Lennard-Jones
plus Various Anisotropic Potentials. p= 0.85,
kT/e = 1.273

46
The anisotropic overlap and dispersion results are from the second order
F F
theory, using expressions for Yo» and Yor> given in Appendix D.
ZA ZB
3.3 Calculation of the Superficial Excess Internal
Energy for Model Fluids
To obtain the surface excess internal energy from the Padé
expression (2-96), the temperature derivatives of the anisotropic
contributions to the surface tension are required. Equations (3-12),
(3-13), (3-14), and (3-15) give, respectively:
3y^(A;ss')
* = Yo*(A;ss')
9T 2A
9£n J
n +n ,
s s -1
3T
(3-27)
9T
ZD
9£n L3(£-;n n ,)
1 s s
9T
(3-28)
9YÍL(AA'A";ss's")
— * = y;a(AA'A";ss's")
3T 3A
9£n J
n +n ,+n ,, 1
s s s -1
9T
(3-29)
9Y^(AA'A";ss's")
* = Yor (AA' A"; ss ' s")
9T
9£n K^(££'£";n n ,n ,, „
s s s 2
9T
(3-30)
In Table 2 values for ug from (2-95) and (2-96) are compared
with computer simulation results for Lennard-Jones plus quadrupole
model fluids (see Chapters 6 and 7). The reference fluid surface
excess internal energy, was obtained from the Fowler model
expression:

47
TABLE 2
Test of the Gibbs-Helmholtz Equation in the Fowler Model
Perturbation Theory for Lennard-Jones Plus Quadrupole Fluids
Q/(ea5)1/2
pa3
kT/e
Padé
F 2,
u o /e
s
MD
0.5
0.85
1.277
1.913
1.923 ± .016
0.707
0.931
0.765
2.670
2.656 ± .012
1.0
0.85
1.294
2.505
2.475 ± .019

48
ulo02/e = 2ttP*2 [J5 - Ju] (3-31)
Y
where the J integrals are defined by (3-16). The values for u were
n sa
determined in the simulation by evaluating the ensemble average given
in Chapter 5.
In view of the demonstrated inapplicability of Equation (2-93) in
the Fowler approximation for Lennard-Jones fluids [31], the agreement
between the theory and computer simulation shown in Table 2 is
surprising. Since the inconsistency in the Fowler expressions for y and
u is not limited to spherical potentials [6], the results in Table 2
s
must be fortuitous. The agreement may, in part, be attributed to the
high density state conditions considered, wherein the Fowler approxima¬
tion is more accurate. Much of the agreement, however, must be due to
cancellation of errors between the y and dy/dT terms in (2-93).
3.4 Surface Tension Calculations for Real Fluids
The Padé perturbation theory developed in Chapter 2 has been used
to predict pure liquid surface tensions for CC^, C^H^, and HBr. In these
calculations the reference fluid was taken to be a Mie (n,6) fluid. The
anisotropies considered were the multipole interactions up through the
quadrupole-quadrupole term, as well as anisotropic dispersion and
overlap contributions.
3.4.1 Mie (n,6) Reference Contribution to Surface Tension
The Mie (n,6) fluid was taken as the reference in the perturbation
theory calculations since Twu has determined values for e, O, and n by

49
fitting perturbation theory calculations of liquid densities and pressures
to experimental values along the orthobaric line for the fluids considered
here [37]. It is felt that the test of the Pad! theory for surface ten¬
sion is strengthened by using these independently determined potential
parameters.
The Mie (n,6) potential u^n’^(r) is given by [41]:
u(n>6)(r)
ne
(n-6)
6(n-6)
-1
(3-32)
To determine the surface tension for this potential, the Helmholtz
free energy of the nonhomogeneous, two phase, (n,6) fluid is expanded to
first order in powers of n ^ about the Lennard-Jones (12,6) fluid free
energy. The surface tension is obtained by applying the thermodynamic
definition (2-1). Then, the Fowler approximation is made in order to
obtain a form amenable to calculation.
The expansion of the (n,6) free energy about the (12,6) may be
done in two ways. In one method, the values of £ and a are taken to be
F C 6 ^
the same for the two fluids. The expression for Yq ’ resulting from
this expansion is:
F(n,6) 2 F(12,6) 2
--°-£ --a - Y° £ ° - 48,p*2 [In 2U«2-6> - J<12-6)} + 6 H<[2’6>]
1 _ 1_
n 12
(3-33)
In (3-33) is the single integral:
h¿Í2’6)(p\t*) e
* *-(n'_2) * (12,6). *, ...
dr r '¿nr gv ’ 7(r ) (3-34)
0

50
Values of this integral for n' = 11 are tabulated in Appendix K and
* *
have been fitted to an empirical equation in p and T .
In the second method of doing the expansion, the values of £
and r^ are taken to be the same for the (n,6) and (12,6) fluids. Here
r^ is the value of r where u(r) = -£. The expression for
resulting from this second expansion is:
F(n,6) 2
Y ’ a
o
F(12,6) 2
Yo ° *2 r n T (12,6)
48fTp [ (l-£n2) *
1_
12
1 .(12,6) , , „(12,6),
2 J5 + 6 H11 J
(3-35)
When the (n,6) fluid is used as the reference, the second and third
tion 1
(n, 6)
order terms and y^ in the perturbation theory involve integrals over
the (n,6) pair correlation function g^11,u/(r). These (n,6) correlation
functions can be related to Lennard-Jones g^^’^^(r) functions (for which
there are molecular dynamics results [32]) in the following way [42,43]:
and
g(n,6)(r) s g(12,6),rep = e~0u( n’6)’rep yHS(d(n,6)}
g0(12’6)(r) - g¿12’6)’rep , e-BuU2’6)’rep yHS(d (12,6)}
(3-36)
(3-37)
where the superscripts rep and HS indicate the repulsive and hard sphere
potential contributions, respectively. The function y is defined by:
, s. 3u(r) . s
y (r) = e g (r)
(3-38)

51
In (3-36) and (3-37) the hard sphere diameter d is taken to
be [42]:
d
[1 - e-^Wjdr
Further, assuming
(3-39)
yHS(d(n-6)) * yHV12-6))
(3-40)
(3-36), (3-37), and (3-40) give:
g(n,6)(r) , g(12,6)(r) e-6[u(n-6)>rel> - uU2,6),r«Pj (3-41)
where (u(n-6,'rcp - uU2,6),rep vanlshes £or r > r .
in
Using (3-41) with the Mie (n,6) potential in the integrals J^,,
Equation (3-16), and K(££'£";nn'n"), Equation (3-24), Twu has found the
resulting values to be negligibly different from those values obtained
for the Lennard-Jones (12,6) potential [37], at least for values of n
close to 12. Hence, in the calculations reported here, the Lennard-Jones
(12,6) pair correlation function has been used in evaluating the terms y^
and in the surface tension expansion.
In the calculations reported here, Equation (3-35) was used to
determine the reference fluid contribution. Values for the Lennard-Jones
term y^(12,6) were obtained from a corresponding states plot of surface
tension for simple fluids, Figure 8. For the temperature range
0.6 ¿ kT/e < 1.24, the curve in Figure 8 obeys:

52
Figure 8. Corresponding States Plot for Surface Tension
of Simple Liquids

53
(12 6) 2 . *2 *
y¿ ’ 'ü /£ = 0.8950T - 3.5177T + 3.0166
(3-42)
3.4.2 Anisotropic Contribution to Real Fluid Surface Tension
The calculations presented here include anisotropic dispersion,
overlap, and multipole contributions up to the quadrupole-quadrupole
term:
u =
Uj. (202) + u(022) 4- u,. (224) + u (202) + u (022)
dis dis dis over over
+ u (202) + u .. (022) +u, . „(224)
over-dis over-dis dis-Q
+ u .(112) + u . (123) + u .(213) + u .(224)
mul mul mul mul
(3-43)
wherein the subscripts dis, over, Q, and mul refer to dispersion, overlap,
quadrupole, and multipole, respectively. Appropriate parts of this model
have been used by Flytzani-Stephanopoulos e_t ad. [33] and Twu [37] in
studies of bulk fluid thermodynamic properties. The multipole contribu¬
tions to surface tension for this model potential are obtained by combining
(3-8), (3-9), (3-10), and (3-11) with (3-12), (3-13), (3-14), and (3-15),
respectively:
Y2A = Y2A(112) + 2 W123) + Y2A(224)
Y3A = 3 Y3*(H2;112;224) + 6 Y3¿(112;123;213)
+ 6 y3^(123; 123; 224) + y^(224;224; 224)
Y3B = Y3B(H2;112;112) + 3 Y3B(112;123;123)
+ 3 Y^g(123;123;224) + (224;224;224)
(3-44)
(3-45)
(3-46)

54
Expressions for the terms on the right side of (3-44), (3-45), and (3-46)
F*
are given in Appendix D. The term is zero for multipoles since only
terms with l ¿ 0 spherical harmonics occur in the multipole potentials.
The dispersion and overlap anisotropies in (3-43) have been
included in only the second order term y^ in calculating the surface
tension from the Padé perturbation theory (2-92). The inclusion of
dispersion and overlap contributions to the third order term y^ requires
evaluation of difficult multibody terms. Expressions for the y and
Ací.
y2g terms for anisotropic overlap and dispersion are given in Appendix D.
3.4.3 Results for Real Fluids
Figure 9 compares the Padé predictions of surface tension for
CC>2 with experiment, while Figure 10 shows a similar comparison for
C2H2 and HBr. The experimental values for surface tension of C02 and
C2H2 were taken from the compilation by Jasper [48]. The corresponding
values of saturated liquid densities for C02 and C2H2 were taken from
Vargaftik [49]. Experimental values of surface tension and saturated
liquid density for HBr were obtained from Pearson and Robinson [50].
Values of the potential parameters for C02> C2H2, and HBr were taken
from Twu [37] and are listed in Table 3.
The deviations in surface tension between theory and experiment
are less than 10% for C02 and less than 12% for C2H2 and HBr for the
temperatures shown in the accompanying figures. The consistent deviations
between theory and experiment, especially for C2H2 and HBr, suggest that
adjustment of the potential parameters would improve the agreement. It
is not a very informative test of the theory, however, to adjust potential

7ct2/€
55
0.8 1.0 1.2
kT/e
Figure 9. Surface Tension for CO2 comparing Perturbation
Theory Calculations (points) with Experimental
Values (line)

y o~21$
56
Figure 10. Surface Tension for C2H2 and HBr comparing
Perturbation Theory Calculations (points)
with Experimental Values (lines)

57
TABLE 3
Potential Parameter Values Used in
Calculating Surface Tension
Fluid
e/k
(K)
a
(A)
n
M(1018)
(esu cm)
QUO24)
(esu cm )
6
K
o
o
244.31+
3.687
16
—
-4.30 [51]
-0.1
0.257
c2h2
253.66
3.901
16
—
5.01 [52]
0.3
0.270
HBr
248.47
3.790
12
0.788 [51]
4.0 [51]
—
—
t
All values for E
'5 @ 5 ^ 5
6 and
K are taken
from Twu [37]

58
parameters using surface tension in order to calculate surface tension.
The deviations between theory and experiment shown in Figures 9 and 10
do not increase much with temperature, and, in fact, for CC^ the devia¬
tions decrease. This is in contrast to the results found for simple
Lennard-Jones fluids wherein surface tension calculated in the Fowler
model show rapidly increasing disagreement with experiment as the
temperature is increased [2,31]. Further, the Fowler model values of
surface tension for Lennard-Jones fluids are generally larger than the
experimental values at the same temperature. It may be that, in addi¬
tion to the questionable adequacy of potential parameter values used
here, the Padé approximant in some way corrects for errors introduced
in using the Fowler model for the interface. There is some evidence
for this from the Padé values for the surface excess internal energy
presented in Section 3.3.
3.5 Correlation of Surface Tension for
Pure Polyatomic Liquids
The perturbation theory for surface tension presented in Chapter 2
has been used as a basis for correlating surface tensions of a large number
of polyatomic liquids. The perturbation theory gives the surface tension
in the Fowler approximation as:
Y * Y0 + Y2A + Y2B + Y3A + Y3B (3-47>
A simple correlation for y may be obtained by making the van der Waals-
type assumption that the reference fluid radial distribution function is
a constant:

59
;oL(r12} = C
(3-48)
Using (3-48) in (2-87), (2-88), (2-89), and (2-90), together with a
similar approximation for the triplet correlation function, gQ(r^2r13r23^’
(3-47) becomes, in reduced form:
*
Y
, *2 , *2 , *3 , *4
. a p a'p a'p a'p
= y + 1 L + -2- -L- + 3 L +
’o * * *2 *2
(3-49)
where
a' = I
al “ 4
= —
2 2
a A3 a?
dr „ r ]f
12 12 a a)lt02
(3-50)
CO
I
CO
* *
A A
dr12 r12
drl3 rl3
0 0 | r
A A
*
* A
-4- r
12 13
AAA A
dr0- r0„
A 23a 23 a a W1W2W3
12 rl3'
x (r 4- r 4- r )
k 12 13 23;
(3-51)
a. = -
7T
12
A A3 AT
dr r
12 12 a ^1^2
A A
drl2 rl2
A A
dr13 r13
r* + r*
12 13
A A
,dr23,r23
r - r I
12 131
(3-52)
AAA AAA
X (r10 + r..., + r„~)
a a' av a)l(jJ23 12 13 23
(3-53)
Equation (3-49) can be written in an equivalent form using the
critical constants, T^, V , and p^ as reducing parameters rather than
the potential parameters, e and O:

60
y +V£ + ^ + !A + Ví
roR T„ T 2 2
R R
R R
where
Yr = y(vc/na)2/3 (ut.)'1
(3-54)
(3-55)
PR = P/Pc (3-56)
Tr = T/Tc (3-57)
In transforming (3-49) into (3-54), the proportionality of the potential
parameters to the critical constants is obtained by the usual correspond¬
ing states method [41]- Here, however, the potential is for polyatomic
fluids and, therefore, contains parameters in addition to the energy and
distance parameters £ and O, e.g., the multipole moments, y, Q, and
anisotropic polarizability, K, and overlap parameters 6. For such an
intermolecular potential, if the usual derivation of corresponding states
theory is followed [41], one finds:
* c * *
Tc = ~~ = cx(y ,Q ,6,k, • • •)
* 3 * *
Pc = pca = c2(y ,Q ,6,k,•••)
p a3
* C * -k
(3-58)
(3-59)
(3-60)
where c^, c^, and c^ are constants only if the anisotropic potential para-
& *
meters y , Q , 6, k, ••• are kept fixed. However, these "constants" can
be absorbed in the terms a^, a2, etc., as has been done in (3-54). Thus,
2
= a|c2/c^, etc. The transformation from the potential parameters £
and a as reducing parameters to critical constants as reducing parameters
may then be made in the usual way [41].

61
Equation (3-54) has been used as a basis for correlating surface
tension by treating the parameters a^ as semiempirical constants. Various
truncated forms of (3-54), including its Padé form analogous to (2-92),
have been tested against experimental surface tensions for numerous poly¬
atomic liquids. The form giving the best comparison with experiment was
found to be that terminated after the y term:
ZB
YR YoR + T ^1 + a2PR)
R
(3-61)
Values for a-^ and a^ for use in (3-61) have been determined for numerous
polyatomic liquids by least squares fitting to available experimental
data. The resulting values are listed in Table 4. In these calculations,
the reference contribution was obtained from a fit of reduced surface
tensions of the inert gases and methane, analogous to the curves in
Figure 8.
Y _ = 2.4724T^ - 7.5918TD + 5.0748 (3-62)
OK K K
which applies for 0.4 < T £ 0.95.
R
The validity of the correlation (3-61) may be tested by determining
2
whether experimental data give a linear relation between (y - y )T /p
R oR R R
and p , as implied in (3-61). Such a test has been conducted for several
K
polyatomic liquids, and results for carbon dioxide, acetic acid, and
methanol are shown in Figures 11, 12, and 13, respectively. These
figures are typical results for small to moderately large polyatomics
and indicate a satisfactory correlation. The corresponding comparisons

62
TABLE 4
Values for the Parameters and a^ in the
Surface Tension Correlation of Equation (3-61)
Substance
Range
a1(10^)
a2(icn
References
T
r
P Y
Paraffins
Ethane
.43-
.59
6.880
-2.246
49
48
Propane
.50-
.77
14.17
-5.606
53
48
n-Butane
.50-
.57
8.548
-2.469
49
49
i-Butane
.52-
.60
7.749
-2.082
49
49
n-Pentane
.54-
.67
6.504
-1.498
49
49
i-Pentane
.59-
.66
6.250
-1.451
49
49
n-Hexane
.54-
.93
-2.398
1.918
49
49
n-Heptane
.54-
.95
-1.860
1.910
49
49
n-Octane
.48-
.90
1.335
0.7617
49
49
i-Octane
.50-
.67
8.536
-1.942
49
49
n-Nonane
. 46-
.63
7.340
-1.230
49
49
n-Decane
.44-
.60
8.699
-1.576
49
49
n-Dodecane
.41-
.57
11.42
-2.262
49
49
n-Tridecane
.40-
.55
12.06
-2.360
49
49
n-Tetradecane
.41-
.54
12.21
-2.339
49
49
n-Pentadecane
.40-
.53
12.83
-2.452
49
49
n-Hexadecane
.41-
.55
11.59
-1.893
49
48
n-Heptadecane
.41-
.54
11.10
-1.720
49
48
n-Octadecane
.40-
.52
10.25
-1.439
49
48
n-Nonadecane
.41-
.52
11.15
-1.596
49
48
n-Eicosane
.40-
.51
11.15
-1.539
49
48
Cycloparaffins
Cyclopentane
.55-
.61
5.689
-1.339
49
48
Methylcyclo-
.53-
.59
5.973
-1.288
49
48
pentane
Ethylcyclo-
.50-
.55
10.65
-2.832
49
48
pentane
Cyclohexane
.51-
.62
6.249
-1.351
49
48
Methylcyclo-
.48-
.65
2.885
-0.4730
49
49
hexane
t
Reduced temperature
range over
which a.
and a„ were
fitted.
Sources of experimental data for liquid densities p and surface
tensions y.

63
TABLE 4 (Continued)
Substance
2 2
Range a (10 ) a2(10 ) References
Tr P Y
Olefins
Propylene
.53-.67
7.143
-2.173
49
48
1-Butene
.48-.70
4.752
-0.9880
49
48
2-Butene
.51-.65
3.606
-0.8401
49
48
1-Hexene
.54-.64
8.630
-2.142
49
49
1-Octene
.47-.56
8.047
-1.876
49
49
Cyclopentene
.56-.62
5.445
-1.195
49
49
Aromatics
Benzene
.48-.93
0.4634
0.5432
53
11
Toluene
.46-.63
8.130
-2.080
49
49
Ethylbenzene
.44-.60
9.602
-2.384
49
49
Isopropylbenzene
.46-.57
8.175
-1.851
49
49
Alcohols
Methanol
.53-.92
11.60
-4.971
49
49
n-Propanol
.55-.68
25.56
-8.854
49
48
i-Propanol
.56-.60
33.13
-10.98
49
49
n-Butanol
.48-.54
20.35
-6.624
49
49
Organic Halides
Methyl Chloride
.68-.73
1.215
-0.1476
53
48
Ethyl Bromide
.56-.60
15.00
-4.565
53
48
Carbon Tetra-
.49-.89
0.9412
0.4869
49
49
chloride
Chlorobenzene
.43-.88
2.138
-0.000452
49
49
Oxides
Carbon monoxide
.61-.68
8.502
-2.748
49
48
Carbon dioxide
.71-.95
-1.587
1.395
49
49
Water
.44-.58
34.75
-11.90
54
48
Others
Acetic Acid
.49-.86
5.984
-2.906
49
49
Acetone
.54-.69
6.178
-1.783
49
49
Ammonia
.49-.58
6.878
-2.269
53
48
Aniline
Carbon disulfide
.39-.65
.51-.58
13.38
5.968
-3.976
-1.869
49
49
Chlorine
.47-.57
4.103
-1.340
49
48
Diethyl ether
.59-.95
-0.4972
1.227
49
49
Ethyl acetate
.52-.90
0.08686
1.058
49
49

64
Figure 11. Test of Surface Tension Correlation (line)
for CC>2

65
Figure 12. Test of Surface Tension Correlation (line) for
Acetic Acid

66
Figure 13. Test of Surface Tension Correlation (line) for
Methanol

67
of predicted surface tensions with experimental values for these and
several other liquids are shown in Figures 14 and 15.
The linear relation suggested by (3-61) is not obeyed by experi¬
mental data for long chain hydrocarbons, however. Typical plots are
shown in Figure 16. In spite of the poor correlation for these
substances, the predicted surface tensions using (3-61) were usually
within 3% of the experimental values as shown in Figure 17.
Generally, the correlation of (3-61) reproduced the experimental
data for substances tested here within 3% for values of T < 0.92. In
K
order for the correlation to apply in the critical region, we must have:
C.P. C.P.
0
(3-63)
hence, from (3-61):
(3-64)
Then
’ yr ■ YoR+ aÍT (1 - V
K
(3-65)
in the critical region. This relation was not obeyed by many of the
liquids in Table 4.

68
Figure 14. Comparison of Surface Tension Calculated from
the Correlation (lines) with Experimental Values
(points) for Several Polyatomic Liquids

69
Figure 15. Comparison of Surface Tension Calculated from the
Correlation (lines) with Experimental Values
(points) for Several Polyatomic Liquids

70
Figure 16. Test of Surface Tension Correlation (lines) for
n-Hexane and n-Octane

71
0.4 0.6 0.8
t/Tc
Figure 17. Comparison of Surface Tensions Calculated from the
Correlation (lines) with Experimental Values (points)
for Several Hydrocarbons

CHAPTER 4
VAPOR-LIQUID DENSITY-ORIENTATION PROFILES
Calculation of surface tension for polyatomic fluids from the
general Kirkwood-Buff equation (2-20) requires knowledge of the
interfacial pair distribution function f (z^r^^jk^ ’ This function
may be written in terms of the interfacial singlet and pair correlation
functions, p(z^03^) and g(z^—i2(jJlÜJ2^ ’ resPect:‘-ve-*-y•'
f(Zl-12t°lü)2^ = p^ziwi^ P(z2t02^ g(-zl-12C°ltú2^ (4-1)
Thus, (4-1) with (2-20,21,22) indicate that the surface tension of
polyatomic fluids is a function of, among other things, the concentra¬
tion and orientation of molecules in the interfacial region. As
indicated in Chapter 1, much effort has been expended in determining
the nature of the interfacial density profile p(z^) for atomic fluids.
Almost nothing has been done in determining the corresponding profile
p(z^oj^) for more complicated molecules. In this chapter a first order
perturbation theory is developed for p(z^üJ^) and calculations are
presented for model fluids interacting with anisotropic overlap and
dispersion forces.
4.1 First Order Perturbation Theory for p(z^QJ^)
4.1.1 Development of the General First Order Term
For a pure liquid with N nonspherical molecules interacting with
72

73
N N ...
total potential U(_r w ) at temperature T, the singlet distribution
function is:
f (£1w1)
N
Z
N N
. N-l , N-l -3U(r W )
dr dw e —
(4-2)
Z is the configurational integral of (2-4), 3 = 1/kT, and 0) = {<}>0x} is
a set of Euler angles giving the molecular orientations.
Now consider the pure liquid to be in equilibrium with its vapor
and to have a planar interface between the vapor and liquid phases.
Assume a space-fixed coordinate system oriented such that the xy plane
lies in the plane of the interface and the positive z-axis is directed
into the vapor phase. If the fluid at any height z is considered to be
uniform in the x and y directions, then only the z-component of (4-2)
applies:
pU-jü^)
N N
, N-l , N-l -3U(r W )
dr dw e —
(4-3)
(4-3) gives the unnormalized probability of finding a molecule with
orientation at height z^ in the fluid.
The perturbation theory for p(z^io^) is developed by writing the
potential as a sum of isotropic reference plus anisotropic perturbation
terms:
... N Nv i» / N. ,,, , N N.
U(r to ) - U (r ) + Au (r U) )
— o — a —
(4-4)
where A is a perturbation parameter as in (2-27). Now expand p(z^OJ^)

74
in a Taylor series with respect to the parameter A about the reference,
i.e., about A = 0:
pCz^) =
p(z 1 )
Bpízjü^)
+ x2
r~2 . .'i
9 p(z1üJ1)
n A
9A
, 4
+ 2
A=0
3A2
+ • • • (4-5)
A=0
Here we consider only the first order theory:
Po(zl}
pCz-jt^) = ^ + p1(z1u1)
(4-6)
where
p^ZjW^ - A
8p(z1to1)
3A
(4-7)
A=0
Taking the derivative of (4-3) with respect to A and assuming the poten¬
tial to be pairwise additive, as in Equation (2-8), we find:
Spiz^)
6
3A
Q
k <
A=0
di2 fo(zlil2) 6_
dr2 dr3 fQ(Zlr12r13) <^(23)^
+ T P„(zi)
Z O 1
dr dr f (z r ) (4-8)
—1 —2 o 1—12 a ^1^2
where the angle average <•••> is defined by (2-15).
If the reference fluid is now chosen to be a Pople reference,
(2-46), then (2-48) causes the last two terms in (4-8) to vanish,
leaving:

75
Spíz^)
-6
dx
x=o Q
dr. f (znr )
—2 o 1—12 a 0),
(4-9)
For anisotropic potentials having £ / 0 order spherical harmonics
= 0
w2
(4-10)
Thus, (4-9) vanishes for multipolar interactions, and the density-
orientation profile is, to first order, just the reference profile,
Po(zl}-
Anisotropic overlap and dispersion interactions contain £=0
spherical harmonics, however, so (4-9) does not vanish for these
potentials. Combining (4-9) with (4-6) and (4-7) and setting X = 1
to regain the real fluid:
pU-jW )
Po(zl}
Q
6
dr_ f (12)
—2 o a to
(4-11)
4.1.2 Evaluation of p^(z^Ol)^) for Anisotropic Overlap and Dispersion
The anisotropic overlap potential is approximated by the first
two terms in a spherical harmonic expansion:
u
over
u (202) + u
over over
(022)
(4-12)
Using the general spherical harmonic expansion for u given in Appendix C,
3.
with coefficients E from Table Cl, (4-12) becomes:
over

76
r a/2
f \
u =16
a
over
|5j
*r12^
12
1 C(202;m^m2m)
x Dlo<“l> D! S("2> Y2m<“l2> + l
1 2 m^n^m
* C(022;ni„2») D^0*«ü2) ^2m(^i2>
(4-13)
But
= 0
m20 2' Cl>2
(4-14)
Thus, on angle averaging over co^, (4-13) reduces to:
=16
over u)2
VI1/2
TT
6e
'•r 12y
12
£ C(202;m^m2m)
m^m2m
X Dz . (OJ.) Y (w10)
m^O 1 m20 2' w2 2m 12
(4-15)
Using the properties.of the representation coefficients and spherical
harmonics [38], (4-15) reduces to:

over co2
86e
P2 (cos 0^) P2(cos 0^2)
(4-16)
where P2 is the second order Legendre polynomial.
To obtain the expression analogous to (4-16) for anisotropic dis¬
persion, we proceed in a similar manner. The model for the anisotropic
dispersion potential is based on London's polarizability approximation
and is written:
-1 j •
dis
= u (202)
dis
+ u.. (022)
dis
+ “dis<224)
(4-17)

77
The terms on the right side of (4-17) are evaluated from the expansion
for in Appendix C and the expressions for in Table Cl. Taking
the angle average of (4-17) over u^, the second two terms vanish due
to arguments analogous to (4-14). Then, carrying out steps analogous
to those cited above for overlap, we obtain:
= - 4kg
dis
P (cos 6 ) P (cos 0 )
r12 ¿ ¿
(4-18)
Combining (4-16) or (4-18) with (4-7) and (4-9) and noting that
for linear molecules to^ = {0^} only,
?!(zl®l) = C p2(cos P0(zl)
*-r
d-12 rl2 P2^cos ei2^ Po('Z2') go^zl-12^
(4-19)
where
f
o
(zl£l2) = P0(zi) p0(z2^ 8o

(4-20)
has been used, and for overlap
C = - 88Se/ft , n = 12 (4-21)
while for dispersion
C = 46KE/Í7 , n = 6 (4-22)
To obtain the interfacial density profile p(z^), Equation (4-6)
is integrated over the orientation to^. Integration of p^(z^0^) over
0)-^ vanishes when p^(z^0^) is given by (4-19). Thus, for the weakly

78
anisotropic potentials for which the first order perturbation theory
is expected to apply, the interfacial density profile p(z^) is just
the reference fluid profile pQ(z^). Thus, there is no layered structure
in the interface of fluids which interact with these weakly anisotropic
potentials. Moreover, the Gibbs dividing surface is the same (to first
order) for the real and reference fluids. However, it is clear from
(4-19) that these fluids will exhibit preferential orientation of the
molecules in the interfacial region.
4.2 Calculations of p(z^ca^) for Overlap and Dispersion
4.2.1 Calculational Form for p^(z^OJ^) using Toxvaerd’s Model for
In order to avoid the integration over 0^ in (4-19), cylindrical
coordinates are introduced, giving:
Pl(Zl0l) = C P2^COS 91^ p0(zl)
dz12 Po(z12+ 21>
dr12 r12
!12l
0
n
P„
i"12]
irl2J
2
^12'
8o(zlz12r12>
2tt
d12 (4-23)
0
Performing the integration and writing in dimensionless form,
P1^Z191^ = 27TC P2^cos V Po^Z1^
dr* r*^1 P
'12 12 2
I
"12'
4* 4* 4* 4*
dz12 po(z12+ zl>
— OO
£
"12
k k k
8o(zlz12r12)
(4-24)
12
To proceed further, an approximation must be made for the unknown inter-
k k k
facial correlation function g (z,z,„r,„). We choose the model used by
o 1 12 12

79
‘V k k
Toxvaerd [14], in which -j^) assumec* to be a density
weighted average of the bulk fluid correlation functions:
* * *
8o(zlz12r12>
* *
* *
= A(Z!Z12) goL(r12PL) + B(Z1Z12} goV(rl2PV) (4"25)
where
* *
A(z1z12)
"k "k k k
ap (zx) + (1-a) p (z2)
PL PV
" P,
(4-26)
* k
B(Zizi2)
Pt -
ap*(z*)
- (1-a)
* *
P (z0)
Pt - P,
(4-27)
and a is an adjustable parameter. It has been pointed out that the model
(4-25) satisfies the symmetry requirement of gQ (z^z^r-^) ^,e:'-n8 invariant
under a renumbering of molecules 1 and 2 only when a = 1/2 [55]. Toxvaerd
found .72 < a < .8 when solving the Born-Green-Yvon (BGY) equation for
p(z^) for a Lennard-Jones fluid [14]. In that calculation the asymptotic
limit exp (-3u) was used for the vapor phase g^(r) while the Percus-Yevick
result was used for g (r). We have used the same vapor phase g(r); how-
J-i
ever, Verlet's molecular dynamics results have been used for g (r) [32].
L
Our solution of the BGY equation for the Lennard-Jones p(z^) using
Toxvaerd's method required 0.2 < a < .4. The resulting profile p(z^)
is very nearly the same as Toxvaerd's results, as shown in Figure (4-1).
Further, Toxvaerd has been able to reproduce the results for p(z^) shown
in Figure 18 by a method which does not involve the model of (4-25) [16].

80
-4-2 0 2
Z/cr
Figure 18. Interfacial Density Profile for Lennard-Jones Fluid

81
Thus, we conclude that while the model of (4-25) is open to criticism,
it seems to provide the correct result.
Combining (4-25) with (4-24) gives:
p^X)
& k
2ttC P0 (cos 0,) P (zi)
z 1 o 1
AAA A
dZ12 Po
dr*2 r*»1-") P2
■«I
rz, „\
12
[A(Z*1Z12} goL(r12)
12
+ B(z*z*2) goV(r*2)]
(4-28)
P1 (Zl^l) = 2lTC P2^cos 61^ Po^Zl^
AAA A
dz12 po(z12+ zl>
A A
A A
[A(z1z12) Kl(z12) + B(z1z12) Kv(z12)]
(4-29)
where
Vz12> =
dr* *(l-n)
.12 12 2
k *
'12 *
'12
12
8oL
(4-30)
KV(z12) ~
dr* *(1-n) p
12 12 2
'12 1
vr
'12
A
12
*oV(r12)
(4-31)
4.2.2 Calculation Procedure for p(z^6p) for Anisotropic Overlap
or Dispersion
1. For a given temperature T and liquid and vapor bulk phase
•k k
densities p T and p calculate the Lennard-Jones reference fluid inter-
oL
oV’
k k
facial density profile PQ(zp) by solving the BGY equation:

82
w
*
dz,
= 6
* * *
* * * * * * * * ^rl2^ z12
d—12 po(zl} Po(zl} 8o(zlz12rl2) = “
(4-32)
dr
12
12
k k k
The model (4-25) is used for 80(z^z22rl2^ anc* t*ie vaPor P^ase correlation
function is approximated by e ^u(r12^. Verlet's molecular dynamics results
k
are used for g (r19) [32]- Details of the solution method are given in
O J-» _L Z
[14].
2. Calculate the integrals and K^, (4-30) and (4-31), as
k
functions of z^- these equations n is given by (4-21) and (4-22)
for overlap and dispersion, respectively.
3. For a given 6 value using (4-21) for overlap or a given K
k
value using (4-22) for dispersion, fix either z^ or 0^ and solve
* k
(4-29) for as a function of 0^ or z^, respectively.
k k k k
4. With p and p.. determined, obtain p (z.,0,) from (4-6).
o 1 11
4.2.3 Results for p (z^8^) for Dispersion
k k
Figure 19 shows the p (z^0^) surface calculated by the above
procedure for molecules interacting with Lennard-Jones plus anisotropic
dispersion forces with K = .25. Figures 20 and 21 are cross-
sectional slices of the volume shown in Figure 19. For these
calculations the z = 0 plane locates the Gibbs equimolar dividing
surface. These figures indicate a preference for these linear
molecules to orient themselves in the liquid side of the interface
with their axes at an angle 0 = u/2 to the normal to the plane of
the interface. That is, the molecules prefer to lie in the plane of
the interface on the liquid side. Away from the interface into the
bulk liquid, all orientations are equally probable, as they should be.

83
Figure 19. Interfacial Density-Orientation Surface for a Fluid
of Axially Symmetric Molecules Interacting with
Lennard-Jones plus Dispersion Model Potential.
kT/e = 0.85, K = 0.25

84
Figure 20. Interfacial Density-Orientation Profiles for a Fluid
of Axially Symmetric Molecules Interacting with
Lennard-Jones plus Dispersion Model Potential.
kT/e = 0.85, k = 0.25

85
Figure 21. Interfacial Density-Orientation Profiles for a Fluid
of Axially Symmetric Molecules Interacting with
Lennard-Jones plus Dispersion Model Potential.
kT/e = 0.85, K = 0.25

86
On the vapor side of the interface, the calculations indicate a slight
preference for the opposite orientations, i.e., the molecules tend to
stand perpendicular to the plane of the interface.
A plausible explanation for this change in most probable orienta¬
tions from the liquid to the vapor phases can be obtained by considering
the difference between the normal and tangential components of the stress
tensor, {p^ - p,j,(z)}. Toxvaerd has calculated {p^ - p^Cz)} as a function
of z by two different methods [14,56]. In both calculations he finds
ÍPn ~ PT(z)} to be positive in the liquid side of the interface, cor¬
responding to a surface tension in the liquid. On the vapor side, however,
Toxvaerd finds the difference {p^ - p^(z)} to be negative, corresponding
to a surface compression of the vapor; see Figure 22. Hence, the
preferred orientations found for linear molecules interacting with
dispersion forces may be interpreted as follows. The surface tension
in the liquid tends to pull the molecules toward orientations in which
the molecules lie in the plane of the interface. On the vapor side, the
surface compression tends to push the molecules together, forcing them to
stand in the interface.
Calculations for values of the strength constant K other than 0.25
show the anticipated results, i.e., for K < .25 the probabilities for the
preferred orientations are weaker than those for K = .25. For K > .25,
the corresponding probabilities are stronger. The general character for
all these calculations is the same as shown in Figures 19, 20, and 21.
4.2.4 Results for p*(z^9^) for Overlap
Calculations have been performed for the Lennard-Jones plus overlap
potential using both positive (rodlike molecules) and negative (platelike

Figure 22. Difference in Normal and Tangential Components of Stress Tensor for
Lennard-Jones Fluid. (From Toxvaerd [14,56].)

88
molecules) values for the overlap parameter 6. As shown in Figures 23
and 24, the behavior of p*(z*0^) for 6 > 0 is opposite to that found
for the dispersion interaction. The molecules tend to stand perpendicular
to the plane of the interface in the liquid. On the vapor side there is a
slight preference for the molecules to lie in the plane of the interface.
Results for 6 < 0 are shown in Figures 25 and 26. These cal¬
culations show the opposite features to those for 6 > 0; however, for
platelike molecules the symmetry axis is perpendicular to the plane of
the molecule. Thus, in the liquid side of the interface the platelike
molecules prefer an orientation with the symmetry axis parallel to the
interfacial plane. The molecules themselves tend to stand perpendicular
to the interfacial plane. An analogous argument holds for molecules in
the vapor side of the interface.

89
Figure 23. Interfacial Density-Orientation Profiles for a Fluid
of Axially Symmetric Molecules Interacting with
Lennard-Jones plus Anisotropic Overlap Model Potential.
kT/e = 0.85, 6 = 0.1

90
Figure 24. Interfacial Density-Orientation Profiles for a
Fluid of Axially Symmetric Molecules Interacting
with Lennard-Jones plus Anisotropic Overlap Model
Potential. kT/e = 0.85, 6 = 0.1

91
Figure 25. Interfacial Density-Orientation Profiles for a
Fluid of Axially Symmetric Molecules Interacting
with Lennard-Jones plus Anisotropic Overlap Model
Potential. kT/e = 0.85, 6 = -0.1

92
Figure 26. Interfacial Density-Orientation Profiles for a
Fluid of Axially Symmetric Molecules Interacting
with Lennard-Jones plus Anisotropic Overlap Model
Potential. kT/e = 0.85, 6 = -0.1

PART II
COMPUTER SIMULATION STUDIES

CHAPTER 5
MONTE CARLO SIMULATION OF MOLECULAR FLUIDS
ON A MINICOMPUTER
5.1 Introduction
The Monte Carlo method used in the study of fluids is a
form of the standard Monte Carlo technique for evaluating multi¬
dimensional integrals. Here the integrals of interest are those
which arise in statistical mechanics for the time-independent
properties of fluids [41]. In the study of fluids the method
is often referred to as computer simulation since the integrations
are over the positions of the molecules in the system. Thus, the
random generation of the variables of integration in the Monte
Carlo method may be considered to represent motion of the
molecules in the system. Even for a fluid system of a few hundred
spherical molecules, lthe integrals to be evaluated approach 1000-fold
in size. Consequently, sampling over several hundred thousand or,
often, a few million values for the independent variables is required
in order to obtain reasonable statistical precision for values of
the integrals (< 3-5%). Such calculations, therefore, require a
significant amount of computer time. The amounts of time vary greatly
from one machine to another, of course. For illustrative purposes,
one Monte Carlo calculation for a simple Lennard-Jones fluid of 256
particles requires a few minutes of CPU time on a CDC 7600, a major
portion of an hour on an IBM 370/165, and a few hours on an IBM 370/155
94

95
or Honeywell 635. Calculations for more complicated fluids (non-
spherical molecules) require significantly longer times (usually by
a factor of five or six). Such a requirement of computer resources
has a constrictive effect on use of simulation in fluids research.
A relatively small number of researchers have access to machines for
such lengthy calculations.
The value of computer simulation in the study of fluids cannot
be overstressed. Since the position (and, in the case of nonspherical
molecules, orientation) of each molecule is known throughout the
calculation, computer simulation provides a degree of detail which is,
as yet, unattained by either theory or laboratory experiment. Such
detail allows the study of local structure in fluids, as well as the
effects of potential models, anisotropic potential strengths, molecular
shapes, etc., on fluid properties. Computer simulation is the only
source of "experimental" data for model fluids which are used in
developing theories of fluids. In addition, computer simulation
provides a valuable method for studying intermolecular potential
functions applicable to real fluids.
In light of the value of machine simulation and the restricted
availability of large computers, it was felt worthwhile to explore
the possibility of using a minicomputer for performing Monte Carlo
calculations. Many researchers, it is felt, have access to mini¬
computers and, further, these machines are often only in use for
some fraction of a 24 hour day. The minicomputers currently
available are certainly slower than the big machines, and so they
could not be applied to many simulation problems. But though the

96
problems the minicomputer can potentially solve may be less glamorous
than some presently being studied, they may turn out to be no less
important.
This chapter describes a method of performing Monte Carlo
simulations of linear molecules developed for a NOVA 2 minicomputer.
The areas of discussion include: an outline of the Monte Carlo method
for nonspherical molecules, description of the NOVA 2 computer, details
of the Fortran program developed for the NOVA, and comparison of results
from the NOVA with those obtained on a CDC 6600.
5.2 Monte Carlo Method for Nonspherical Molecules
5.2.1 Intermolecular Potential
The Monte Carlo method for a system of spherical molecules is
well documented [57,58], so only an outline of the procedure is given
here. The emphasis in this work is on systems of nonspherical molecules,
wherein the intermolecular potential U depends on the orientations w of
the molecules in addition to their locations _r. The orientations are
usually specified by a set of Euler angles U) = {00x) between a body-fixed
frame and a reference frame fixed external to the system. The full
potential for a system of N molecules is, then:
U = U( A)N) (5-1)
which is assumed to be a sum of pair interactions:
U(r\)N) = I y u(r..U).w.)
i (5-2)

97
Here we consider pair potentials of the form:
u(r . ,(*) .0). ) = uTT(r..) +u (r . .to.ü).)
-il i j LJ i] a —ij i j
(5-3)
In (5-3), u^j is the Lennard-Jones pair potential of Equation (3-2)
and u is an anisotropic potential, e.g., dipole, quadrupole, aniso-
cl
tropic overlap, or anisotropic dispersion. Expressions for these
potential models are given in Appendix C.
5.2.2 Monte Carlo Procedure
Monte Carlo simulations may be done in any of several ensembles;
here we use the canonical ensemble. A number of particles N, system
volume V, temperature T, and form for the intermolecular -pair potential
(5-3) are chosen. Initial positions and orientations are assigned to
each of the N particles. The nature of this initial configuration is
arbitrary, i.e., it may be randomly generated, pathological (e.g.,
FCC lattice), or the last configuration from a previous calculation.
The position of the i*"*1 particle is given by three components of a
vector _r locating its center of mass. Rather than specify the
orientation of the molecules by Euler angles OK = it is
more convenient computationally to use the direction cosines of the
molecular axes. For linear molecules the orientation is completely
specified by two independent variables, though in practice all three
of the direction cosines are stored for each molecule.
The simulation then proceeds as follows:
1. For the current system configuration, denoted by superscript
(1), the system energy is calculated:

98
,(D
U = y y u (r. .OJ.OJ.)
11 11
1 (5-4)
2. One particle, having position r_í^ and orientation ü)í^,
is selected, either cyclically or at random, and a new position
(2) (2)
r\ and orientation are proposed by:
rf2>
—i
4“ +
(5-5)
(2) _ (2)
U). - 0). + t,. . dn
i i :zk+l W
(5-6)
where ¿ is a vector of random components uniformly distributed on
(-1,1), while d and d^ are maximum allowable steps lengths for the
translational and rotational motion of the molecules, respectively.
3. The energy of the proposed new configuration is calculated:
,(2)
- I l
i (2) , ,
u (r . .w.o). )
iJ i]
(5-7)
4. The proposed new configuration is accepted or rejected,
based on the relative probabilities of occurrence of the two con¬
figurations. The probabilities are proportional to the Boltzmann
factor:
P(1) = exp [- U(1)/kT] (5-8)
P(2) = exp [- U(2)/kT]
(5-9)

99
The criteria for accepting or rejecting the proposed configuration is:
(2^ (1)
a) If Pw > P , accept the new configuration.
b) If P^ < do not reject the new configuration
out of hand, rather accept it with probability proportional to the
Boltzmann factor. This may be accomplished by generating a random
number £ on the interval (0,1), and:
If p(^)/p^) > accept the new configuration.
If p^^'pvl) < reject the new configuration.
5. If the new configuration is rejected, the old configuration
is counted as the new configuration, and the properties of interest
are determined for this new configuration.
6. Steps 1-5 are iterated over the length of the calculation.
At the end of the run, ensemble average property values are obtained
by averaging the property values obtained for each configuration.
Relations for several equilibrium properties in terms of ensemble
averages are given in Table 5. In Table 5, the angular correlation
parameters G are measures of the correlation of orientation in the
J-i
fluid. When fluctuations in the collision-induced dipole moments are
neglected, the parameters G are. related to the dielectric and Kerr
L
constants (F^ and F^) for linear molecules. The parameters G also
arise in theories of depolarized light scattering [59]. (There seems
to be no standard notation for these angular correlation parameters,
cf. Cheung and Powles [60].)

100
TABLE 5
Equilibrium Properties in the Form
of Ensemble Averages
Configurational
Internal Energy
= m I u(r. ,w.U).)>
N N 11 i i
i (5-10)
Residual Heat
Capacity
^ = -4 [ - 2]
NT
(5-11)
Pressure
Fowler Model
Surface Tension
pkT
= 1 -
, 9u(r..w.w.)
1 .v t i.l i .1
<£ l
i 3NT — ij
9r. .
ij
_ „ 9u(r . .co.o).)
/ = _e_ ' 16N ¿4 ij 9r..
i > (5-12)
(5-13)
F — Q v V
Fowler Model Surface U„ = -ttt <) ) r.. u(r . .oo.UJ.) •
T7 S 4N ij ij i ]
Excess Energy i (5-14)
Angular Correlation
Parameters”
CT = < l PT (cos y )>
jf*l J
(5-15)
Mean Squared Force
<Â¥ > = < l (V u(r 0) u)
j^l 1 1 J
,))2>
(5-16)
Mean Squared Torque
= <
lVu(rij“i“j)}'
J^l 1
(5-17)
is the angle between the axes of molecules 1 and j . P^ is the
order Legendre polynomial.

101
During the first few hundred configurations of the calculation,
the system is relaxing from the arbitrary starting configuration to an
equilibrium state, which is reflected by the system energy fluctuating
about a minimum value. These initial configurations are "discarded"
in that they are not used in calculating the average values for the
properties. At regular intervals throughout the calculation, the step
lengths in (5-5) and (5-6) are adjusted to maintain acceptance of about
half the proposed new configurations. In order for the small number
of particles (here N = 128) to adequately mimic a bulk fluid system,
so-called periodic boundary conditions are used. Moreover, the potential
is set to zero beyond some cut-off distance r^, taken to be the radius
of the largest sphere which can be inscribed in a cube of volume V.
More detailed descriptions of the method can be found in [57,58]*
5.3 Description of the Minicomputer System
The computer used in this study was a Data General NOVA 2/10 with
optional 32K 16-bit words of core storage and 1000 nsec memory cycle
time. The CPU included the following options: power monitor, auto
restart, auto program load, real time clock, hardware multiply/divide,
and high performance hardware floating point processor. The CPU serviced
the following peripherals: two moving head disc units with controller
for a total of 2.5 megawords of storage, serial matrix line printer
(165 cps), fast paper tape reader and punch, and teletype.
The computer was operated under the Data General real time
disc operating system, RDOS, revision 3.02, which handles task
scheduling and system maintenance. The executive remained core
resident and occupied about 3K words, so about 29K word locations

102
were available for computation. The available software included
Data General's Fortran IV, Fortran V, Basic, and Algol. The program
for this work was written in double precision arithmetic for the
Fortran V computer. The Data General Fortran V is a code optimizer,
in that redundant operations are eliminated and floating point opera¬
tions are optimized for effective use of the floating point hardware.
5.4 Monte Carlo Program for the NOVA
5.4.1 Calculation of System Energy
The major time-consuming calculation in Monte Carlo simulations
is evaluation of the system energy U in (5-4) and (5-7). The full
double sum in (5-4) and (5-7) does not have to be evaluated, however,
since the particles are moved sequentially. Thus, when particle i
is moved from configuration 1 to configuration 2, only (N-l) pair
energies change. The new system energy can be found, therefore, from:
ü(2) = U(1) + AU. (5-18)
i
where
AU. = l u(r^2)ojf2)o>.) - l u(
1 iH 1J 1 J j?i
r^Wo.)
ij i 3
(5-19)
In the usual Monte Carlo calculation each of the pair energies
u(r„üK0J.) for the starting configuration is calculated and stored
in core. The total system energy for the starting configuration is
,(D
obtained from (5-4) and stored as U
For each subsequent con¬
figuration generated, the system energy is determined from (5-18)
and (5-19) rather than (5-7). For this calculation the pair energies

103
for the first term on the right side of (5-19) are calculated, while
the pair energies for the second term are obtained from storage.
Whenever a proposed configuration is accepted, the pair energies
. (1) (1) \ . . , , (2) (2) .
u(r.. co. ui.) in storage are updated to the new values u(r.. 0). ui.)
ij i J ij i j
and the new system energy U is stored as This procedure
significantly decreases the amount of calculation required, but places
a heavy demand on core storage. Even though the matrix of pair energies
is symmetric, i.e.,
u(r,.ui.üi.) = u(r..01.(0.)
ij i J Ji J i
(5-20)
y N(N-l) values must be stored just for the energy. If other properties
are being calculated, similar storage must be provided for each of them.
This large storage requirement is avoided in the program for the
NOVA by calculating both sums in (5-19) for each configuration generated
and, therefore, storing none of the pair properties. Execution time
is increased accordingly. Alternatively, the pair energy matrix could
be stored on disc and transferred into core column by column as it is
needed. When a move is accepted, however, a major portion of the matrix
must be brought through core in order to update the changed elements from
u(r f^oi^^ai.) to u(r Í ?^uif ^^oi.). On the NOVA 2 system, for 128 particles,
we found this transfer to be slower than recalculating the needed matrix
elements as indicated above. Hence, the program described here does not
use core-disc data transfers during execution.

104
5.4.2 Program Structure
Figure 27 shows a simplified schematic flow diagram of the
program written for the NOVA 2. The Monte Carlo calculation is done
by the main program labeled 1, which is supported by subroutines INITIAL
and ENERGY. Subroutine ENERGY consumes the major portion of execution
time as it forms the sums analogous to Equation (5-19) for all properties
of interest. Disc file 1 holds the starting configuration's location
vectors and direction cosines. Disc file 2 holds intermediate con¬
figuration particle location vectors, direction cosines, and property
values, all of which are dumped at about half-hour intervals throughout
the calculation. This periodic data dump serves as a back-up against
possible power failure and allows other system users to interrupt
program execution. The entire program for 128 particles resides in
core throughout the calculation and requires about 27K words of NOVA 2
memory. A further saving of 3-4K words could be attained by making
the subroutine INITIAL a separate main program, as it is only used at
the start (or restart) of a calculation. The main program labeled 2 takes
the raw data generated by program 1, scales property values, and estimates
statistical precision of the results.
5.4.3 Program Execution Speed
Speed of execution of the Monte Carlo program was found to be
largely a function of the number of particles used in the simulation.
The complexity of the potential model used and the number of properties
calculated were found to have only slight effect on program execution
time. These findings are summarized in Table 6 which compares the

105
Figure 27. Simplified Schematic Flow Diagram of FORTRAN Monte Carlo
Program Developed for NOVA 2

106
TABLE 6
Approximate Number of Monte Carlo Configurations
Generated per Hour on the NOVA 2

107
number of configurations generated per hour by the Fortran program for
64 and 128 particles using the Lennard-Jones potential (5-4) and the
Lennard-Jones plus quadrupole potential of Equations (5-3) and (C28).
Table 6 indicates that doubling the number of particles roughly doubles
execution time. In addition to the results shown in the table, the
Lennard-Jones plus dipole model of Equations (5-3) and (C27) executes
about 2% faster than the Lennard-Jones plus quadrupole model. The
Lennard-Jones plus overlap model, Equations (5-3) and (C30), in turn,
executes about 2% faster than the Lennard-Jones plus dipole model.
These program execution speeds on the NOVA may be compared with speeds
attained on IBM and CDC machines. A Fortran program for simulating
the Lennard-Jones plus overlap model fluid, using 64 particles and
calculating only the system internal energy and the angular pair
4
correlation function at seven orientations generates about 2(10 )
configurations per hour on an IBM 370/155 and about 1(10^) con¬
figurations per hour on a CDC 6600. Each of these programs tested
was written in double precision arithmetic.
5.5 Comparison of NOVA Results with Full-Size
Computer Results
Equilibrium property values are obtained in the Monte Carlo
method by forming ensemble averages over appropriate functions of the
positions and orientations of the molecules, as shown in Table 5.
In addition to the properties listed in the Table, the program written
for the NOVA determines the center-to-center pair correlation function
gc (r12^ an<^ values for the angular pair correlation function g (r^^j^^
at five specific orientations.

108
Property values calculated on the NOVA for a Lennard-Jones
plus quadrupole model fluid are compared in Table 7 with results
previously obtained on a CDC 6600 [61,62,63]. The value for the
energy U reported in reference [61] does not include a long-range
correction for values of the radial component r greater than the
value at which the potential is set to zero, r . For the sake of
consistency in the comparison, no long-range correction has been
made to the minicomputer value for U shown in Table 7. Further,
2
the values for the mean squared force and mean squared torque
2
reported in [62,63] were obtained by integration over a histogram
of Monte Carlo-determined g(rvalues* Determination of such a
histogram is difficult to accomplish since: (a) long runs are required
in order to obtain statistically reliable sampling of less probable
configurations (especially in angle space), and, (b) small radial and
angular increments are required in order to obtain sufficient data
2 2
on g(r^2W^2^ f°r accurage integration. The values for and
calculated on the NOVA were obtained by direct evaluation of the averages
given in Table 5. It is felt, therefore, that the NOVA property values
are more reliable than those obtained from the CDC machine.
Further comparisons of NOVA results with CDC results are made
in Figures 28 and 29. Figure 28 shows values for the center-to-center
pair correlation function g (r.^) f°r a Lennard_Jones plus quadrupole
fluid. The CDC values for g^ were obtained for a system of 64 linear
4
molecules from a Monte Carlo chain of some 8(10 ) configurations after
the system had reached equilibrium [61]. The minicomputer results are
from a run of similar length on a system of 128 molecules.

109
TABLE 7
Comparison of NOVA and CPC Results for Property
Values of Lennard-Jones + Quadrupole Model
Fluid. kT/e = 0.719,
pa3 =0.80
9
Q/ 1
Property
NOVA 2
CDC
6600
U/Nc
- 8.642 ± 0.197
- 8.483
± 0.131
f 61]
02/e2
744.0 ±34.0
772.0
±15.0
[62]
/e2
31.2 ± 1.09
31.0
± 1.5
[63]

110
Figure 28. Comparison of CDC (line) and NOVA 2 (points) Monte Carlo
Results for the Center-Center Pair Correlation Function
for a Lennard-Jones plus Quadrupole Fluid

Ill
Comparison of CDC (line) and NOVA 2 (points) Results
for the Angular Pair Correlation Function for a
Lennard-Jones plus Quadrupole Fluid for Molecular
Pairs in the Tee Orientation
Figure 29.

112
Figure 29 makes a similar comparison for the angular pair
correlation function g(r f°r t^e Lennard-Jones plus quadrupole
fluid. The values of gír^aJ-jü^ shown are for molecular pairs in the
tee orientation, i.e., 0^ = tt/2, 02 = 0, undefined, where 0^, d^, and
4>^2 are the orientation angles relative to the intermolecular axis, as
shown in Figure Cl. The CDC results are from a chain of about 4.1(10“*)
configurations for a system of 64 particles. The minicomputer results
are from the same run as that for Figure 28. To achieve consistency in
the comparison, the same angular increment of ± 15° was used in the NOVA
calculation as was used in the CDC calculation [64].
5.6 Conclusions
The above results indicate that the NOVA 2 is capable of reliable,
sustained operation of sufficient magnitude to produce useful results
by the Monte Carlo method for simulating fluids. The major limitation
in using the NOVA for such calculations was found to be speed of program
execution. This, in türn, imposes a limitation on the class of problems
for which one might use a minicomputer. An upper bound on the time one
would be willing to invest in a single run is probably of the order of
two weeks of computer time. Such a calculation would yield 5-10(10^)
configurations of a Monte Carlo sequence, depending on the number of
particles, potential model, etc. Thus, one would not consider using
present minicomputers for specialized studies which require significantly
longer runs than calculation of bulk fluid properties, e.g., study of
phase transitions or critical phenomena.

113
A major improvement in speed of program execution can be attained
by use of faster hardware than is available on the NOVA. Thus, Freasier
et_ al. [65] have recently reported Monte Carlo results for a hard dumb¬
bell fluid using a PDP 11/45. The calculations were done for 108 particles
and about 10 configurations were generated for each state condition studied.
The code was written in Fortran, except the time consuming routines for
which assembler language was used.

CHAPTER 6
MOLECULAR DYNAMICS METHOD FOR AXIALLY
SYMMETRIC MOLECULES
6.1 Introduction
A second method used in the computer simulation of fluids (as
opposed to the Monte Carlo technique described in the previous chapter)
is the molecular dynamics method. This simulation method is in many
ways similar to the Monte Carlo technique: both methods are applied to
small, finite systems4 both use periodic boundary conditions, and both
usually assume a pairwise additive intermolecular potential. The major
distinction between the two is that in molecular dynamics, generation
of states accessible to the system is accomplished by solving Newton's
equations of motion for each of the particles; whereas, in the Monte
Carlo method accessible states are obtained by random sampling of
configurational phase, space. Sequential solution of Newton's equations
of motion for a system of particles implies development of the time
evolution of the system on the molecular level. This time evolution
of the system provides new information not attainable in Monte Carlo.
Hence, molecular dynamics may be used to study various time dependent
properties of the system, in addition to configurational properties
(those, e.g., listed in Table 5 ). The former include transport
coefficients, such as viscosity and diffusion coefficients, as well as
a host of time correlation functions. The time correlation functions
114

115
include the force and translational velocity autocorrelation functions,
and, for nonspherical molecules, rotational velocity, reorientation, and
torque autocorrelation functions. The time correlation functions obtained
from computer simulation are of interest in testing theories for time
dependent properties and relaxation processes in fluids. Many of the
time correlation functions are experimentally accessible via various
radiation scattering experiments [66,67]; hence, these functions provide
a means for studying the intermolecular potential of real fluids.
In molecular dynamics the connection between the configurational
and time dependent aspects of the simulated system is made by requiring
that the force _F on a molecule be given by both the negative gradient
of the intermolecular potential:
Fx U(r a) ) (6-1)
and by Newton's second law. For (6-1) to hold, the system must be
conservative, i.e., the mass, momentum, and total energy are constants
of the motion. Thus, with the system volume fixed, molecular dynamics
calculations are performed in a modified microcanonical ensemble, in
contrast to the usual microcanonical ensemble wherein only the mass,
volume, and total energy are constants. This may seem restrictive
compared to the situation in Monte Carlo simulations where any of
several ensembles may be utilized; however, in this work, the
modified microcanonical ensemble introduces only the irritation
that the system temperature is difficult to set beforehand. (In
studying thermodynamics of mixtures and phase equilibria it is often

116
convenient to use isobaric or grand canonical ensembles, in such cases
the Monte Carlo method may be preferable.) Comparison of configurational
property values from the "molecular dynamics" microcanonical ensemble
with Monte Carlo results in the canonical ensemble agree within
statistical fluctuations [57,68].
This work has involved molecular dynamics simulations of fluids
of axially symmetric (linear) molecules interacting with pair potentials
of the form given in (5-3). The immediate goals have been: a) comparison
of simulation results with perturbation theory predictions for equilibrium
property values, and b) study of the molecular structure in the fluid
via the angular pair correlation function. More long range goals
include: c) study of time dependent properties, and d) comparison of
results for various model potentials in an attempt to gain insight
into the nature of the intermolecular potential for real fluids.
Initial simulation studies of structure in fluids of nonspherical
molecules were done by direct evaluation of the angular pair correlation
function, g(r u) o^) (as> e-g-> reported in Chapter 5). This direct
evaluation of a multidimensional function was found to have severe
limitations in that small angular increments are required for accurate
representation of g(rq2Wl large amounts of computer memory and long simulation runs to reduce
statistical error, especially for less favored molecular pair orienta¬
tions. These problems can be significantly reduced by expanding
g(r^2W-^W2^ an terms °f spherical harmonics and evaluating the expansion
coefficients, g^ ^ t*ie simulati°n rather than g(r^cú-^u^) [ 69 ] •
In addition, various combinations of these expansion coefficients are

117
related to equilibrium properties, thus affording a consistency
check on the property values determined in the course of the
simulation.
This chapter presents the method of performing molecular dynamics
simulations of fluids containing axially symmetric molecules. Section
6.2 develops general expressions for obtaining the force and torque
from the intermolecular potential for such molecules. Section 6.3
indicates the method used for solving Newton's equations of motion and
outlines the molecular dynamics algorithm. In Section 6.4 the spherical
harmonic expansion for the angular pair correlation function is developed
and the method of determining the center-center and site-site pair cor¬
relation functions is described. Relations between the expansion
coefficients g^ ^ m^r12^ an<^ vari°us equilibrium properties are
developed in Section 6.5.
6.2 Expressions for Force and Torque for
Axially Symmetric Molecules
A nonspherical molecule exhibits translational and rotational
motion described by, respectively:
*1
= m
~2
5 ^i
9t2
^1
= I
d2Q
¡7
(6-2)
(6-3)
In (6-2) F, is the force exerted on molecule 1 by all other molecules
in the system, m is the molecular mass, and _r is a position vector
locating the center of mass of molecule 1 relative to some arbitrary

118
space fixed axis. In (6-3) _T is the torque applied to molecule 1 by
all other molecules in the system, I is the molecular moment of inertia,
and 9 is the angular displacement of the molecular axis due to rotation
about its center of mass.
In a conservative system, I? and are also given by, respectively,
the negative radial and angular gradients of the intermolecular potential,
taken to be a sum of pair contributions:
Fi = a
—1 oi
— I u(r 0) to. ]
II ¡Ln -U 1 3
(6-4)
T, = -
9
Su,
— I u(r..ii)1iD.)
m 1J
(6-5)
In (6-5) 9/9w^ represents the angular gradient in terms of Euler angles
and oi is the orientation of molecule 1.
Evaluation of the force and torque on each molecule in the system
from (6-4) and (6-5) at regular intervals throughout a molecular dynamics
calculation is a major time consuming procedure. It is important, there¬
fore, to develop efficient methods of evaluating (6-4) and (6-5).
Significant improvement in program execution speed may be realized by a
proper choice of the manner in which molecular orientations are specified.
6.2.1 Specification of Molecular Pair Orientation
The orientation of an axially symmetric molecule is completely
specified by two independent variables. As shown in Figure 30, the
orientation ok of molecule i may be given by the polar (B.) and
azimuthal (ot^) angles between the molecular axis and a space fixed
frame, x y z . A second method of giving the orientation oj. is by

119
Figure 30. Methods of Specifying the Orientation of an
Axially Symmetric Molecule

120
a unit vector fi aligned along the molecular axis. The components of
fL are related to angles a^,8_^ by:
h. = sin
IX
8. cos a.
i i
(6-6)
h. = sin
iy
8 . sin a.
i i
(6-7)
h. = cos
a.
(6-8)
Obviously, only two of the components of JL are independent.
For intermolecular potentials which are taken to be a sum of
pair terms, the orientations of both molecules are needed. A possible
choice for specifying the orientations is use of the angles {3^,01^} for
each molecule i, shown in Figure 31. Use of such a system is cal-
culationally prohibitive, however, due to: a) the large number of
terms which arise when the pair potential is written in terms of these
angles, b) the necessity for including the orientation of the inter-
molecular axis r_ i-n the potential.
Simplification of the pair potential results if the reference
frame is chosen to be aligned relative to the pair of molecules.
One such intermolecular reference frame which is often used has its
z-axis aligned along the intermolecular axis _r , as sh°wn in Figure 32.
Using this frame, the orientation dependence of _r^ in the potential
vanishes. Further, the dependence of the potential on azimuthal angles
4^ and $2 occurs in terms of the difference ^2* Expressions
for several anisotropic model potentials in this set of variables are
listed in Appendix C. This intermolecular frame also has the advantage

121
Figure 31. Orientation Angles for Axially Symmetric Molecules
in an Arbitrary Space Fixed Frame

122
ZS
Figure 32. Orientation Angles for Axially Symmetric Molecules
in the Intermolecular Frame

123
that the polar angles 0^ may be found from simple dot product relations
between a unit vector r.. along r.. and the unit vector fi. along the
-ij -ij -i
molecular axis:
cos 0. = fi. • r. .
i -l -ij
(6-9)
The angle law of cosines of spherical trigonometry:
cos (I) . .
ij
cos Y - cos 0.
i
sin 0. sin
i
cos
0.
J
e.
j
(6-10)
where y is the angle between the axes of molecules i and j, given by:
cos Y = * fij (6-11)
There remain difficulties in using this set of intermolecular angles.
In particular, Cheung [ 70 ] has pointed out that use of the angles
0^, 0j, products to determine the direction of the torque on a molecule.
Further, as can be appreciated from (6-10), there are computational
difficulties in the neighborhood of 0^ = 0.
Equation (6-9) implies that <|> may be eliminated in favor of
the angle y and that, then, the cosines of 0^, 0 , and y may be used
as the independent variables specifying the molecular pair orientation
rather than the angles themselves. Use of the set cos 0^, cos 0^, cos y
removes the above mentioned difficulties associated with the intermolecular
reference frame. As is shown below, using these angles allows determination

124
of the direction of the torque by evaluating a single vector cross
product. Use of (6-9) and (6-11) also avoids the problem arising
in (6-10) around 0^ = 0. Thus, in the molecular dynamics work
reported here, we have used the pair potential in the form:
u. .
ij
(6-12)
or, equivalently,
u. .
ij
u(r..jfi.-r.. ,fi . • r . . ,fi. .)
ij ’-l -ij ’-j -ij ’-x -j
(6-13)
Expressions for several anisotropic potential models in the form of
(6-12) are given in Appendix C.
6.2.2 General Expressions for the Force
In this and the next subsection, derivations are outlined for
the force and torque,on axially symmetric molecules when the molecular
pair orientations are given by (6-9) and (6-11) and the pair potential
is written in the form of (6-12) or (6-13).
Equation (6-4) for the force on a molecule may be written as:
F, - - I ,2—
_1 m
u(r^^co^(.o_.)
(6-14)
where
(6-15)
has been introduced. Writing the gradient operator x—— in terms of
the spherical coordinate angles ($,a) of Figure 32 for the intermolecular
axis, the components of _F are:

125
lx
I
3*1
-sin B cos a
3uii
3ru
O 9u, •
cos B cos a Jj
9u .
sin a lj
lj
9B
r^. sin B 9a
(6-16)
iy
I
-sin B sin a
9u, . „ . 9u, .
lj cos B sin a lj
9rli
9u_. .
cos a lj_
lj
3B
r^ sin B 9ot
(6-17)
Flz = l
3*1
-cos B
9u. . a 3u .
Li , sin B 1¿
9rn . Tt. 3B
lj lj
(6-18)
To transform (6-16), (6-17), and (6-18) to the angles 0^, 02, and
Y in the intermolecular frame, we use the following relations given by
the law of cosines of spherical trigonometry:
cos 0. = cos B. cos B + sin B. sin B cos (a. - a)
i i l i
(6-19)
cos y = cos B.
i
cos B. + sin B. sin B. cos (a. - a.)
J i J i J
(6-20)
Applying the chain rule of partial differentiation to (6-16), (6-17), and
(6-18) gives:
Flx = l
3*1
3u . „
„ . li cos B cos a
-sin B sin a tt—— -
9r,. r, .
lj lj
3 cos 0, 3u, .
1 Ll
3B 9 cos O..
+
3 cos 0. 9u,.
Jl Ll
9B 9 cos 0.
J
sin a
r, . sin B
lj
3 cos 0, 9u,.
1 i1
9a 9 cos 0,
9 cos 0. 9u, .
1 Ll
3a 3 cos 0 .
J
(6-21)

126
I
3*1
-sin 6 sin a
3u
3r
M
lj
cos (3 sin a
cos a
3 cos 0^
3u . 3
1.1 ,
cos 0 .
J
3um '
r, . sin 3
lj
3a 3
cos 0^
3a
3 cos 0.
J.
(6-22)
F
lz
l
3*1
i „ ^Ulj , sin 3
I -cos 3 -—— +
3r
lj
lj
3 cos 0^
3u . 3
cos 0 .
J
3\i ‘
33 3
cos 0^
33
3 cos 0.
J.
(6-23)
3 cos 0. 3 cos 0.
Evaluating the derivatives — and from (6-19) and
(6-20), using the relations (6-6), (6-7), and (6-8), performing some
algebraic manipulation, and using the trigonometric relation:
sin x cos (x-y) - cos x sin (x-y) = sin y
(6-24)
(6-21), (6-22), and (6-23) can be recast in the vector form:
II = l
3*1
3u
-r —ll + —i— fr (fi *r )
-lj 3r, . r . L-ljV-l-lj;
lj lj
3u
- fij
n
—1 9 cos 0,
9u
- fij
u
â– j 3 cos 0 .
JJ
(6-25)
Equation (6-25) is the general expression for the force exerted on
molecule 1 by a system of axially symmetric molecules when the pair
potential is written in the form of (6-12). It is readily apparent

127
that for an isolated pair of molecules, (6-25) gives ]? = - •
Expressions for the derivatives in (6-25) for several anisotropic
potentials are given in Appendix C.
6.2.3 General Expression for the Torque
The derivation for the general expression for the torque proceeds
in an analogous fashion to that for the force. To obtain (6-5) in terms
of angles in the space fixed reference frame, we use the fact that the
components of the angular gradient in (6-5) can be shown to be propor¬
tional to components of the angular momentum operator L [ 38 ], then,
ii ‘ - 1 h l u J^l
where, in the present notation,
(6-26)
(6-27)
The components of in the spherical coordinates of Figure 32 are
given by Rose [ 38 ]. Hence, we find the components of _T^ in angles
relative to a space fixed frame to be:
T
lx
T
iy
T
lz
l
j/1
1
l
jtfl
icos 6,
cos
a, 8u, .
j kl
1 lj
sin
B1
3a^
[cos 6,
sin a,
3u, .
1
1
lj
sin
^1
3a^
du
4- sin a
ij
1 36,
9u
cos a
1 36,
(6-28)
(6-29)
(6-30)

128
To transform (6-28), (6-29), and (6-30) to forms involving
Qj, 9j> Y> apply the chain rule:
Tlx - .1
3Tl
COS 8-^
8, cos a^ 8 cos 0^ 8 cos 0^
~r> ^ + sin ai Ko~~
8-j^ 90]^ 1 381
8u
8 cos 0-,
cos 81
cos a^
8 cos Y
8 cos Y
8ulj 1
sin
gl
r\ 1 sin cx—
9a^ 1
881
8 cos Y|
(6-31)
ly
= l
j/1
cos 8-^ sin a^ 8 cos 0^ 8 cos 01
:—ñ — - cos a,
sin 8,
8a,
1 98,
9u, .
8 cos 0,
+
cos 8,
sin a, „
8 cos Y
8u .
1
1 8
cos Y
lj
sin
B1
8a^
COS Ct i n q
1 dp^
8 cos Y
(6-32)
Tlz = ^
j^l
8 cos 0, 3u,.
1 Jj_
801 8 cos 0^
~ 8u .
8 cos y lj
88x 9 cos Y
(6-33)
As for the force, evaluate —~
8 cos 0, 9 cos 0, n.
1 18 cos Y
a.
98,
8a,
, and
d ^ from (6-19) and (6-20), use (6-6), (6-7), (6-8), and (6-24),
dp-j^
and write in the vector form:
8u1. 8u .
T = - 'i -(r 3 t fi. 3 |
—1 —1 . ^ 1—13 8 cos 0^ —J 8 cos y|
(6-34)
Equation (6-34) is the general expression for the torque exerted on
molecule 1 by a system of axially symmetric molecules interacting with
a pair potential expressed in the form of (6-12). Note that the vector
cross product in (6-34) is taken after performing the summation over the

129
other (N-l) molecules in the systems. This results in a considerable
saving of execution time. For an isolated pair of molecules, it is
straightforward to show that (6-25) and (6-34) satisfy:
(6-35)
i.e., the angular momentum of the system is conserved. Expressions for
the derivatives in (6-34) for various anisotropic model potentials are
given in Appendix C.
An independent derivation of (6-25) and (6-34) has recently been
given by Cheung [ 70 ]•
6.3 Method of Solution of the Equations of Motion
and the Molecular Dynamics Algorithm
6.3.1 Method of Solution of the Equations of Motion
Several different methods have been used to solve the translational
and rotational equations of motion (6-2) and (6-3) for axially symmetric
molecules [66,71]. The method used here is that due to Cheung and Powles
[ 60 ]. A major advantage of this method is that problems associated with
solving the second order differential equation (6-3) for the molecular
orientations in the intermolecular frame are avoided by solving, instead,
the corresponding first order differential equation for the angular
velocity
(6-36)
The difficulties which arise in solving (6-3) are: a) the second order
equations for the polar and azimuthal angles must be solved separately,

130
thus, more computation Is involved, b) the second order equation for
2
the azimuthal angle cp contains a term involving 1/sin 0. Such a term
introduces computational difficulties in the region around 0=0 [71 ].
Once the angular velocity is obtained from (6-36), there remains
the determination of the molecular orientation _fi . The orientation may
be found by realizing that the angular velocity and angular acceleration
are mutually perpendicular to one another and to the molecular axis [60 ].
Since the angular acceleration is proportional to the torque obtained
from (6-34), the unit vector may be found by:
fi.
—i
—i 1 —i
(6-37)
, (n) . ,. , th
where x indicates the n time derivative of x.
The method used for solving the second order translational equations
of motion (6-2) and the first order rotational equations of motion (6-36)
is the predictor-corrector algorithm of Gear [72 ]• The method involves
three steps: prediction, evaluation, and correction. In the predictor
step, the position _r^ and orientation fi of each molecule and their first
five time derivatives at time t + At are predicted from their values at
time t by a Taylor's expansion. The angular velocity fi of each molecule
and its first four time derivatives are predicted in the same manner:
r\ (t+At) = r.(t) + r(1)(t)At + r(2)(t)(At)2/2! + ••• + r(5)(t)(At)5/5!
rP(1)(t+At) =rj1}(t) t+ rJ2)(t)At + ••• + r[5)(t)(At)4/4!
rP(2)(t+At) =r^2)(t) +r^3)(t)At+ ••• + r|5)(t)(At)3/3!
(6-33)
rP(5)
(t+At)
rP(5)
(t)

131
nJ(t+At) = n (t) +^x)(t)At + ••• + n^4)(At)4/4!
gP(4)(t+At) =nP(4)(t)
(6-39)
fiP(t+At) = fi.(t) + fia)(t)At
—i —i —i
fiP(5)(t+At) = fi^5)(t)
+ fij5)(At)5/5!
(6-40)
In the evaluation step, the force (translational velocity) and
torque (angular velocity) are determined at the predicted positions
P P
r_. and orientations _fi. from (6-25) and (6-34).
p
In the correction step, the predicted positions r_. and angular
p
velocities and their derivatives are corrected by:
rfn)(t+At) = rP(n)(t+At) +a Ar. [(At)n/n!]-1
—i —i n —i
(6-41)
f^n)(t+At) = ftP(n) (t+At) + 6n Afi± [ (At)n/n! ] 1
(6-42)
where the correction terms in (6-41) and (6-42) are proportional to the
difference between the predicted and evaluated accelerations:
Ar.
—l
[rf }(t+At)
- rP(2)(t+At)](At)2/2!
(6-43)
AÍ2.
—1
[n|1}(t+At)
fiP(1)(t+At)]At
(6-44)
The parameters a and 6 in (6-41) and (6-42) are chosen to maintain
n n

132
stability of the solution and depend on the order of the differential
equation and the degree of expansion in the Taylor series predictor
step. For the problem described here, Gear [ 72 ] gives the values:
a0’ ' ' ' ,a5 = 3/16, 251/360’ 1> 1;L/18> i/6’ i/60 (6-45)
= 251/720, 1, 11/12, 1/3, 1/24 (6-46)
This predictor-corrector method is not self-starting; consequently,
we use the following startup procedure:
a) For the initially assigned molecular positions, velocities,
orientations, and angular velocities, calculate the translational and
angular accelerations from (6-25), (6-36), (6-34) and (6-2), and set
the higher derivatives of r. and to zero.
—i —i
b) From the initially assigned orientations, evaluate the
derivatives of fi. to by repeated differentiation of:
—x —i
fif1) = - fi. „ Í2. (6-47)
—l —i —i
The calculation then proceeds with the predictor step above. Setting the
higher derivatives of r_ (0) and fi^(0) to zero introduces a slight error
into the first few solutions of the equations of motion. However, the
algorithm corrects itself to the proper solutions after a few time steps
have evolved.
It should be emphasized that no comparison of various methods for
solving the equations of motion have been made in this study. The method

133
described above has been found to give stable solutions and maintain
-14 -15
conservation of energy and momentum when time steps of 10 to 10
seconds are used; however, more economical and efficient methods need
to be developed. In this regard, Beeman [73 ] has recently compared
several methods applied to the molecular dynamics simulation of a
Lennard-Jones fluid. He finds a simple third order predictor-corrector
method in which only the positions _r are predicted and the velocities
r are corrected to be superior to any of the other methods tested.
It would be useful to test this Beeman algorithm on angle dependent
potentials such as those studied herein. For calculations in which
one is not interested in time dependent properties, Bennett [74 ] has
developed a method of speeding up the dynamics of the system by scaling
the particle masses. This method, likewise, has only been tested on
fluids of spherically symmetric molecules.
6.3.2 Molecular Dynamics Algorithm
The molecular dynamics calculations reported here were done for
256 particles interacting with pair potentials of the form given in
(5-3). The basic cell for the system was of cubic shape and periodic
boundary conditions were used to negate surface effects. In order to
solve the rotational equations of motion (6-36), the molecules were
assumed to be homonuclear diatomics and the bond length corresponding
to the nitrogen molecule £/o = 0.329 was used [ 71 ]. The choice of bond
length can have no effect on the equilibrium properties since potentials
of the form (5-3) are independent of bond length. In these calculations
the potential was cut off at either 3.2a or 3.0a. The unit of length

134
was taken to be the molecular diameter 0 and the unit of mass was taken
to be the molecular mass. The calculation procedure is as follows:
1. Choose a particular potential model whose form is given in
* 3
Appendix C, a fluid density p = pp , and a time step At at which to do
k
the simulation. Choose an approximate temperature, T = kT/e for the
system.
2. For the initial positions r\(0) of the molecular centers of
mass use either an FCC lattice structure or take the positions from the
end of a previous calculation.
3. Assign initial orientations ifu(O), translational (0), and
rotational fi^(0) velocities to the molecules from random numbers uniformly
distributed on (-1,1).
4. Scale the velocities so there is no net translational or
rotational drift on the system:
i1} (0) - | l ^(O)
(1)
i=l
(6-48)
n. (o) = si.(o) - 4 y n. (o)
—i —i N . , —i
1=1
(6-49)
5.Scale the velocities to give approximately the correct temper-
k
ature T :
r^1}(0)
k
3T N
l r;
. — i
i
(1)2
1/2
n.(o)
fi±(0)
*
2T N
I l Í22
. —i
(6-50)
1/2
(6-51)

135
where the moment of inertia is taken to be
I
m
m
(6-52)
6. Begin the predictor-corrector algorithm using the startup
procedure given in Section 6.3.1.
7. Predict the molecular positions, angular velocities, orienta¬
tions and their derivatives at time (t+At) by (6-38), (6-39) and (6-40).
8. Using the predicted positions and orientations, evaluate the
force and torque on each molecule using (6-25) and (6-34), and, hence,
get the actual translational and angular accelerations from (6-2) and
(6-3). Also, use the predicted positions and orientations to evaluate
the ensemble averages for the equilibrium properties given in Table 5.
Determine the pair correlation functions from the predicted jr and fL
by the method given in the next section.
9. Correct the predicted positions and angular velocities using
(6-41) and (6-42).
10. Calculate the molecular orientations at (t+At) from (6-37).
11. Check that each particle is in the basic cell based on the
corrected positions. Use the image having the "minimum image distance"
for any particle no longer in the cell [57 ] .
12. During the initial time steps for which the mean squared
displacement of the molecules is less than some e value, the system
is not considered to be at equilibrium and the velocities are rescaled
after each time step using (6-50) and (6-51). When starting from an
FCC lattice structure, E is taken to be about 10% of the molecular
radius, which is a corruption of Lindemann's law of melting [ 75 ].

136
The breakdown of the FCC lattice usually requires 200-400 time steps.
Contributions to property averages are not calculated during this
portion of the calculation.
13. When the mean squared displacement exceeds the desired £
value, steps 7-11 are iterated over the length of the run, usually
2000 time steps.
14. During the last 1000 time steps of the calculation,
positions, orientations, translational and rotational velocities
and accelerations of the molecules are transferred to magnetic tape
at intervals of 10 time steps. This data is for subsequent analysis
of time correlation functions and van Hove distribution functions.
The program was written in Fortran using single precision
arithmetic. For 256 particles interacting with a Lennard-Jones plus
quadrupolar potential cutoff at 3.2a, the program generates about
60 time steps per hour on the Honeywell 635 at the U.S. Military
Academy. For the Lennard-Jones plus overlap potential cutoff at
3.00, the program executes about twice as fast.
6.4 Evaluation of Pair Correlation Functions
6.4.1 Definitions of Various Pair Correlation Functions
(This subsection of Chapter 6 is taken from references [40], [41],
[76]-)
The angle dependent distribution functions for molecules inter¬
acting with potentials of the form of (5-3) are defined in a manner
analogous to that for distribution functions for spherical molecules.
The generic distribution function of order h f(.r , ...r ü) . . .w, ) is

137
defined such that f (r\j^) dr^dui^ is proportional to the probability
that given a molecule 1 with position r\^ in d.r^ and orientation in
doj^, molecule 2 is at in dr\^ with orientation oo^ in do^, etc., up
to molecule h, irrespective of the positions and orientations of the
remaining N-h molecules [40]:
h h. N!
f(r (*) ) =
„ -6U(rNuN) , N-h , N-h
e — dr dcú
(S-h)! ' -BU(rV) N , N
e — dr did
(6-53)
An angular pair correlation function gCr^ca*1) can be defined in terms of
f (r^to*1) analogous to the definition of the radial distribution function
for spherical molecules:
f(r%h) = fCr^) f (_r2“2) *''f (ihwh^ gC^V1)
(6-54)
For an isotropic, homogeneous fluid, the value of the singlet angular
distribution function f (_r^üJ^) is independent of r, and U)^; hence, putting
h = 1 in (6-53) gives:
fO^co-^) = p/ft (6-55)
where ft is the integral over the angular volume element dw; ft = 4tt for
2
linear molecules and ft = 8ir for nonlinear molecules. Thus, for isotropic,
homogeneous fluids, (6-54) reduces to:
f(r u ) = (p /ft ) g(£ )
(6-56)

138
The distribution of molecular centers, independent of molecular
orientation, is obtained by integration of f(_r^tO^) over orientation
f(rh) =
c , h hN h
f(r to )dto
(6-57)
Using (6-56),
f(rh) =
h^ , h h,,
p
to
(6-58)
or,
, hN „ , h hw
gc(£ ) = h
to
(6-59)
The pair correlation functions are studied in detail here. From
(6-53) and (6-56) the angular pair correlation function is given by:
, . N(N-l)fi
g(r12WlW2) 2
-8U(rNtoN) . N-2 , N-2
e — dr dto
N N
-6U(r to ) , N , N
e — dr dto
(6-60)
Likewise, the center-to-center pair correlation function can be
obtained from (6-59) :
gc(rl2) <8(rl2a)lW2')>to1to2
(6-61)
Less formally, g^(r) may be defined as the ratio of the local
number density of molecular centers of mass at distance r from the

139
center of mass of a given molecule, independent of the orientations
of the molecules, p^(r) to the bulk fluid number density p:
P (r)
8c(r) = “7“ (6_62)
It is also of interest to study so-called site-site pair cor¬
relation functions g D(r), where a,3 are sites located on a molecule.
ctp
Usually the molecular sites of interest are the atomic centers on a
polyatomic molecule. The g^(r) is, then, proportional to the prob¬
ability of finding the 3-site of some molecule at a distance r from
the a-site of some different molecule [76 ]. The function 8ag(r) may
be obtained from the angular pair correlation function by:
WW ' (6-63)
where r. is the vector from the center of molecule i to site a.
—i
Less formally, 8ag(rag) can defined as the ratio of the local
number density of 3-sites at distance r_^ from the a-site of a given
molecule, independent of molecular orientations and excluding the 3~site
of the given molecule, to the bulk fluid number density:
= P3(ra3}
8a3^a3^ p
(6-64)
6.4.2 Evaluation of Center-Center and Site-Site Pair Correlation
Functions
The center-center pair correlation function gc(r) can be determined
in a molecular dynamics calculation by using the definition (6-62). The

140
procedure is to first divide the volume of the cubic system of side
into spherical shells of thickness Ar. In this work Ar was set
equal to 0.025a. During the evaluation step of the molecular dynamics
procedure, the distance r_ between centers of each pair of molecules
is determined, using the predicted positions. For each r„ , the spherical
shell in which molecule j lies relative to molecule i is determined and
a counter I (r..) is incremented. After n time steps,
cv ij K
21 (r. .)
c iJ =
n
Average number of molecular centers in
spherical shell having boundaries
. , Ar
at r ±
(6-65)
The factor of 2 arises in (6-65) to include the contribution of the
shells due to r.., which are not included in I(r..).
JJi ij
If the fluid were of uniform density p for all r regardless of the
location of any given molecule, the average number of molecular centers
in any spherical shell would be given by NpAV(r), where AV(r) is the
volume of the shell. Then from (6-62):
gc(r)
21 (r)/n
c
Np AV(r)
(6-66)
where
AV(r) =
2tt
dtp
r +
Ar
d(cos 6)
-1
r -
2 ,
r dr
Ar
(6-67)
4tt .
= 3- Ar
3r2+IAl)J
(6-68)

141
Determination of the site-site pair correlation functions
g Q(r D) is accomplished using (6-64) in the same manner as for
Ctp Cip
gc(r) described above. The resulting relation is:
sa3(raB
) =
2IaB(raB)/n
Np AV(raB)
(6-69)
To determine the distances r D for a pair of molecules, consider the
CXp
geometry for a pair of diatomics with sites A and B separated by distance
£ on each molecule, as shown in Figure 33. In the molecular dynamics
calculation, the locations of the centers of mass of each molecule
relative to some space fixed frame r^ are known. The orientations of
each molecule are also known in terms of unit vectors fi aligned along
each molecule. Thus, with Figure 33 and some simple vector addition,
the distances r n are found to be:
r
tx8
(6-70)
where the first sign on the right hand side is + if a = A, - if a = B,
and the second sign is + if a ^ 8, - if a = B> and r^ is given by (6-15).
Note that for homonuclear diatomics, g^ = g^ due to symmetry.
6.4.3 Spherical Harmonic Expansion for g(r^2—x—2-i-
As pointed out in Section 6.1 it is both inaccurate and inefficient
to attempt direct evaluation of the angular pair correlation function
®^r12°JlÜJ2^ '*'ri comPuter simulation studies. The better method is to
expand g^^^t^) •’■n spherical harmonics and evaluate the expansion

142

143
coefficients in the simulation. The expansion considered here is
an infinite series in terms of products of spherical harmonics of
the molecular orientations in the intermolecular frame of Figure
32:
g('r12“la)2^ = 4?r
(6-71)
where
m
m .
(6-72)
[38]. The gp j, m(r^2^ are t^ie exPansi°n coefficients to be determined
in the simulation. For molecules with a plane of symmetry perpendicular
to the molecular axis, the g„ „ (r,„) are zero unless both and £„ are
X, x, m 12 12
even (homonuclear diatomics, e.g.). For heteronuclear diatomics g0 „
± zm
with both odd and even values contribute.
Multiplying both sides of (6-71) by Y (oo„) , where *
„m 2
indicates complex conjugate, and integrating over 0)^ and :
(6-73)
The orthogonality of spherical harmonics gives [ 38 ] :
(6-74)
So (6-72) becomes:

144
8¿1Jl2m(r12) 4tt
dü3i dw2 Y£1m(a)l) Y£2m(W2) g(r12°Jlü)2)
(6-75)
Dividing both sides of (6-75) by
du)
1 do)2 g(r12w1w2)
S£1£2m(rl2)
dw1 dw2 g(r12w1<*)2)
1_
4tt
d“l d“2 Y21m("l) Yi2m(“2)
dü)1 dw2 g(r12co1(J2)
(6-76)
The ensemble average of a property X (r^co^o^) in a spherical shell of
radius between r^2 and r^2 + dr^2 is defined by:
8hell
dco1 dw2 x(r 2(11^2) g(r12w1w2^
dw1 dw2 g (r 2oj oj2)
(6-77)
Further, from (6-75),
8000(r12) „ ,2
(4tt)
dw, dco2 g(r12u1iJ2)
(6-78)
Combining (6-76), (6-77), and (6-78) gives:
8£1£2m(rl2) 47T 8000(r12) Y£2m(ü32)>shell
(6-79)
Note that comparison of (6-78) and (6-59) gives:
8000('rl2^ gc^r12')
(6-80)

145
Expressions for all ^ m^rl2^ terms having £^ and even up to
{£^£2m} = {444} and the additional terras with £^ and even, m = 0
to {£^£2m} = {800} are tabulated in Appendix F.
Some authors prefer to expand 8(r^2ÜJla>2^ terms °f products
of spherical harmonics of the molecular orientations in the space fixed
frame of Figure rather than the intermolecular frame [77 ]. This
expansion is:
S(rl2WlW2)
l I g(^1^2i,’ri2-) C(£1£2£;m1m2m)
£^£2£ m^m2m
x Y (to ) Y (oa )
X/^ni-^ 1
yL(w)
(6-81)
Here w = {3»ot} is the orientation of the intermolecular axis, as in
Figure 31; Cjm^n^m) is a Clebsch-Gordan coefficient in the
convention of Rose [ 38 ] .
The coefficients of the intermolecular frame expansion (6-71),
^ m^rl2^’ Can determined from those of the space fixed frame
expansion, (6-81), g (£j£ £; r12), by:
J3
g£1£2m(rl2) = (47T)~ 2 \ (2£+1>1/2 g(^2¿;r12) C(¿^jm^O) (6-82)
Conversely, the g(£^£2¿;r^2) may be found from the g^ ? m^rl2^
3/2
g(£1£2£;r12) * *i/2I
.1/2 L &£ £ m.. (r12) c(i'1£2£;mlH1°)
(2xrKL) 12 1
(6-83)

146
6.5 Equilibrium Properties from the g^ „ ^(r^^)_
The coefficients ^ m^ri2^ Provide useful information about
local structure and molecular orientations in fluids when recombined
in the series (6-71) to obtain the angular pair correlation function
g(r12C°l(JJ2) ’ Edition, integrals over various combinations of the
gf, £ m^rl2^ 8ive t'ie ecluilibrium properties of the fluid. In the
context of machine simulation studies, such relations provide useful
consistency checks between the direct evaluation of the properties
via the ensemble averages of Table 5 and the determination of the
go o (r..„) coefficients. In this section the derivation of the
relation between the g^ ^ m(r^ 2) and the configurational energy U
due to a Lennard-Jones plus quadrupolar model potential is given.
Derivations of the corresponding relations for other potential models
and other properties are accomplished in an analogous manner; therefore,
only the resulting expressions are presented in tabular form.
The configurational energy for polyatomic fluids is given by:
dr
(1)^2 12
(6-84)
where X(12) = Xir^a^t^) anc* <•**> is defined by (2-15). Using for u(12)
the model of (5-3) gives:
U/N = ULJ
+ U
a
(6-85)
ULJ(r12) g000(r12-) d-12
where
(6-86)

147
and
U =£
a 2
Wia)2 dr12
(6-87)
Equation (6-86) has been obtained by using (6-78). Substituting the
Lennard-Jones potential (3-2) into (6-86) gives:
ULJ - 8710 e
*-±2 *-6 *2 *
8000(rl2') ^r12 “ r12 ^ r12 dr12
0
&..£„m . .
12 * "k
Defining the integrals (p ,T ) by:
(6-88)
J
n
, x *-(n-2) , *
8¿1¿2m(rl2) r12 dr12
(6-89)
(6-88) can be written:
,, o * r T000 ,000.
ULJ ' 8,1(5 c IJ12 - J6 1
(6-90)
To evaluate U from (6-87), first obtain an expression for
3.

a &
Consider u (12) to be the quadrupole-quadrupole
cL
potential. Then from Appendix C, Eqs. (C5) and (C8) give the expansion
for Uqq(12) in terms of spherical harmonics of the molecular orientations
in the space fixed frame:
,, 9x _ 8rr .1/2 Q_
UQQ^12^ 15 (70tt) 5
r12 mlm2
l C(224;„1m2„) Y^ÍUj)
(6-91)
Choosing the space fixed z-axis to lie along the intermolecular axis gives
* 1/2
9 = 0, Y4m(w) = 3/(4tt) ó q, and m2 = m^, hence (6-91) reduces to:

148
V12)
4tt
5
,2 2
(70)1/^2 £ C(224;m1m10) (w ) Y (u ) (6-92)
r12 ml=- 1 ~1
Combining (6-71) and (6-92)
^QC/12^12^
5 (70) Q5 f Sl a m(r12* ^ cC224;m1m10)
r12 í'ií'2in 12 ml
X Y2m1(“l) h¿SU2> Y2m1(W2)>V2 (6’93)
Using
(6-94)
and the orthogonality condition (6-74), (6-93) reduces to:

QQ 0)^2
1/2 2 2
I £ g99 (r 9) £ C(224;m1m10) ó 6
5 5 L 22m 12 1—1 mm, mm,
r^2 m m^=2 — 1 —1
(6-95)
Using standard tables to evaluate the C(224;m1m^0) [35] and with the fact
that since the g£ , (r^)
are real,
8¿1£2m(rl2) 8£1£2m(r12)
(6-96)
(6-95) becomes:
(Jü1tü2
5 5 [g220(r12) + 3 8221(rl2) + 3 8222(rl2)]
r12
(6-97)

149
An analogous procedure gives ^ ^ for other potential
models. Results are given in Table 8.
Combining (6-87) and (6-97), and using (6-89),
12tt
5
* *?
p Q e
[J
220
5
222.
5 J
(6-98)
Results for u for other potential models are given in Table 9.
SL
Tables 10, 11, 12, 13, 14, and 15 give expressions relating the
^1^2m * *
integrals Jn (p ,T ) over the coefficients g^ ^ m^rl2^ t0 tlie Pres_
sure, Fowler model surface tension, Fowler model surface excess internal
energy, mean squared force, mean squared torque, and angular correlation
functions G , respectively.
Lj

150
Expressions
TABLE 8
for in Terms
a ° 0)10)2
) for Various Model Potentials
u) co 3 3 [8110(r) + 8lll(r)]
12 r
(6-99)
üo to 5 Q5 [g220(r) + 3 8221(r) + 3 g222(r)] (6 100)
1 2 r
to1to2 g120(r) ^ g210(r) + 2g121(r) 2g211(r)]
(6-101)
= — 6e
over 6 wxw2 ^
r il2
o
g2oo(r^
(6-102)
= - — ke
dlS (JJ^O)2
0_
r
g200^r')
(6-103)
432 2
175 K £
[g220(r) + 3 §221^r^ + 3 g222^r^
t
D,Q,over,dis = dipole, quadrupole, overlap, and dispersion, respectively

151
TABLE 9
Expressions for the Configurational Energy in Terms
of ^ ra(r^n) for Various Model Potentials
U/N = U , + U
LJ a
(6-104)
// o * r,000 ,000,
ULJ = 8iTp £ [J19 - JA ]
12
(6-105)
_ 4tt * *2 , T110 , ,111,
UDD 3 P 0 £ [J3 + J3 ]
(6-106)
_ 12tt * *2 r 220 ^ 4 ,221 ^ 1 ,222,
UQQ 5 P Q £ [J5 + 3 J5 + 3 J5 1
(6-107)
„ 2tt * * * . f- ,120 R ,210 . .,121 0,211,
% = — P 0 Q £ [/3 J4 - /3 J4 + 2J4 - 2J4 ]
/5
(6-108)
over
3 2tt * ,200
p óe J,o
/5 12
(6-109)
U,. =
dis
16tt * ,200
P <£
/5 6
864, * 2 ,,220 , 4 ,221 , 1 ,222. ,,
175“ p K £ [J6 + 3 J6 + 3 J6 1 <6_110)
175

152
TABLE 10
Expressions for the Pressure in Terms of
gn n (rio) for Various Model Potentials
x. 2m—^
P
pkT
(6-111)
16ttp r 000 _ 000
LJ kT 1 6 12
(6-112)
P
a
—2— U
3kT a
(6-113)
where ng = 6,12,(£+1) for dispersion, overlap, or
‘multipole, respectively
and expressions for U are as given in Table
3.

153
TABLE 11
Expressions for the Fowler Model Surface Tension
in Terms of ^ n/r12 ^or ^ar-‘-QUS Model Potentials
F 2, F 2, , F 2,
Y a /e = yLJa /e + yao /e
F 2, „ *2 rT000 „T000n
YLJa /e = 3Tip 12T5 - 2JU ]
F 2, tt *2 *2 rT110 , Tlll1
TDD° ,£’ F 2, 3 *2 *2 ¡ t220 , 4 T221 , 1 ,222,
YQQ° /£ = " 4 ^ Q [J4 + 3 J4 + 3 J4 ]
F 2, TT * *n* r To t120 75- T210 , ..121 0,211,
Ynrio /e p y Q [/3 J - /3 J„ + 2J- - 2J, ]
DQ 2/5 3333
F 2,
Y a /e =
over
24tt *2- t200
—- P 6 J
/5 11
F 2. 6tt *2 t200 , 324 *2 2 r T220 , 4 ,221 , 1 ,222,
ydisa /E - ^ P 15 J5 + 175 P K [J5 + 3 J5 + 3 J5 1
(6-114)
(6-115)
(6-116)
(6-117)
(6-118)
(6-119)
(6-120)

154
TABLE 12
Expressions for the Fowler Model Surface Excess Internal
Energy in Terms of g^ ^ m^lZ^ For ^ar:‘-ous Model Potentials
UgCr2/£ = Ug a2/e + Ug a2/e
LJ a
7TF 2, *2 n000 T000!
U a /£ = 2ttp [J - J . ]
bLJ 5
„F 2, tt *2 *2 , T110 , ,111!
U a /e = t p p [J + J ]
DD J ¿ ¿
„F 2, 3tt *2 *2 ,720 , 4 T221 , 1 t222i
U a /e = - — p Q [j + — j, + — j ]
QQ P 4 3 4 3 4
..F 2, TT *2 ‘* * r/T 120 pr 210 t121 ot2111
U a /£ = p p Q [/3 JQ - /3 J, + 2J0 - 2J0 J
SDQ 2/5 3333
„F 2. 8tt *2x 200
us 0 /e - - - p 6 J
over vd
ttF 2, 4tt *2 200 , 216 *2 2 r 220 , 4 221 , 1 222.
U CT/£- — p < J5 +175TP K [J +IJ5 +3J5 ]
dis /5
(6-121)
(6-122)
(6-123)
(6-124)
(6-125)
(6-126)
(6-127)
175

155
TABLE 13
Expressions for the Mean Squared Force in
Terms of ^ m^r12^ ^or Var:*-ous Model Potentials
1 2
2 2-2 * * onn non
LJ o e = 96TTP T [22 J~u - 5Jg]
2 2-2
a e =0
1 MULTI
2 2-2 16 * * 200
o e = (0.504TT) — p T 6 jr,
1 over /5 ^
(6-128)
(6-129)
(6-130)

TABLE 14
for the Mean Squared Torque
in Terms of
Various Model Potentials
- - 4 T* Vs <6-133)
/e2 =
1 over
- 6 T* U /e (6-134)
over
. 2 ,2
,. /e = -
1 dxs
6 T* U /e (6-135)
dis
(6-135)

157
TABLE 15
Expressions for the Angular Correlation
Functions in Terms of g^ ^ ü/r12^ ^or
Various Model Potentials
â– k
4fTp y . .m LLm
L “ 2L+1 ¿ C ) J0
m
(6-136)
e.g.,
4rrp* r 110
1 3 LJ0
G
2
4?rp 220
5 N0
2jJX1] (6-137)
2Jq21 + 2Jq 2 2] (6-138)

CHAPTER 7
MOLECULAR DYNAMICS RESULTS
This chapter presents the equilibrium properties obtained by
the molecular dynamics method described in Chapter 6. In Section 7.1
analysis is made of the potential models considered in this study.
Section 7.2 gives values for equilibrium properties obtained from
molecular dynamics and comparison with predictions from perturbation
theory is made. Values for the spherical harmonic coefficients
£ n/rl2^ are Presented i-n Section 7.3. Also, values for the
J integrals are given and equilibrium properties obtained from
the are compared with values obtained by direct evaluation in
the course of the simulation. In Section 7.4 the angular pair cor¬
relation function g(r^2U^W2^ obtained by recombining the spherical
harmonic expansion is studied. The site-site pair correlation
function is discussed in Section 7.5. In Section 7.6 a method for
producing filmed animations of the molecular motions from a molecular
dynamics simulation is described.
7.1 Potential Models
The model intermolecular pair potentials considered here are
of the form in (3-1), i.e., isotropic Lennard-Jones plus an aniso¬
tropic contribution. The anisotropic contributions considered
include dipole, quadrupole, anisotropic overlap and anisotropic
dispersion. Specific expressions for these anisotropic potentials
158

159
are given in Appendix C. In this section we explore the nature of
these potential models for various molecular orientations and as
a function of the strength constant (y,Q,6 or <) associated with each.
A dégree of ambiguity arises in trying to resolve a physical
description of the molecule associated with potentials of the form
in (3-1). The potential contains a spherically symmetric repulsive
core (due to the Lennard-Jones contribution) plus a nonspherical,
axially symmetric anisotropy. In the case of multipoles, the
anisotropy may be interpreted as arising due to point charges
imbedded in the core. The resulting force field is, therefore, axially
symmetric about the molecule. Geometrically, the molecule may be
interpreted as axially symmetric. For dipole, quadrupole, dispersion,
and overlap (with 6 > 0) anisotropies, the molecules are considered
linear. For the overlap model with 6 < 0, the molecule is platelike
(e.g., benzene). When the anisotropy is quadrupolar or overlap with
6 > 0, the molecule can be considered to be prolate ellipsoid (e.g.,
homonuclear diatomic). If the anisotropy is dipolar, the molecule
can be considered to be heteronuclear diatomic. These distinctions
are important, e.g., in recognizing which terms contribute to the
spherical harmonic expansion for g . In solving the equations
of motion in the molecular dynamics simulation, the moment of inertia
appropriate to the shape of the molecules is required (see Section 6.3.2).
For linear molecules, the moment of inertia depends on the length of the
molecular axis (considered to be the bond length in diatomics), as in
Equation (6-52). However, the potential model (3-1) is independent of
bond length. Thus, equilibrium property values obtained in the simulation

160
are independent of bond length for the model of (3-1). Time dependent
properties, on the other hand, remain a function of bond length, through
the moment of inertia and the rotational equations of motion. The value
chosen for the bond length will have an effect on the rate of con¬
vergence of static properties to their equilibrium values (i.e., the
number of time steps required to obtain statistically meaningful results).
Thus, for a given value of the torque on a linear molecule, Equations
(6-36) and (6-52) indicate that smaller bond lengths correspond to
higher rotational velocities which promote more extensive sampling of
angular phase space compared with that resulting from longer bond
lengths.
7.1.1 Lennard-Jones Plus Dipole Potential
This potential model is obtained by combining Equations (3-1),
(3-2), and (C20). Figure 34 shows values of the pair potential for
various molecular pair orientations as a function of the separation of
their centers of mass, r. The six pair orientations shown in Figure 34
are the "primary" orientations, i.e., for the models considered in
this study, one of these six has always been found to be the most
probable orientation in terms of both minimum energy and maximum
value of the angular pair correlation function g(r^^j^^ ’ Definitions
of these orientations in terms of the relative angles in the inter-
molecular frame of Figure 32 are givei> in Table 16. Figure 34 in¬
dicates that the pair orientation having the minimum configurational
energy is the head-to-tail endon orientation; while that having the
largest energy is the head-to-head orientation. Note that the dipole

161
Figure 34. Pair Potential for Lennard-Jones plus Dipole Model
Fluid at Primary Pair Orientations

162
TABLE 16
Primary Orientations for Pairs of Linear Molecules
61
61
*12
Y
0.
0.
J-
1
head-to-tail
endon
0.
180.
head-to-head
endon
t i
90.
90.
0.
0.
parallel
t 4-
90.
90.
180.
180.
antiparallel
1 -
90.
0.
• * '
90.
tee
+
90.
90.
90.
90.
cross
t
indicates
the angle
is undefined.

163
potential makes no contribution when the molecular pair is in the
cross or tee orientations. As the strength of the dipole moment
is increased, the well in the potential energy curve is found to
deepen for the head-to-tail and antiparallel orientations. Whereas,
the depth of the well for the head-to-head and parallel orientations
decreases.
The characteristics of the potential models presented in this
chapter do not necessarily generalize to other, similar models. For
example, the Lennard-Jones diatomic plus dipole model differs signif¬
icantly from the Lennard-Jones atom plus dipole model described above.
The Lennard-Jones diatomic potential is given by [71]:
u(rl?UJ1 w?) = 4e I
k=l
12 6
(f) - (f-)
k k
(7-1)
where the r^ are the distances between atom centers for a pair of di¬
atomic molecules, as shown in Figure 33. The orientation for the
Lennard-Jones (LJ) diatomic plus dipole potential having the minimum
energy is the antiparallel, for y =1 and a molecular elongation of
.5471a. The differences in the LJ diatomic plus dipole and LJ atom
plus dipole potentials arise from: (a) the competition in orientations
between the anisotropic LJ diatomic and the dipole potential; whereas,
the LJ atom potential does not contribute to orientational effects, (b)
geometric effects which occur in the diatomic potential due to its
dependence on molecular elongation, which do not occur in the LJ atom
potential.

164
7.1.2 Lennard-Jones Plus Quadrupole Potential
This potential model is obtained by combining Equations (3-1),
(3-2), and (C21). Figure 35 shows values for this model for various
molecular pair orientations. The quadrupolar energy cannot differentiate
between the two endon or parallel orientations due to symmetry of the
molecule about a plane perpendicular to the molecular axis. Hence, there
are only four primary orientations. The orientation having the minimum
energy is the tee. As the quadrupole moment is increased, the well
depth of the tee orientation drastically deepens at the expense of the
other primary orientations. The change in the potential energy curve
for the tee as Q = Q/(ea ) is increased is shown in Figure 36.
7.1.3 Lennard-Jones Plus Multipole Potential
Here we consider the more general multipole model:
u(r12“l“2> *
LJ
+ UDD + U
DQ
+ u
QQ
(7-2)
where u^j is the Lennard-Jones potential of (3-2), u^ is the dipole-dipole
interaction of (C20), u^ is the dipole-quadrupole potential of (C22), and
u is the quadrupole-quadrupole potential of (C21). For this model the
nature of the potential depends on the relative strengths of the dipole
and quadrupole moments. Thus, if the quadrupole moment is stronger than
* *
the dipole, as shown in Figure 37 (wherein Q = 1.75, y =1), the tee
orientation exhibits the minimum potential energy curve, followed by
the cross and antiparallel orientations. On the other hand, if the
relative strengths are reversed, as in Figure 38 , then the minimum

165
Figure 35. Pair Potential for Lennard-Jones plus Quadrupole
Model Fluid at Primary Pair Orientations. (Key
as in Figure 34.)

Figure 36. Surface of the Lennard-Jones plus Quadrupole Pair Potential for
the Tee Orientation as a Function of the Quadrupole Strength
166

167
Figure 37. Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. y/(ea3)l/2 = p.o, Q/(ea5)l/2 = 1.75.
(Key as in Figure 34.)

168
Figure 38. Pair Potential for Lennard-Jones plus Dipole, Dipole-
Quadrupole, and Quadrupole Model Fluid at Primary Pair
Orientations. p/(ea3)l/2 = 1.75, Q/(ecj5)l/2 = ]_. (Key
as in Figure 34.)

169
potential energy curve corresponds to the head-to-tail endon, followed
closely by the antiparallel orientation. In this model potential,
the strength of the dipole moment must be significantly greater than
that of the quadrupole moment in order for the endon orientation to
be favored over the tee. Thus, the depth of the potential wells for
the endon and tee become equal at about p =1.5 when Q =1.
7.1.4 Lennard-Jones Plus Anisotropic Overlap Potential
The anisotropic overlap potential is taken to be the sum of the
two terms A = £0& and A = 0&& in the spherical harmonic expansion of
the general anisotropic potential, as given in Appendix C. For this
model, the appropriate 2. values are those of the first contributing
term allowed by symmetry. Thus, for axially symmetric molecules [76]:
u = u (202) + u (022)
over over over
(7-3)
Using Equation (C5) for each of the terms on the right of (7-3), with
expressions for the expansion coefficients E(202;00;r). and E(022;00;r)
from Table Cl, and with the help of the spherical harmonic addition
theorem [38], Equation (C18) is obtained for u . For the total
over
overlap potential to be positive, the overlap parameter 6 must lie
in the range -0.25 £ 6 £ 0.5. Further, rodlike molecules (linear) have
6 > 0, whereas platelike molecules have 6 < 0. Figure 39 shows the
Lennard-Jones plus anisotropic overlap model for both a positive and
negative 6. For 6 < 0 the orientations angles 0^ and 02 are with

170
Figure 39. Pair Potential for Lennard-Jones plus Anisotropic
Overlap Model Fluids at Primary Pair Orientations

171
reference to the symmetry axis which is perpendicular to the plane of
the molecule. The primary orientations of Table 16 refer to the sym¬
metry axis and not the molecules themselves, when 6 < 0. Thus, an
endon orientation has the platelike molecules in parallel when 6 < 0.
Note from Equation (C18) that the overlap potential is independent
of the angle

and cross orientations cannot be distinguished. Also, the model
cannot distinguish between the two endon orientations due to symmetry
of the molecule about a plane perpendicular to the molecule. Thus,
there are only three primary orientations for this model. From
Figure 39 , the minimum potential energy curve is due to the parallel
(cross) orientation for 6 > 0 and is due to the endon (molecules
parallel) for <5 < 0.
7.2 Equilibrium Properties
7.2.1 Time Development of the Simulation
The development of a molecular dynamics calculation may be
monitored by following a number of system variables: temperature, total
energy, translational and rotational kinetic energy, and mean squared
molecular displacement. The mean squared displacement of the molecular
center of mass is of value since: (a) it gives an indication of whether
the system is liquid or solid, and (b) it may be used to estimate the
-13
mass diffusion coefficient. The short time behavior (t 1 2.5 (10 )s.)
of the mean squared displacement has been found to be the same for both
the solid and liquid states in the case of the Lennard-Jones potential
model [66]. For long times, however, the displacement becomes bounded

172
for the solid, but increases monotonically with time for the liquid.
Figure 40 shows a typical plot of the mean squared displacement of the
center of mass <{Ar^m(t)} > for a Lennard-Jones plus quadrupole liquid.
Further evidence for solidification of a liquid system may be obtained
by determining (essentially) the structure factor [66], or the centers
pair correlation function. The latter exhibits behavior similar to the
radial distribution function for spherical molecules [78] when solidifica¬
tion occurs. The mean squared displacement of centers of mass is related
to the velocity autocorrelation function by;
t
(t-t')dt' (7-4)
0
and, through the Green-Kubo relation, to the diffusion coefficient [66]:
<{Ar (t)}2> = 6Dt + C (7-5)
—cm
<{Ar (t)} > = 2
—cm
where C is a constant which allows for initial coherent motion of the
molecules.
Other system parameters which may be followed are related to the
system energies. The total kinetic energy may be monitored through the
kinetic temperature, which, for translational and rotational motion,
is given by:
1
5Ne
+ I
N
l n
2
i
kT/e =
m
(7-6)

173
Figure 40. Mean-Squared Displacement of Molecular Centers of
Mass for Lennard-Jones plus Quadrupole Fluid

174
Figure 41 shows an example of fluctuations of the kinetic temperature
for a Lennard-Jones plus quadrupole fluid. The time t = 0 in the figure
corresponds to the start of the calculation after equilibration has been
performed by scaling the velocities in the manner described in Section
6.3.2. It is also of interest to follow separately the translational
and rotational contributions to the kinetic energy, since the theorem
for equipartition of energy requires the ratio of translational to
rotational kinetic energy to be 3/2 for classical fluids. Figure 42
shows fluctuation of this ratio about the value 3/2 for the Lennard-Jones
plus quadrupole fluid. Finally, following the system total energy should
give no fluctuations since the molecular dynamics microcanonical ensemble
conserves the total energy.
7.2.2 Comparison of Results with Berne and Harp
Berne and Harp [66] have done a molecular dynamics study of
nitrogen using the Lennard-Jones plus quadrupole potential. Their
results have been reproduced as a check on the program used in this
work. Values of the internal energy, mean squared force and torque,
and diffusion coefficient obtained in this study are compared in
Table 17 with values obtained by Berne and Harp. In the table, the
internal energy values attributed to Berne and Harp are their published
values plus a long range correction (described below) for molecular pairs
separated by distances greater than the potential cutoff, r^. Berne and
Harp used r^ = 2.25O. The value for the diffusion coefficient D given
by Berne and Harp was obtained from the Green-Kubo formula:
D =
1
3
OO
dt
(7-7)
0

175
Figure 41. Fluctuation in Temperature for Lennard-Jones plus
Quadrupole Model Fluid. Solid line gives instan¬
taneous values of kT/e; broken line is average
value of kT/e. pa^ = 0.85, Q* = 1.0

176
Figure 42. Fluctuation in the Ratio of Translational to
Rotational Kinetic Energy for Lennard-Jones
plus Quadrupole Fluid

177
TABLE 17
Property Values of a Lennard-Jones Plus Quadrupole
Fluid Obtained in this Work and Compared with
those given by Berne and Harp [66]
Property
This Work
Berne and Harp
No. Particles
N
256
216
LJ Parameters
e/k
87.5K
87.5K
a
3.702A
3.702Á
Elongation
SL/o
0.2955
0.2955
Q Moment
Q/(eoV/¿
0.707
0.707
Density
po3
0.931
0.931
Time Step
At
4.6(10_15)s
5.(10_15)s
Length of Run
M
2000At
600At
Temperature
kT/e
0.765 ± .020
0.758 ± 0.024
Potential Energy:
total
U/Ne
- 7.727 ± .05
- 7.635
LJ
Aj/n" ,
- 6.327
- 6.146
Mean Squared Force
oz/eZ
931.2 ± 16.7
959.2 ± 90.1
Mean Squared Torque
/e2
12.8 ± 0.3
12.5 ± 0.9
Diff. Coefficient
D(m/e)1/2/a
0.0189
0.0193

178
Whereas, in this work, D was estimated from the mean squared displace¬
ment of the molecular centers of mass by (7-5). The good agreement
between the two values for D is, to a certain extent, fortuitous.
The state condition which Berne and Harp chose to study is near the
melting point of nitrogen. When starting the simulation from an FCC
lattice structure, it was necessary to raise the temperature of the
system for the first several hundred time steps in order to "melt"
the lattice. The temperature was then lowered to the value given in
Table 17 by scaling the velocities as the system evolved for a few
hundred additional time steps. During the 2000 time step production
phase (evaluation of properties) of the calculation, however, the mean
squared displacement occasionally remained unchanged for periods of
100 to 300 time steps, evidencing the proximity of the phase boundary.
In Table 17, and for results reported below, the standard
deviation of some property F from its mean value was determined by:
a
F
1
M—1
l (F - M)2
i=l
1/2
(7-8)
which can be shown to reduce to:
aF =
M-l
M
l FT
i=l 1
- M
1/2
(7-9)
where
M
= ¿ If.
M i=i *
(7-10)

179
and M is the number of observations of the property F, i.e., the number
of time steps in the production phase of the calculation.
The slight deviations between the property values of Berne and
Harp and this work shown in Table 17 can be attributed to the difference
in temperature, number of particles and length of run between the two
calculations.
7.2.3 Results for Lennard-Jones plus Quadrupole Fluid
A system of 256 linear molecules interacting with the Lennard-
Jones plus quadrupole model fluid of Section 7.1.2 has been studied
for three values of the quadrupole moment, Q = Q/(ea'*)^^ = 0.5, 0.707,
1.0. A molecular elongation corresponding to the nitrogen molecule,
l/o = 0.3292 [70], and a time step of about 5(10 "^)s were used in the
calculation. In each case the production phase of the calculation was
of 2000 time steps duration. For this model, the potential was set to
zero for molecular pairs separated by distances greater than r^ = 3.2a.
In the calculation of. the internal energy, U, pressure, P, Fowler model
F
surface tension, y , and Fowler model superficial excess internal energy,
F
U , a long range correction has been included to account for contributions
due to Lennard-Jones interactions at molecular separations greater than
r^. This long range correction was estimated by evaluating the appropriate
integral which arises in the radial distribution function theory of fluids
in statistical mechanics [41], with g(r.j^) approximated by unity for
r^ > r . Thus, the corrections were given by:
OO
u
*
* *2 *
uLj(r ) r dr
(7-11)
X
r
c

180
P
cor
pkT
OO JL.
dULJ(r }
*
J * dr
r
c
*3
r
*
dr
(7-12)
F
Y
7T *2
8 P
CO *
dULJ(r }
*
J * dr
r
c
*4
r
*
dr
(7-13)
G
cor e
*
Yj(r >
(7-14)
The quadrupolar contribution to the long range correction was estimated
using the second order perturbation theory of Ananth et_ £il. [27]. How¬
ever, the quadrupolar contribution was found to be negligible compared
to the Lennard-Jones terms, so no quadrupolar long range correction
was included in the molecular dynamics property values. (A long range
correction would be significant in the case of dipoles.)
&
The full equilibrium property values obtained for Q =0.5, 0.707,
and 1.0 are given in Tables 18 , 19 , and 20 , respectively. The
properties were evaluated from the ensemble averages in Table 5, and
standard deviations were determined over the entire 2000 time step cal¬
culation using Equation (7-9). As indicated in the figures, the mean
squared force and torque have each been evaluated by two different
2 2
methods. The values labeled with the subscript L ( and ) are
X i_j 1 L
the "literal" values obtained by direct evaluation of Equations (6-25)
and (6-34) for the force and torque, respectively. The second values
of the force and torque were calculated from the expressions [62,63]:

181
TABLE 18
Equilibrium Properties for Lennard-Jones plus Quadrupole
Fluid at pg3 = .85, Q/(ea5)1/2 = 1/2
_ +
Property
Molecular Dynamics
Perturbation Theory
kT/e
1.277 ± .028
1.277
il/Ne
-5.736 ± .071
-5.808
P/pkT
3.098 ± .297
2.806
F 2,
Y a /e
0.502 ± .058
0.457
F 2,
usa /e
1.923 ± 0.016
1.913
c“/Nk
0.782
0.854
G1
0.048 ± 0.57
0.
G2
-0.040 ± 0.42
-0.027
a2/e2
1456. ± 43.
. • .
1850. ± 185.
2 2
/e
3.70 ± 0.47
3.79
3.76 ± 0.25
3.79
f r
Symbols defined in Tables 5 and 17.

182
TABLE 19
Equilibrium Properties for Lennard-Jones plus Quadrupole
8 s 1/2
Fluid at po = 0.931, Q/(eo ) = 0.707
Property Molecular Dynamics Perturbation Theory
kT/e
0.765
.020
0.765
U/Ne
-7.727
+
.048
-7.914
P/pkT
1.809
+
.309
0.631
YFc2/e
0.995
+
.040
0.847
UFa2/e
S
2.656
+
.012
2.670
CR/Nk
V
1.027
2.510
G1
-0.073
+
.64
0.
G2
-0.165
+
.44
-0.589
a2/e2
931.2
±16.7
La2/£2
1183.
±89.
/e2
12.85
+
0.32
13.83
„ 2^ , 2
L/e
12.73
+
0.84
13.83

183
TABLE 20
Equilibrium Properties for Lennard-Jones plus Quadrupole
——*
Fluid at pg = 0.85, Q/(ectj) ' =1.0
Property Molecular Dynamics Perturbation Theory
kT/e
1.294 ±
0.032
1.294
U/Ne
-8.134 ±
0.080
-8.339
P/pkT
1.197 ±
0.339
0.529
F 2,
Y o /e
0.961 ±
0.068
0.791
„F 2,
U o /e
S
2.475 ±
0.019
2.505
Cy/Nk
0.986
1.817
G1
-0.031 ±
0.58
0.
G2
-0.110 ±
0.44
-0.421
a2/e2
1725. ±59.
2 2 2
2036. ±214.
' * ’
/e2
46.64 ±
1.36
48.44
/e2 46.14 ± 2.61
J. JLi
48.44

184
2,2
>a /e
(kT/e) 24< J
22
' ’
G
3*1
kjJ
(7-15)
/z2 = - (kT/e) < l l JL(JL+1) u (A)>
j/l A 3
(7-16)
Expressions for u (A) in (7-16) are given in Appendix C.
cl
Also shown in Tables 18, 19, and 20 are values of the prop¬
erties calculated by thermodynamic perturbation theory. Values for the
internal energy, pressure, and residual heat capacity were obtained
using the Padé theory of Flytzani-Stephanopoulos jet al. [33]. Values
for the mean squared torque were determined from the Padé theory of
Twu et al. [79]. The Fowler model surface tension and surface excess
internal energy were calculated using the Padé theory of Chapter 2,
above. Values for the angular correlation parameters G were found
Li
using the second order perturbation theory of Gray and Gubbins [80].
(Values for the triple integral 1,.,.^ used in determining the three-body
term in the perturbation theory for G^ for quadrupolar fluids have been
* *
calculated and fitted to an empirical equation in p and T . Results
are given in Appendix G.) Tables 18, 19 , and 20 show good agree¬
ment between the molecular dynamics results and perturbation theory
predictions for those properties which are directly related to the
intermolecular potential: internal energy, surface excess internal
energy, and mean squared torque. Significant deviations in the results
from the two methods are evident, however, for those properties which
are related to derivatives of the potential: pressure, surface tension,
and heat capacity, especially at the higher quadrupole moments.

185
The disparate values for the heat capacity, C^, found by simula¬
tion and perturbation theory are not surprising since it is well known
that the relaxation of fluctuations in the heat capacity require
significantly longer times than can generally be attained in the
simulation of even a simple Lennard-Jones system [68]. As a con¬
sistency check on the simulation value for C^, Lebowitz et al. [81]
have shown that the heat capacity may be obtained from the fluctuation
in the kinetic energy in molecular dynamics, since the total energy
is conserved in the molecular dynamics microcanonical ensemble. Hence,
the heat capacity can be obtained from the fluctuation in the temper¬
ature due to (7-6). Combining (5-11), (7-6), and (7-9) gives:
Cv _ N c5 -) 2
Nk t*2 l2 T>
(7-17)
where O^ is the standard deviation in the temperature. The values for
aT and given in Tables 18, 19, and 20 satisfy (7-17), when
allowance is made for the roundoff of the values given in the tables.
The relatively large standard deviations and disagreement between
the molecular dynamics and perturbation theory results for the pressure
and surface tension are less clearly resolved. Both these properties
arise from cancellation between a negative contribution for molecular
pairs separated by small r distances and a positive contribution of the
same order of magnitude at larger r distances (see Section 7.4.3 below).
Hence, one might expect some difficulties in obtaining accurate computer
results for these properties. McDonald et_ al., in fact, have found it

186
necessary to perform 2-4 million Monte Carlo configurations on a
system of 108 particles interacting with the Lennard-Jones plus
dipole potential, in order to obtain reliable values for the Fowler
model surface tension [39]. On the other hand, 0.3-1 million Monte
Carlo configurations were sufficient to determine the pressure for
the same system [82]. In spite of the large fluctuations, the values
for the surface tension and pressure in Tables 18 , 19 , and 20
are consistent with values obtained by integrating over the appro¬
priate spherical harmonic coefficients, g^ ^ m^ri2^ (see Section 7.3.3
below).
The large uncertainties associated with the angular correlation
functions and confirm the difficulties experienced by others
in the study of these properties [60,83]. Cheung [83] has indicated
that for the Lennard-Jones diatomic potential fluid contributions
to G2 seem to occur for molecular pairs with separations in the range
2a £ r ¿ 3.2a. The argument is based on the expressions for G2 in
Table 15, in which the contributions to G2 are found to cancel for
r < 2a. In fact, a significant contribution to G2 may occur for
molecular pairs with r > 3.2a, since Cheung has found that when the
potential cutoff distance r^ is extended by increasing the number of
particles in the system from 256 to 500, the resulting values for G2
are systematically increased, as well. The r dependence of the
contributions to G2 for the fluids studied here is considered in
Section 7.3.3 below. Recent Rayleigh light scattering experiments
by Bruining and Clarke [84] give values of G2 as -0.15 ± 0.20 for
carbon monoxide and + 0.30 ± 0.20 for nitrogen. The negative value

187
for CO is attributed to the effect of the quadrupole moment. The
present molecular dynamics and perturbation results tend to confirm
that the quadrupole potential leads to negative values for Ghowever,
the large uncertainties in both the simulation and experimental results
prevent confidence about even the sign of G2.
In Tables 21, 22, and 23 the purely anisotropic contributions
to the equilibrium properties are given. These anisotropic contribu¬
tions were obtained by subtracting the isotropic Lennard-Jones part
of each property from the total property value given in Tables 18,
19 , and 20 . The isotropic contributions to the internal energy
and pressure were obtained from the Monte Carlo study of Lennard-Jones
fluids by McDonald and Singer [85]. The isotropic contribution to
the other properties were obtained by evaluating the appropriate
integrals (analogous to Equations (7-11) to (7-14)) in the radial
distribution function theory for simple fluids, using the molecular
dynamics results of Verlet for gir.^) [32].
7.2.4 Results for Lennard-Jones plus Overlap Fluid
A system of 256 linear molecules interacting with the Lennard-
Jones plus anisotropic overlap model of Section 7.1.4 has been studied
for two values of the overlap parameter, 6 = 0.10 and 0.30. The mole¬
cule and system parameters used were the same as those for the quadru-
polar fluid, save that the potential was set to zero for molecular
pairs separated by distances greater than r^ = 3.0o. Long range
corrections for distances greater than r^ were made for the Lennard-Jones
contribution using Equations (7-11) to (7-14). No correction was included
for the long range anisotropic overlap contribution to property values.

188
TABLE 21
Anisotropic Contributions to Equilibrium Properties for
Lennard-Jones plus Quadrupole Fluid at
po3 = 0.85, Q/(ea5)1/2 = 1/2
Property
Molecular Dynamics
Perturbation Theory
kT/e
1.277
1.277
U /Ne
a
-0.149
-0.221
P /pkT
a
0.
-0.203
F 2,
YaO /£
0.074
0.029
U* az/e
sa
0.062
0.052
CR /Nk
va
0.051
0.123

189
TABLE 22
Anisotropic Contributions to Equilibrium Properties
for Lennard-Jones plus Quadrupole Fluid
at pa3 = 0.931, Q/(eo5)1/2 = 0.707
Property
Molecular Dynamics
Perturbation Theory
kT/e
0.765
0.765
U /Ne
a
-1.164
-1.350
P /pkT
a
-1.338
-2.516
F 2.
V /E
0.346
0.198
az/e
sa
0.331
0.344
CR /Nk
va
-0.493
0.990

190
TABLE 23
Anisotropic Contributions to Equilibrium Properties
for Lennard-Jones plus Quadrupole Fluid
3 5 i/?'
at po = .85, Q/(eg ) = 1.0
Property
Molecular Dynamics
Perturbation Theory
kT/e
1.294
1.294
U /Ne
a
-2.559
-2.764
P /pkT
a
-1.838
-2.507
F 2,
V /€
0.546
0.376
UF a2/e
sa
0.617
0.648
CR /Nk
va
0.280
1.111

191
Full equilibrium property values and the corresponding anisotropic
contributions are given in Tables 24 and 25 . A perturbation
theory including three body terms applicable to this model potential
has only recently been worked out and, therefore, perturbation theory
results are not yet available for comparison.
7.3 Spherical Harmonic Coefficients,
■¿1¿ ^nr^—L 2—
The coefficients g„ „ (r „) in the spherical harmonic expansion
a. ¿
for the angular pair correlation function gCr^unu^ have been deter¬
mined from the ensemble average of (6-17). For the quadrupole and
overlap fluids, only the coefficients with even values for and Ü^
are nonzero, due to symmetry of the molecule. All such coefficients
up to 8444(^2)» and the §6£ 0^rl2^ and g800^r12^ coefficient:s have
been evaluated. Explicit expressions for each of these are given
in Appendix F.
7.3.1 Spherical Harmonic Coefficients for Lennard-Jones plus
Quadrupole Fluid
7.3.1.1 Center-center pair correlation function. Figure 43 shows
that for the strengths of the quadrupole moment considered in this
work, the addition of the quadrupole potential has little effect
on the center-center pair correlation function, g^r^)» Within
the statistical precision of the simulation, the curves in Figure 43
for the Lennard-Jones and Lennard-Jones plus quadrupole fluids are
the same. The slight difference in the first peak heights of the
two curves could be due to the temperature difference between the

192
TABLE 24
Equilibrium Properties for Lennard-Jones plus
Overlap Fluid at pg2 = 0.85, 6 = 0.10
Molecular Dynamics
Property Full Property Anisotropic
kT/e
1.291 ±
.030
U/NE
-5.564 ±
.074
0.013
P/pkT
3.255 ±
.353
0.224
yFa2/e
0.460 ±
.069
0.042
uFa2/e
S
1.880 ±
.016
0.022
C®/Nk
0.844
0.133
G1
0.082 ±
OO
G2
0.045 ±
.32
La2^e
1861. ±210
•
/e2
1.145 ±
.255
^ 2. . 2
1 l'
1.203 ±
.144

193
TABLE 25
Equilibrium Properties for Lennard-Jones plus
Overlap Fluid at pg2 = 0.85, 6 - 0.30
Molecular Dynamics
Property Full Property Anisotropic
kT/e
1.287
+
.025
U/Ne
-6.301
+
.063
-0.721
P/pkT
1.996
+
.280
-1.029
yFa2/e
0.741
+
.054
0.320
UFa2/e
S
2.028
+
.013
0.169
c>k
0.618
-0.099
Gi
-0.019
+
OO
C2
0.037
+
.50
l°2/e
1805.
±197.
/e
16.35
1.06
.2,2
l/£
16.61
+
1.64

194
Figure 43. Effect of Quadrupole Moment on the Center-Center
Pair Correlation Function at pc?3 = 0.85

195
two calculations. The plot of g (r^) ^or t*ie 9uadrupole fluid with
5 1/2
Q/(ea ) =0.5 falls essentially on the Lennard-Jones curve in
Figure 43 , with its first peak falling between the two peaks
shown in Figure 43 , since the average temperature of the
5 1/2
Q/(ea ) = 0.5 simulation was kT/e = 1.287 (between 1.294 and
1.273).
Patey and Valleau [86 ] have conducted Monte Carlo studies of
hard sphere plus quadrupole model fluids. They find distortion of
the center-to-center pair correlation function occurring at high
5 1/2
values of the quadrupole moment (i.e., at Q/(kTR ) = 1.291, where
R is the hard sphere diameter). This distorted g (r^) seems to be
evidence for an FCC-type local structure forming in the fluid. In
the present work, the highest quadrupole moment studied was
Q/ (ea ), = 1, and, as shown in Figure 43, no distortion of gc(r.^)
has been found.
7.3.1.2 Other spherical harmonic coefficients. Results for the
remaining spherical harmonic coefficients are shown in Figures 44 ,
5 1/2
45 , 46, and 47 for the quadrupolar fluid having Q/(eo ) =1.
5 1/2
The coefficients for the fluids with Q/ (ea ) =0.5 and 0.707 are
not shown for brevity; however, their results are qualitatively similar
to the coefficients shown in the following figures. In general, the
magnitudes of the g„ 0 coefficients are found to decrease as: (a)
or &2 is increased, (b) ra is increased at fixed and SL^t (c) the
fluid density is decreased, or (d) the strength of the anisotropic
potential is decreased. These conclusions are demonstrated by the

196
Figure 44. Spherical Harmonic Coefficients g„„ for Lennard-
2.36
Jones plus Quadrupole Fluid at pcf3 = 0.85, kT/e =
1.294, Q/(eo5)l/2 = i.o

197
Figure 45.
Spherical Harmonic Coefficients g.„
/ < 4X.„m
of Figure 44 4
for the Fluid

198
r/cr
Figure 46. Spherical Harmonic Coefficients
of Figure 44
for the Fluid

199
r /cr
Figure 47. Spherical Harmonic Coefficients g,„ _ for the Fluid
of Figure 44 2

200
magnitudes of the first peak of the ^ m^r12^ coeffic:*-ents given in
Table 26. The exception to these general observations is the quadru-
polar fluid, wherein the magnitude of the &220 coeffieient dominates
the other coefficients, including the §200’ From Table 26, the
magnitudes of the first peak in the g„ 0 for the quadrupolar fluids
3 SI/? \ ^ SI/?
at pa = .85, Q/(ea ) = 1/2 and pa = .931, Q/(eo ) ' = .707 are
roughly the same, indicating that the effect on the g„ n of de-
1 zm
creasing the quadrupole moment is largely cancelled out by the
increase in fluid density.
Another effect on the g^ ^ m^r12^ increas:>-ng the strength
of the anisotropic potential is to decrease the degree of randomly
scattered values of the coefficients beyond the first peak. This
scatter in the g^ £,m^r12^ ^ata, especially at r^ values beyond the
first peak occurs, in the case of weakly quadrupole fluids, because
of the short range correlation of orientation of molecular pairs.
In the case of strong quadrupole fluids, the scatter is largely due
to strong correlation of particular orientations of molecular pairs,
so that less probable pair orientations which would contribute to
certain g^ ^ m^r12^ terms only rarely occur. The scatter has been
made systematic by smoothing the g^ ^ m^ri2^’ usan§ a seven-point,
third degree smoothing formula [87] applied three times to the
g^ £ n/rl2^ va^ues beyond the first peak. The curves in the
accompanying figures are the smoothed g„ g (r „). The high fre-
quency oscillations shown in a few of the g^ ^ m^rl2^ reflect the
smoothing of nearly randomly scattered data and should not be

201
TABLE 26
Effect
of Potential
Model and State
Condition
on the First Peak Height of the g„ „
Coefficients
12
Quadrupoles
Overlap
£„m
(a)
(b)
(c)
1 2
g200
+0.15
+0.22
-0.88
8220
-1.65
-1.68
+0.62
S221
-0.64
-0.63
+0.09
g222
-0.21
-0.15
+0.04
S400
+0.29
+0.40
+0.36
S420
+0.24
+0.23
-0.30
8421
+0.08
+0.10
-0.02
S422
-0.03
+0.03
-0.02
S440
+0.36
+0.54
+0.17
S441
1 +0.24
+0.33
-0.02
g442
+0.13
+0.14
+0.02
g443
+0.06
+0.06
+0.02
g444
-0.03
-0.02
+0.02
g600
-0.07
-0.03
-0.16
g620
-0.11
-0.18
+0.14
g640
-0.08
-0.11
-0.10
(a) po3= .
931, kT/e = .
765, Q/(ea5)1/2
= .707
(b) pa3 =
.85, kT/e =
1.294, Q/(ea )1/2 = 1.0
(c) pa3 =
.85, kT/e =
1.287, 6 = 0.3
1.0

202
interpreted as significant long range character for the coefficient.
The original, unsmoothed g^ ^ m^ri2^ data f°r t^e systems studied are
tabulated in Appendix H.
7.3.2 Spherical Harmonic Coefficients for Lennard-Jones plus
Anisotropic Overlap Fluid
7.3.2.1 Center-center pair correlation function. Figure 48 shows
the center-center pair correlation function g (r^) obtained from
simulation of the anisotropic overlap fluids. For the value of the
overlap parameter 6 = 0.1, the g^Cr.^) curve falls essentially on the
simple Lennard-Jones g(r^) plot of Figure 43, except that the height
of the first peak is decreased somewhat. This difference in peak heights
could be due to the slight difference in temperature between the two
calculations. However, when 6 = 0.3, there is a definite lowering of
the first peak height, which is consistent with interpretation of the
repulsive overlap potential as tending to restrict close approach of
molecular pairs.
7.3.2.2 Other spherical harmonic coefficients. Figures 49, 50, 51,
and 52 show the smoothed spherical harmonic coefficients for the
anisotropic overlap fluid having 6 = 0.3. The g0 Q coefficients
for the fluid with 6 = 0.1 are qualitatively similar, and the general
statements concerning the factors affecting the magnitudes of the
coefficients made in Section 7.3.1.2 apply to the overlap fluid, as
shown in Table 26.
Also seen in Table 26 is that the quadrupole and anisotropic
overlap fluids show considerable differences in the nature of their

203
Figure 48. Effect of Anisotropic Overlap Parameter on the
Center-Center Pair Correlation Function at
po3 = 0.85

204
kT/e = 1.287, 6 = 0.3. (Key as in Figure 44.)

205
in magnitude than g^l f°r aH r anc* has been
omitted for clarity.)

'44m
206
Figure 51. Spherical Harmonic Coefficients g,. for the Fluid
of Figure 49. (Key as in Figure 46. g^3 and g^
are smaller in magnitude than f°r a4 r and
have been omitted for clarity.)

207
Figure 52. Spherical Harmonic Coefficients g,0 n for the
bx^U
Fluid of Figure 49. (Key as in Figure 47.)

208
g5 0 coefficients. Of the coefficients evaluated in this study
l zm
(excluding 8oo0^rl2^’ about two-thirds of them exhibit sign changes
in the first peak region when comparing quadrupole with overlap fluids.
(The coefficient g nn(r10) was found to be negligible for both quadrupole
and overlap fluids.) Hence, one would expect significant differences
in local orientational structure between quadrupole and overlap fluids.
Such differences in structure have been foreshadowed by study of the
pair potentials themselves in Section 7.1. Detailed study of the
go n (t.„) coefficients would indicate the character of the orien-
^ 2^ J- Z
tational structure; however, it is much more straightforward to
recombine the coefficients into the spherical harmonic expansion and
study structure via the angular pair correlation function g(r^^k^ *
This approach is taken in Section 7.4.
7.3.3 Equilibrium Properties from the g^ ,, ^(r^
The gff p (r „) coefficients provide detailed information on
2^ J- z.
local structure through g(r^^-j^^ ' Of potentially more value [88]
is the determination of equilibrium properties from these coefficients.
The development of relations between equilibrium properties and the
2m
integrals J over the gn n (r,„) has been described in Section 6.5.
n x, Jo m Iz
s^m
The integrals (6-89) for the quadrupole and anisotropic overlap
fluids studied here have been evaluated through n = 24 and are tabulated
in Appendix I. These integrals have been used with the relations in
Section 6.5 to calculate those equilibrium properties which have also
been determined by direct ensemble averaging in the molecular dynamics
simulation. Thus, the relations in Section 6.5 provide a consistency

209
test between the equilibrium properties and the gn „ (r, J coefficients
1
determined in the simulation. Comparisons of the equilibrium property
values obtained directly from the simulation with values obtained via
m
the integrals are made in Tables 27 and 28 for the quadrupole
fluids, and in Tables 29 and 30 for the overlap fluids. There is
remarkably good agreement between the two methods of evaluation for
all the properties and each fluid studied. The variations in the
values which do occur are probably due to statistical fluctuations
in the ensemble averages of both the properties and the g^ ^ m^rl2^
coefficients. One should not infer from these results that these
property values are "correct" compared to, say, the perturbation
theory results given in Section 7.2. Rather, the agreement in
Tables 27 to 30 indicates no significant errors have been introduced
in the method of determining either the properties or the g„ „ (r10)
l /m
coefficients. More importantly, these results demonstrate the ability
of the g^ ^ m^r12^ coeffieients to give values for equilibrium properties.
The property showing the worst agreement in Tables 27 to 30
is the angular correlation function G2. Some of the difficulties in
evaluating this property have been suggested in Section 7.2.3. Armed
now with values for the g^ ^ m^rl2^ coeffieients, further insight into
G2 may be attempted. In Figure 53 the integrand of Equation (6-138),
which gives G2 from the appropriate g^ ^ ^(r^) coefficients, is plotted
as a function of the molecular pair separation r^2’ f°r a Lennard-Jones
plus quadrupole fluid. This figure indicates two problems in determining
G2= (a) the value of G2 is the result of cancellation between positive
and negative contributions of nearly equal magnitude, (b) the function

210
TABLE 27
Comparison of Molecular Dynamics Results for
Equilibrium Properties with Values Obtained
^1^2m
from J Integrals for Lennard-Jones plus
Quadrupole Fluid with p= 0.85, kT/e = 1.277,
Q/(£q5)1/2 = 0.5
Property
Molecular Dynamics
Í, £ m
J
n
ulj/Ne
-5.495
-5.474
uqq/N£
-0.241
-0.242
PLJ/pkT
-2.413
-2.477
V"
0.315
0.316
F 2,
ylj /e
0.433
0.418
YQQ°2/e
0.069
0.069
"sLja2/E
1.867
1.861
usqq°2/£
0.055
0.055
G2
-0.040
-0.057
/e2
3.698
3.707
a2/e2
1456.6
1464.1

211
TABLE 28
Comparison of Molecular Dynamics Results for
Equilibrium Properties with Values Obtained
from J Integrals for Lennard-Jones plus
Quadrupole Fluid with pQ2 = 0.85, kT/s = 1.294,
Q/(£q5)1/2 = 1.
Property
Molecular Dynamics
£ £ m
J
n
ULJ/NE
-5.129
-5.113
UQQ/N£
-3.005
-3.016
PLJ^pkT
-4.068
-4.142
PQQ/pkT
3.871
3.884
F 2,
/e
0.102
0.092
F 2
yqq° /£
0.858
0.861
uP a2/e
Slj
1.788
1.787
UP a2/c
SQQ
0.687
0.689
G2
-0.110
-0.152
/£2
46.64
46.83
o2/e2
1725.8
1738.5

212
TABLE 29
Comparison of Molecular Dynamics Results for
Equilibrium Properties with Values Obtained
from J Integrals for Lennard-Jones plus
n ° ~" 3 *
Overlap Fluid with pa = 0.85,
kT/e = 1.291, 5 = 0.1
S, £,„m
Property
Molecular Dynamics
J 1 2
n
ulj/Ne
-5.417
-5.411
U /Ne
over
-0.148
-0.149
PLJ/Pkl
-2.712
-2.753
P /pkT
over
0.457
0.461
F 2.
ylj° /e
0.370
0.369
yF o2/e
over
0.090
0.091
„F 2,
U„ 0 z
SLJ
1.850
1.854
„F 2 ,
U 0 /e
"’over
0.030
.030
G2
0.045
0.004
/e2
1.142
1.154

213
TABLE 30
Comparison of Molecular Dynamics Results for
Equilibrium Properties with Values Obtained
from J Integrals for Lennard-Jones plus
Overlap Fluid with pq~* = 0.85,
kT/e = 1.287, 6 = 0.3
&,£0m
Property
Molecular Dynamics
J 1 2
n
ulj/Ne
-4.184
-4.172
U /Ne
over
-2.117
-2.123
PLJ/t>kI
-7.573
-7.640
P / pkT
over
6.576
6.598
F 2 ,
V /c
-0.518
-0.527
F 2.
Yover° /£
1.259
1.263
Ug a2/e
bLJ
1.608
1.608
,,F 2 ,
Uc: a /e
sover
0.420
0.421
G2
0.037
.092
/e2
16.35
16.39
a2/e2
2271.
2281.

214
2
Figure 53. Integrands [g22Q(r) - 2g221(r) + 2g222(r)]r and
[g22o^r^ + ^3g221^ + 1/,38222^r^r_3 for G2 and
U , respectively, for Lennard-Jones plus Quadrupole
3 3 S 1 / 2
Fluid. poJ = .85, kT/e = 1.294, Q/(ecT) 1 =
1.0

215
is certainly long ranged compared with, for example, the quadrupole
contribution to the internal energy. The integrand for gives no
indication of approaching zero, even at r^ = 3.2a. This finding
supports the long ranged characterization of suggested by Cheung
[83] for the Lennard-Jones diatomic potential. Hence, to obtain
reliable values for G^ from simulation of the potential models
considered here, one must perform much longer calculations than
those required to obtain the energy, with significantly more particles
than the 256 used in this work. Some of the disagreements between
values for G„ obtained from direct ensemble averaging and from the
í-1¿2 m
J integrals must be due to this long range contribution. A long
n
range contribution has been included in the evaluation of the J
n
integrals; whereas, no such correction has been made to the ensemble
averages for G^.
7.4 Angular Pair Correlation Function
7.4.1 Convergence of the Spherical Harmonic Expansion
The smoothed gn „ (r,„) coefficients described in Section 7.3
x.2*
have been recombined in the spherical harmonic series (6-71) to obtain
the angular pair correlation function g (r^^1*^ ' Here we consider
how well the series reflects the multidimensional gir^^M*^) by
obtaining values for gCr^^-j^^ us:*-ng different numbers of g^ ^ m^ri2^
terms in the expansion. A total of 19 coefficients (including 8Qoo^r12^
have been explicitly evaluated in the molecular dynamics calculations
for the quadrupole and overlap fluids studied here. Expressions for
these coefficients are given in Appendix F and the resulting values

216
are given in Appendix H. In actuality, twice that number of coefficients
have been determined, since, due to symmetry of the molecules:
8£1¿2m(rl2) 8£2J¿1m(r12)
(7-18)
However, we will refer to a total sum over, say, 19 coefficients, it
being understood that both g„ n (r,„) and gn . (r.„) terms have been
included in forming (6-71). For each of the fluids studied, the §goo*‘rl2^
coefficients were found to be negligible and were not included in forming
g (r^co^o^) . In addition to forming the sum over 18 coefficients, sums
for g(r^2tü^w2) over ten and six terms have been determined by including,
respectively, the terms containing: g000<-r12^’ g20(/r12^ ’ g220^r12^’
g400(rl2)’ g420(r12)’ g440(rl2)’ 8600(r12)’ g620(rl2)’ g640(rl2)’
g660(rl2) and g000(rl2)’ 8200(rl2)s g220(rl2')’ g400(rl2)’ g420<‘r12)’
®440^r12^‘ These m = 0 coefficients are those which are independent
of the angle cj), where (p is the difference in azimuthal angles of a
pair of molecules in the intermolecular frame of Figure 32. It is
of interest to study the diatomic potential models exhibit weak

[69].
7.4.1.1 Lennard-Jones plus quadrupole fluid. Plots of the angular pair
correlation function obtained from 17, 9, and 6 terms in the expansion
(6-71) are shown in Figures 54 and 55 for the tee, cross, and
parallel pair orientations, respectively. (Due to a programming error,
the coefficient was incorrectly determined for the Q/(eo ) =1.

g (12)
217
Figure 54. Angular Pair Correlation Function for the Lennard-
Jones plus Quadrupole Fluid of Figure 44 for the
Tee Orientation (0^ = 90°, 02 = 0, cj) undefined)

g (12)
218
Figure 55. Angular Pair Correlation Function for the Lennard-
Jones plus Quadrupole Fluid of Figure 44 for the
Cross and Parallel Orientations (0^ = = 90°)

219
fluid and has not been included in the summation for g' For
the tee orientation, the angle

contribute to the sum. Hence, both the 17 term and 9 term sums give
the same gCr^^c^). As one would expect, convergence of the expansion
is good for this highly probable orientation, there being distinguish-
ability of the 17 and 6 term sums only around the first peak in
g(r12UlU2)-
Figure 55 shows gCr^^ü^) ^°r t^ie cross ant^ parallel orienta¬
tions obtained from a sum over 17 spherical harmonics. The 9 and 6 term
sums each give the same curves for these two pair orientations since
only the angle changes in going from a parallel to a cross orientation.
This figure indicates a nonnegligible dependence of the angular pair
correlation function. Evidence for the by considering "skewed" pair orientations, i.e., orientations other
than the primary ones, as defined in Table 16. Thus, for the skewed
orientation of Figure 56, there is substantial variation in the values
for g(r or) obtained from the .17 and 9 term sums.
JL ¿ 1 ¿
The convergence of the expansion for the least probable orienta¬
tions, e.g., the endon, is not very good even when 17 terms are included
in the sum. Thus, especially at high values of the quadrupole moment,
spuriously strong peaks or unphysical, negative values may be obtained
for g(r^ ^or -^east: ptobable orientations. To overcome this
problem, significantly longer molecular dynamics calculations are
required.
Convergence of the spherical harmonic expansion for the other
5 1/2
quadrupole fluids studied (Q/(ecT ) = 0.5 and 0.707) is qualitatively

220
Figure 56. Angular Pair Correlation Function for the Lennard-
Jones plus Quadrupole Fluid of Figure 44 for a
Skewed Orientation (9^ = = = 45°)

221
5 1/2
similar to that shown here for the Q/(ea ) =1.0 fluid. The degree
of variation between the gír^^^^ determined from the different
number of terms is not as strong for the weaker quadrupole strengths.
7.4.1.2 Lennard-Jones plus anisotropic overlap fluid. Convergence
tests for the spherical harmonic expansion of g (r-j^-J*^ ^or t^ie
Lennard-Jones plus anisotropic overlap fluid have been conducted in the
same manner as described above for the quadrupole fluid. Results are
shown in Figures 57 and 58 for the overlap fluid with 6 = 0.3 for
the parallel and endon pair orientations, respectively. Calculations
of g(r12^1^2) ^or vari°us pair orientations shows the most probable
orientation to be the parallel, as suggested from the potential energy
curves of Figure 39. Thus, the series converges well for the most
probable orientation, but for less probable orientations, the full
18 terms are necessary for accurate results. (The high frequency
oscillations after the first peak in the plot of g (r^^-^^ f°r t^e
endon configuration, Figure 58 , reflect statistical fluctuations
in the sampling of this low probability orientation.) The behavior
of the series for g (r-^^u^) for the overlap fluid with 6 = 0.1 is
qualitatively similar to that shown for the 6 = 0.3 fluid; faster
convergence of the series is found for the 6 = 0.1 fluid. The question
of potential is independent of the angle c}> (see Equation C3, Appendix C) .
7.4.2 Local Structure from g(r^2^—2—
The angular pair correlation function gír-j^üljü^) as particular
value in elucidating the nature of molecular pair orientations in the

g02)
222
Figure 57. Angular Pair Correlation Function for the Leonard-
Jones plus Anisotropic Overlap Fluid of Figure 49
for the Tee Orientation (0 = 90°, 0=0, un-
defined)

gd2)
223
Figure 58. Angular Pair Correlation Function for the Lennard-
Jones plus Anisotropic Overlap Fluid of Figure 49
for the Endon Orientation (9^ = ~ $ = 0)

224
fluid. A major deterrent to detailed study of gCr^^t^) is its multi¬
dimensionality. Thus, for simple linear molecules, gir^^j^) depends
on the distance of separation of the centers of mass of a molecular pair,
rl2> and three angles, 0^, 0^, and tion of the pair. A five dimensional space is required to pictorially
represent the entire gCr^u^o^) function for linear molecules. In this
section, three dimensional surfaces cut from the gCr^O1^^ hypersurface
for the quadrupole and overlap fluids are presented for study. The
objectives of the study are: (a) to discover the relative probability
of occurrence of the primary pair orientations, and (b) to determine
the effect of the anisotropic strength constant (quadrupole moment Q
or anisotropic overlap parameter 6) on the occurrence of pair orienta¬
tions in the fluid.
The three dimensional figures have been generated from the
recombined spherical harmonic expansion (6-71) using the smoothed
9 m^rl2^ c*ata> The resulting g(r curves have not been
12 .
smoothed, so that a couple of the surfaces shown contain small
statistical "flaws," which are pointed out in the text. Each sur¬
face has been generated by fixing two of the relative orientation
angles (0^, 0^, or ) while rotating the third angle through 0 to tt
for all r^2 values in the range 0.5 ¿ ^ 2.5. For linear molecules,
0^ and @2 are restricted to the range (0,tt), and gír^w^ü^) as sy^etric
about

dimensional plot a numerical scale is shown which applies to the z-axis
(i.e., the axis). The numbers on the left side of the scale
refer to the physical height of the drawing on the page, in inches.

225
The numbers on the right are the corresponding values for g(r
at that height. The top number on the right of the scale is the
maximum value of gir^^^) which occurs on the plot. The plots are
presented in pairs; the second plot in each pair is for the same
orientations as the first, but at a higher value of the anisotropic
strength constant.
7.4.2.1 Lennard-Jones plus quadrupole fluid. Figures 59 and 60
show the g(r^utja^ surface generated with 0^ = 90, 5 1/2
rotated from 0 to TT, for the Q/ (£ü ) =0.5 and 1.0 fluids, respec¬
tively. The relative pair orientations are thus rotated from the tee,
through the parallel, and back to the tee, as 6^ changes from 0 to tt.
Figure 59 shows that, for the weakly quadrupole fluid, the tee
orientation is about twice as probable as the parallel in the region
5 1/2
of r^ around the first peak in 8Íri 2t*V*)2^ ' When Q / (ea ) is in¬
creased to 1.0, there is a dramatic shift in probability, favoring
the tee over the parallel, as shown in Figure 60 . The height of
5 1/2
the first peak in g (r 2ÜJ1Ü°2 ^ ^or t*ie tee orientat:'-on f°r Q/ (GO ) =1.
is more than an order of magnitude larger than that for the parallel
orientation, and is about three times larger than for the same tee
5 1/2
orientation with Q/ (ea ) = 0.5.
Figures 61 and 62 show the gsurface generated with
5 1/2
0^ = @2 = 90° and rotated from 0 to tt, for the Q/ (ea ) = 0.5 and
1.0 fluids, respectively. The relative pair orientations are thus
rotated from the parallel, through the cross, and back to the parallel.
For the weakly quadrupole fluid, Figure 61 indicates a weak, but discern-
able (f> dependence in gir^^0-^) in t*ie ^arst Paak region and very slight

?. so
2.50
-t ftft
L. r ÜÜ
1.50
1.00
0.50
0.00
y.oo
3.93
3.]!1
2.36
3,57
0.79
0.00
ro
N>
O'
Figure 59.
Surface of the Angular Pair Correlation Function for Lennard-Jones plus
Quadrupole Fluid for 0^ = 90, cj> = 0, with Q* = 0.5, T* = 1.277, and p* = 0.85

0.5
2.60 .
¡0.93
2.50
¡0.51
2.00
s.yj
1.50
6.31
1.00
y.2i
0.50
2.10
0.00
0.00
Figure. 60. Surface of the Angular Pair Correlation Function for Lennard-Jones plus
Quadrupole Fluid for 9 = 90, (J) = 0, with Q* = 1.0, T* = 1.294, and
p*= 0.85
227

2.60
2.39
2,50
2.30
2.00
1.6«
!, 50
J ,38
1.00
0.32
0.50
0.U6
0.00
0.00
Figure 61. Surface of the Angular Pair Correlation Function for Lennard-Jones plus
Quadrupole Fluid for 0. = 90, 0„ = 90, with Q* = 0.5, T = 1.277, and
p*= 0.85
228

Figure 62.
229

230
5 1/2
p dependence in the second peak region. When Q/ (eo ) is increased,
Figure 62 shows that the ^^rl2tJlÃœJ2^ stren8t^iene^ favor of the cross orientation. The (p
dependence in the second peak region remains weak. Note that on in¬
creasing the quadrupole moment, the probabilities for finding molecules
in the first peak region are reduced compared to those in the second
peak region. Hence, the probability of finding a molecular pair in the
first peak region with 0^ = 9^ = 90, irrespective of the value of p,
is reduced when the strength of the quadrupole moment is increased.
(The "blip" on the small r^ side of the first peak in Figure 62 is
a statistical fluctuation and is not indicative of fluid structure.)
Figures 63 and 64 show the gCr^^j^^ surface generated
for a set of skewed pair orientations; specifically, = ®2 = an<^
5 1/2

tively. The relative pair orientations are thus rotated from a quasi-
tee, through a quasi-cross, to a quasi-parallel. These figures suggest
that molecular pairs in orientations close to the tee are favored over
other orientations and that the tee, again, is more favored at high
values of the quadrupole moment.
In decreasing order of likely occurrence, the primary orientations
for the Lennard-Jones plus quadrupole fluid are found to be: tee, cross,
parallel, and endon. This order correlates well with the depth of the
pair potential well for these orientations. There is also a general
correlation between the location of the first peak heights in g(r^^j^)
for these orientations (Figures 54 and 55 ) with the location of the
minima in the corresponding uir^^j^) (Figure 35 ), as shown in

0.5
Figure 63. Surface of the Angular Pair Correlation Function for Lennard-Jones plus
Quadrupole Fluid for 9. = 45, 0„ = 45, with Q* = 0.5, T* = 1.277, and
p*= 0.85
231

232

233
5 1/2
Figure 65 for the Q/(ea ) =1. fluid. These results, also, con¬
firm earlier Monte Carlo studies of the Lennard-Jones plus quadrupole
fluid [64].
7.4.2.2 Lennard-Jones plus anisotropic overlap fluid. Figures 66
and 67 show the gCr^^t*^) surface generated with 6^ = 90, = 0,
and rotated from 0 to 7T, for the 6 = 0.1 and 0.3 fluids, respectively.
The relative pair orientations are thus rotated from the tee, through
the parallel, and back to the tee, as changes from 0 to TT. Figure 66
shows weak 0^ dependence at these relative pair orientations for the
small 6 value fluid. The preference of the parallel orientation over
the tee is slight. Note the hint in Figure 66 of the effect of in¬
creasing 6, namely, the predominance of the parallel over the tee, and
the slight shift of the first peak to smaller r^ values, as shown in
Figure 67.
The order of likely occurrence of the primary pair orientations
for the Lennard-Jones plus anisotropic overlap fluid with 6 = 0.1 is
found to be: parallel, tee, and endon. This is in agreement with the
relative depths of the potential wells for overlap fluids with 6 > 0
shown in Figure 39 . For the 6 = 0.3 fluid, the endon configuration
is found to be slightly more probable than the tee. This does not
correlate with the corresponding potential well depths of Figure 39.
It is felt that this (small) discrepancy is due to poor convergence
of the spherical harmonic expansion for gCr^^^^ caused by infrequent
sampling of almost equally low probability pair orientations in the
course of the molecular dynamics calculation. A longer molecular
dynamics calculation would show whether this is the problem.

g (12) u (12)
234
Figure 65. Comparison of Peak Heights in the Angular
Pair Correlation Function with Well Depths
in the Pair Potential for the Lennard-Jones
plus Quadrupole Fluid with Q/(eo5)l/2 = p.

0.5
2.60
2.50
2.00
1.50
1.00
0.50
0.00
_ 2.65
.. 2.75
.. 2.20
.. 1.65
.. 1,10
.. 0.55
.. 0.00
Figure 66. Surface of the Angular Pair Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid for 0-, = 90, tj) = 0, with 5 = 0.1, T* = 1.291,
and p*= 0.85 1
235

0.5
Figure 67. Surface of the Angular Pair Correlation Function for Lennard-Jones plus
Anisotropic Overlap Fluid for 9. = 90, <}> = 0, with 6 = 0.3, T* = 1.287,
and p*= 0.85
236

237
7.4.3 Orientational Contributions to Equilibrium Properties
In the book by Egelstaff [89] a study is made of the combined
pair potential and distribution function contributions to various
equilibrium properties of monatomic fluids by considering the property
integrands which arise in the radial distribution function theory of
fluids. We have conducted a similar study for the fluids considered
here, wherein orientational contributions must be included. The
properties readily accessible for such study are the internal energy,
pressure, and Fowler model surface tension, which are given, in the
pair theory of fluids, by:
U_ - £l
Ne 2tt
•
’
dti),
dui„
1
2
^2 ^ ^
rl2 u(rl2aJlü)2) g(r12a)lW2) drl2
(7-19)
pkT
ólfi
â– 
'
dtú-
doo„
1
2
CO k
*3 9u(r12w1w2) * *
rl2 7* g(r12“l“2)dr12
(7-20)
9r
12
F 2 *2 r r
Y a = P
e 32ft
dio,
dcü„
O k
*4 9u(ri2a,iaJ2) * *
r12 „ * g(r12U)ia)2) dr12
(7-21)
9r
12
In addition, the mean squared torque is directly proportional to (7-19)
by the relations in Table 14.
Using the recombined spherical harmonic expansion for g(r^ui^o^),
k
the integrands under the r^2 integrals in (7-19), (7-20) and (7-21) have
been calculated for the primary pair orientations for both the quadrupole
and overlap fluids. Such calculations provide insight into the effect
of orientation on property values. When coupled with similar calcula¬
tions for other model potentials, they provide a tool for distinguishing

238
among the various models and could, potentially, suggest experimental
work which would illuminate the nature of the potential in real fluids.
For both the quadrupole and overlap fluids, the integrands in
(7-19) to (7-21) were found to be dominated by the most probable
orientations, tee and parallel, respectively. The integrands for
the energy and pressure for quadrupole fluids in the tee orientation
are shown in Figures 68 and 69 , and for the overlap fluids in the
parallel orientations in Figures 70 and 71 . The curves for the
Fowler surface tensions are not shown, but are of the same shape as
the curves for the pressure, with slightly higher magnitudes in the
large r^ portion of the plots. Note that the curves for the energy
and pressure have the same shape for two different potential models
and different pair orientations.
The curves for contributions to these properties from other
pair orientations do not conform to the shapes shown here for the
most probable orientations. The range of values for the integrands
for pressure and energy for the primary orientations of the fluids
studied are given in Table 31.
7.5 Site-Site Pair Correlation Functions
The site-site pair correlation function g „(r n) has been
a8 a3
determined in the molecular dynamics calculation by the method out¬
lined in Section 6.4.2. The linear molecules studied here were
considered to be homonuclear diatomics for which the sites of interest
are the atomic nuclei. The sites were taken to be half the atom bond
length, Z, from the molecular center of mass. The bond length for the
nitrogen molecule was used in the calculation, i.e., Z/o = 0.3292 [71].

239
Figure 68. Integrand for Internal Energy,
r^u(12)g(12), for Lennard-Jones plus
Quadrupole Fluids for the Tee Orienta¬
tion

r3 du(12)/dr g(12)
240
O
Figure 69. Integrand for Pressure, r du(12)/dr g(12), for
Lennard-Jones plus Quadrupole Fluids for the Tee
Orientation

u(12) g(12)
241
2
Figure 70. Integrand for Internal Energy, r u(12)g(12), for
Lennard-Jones plus Anisotropic Overlap Fluids for
Parallel Orientation

242
Lennard-Jones plus Anisotropic Overlap Fluids
for Parallel Orientation

243
TABLE 31
Range of Values for Orientational Contributions to Property
Integrands for Quadrupole and Overlap Fluids
Property
U/Ne
P/pkT
Fluid
Strength
Constant
Pair
Orientation
Range of Values
for Integrands
LJ + QQ
*
Q = 1.0
tee
0.2
to
-43.
cross
0.7
to
- 1.
parallel
1.2
to
- O.i
endon
0.4
to
- 5..
LJ + Over
6 = 0.3
tee
0.9
to
- 6.
parallel
0.4
to
-14.
endon
3.6
to
- 1.:
LJ + QQ
*
Q =1.0
tee
80.
to
-160.
cross
4.
to
-20.
parallel
0.4
to
-19.
endon
54.
to
- 3.
LJ + Over
6 = 0.3
tee
108.
to
-33.
parallel
26.
to
-88.
endon
3.
to
-72.

244
7.5.1 Lennard-Jones plus Quadrupole Fluid
Figure 72 shows the site-site pair correlation functions found
for the Lennard-Jones plus quadrupole fluids. For both the weak and
strong quadrupole strengths, the first peak is suppressed and broadened
compared to the corresponding center-center pair correlation functions
of Figure 43 . The site-site function is found to go to zero at
smaller separations r^ than the center-center function, due to
geometry. In addition, the stronger quadrupole fluid has developed
a shoulder on the low r^ side of the first peak. This phenomenon
is distinctly different from that found to occur for Lennard-Jones
diatomic potential models, wherein the first peak of §ag(rag) broadens
in the large r^^ direction [60,71].
An explanation for the nature of this g^ curve at high quadru¬
pole strength can be suggested based on molecular structure. Consider
the Lennard-Jones plus quadrupole molecule to be a repulsive spherical
shell with embedded point charges such that an attractive, axially
symmetric quadrupole field exists about the spherical shell. Located
on the linear axis of the quadrupole field are sites A and B, centered
about the center of the sphere, and separated from one another by about
a third of a molecular diameter (£ = 0.3292a). From the analysis of
the pair potential in Section 7.1 and the angular pair correlation
function in Section 7.4, pairs of these quadrupolar molecules tend to
align in a tee orientation. Let us assume that the quadrupole strength
is sufficient so that not only pairs of molecules tend to contribute
to structure, but that clusters of at least four molecules form. In
such a cluster, each pair of molecules is in a relative tee orientation

245
Figure 72. Site-Site Pair Correlation Function for Lennard-
Jones plus Quadrupole Fluids

246
and is near the closest point of approach (limited by the repulsive
shell), as in Figure 73 . When the clusters contain four molecules,
each molecule has 12 site-site distances r _ between its two sites
aB
and the six other sites in the cluster. When the distance of separation
of sites on a molecule is about a third of the molecular diameter, and
the sites are indistinguishable, two groups of these 12 site-site
distances are equal, as indicated in Figure 73:
rlA3A rlA3B rlA2B rlB2B rlB4A
(7-22)
rlA2A rlB2A rlB3A rlB3B
(7-23)
The first connection between the proposed cluster structure in Figure 73
and the g 0 in Figure 72 is that the number of equal distances in
otp
(7-22) and (7-23) are in the ratio: 5/4 = 1.2, while the ratio of the
main peak height to the height of the shoulder in Figure 72 is (from
Appendix J): 1.523/1.256 = 1.212. Further, there is some correlation
between the distances in (7-22) and (7-23) and the r^ values at which
the shoulder and first main peak occur. The distances ^~22)
are found to be ~ 1.16a, while (from Appendix J) the first peak maximum
occurs at r „ = 1.175a. The distances r1AOJ of (7-23) are ~ 0.84a,
aB 1A2A
while the shoulder occurs at ~ 0.97a. The r 0 locations of the shoulder
aB
and first peak would be expected to be at slightly longer distances
than those in (7-22) and (7-23), since the minimum energy occurs just
beyond r^ = 1.050 for the most probable orientation. Therefore, a
cluster of molecular pairs exhibiting square packing with interlocking

247
Figure 73. Possible Square Packing of Lennard-Jones plus
Quadrupole Molecules for Interpreting g „(r)
Cip

248
tee orientations is consistent with the form of found for Lennard-
Jones plus quadrupole fluids having strong quadrupole moments, and
l/o = 0.3292. It is interesting to note that Harp and Berne, in a
study of carbon monoxide using a Lennard-Jones plus dipole-dipole,
dipole-quadrupole, and quadrupole-quadrupole potentials, find the
three site-site correlation functions, g _ , g , and g , each
have forms similar to that found here. Whereas, if the simulation
for carbon monoxide is done using only the Stockmayer potential,
the structure in the g^g does not appear [90]. Test of the above
explanation of g^ for the Lennard-Jones plus quadrupole fluid may be
made by determining g „ for various molecular elongations. As the
Qtp
elongation is increased these structural arguments suggest the main
peak and shoulder will become separated from one another since the
difference in values of the distances of (7-22) and (7-23) would
increase. There would also be slight changes in the relative heights
of the peak and shoulder with elongation, as various terms would
satisfy (7-22).
7,5.2 Lennard-Jones plus Anisotropic Overlap Fluid
The g^g(r) for the Lennard-Jones plus overlap fluids are shown
in Figure 74. As in the case of the quadrupole, the first peak in
g 0 for overlap is shortened and broadened compared with the center-
OCp
center pair correlation function of Figure 48. No special features
are found on the overlap g^ curves, though there is a hint of a
shoulder forming near the high r^ side of the first peak for the
6 = 0.3 fluid.

249
Figure 74. Site-Site Pair Correlation Function for Lennard-
Jones plus Anisotropic Overlap Fluids

250
Values for gag(r) for both the quadrupole and overlap fluids
at the conditions studied in this work are tabulated in Appendix J.
7.6 Filmed Animation of Molecular Motions
7.6.1 Introduction
A principal advantage of performing computer simulation of
fluids by the molecular dynamics method, rather than by Monte Carlo,
is that in the course of the simulation the classical time develop¬
ment of the system is evolved. By saving (usually on magnetic tape)
the positions and orientations of the particles at discrete time steps
for which the equations of motion are solved, one obtains a permanent
record of the evolution of the system. This record (data) may be
transformed into a motion picture in which the time evolution of the
system is visually displayed. Such movies have been produced by several
workers as a medium for study of the microscopic nature of fluid systems
[91,92,93]. These motion pictures are of value in gaining insight into
the character of molecular diffusion, the frequency and nature of
molecular collisions, and the presence of molecular scale structure
in the fluid. They provide a means of differentiating among suggested
ideas about molecular scale behavior, e.g., the nature of molecular motion
and collisions are postulated, in turn, by Enskog [94], Eyring [95], and
Rice and Allnatt [96]. With movies of fluid mixtures one can study the
molecular environment of a molecule of a given species under a variety
of molecular geometries and interaction potentials. Such studies would
provide insight into the validity of mixture theories, such as regular
solution theory [97]. For fluids of nonspherical molecules, a movie

251
would be a means of comparing rotational and translational motion and
would indicate the degree of hindered rotation experienced by the
molecules. Certainly, much effective study of these areas can be
done in a more quantitative manner through theory, experiment, and
simulation. The movie, however, provides a medium for communicating
detailed aspects of a system, especially, to those audiences who
have an interest in the problems, but who have little time or inclina¬
tion for delving deeply into the theoretical or experimental details.
In this connection, the motion picture has obvious pedagogical
advantages. On a molecule by molecule basis, graphic portrayal of
a system by such means as movies provide information that may, otherwise,
only be available by inference, if at all. For example, attempting
to characterize the type of local structure in a fluid system by a frame by
frame study of a movie would be a straightforward method of attack.
Whereas, attempting to solve the same problem by study of distribution
functions is more cumbersome, since the distribution functions are
averages over time [93]; e.g., the discussion concerning the site-site
pair correlation functions in Section 7.5.1.
In this section we report development of a technique for producing
filmed animations of molecular motions generated from a molecular dynamics
calculation. The particular subject of this initial film was a simple
Lennard-Jones fluid. The emphasis here was not to study Lennard-Jones
fluids which are, indeed, largely understood; rather, the objective was
to develop a method of filming molecular motions. The Lennard-Jones
fluid offers several simplifications which, in this first attempt,
allow concentration on the mechanics of the method without the added
difficulties of a complicated fluid system.

252
7.6.2 Fluid System Filmed
Data for a system of 256 Lennard-Jones molecules were generated
by the molecular dynamics method described in Chapter 6. The state
3
condition chosen was kT/e = 1.273 and pa =0.85. This state condition
has been studied previously by Verlet, and the values for the internal
energy, pressure, and radial distribution function found here agreed
with those given by Verlet [32,98]. The calculation was started from
an FCC lattice structure, and a total of 2000 time steps were generated,
-14
each of 2.16(10 ) second, when the e,a parameters for argon are used,
O
i.e., e/k = 119.86K, a = 3.405A. The three coordinates locating the
center of mass of each molecule were saved on magnetic tape after
each time step.
For the density and number of particles used here, the system
was a cube of side 6.7a. The subject of the film was taken to be any
molecule whose center of mass was within a volume element la thick,
taken from the center of the cubic system, parallel to the YZ plane,
as shown in Figure 75. The line of sight was taken to be perpendicular
to the YZ plane. At any given time step in the simulation, from 30 to
50 molecules were found to have their centers of mass in the volume
element. Each molecule was visually displayed by drawing a circle on
a plane surface which represented the projection of the spherical
molecule onto the YZ plane. This procedure introduces two difficulties
into the animation:
(1) The volume element is an open system in the ± X' direction.
Molecules which enter and leave the volume element become recorded as
appearing and disappearing on the film.

253
Figure 75. Box Representing the Molecular Dynamics
System with the Volume Element Sampled
for the Filmed Animation Indicated.
(Arrow indicates the line of sight in
the movie.)

254
(2) A volume element thickness of one molecular diameter allows
the centers of mass of two molecules to be nearly aligned with one
another along a line parallel to the x-axis and both be within the
volume element. The projection of two such aligned molecules onto
the YZ plane gives the appearance of large overlap of the molecules,
which, in fact, has not occurred. This problem can be overcome by
reducing the thickness of the volume element, which, also, reduces
the number of particles found in the element at any instant.
7.6.3 Equipment
The circles representing the molecules were drawn with software
associated with a Gould 5100 electrostatic plotter and were displayed
on a Tektronix 4013 graphics terminal. The photography was done with
a 16mm Bolex H-16 SBM camera using a Soligor ITV, 25mm lens. The
film was Kodak black and white TRI-X (7/7278) and an exposure setting
of 0.95 was used.
7.6.4 Film Production Technique
The procedure for making the film was to first analyze the
magnetic tape containing the positions of the molecules to find the
locations of all molecules whose centers of mass were within the
volume element at each time step in the calculation. For each time
step, code for drawing the circles representing the molecules in
the volume element was generated from the Gould plotter software.
The code was stored on disc in the sequence corresponding to the
time evolution of the system. The disc file containing the "drawings"
was then accessed from the Tektronix terminal and the drawings for

255
each time step were displayed and photographed one at a time. To
create a slowly evolving, yet smooth animation, four movie frames
were photographed of each time step drawing. When projected at a
speed of 24 frames per second, this corresponds to six time steps
per second of system evolution. Hence, 1000 time steps gave about
2.8 minutes of film time.
Figures 76 , 77 , and 78 show the frames for the first,
s t
sixth, and 101 time steps. In these figures (as in the movie),
the lengths of both sides of the box have been lengthened by one
molecular diameter (la), so that any circle near the edge of the
box will appear to be wholly within the box. Figure 76 shows
the FCC starting configuration for the volume element. Figure 77
shows the beginning of breakdown of the FCC structure due, at this
point, largely to the effect of periodic boundary conditions. Note
that several pairs of molecules are colliding after only six time
steps. Figure 78 shows the FCC lattice to be dissolved after
100 time steps, and illustrates the misleading overlap of molecules
discussed in Section 7.6.2.

256
Figure 76. Initial FCC Lattice Configuration of Lennard-Jones
Molecules in the Volume Element Sampled in the
Filmed Animation

257
Figure 77. Frame from the Filmed Animation of Lennard-Jones
Molecules Corresponding to the Sixth Time-Step
in the Molecular Dynamics Calculation

258
Figure 78. Frame from the Filmed Animation of Lennard-Jones
Molecules Corresponding to the 101st Time-Step
in the Molecular Dynamics Calculation

CHAPTER 8
CONCLUSIONS
8.1 Theory for Surface Tension of Polyatomic Fluids
A general, first order perturbation theory for the surface
tension of polyatomic fluids has been developed. Upon introduction
of a Pople reference, the difficult multibody effects in the higher
order terms, as well as the first order term, vanish, allowing
extension of the theory to third order. The nonvanishing terms
in this theory involve the unknown interfacial pair and triplet
correlation functions g (z1r-ir>) and g (z,r,„r,~)» In order to
°o 1—12 o 1—12—13
perform calculations, the Fowler model of the interface is introduced.
The theory has been tested against Monte Carlo calculations of the
Fowler model surface tension for a Stockmayer fluid. The behavior of
the second and third order perturbation theories for surface tension
is much the same as that found for bulk fluid thermodynamic properties.
The second order theory is found to work well for weak dipole strengths
(y/(CO ) £0.6), but both the second and third order theories give
poor results at high values of the dipole moment, compared to the
Monte Carlo work. When the third order theory is recast in the form
of a [1,2] Padé approximant, however, the theory agrees with the
3 1/2
Monte Carlo results up to y/(ea ) =1.75. As is the case for bulk
fluid thermodynamic properties, the reason for the success of the Pade
is obscure; it gives correct results in the limits of high and low
anisotropic strength, and apparently interpolates correctly between
these limits.
259

260
This Padé theory has been used to predict surface tensions for
pure polyatomic fluids. A simple to use correlation for pure fluid
surface tensions has been developed based on the perturbation theory.
Future work to be done on the pure fluid theory is the replacement
of the Fowler model with more realistic expressions for the reference
fluid distribution functions. An obvious, realistic choice would be
the use of interfacial distribution functions obtained from computer
simulation studies. The singlet distribution function Pq(z^) has been
obtained for a Lennard-Jones fluid [20,21], the reference fluid used
in this work. However, the interfacial pair correlation function
g (z^r^) presents more difficulties due to its higher dimensionality
and the slow relaxation times inherent in interfacial phenomena. The
theories for pure fluid surface tension are at a stage where expres¬
sions for an interfacial pair correlation function, g^z^r^) or
Cq(z^rv^) > would be of great value.
To extend this pure fluid theory to mixtures, one is faced
immediately with the problem of obtaining the interfacial density
profiles for each component in the system. For multicomponent
systems this is a formidable problem requiring the simultaneous
solution of integral or, even, integro-differential equations.
Even in the (usually) simple conformal solution theory approach,
the average "composition" of the interfacial region is required.
On the experimental side of the mixture problem, there is a paucity
of experimental surface tension data for even binary mixtures over
the whole composition range and much of the phase diagram.

261
8.2 Theory for the Interfacial Density-Orientation
Profile of Polyatomic Fluids
A first order perturbation theory has been developed for the
interfacial density-orientation profile p(z^o^) for polyatomic fluids.
Upon introduction of a Pople reference, the first order term
vanishes for multipolar anisotropies, but does not vanish for aniso¬
tropic overlap or dispersion potentials. Calculations of p(z^a^) for
axially symmetric molecules interacting with overlap or dispersion
forces have been performed, using a Lennard-Jones reference fluid, and
the model for the interfacial pair correlation function used by
Toxvaerd [14]. For even the weakly anisotropic potentials accessible
by first order perturbation theory, the axially symmetric molecules
exhibit preferred orientations in the interfacial region. Molecular
adsoroption in the interface cannot be obtained from the first order
theory, however. It would be useful to have computer simulation results
for p(z^co^) for these simple anisotropic fluids in order to test the
theory. ‘
To attempt prediction of p(z1w1) for multipolar fluids, the
theory must be extended to higher order. There are difficulties in
extending the expansion, however. To obtain the interfacial density
profile, p(z^) from the density-orientation profile, pCz^o^)» one
simply integrates p(z^aj^) over the orientation. When this is performed
on the first order theory of Chapter 4, the result is merely the reference
fluid profile pQ(z^). It appears, however, that the density profile
obtained from the higher order theory will not be the reference profile;
i.e., the profile may be displaced from the reference profile PQ(z^).

262
It is not clear, at this point, how to best account for shifting of
the density profile when carrying out the expansion.
8,3 Computer Simulation of Polyatomic Fluids
Much effort in the simulation work reported herein has been
directed towards developing methods: application of a minicomputer
to Monte Carlo calculations, filmed animations of molecular motions
from molecular dynamics, and development of expressions for efficient
evaluation of the force and torque in molecular dynamics simulations
of linear molecules. The minicomputer is rapidly making inroads into
areas of computational chemistry and physics once felt to be solely
the domain of the biggest and fastest machines available [99]. Both
the advantages and disadvantages to using minicomputers in extended
CPU-time calculations are clear-cut. The problems are: (a) the slow
execution time compared to large machines, hence (b) the need for a
minicomputer "dedicated" to simulation work, (c) the necessity for the
user to be more than just a programmer; he must be programmer, operator,
and hardware technician. The overriding advantage is economics. Given
the present cost of CPU-time on large machines, it is easy to visualize
situations wherein the choice is between doing the calculation on a
minicomputer, or not at all. As the minicomputer industry continues
to mature, the above listed disadvantages will diminish.
Motivation for producing movies from molecular dynamics results
have been given in Section 7.6. Future studies in this medium could
entail elaborate three dimensional portrayals of systems of non-
spherical molecules using holography, for example.

263
Because of the large amounts of computing time required in
molecular dynamics calculations, any effort which can even frac¬
tionally improve speed of execution is worth pursuing. In addition
to the possible techniques mentioned at the end of Section 6.3.1 for
speeding up the calculation, a particularly promising method seems
to be one which takes advantage of the fact that the force on a
particle can be divided into two parts which operate on different
time scales. Such a method has been successfully applied to studies
of the time evolution of stellar systems [100], which are calculations
analogous to molecular dynamics, save that the. interaction is attractive
only.
The method of molecular dynamics has been applied to the study of
systems of axially symmetric molecules, in particular, Lennard-Jones plus
quadrupole and Lennard-Jones plus overlap models. Numerous equilibrium
properties have been determined by direct ensemble averaging in the course
of the calculation. Values for those properties directly related to the
potential: internal energy, surface excess internal energy, and mean
squared torque are found to be in good agreement with the Padé perturba¬
tion theory, for the quadrupolar fluids. Values of properties related to
derivatives of the potential: pressure, surface tension, and heat capacity
are not in such good agreement with the theory. A question to be faced in
the future of this work is whether knowledge of the derivative properties
will be of sufficient value to justify the longer calculations with
more particles than this study indicates are required. The angular
correlation function is not directly related to the potential or
its derivatives, but it seems to be difficult to study by theory,

264
experiment, or simulation. The evidence here indicates that accuracy
in simulation values for will, also, require longer calculations
with more particles.
In addition to these properties, the coefficients g„ . (r „)
-LZ
in an expansion for the angular pair correlation function
in terms of products of spherical harmonics of the molecular orienta¬
tions in the intermolecular frame have been determined. These
^1^2m * *
g0 p (r ) have: (a) been related through integrals J (p ,T )
1 zm ^
to the equilibrium properties discussed above, for several anisotropic
potentials, and (b) been recombined in the spherical harmonic expansion
to obtain gCr^u^u^’ which, in turn, was used to study local fluid
structure. The g^ ^ m^ri2^ coefficients provide a new avenue for
exploration of equilibrium properties and, hence, are of considerable
value even if the spherical harmonic expansion for gCr^^t^) did not
converge [88]. The series for i-n ^act> seems to converge
well for the most probable pair orientations, though the convergence
is questionable for the least probable orientations. Study of
®^r12ü)ltJ2^ s^ows the most probable pair orientations in the first
peak region to be the tee for quadrupolar fluids and the parallel
for anisotropic overlap fluids with positive 6 values. These most
probable orientations have been found to make the most significant
contributions to the internal energy, pressure, and Fowler model
surface tension. Study of molecular structure for the quadrupolar
fluid based on interpretation of the features of the site-site pair
correlation function suggests that the strongly quadrupolar molecules

265
tend to exhibit square packing reinforced by the tee orientation
between adjacent molecular pairs.
The availability of values for the gg 0 (r ) coefficients
' 1 2m
will undoubtedly spawn new theoretical efforts in liquid physics,
as indeed, it already has [77,88]. It is also likely that other
expansions, appropriate for more strongly anisotropic molecules
than those considered here, will be studied; examples are expansions
in ellipsoidal harmonics [101] and expansions of the site-site pair
correlation function.
Finally, the analysis of the molecular dynamics calculations
presented here is only half the story. The appropriate data are
stored on magnetic tape for future analysis of time dependent
properties; including, a host of time correlation functions, the
van Hove distribution function, and the dynamic structure factor.
Study of these time related properties may prove to be of more value
than the equilibrium properties, since many of these are more sensitive
to the interaction potential than any equilibrium property.
The potential, then, brings us back to square one, for it is
the objective of statistical mechanics to characterize fluids in terms
of contributions from the distribution and interaction of the molecules
in the system. The spherical harmonic expansion, in the pair theory
of fluids, at least, gives a firm handhold on the pair distribution
function for polyatomic fluids. The interaction potential, however,
remains elusive. It will require a gentle marrying of theory, experiment,
and simulation (Figure 1) to reveal and quantify the potential from a
statistical physics approach.

APPENDICES

APPENDIX A
EXPRESSIONS FOR THE ANGLE AVERAGES IN
EQUATIONS (3-4) TO (3-7)
The angle averages , , ,
a 0)^2 a (1)^2 av av U1CJ2W3
"k "k "k
which occur in Equations (3-4) - (3-7) may be
av a a (jJi°J2003
evaluated by substituting for u its spherical harmonic expansion, (C2), and
3.
performing the angular integrations over spherical harmonics. The properties
of spherical harmonics [38] simplify the procedure. Details can be found
in reference [33], The resulting expressions are:
*2

a 0)^2
1 V (2£+l) y y f \ *
4tt l (2£.+l)(2£„+l) ¿ K Es(A;nln2) r12
A 1 2 npn2 ss
.-n
* s
X E*,(A;nin2) r*2 S
(Al)
*3

a u),w
12 (4tt)j,í AA'A
ñ77 I (2£+l)(2£'+l)(2£"+l)
z A A t A"
£" £' £
0 0 0
x
o” S' £
*1 1 1
*2 2 z
£" £' £
I
nln2nl ss's"
_ I It H
n2n-j
Í£"
*1
£ij
Í£"
*2
-
7 '
z
k
*i-
—2
_ i
-2
-2-
*_(ns+ns,+ns") 3
X Es(A;n1n2)Es,(A';n|n’)EsI,(A";n^) r^ /eJ
(A2)
267

268
tü a u 4tt Jj, 6£ £'6£ £Ó£ £,6£ 06£I0 ^
1 ¿ 3 ¿VA 111 1 2 3 n^ ss
,-n
x (-) Es(¿10£1;n10) r12 Eg, (^O^j^O)
* ns f 2
X r13 P£ ^cos ai^£
(A3)

3 a 3 “l“2“3 AA'A"
£+£'+£"
I (~) *->£ o'^o £"^£'£"
1 1 2 2 3 3
[(2£+l)(2£'+l)]1/2
C2£1+l)(2£2+l)(2£^+l)
o'
3
£ £' £"
£2 £j_
^££'£"(ala2a3)
nln2n3 ss's"
n..+n„+n'
I (-) Eg (A J n1n2) Eg,(A,;n1np
* ns * ns' * ns" 3
X Es"^A ’—2—3^ r12 r13 r23
(A4)
In these equations the notation is the same as that for (3-12) to (3-15)
and is described under (3-11) and (3-15).
In Equation (A4) the function is defined by the expansion
in Equation (3-19). Expressions for the have been evaluated for
multipoles and are given by [33]:
4»
222
5_
8tt
14tt
1/2
(1 + 3 cos cos c¿2 cos a^)
(A5)
4>
233 256tt
3tt
1/2
[9 cos ctj - 25 cos 3 + 6 cos (a^- a
(3+5 cos 2 a^)]
(A6)

269
334 512tt
22tt
1/2
[3(cos + 5 cos 3 a^) + 20 cos (a^- a^)
(1-3 cos 2 a^) + 70 cos lia^- a^) cos a^]
(A7)
4>
81
444 1024tt
2 0 0 277
1/2
[-27 + 220 cos cos a^ cos
+ 490 cos 2 cos 2 cos 2 + 175 {cos 2(a^- «2)
+ cos 2(a^~ a^) + cos 2(a^- a^)}]
(A8)

APPENDIX B
COORDINATE TRANSFORMATION AND INTEGRATION OVER EULER
ANGLES TO OBTAIN EQUATIONS (2-89) AND (2-90)
F F
The Y2g and terms of (2-85) and (2-86) are three body terms
involving molecules 1, 2 and 3. The integrations over 0)^ and in
these terms cannot be done in a straightforward manner due to the angle
dependence of the integrands. In the Fowler model, however, the inte¬
grands are independent of motions of the rigid triangle formed by r.^>
r^^ and ^23" Thus, we transform the angles and to a set of
angles which include Euler angles {({>0x} that specify the orientation of
triangle 123. Integration over the Euler angles may then be performed.
B.l Choice of Euler Angles
The body-fixed frame is chosen to have its origin on molecule 1,
its x-axis along r^, and its z-axis perpendicular to the plane of the
123-triangle. Euler angles {4>0x} between this body-fixed frame
(x y z ) and some arbitrary space-fixed frame (x„y„z„) are defined
D IJ D bob
by the following rotations (see Figure Bl):
(a) is the angle of rotation of the x y plane about z ,
OO O
aligning x on the projection of z onto the x y plane,
b d b b
(b) 0 is the angle of rotation of the z'x' plane about y',
O O D
aligning z' onto z .
b d
(c) X is The angle of rotation of the x’’y'' plane about z"
u u J
aligning x" onto xD.
0 B
270

271
Figure Bl. Rotations Defining the Euler Angles {0x}

272
The Euler angles {0X^ thus specify the orientation of the three body
system in the fluid. A fourth angle is required which, with the lengths
rl2> r^ and specifies the shape of the triangle itself. We choose
the interior angle at molecule 1 to be this fourth angle. Hence, we
consider the transformation:
du)^2 dü)^3 = d(J>^2 d cos 0^ -^3 d cos 0^ = J d(f> d cos 0 dy d cos
(Bl)
The Jacobian in (Bl) can be shown to be unity.
Equations (2-84) and (2-86) thus become:
2B
3 3
2 PL
2
2
dr12 r12
J J
drl3 r13
-1
d cos a,
1 a a (jJ1üj2íjJ3
x g (r r r ) X
goI> 12 13 23' z
(B2)
yf =^p3
'3B 6 PL
dr12'r12
drl3 rl3
d cos a
1 a 'a' av wl“2W3
goL(rl2r13r23) Iz
(B3)
where
I
z
2ir 1
d(j)
J 4
0 -1
d cos 0
2tt
dX
z
max
0
(B4)
and z is the value of z, when one of the molecules 1, 2, or 3 first
max 1 ’ ’
cuts the z = 0 surface, when the rigid triangle 123 approaches the
surface from the liquid side.

273
B.2 Evaluation of Integral I
Consider the triangle of molecules 123 to be located on the liquid
side of the vapor-liquid interfacial region of a Fowler model two phase
system. The space-fixed frame is oriented such that the x y plane lies
Ü J
in the interfacial plane and the positive z -axis is directed into the
vapor phase (see Figure B2). Values for z in (B4) are given by the
max
manner in which triangle 123 first cuts the z = 0 plane as the triangle
is rotated, in turn, through each of the Euler angles.
Figure B2 show immediately that z is independent of (J) since
rotation of the triangle about z^ does not change the z-position of any
part of the triangle. Hence, (B4) reduces to:
I
z
2tt
1
d (cos
0)
-1
2tt
dX
0
z
max
(B5)
Rotation of the triangle through X gives three possible values
for z , depending on which molecule first cuts the z = 0 plane:
max
0
when
molecule
1
cuts
z = 0
first
z = â– 
max
r12
cos
e!2
when
molecule
2
cuts
z = 0
first
(B6)
â–  _r13
cos
ei3
when
molecule
3
cuts
z = 0
f irst
Thus, (B5) becomes:
1
I = 2tt
z
d cos 0
-1
dX (~ri2 cos 0i2^ +
dX (-rx3 cos 013)
*2
(B7)
where x^> X2’ and X3 are tlie values of X when molecule 1, 2, or 3 first
cuts the z = 0 plane, respectively.

274
Figure B2. Rotations of the Triangle 123 in the Fowler
Model Interface to Define Values for z
max

275
The spherical coordinate angles 0^ and 0^3 are related to the
Euler angles by the law of cosines of spherical trigonometry:
cos 0^2 = " sin 0 cos (y + a^)
(B8)
cos 0^2 = - sin 0 cos y
(B9)
Evaluating the integration limits y^, y£, and y^ in (B7) for 0 ^ 0, tt gives
X1 " 2 11
(BIO)
x2 =
7T
2 al
5tt
2 ai
if al < 2
i£ “l ' 2
(B11)
X3 = -cos
-1
-1
1 +
rl2 C0S al " r13
r^2 sin
(B12)
The negative sign in (B12) arises because y^ must lie between tt/2 and tt
when r^2 cos ot^ < r^ and between tt and 3tt/2 when r^ cos ot^ > re¬
using (B8) to (B12) in (B7) gives:
1
d cos 0 sin 0[r12 sin (X3+ al^ " ri3 sin X3~ r\2~ r13^ (B13)
I = 2fr
z
-1
Integrating over 0 gives:

276
!z = tt tr12 sin (x3 + o^) - r13 sin x3 ~ r12 “ r13l
(B14)
Using (B12) and the law of cosines of plane trigonometry, (B14)
reduces to:
Iz 77 (rl2 + r13 + r23)
(B15)
Combining (B15) with (B2) and (B3) gives:
OO 00
Y
2B
B 2 3
2 77 PL
drl2 ri2
drl3 r13
0
0
-1
d cos a
1 a a ü)i(jl)2ü)3
X goL(rl2rl3r23)(rl2 + rl3 + r23)
(B16)
3B
B 2 3
6" 77 PL
2
2
dr12 rl2
dr13 r13
0 0-1
X goL(r12r13r23)(rl2 + r13 + r23}
d cos a,
1 a a a a)iU2U3
(B17)
The integration over cos may be transformed to an integration
over r23' The law of cosines gives:
r12+ r13
d cos =
dr
23
-1
r12 r131
r12r13
(B18)
Substituting (B18) into (B16) and (B17) gives (2-89) and (2-90), respectively.

APPENDIX C
MODELS FOR ANISOTROPIC POTENTIALS OF
LINEAR MOLECULES
The anisotropic potential u can be expanded in terms of products
cl
of spherical harmomics of the molecular orientations. The expansion is
a sum over all terms in Z^, Z^, and Z except Z^ = Z^ = Z = 0 and has
the form [27,38]:
u
a
(r12“i“2) " { ua(A)
(Cl)
ua(A)
^
E(A;nin2;r12) CCAjm^m) D C^)
nln2 1 1
m^m2m
Z
where A = Z^Z^Z, u) = orientation of vector _r^2 along the line of centers
from molecule 1 to molecule 2, C(A;m^m2m) = a Clebsch-Gordan coefficient,
Z
D (to) = a representation coefficient, and Y„ (go) = a spherical harmonic,
mn Zm
each in the convention of Rose [38]. The superscript * indicates a complex
conjugate.
The coefficient E is a strength constant taken to be a sum over
various interactions, E^, where s represents multipole, overlap or dis¬
persion. The r dependence of such interactions is usually assumed to be
of a power-law form:
277

278
E(A;n;Ln2;r) = £ E^Ajn^)
s
(C3)
where ng = (£+1), 12 or 6 for multipole, overlap, or dispersion, respectively.
For axially symmetric molecules Es(A;n^n2) vanishes except for n^ = n2 = 0.
Expressions for Eg for particular potential models are given in Table Cl.
£
Since n. = n„ = 0 for axially symmetric molecules, D in (C2)
1 / mn
simplifies to:
DmO(*0X) =
4tt
2£+l
1/2
(C4)
and (C2) reduces to a sum over three ordinary spherical harmonics:
ua(A) =
4tt
[ (2£^+l) (2£2+l) ] mim2
Yj2 1 E(A;00;r12) CCAjm^m)
x Y (4 0 ) Y (cj) 0 ) Y, (4)0)
£-^m^ 1 1 ^2m2 ^ 2 £m
(C5)
where m^ + m2 = m.
With the equations from Table Cl and the properties of spherical
harmonics [38], equation (C5) gives expressions which are amenable to
calculation. Expressions for various anisotropic model potentials in
the intermolecular frame of Figure 32 are given in Table C2. Expressions
for the same models but using the angle y of Equation (6-11) rather than
(J)^2 from (6-10) are tabulated in Table C3.

TABLE Cl
Expressions for the Expansion Coefficients E for Various Interaction
Potentials for Linear Molecules [28,38]
General Multipole
Dipole-Dipole :
Quadrupole-Quadrupóle:
Dipole-Quadrupole :
Dispersion-Dispersion:
Overlap-Overlap
(-)
Emult(¿l2'2i¿;00;r) (2«,+l)
4tt(2£+1) !
(2Í¿1) ! (U2) !
1/2 QL Q£
*1 ^2
l+l
Edipole(112;00;r) = " 2(6lT/5)172 ^2/r3
Equad(224;00;r) = f (70tt)1/2 Q2/r5
EDQ(123;00;r) = 2(15rr/7)1/2 yQ/r4
E,. (202;00;r) = - 8(tt/5) 1/2 ei<(a/r)6
ais
= E (022;00;r)
dis
Edis(224;00;r) = " 48(2tt/35)1/2 e<2(a/r)6
E (202;00; r) = 16(tt/5) 1/2 6e(a/r)12
over \ /
= E (022;00;r)
over
(C6)
(C7)
(C8)
(C9)
(CIO)
(Cll)
(C12)
(C13)
(C14)
y, Q, K and 6 are the dipole moment, quadrupole moment,
and dimensionless anisotropic overlap parameter, respectively
dimensionless
Qn and Qp
*1 x.2
anisotropic polarizability,
are general multipole moments.
279

280
TABLE C2
Expressions for Anisotropic Potential Models
in the Intermolecular Frame of Figure 32
UDD(1J) ' [3iSjC - 2ciCj>
ij
..+ 3Q2 ,1 ,2 2
uqq(12) * - 5ci - 5cj
ij
,..s* 3 yQ r . 2 ,.
UDQ(1J) * 2 -TT [ci(3cj- X) -
ij
12
u (ij) = 46e
over
r. .
v iJ"
2
[3cf +
i
Udis(ij) = " 2£K
r.
v ij'
2
[3cf +
i
54 2
35 K £
r. .
v ij'
fl -
- 15c?c? + 2{s.s.c - 4c.c.}2]
i J i J i 1
c.(3c?- 1) - 2s.s.c(c. - c.)]
j i J j i
3c2 “ 2]
3c? - 2]
J
,2 ,2 .. 2 2 . _r
5c.- 5c.- 15c.c. + 2ts.s.c -
i J i 3 1 J
(C15)
(C16)
(C17)
(C18)
(C19)
4c.c.}2]
i J
D = dipole, Q = quadrupole, over = overlap, dis = dispersion. C_^ = cos 0^,
s^ = sin 0_^, c = cos .<(> . y, Q, 6, and k are the dipole moment, quadrupole
moment, overlap parameter, and anisotropic polarizability, respectively.
^Eq. (1.3-10) in ref. [102] is too small by a factor of 4.
^Eq. (178) in ref. [66] is too small by a factor of 2 (cf. Eq. (3.10-19)
in ref. [102]).

281
TABLE C3
Expressions for Anisotropic Potential Models in the
Intermolecular Frame, using y rather than (p. .
2
V [c(Y) -
3c.c . ]
i J
ij
3Q5 n
4r?.
ij
q 2 c 2 ..22
- 5c.- 5c.- 15c.c.
i J 1 J
3 yQ [(c
2 4 u i
r. .
ij
- ci) (1 + Sc-jCj
- 4óe [ 0
r. .
12 2 2
[3cT + 3cf -
i J
'
- 2k£
0
r. .
ij
[3cf + 3c7 -
J 1 J
54 2 f
a ] 6 ri c2
uDB(lj>
QQ
DQ
over
dis
35
k e
r. .
v ij
(C20)
+ 2{c(y) - 5ciCJ2] (C21)
- 2c(y))] (C22)
2] (C23)
2] (C24)
5c?- 15c?c? + 2{c(y) - 5c.c.}2]
J i J i J
D = dipole, Q = quadrupole, over = overlap, dis = dispersion,
c. = cos 0., c(y) = cos Y> y and Q = dipole and quadrupole
moment, 6 = overlap parameter, K = anisotropic polarizability

TABLE C4
Derivatives of Various Anisotropic Potentials for Evaluating
the Force and Torque from Eqs. (6-25) and (6-34)
Pair
Interaction
9r. .
13
9u
9c.
1
9c (y)
DD
QQ
3u
DD
5u
QO
r. .
ij
3y
c.
„3 3
15 _Q_ r , , 2
~ 5— L~c . + 7c .c .
2 5 1 it
r . . J
ij
2cj c(y)
3
r. .
ij
^ [c (y) - 5c c ]
r. .
ij
DQ
. 4udQ
r. .
ij
3yQ
2r7.
ij
. [2c(y) + 5c. - 10c.c.- 1]
4 1 1 j
- [c _ c ]
4 L i iJ
r. . J
ij
over
12u
over
r. .
ij
246e
r. .
v i3y
12
c.
1
dis
6u,.
dis
r. .
13
- 12ke
'â– rij
c.
1
216 2
35" £K
r ^6
a
'rij'
[c(Y) -
5c .c . ]
1 3
108 2
< £
r. .
' 13'
[ —c . + 7c . c .
1 13
2Cj c(y) ]
7
282

APPENDIX D
EXPRESSIONS FOR y^, Y^g, YgA AND y^g FOR VARIOUS
ANISOTROPIC POTENTIALS FOR AXIALLY SYMMETRIC MOLECULES
F F F F
Expressions for the perturbation terms Y2A> ^2B’ ^3A’ anc* ^3B
amenable to calculation are obtained by substituting the strength
coefficients from Table Cl into (3-12), (3-13), (3-14) and (3-15),
respectively. The resulting expressions for a variety of anisotropic
potentials are given in Tables Dl, D2, D3, and D4. In these tables
the reduced dipole and quadrupole moments are defined by y = y/(ea^)^ ^
and Q = Q/ (ea ) , respectively.
283

TABLE DI
Expressions for y for Various Anisotropic
Potentials for Linear Molecules
Dipole-Dipole
Quadrupole-Quadrupole:
Dipole-Quadrupole
Overlap-Overlap : y A(202)
¿-Pi.
Dispersion-Dispersion: y2A(202) + y2A(022)
Dispersion-Quadrupole:
Dispersion-Overlap : y (202)
¿.Pi.
*2
PT
TT PL *4
6 * y
T
J5
(Dl)
*2
Y^(224)
7tt PL *4
' io T* q
J9
(D2)
*2
F*
^2A(123)
TT PL *2
= 4 * y
T
*2
Q J7
(D3)
*2
yF2*(°22)
32tt PL .2
= 5 * 6
T
: J23
(D4)
*2
4(224)
8tt PL 2
= -z 3— K
5 *
T
J11 i1
, 324 2,
+ 175 k ]
(D5)
W224)
*2
72tt PL
25 T*
2 o*2
K Q
J10
(D6)
*2
IT*
y2a(°22)
32TT PL
5 T*
5k J^7
(D7)

285
TABLE D2
F
Expressions for y^A for Multipole Potentials
for Linear Molecules
F*
Y (112;112;112) = 0
(D8)
Y32(112;123;123) = 0
(D9)
*2
F* 2tt PL *4 *2 * *
Y3A(112;H2»224) - - 25 T*2 y Q J10(p ’T }
(DIO)
*2
F*, 2tt PL *4*2 * *
Y3A(H2 5123; 213) = -yj-^y4Qz J1Q(p ,T )
(Dll)
*2
F* 2tt PT *2 *4 * *
Y 3A (123 ;123; 224) = 'f Q J12(P ,T )
L
(D12)
*2
F* 36tt PL *6 * *
Y3A(224;224;224) = - ^ Í2 Q J^(p ,T )
(D13)

286
TABLE D3
F
Expressions for yo„ for Various Anisotropic
4 J~$ *
Potentials for Axially Symmetric Molecules
Overlap-Overlap
y2b(202)
Dispersion-Dispersion:
y2b(202)
Overlap-Dispersion
W202)
*3
32tt2 PL r
— — .0
5 *
T
*3
8tt2 PL 2
— — K
T
*3
16-it2 PL
5 T*
2 LY(2;12,12)
LY(2;6,6)
6k LY(2;6,12)
Multipoles
0
(D14)
(D15)
(D16)
(D17)

287
TABLE D4
Expressions
for y
F
3B-
for Multipole Potentials
for Linear Molecules
^3b (ii2; HZ; 112)
/ 3
4tt
135
*3
14tt'| 3-/2 p
^ y KY(222;333)
l 5 J T*2
(D18)
*3
F*
^3B (112;123;123)
8tt^
315
1/2 *4 *2 Y
(3f)±/Z y 4 Q Z (233; 344)
T
(D19)
(112;112;224)
= 0
(D20)
Y^g(112;123;213)
= 0
(D21)
^3B(123;123;224)
*3
4tt3 Í22f)‘*'^ ^L *2 *4 Y
= ~ y Q KY(334;445)
Uj ! J T Z
(D22)
*3
^3B(224;224;224)
4tt^
2025
1/9 *(S V
(2002tt)1 Z Q ° K'(444;555)
T Z
(D23)

APPENDIX E
THE INTEGRALS KY(M'A";nn'n") AND LY(£;nn')
Tables El and E2 show values of KY and LY for the state
conditions studied by Verlet [32]. To facilitate interpolation
Y Y
between these state conditions these results for K and L have
been fitted to an empirical equation of the form:
£n|KY(££'£";nn'n")I = A p*2 £n T* + B p*2 + C p* £n T*
1 1 n n n
+ D p* + E £n T* + F (El)
n n n
y
with a corresponding equation for L . On the right side of (El) n
signifies (nn'n") in the case of KY and (nn1) for LY. |kY| is the
Y Y
magnitude of K ; the K take on positive values, except for
Y Y
K (334;445), which is negative, while all the L used here are
negative. Equation (El) was previously used by Gray and Gubbins [10 ]
and by Flytzani-Stephanopoulos ^it al. [33 ] for and K(££'£";nn'n")
and values of the constants for these have been tabulated. The constants
in (El) have been determined by a least squares fit and are given in
Table E3. (The integrals KY and LY for p = 0.85, T = .658 and
* *
p = 0.65, T = 1.827 do not fall on any smooth curve through the
others and are therefore omitted from Tables El and E2 and in fitting
the constants in Table E3.) The maximum deviation for the predictions
of Equation (El) is less than 1.8% of the values in Table El for the
Y Y
K integrals and less than 2.4% of the values in Table E2 for the L
integrals.
288

289
TABLE El
*
p
The
Integrals K^(££'£";nn'n")
for Pure Fluids^
*
T
kY333
222
Y344
233
y445
334
kY555
K444
.88
1.095
.13376
.06811
-.06007
.05321
00
00
.936
.13288
.06727
-.05895
.05191
00
CO
.591
.13070
.06583
-.05694
.04899
.85
2.888
.13758
.07072
-.06383
.05909
.85
2.202
.13601
.06919
-.06162
.05618
.85
1.273
.13287
.06666
-.05817
.05129
.85
1.127
.13171
.06583
-.05713
.04997
.85
.880
.13066
.06506
-.05605
.04846
.85
.786
.13020
.06466
-.05555
.04797
.85
.719
.12982
.06432
-.05506
.04734
.824
.820
.12909
.06329
-.05378
.04607
.75
2.845
.13094
.06405
-.05518
.04912
.75
1.304
.12676
.06059
-.05052
.04297
.75
1.070
.12603
.05974
-.04945
.04173
.75
.827
.12356
.05952
-.04891
.03912
.65
3.669
.12513
.05921
-.04956
.04358
.65
1.584
.12249
.05624
-.04528
.03768
.65
1.036
.12166
.05521
-.04383
.03587
.65
.90
.12204
.05561
-.04416
.03589
. 6
1.360
.11900
.05195
-.03973
.03163
.45
2.934
.11453
.04932
-.03778
.03089
.45
1.710
.11865
.05130
-.03897
.03104
.45
1.552
.11740
.05058
-.03827
.03033
YIn the table,
subscripts
on are SLÍL'
and superscripts are nn'n"
Thus
k222 = kY(222;333)>
etc.

290
TABLE E2
The Integrals LY(£;nn') for Pure Fluids
*
p
*
T
LY(2;6,6)
LY(2;12,12)
.88
1.095
-.094590
-.034505
.88
.936
-.088000
-.028847
.88
.591
-.074936
-.018479
.85
2.888
-.13374
-.088380
.85
2.202
-.12238
-.067320
.85
1.273
-.10148
-.040432
.85
1.127
-.096296
-.034552
.85
.880
-.088666
-.027976
.85
.786
-.084445
-.024443
.85
.719
-.082117
-.022472
.824
.820
-.087964
-.026075
.75
2.845
-.13151
-.077086
.75
1.304
-.10516
-.038798
.75
1.070
-.098936
-.032645
.75
.827
-.092278
-.026591
.65
3.669
-.13692
-.083730
.65
1.584
-.11152
-.042113
.65
1.036
-.10116
-.030680
.65
.90
-.097834
-.027544
.5
1.360
-.10987
-.034114
.45
2.934
-.11942
-.051890
.45
1.710
-.11520
-.038650
.45
1.552
-.11201
-.036003
Y(2;6,12)
-.054839
-.048249
-.035915
-.10407
-.086940
-.061476
-.055242
-.047642
-.043532
-.041232
-.045886
-.096764
-.061397
-.054545
-.047789
-.10318
-.066088
-.053592
-.049949
-.059170
-.076157
-.064651
-.061532

TABLE E3
The
Constants in Equation El
+
Integral
A
n
B C
n n
D
n
E
n
F
n
„(y)333
222
-.53198
.87067 .91913
-.91804
-.35300
-1.8785
v(y)344
233
-.77071 r
1.2988 1.2821
-1.1003
-.46537
-2.7264
i (Y)445i
>334 1
-1.0035
1.6946 1.6283
-1.2575
-.55474
-3.0226
T.(y)555
K444
-1.3905
2.1902 2.1693
-1.6278
-.67943
-3.2038
|l(y)(2;6,6)|
-.036717
-.22474 .67271
-.081620
-.19173
-2.1492
|L(Y)(2;12,12)
.25967
-.36137 .63695
.77312
.25115
-3.8612
|L(y)(2;6,12)|
.025587
-.25070 .76463
.26433
-.004298
-3.0081
Tk222333 “ K(Y)(222;333)
etc,
291

APPENDIX F
Expressions for the Spherical Harmonic Coefficients
g& £ m(r) in Equation (6-77)
12
g200^
2 ^ 8000^
g220^r')
45 , w 2 lw 2 1..
4 8000('r') ^C1 3 C2 3 >
g221^r)
2 g000^r^
g222^
8 8000('r^
g400(r)
105, (r)
8 g000v ' 1 7 1 35
g420(r)
315/5 , w 4 6 2,l_w2 1..
16 8000(r) (cl 7 C1 + 35 ^C2 3 >
g421(r)
105/3 , . . r 2 3,
“ —/ 2 8000(r)
g422(r')
g440(r)
105 , w 2 , 2 1N 2 /r^ 2 i \ ^
16" ^ B000(r) S2 <2C ’ X)>
11025 , w 4 6 2 . 3_w 4 6 2 3,.
64 8000(r) C1 7 C1 + 35 °2 7 C2 + 35}
g441(r)
2205 ( . . ,23. , 2 3.
16 8000^r') Slcl(cl 7 S2C2 c2 7 C
g442(r)
g443(r)
2205 ,..2,2 I. 2,2 1 . 2 ...
32 8000^ “ l6~ 8000(r')
g444(r)
315 M 4 4 4 „ 2 ..
128 g000(r) c .
1
cos 0., s. = sin 0., c = cos é
i i i’
292

293
g600(r)
g620(r)
g640(r)
g660(r)
g800(r)
= /l3 g000(r)<231cj - 315 cj + 105 - 5>
= g000(r)<(231cj - 315 cj + 105 - 5) (c* - j)>
= —^ g000(r)<(231cj - 315 cj + 105 cj - 5) (cj - y cj + |j)>
= ^ gQ00(r)<(231cJ - 315 cj + 105cJ - 5)(2310^ - 3150^ + 105^
= gQ00(r)<6345cJ - 12012cJ + 6930cJ - 1260c^ + 35>
- 5)>

THE INTEGRAL I
APPENDIX G
USED TO CALCULATE THE ANGULAR
554
CORRELATION PARAMETER G2 FOR QUADRUPOLES
Table G1 gives values for the integral which arises in the
second order perturbation theory for the angular correlation parameter
* "k
G„ for quadrupolar fluids [ 80 ]• The integrals I ,„(p ,T ) are defined
z nn X/
by:
â– JL k
I ,„(p ,T )
nn 3c
k k
drl2 r12
dr
*-(n-1)
13
0
* * *
* *
r + r
12 13
• * *_(n'-l)
23* 23
rl2 r13'
X go(r12r23r13) P£(cos ^
(Gl)
where P^ is the order Legendre polynomial, and is the interior
angle at molecule 3 in the triangle formed by molecules 1, 2, and 3.
In calculating I from (Gl), the superposition approximation is
employed for the triplet correlation function and Verlet's molecular
* ^ *
dynamics results are used for the pair correlation functions gQ(r ) t 32 ]
To facilitate interpolation between the.state conditions in Table Gl
the results for have been fitted to an empirical equation of the
form:
£n |l I = A p*2 £n T* + B p*2 (G2)
k k k
+ Cp £n T + Dp
k
+ E £n T + F
294

295
|l represents the magnitude of I,.^ since 1,.,.^ is negative. The
constants in (G2) have been determined by a least squares fit and are:
A,B,C,D,E,F = - 1.41842, 2.29308, 2.07360,
- 1.45408, - 0.89302, - 3.70178
(G3)
The maximum deviation for the predictions of Equations (G2) and (G3) is
•k
less than 1.7% of the values in Table Gl. (The integral I for p = 0.85,
* * *
T = 0.658 and p
through the others
= 0.65, T = 1.827
and are therefore
do not fall on any
omitted from Table
smooth curve
Gl and in fitting
the constants in (G3).)

296
TABLE G1
The Integral 1^, for Pure Fluids
*
p
*
T
*554
.88
1.095
- .03987
.88
.936
- .04094
.88
.591
- .04438
.85
2.888
- .03214
.85
2.202
- .03340
.85
1.273
- .03594
.85
1.127
- .03698
.85
0.880
- .03820
.85
.786
- .03948
.85
.719
- .03995
.824
.820
- .03674
.75
2.845
- .02632
.75
1.304
- .02904
.75
1.070
- .02984
.75
.827
- .03128
.65
3.669
- .02113
.65
1.584
- .02333
.65
1.036
- .02478
.65
.90
- .02613
.50
1.360
- .01986
.45
2.934
- .01561
.45
1.710
- .01818
.45
1.552
- .01817

APPENDIX H
VALUES FOR THE g£ ¿ m(r12)
COEFFICIENTS

298
TABLE HI
Values of gQ00(r12) ~ for Lennard-Jones
plus Quadrupole Fluid at KT/e = 1.277, po^ = 0.85,
Q/(ca5)1/2 = 0.5
R/SIGMA G1000) G(200) G(220) G{221) G(222) G(4C0)
0.800
0.825
0.850
0.875
0.900
0.925
0.950
0 .975
1 . 000
1 .025
1 .050
1 . 075
1.100
1.125
1.150
1 . 1 75
1.200
1 .225
1 . 250
1.275
1 . 300
1 . 325
1 . 350
1 .375
1.400
1 . 425
1 .450
1 .475
1 .500
1.525
1 .550
1 .575
1.600
1 .625
1 .650
1 .675
1 .700
1 . 725
1 .750
1 . 775
1 .800
1 .825
1 . 850
1.875
1 .900
1.925
1 .950
1 .975
2.000
2.025
2.050
2.075
2.100
2.125
2.150
2.175
2.200
0.0
0.0
0.0
0.002
0.027
0.184
0.592
1.213
1.916
2.392
2.627
2.60 1
2.437
2.255
1.998
1.757
1.566
1.394
1.248
1.111
1.023
0.928
0. 838
0.789
0.749
0.720
0.706
0.692
0.664
0.664
0 • 666
0.665
0.67 1
0.689
C . 71 0
0. 732
0.769
0.80 1
0.834
0.877
0.933
0.988
1.026
1.071
1.128
1.16 1
1.167
1 . 1 «2
1.206
1.223
l . 22 8
1.234
1.222
1.188
1.169
1.130
1.113
0.0
0.0
0.0
0.000
0.003
0.006
-0.000
0.007
-0.003
0.003
0.002
0.003
-0.OC1
0.007
0.001
-0.015
-0.003
-0.007
0.004
-0.009
0.003
-0.004
-0.004
-0.009
-0.011
-0.004
0.004
-0.000
-0.003
0.002
0.003
0.005
0.005
-0.001
-0.003
0.002
0.009
0.006
0.012
0.006
-0.000
0.009
0.003
-0.000
0.006
0.002
-0.014
-0.007
0.001
-0.000
-0.004
-0.000
-0.004
0.005
0.007
0.002
0.002
0 .0
0.0
0.0
-0.001
-0.011
-0.060
-0.151
-0.290
-0.434
-0.488
-0.453
-0.414
-0.357
-0.284
-0.243
-0.168
-0.142
-0.112
-0.085
-0.076
-0.077
-0.058
-0.038
-0.025
-0.019
-0.017
-0.020
-0.020
-0.021
-0.011
-0.001
-0.005
-0.019
-0 .007
-0.004
-0.014
-0.021
-0.019
-0.012
-0.007
-0.005
-0.014
-0.023
-0.016
-0.026
-0.035
-0.018
-0 .010
-0.008
-0.002
-0.002
-0.005
-0.012
-0.007
-0.008
0.017
0.007
0.0
0.0
0.0
-0.000
-0.005
-0.031
-0.090
-0.189
-0.252
-0.284
-0.303
-0.231
-0.181
-0.137
-0.093
-0.086
-0.076
-0.072
-0.057
-0.051
-0.042
-0.029
-0.032
-0.012
-0.014
-0.016
-0.013
-0.018
-0.007
-0.008
-0.007
-0.010
-0.011
-0.006
-0.009
-0.008
-0.007
-0.009
-0.006
-0.006
-0.005
-0.003
-0.003
-0.005
0.001
-0.012
-0.007
-0.004
-0.002
-0.002
0 .004
-0.005
-0.005
-0.009
-0.005
-0.019
-0.004
0.0
0.0
0.0
-0.000
0.001
-0.006
-0.018
-0.019
-0.054
-0.047
-0.061
-0.071
-0.084
-0.079
-0.041
-0.025
-0.032
-0.024
-0.013
-0.005
-0.028
-0.029
-0.016
-0.013
-0.011
-0.003
-0.008
-0.002
0.002
-0.002
0.004
-0.000
0.003
-C. 000
-0.001
0.004
0.001
-0.007
0.005
0.008
0.001
-0.003
-0.007
-0.015
-0.007
-0.003
-0.007
-0.012
-0.004
-C.003
-0.003
0.006
0.007
0.008
-0.005
-0.004
0.003
0 . C
0.0
0 . C
0 . COO
0 . C04
0 . CO 5
0.011
0 . C01
0.09
0 . Cl 5
0 . Cl 6
0 . C09
0.014
0.030
-0.017
-0.C04
-0.000
-0.COO
0.001
0.001
-0.CO 3
-0.C07
0.002
-0 . C03
-0.005
-0.Cl 4
-0.CO 7
-0.010
-0.C02
0.000
-O.C02
-0.003
-0.001
0.000
-0.006
-0.C07
-0 . CO9
-0.003
-0.003
-0.CO 4
0.006
0.013
0 . CO6
0 . CO 3
0 . C04
0 . C09
0 . CO4
0.0
0 . Cl 1
0 . COO
0.001
0 . COI
0 . C05
0.000
-0 . C04
-0.CO 2
0.0

299
TABLE HI (Continued)
9/SIGMA G(000) G(200) G(220) GÍ221) G(2?2) G(4 CO)
2.225
2.250
2 .275
2.300
2.325
2.350
2.375
2.400
2.425
2.450
2.475
2.500
2.525
2.550
2.575
2.600
2.625
2.650
2.675
2.700
2.725
2.750
2.775
2.800
2.825
2.850
2.875
2.900
2.925
2.950
2.975
3.000
1 . C66
-0.001
0.002
1.022
0.006
-0.007
1.002
0.006
-0 .00 1
0.976
-0.008
0.001
0.95 0
-0 .005
-0.004
0.932
-0.005
-0.003
0.922
-0.007
-0.013
0.906
-0.001
-0.012
0. 900
-0.OC1
0.003
0.890
-0.009
0.006
0. 880
-0.000
0.003
0.876
-0.003
0.005
0.832
-0.001
-0.002
0.882
0.002
-0.004
0.887
-0.000
-0.010
0.901
-0.002
-0.006
0.918
-0.002
0.001
0.934
-0.001
-0.007
0.945
0.000
-0.008
0.963
0.001
-0.000
0.972
0.005
0.002
0.998
0.003
0.004
1.004
0.008
0.005
1.029
0.014
0 .00 1
l . 039
0.016
0.003
1.054
0.009
-0.001
1.064
0.005
0.002
1.068
-0.006
-0.000
1.07 1
0.002
0.002
1.064
0.002
-0.003
1.071
-0.000
-0.000
1.073
-0.002
0.004
-0.002
0.007
0.008
-0.006
-0.001
0.005
-0.004
-0.000
-0.002
-0.003
0.003
-0.001
0.005
0.000
-0 . CO4
0.001
-0.002
-0.C05
-0.000
-0.002
-0.C06
-0.001
-0.002
-0.005
0.002
-0.005
-0.C04
0.005
0.00 1
0.003
0.001
0.006
0 . CO 3
-0.002
0.004
0. COI
-0.004
0.003
0. C04
0.003
0.003
0.000
0.002
0.005
-0.002
-0.001
0.003
0.006
0.001
-0.003
0 . C02
-0.004
-0.005
0 . CO 5
-0.004
-0.006
0 . C02
-0.010
0.003
-0.004
-0.004
0.006
-0. C 0 4
0.002
0.009
0.00 0
0.005
0.007
0.006
0.004
0.001
-0.COO
-0.001
0.009
-0.003
0.001
0.003
-0.C02
0.002
-0.007
0.001
-0.000
-0.005
-0 . COO
0.005
-0.002
-0.001
0.002
0.002
-0.C05
0.000
-0.004
0 . C02
-0 .002
-0.010
0.00 2

rvivr\.fvM^fvivt\j>->“>-*-*->->->->- — >->->->->->-.->-oocoooco
i^h.-->-oodo>t;>üií.>nao3(ijai'js'JsoO'(ja'ü)Uiaiau>^^^uuiuuM\)i\)N'",->'>'Oooon;'io>fJií)aa>a)aj
o-vUTKONOiMoiuiivoNUirjONO, K:oNoitvo->iyiivo'vjoir^O'gu*i\JO->iuiroo--oui|^ONOiivc>'>imK)o^uiivo
ouiouiouiouiouiouiou'iocnocroUioUiooiouiouiouiOLnoinouiou'otnouiouiouiouiou'ioU'OUiouio
71
\
in
t—i
O'
?
>
III I I I I III II III I I I
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOO
000000 c.' 0000 >-000000000000 ooo»-rvooooo>-ooooo>-urv;>-oo>-.->->->-ooo
-j>-oNf'j*--i^j>-->if\:'>-rv>-pa;-Ni>-^j>->-o>j>~*-ucruoi\>>-GPor\;ruorvoi>->-tv*-aPO'0'0'P>-u.O'o>-o
o
p
rvj
o
II III I I I I I II III I I I I Í I
oooooocooo ooooooooooooooocoooooooooooooooaooooooooo
o o o o o o
CD
cooogoc. ooooooooocogcoooooooOoocoocOgogcoooooooqoooooooooo
OOOOOOOOOOOOOOOOOOOOOOOOOOOOoOOOOCOOOOOOOO>-O>-Or0>-fV)K:OOOO
ro^iv>-o>->-oj.f:>--oO'CT'>->-wffi>-^.c>>“Uiai>->-ou. >-o — ui>-^iv'^i(xuio(jiO'vnvcroov£)>j^-viuit>-or\)o
p
III I I I I I I I I II II I I I I I I I I I I
oooooooooooooooooooooooooooooooooooooooooooooooaoooooocoo
cOooooOooooooooooooooooooooooooooooooooooooooooooOooooooo
00000000000000000000000000000000^-00000000000l\/0>-0000000
â– PUi>-u>o>->-i\)0'>->-ot>juioivc7'rocjoiv(jiP'>-a3.Po>-p>>-0'L>Joo'avOCDroi\)\oCu'*n>iuiO'4ui-f>-Nia50J*->-o
c,
p
rj
ru
I I I I I I III I
OOOCOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOCOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOO
ooooo ooooooo>*>*oo>-oooooooooooooo>*oowi\)i\jo>*MoiuM>uc>aistMooo
cr. n •- od a piv>p*i\;fouc>-i\3 0'-PUicruiuiu!U)outj-jp-p>-os>-o->j'^oo>-o>-Uj'-rv)rv.ui^i^uiuioop-tvo
CD
p
p
O
I I I I I lililí | II
OOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOGOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
o>-ooooo>-ooo>-oo>-oooooooooooo>-ooooooo>->-o>->-oOoooro-PPUiui\;ooo
ai>->-rooui'>i-PO'0'^ou'0'o-Pa->j>-o.>-ojívujoojO'Ovoc»io>-NOJCM\;C7'voo'PO''n'Po.ppoiuiüio>-tJ>-o
CD
p
p
I I III I I I I I I I I I I I I I I I I I I I I I I I I I II
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
CD
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
oooooo*-ooooooooooooooooooooooooooooaoo>-o>-ooo>->-Go'-o>-ooo
OPUIS-*ISKP>-l\iiO>-OPPO'PPSOlOJOJUSU)í>0'UIOü',“>->-Oity>ID"JUO''-WP4-OPlüalAlS>-UI>'>-
p
p
r\j
Values of R420(rl2) - for the Fluid
of Table HI

r\j iv fv> rv ro ro rv r\
301
TABLE H2 (Continued)
k/SIGWA
2. 2¿ 5
2. ¿5 0
2*275
2.300
2.325
2.350
2.375
2.40C
2. 425
¿.450
2.47 5
2.50 0
• 5
• 55 0
. 575
. óO 0
• 625
• bo 0
.6 75
.70 0
2*72 5
2. 75 C
2.775
2.600
2*625
2.35 C
2.375
2.900
2.325
2.950
2.975
5.00 0
G(420J
0 . 003
-0.001
-0.003
-0.001
0.005
0.003
-0.001
0.001
0.001
0.001
0.003
0.006
0.008
0 .005
C .002
0.005
-0.001
0 .00 1
-0.002
0.004
0.006
0.003
0.005
0.002
0.001
0.001
0.003
C .00 1
0.002
0.005
-0.001
-0.006
G( 42 1 )
0.002
-0.005
0.003
-0.001
0.006
0.005
-G .000
0.002
0.002
-0.001
-0.002
0.000
-0.002
0.002
0.002
0.003
-0.002
-0. OQcl
-0.001
-G .002
0.008
0.002
-0.001
-0.001
-0.002
-0.001
-0.002
0.001
0.001
0.00-+
0.000
o.ooi
G(4220
-0.001
-0.001
0.001
-0.000
-0.001
-0.002
-0.007
- 0.005
-0.001
0.001
-0.000
-0.004
-0.003
-0.003
-0.004
0.001
-0 . OC1
-0.000
0.001
0.003
0.003
0.003
C . 000
-0.003
-0.002
-0.001
0.002
0.001
0.000
-0.001
0.002
-0.007
G( 440 )
0.003
0.011
-0.C02
0.003
-0.002
-0.004
0 .000
0.003
0.001
-0.003
0.006
-0.001
0.002
0.006
0.010
0.006
0 . C 0 5
0.006
0.000
-0.003
0.000
0.003
0.006
-0.002
-0.000
-0.003
-0.001
0.004
0 . 002
-0.003
-0.006
-0.010
G ( 4 <4 1 )
-0.008
0.002
0.002
0.006
0.000
-0.000
0.003
-0.005
-0.004
-0.005
-0.003
-0.000
0.002
0.001
-0.001
0.004
0.011
-0.001
-0.000
-0.000
-0.001
0.000
-0.000
0.004
0.006
0.002
0.001
0.003
0.0 00
0.000
-0.003
0.002
G(442)
0.004
0.003
0.000
0.004
C .000
0.002
-C.004
-0.004
-0.002
-0.002
-O.OOd
0.002
-0.008
0.001
0.000
-0.002
0.001
-0.004
0.001
-0.003
-0.006
-0 .004
-0 .OOo
-0 .003
C .001
0.0
-0.001
0.001
0 .001
0.0 08
0.0 01
-0.003

o
»X)
vO
'-iro-isi'0'-<-'','>'\Ji'Os
000-‘0-<-'0'r)0M-<-i00000OO00O0 "«OOOOOOOOOOOOOOOOOO-iOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
o
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
I ( I I I II I lili I I I I I lili II I
T3
•H
a
r—t
fe
J=
S-i
o
o
s
o
o
O-iMO(0'00'-<'0-CvJ^Nrv|i0^O-if0í\IO-if\J(M<>-<)-<ívt(\JC\J-«Oif)s}-Ov0-''-,í> 000-<0-.rjO'-<0'-<00*H-<0000000000000000000000000000-<000->000000
000000000000000000000000000000000000000000000000000000000
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
I I I I I I I I lili I I I lililí III
o
vO
0(Mro*"OM^’y1N.0-*-.t0^0O<}-(T»'0Or<)'0'0"0v0'0<>--\J'AJS
OOOOOO-c-t-'-i-.OOOO-i-iO-.OO OOOOOOOOO—OOOOOOOOOOO — O-iOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
o
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
III I I I I I I I I I I I I I III III II III III I
CSI
r—I
—
CO
d
b
rn
vD
M
5
1—1
CM
w
60
O
03
CO
CQ
1
H
H
CM
O
I—I
sr
bC
o
o
vO
KD
OO--»C\JC\J-irvJ'y)s'\JN
ooo-«ooo-«n-H-»oooooo-*^oooooooooooooo- OOOOOOOOOOOQOOOOOOOOOOOOOOOOOQOOOOOOOOOOOOOQOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
I I I I t III I I I I I I I I I I I I I I I I I I I I I
<*
0(MuDO'lí)-><ífvlOOlf)',l^i/)vONvOrO'£)CO->-<''\inif)^)- 000000"0^0--00 — 00000 0j000-<00000000000o0000-<00'-i-‘00-<0000000
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
II I I I I I I I I I III II II I • I I I I I I 1 I I I I I
-n
-S'
S
v5
o n ii) n '0 o o n to s o rj in "o ^ -i o o "i -< o o o cm ro n in -o ir> m -< t) cmo rn — -o n o n o n o s -< to n
O OOOn^OCMO f\jo-- ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
II I I I I I I III I III
<
X
u>
►—<
s
LX
oinoinoinoinotnoinoinouioinoinoinoinoinoinoinoinoinoinoinoinoino'noinoinoinoinoinoiiio
ONinNO!Mlf)NONinNOMinNOiMinNOMIi)NOMflNON:ONO'MinSOMIfiNOWinSOMinNO(MinK)NinSO
fOfflcO'l)(M)'!>l7'OOOOH-1_-ifuMr\|M^.'Ol»)n
Oj iv r rv tv rv rv rv rv rv rv tv iv rv tv rv rv tv tv tv tv tv tv rv rv rv rv tv tv rv tv tv
J)
N
O'.
0'U'OvO>oaaaja-sj-M'siNO'000'ü'iuiuiui-^4>'í>-^uUiUCj(vtvK!
*-«
o -'t oí tv o -vj oi tv o s u iv o c. iv o -ni ui tv o 'j ai r: o -g ui ^ o -si cti tv
G
ooiocriooiocnooiouiouiooiou'ocnouiooiooiouioooai
>
ill i i i i ill i
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
G
OOOOOOOOCOOOOOOOOOOOOOOCOOOOOOOO
CGCOOÃœOOOOGOÃœChOO o c O O CO oooooooc
e
N-(>utvrvi-otvou>-rv»-oioiooo'rvotv'“0'iO rv O'«- ro rv v. O'
I 1 I 1 1 • I 1 i 1 1 1 1 1 II 1 1 1 1 1 (III
ooaooooooooooooooooooooooooooooc
G
OOOOOGOOOOOOOOOOOOOOOOCOOOOOOOOO
0000000000000000>-000000>-0000000*-
â– 1*
U*-C'-s|CDU!tVtV0J>“^>-0''--trv>-*->-OO0JfV'Ci-vj4;'0JU'“,-UUl
1 III III II 1 1 i 1 1 1 1 1 1
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
G
O'
o
oooc&ooooGooooooooooocoooooooooo
oooooooooooooooooooooooooooocooc
o
>-o-rvoj>-CD>-oojui'-^rvuicr*-cno>--^-^>“»“^'i>o>-'vj>->-o>-
w
1 1 II 1 1 1 1 1 1 1
OOOOOOOOOOOOOOGOCOOOOOOOOOOOOOOO
G
a
rv
OOOOOOOOOOOOOOOOOOOOGOOOOOOOOOOO
OOOOOOOGOOOOOOOOOOOOOOOOGOOOOOOO
o
0Jcno<-*-0'.uoutvrvoi->“o u-p-rvrvo'oo'- -^oirorvivroa-p-tv
V-*
1 III 1 III III III ill
OgOOOOOOOOOOOOOOCOOOOOGOOOOOOCOO
G
»-■»
O'
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
4>
OOOOOOOGOOOOOOOGOOOOOOOOOOOOOOOO
o
•f><-oJurvOi-ouLj*-1-H-u¡-P-rv*-^'-oojuioiv*-iv»-oj.prvi*isa)
1 II 1 1 1 1 1 1 III
OOOOGOOOOOOOOOOOOOOOOOOOOOOOOOOO
G
O'
0
OOGGOOOOOOOOOOOOOOOOOOOOOOOOOOOO
OGO O OO O O O O O O Go *- o O O O O G O O O OGOOOO
a
vGCjO h-01r-rv*->-|V^->-4>vUOGtnU.'-U,-'O''IG U 00 -ti- O' CO TABLE H3 (Continued)

304
TABLE H4
Values of g
(rin) -
g/.nn<>io) for
Lennard-
Jones
UVJU _L¿ 4UU 14
plus Quadrupole Fluid with kT/e
= 0.765,
po3
“5 1/2
= 0.931, and Q/(ea ) '
= 0.707
R/SIGMA G{0 00 >
G(200 )
G( 220 )
G(22 1 )
G(222)
G( 4 CO)
0.800
0.0
0.0
0.0
0.0
0.0
0 . C
0.925
0.0
0.0
0.0
0.0
0.0
0. C
0.850
0.0
0.0
0.0
0.0
0.0
0.0
0.875
0.0
C . 0
0.0
0.0
0.0
0.0
0.900
0.00 3
0.001
-0.002
-0.001
0.000
0. C01
0.925
0.049
0.007
-0.040
-0.020
0.000
0.005
0.950
0.319
0.048
-0.260
-0.108
-0.007
0 . C60
0.975
0.99 3
0.120
-0.745
-0.297
-0.032
0.154
1 .000
1.971
0.154
- 1.309
-0.491
-0.086
0.268
1 .025
2.800
0.123
-1.650
-0 .628
-0.160
0.286
1 . 050
3.165
0.087
-1.651
-0.637
-0.183
0.226
1.075
3.257
0.049
-1.496
-0.622
-0.212
0.103
1 . 1 00
2.989
-0.002
-1 .245
-0.476
-0.194
0 . C64
1.125
2.555
-0.028
-0.939
-0.376
-0.158
0. Cl 2
1.150
2.178
-0.056
-0.709
-0.274
-0. 146
-0.C02
1.175
1.817
-0.067
-0.525
-0.226
-0.094
-0.013
1 . 200
1.512
-0.053
-0.395
-0.160
-o.oe4
-0.C34
1 . 225
1.291
-0.051
-0.271
-0.134
-0.059
-0.C22
1 . 250
1 . 095
-0.042
-0.184
-0.112
-0.058
-0.027
1 .275
0.952
-0.037
-0.138
-0.1 l 1
-0.042
-0.C27
1 . 300
0.854
-0.027
-0.120
-0.107
-0.025
-0.030
1 . 325
0.772
-0.014
-0.091
-0.086
-0.031
-0.036
1 . 350
0.673
-0.020
-0.063
-0.073
-0.030
-0.027
1 . 375
0.635
-0.011
-0.052
-0.065
-0.024
-0.031
1 . 400
0.608
-0.015
-0.032
-0.070
-0.020
-0.035
1 .425
0.554
-0.019
-0.024
-0.056
-0.013
-0.024
1.450
0.539
-0.004
-0.017
-0.059
-0.018
-0.033
1 .475
0.525
-0.012
-0.009
-0.065
-0.012
-0.030
l .500
0.510
-0.002
-0.002
-0.061
-0.004
-0.026
1 .525
0.523
‘ 0.009
-0.008
-0.061
-0.007
-0.016
1 .550
0.539
0.001
-0.013
-0.054
-0.002
-0.006
l .575
0.572
0.002
-0.020
-0.061
-0.009
-0.021
1 .600
0.603
0.006
-0.018
-0.060
-0.009
-0.026
1.625
0.643
0.015
-0.023
-0.066
-0.009
-0.029
1 .650
0.671
0.020
-0.037
-0.064
-0.003
-0.028
1 .675
0.730
0.009
-0.035
-0.064
-0.005
-0.C20
1.700
0. 785
0 . C02
-0.038
-0.056
-0.010
-0.009
1 . 725
0.838
0.007
-0.043
-0.050
-0.008
-0.003
1 .750
0.898
0.007
-0 .049
-0.046
0.001
-0.002
1 . 775
0.96 3
0.011
-0.075
-0.036
0.004
-0.CO 3
1 .800
1.033
0.018
-0.083
-0.036
-0.008
0 . CO 3
1 . 825
1.094
0.012
-0.101
-0.034
-0.005
0.004
1 .850
1.153
0.020
-0.096
-0.028
-0.009
0 . CO9
1 .875
1.19 1
0.015
-0.100
-0.027
-0.007
0 . CO 3
1 .900
1.22 1
0.004
-0.090
-0.021
-0.004
0.016
1 .925
1.259
0.002
-0.079
-0.009
-0.002
0.023
1 . 950
1.283
0.014
-0.051
-0 .006
-0.004
0.018
1 .975
1.308
0.008
-0.036
-0.008
-0.006
0 . Cl 9
2.000
1.308
-0.007
-0.019
-0.020
-0.005
0 . Cl 1
2.025
1.317
-0.002
-0.009
-0.026
-0.001
0.023
2.050
1.322
-0.015
0.006
-0.036
0.0C4
0.013
2.075
1.288
-0.003
0.017
-0.025
0.004
0 . 028
2.100
1.270
-0.003
0.034
- 0 . C 2 3
0.003
0.027
2.125
1.236
-0.013
0.022
-0.015
-0.001
0 . 021
2.150
1.190
-0.006
0 .020
-0.013
-0.008
0.009
2.175
1.130
-0.005
0.023
-0.010
0.005
0.010
2.200
1.073
-0.009
0.021
-0.011
0.007
0 . 002

305
TABLE H4 (Continued)
P/SIGVA G(000) G(2 0 O) G( 220) G(221) GC222) G(4C0)
2.225
2.250
2.275
2.300
2.325
2.350
2.375
2.400
2.425
2.450
2.475
2.500
2.525
2.550
2.575
2.600
2.625
2.650
2.675
2.700
2.725
2.750
2.775
2.800
2.825
2.850
2.875
2.900
2.925
2.950
2.975
3.000
1 . 030
0.001
0.019
0.976
-0.016
0.003
0.932
-0.007
0.017
0.883
-0.003
0.005
0.847
-0.006
0.002
0.814
-0.001
0.005
0.79 1
-0.003
-0.001
0. 784
-0.0C1
-0.001
0.787
-0.000
0 .002
0.788
-0.000
-0.006
0.797
0.004
-0.005
0.795
-0.004
-0.003
0.818
-0.005
0.005
0.851
-0.005
0.000
0.88 1
-0.002
-0.003
0.901
0.001
-0.005
0.930
0.011
-0.010
0.959
0.013
-0.012
0.998
0.008
-0.014
1.028
0.008
-0 .002
1.052
0.007
-0.012
1 . 069
0.006
-0.014
1.096
0.005
-0.010
1.102
-0.002
-0.001
1.111
-0.008
-0.001
1.121
-0.000
-0.006
1.130
-0.002
0.007
1.132
0.001
0.000
1.130
0.006
0.000
1.124
0.005
0.006
1.109
-0.002
0.009
1.104
-0.004
0.008
-0.016
0.009
-0 . COO
-0.009
0.008
-0.C04
-0.008
0.001
-0.011
-0.006
-0.003
-0.Cl 4
-0.007
-0.005
-0.013
-0.010
-0.003
-0.C06
-0.009
-0.003
-0.004
-0.006
-0.002
-0.001
-0.002
0.000
-0.C08
0.005
-0.002
-0.008
0.006
-0.001
-0.007
0.001
-0.004
-0.014
0.006
-0.003
-0.CO 7
0.007
-0.001
-0.003
0.008
0.005
-0.C08
0.008
0.001
-0.010
0.000
-0.002
0.00 1
0.005
-0.009
-0.004
0.008
-0.009
-0.C09
0.008
-0.004
-0.CO 2
0.003
-0.001
-0.CO 4
0.002
0.00 1
-0.C07
0.008
0.002
-0.004
0.002
0.0C9
-0.C03
0.009
0.004
-0.003
-0.002
0.002
0.007
-0.004
0.004
0 . C09
0 .002
-0.001
0 . C09
0.005
-0.004
0 . Cl 3
-0.005
-0.008
-0.C03
-0.001
-0.000
0.00 3
0.005
-0.002
-0.001

(^^-►-^.►-ocooíOvü\ovf)a)Oüao3-vi'vi>)-«joO'0-'0'UUjiaicn-p.^í>-t-(>iUuojivirororv*-'->->-oooovoví)vL)'CCBa;üJa
o-sjuiivo-viiT. i\jo->n/iiV'o-sicíirooNu':r\)o^4Uiroo-Nicnr\;osuiMo->j(;iiv.o-gu'r^o-^U)iviO'JU''r'jO-vi'-^r'-io^tcnr\;o
ouioirouioaiouiotfiou'ou'oaoo'oüioüioiiioaouio'jiotroaoüiouiouiouiociioirouiouioiíotro
31
S.
C.
>
lililí I i I I I I l I I I I I I
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOO
oooooooooooooooooc2oooooooooooooooooooooooo>- — — rviMroi-oooooooc
ooooo>-oooHi“Oi-HOPoo^oi-ooooi-oooococooHM'-(i-oo
>"í>o^-e'N)>íja,Ki\.rv->j>-o^-p-rv'>-ouo-siOjU;croiT'0’rv>-f't>u-Ni>-^fv,vü.pMüiuttau>i'f'Cr-<:uia^c¡
o
â– p-
IV
o
I I I I I I I 1 I I I I I I
OOOOOCOOCOOOOOC'OOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOCOOCOOC
OOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOCOCOOOOOOOOOOOOOOOOOOCO
ocooocooocooooo>->“'-'“Oooo^*O'-oooooooooo*-ooo*-*-rv).^a^i'^a)0Ui*-oo
vij-^>-.p.f>uitra*-^'-MU)*viotv'-aio>f!Ui(>j'-ocro-«jr\.'o*-'Ca5'4-p-0'>ofvivot*Jooc'iV)CDiv'(>J(jj'f>Morv'(>jO
c,
p
ro
i i ii i i i i i i i i i i i i i i i i i i i i
oooooooooooaooooooooooooooooooooooooooooooooooooooooooooo
oüooooooooooooóooooooooooooooooooooooooooooooooooooooootoo
OOOi-OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCIVrVi*-IV'IV>“>->—►-oo
ODUi-t'»-a)\ouiva)^ioi^uia'UO-f>Mfv)'-'-^iUirvutn>-'-F>OjUiun>JUi-p-i\>OJi-i\)uie-c)ouii>a3U'UiO'gi-^j>-
Ci
&
rv>
<
03
h-*
c
ro
en
o
»-h
0Q
-o
N>
?~
I—*
ls>
H
1
H
03
>
cr
0Q
w
UJ
h-*
4
r1
0)
•O
N3
m
O
O'
oc
PC
f-t
Vn
tv>
I II I I I I I I I I I I I I I
ooooooooooococooooooooooooooooooooooooooooooooooooooooooo
oooooooaooooooooooeoooooooooocooc, ocoooooooo«-'~fvLj(vLí\jooooooo
OOOOOOOOOOi"-HHOOOOOOOOO'“OOOOOOOOi->-0'-0>-cMI>(ÍUlNillUi-CIMwa'“0
í-«c>-M>-Ma,(jlíiCBí'^rüHt.'-0''UH>iou.uaia'a^Q3(í'MsUMvi)^aoui'-Mu'0'a'-osui*0'N
c
p
p
o
lili I I i I I I I I I I I I I I
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOGGOOGOOOOOOOOOOOOOOOOOOO
oooooooooooooooooooooooooooooGoooooooooooooo>->-Mfvfv>-ooooooo
0000 0>-C»-C0000000'-G000,-0000O00O0<-0>->-G00>->-r\j-(>'^'^< -J V. IV O C oo
GMro»ocoucno'ji\)GUioDroO'orv)^*ü'*-C'*-ün;¡ojMi\3^-f>>-r'ooo-i>o4i-G->iroa)-^o>o>J'p'i\)üu*ja>vOí>'0^
O
4>
i-n
o
â– y
(II
*ri
I—1
c
M-
cv
o
i-n
i I I I I I I I III lili I
OOOOOOOOOOOOOOOOOOOCOOOOOOOOCOOGOOGOOOOOGOOOOOOOOOOOOOOOO
G
OOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOCOOOOOOOOOOOOO'-G'-OOOOOOOOO
0>'OOOoOoGOOOGOOOOOOGOOGOOOOCHHGOCOCOi-OHh.M^uUHI»li(3fU coü'CuM*->-->iüiüirooia/>-í>,-GU-f:',-'-'üuiM>-S'-orv>-orjO''(juCD»-ou'úCíi'0-i!-iv-f>uj>*Jrvo'0>j-P'-'i>-uio
•(>
IV

u IV) iv i\ rv rv iv iv n 1 iv iv rv
IV
ro
IV
IV
ro
rv
ro
rv
rv
IV
ro
ro
rv
rv
rv
rv
rv
rv
rv
IV
*
*
•
»
•
%
•
•
•
•
•
*
•
V-
•
•
•
•
»
O'0G'0'C/C0acCCCS.-'i>J~v
0
o
O'
O'
U!
Ol
0Í
U)
p
Ui
Oj
Ui
Oo
r
ro
ro
o -xi iU‘ rv o -v cn ro
o
01
rv
o
O'
rv
o
~vl
cn
M
o
•vl
U1
ro
o
-nJ
cn
ro
OUIOWOUIOUIOOIOO
o
cn
o
cn
o
cn
c
U1
o
cn
c
cn
o
cn
o
cn
o
cn
o
cn
II lililí I III III I I I I I III
OOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOcjOOOOOOOOOOCCOOOOOOOOOOOOOOOC
0'P>-rvfvcn,-cn-Nicnpprvua>Pivrvuoivpi\>>--NiP~>iPrv>-cna'
XI
\
U!
►“>
c
>
Cl
p
rv>
o
i i i i lili lili I i l l i
ocooooooaooooooooooocoooaooooooo
ooooooocoooooooooooooooooooooooo
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
ojtno.~i-rvl>jrvoincnpc*ju!rvivrvu*-rv)>-po»-rv'uivcnu.orup
e
p-
M
I I I I III I III I
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
c
OOOOOOOOOOOOOOGOOOOOOOOOOCOOOOOO
ooooooooo ooooccoooooooooooooooo
oiroro*-(>jrv>'-'Oi>iuiororviN>rvrv)iv'*-'-u.oj'“
p
K)
fv
III I I I I I I III I I I I I I I I
OOOOOOOOOOOOOGOCOOOOOOOOOOOOOOOO
cooooooooooooooooooocooooooooooo
oooogo>->-oooooogooocoooocooooo>-oo
CD>-rvvCCnU'0'Ua>,--P‘'J(jJCnt»J-''JUPfVPCniVP*-''J>->-a'\OP~'JP
c>
p
p
o
I I I I I III I I I I I I i I I I III I
GOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOO
Cl
OOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOI-OOO
p
p
I I I I I I I 1 III I II
oooooooooaooooooooeaoooooooooooo
n
ooocoooooccooooooooooooooooooooo
GGOOOOOO^-OOGOOOOOOOOOOOOOOO OOOO
UUH'-i-Ul'-M'-OOMUPMUlUlO'UWM'JPPPOiPW ►— ~>l CD •—
p
p
V
TABLE H5 (Continued)

(vj^.H.^-^-oooovcuO'í. 'Oaa¡a<03-^N->j-Ni0'ff'O'auitJiaiui^-^-*í'^u;U:uurv.ivNir'j'-'- — ►-oooov<)>0'0>í;cx)Obodc
o-vjuii\joNUiiv.O'jü:wo->)0^. 0'vi(jirvo*vi(ji(\-o-Nicni\.o-v|üiiv;0'vjUi(\ic
o en o en o en o en o en o en o en o 01 o 01 o a o oí o u- o en o en o ai o ® o en o X'
N
01
*—»
2
>
I II III lili lili III
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
©OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
ooooooooaoo>-ooooooooaooooooooooo oohhoooooloooooiuoooo
■-CDo->jocr*“>i!-C'Ui,“í-'ivNocr-^X'uivorv>-o.'-oiO'ooiu-vioJo1 n. o> pi p >“ o ■—sj en a* cd en o rv iv e* -vi ro rv> o
c.
p
p
OJ
III 1 I I I I I I I I I I I I III I I II
o o o o o o o o O G o O o O o o O O O O O O O O O O C C O O O O O O O O O O O O O O O GO O o o o o o o o o o o. o
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOO
ooooooooao*-o>-oooooo^oooooooooooo OOOOOO 0^0>-0|\;00000
ojNr\:a:voooGJnot¿rvtn'Go*-o,--i>cn-^o^io(JjM,i>>-ro>“,_eni\.ui >- en ® cr p cu i'jroPcnrvccaapO''-
p
p
p
I II lili I II I I I I I I I I I I I I I I I I I I I
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
OOOCOCOOOOOOOOGOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOO
OOOI“OOOOOOOOOOOOOh»>-ü>-oOOGOOOOOOOOOOOOOOOOMuPSÜ(JB1KOOO
->iPUi»-e*pencrcn'jei.'t\)ocn'0-vji'ooiNiOvC'---'ií\)a>enr\>r'jru-genvoppf'jeorv>0'PPoe>irusenekjOop*-enouCjo
c
O'
©
o
I l l lili lililí lili i l l i l I i I
COOOOOOOOOOCOCOOOGOOOCOOOOOOOOOOOCOOOOOOOOOOOCOOOOOOOOOOO
ooooooooooooooooooooooooooooooooooooooooooooooo'-oooooooao
oooooooooooooooooo»-ooo>-oooooooo>-o*-ooooooo>-or\;ooen'-cBen**oo
pujC'or\.Proi-rvpajP(j¿0''>-oji\;cr'>-'-oaocn'“Uj,-'“r\jUiOouiu'0'ro[vC'0'>-foena>crPON'- (Ti
en
M
O
I i I I i I I I III lili I I I l I I I I I I I I I i
ooooooooooooooooooooooooooooooooGGOoooooooooooooooooaoooo
OOOOOOOOOCOOOOOOOOOOOOCOOOGOGOOOOOOOOOOOOOOOOOOOOOOOOOOOO
oooooo ooooooooooooooooooooooooooooo»-ooP->-*r\oW(\jenNO'~g'jP^oo
iíms>imo U'-P'0'JP®oj*-rvj^i,-ooo'üo.;>-0'ü'>“Oi>-',“Pcjf\:'ÍJÜio>-uioiPüiocDooojccO'OoO'en>-
0'
o-
o
<
h-1
e
0)
C/5
o
f-h
OQ
4S
Tí
I—1
no
H
CU
cr loo
i—1 cñ
(D cp
H
>
co
©<
m
re
CP
OJ
o
00
I I II II I lili I I I I I II III lililí I I I I I I I
oooooooooooooooooooooooooooooocoooooooooooooooooooooooooo
ooooooooooooooooooooooooooooooooooooooooooocoococoooooooo
ooo>-*-ooooo>-ooo*-oooo>-oOf-oooooGOCcoo>-oooorooo*-f\j>-'-t'oPuiro^*oo
ppprooiOPcvü'0>-->iusp'OU:®r'je>.ü'-'inoopoaP'p->iofooo'-MuiMO'®CjO"ororo>-oiPM,-aooo
en
O'
a
o

309
TABLE H6 (Continued)
R/SIGMA
2.225
2*250
2» 275
2* 300
2* 325
2 * 3 o C
2.375
2.400
2. 425
2.450
2.475
2.50 0
2.52 5
2.550
2. 575
2.60 0
2.625
2. 650
2.675
2.700
2*725
2.75 0
2.775
2.6 0 C
2.92 5
2 * 85 0
2» 875
2.30 0
2 • 92 5
2*350
2.975
3.00 0
G( 443 )
-0.002
0.001
-0.001
0.005
-0.002
0 .002
0.001
C . 0 0 4
0.000
0.0 12
0.004
0.005
0.002
0.002
-0 .006
-0.004
-0.009
0.001
-0.002
-0.000
-0.000
-0 .005
0.003
- 0.0 0 d
-0.001
0 .002
-0 .003
-0.000
0.005
-0 . 0 0 fc
-0.001
0.006
G{ 444 )
-0.004
-0.001
-0.004
0.001
-0.007
-0.010
-0.004
-0.005
-0.006
0.001
0.002
0.001
0.006
0.003
0 .004
-0.000
-0.002
-0.007
-0.006
0.000
-0.002
-0.002
0.002
- C . 0 0 3
0.000
0.005
0.004
-0.003
-0.000
-0.003
-0.003
0.002
G(600 )
-0.002
-0.005
0.000
0.003
0.003
0.002
-0.002
- 0.004
-0.003
0.004
-0.000
-0.001
-0.002
-0.004
0.001
0.001
0.005
0.001
0.005
0.00 1
0.0
0.008
0.005
- 0.004
0.0
-0.002
-0.001
0.002
0.001
-0.002
-0.002
0.002
G(620)
0.007
0.001
0.002
0.000
0.0 04
0.003
0.004
-0.000
0.002
-0.002
0.004
-0.003
-0.002
0.005
0.005
0.006
0.010
0.004
0.010
0.002
-0 .005
0.002
-0.00 1
— 0 • OOo
-0.007
0.000
-0.001
0.004
0.005
0.001
0.003
0.006
G(640)
0.005
0.001
0.005
0.002
0.003
-0.004
-0.002
-0.008
-0.005
-0.002
-0.003
0.005
0.002
-0.000
-0.003
-0.003
0.000
-0.001
-0.003
-0.002
0.002
0.006
0.003
0.002
-0.004
-0.008
-0.003
-0.002
0.001
-0.004
0 . C04
0.001
G(660)
0.006
-0.003
-0.001
-0.001
0.001
-0.000
-0.003
-0.003
-0.001
0.010
0.004
-0.003
0.004
0.005
0.002
0.006
-0.000
0.002
0.006
-0 .002
-0.001
-0.009
0.001
0.00a
0.003
0.008
0.005
0.006
-0.000
-0.004
-0.001
-0 .002

310
TABLE H7
Values of SqqqU^) ~ for Lennard-Jones
plus Quadrupole Fluid with kT/e = 1.294, p= 0.85
and Q/(£Q5)1/2 = 1.0
R/SIGMA G(000) G(200) G(220) G(221) G(222) G(4C0)
0.800
0 .825
0.850
0.875
0.900
0 .925
0.950
0.975
1 .000
1 . 025
1.050
1 .075
1.100
1.125
1.150
1.175
1 .200
1 . 225
1 . 250
1 .275
1 .300
1 . 325
1 . 350
1 . 375
1 .400
1 . 425
1.450
1 .475
1.500
1.525
1 .550
1 .575
1 .600
1.625
1 .650
1 .675
1.700
1 .725
1 . 750
1 . 775
1 .900
1.825
1 .850
1 .875
1 .900
1 .925
1.950
1 .975
2.000
2.025
2.050
2.075
2.100
2.125
2.150
2.175
2.200
0.0
0.0
0.0
0.0
0.000
0.000
0.005
0.001
0.064
0.015
0.326
0.061
0.872
0.1 46
1.603
0.213
2.167
0.218
2.48 1
0.173
2.575
0.112
2.440
0.054
2.250
0.008
2.057
-0.024
1.84 3
-0.040
1.626
-0.054
1.425
-0.068
1.295
-0.064
1 . 156
-0.063
1.040
-0.066
0.997
-0.064
0.897
-0.050
0.853
-0.049
0.807
-0.039
0.76 7
-0.023
0.747
-0.017
0.713
-0.014
0.691
-0.010
0.686
-0.008
0.694
‘ 0.000
0.690
0.002
0.696
0.004
0.724
0.012
0.72 5
0.019
0.751
0.013
0.785
0.0 10
0.818
0.019
0.859
0.021
0.885
0.015
0. 928
0.021
0.952
0.022
0.983
0.023
1.033
0.026
1.074
0.018
1.096
0.014
1.128
0.007
1.174
0.005
1.185
0.002
1.197
-0.009
1.201
-0.009
1.204
0.002
1.202
0.001
1.169
-0.008
1.145
-0.008
1.124
-0.012
1 . 088
-0.013
1.077
-0.013
0.0
0.0
0.0
0.0
-0.000
0.0
-0.007
-0.001
-0 .067
-0.026
-0.318
-0.124
-0.823
-0.301
-1.372
-0.507
-1 .667
-0.617
-1 .685
-0.630
-1.584
-0.599
- 1 .305
-0.496
-1 .046
-0.416
-0.850
-0.344
-0 .649
-0.287
-0.500
-0.247
-0.388
-0.203
-0.306
-0.172
-0.228
-0.144
-0.176
-0.131
-0.151
-0.118
-0.112
-0.1C8
-0.097
-0.095
-0.074
-0.093
-0.065
-0.084
-0 .054
-0.079
-0.037
-0.082
-0.034
-0.078
-0.021
-0.082
-0.031
-0.080
-0.034
- 0.082
-0.036
-0.082
-0.019
-0.088
-0.036
-0.071
-0.035
-0.064
-0.041
-0.059
-0.056
-0.054
-0.071
-0.048
-0.074
-0.053
-0.087
-0.037
-0.091
-0.030
-0.081
-0.018
-0.081
-0.012
-0.079
-0.008
-0.058
-0 .007
-0.055
-0.008
-0 .04 1
-0.013
-0.026
-0.008
-0.020
0.001
-0.011
-0.004
-0.005
-0.013
0.015
-0.014
0.007
-0.012
0.015
-0.015
-0.003
-0.018
-0.019
-0.015
0.003
-0.011
0.0
0 . C
0.0
0. C
0.0
0.000
0.000
0. CO4
0.002
0 . C22
0.005
0.092
-0.008
0.250
-0.054
0.382
-0.106
0.402
-0.117
0.245
-0.137
0.264
-0. 147
0.178
-0.138
0.100
-0.124
0 . C67
-0.113
0. Cl 0
-0.103
-0.001
-0.095
-0 . Cl 5
-0.085
-0.004
-0.061
-0.C35
-0.050
-0.C45
-0.048
-0.C47
-0.037
-0.C53
-0.045
-0.C55
-0.033
-0 . C43
-0.035
-0.C52
-0.036
-0.C51
-0.028
-0.C47
-0.032
-0 . C49
-0.016
-0.045
-0.012
-0.035
-0.008
-0.031
-0.003
-0.028
-0.015
-0.C34
-0.010
-0.020
-0.005
-0.012
-0.005
-0.024
-0.005
-0 .C21
-0.006
-0.Cl 5
-0.003
-0.C 0 2
-0.002
-0.C09
-0.0C4
0 . CO 1
-0.006
-0.CO 1
-0.000
-0.003
-0.002
0 . C06
-0.016
0.019
-0.010
0 . C 2 0
-0.007
0 . C33
-0.011
0.024
-0.004
0 . C 3 3
-0.000
0.026
-0.010
0.024
-0.009
0.025
-0.001
0.023
0.007
0 . Cl 7
0.007
0.021
0.004
0.004
-0.003
0 . C

311
TABLE H7 (Continued)
R/SIGMA
2.225
2.250
2.275
2.300
2.325
2.350
2.375
2.400
2.425
2.450
2.475
2.500
2.525
2 .550
2.575
2.600
2.625
2.650
2.675
2.700
2.725
2.750
2.775
2.800
2.825
2.850
2.875
2.900
2.925
2.950
2.975
3.000
G(000 )
1.046
1.016
0.999
0.971
0.957
0.941
0.921
0.916
0.902
0.902
0.897
0.908
0.953f
0.96 0
0.967
0.974
0.993
0.998
1.008
1.021
1 . C39
1 .057
1.067
1.082
1.086
1.091
1.093
1.106
1.107
1.110
1.114
1.118
G ( 2 0 C >
-0.009
-0.009
-0.013
-0.010
-0.007
-0.004
-0.001
-0.000
-0.002
-0.001
0.003
0.002
0.002
0.003
-0.000
0.004
0.005
0.006
0.011
0.006
0.006
0.001
0.002
0.006
-0.000
-0.004
0.002
0.003
-0.002
-0.000
0.004
-0.006
G(220 )
-0.001
-0.001
-0.006
-0.005
0.002
-0.012
-0.013
-0.009
-0.000
-0.004
-0.003
-0.005
-0.009
-0.007
-0.016
-0.015
-0.006
-0.002
-0.010
-0 .025
-0.009
-0.001
-0.001
-0.004
0.005
0.008
0.007
0.0
-0.004
-0.005
-0.001
-0.007
G(22 1 )
-0.010
-0.004
-0.001
-0.002
0.003
-0.004
0.002
-0.000
-0.002
0.000
-0.006
-0.007
-0.006
-0.007
-0.010
-0.002
-0.004
-0.004
-0.010
-0.010
-0.011
-0.007
-0.013
-0.003
-0.006
-0.007
-0.002
-0.000
-0.007
-0.008
-0.008
-0.010
G(222)
-0.003
0.006
-0.000
-0.001
0.000
-0.003
-0.002
0.001
0.001
0.003
0.001
-0.002
-0.006
-0.008
-0.005
-0.003
-0.001
-0.008
0.002
0.004
-0.001
-0.005
-0.003
0.003
0.002
0.005
0.004
0.001
0.003
0.003
0.003
-0.007
G(4 CO )
0 . C
-O . C06
-0.CO 5
-0.004
-0.012
-0 . Cl 2
-0.C08
-0.015
-0.015
-0.C05
-0.CO 7
-0.010
-0.013
-0 . Cl 0
-0.C04
-0.C07
-0 . CO6
-0.003
-0.CO 3
-0.CO 6
-0.000
-0.CO 1
0 . COI
0 . COO
o. coe
0.004
0 . C 0 5
0 . C08
0 . C09
0 . C 0 4
0 . C06
0.008
T
Due to a programming error the values of §QQQ(r) i-n this table for
r > 2.5a are in error.

roi\)i\ji\) fvf'jror'jfv •->—>—<-»->-occooooo
fja.a-Ki-ocoo^iüC'üoittüaas-'issO'O O' o m in ui 01 p p pp'UOJUiUj|vivivrv,-'--,->-ooooi£MO'OvocDa>aa
o -vj u¡ rv o ^ en iv o -nj on iv o vj 01 iv o o' iv o o< ro o "J in iv o •'■j ir iv o •vi on iv o ~>i m iv o -%j in rv o '•j in iv o *>j in iv o ~*j u» n o
o X
s
►—<
Ci
s:
>
â–  i i i i i i i i i i i i i i i l i i i i i i i i i i i
OOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
c
oooooooooooooooooooooooooooooooooooooooooc*->-'-'>-r\)iv)*“>“Ooooooo
’-oooocoooooooooooooooci-ooooooooooo'-iiruitfUKMPiuuiO'tuoffiNiO'K-oo
nin'ju-'jporvGcaO'prvruininivivui-cnovoaujin'fluioou'ivvoccpo. -vitva.Ui'-fVi-purjvfjrvir. puuo
p
IV'
o
I I I I I I I I I I I I II II
OOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOO
OOOOOCOOOOOOOOCOOOOOOOOOOOOCOOOOOCOOOOOOOOOOOOCOOOCOOOOOO
0000000o0 0c0000000000*-00000000000000oco00'-rvirv010'vü'0v£)-vl-p,-0
tv-Nivoc¡'rv>--rv>--iniv>rv!inu:'-puin»oinci'OUiop-vipini-opoiv>-r'oivinrv'>-ovO'oao'-vi^j>-p c
p
IV
I I I I I I I I I I 1 I I I I I I II III II
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooocooooooooooooootoooooooooooooooooooooooo
oooooooocoooooooo»-oooooooooooooooooo>-oooooooo>-oo>-ojUj>-ooo
0'Wo-f>o-Pfv-uoivfvou>.i-p'CD\0'-cn'-oOj'-‘'-*-'^noa)~JO'0'0'rvCT'-si-vii“CnojO'ai'“'p'^-siou>JUi-t>tj'-ui-poo
c
p
(v
fv
I II I II II II III
OCOOOOOOOOOOCOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOO ooo
OOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOO>-a-r\)0jPinPll*-OOOOO
00000'“000>“0000'-0>- O O 00>-0'-'0>-IV>“>-tv»-0'-'00'-rv|V*-CJliJC(X>IV(X>>oP>-'JP'0,-UU<00
uo.ufv->jfV'>j'>j>-'4i—jiv>-uj>-o-vic;ioojp>'Sosuiiv-P'0''0^'(jiNcni.jrvo''OiP'tna-t>cocDo-siivu»-rooi>“-p--Po
o
p
p
o
I I I I I I I I I I I
OOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO'-'-'-NfoOJUrvOOOOOO
OOOOOOOOOOOOOOOOOOO*-'“OOOOO>-«“OOOoOOOIV*-«-—|v0:0'CL>-IV-vlUv0FVOOa)rvC
►-OjUiinpC'roo->iO'OU’uiP-^ouJ0'UP'-POJinina>uininarv*-POj'00'-i->i'0~>iooii' — oprv-a'pininoiv
CD
p
p
II I I I 1 I I I I I
ooooooooooooooooooooooooooooooooooooooooooooooooooooooooo
ooooooooooooooooooocoooooooooooooooooooooooooo>-a-'->-oocoooo
oooooooooooooo>-p-oooooooo'-'-ooo*-oo'-*->-^o>“0>-rv'-u.oicncDoouu;-Niojooo
p'CMVk->juuiuuvru->-oiv>-rv>->-xop^^p-p-vouiuroN'OP‘*-'jo-^vua'OOj->i^a)-pa'Oui^o
c
p
p
IV
Values of 2Q(r12) - ^2_(rx2) for the Fluid
of Table H7

313
TABLE H8 (Continued)
R/SIGMA
2 . 22 5
2.250
2.2 75
2. 300
2. 325
¿•350
2.3 75
2.400
2.425
2.450
2.475
2b 500
2. 525
2.550
2.575
2.600
2.625
2 . 65 C
2.675
2.700
2. 725
2. 75 C
2.775
2. SC 0
2.325
<;.85C
2.875
2. 900
2. 925
2.950
2.975
3.000
G( 42 0 )
-C .010
0.000
-0.003
-0 .004
-0.000
-0.006
-0 .002
-0.000
-0.002
-0.002
-0.002
0.001
-0.002
-0.004
0.003
0 .002
0 .00 1
0 .005
0.007
0 .002
-0.002
0.003
C .004
0.001
0.003
-0.001
0.004
0.003
-0.002
0 . CC5
0.004
0.000
G< 42 1 )
-0.003
-0.000
-0.003
-C .004
-0.004
-0.002
0.001
0.0 02
-0.000
0.001
-0.002
0.001
0.0 02
-0.000
-0.000
0.005
O.OOi
0.000
0.002
-0.002
0.002
C . 003
-0.001
-0.000
0.000
-0.002
-0.001
-0.003
-0.002
0.003
0.004
0.005
6(422)
0.004
0.004
0.004
C .002
0.002
0.002
0.004
0.004
-C .000
0.001
0.006
0.002
0.003
0.001
-0.001
-0.002
-0.002
0.002
0.003
0.003
0.003
0.002
0 .004
0.0
-0 .004
0.001
0.002
-0.000
-0.004
-0.001
— 0.0 0 «5
-0.010
G( 440 )
-0.001
-0.006
0.003
0.002
-0.006
-0.003
0 .006
-0.000
0.008
-0.002
-0.002
-O.006
0.006
0.011
0.010
0.008
-C .00 1
-0.002
-0.005
0.004
-0.009
-0.004
-0.003
-0.000
0.004
0.004
0.001
-0 . C03
0.003
0 . C07
0.004
0.002
G(44 1 )
0.000
-0.001
-0.002
-0.003
-0.000
0.001
-0.000
0.002
-0.000
-0.003
0.002
-0.006
0.000
0.006
0.003
0.002
0.001
-C.002
0.006
0.005
0.001
-0.005
0.004
0.001
0.002
0.001
-0.008
-0.006
0.001
-0.004
-0.004
-0.007
G(442)
-0.009
-0.003
-0.003
-0.003
-0.014
-0.002
0.002
0.003
0.006
0.001
0.001
-0.004
-0.001
0.007
0.005
0.000
0.002
0.003
0.002
-C .000
0.001
0.006
0.000
-0.003
-0.003
-0 .002
0.002
0.002
-0 .000
-0.004
0.004
0.007

Values of g^oC^o) - g6AO(.r-12> for the F.1.u-i-d.
of Table H7
o
â– it
o
O ONNiflinH.M^HM)-, m D ifi 0 Aj u> AJ -it .0 n- if) 'T i h^IhO S'lOJ'O l") O —• g) \l —« 0 fvl O -< 0 m) r0 ;0 ID --t —.
r> o e* rvi :n O' *-• o m *n o o o o o o o o o r> o —«o o o o o o o o o o o o o o o o o o o o o
O O O O O O O O — O O O O O C> O O O O O C) O O O O O O C'’0 O O O O O O O D O O O ' ' O O O O O O o o o o o o o o o o
o
O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o
t I I I I I I I I I I I I I I lili II III II II I I I I I I I I I
c -o -g n in in r—iMOifNinoHOiocun o ui o (.i a u n •/» o o n o -< m
o o o -t n m c\j a) cm M o o -i -> o o o o o o o — -i o o o o o o o o o o o o o o o o o o -• o o o o o o o o o o
eg oocjooo^^. — -.-ic'ocjooooooooooooooouooocjooc'oooooooooooooooooot.ju
in
o o o o o o o o o r> o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o
LO I I I I I I I I I I I I I I I I I I I II I I I i I I I III
o
o
vii
CO
- ro o M n ai tM in o O' in i0 -i s in 'O -í -o o o o -h -•« eg eg rg ro eg rg -o o o o o o o o o o o o -h o o o o o o o o o o o o o o o o o o o o o o o o o o o
oooooooooooooooooooooooooooooooooooooooooooooooooooooooOo
oooooooooooooooooooooooooooooooooooooooooooooooc>ooooooooo
I I I I I I I I I I I I I I I I I I III II I I I I I I
â– g
g
g
Ia
o -< aj <\j -o g g n -< n o in o '0 o ,g in g —> run n o «ii in 1/1 ..g aj o in in oj o o o o n o —i —• o o o —■ o o —• o f' o o o —• o o o o o o o o o o o o o o -j o o o o o o 0 0 o o _ _ o o o o o
o o o o o o o o o o o o o o o c> o o o o o o o o o o o c > o o o o o o ci r> o o o o o o o o o o o o o o o o o <.' o o o
O O O O O O O O O O O O O O r3 O v") O O O C3 O f) O â–  3 l > O CD O O I 3 O O O O O O O O O O O O ( ) o o <â–  > O (_) O O O O O O ."> o
I I II I I I I I I -I I I I I I I I I I I I I I I I I I I II
n
g
g
co
OHHO'íi0 o o o o -i m in in gun m o -< eg -• o o o o o -i o o o o o a c o o o o
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
O O O O o CD O O O O o O o O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o CD O O O O C D o o o o o
â–  I I I I I I I I I I I I I
<
V
10
o in o in o m c> moinonoino in o in o in o in o in o n o m o in o in o m o m o ir> o m o m o n c> in o in ci m o in o in o in o
o m m f- o eg in n o (\j in n o rg m N- o m m i^- o nj in n- o \i in n o "g in n o rg in n o .o n o ni io n- cj nj >n n o eg m n o m m o
i.o in m in ni o n\ n- o o o o -i — rg nj eg eg m n n in ^ mi m ui in >o '0 n if' n i^ n- n- en to Ti n O' > o' o' o o o r> —■ —■ —i -h nj
n
v
IT
O O o o fo o o o
rg M eg CM M 'M 'M (M M

M 0.1 (M OJ 'I OJ M OJ -\J <\j CM <\J r\i OJ OJ OJ oj m ^
315
TABLE H9 (Continued)
/SIGMA
G( 443 )
G( )
G(600 )
G ( 6 2 0 )
G( 64 0 )
2. ¿¿5
0 . 0
-C.009
-0.003
-0.001
-0
.0 02
2.250
0.001
0.002
- 0.001
-0.003
-
r-\ -
• O ■> £_
2. 275
-0.009
0.0 0 2
0.002
0.005
.00 5
2.300
-0 .00 4
- 0 . C 0 1
-0 .002
-0.003
-0
.003
2. 32 5
-0.001
0.0 0 0
-0.004
0 .COo
.003
2.350
C . 0 0 0
0.005
- 0 . C 0 9
G . 0 C 2
.00 1
2.375
0 . C 0 0
- 9 .002
-0.007
- 0 . C 0 0
o
.005
2.400
0.002
-0.004
-0.010
-0.001
- 0
• 0 0 3
2.4-25
-0 .002
-0.005
-0.000
0.002
-0
. CCl
2 . 4 5 C
0.003
-0.002
0 . OC 4
0 .0 06
0
.004
2.475
0.003
- 0 . C C 3
-0.002
0 . C 0 0
0
.002
2.500
0.007
-0.009
0.002
o . o o a
__ r-
.004
2 . d 2 5
0.002
-C.0 0 7
C . 007
0.004
-0
.00 5
2.550
-0.000
- C . 0 G 2
0.004
0.001
-0
. occ
2.575
0.000
-0.000
-0 .00 1
0.000
-0
.000
2.50 0
0.0C3
C. 0 03
0.002
0 . OO'j
_ 'A
.007
2.525
0.004
C • 0 ^ 3
-0.002
0 . 0 0 3
-0
.000
2. 550
0.007
- 0 . C 0 2
-0.002
0 .0 02
0
.094
2.575
0.004
-0.001
0.003
0 . C 0 1
-0
.0 02
2.700
-0.004
-0.002
-0.002
- 0 . C 0 1
0
.000
2.725
0.001
0 . 002
C . 004
-0.002
0
.004
2.750
-0.001
C . OCO
0.004
C . 00 1
o
.00 1
2.775
-0.001
- 0 .005
0.002
0.003
0
.0 02
2 . cO 0
-C .000
-0.001
0.006
-0.006
0
.004
2.325
0.002
0.001
0.0
-0.003
0
.0 03
? . 950
0.0
0.000
0.00 1
-0 .00 1
0
.000
2.875
0.002
-0.006
0.002
0. ooc
0
. ooc
2.900
0.005
-0.003
-0.00 1
-0.002
n
.003
2. 925
0.00 1
0.00^
-0.003
0.004
.004
2.950
0.002
C . 0 05
0.005
-0.C01
6
.00 1
2.975
0.009
3 • 0 C 3
0.004
-0 . C02
0
• 300
3.000
0.012
-0.007
-0.003
- 0.0 0 o
-0
.012

316
TABLE H10
Values of gpop^J ~ g^pp^jp) for Lennard-Jones
plus Anisotropic Overlap Fluid with kT/e = 1.291
per"* = 0.85, and 6 = 0.10
9/SIoMA 5(000
0.30 0
0. 5 2 5
0.3 -5 C
0.375
0.900
C . 5 2 5
0. 95 0
0.975
1 . 00 G
1.025
1.050
1.075
1. 10 0
1 . 1 2 =
1 . 15 0
1.17 5
1.200
1 . 22 5
1.230
1.275
1.300
1.3 si 5
1.3 = 0
1 • 37 5
1.400
1 . 4 2 =
1.450
1.475
1 . 5 0 0
1.525
1.5 5 C
1.575
1 . 50 0
1 . 62 5
1.650
1.575
1.700
1.725
l . 7 5 C
1.775
1.500
1.625
1.3 = 0
1.375
1.90 C
1 . 9 2 3
1.950
1 .975
2.0 c c
2.02 =
2 . 'j 5 C
2.075
2.100
2 . 12 5
2. 150
2.175
2.200
0.0
0.0
0.000
0.004
0.041
0 .225
0.644
1.273
1.943
2.370
2.564
2.5c 1
2.43 =
2.205
1 .971
1.74 =
1 • 54 3
1 . = c 2
1 .243
1.113
1 . 007
0.923
0.36 3
0.31 =
0.7 6 3
0 . 70 *
0.676
0.661
0 . = 7 1
0.661
0.667
0.67 =
0.55'=
0.6*7
0.725
0.754
3.767
0 . 32 =
0.5=4
0 • 9 C 3
0.945
0.9 7 ;
1 .027
1 .0=9
1.104
1 . 1 3o
1.1 = 1
1 . 1 90
1.13 3
1.20 7
1.214
1.214
1.2 = 4
1.1=2
1.1=3
1.113
1 .0 93
C)
rvi
o
o
G(220 )
6(221.)
o ( 2 c. ¿ )
G ( 40 0)
u • 0
0.0
0.0
0 . c
o
.0
0 . c
0.0
0 .c
J • J
Q
rs
• J
- 0.0 0 0
0.000
0.0
0 • c
r
.c
-0.00 3
0.002
0.0 C 0
0.0C 1
Q
.001
-0.023
0 . Cl 3
- 0 . C G 0
-0.002
c
.00 5
-0•099
0.045
0 . C 0 3
C . C 0 6
0
.022
-0.212
0.03 =
0 . C 2 1
- 0.0 C 1
o
.0 36
-0.310
0.105
C . C 0 s
0.013
r-
.0 17
-0.313
0.060
-0.006
- 0 . C 1 2
0
.007
-0.228
0.0C3
C .02 =
-0.027
6
.002
-0.137
-0.010
-0.001
0 . C 0 3
— (J
. C 4 4
-0.0 72
-C . 024
-0 .00 =
C .004
0
• 0 0:3
C . 002
-0.005
-0 .009
C . 007
-0
.0 10
0.0 53
C . 0 1 1
0.010
0.017
— J
.017
0 • C 7 2
- 0.0 2 C
j • 0 d 5
-0.016
c
.011
0 . 0 7 7
-0.011
0.013
-0.019
c
• C 0 b
0.0*3
-0.005
-0 .003
-0.0)7
-0
.0 02
0.0 = 5
0.02 =
-0.014
-0.000
-A
.0 10
C • 0 = 5
-0.011
-0.009
” 0 « 0 1 j
c
.011
0 .063
0.004
- o. o i;
0.002
Q
.coo
0.063
- 0.001
-C . 0 1 5
-0.001
J
. 0 0 1
'J • O ^
0.015
__ -= r j />,
0.003
c
.coo
0.043
C . 0 1 4
- 0 . C 0 7
-o .: i:
- c
• C ) d
0.037
-0.002
-0.00=
-0.014
-c
.011
0 . 0 34
-0 .Cl 4
-0.010
0 • j 0 d
-c
.0 02
• 0 ¿ 5
- 0.0 C t
-: .ce*
C . C 03
.001
• u 2
- 0.0 C 6
u .004
C • 0
- c
.005
0.017
C . 0 - e>
- .003
-0 . C C 3
0
.00 4
C . 0 1 5
0 • C 0 9
-0 .00 =
-0.0 03
0
• V = U
C • 0 ¿J
0.004
- 0 . c 0 3
-0.002
0
.002
0.0 1 3
0.014
- 0 .00 =
- 0.000
"" J
.004
0 . 00 *
0.017
-:.coc
0 • o 0 d
0
.001
0 . 0 0 5
0.011
- C . c 0 =
-0.000
o
.0 0 =
-0.C0=
0.0 C c
0.002
0.0 01
- o
.0 0 =
- 0.0 C 3
0.017
0 . C 0 3
-0.004
-6
.00 =
-C .012
G . C 1 1
-C .0 0 3
-0.001
_ r
.005
- C . 0 1 1
0.012
- C . C 01
-0.004
- 'J
.001
—J•Odl
-0.03 C
-0.002
0.00 =
-c
. 0 05
-0.016
0 . CO 1
- 0 .004
-> ^ a or
j • ^ j j
- 0
• C3J
- 0.0 2 1
-0.005
- 0 . C 0 1
0.00 5
-0
. CO =
-0.013
0.003
-0 .003
0 . C 06
(“
.007
-0.027
0.01 =
0 . C 0 2
0 . C 0 1
.0 02
-0.021
0.01 =
C . U 0 1
-0.001
-0
.001
-0.026
0.011
0.007
n n n
^ • 0/ J c.
— 0
. 0 0 6
-0.032
-0.004
C.COo
-0.002
rj
.C 01
- J • C 2 Ó
-0.003
- 0 . 0 0 8
0.005
0
.0 02
- 0 • 0 d 1
- C . 0 0 =
-0.003
0 . C 1 =
c
. 0 07
- 0. 0 2 5
-0.001
- U . 0 0 4
0.011
w
.003
-C. 0 1 *
0.003
-0.0 02
'A ^ -
• = o -J1
.0 11
-0.013
0 . C 0 J
-0.00=
-o . oc:
0
.0 10
-0. CC 7
- C . 0 0 1
-,.009
-0.005
0
.010
- 0.0 C 4
-0.003
- 0 .00 =
_ (=> ' 7
-/ • = - '
_ r
.000
C . 0 0 4
-0.006
0.00 4
-0.COc
c
.000
c . 0 07
-0.013
0.003
-0.002
- 0
.0 04
0.007
- 0.CC 7
0 .00 =
C . 0 0 5
0
. 0 0 3
C . 0 1 7
-o.oce
0 . C 0 =
0.0 0 C
0
r\ r • <= = =
0.015
- C . 0 C 4
C . 0 0 7
- 0.0 C 3
r
n o 7
• J

317
TABLE H10 (Continued)
3/SIGMA G(OOO) G 1200) G1220) G1221) G(222) G14C0)
2.225
2.250
2.275
2.300
2.325
2.350
2.3 75
2.400
2.425
2.450
2.475
2.500
2.525
2.550
2.575
2.600
2.525
2.650
2.6.75
2.700
2.72 5
2.750
2.775
2.800
2.92 5
2.850
2.375
2.900
2.925
2.950
2.975
3.000
1.06 9
0.012
-0.009
1.028
0.012
-0.010
1.007
0.016
-0.017
0. 980
0.014
-0.017
0.961
0.017
-0.010
0.95 4
0.0 16
-0 .003
0. 931
0.0 19
-0.002
0.899
C . 008
0.002
0. 888
0.011
-0 .006
0.887
0.006
-0.001
0.88 1
0.006
-0.003
0. 890
0.008
0.006
0.902
0.006
0.008
0.899
0.006
0.003
0.909
0.001
0.006
0.918
0.005
0.001
C. 930
-0.002
-0.001
0.938
-0.003
-0.004
0.953
-0.012
0.003
0.969
-0.013
0.001
0.985
-0.012
-0.001
0.9°4
-0.012
0.003
0.998
-0.016
0.007
1.018
-0.0 11
0.006
1 . C31
-0.004
0.012
1.040
-0.011
0.007
1.050
-0 .006
0.006
1.05 4
-0.004
-0.002
1 . 059
-0.004
-0.004
1.062
0.000
-0.005
1.065
C . 004
-0.008
1.053
0.002
-0.011
0.011
0.003
-0.C02
0.004
0.001
-0.004
0.005
-0.004
-C.C 07
0.001
0.002
-0.004
0.012
-C.000
-0.C05
0.005
0.002
0 . CC6
0.004
-0.006
0 . C02
0.003
-0.000
-0.C06
0.002
0.004
-0.003
0.001
-0.OOC
0.003
0.002
-0.000
C. COO
-0.007
0.000
0 . C02
-0.002
-0.0C4
-0.CO 3
-0.001
-0.005
0. COO
0.005
-0.003
C . 003
-0.001
0.001
0 . CO2
0.001
-0.007
-0.003
-0.001
-0.002
-0.C 0 4
-0.002
0.002
-0.CO 3
-0.006
0.003
-0.C05
-0.002
0.001
-0.C02
0.001
C .003
-0.001
-0.004
-0.002
0.00 1
-0.002
-0.003
-C.COO
0.001
0.003
-0.003
-0.003
0.006
0. COI
-0.003
0.002
0. COI
-0.007
-0.000
0.003
-0.005
C. 0C2
-0.C 09
-0.008
0.005
-0.COO
-0.003
0.005
0 . C02
0.000
0.003
0 . CC6

318
TABLE Hll
Values of g42Q(r12) ~ g/,,, oil x 2) for the Fluid
of Table H10
9/SIGMA G(420) G(421) G(422) G(44C) G<441> G(442)
0.900
0 .925
0.950
0.875
C .900
0.925
0 .950
0 .975
1 . 000
1 .025
1 .050
1 .075
1.100
1.125
1.150
1 . 1 75
1.200
1 . 225
1 .250
1 . 275
1 .300
1 . 325
1 .350
1 .375
1.400
1 .425
1 .450
1 .475
1 .50 0
1.525
1 .55 0
1 .575
1.600
1 .625
1.650
1 .675
1.700
1 . 725
1.750
1.775
1 .800
1 .825
1.850
1 .875
l . 900
1.925
1 .950
1.975
2.000
2.025
2.050
2.075
2.100
2.125
2.150
2.175
2.200
0.0
0.0
0.0
- 0.00 1
-0.004
-0.014
-0.024
-0.010
0. 006
-0.005
-0.023
-0.007
-0.014
0. 004
-0.003
-0.001
0. 005
0.017
0.020
0.005
-0.008
-0. CCS
0.004
0.001
0.000
-0.010
-0.002
0.002
0.007
0. 002
0.008
-0.003
-0.006
-0.00 3
-0. 013
-0.001
-0.001
0. 006
0.007
0.003
-0.006
-0.011
-0.0C6
-0.003
-0.004
0.001
0.001
-0.002
-0.012
-0.009
-0. 00 2
-0.0^4
-0.00 2
0.003
0.003
-0.002
0.00 1
0.0
0.0
0 .0
-0.000
0.000
-0.004
-0.005
-0.004
-0.002
0.0
0.009
-0.019
-0.020
-0.017
-0.014
-0.003
-0.011
0.002
-0.001
-0.002
-0.004
-0.002
0.005
0.009
0.009
0.012
0.014
0.0C8
0.005
0.004
0.002
-0.003
-0.005
-0.001
-0.004
-0.005
0.002
-0.007
-0.003
-0.000
0.002
0.001
0 . OCfe
-0.002
-0.005
-0.007
-0.006
0.006
0.001
-0.002
0.001
-0.000
-0.007
-0.003
0.004
0.001
0.001
0 .0
0.0
0.000
-0.000
0.001
-0.003
-0.006
-0.021
-0.004
-0.013
-C.004
-0.002
-0.009
-0.012
0.001
0.005
0.003
-0.01 1
-0.021
0.002
0.006
0.003
0.002
0.001
0 . ooc
-0.000
-0.005
-0.004
0.005
0.000
-0.001
0.000
-0.000
0.000
-0.003
0.004
-0.002
-0.004
-C . 007
0 .0
0.000
0.002
0.003
-0.006
0.001
-0.000
-0.000
0.001
0.004
-0.002
0.003
0.001
-0.000
-0.000
-0.002
-0 .000
0.002
0.0
0.0
0.000
0.000
0.002
0.003
0.008
0.006
-0.023
-0.014
-0.001
-0.021
-0.005
-0.009
-0.003
0.023
0.013
0.007
-0.010
-0.001
-0.009
-0 .0 1 t
-0.019
-0.022
-0.015
-0.004
-0.000
-0.001
0.001
-0.003
-0.012
-0.008
-0.012
-0.003
-0.012
0.006
0.002
0.014
-0.004
-0.006
0.004
0.002
0.006
0.005
0.025
0.014
-0.007
-0.003
0.003
-0.003
-0.020
-0.000
0.005
-0.002
-0.007
-0.009
-0.006
0.0
0.0
0.0
0.000
-0.000
0.001
0.006
-0.000
0.017
-0.001
-0 .020
-0.014
-0.023
-0.009
-0.014
-0.001
-0.000
-0.009
-0.021
-0.003
-0.001
0 .003
0.002
0.005
0.000
- 0.OC1
0.003
0.009
-0.005
-0.002
-0.000
-0.001
-0.002
-0.001
-0.000
-0.002
-0.004
-0.002
0. Cl 2
0.008
-0.001
0.008
0 .002
O. 0C4
0.009
0.005
O. CC5
0.003
-0.002
-0.014
-0.007
-0.004
0.007
-0.001
0.002
0.004
0.004
0.0
C. C
-C.COO
0.000
0 . COI
0.001
0.004
0. C 0 8
0.008
-0.CO 2
-0.018
-0.C29
-0. C 37
-0.Cl 5
-0.010
-0.C 0 4
-0.C09
0 . C03
-0.010
0.003
C. COO
0.006
0 . C05
0 . C 0 2
-0.005
0 . C03
-0.005
-0.005
-0.C06
0.003
O. C07
O. C08
0.00 1
-0.C02
0.001
O. C04
0 . C04
0.016
0. C07
0.004
0.002
0 . C02
0.008
0. CO 7
0 . C06
-0.005
-0.COI
0.003
0. Cl 0
0. C14
0.005
-0.CO4
0 . CO 3
0 . Cl 0
-0.C05
0 . C08
0 . C03

cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cm cv n
319
TABLE Hll (Continued)
3/SIGMA G(4 2 O) G( 4 2 1 ) G( 422) G(440> G(44t) G[442)
2.225
- 0. C06
-0.000
0.002
0.001
0.011
0 . COO
2.250
-0.004
0.003
-0.001
-0.001
0.010
0.004
2.275
-0.004
-0.001
-C .003
-0.003
0 .004
0 . CO6
2.300
-0.CO 2
-0.001
-0.001
-0.007
0.001
0.005
2.325
0.001
-0.002
0.002
-0.005
0.003
0. CC4
2.350
0. 003
-0.001
0.004
0.002
0.004
0.001
2.375
-0.005
-0.002
-0.901
0.011
-0.003
-0.C06
2.400
0.002
-0.007
0.002
-0.001
- o . o o e
-0 . C02
2.425
0.00 1
-0.001
0.005
0.002
-0.006
-0.00 1
2.450
0.001
0.004
0.007
-0.008
-0.006
0 . C07
2.475
-0.001
0.006
0.005
-0.001
-0.007
-0.003
2.500
-0.000
-0.000
-0.000
0.004
-0.000
-0.COO
2.525
-C.001
-0.006
-0.002
0.002
0.0C9
0 . C03
2.550
- 0. 002
-0.003
-0.001
-0.003
0.006
0 . CO4
2.575
-0.001
0.003
0.000
-0.001
C. 00 l
0. COI
2.600
0.003
0.005
-0 .004
O.OC2
-0.004
-0.005
2.625
-0.002
0 . C 0 3
-0.002
0.005
-0.004
-C.00l
2.650
0.000
0.000
0.001
-0.004
-0.008
0 . CO 1
2.675
0.00 1
0.000
-0.001
-0.003
-0.005
0 . CO 3
2.700
0.004
0.001
0.001
0.00 1
-0.004
0 . C05
2.725
0.007
0.001
-0.002
-0.004
0.001
0.006
2.750
0.002
C . 0
-0.000
-0.005
0.004
0 . C03
2.775
-0.003
-0.002
0.002
-0.006
-0.002
0. C04
2 . SCO
- 0.002
- 0.000
-0.000
-0.007
-0.001
0 . CO4
2.825
0.002
0.001
0.001
-C.003
-0.003
0. C07
2.850
0.002
0.001
-0.003
0.004
-0.000
0. C04
2.875
0.009
0.002
-0.003
0.003
-0.003
0 . C02
2.900
0.00 4
0.003
0.002
0.009
-0.005
0. C05
2.925
0.004
-0.001
-0.002
0.005
-0 .008
0 . C05
2.950
0.001
-0.002
-0.005
0.005
-0.001
-0.C02
2.975
0.000
-0.004
-0.009
-0.006
0.001
-0.C02
3.000
0. OG3
-0.002
-0.003
-0.007
-0.002
-0 . C04

320
TABLE HI2
Values of
for the
Fluid
-“443 1 Z
of
-wooU 1Z
Table H10
R/SIG^A
G(443 )
G(444 )
G( 600 >
G(620)
0( 640 )
G ( 6 € 0 )
0.800
0.0
0.0
0.0
0.0
0.0
0 . C
0.825
0.0
0 .0
0 .0
0.0
0.0
0 . c
0.850
0. 000
0.000
0.000
0.0
-0.000
0.000
0.875
0.0
-0.000
-0.000
0.000
-0.000
0 . CO 1
0.900
-0.003
0.002
-0.001
0 .000
0 .0
0 .coo
0.925
0.002
0.003
-0.007
0.003
0.004
-0.006
0.950
0.003
0.002
-0.009
-0.015
0.011
0. CO4
0.975
- 0 . 0 C 4
-0.001
-C .01 2
0.001
-C . 005
-0.013
t . 000
-0.026
0.024
-0.036
0.007
0.012
-0. Cl 4
1 . 025
0.013
-0.011
-0 .00 4
-0.001
0.001
-0.C41
1.050
-0.010
-0.010
-0.002
-0.001
0.003
-0.C35
1 .075
-0.005
-0.001
-0.006
-0.004
-0.0 08
0. CIO
1.100
-0.006
-0.008
-0.001
-0.004
0.004
-0.C07
1.125
-0.010
0.011
-0.021
-0.015
0.011
0. Cl 2
1.150
-0.009
-0.00 l
-0.032
-0.008
0.006
C. C3C
1.175
0.013
-0.002
-0.002
-0.012
0.002
-0.Cl 1
1 .20 0
0.009
0.001
-0.015
-0.006
0.000
-0 . Cl 8
1 . 225
-0.014
-0.004
-0.004
-0.004
-0.011
-0.C20
1 .?50
-0.003
-0.005
0.005
- C . 0 0 3
0.005
-0.C07
1 . 275
-0.002
0.001
-0.005
-0.002
0.0C8
-0.C03
1 . 300
0.015
0.006
-0.002
-0.002
-0.010
-0.007
1 .^25
0.007
0.001
-0.019
-0.0C5
-C.004
0 . C03
1 .350
0.00 3
C .003
-0.010
0.003
0.010
0 . C08
1 . 37o
-0.011
0.007
-0.012
0.004
0.003
-0.C04
1 .*00
-0.005
0.005
-0.011
-0.002
-0.004
-0.C04
1 . 425
-0. 004
0.008
-0 .006
-0.002
-0.007
0 . C03
1 . 450
0.006
0.002
-0.003
0.003
-0.005
-0.CO 3
1 .475
-0.0C5
-0.003
-0 .002
-0.007
-0.003
0 . CC2
1.500
0. 002
0.001
-0.002
0.004
0.00 1
-0.CO 4
1 .525
0.002
• 0.001
0.003
-0.006
0.001
0. C02
1 . 550
0.006
C . 000
0.003
-0.003
-0.003
-0.C07
1.575
0.00 7
-0.000
-0.004
-0.001
0.001
0 . CO 7
1 .600
0.00 7
-0.006
0.000
0.009
0.004
-0 . C04
1.625
0. C09
0.003
0.001
-0.002
0.003
-0.007
1 .650
0.00^
0.004
-0.004
-C.000
0.003
0 . C04
1 .675
0.003
0.011
-0.003
-0.002
0.006
-C.COO
1.700
-0.001
0.000
0.002
-0.004
-0.000
-0.004
1 . 725
-0.004
0.0
0.002
0.004
-0.004
-0 . CO5
1.750
-C. 003
-0.006
0.018
0.003
-0.005
-0.C08
1 .775
-0.005
-0.002
0.011
C. 000
-0.004
-0.007
1 .800
0.003
-0.003
0.006
0.000
-0.004
-0. C 07
l . 825
0 . Cl 2
-0.0C4
0.008
0.006
0.005
0 . C06
1 . 850
-0.003
0.006
-0.002
0.009
0. CCO
-0.CO 1
1 . 875
0.003
0.005
0.006
0.000
0.006
C. COO
1.500
0.006
0.005
0.006
-0.001
0.002
0.002
1 .925
0.002
0.002
0.007
-0.003
- 0.CO 1
-0.CIO
l . 950
0.008
0.004
0.000
-0.000
-0.007
-0.002
1 .975
0.002
-0.008
0.00 1
-0.003
0.004
0 . C06
2.000
-0.005
-0.004
0 .005
-0.002
0.006
0 . Cl 1
2.025
0. 001
0.002
0.005
-0.002
0.000
0.005
2.050
-0.001
-0.000
0.000
0.001
0.003
-0.C 0 7
2.075
-0.0C5
0.006
-0.001
0.013
-0.009
-0.013
2.100
-0.003
0.005
-0.001
0.0 0 1
-O.OCl
-0.Cl 2
2.125
-0.002
0.000
-0.008
-0.000
0.006
-0.C02
2.150
0.000
0.000
-0.009
-0.001
-0.004
-0.C02
2.1 75
-0.008
0.007
-0.003
-0.011
0.004
0. COO
2.200
- 0.002
0.000
0.001
-0.006
0.005
-0.002

321
TABLE H12 (Continued)
9/SlG^A G ( 4 4 3)
2.225
2.250
2.275
2.300
2.325
2.350
2.375
2.400
2.425
2.450
2.4 75
2.500
2.525
2.550
2.575
2.600
2.625
2.65 C
2.675
2.700
2.725
2.750
2.775
2.800
2.825
2.850
2.875
2.900
2.925
2.950
2.975
3 .000
-0.002
- 0. 004
-0.002
-0.003
- 0. 012
-0.008
-0.013
-0.004
-0.00 1
0. 00 1
-0.003
0.002
-0.005
-0.002
C. 000
-0.008
-0.003
- 0. 003
0.00 1
0.001
0.000
-0.000
0. 003
-0.009
-0.004
0.0^1
-0.000
0.004
0.003
0.002
0. 00 1
0.006
G(444 )
-0.000
0.001
-0.003
-0.007
-0.004
-0.004
-0.001
-0.009
0.001
-0.001
0.0 0 l
0.003
-0.001
0.003
-0.003
-0.005
-0.006
-0.004
0.002
-0.001
-0.001
-0.000
0.002
0.004
0.002
0.009
0.006
0.001
- 0.000
0.001
-0.003
-0.005
G(600 )
0 .002
-0 .007
-0.006
0.005
0.003
-0.001
-0 .009
-0.002
-0.001
0.005
0.002
0.000
0.004
-0.003
0.002
-0.001
-0.004
-0.003
-0.008
0.003
0.001
-0.003
0 .006
-0.001
0.002
-0.002
0.001
0.003
-0.002
-0.005
0.004
0.002
G(620 )
-0.001
-0.004
0.005
0.005
0.003
0.004
-0.009
-0.004
- 0.00l
-0.004
-0.003
-0.003
0.001
-0.004
0.001
-0.003
0.003
-0.001
-0.005
-0.007
-0.005
-0.003
C .000
0.002
-0.006
-0.003
-0.002
-0.003
-0.003
-0.003
0.002
0.000
G(640 )
0.010
0.003
0.004
0.001
0.004
C . 002
0.002
- 3.004
-0.002
-0.006
0. C 03
-0.001
-0.001
-0.003
0.002
0. OOC
-0.0C8
-0.005
-0.005
0.001
0.004
0.005
0.001
0.004
0.003
0.005
0 . C C 2
0.001
-0.001
0.005
0.005
-0.001
G ( 6 60 )
c. COO
0 . CO 1
0 . C03
O. C07
0 . C08
0. C07
0.001
0 . CO 1
-0.C 07
-0.CO 6
-0.C 0 7
-0 . CO5
-0.006
-0.C09
0 . C07
0 . C02
C. Cl 0
0.005
O. C04
-0 . COO
0 . C03
-C.C02
-0 . C05
0.003
-0. coo
0 . CO3
0. C04
0 . C07
-0.000
C. C02
-0 . C07
0 . Cl 1

TABLE H13
Values of g^,
plus Anisotr
J - í,„n(r,J for
Lennard
-Jones
IUU 1¿ 4 UU
opic Overlap Fluid with kT/e =
1.287,
pü^ = 0.85,
and 6=0.
3
R/SIGMA
6(000)
G( 200 )
o(220 )
G(221)
u(222)
G ( 4 C 0 )
0- 80C
C .0
0.0
0.0
0.0
0.0
0.0
o. 32 5
0.003
-0.004
0.004
-0.000
0.001
0.003
C* 850
0.051
C • 0 S ¿
0.055
0.001
0.001
0.043
0 • e 7 5
0.194
-0.191
0.169
-0.001
0.008
0.145
C* 90 0
0 .452
-0.417
0.385
0.002
0.027
C .273
0. 325
0.796
-0.6o2
0.552
0.008
0.041
0 .359
0» 95 Ó
1 .19 7
-0.860
0.618
0.012
0.035
0.364
0. 975
1.547
-0.6 32
0.496
0.026
0.037
0.263
1 . OOC
1 .808
-0.761
0.500
0.040
0.044
0.151
1.025
2 • 05 9
-0•559
0.-121
0 • 056
0.020
0.092
1.050
2.144
- 0.5 5 1
0.027
0.075
0.021
0.029
1.075
2.159
-0.148
-0.006
0.037
0.049
0.004
1.100
2.064
C . 01 9
-0.01e
0.072
0.010
-0 .003
1 . 1 ¿5
1 .92 9
0.125
0.006
0.062
0.009
-0.015
1.150
1 .792
0.185
0.019
0.044
0.002
-0.02o
1.175
1 .65 5
0.228
0.037
0.042
0.006
-0.019
1.200
1.513
0.242
0.034
0.060
0.004
-0.023
1.22 5
1 .363
0.230
0.045
0.052
-0.015
-0.029
1.250
1 .252
C . 22 1
0.056
0.032
-0.005
-0.017
1.27=>
1.134
0.211
0.045
0.036
0.007
-0.025
1*500
1 .033
0.203
0.062
0.035
-0.011
-0.009
1.525
0 • 95o
0.179
0.059
0.025
0.004
-0.029
1.550
0.897
0.158
0.041
v . 024
-0.004
-0.023
1.375
0.814
0.141
0.036
0 . 028
-0.004
-0.033
1.400
0.7 76
0.111
0.025
0.015
-0.000
- 0.0 31
1.425
0.747
0.106
0.034
0.017
0.001
-0.031
i • Q b 0
0.7¿6
0.0 93
0.015
0.020
0.005
-0.025
1.475
0.71 ¿
0.087
0.012
0.01 5
-0.004
-0.031
1*500
0.691
0.064
C .016
0.014
0.012
-0.019
1.525
0 . óüj
. 0.C52
0.030
0.009
-0.001
-0.021
1.55C
0 .6 77
0.039
0.027
0.002
0.004
-0.024
1.575
0.697
0.054
0.042
0.005
-o.coo
-0.022
1.600
0.706
0.022
0.035
0.003
-0.007
-0.025
1.625
0.736
0.012
0.019
0.004
-0.001
-0.021
1.650
0.762
-0.012
0.025
0.006
-0.002
-0.014
1.675
0 .79o
-0.025
0.025
0.007
-0.013
-0.004
l . 70 C
0.637
-C.056
0.034
-0.001
-0.012
-0.000
1.725
0.856
-C.046
0.025
-0.002
-0.007
0.003
1.750
0.900
-0.052
0.018
O.COb
-0.006
-0 .000
1.775
0.949
-C.064
0.018
0.005
-0.011
-0.001
1 . dO 0
0.968
-0.074
0.017
-0.005
0.002
0.001
1*625
1 .035
-0.082
0.020
0.002
0.004
0.006
1.850
1 . 075
-0.06 3
0.011
0.003
0.001
0.015
1.675
1.10 7
-0.061
0.012
0.007
0.003
0.020
1 » 90 C
1.127
-0.076
0.016
-0.001
-0.002
0.017
1.925
1.14 1
-0.075
0.008
-0 . C09
0.010
0 .020
1.9oG
1.156
-0.061
-0.000
-0.001
0.005
0.0 1 5
1.97 5
1.168
—0.049
-0.001
-0.000
0 • 0 0 b
0.0 12
.2. 00 C
1.185
-0.042
-0.020
-0.003
0.004
0.010
2. 025
1.176
-C.033
-0.017
-0.002
0.003
0.008
¿.050
1.163
-C.021
- 0.028
- 0 . C 0 3
0.009
0.007
2.07 5
1.142
-0.015
-0.026
-0.00 3
-0.001
0.005
2.100
1 . 1 25
-0.002
-0.032
0.000
0.001
0.015
2. 1 25
1.110
0.019
-0.016
0.000
-0.007
0.010
2 . 1 d 0
1 . 062
0 • C cL b
-0.031
0.004
-0.002
0.013
¿.175
1 • Oo 5
0 . C 1 9
-0.022
0.00o
-0.003
0.0 08
2.200
1 . 050
0.022
-0.024
0.004
-0.005
0.001

323
TABLE H13 (Continued)
R/SIGMA
2* 22 £
2 « 2d 0
2.2 75
2. 300
2i 325
2.3 d 0
2. 375
2.400
2. 425
2.450
2.47 5
2. 50 0
2*52 5
2. 55 0
2 • 5 7 5
2 4 600
2. 625
2*650
2.675
2.70 0
2. 72 5
2» 75 C
2.775
2*60 0
2. 62 5
2.85 0
2*87 5
2*90 0
2.92 5
2*950
2.975
3.000
G( 000 )
1.015
1.014
1 .002
0.973
0 .954
0.946
0.926
0.918
0.909
0.915
0.911
0.915
0.919
0.930
0.950
0.951
0.946
0.971
0 • 960
0.969
C .990
1.006
1.016
1.026
1.055
1 .043
1.041
1 .055
1 .OoO
1.045
1 .040
1 .036
G(200)
0.032
0.041
0.042
0.041
0.032
0.036
0.026
C .028
0.027
0.026
0.019
0.015
0.012
0.007
0.001
-0.001
-0.007
-0.013
-0.012
-0.013
-0.018
-0.015
-0.017
-0.021
-0.018
-0.017
-0.013
-0.015
-0.010
-0.011
-0.009
-0.005
G(220)
-0.015
-0.006
0.000
-0.006
-0.007
-0.01 1
-0.011
-0.000-
0.005
0.005
0.012
0.014
0.017
0.014
0.006
0.025
0.013
0.009
0.010
0.004
0.006
0.009
-0.001
0.001
0.003
-0 .005
-0.013
-0.003
0.006
0.006
-0.011
-0.008
G(221)
0.008
0.003
0.0 0 d
0.004
-0.005
0.011
0.009
0.001
0.000
-0.001
-0.001
-0.003
0.001
0.002
0.006
0.00 9
0.002
-0.003
-0.003
0.000
0.003
0.007
-0.002
-0.003
-0.006
0.003
0.004
0.001
0.003
0.002
-0.001
0.001
G(222 )
0.001
0.008
0.006
0.003
-0.003
-0.005
-0.C03
-0.004
-0.001
0.002
0.001
0.005
0 . COI
-0.001
-0.001
0.002
-0.000
-0.005
0.003
-0.005
-0.005
0.000
-0.002
-0.003
-0.0 Od
-0.002
-0.000
-0.001
0.000
-0.003
-0.002
-0.000
G( 400 )
-0.00 7
-0.015
-0.009
-0.011
-0.013
-0.0 12
-0.011
-0.004
-0.009
-0.009
-0.006
-0.003
-0.003
-0.007
-0.006
-0.002
-0 .00 l
-0.001
0.003
0.003
0.005
0.006
0.003
0.011
0.011
0.003
0.002
0.003
0.005
0.0 03
0 .005
0.001

f\j(v)r\-ívr\jr\.rvi\jr\j»—»-o o oooooo
w*-'--'-*'00oovOvr:'C'i}cc.acEou--j'j^j--go'U'd'0'Ui-e--t-'.p>uiuc.urv>rufviv>->-'->-oooo*c'Ou>ciaia)CDUj
O'ju ivo-juii\jo-NjoirY>c-'jo fvo->ju;r\>Gsuifgo^iL;'r\.o-viuir^CNUirvo-'jU)rvo''ju‘'r\jo^joir\ O'J0'!i\iO'jiJir\.o
otpot^otrou'coiouiociouioirouioaocnouioaioc^ocnoi/iouiou'ouiouiouiouiotnocnoaiocriouio
N
r—«
c.
s
>
I I I I I I I I I I i i I I I I I I I III I I I I I I I I I I I I I I I I I I
oooooooooocoooooooooooooooooaoooooooooooooooooooooooooooo
oooooooooooooooocoooooooooooooooooooooooooooooooo>-wNro*-ooo
"-0000000 GO*-G^*OO>-O,-*,-,“GOOOCOOOO>“*-OOOOOOOOOO'-,--*-->->->-OtrifCk0U;.f>'£O
•-a-Niui-f:r^o>-NU'o>kCoovt')üia;(jurva!*-iv<“a^Lnrvi-uico'“ai~-p-oui'->-^'>irv-f>>“,-a'-viujuo>-03.p-i\;-p-u
p
r\;
o
il ii l i i i l i l i i i i l i i i i l i i i i i i i i i i i i i i
ooooocoococooooooooooeooooooooooooooooocooooococooooooooo
OOOOOGOCOOOOOOOOOOOOOOOOOOOOOOOOOOOOOGOCOOOOOOOGCOOOOGOOO
o>-ooooooo oooooo*-ooooooooGoooooooo«-oo*-*-*“*-ooo*-Gr'j*->-"-"-ooooo
wouujoojc*i"-Cí. >-N^iO"oajC(OoMr\jk-^i>-cMv>uJ*“UO'AiwOjCDiv^uro^juujiCj^^ ►-UM>ncooO'N'""0
I I I I I I I II I I I I I I I I I I I I I I I
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
ooOooooooooOooooooooooooooooooooOoooooooooooooooooooooooo
000000000000 00000000000000000000000000>-c00>-0c0'-00>“l\;tv'000
orv>uio>--'joouiojujN cjro*-NOi(XiO'Ui-f>k-->iuiiNj*->-ii-'-ui'-,ro-p-"J-f>r\)C3'Ojrv;o'^->icM\)"-Mr\)UiO)'£)(^rooi>->-
c
p
lu
k
III I III I i I I I I I I I I I I II II
ooooooooooooooooooooococoooGooooooooooooooooooooooooooooo
OOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOCOOOOOOOOOOOO •->—I—>-000
OUOOOOO'-OOGOO*'""i“"OOOOOCOOOO"OOO"OG'-OOW""OI\iliiMOOOOMOUI0'OUO
ojoO'U«-*-uuirV'i>'C'“Ui'4>-tii>i)U'i'-',“Cui>-cayiO'C'ro-t*.trl*-'-'-NaiO'u
O
P
p
o
I II II II I I I I I I I I I I I I I i I I II
OOOOOOGOOOOOOOOOOOOOOOOGOOOOOOOOGOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOGOOOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOGOG — OOoOOOOCG>->-OGOGOGCGOOCOOO>-GOO>->- — Og>-'->-OOOGOGOO
uii\>r\;uivO'r\)vO'0"jpp>-o'f\:o'Ni-"''-ppuiPfv>Pi>.oooJui>-i>jcntr'-rop>-iU\o©a;f\jr\j'-roo
c
p
p
I I I I 1 I I I I I I I I I I I I I I I I I l I I
OOOOOOOOGOOOOOOOOOOOOOOOOOOOOOOOOGOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOCOOOOOOOOOOOOOOOOOGGOOOCOOOOOOOOOOOOOOOOOCOOOOOO
OoOOO"-OOOOOOOOOOOGOOGOOO«-GOOOC"-GOOOOO>-0'-0^>-0-->-OOrj|\jO>->-0 o
U’i"*U>'OUl'-CM\,UUl'“0'>-U'0'"-4l\/~'JOt\J->JOjUJ~'IO-->ir\)'“"'POOgU*JOjO:UJ>-*l\>"-Uif\JUI|\J'NlOOO'UlPU'>v6MUlK> "*
p
p
l\J
Values of 2Q(r ±2)_~_¿¿i^2(r12) for the Fluid
of Table H13

i\j ro ivi rv- rv: iv r\
325
TABLE H14 (Continued)
P./SIGMA
G( 420 1
G{ 42 n
G(422)
2.225
-0 .003
0.000
-0.002
2.250
-0.000
0.002
0 .00 6
2. 27a
0.005
-0.001
0 .00 4
2.300
0 • 0 0 wi
0.002
-0.000
2» 626
0.003
-0.0C3
-0 .005
2.35 0
0.004
0.005
-0.004
6. 375
0.002
0.002
0.000
2. 40 C
0.003
0.006
-0.004
2. 42 5
0.004
0.004
-0.00l
2.450
-0.001
-0.002
0.004
2*47 5
0.004
-0.000
0.003
2.50 0
0.003
-0.000
0.001
2.525
-0 .002
-0.002
-0.005
2 • o5 0
C . 00 1
-0.001
-0.001
2» 57 5
-0.001
-0.006
0.002
2. 60 0
0.008
0.002
-0.002
2.62 5
0.002
-0.001
0.001
2.650
0.0 o o-
-0 .003
-0.002
2. 67 5
-0.002
-0.001
-0.001
2.70 0
-0.004
-Ü.001
-0.002
2.725
0.0
0.001
-C . 002
2.750
-0.00 1
0.004
0.003
£1.775
0.002
0.002
0.000
2.30 0
-0 .002
0.003
0.001
2.82 6
-0.002
-0.C01
-0.002
2.850
-0.001
-0.001
-0.00 5
2.875
0.003
-0.001
-0.004
2.900
0.004
-0.001
-0.005
2i 92 5
-0.005
0.002
-0.004
¿1.95 0
-0.006
-0.003
-0.004
2.975
0.002
-0.001
-0.002
3.000
-0.005
0 .003
0.000
G(440)
0.002
0 . COO
-0.011
-O.OOj
-0.001
-0.003
-0.006
-0.004
0.000
-0.002
-0.005
-0.002
-0.001
-0.003
-0.009
-0.005
-0 . C02
-C.002
-0.004
0.002
-0.003
-0.002
0.003
-0.001
0.003
O'. 007
0.005
0.007
0.003
0.002
— C .000
0.004
Gl 44 l )
0.003
0.003
0.004
0.002
-0.004
-0.004
-0.001
-0.000
-0.003
-0.003
0.0
0.004
0.002
-0.000
0.001
-0.002
-0.006
-0.001
0.005
0.001
-0.003
0.002
0.003
0.002
-0.006
-0.004
0.002
0.00 1
-0.C04
0.0
C .004
0.006
G(442)
-0.002
0.001
-0.003
-0 .001
0.003
0.004
-0.003
0.007
-0 .002
0 .002
-0.005
-0.00 4
-0.C03
-0.001
0.002
0.003
-0.000
0.003
0.001
0.003
-0.002
-0.001
-0.001
0.006
0.007
0.004
-0.000
-0.002
-0.002
-0.003
-0.000
-0.002

(v) rv< Iv tvoooooooo
rv ►- «- — ►-oocoiflO'C». aoocDCL.N^J^^jO'0'0'0'CriuiO'U‘-fí-f-'.f.(>CuUJUiUirvi\jrviv>-'“*---ooooví>'íj'f''t OSUlfViO^KMtViO'>4UiIVOSOiIVO''J(J.|\)O^JUirvo->JUlfVO->|ü:(VONUllVO'v)CnMO'vlUlIVO>JOlKO-vlUl|VO>IOlfVO
oo’oaouiouioyiooiouiouiouiouiouiodiouiouiooiouiou’ouioiíoiíouioijiouioa'ouioaiouiooio
T,
\
w
o
Í'
II III I I II lili I I I I I III I III I I II I I
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
Of-oooeoocooooccjoo>“'“Oocooi-ooooooo>-oc^-ooooi-->-<->-oc>-oo>-oooo
UOUO'-OUIÍ>HOO.OUOONjaoUOOÍllH-IVOUiy'ÍNMJIU!t'NU4M3'uí:N''UG.i-ÍC:CC'-(JO'OK;0>“
c
p
p
u
c
w
I—1
c
0)
CO
o
hti
i I I I I I I I I I I I I I I I I i I I III
OOOOOC.COOOOOOOOOOOOOC'. OOCCOOOOOOOOOOOOOOCOOOCOOOOOOOOOOOOO
OOCOOOOOOOGCOOOOCOOOOOOGOOOOCOOOOOCOOOOOOOOCGOOOOOOOOOOOO
0*-OOoGOOO*-00>-0>-->~<-OOCOOOOOOOC*'0 0<“0000000>-0>->-*C>-i-i~0>-0'->->~0 0
->joivtuui«-c);'“uo>-'íío>-uU)'-.f>i*;0(vi>.>oojivvoo'airv'- rvouiurvfvooivoo'-cN'Ooouiocr'-U'jaoj
Cl
p
p
p
I I I I I I I I I I II I lili I I I I I I I I I I I I I I I I I I I I
oooooooooooooooocoooooooooooooooooooooooooooooooooooooooo
OOOOOOOOCOOOOOOOOOOOGOOOCOOOOOOCOGOOOOOOOOOOOOOOOO>->->-OOGO
GO'-OOOhOOOOGOOOhOhhooOOOOOCOOOOOOOOOI\)I\)N)I\;MM'“>-IU(ií''|\j11;>||\)UUíiüUO
0'ivo0'Ucn*-a>*-uiPQPUi0'©cn0D03-'i-vi-'J>-'~u,-*'tiJtv*'*-u¡o^>“»-crifvP>-'CCriP~>j~'lU'0v0 0JLn>-'0a0.*-iv
Ci
#«**
O'
o
o
TO
p-
P~
It) TO
O'
K O'
ISJ
H
>
w
t-1
w
re
I-*
U1
U>
K>
O'
I I I I I I I I I I I I I I I I lili I I I I I
ooooocoooogoooooocoocooooogggogoooooooooogooogoogoocooooo
ooocooooooooooooooooooooooooooooooooooooooooooooooo>->-oooo
oooooooooooooooo>-ooooo>-o>--*-Goooooooo>-'-i-oo«-oooo>->-oo-paujUGUjo
fv05>-ivuC'''i'PC'^(;io-PtuOCTiv>-o i\jivfvoiv>“p-u!>“,-ojO'\oO'Cuuoui(v>-Nfvvoa.pa^'Jcr->irv-'io>i5>-P iv>
c
O'
rv
o
I I lili I III I I I I I I I I I I I I I I I II II I I I I I I I
OOOOOOGOOOOOOGOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOO
OGOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
GOOGOOOGOOOOOOGGi-OGOGGOCOGGOOOOGGGOO — 0>-0 OOOOGHi-O'-UO'iO'd.O
PUl~gUl''ltV)^-ao©CnvOlV:‘-*'H'''IOO'010*-fVOUU10*'>-OUl''IPO'~>iPO'0'-'Jor\) CCUlOtVCOOCOCM-OPPSCMV)
c
#■»*
C'
p
G
III I I I I I II III I I I I i I I I I I I I I I I
OOOOOOOOOOOOOOOOOOOOOOOOOOOOoOOCOOOOOOOOOOOOOOOOOOOOOOOOO
Cl
OOOOOOOOOOOOOOOOOOOOOOOOOOOGGOOOOOOOOOOOOOOOOOOOOOOOOOOOO
üOO>-i->-oOOOüOüOOOHOOOüOOoOOOOOOOOi-Ot'OpHMOWWOi*(yOO'->‘HOWU)PHO
uirvrv-rvi-uiuuou-'icivppoc-poui'-o'. p-srvc-oi-voo'UUiPODOiouipPM'OuopsuiPouou'-'orv
o
O'
o

ru m rv iv Ki ru ro rv n
327
TABLE H15 (Continued)
k/SIGMA G(443) G(444) G(600) 0(620) G(640) G(660)
2# 22 6
2i 25C
2.275
. JOC
. 325
*350
.o75
.400
. 425
.45 0
.475
. 50 C
» 52 6
¿.55 0
2.575
2.60 0
2. 62 5
2i 650
2. 675
2.700
2. 725
2. 750
2. 775
2.600
2 • 62 5
2 * 850
2. 375
2.900
2.92 5
2. 950
2. 975
3. 00 0
-0.007
0.001
0.002
-0.002
-0 .00 1
0.005
0.001
0.001
-0.002
-0.001
-0.001
-0.006
-0.004
-0.003
-0.001
-0.007
-0.001
-0.001
-0.00b
-0.008
-0 .002
-0.007
-0.001
— 0.00 5
-0.005
-0.007
-0.004
-0 .003
0.001
-0.000
-0.003
0.007
-0.002
-0.002
-0.005
-0.001
C . 001
-0.001
-0.004
-0.000
0.001
C . 007
0.002
0.004
-0.004
-0.000
-0.003
0.000
-C . 003
0.002
0.006
0.005
0.002
0.001
-0.001
0.0 0 o
-0.00 l
-0.000
0.004
0.001
-0.003
-0.001
-0.000
C . 003
-0 . ooc
0.001
0.003
O.OCOi
0.005
-0.002
-0.003
-0.005
-0.002
-0.002
0.001
0.001
0.000
0.00 1.
0.003
0.003
0.001
-0.001
-0.001
-0.003
-0.002
-0.002
-0.002
-0.001
0.001
-0.001
0.003
0.007
0.004
-0.002
-0.004
0.005
0.004
0.012
-0.003
0.0 0'4
0.005
0.005
0.004
0.009
0.003
0.000
-0.004
-0.001
0.002
0.004
0.001
-0.005
0.001
-0.003
-0.001
0.003
0.002
0.006
0 .002
-0.002
-0.003
-0.002
-0.004
-C . 00 1
-0.005
-0.002
C .003
-0.000
-0.010
-0.012
-0.005
-0.005
0.002
-0.006
-0.004
-0.003
0.000
0.003
-0.004
-0.000
-0.003
0.000
-0.002
0.0
-0.002
0.003
-0.006
-0.005
-0.000
-0.001
0.002
-0.003
-0.004
0.003
0.005
0.002
0.005
0.006
0.0
0.005
-0.007
-0.005
0 .005
-0.004
-0.001
0 .003
-0.003
0.000
0.005
0.006
0.005
-0.003
-0.004
0.000
0.003
0.003
0.002
0.004
0.007
-0 .002
-0.001
0 .002
0.006
0.005
-0.006
0.003
0.002
— C .006
-0.003
0.0
0.000
0.001

APPENDIX I
2, S, m
VALUES FOR THE J
n
INTEGRALS

329
TABLE II
The Integrals
222
- J for
a Lennard-Jones
Li ir ~
plus Quadrupole Fluid. pa
.85, kT/e =
1.277,
Q/(eaV/2 = 0.
5
N
J(N;000)+
J(N;200)
J(N;220)
J(N;221)
J(N;222)
0
-0.00257
-0.15423
-0.08598
-0.02223
1
• • • •
-0.00093
-0.12748
-0.07211
-0.01963
2
....
-0.00035
-0.11024
-0.06284
-0.01726
3
....
-0.00012
-0.09836
-0.05632
-0.01529
4
1.26163
-0.00001
-0.08971
-0.05152
-0.01367
5
0.75845
0.00008
-0.08312
-0.04785
-0.01236
6
0.58650
0.00015
-0.07795
-0.04496
-0.01127
7
0.49697
0.00023
-0.07380
-0.04265
-0.01038
8
0.44056
0.00030
-0.07041
-0.04076
-0.00963
9
0.40104
0.00038
-0.06764
-0.03922
-0.00900
10
0.37151
0.00045
-0.06536
-0.03795
-0.00846
11
0.34856
0.00052
-0.06350
-0.03692
-0.00801
12
0.33026
0.00059
-0.06200
-0.03608
-0.00763
13
0.31546
0.00067
-0.06081
-0.03542
-0.00731
14
0.30342
0.00074
-0.05990
-0.03491
-0.00703
15
0.29361
0.00082
-0.05924
-0.03453
-0.00680
16
0.28566
0.00090
-0.05882
-0.03428
-0.00661
17
0.27931
0.00099
-0.05860
-0.03414
-0.00645
18
0.27435
0.00109
-0.05860
-0.03411
-0.00632
19
0.27063
0.00119
-0.05879
-0.03419
-0.00622
20
0.26802
0.00131
-0.05916
-0.03436
-0.00615
21
0.26644
0.00143
-0.05973
-0.03463
-0.00610
22
0.26581
0.00157
-0.06048
-0.03500
-0.00607
23
0.26608
0.00172
-0.06141
-0.03546
-0.00607
24
0.26722
0.00188
-0.06254
-0.03602
-0.00608
4. £, £~m
'j E J(n;£ £„m)
n 1 ¿

O
M
CN
O
M
OJ
i—I
o
W
rn
hJ
1
CT3
CO
m
H
c
o
H
o
<-w
O
ni
Vj
M
0)
n. co >0'0cfaiTin^|r)cvi0>Nin-ivM-H-.ooc\0'ffi0'0'0'0'
-i tn m o c* co n n n o co vo co >0 >o co co -o m m imr> in in
rvj«-._.ooooooooooooooooooooo
OOOOOOOOOOOOOOOOOOOOOOOOO
ooooooooooooooooooooooooo
-* omoMoiN nin f^cnc\i — ooooooooooooooooooc»oo
ooooooooooooooooooooooooo
ooooooooooooooooooooooooo
OOOOOOOOOOOOOOOOOOOOOOOOO
I I I I I I I I I I I I I I I I
cnd-o<<í _'OHO>Mníai~iOOC>'5'XICOtlIT)'OCOB330'a'00
in OOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOOOOOOOOOOOOOOOOOO
OOOOOOOOQOOQOOOOOOOOOOOOO
i-iCMc\ion3a\N-<0'Osrosc0'0Na'c\iinoin-*N'n
cD"nin,no'<í-«cT>cD-i)Oin>n-cn
ONin^-n;nrnnjnj'\j^ji\|M0jnjnj\jr\jr\jAj(\JCMrg-jAj
-iOOOOOOOOOOOOOOOOOOOOOOOO
ooooooooooooooooooooooooo
ooooooooooooooooooooooooo
N
cm
O'
r*
0»
00
in
H
n
OJ
N.
"0
O'
0
"0
4*
0
o>
o
o
in
O
o
:\J
'0
m
n>
0
v£)
N
•o
o
o
CM
n
m
N
O'
m
o
CM
CM
C\J
C\J
CvJ
CM
AJ
OJ
OJ
m
CVJ
nj
CVJ
CM
CM
OJ
n
m
"n
ro
"0
"0
m
O
o
O
o
o
o
o
o
o
o
o
O
o
o
o
o
o
o
o
o
o
O
o
o
o
• *■
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
o
O
a
o
o
OOOOOOOOOOOOOOOOOOOOOOOOO
ic\irno-
c\jrn-iinvoNcoo'0- — —• —• —• oj rvj oj aj r\j

N
O
1
2
3
4
5
6
7
8
9
1 O
11
12
1 3
14
1 5
1 6
1 7
18
19
20
21
22
22
24
331
TABLE 13
441
The Integrals J
T600 * +u
J for the
Fluid
LI II
of Table 11
i ( NJ 44l )
J ( N ; 4 4 2 J
J(N;443)
J(N í 444)
J(N í 600)
0 ,C 124q
-0.00219
0.01159
-0.C1108
-0.00097
0.01059
-0.00169
0.00722
-0.00516
0.00071
0.00914
-0.00129
0.00493
-0.00222
0.00 135
0.00807
-0.00095
0.00369
-0.00071
0.00154
0.00730
-0.00066
0.00299
0.00010
0.00152
0.00674
-0.00041
0.002 o0
0.00056
0.00 142
0.0 0ó33
-0.00021
0.00236
0.00083
0.00128
C.00603
-0.00004
0.0022 1
0.00099
0.00113
0.00560
0 .000 1 1
0.00212
0.00109
0.00098
0,005oJ
0.C 002 4
0.00206
0.00116
0.00083
0.C 058 0
0.00036
0.00203
0.OC120
0.C3069
0.0 0541
0.00047
0.00201
0.00123
0.00055
0.00534
G.00058
0.00200
0.00125
0.00041
0.00530
0.0 0 0 o 8
0.00201
0.00126
0.00029
0.00527
0.00078
0.00202
0.00127
0.00016
0.0 0 52 6
0.C0087
0.00204
0.00127
0.00004
0 .00527
0.00097
0.0020 7
9.001^7
-0.00008
0.0 0529
0.00107
0.00210
0.00127
-0.00C20
0.00532
C.00118
0.00214
0.00127
-0.0C031
0.00537
0.00126
0.00219
0.00127
-0.00043
0 .00542
0.00139
0.00224
0.00126
-0.00054
0 .0 0549
0.00150
C.00230
0.00126
-0.00066
0.00557
0.00162
0.0 0237
0.00126
-0.00077
0 .00 566
0.00174
0.00244
0.00125
-0.00089
0.0 0576
0.00167
0.00252
0.00125
-0.00101

332
TABLE 14
The
_ , t620
Integrals J
T660 p -U
- J for the
Fluid
LI II
of Table 11
N
J(n;62C1
J(NÍ640)
J(N;660)
0
-0.00270
-0.00153
-0.00307
i
-0.00328
-0.00145
- 0.00052
2
-0.00317
-0.00149
0 .0 0 085
3
-0.00265
-0.00158
0.00160
4
-0.00249
-0.00166
0.C 0199
5
-0.00216
-0.00173
0.00220
6
-0.00187
-0.00173
0.00231
7
-0.00163
-0.00182
0 .00237
6
-0.00143
-0.00185
0.00241
9
-0.00127
-0100188
0 .00244
1 0
-0.00113
-0.00191
0 .00246
1 1
-0.00101
-0. 0 0 19*+
0.00249
1 2
-0.00092
-0.00197
0.00252
1 3
-0.00084
-C . 00 20 0
0.00257
1 s-
-0.00078
-0.00204
0.00262
l 5
-0.00073
-0.00203
0.0 C 2 6 8
1 6
— G.00C69
-0.00213
0.00275
1 7
-0.0 0 066
-0.00218
0.00283
16
-0.00063
-0.00224
0.00292
19
-0.00062
-0.00230
0.00303
20
-0.00062
-0.00237
0.003 1 5
2 1
-C.00062
-0.00243
0.00329
22
-0.00063
-0.00254
0.00344
23
-0.00 066
— 0.0C264
0 .00 361
24
-0.00068
-0.00274
0.00380

333
TABLE 15
m1 , T000 t222 , _ , T
The Integrals J - J for a Lennard-Jones
plus Quadrupole Fluid. pa^ = .931, kT/e = 0.765,
0/ = 0.
707
N
. j( n; ooo ) +
J(N;200 )
J ( N
; 220 )
J(N;221 )
J(NÍ222)
0
• • • •
-0.00119
-0.
4 1893
- 0•25450
-0.06191
1
• • • •
0.00486
-0 .
362 65
-0.19627
-0.05209
2
• • • •
0.00510
-0.
321 30
-0.15984
-0.04474
3
• • • •
C . 0 0 6 7 6
-0.
29111
-0.13583
-0.03907
4
1 .2 7916
0.00797
-0.
2o 724
-0.11918
-0.03468
5
0.77617
0.00896
-0.
24810
-0.10708
-0.03095
6
0 .60370
0.00980
-0 .
23236
-0.09793
-0.02795
7
0.51300
0.01052
-0.
2 1916
-0.09077
-0.02545
3
0.45490
0.01113
-0 .
20792
-0.08501
-0.0 2o34
9
0.41325
C.01166
-0.
19825
-0.08029
-0.02153
1 0
0.30127
0.01212
-0.
1 8988
— 0.07 635
-0.01996
1 1
0 .3 55o2
0.01252
-0 .
18269
-0.07303
-0.01660
12
0.3344o
0.0 1289
-0.
1 7623
-0.07021
-0.01741
13
0.31665
0.0132 1
-0.
1 70o3
-0.06780
-0.01637
1 4
0.30149
0.01352
-0 .
1 3583
— 0.06574
-0.C1544
15
0.20647
0.01331
-0.
1616 5
— 0.06398
-0.014b2
16
0.27743
0.01408
-0.
1 58C 2
-0.C6247
-0.Cl 390
17
0.26749
0.0 1436
-0.
1 5490
-C.06120
-0.C1324
1 6
0.2o90 6
0.01463
-0.
1 5226
— 0.06013
-0.01266
1 9
0.25176
C .01491
-0 .
1 50 05
-0.05925
-0.01214
¿0
0.24547
0.01519
-0 .
14824
—0.05654
-0.21 157
21
0.24007
0.01548
-0.
14680
-0.05799
-0.01125
22
0 .23540
0.01579
-0 .
1 4572
-0.05759
-0.01087
23
0.23162
0.01611
-0.
1 4496
-0.05732
-0.01053
^4
0.24843
G . 0 1 644
-0.
1 44 j3
-0.05719
-C.01023
t/lV = J (n;
n

N
O
1
2
3
4
5
6
7
8
9
1 O
11
1 2
1 3
1 4
1 5
1 6
1 7
1 8
19
20
21
22
2 3
24
334
TABLE 16
T _ t400 t440 c , -j
The Integrals - J for the Fluid
of Table 15
j(n;40c> j(n;420) j(n;4211
0 .0 1913
0.01905
0 .02006
0.02149
0 .02238
0 .0 2323
0.02384
0 .02424
0.02449
0.02463
0 .02469
0.02470
0.02468
0.02465
0.02461
0.02459
0 .02458
0.02458
0 .02460
0.02466
0.02473
0.0 2432
0.02495
0.02511
0.0 252 9
C.0459 1
0.04683
0.04120
0.03847
0.0359 1
0.05358
0.03150
0•C2966
0.02302
0.02667
0.02529
0.024 1 5
0.02314
G.02225
0.02145
0.02075
0.02014
0.01960
0.01912
0.01871
0.01835
C .01805
0.01730
0.01759
C.01742
0.01901
0.01714
0.01533
0.01380
0.C125b
0.01157
0.01077
0.C10 12
0.00963
O. 009 13
0.0 06 74
O.C 084 0
0.00812
0.00736
0.OO 7o 4
0.00745
0.00729
0.00715
0.00703
O.C0693
0.00664
0.00673
0.00672
0.00669
0.00667
JÍNÍ4221
0.00034
-0.00132
-0.00204
-0.00230
-0.00234
-0.00226
-0.002 1 3
-0.00199
-0.00185
-0.00167
-0.00152
-0.OC138
-0.OC124
-0.00111
-0.00099
-0.00088
-0.00077
-0.00066
-0.00056
-0.00047
-0.00037
-0.00028
-0. 00020
-0.00C11
-0.COOC 2
J(N;440)
0.05235
0.05291
0.05187
0.05022
0.04 Q43
0.04571
0.04511
0.04369
0.04242
0.04131
0.04035
0.03953
0.03383
0.03824
0.03776
0.G3759
0.03710
0.03691
0.03680
0.03677
0.C3632
0.03694
0.03714
C.03742
0.03776

N
O
1
2
3
4
5
6
7
8
9
1 O
11
12
1 3
1 4
15
1 ó
1 7
1 6
19
20
21
22
¿o
24
335
TABLE 17
The Integrals J
441
,600
of Table 15
for the Fluid
j( n;44 i )
0.0 4233
0.03952
0.03706
0.03499
0.0 3o2o
0.03177
0.03051
0.0 29 4 _>
0 • 02850
0 .02709
0 .02700
0.02640
0•02568
0.02545
0 .0250 9
0.02479
0 . C 2 4 5 o
0.02436
0.02426
0 .0 241 9
0.02417
0.02420
0 . 0 2423
0.02440
0.0245o
J ( N I 4 4 2 )
0.01935
0.02022
0.01980
0.01692
0.01793
C.01698
0.01611
0.01534
0.0 1 4 o 5
0.01405
0.0 1353
0.0150 7
0.01267
0.0 1 2 3 1
0.01201
0.01174
0.01152
0.01132
0.01116
0.01103
0.0 1093
0.01085
0.0 1 079
0.01076
0.01076
J(NJ443)
0.01160
0.00695
0.00739
0.00640
0.00572
0.00522
0.00463
0.00451
0.0 0425
0.00402
0.00361
0.0 0 355
0.0 034 ó
0.00331
0.00317
0.003C*
0 . C 0 2 91
0.00279
0.C 02 68
0.00257
0.00246
0.00236
0.00226
0.00216
0.00206
J(N;444)
-0.00137
0.00027
0.00119
0.00154
0.00165
0.00160
0.00152
0.00142
0.00132
0.00123
0.00115
0.00107
0.00101
0. 00095
0.00089
0.0C035
0.00030
C.00077
0.00073
0.00070
0.0 0 0o8
0.00066
0.00064
0.00062
0.0C060
J ( N ; 6 0 C )
-0.00960
-0.00875
-0.00324
-0.00786
-0.00753
-0.00723
-0.00694
-0.00667
-0.00642
-0.00620
-0.00599
-0.00580
-0.00563
— 0.0 0 54 3
-0.00 535
-0.00523
-0.00 5 1 3
-0.00504
-0.00496
-0.00490
-0.00485
-0.00481
-0.00473
-0.00475
-0.00474

336
TABLE 18
. _ . t620 t660
The Integrals - J
of Table 15
for the Fluid
N
J(Ni620 )
J(N;640)
J(N;660)
0
-0.00247
-0,01178
-0
.00833
1
-0.00538
-0.01233
-0
.00741
2
-0.00673
-0.01232
-0
. 00650
3
-0.00736
-0.01199
-0
.00573
4
-0.00764
-0.0115o
-0
.00524
5
-0.00778
-0.01111
-0
.00434
6
-0.00784
—0¿01063
-0
.00455
7
-0.00788
-0.01029
-0
.00433
8
-0.00790
-0.00994
-0
.0041 8
9
-0.0C792
-0.00963
-0
.00407
1 0
-0.00795
-0.00937
-0
.00399
1 1
-0.00797
-0.009 14
-0
.00394
1 2
-0.00800
-0.00894
-0
.00390
1 3
-0.C08C3
-0,0 0877
-0
.00388
1 4
- 0.00807
-0.00863
-0
.00387
15
-0.00812
-0,00352
-0
.00337
1 6
-0.00817
-0,00344
-0
.00383
17
-0.00823
-0.00837
-0
. 00390
1 8
-0.00830
-0,0 0333
-0
. 00392
19
-0.0 0838
-0.00331
-0
.00395
20
-0.00346
-0.00331
-0
.00398
21
-0•00856
-0 .00333
- 0
.00402
22
-0.00866
-0.00337
-0
.00407
23
-0.0087d
-0,00342
-0
.00411
24
-0.00891
-0.00350
-0
.00417

N
O
1
3
4
5
6
7
8
9
10
1 1
1 2
18
14
15
16
1 7
13
IS
20
21
22
23
24
337
TABLE 19
^ , T000 222 . T , T
The Integrals J - for a Lennard-Jones
plus Quadrupole Fluid. pg2 = .85, kT/e = 1.294,
Q/(eo5)1/2 - 1.0
j(n;oco) j(n;20ü) j(n;220) j(n;2211 j(n;222)
• • • •
• • • •
1 .2 7684
0.77403
0.6041o
0.51730
0.46374
0.42709
0.40041
0.88028
0.3 o4 8 0
0.35282
0.34361
0.3 366 9
0 .33170
0.32340
0•32660
C.32618
0.32703
0 .3290 9
0.33282
0 .3 3 6 68
0.34216
0.01296
0.0 1 2 2 1
C.01333
0.01504
0.01679
0.01847
0.02000
C . 02139
0.02266
0.0 238 2
0.02491
0•02595
0.02696
0.02796
0.C2898
0.0 30 0 1
0.03109
0.03222
0.03341
0.03468
0.03603
0.03747
C.03903
0.04069
0.04249
-0.50332
-0.42230
-0.37142
— 0.83725
-0.31298
-0.29499
-0.28123
-0. 27052
-0.26212
— 0.2 5556
— 0.250 53
— 0.2 4~6 8 1
-0.24423
-0.2426 3
-0.2420 7
-0.24235
-0.24347
-0.24540
-0.24811
-0.251 61
-0.2 5589
-0.26096
-0.26684
-0.27356
-0.28113
-0.28545
-0.21850
-0.17887
-0.15364
-0.13o61
-0.12456
-0.I 1574
-0. 109 14
-0.10413
-0.10031
-0.09740
-0.09523
—0.09366
-C.09262
— 0.09 20 2
-0.09183
-0.09201
-0.09254
-0.09339
-0.09456
-0.09604
-0.09733
-0.09 993
-0.10235
-0.10509
-0.06940
-0.05503
-0.04514
-0.03305
-0.03273
-0.02373
-0.02555
-0.02300
- C . 0 2 0 9 1
-0.01917
-0.01771
-0.01647
-0.01540
-0.01448
-0.01367
-0.01297
-0.01234
-0.01179
-C.01129
-C.01034
-0.01044
-0.01007
-C.00973
-0. 00 94 1
-0.0C912

N
O
1
2
3
4
5
6
7
6
9
1C
11
1 2
13
1 4
1 5
16
1 7
1 8
19
20
21
22
23
24
338
TABLE 110
400 440
The Integrals for the Fluid
of Table 19
J(n; 400 1
0.02318
0.03179
0.03731
0.04113
0 .0 43 a 8
0.04593
0.04731
0.04877
0.04983
0.05078
0.03169
0.05260
0 . 0 53 5 3
0 • 054o7
0.05567
0.0 5 6 6 8
0.05822
0.0 556 9
0.06131
0.0 630 9
0.06504
0.06718
0.06953
0.07209
0.07489
J(N;420)
0.04303
0.04410
C.04239
0.04033
0.03822
0.03620
0.03437
0.03273
0.03130
0.03006
0.02900
0.02810
0.02733
0•0^670
0.02619
0.02578
C.02547
0.0 2526
0.02513
0.02503
0.02511
0.02522
C.02339
0.02564
0.02596
J(N Í 421 )
0.C1371
0.01767
0.01664
0.01574
0.01499
0.01438
0.01339
0.01350
0.01319
0.01294
0.01275
0.01262
0.C1252
0.01247
0.01246
0.01248
0.0125-4
0.01263
0.01276
0.01292
0.01312
0.01334
0.C 1531
0.01391
0.01425
J(NÍ 422)
0.00810
0.00502
0.00577
0.00324
0.00302
0.00294
0.00295
0.00300
0.00506
0.00320
0.00334
0.00350
0.00363
0•0 0 38 o
0.00410
0.00434
0.00459
0. C0486
0.00513
0.00 54 7
0.00330
0.006 16
0.00655
0.00697
0.00741
J(N;4 40 J
0.09781
0.09103
0.08627
0.08279
0.08019
0.07825
0.C 7 68 3
0.07533
0.07520
0.C7489
0.07436
0. 0751 1
0.07560
0.07633
0.07730
0.07851
0.07995
0.C8 163
0.08355
0.03573
0.08817
0.09090
0.09391
0.C9724
0.10090

N
O
1
2
3
4
5
6
7
3
9
1 O
I 1
12
13
14
1 5
16
17
1 3
1 9
20
21
22
23
24
339
TABLE Ill
441
The Integrals J - J
600 r
for the
Fluid
of Table
19
J ( N 5 4 4 l )
J(NJ 44 2)
J(Ni443)
j(n;444 )
' JCNÍ600)
0 .0 6507
0.03501
0.01331
-0.01 1 90
-0.00661
C.059o2
C.02978
0.01360
-0.00549
-0.00693
0.05622
0.0 26 5 0
0.01117
-0.00265
-0.00679
0.05359
0 . 02430
0.00975
-0.00138
-0.C0646
0.05161
0.0227 4
0.00381
-0.00081
-0.00 607
0.05010
0.02162
0.00812
-0.00056
-0.00567
0.04897
0.02079
0.00757
-0.00C45
-0.00530
0 .0431 5
0.02018
0.0071 1
-0.00040
-0.0C49O
0.04760
0.01974
0.00671
-0.00037
-0.00466
0.04727
0.01943
0.00636
-0.00036
-0.00440
0.04715
0.C 1 923
0•00605
-0.00035
-0.0041o
0.04722
0.01913
0.0 05 73
-0.00034
-0.00396
0.04745
C.019 1 0
0.0 0553
-0.00C33
-0.00379
0.047 85
0.01915
0.00531
-0.00031
-0.00363
0.0484 1
0.01927
0.00511
-0 . 00 02 9
-0.00350
0.04913
0.01944
0.00494
-0. 00 0¿7
-0.003o9
0 .05000
0.01968
0.0C477
- 0.O0025
-0.00329
0.051G¿
0.01998
0.00463
-0.00023
-0.00320
0.05220
0.02033
0.0 04oO
-0.CO 02 0
-0.CO 312
0 .0 5354
0.02075
0.00439
-0.0001 7
-0.00306
0.05505
0.02122
0.00428
-0.00014
-0.00300
0.05673
0.02175
0.00419
-0.00010
-0.00294
0 •0536 0
0.02235
0.004 1 0
-0.00007
-0.00289
0.06066
0.02300
0.00403
-0. 000 03
-0.00285
0.0 o 2 9 2
0 • 0 a 3 7 Ó
0.C0397
0. 0000 1
-0.00230

340
TABLE 112
T . t620 640 . r1 . ,
The Integrals for the Fluid
of Table 19
N
J(N;620)
J(N;640)
0
-0.01498
-0.02022
1
-0.01913
-0. C 1 807
2
-0.02094
-0-01891
3
-0.02178
-0.01621
4
— 0.0 ¿222
-0.01573
5
-0.02249
-0.01541
6
-0.02270
-0.01516
7
-0.0229 1
-0.01493
8
-0.02314
-0.01484
9
-0.02340
-0.01476
1 0
-0.02371
-0.01472
1 1
-0.02406
-0.01473
12
-0.02447
-C.01478
1 3
-0.024^4
-0.01437
1 4
-0.02547
-0. 0 1 50 0
1 5
-0.02607
-0.01518
1 6
-0.02674
-0.01539
1 7
-0.02748
- 0•01566
1 8
-0.02830
-0.01596
1 9
-0.02921
-0.01632
20
-0.03021
-0.01672
2 1
- 0.03131
-0.01717
22
- 0. 03252
—0.01768
23
-0.03384
-0.01824
24
-0.03529
— 0.01885

N
O
1
2
3
4
5
6
7
6
9
10
11
1 2
1 3
1 4
1 5
16
1 7
1 8
19
20
21
22
23
24
341
TABLE 113
m1 , 000 t222 . . , T
The Integrals J - J for a Lennard-Jones
plus Anisotropic Overlap Fluid. pcT
kT/e = 1.291, 6 = 0.10
= .85,
J(N;0QOJ J(NJ200.) J(N;220) J(N;221) J(N;222)
• • • •
1.26830
0.76243
0.68975
0.5 005 0
0.44451
C.40534
0 .3 7642
0.354 1 0
0 .33647
0.32236
0.31103
0.30195
0.29478
0.23924
0 .28513
0.28251
0 .2 80oo
0.28011
0.26057
0 .28202
0 .28442
-0.01432
-0.01618
-C .0 1861
-0.02126
-0.02387
-0.02651
-0.02855
-0.03059
-0.03^46
-0.03419
-0.03583
-0.03741
-0.03897
-0.04053
-0.04211
-0.04375
-0.04545
-0.04724
-0.04913
-0.05114
-0.0532 9
-0.05510
-0.05807
—0.06073
-C.06359
0.00475
0.00 697
0.00763
0.00780
0.00788
0.00799
.0.0 08 18
0.00843
0.0 087 5
0.00912
0.00953
0.C099S
O. Oil 0 47
0.01 099;
0.01154
0.01215
0.01276
0.01343
0.01414
0.01489
0.01569
0.01654
0.01744
0.01839
0.01940
0.00002
-0.00071
-0.00062
-0.00026
0.00009
0.00042
0.00068
0.00089
0.00105
0. 00119
0.00130
0.00139
0.00148
0.00155
0.00163
0.00170
0.00177
0.CC134
0.00192
0.00200
0.00208
0.00217
0.00226
0.00236
0.0024 7
-0.00137
-0.00153
-0.00121
-0.00107
-0.00092
-0.00C.79
-0.00063
-0.00059
-0.00051
-0.00045
-0.00040
-0.0C036
-0.00033
-0.00030
-0.00028
-0.00 026
-0.00024
-C . 00023
-0.00021
-0.00020
-0.00019
-0.00018
-0.00013
-0.00017
-0.00017

342
TABLE 114
The Integrals J
400
J
440
of Table 113
for the Fluid
N
j( n; 400 )
J (Ixl ; 420 )
jin;421)
J(NJ422)
J(N1440)
0
0.0 0291
-0.00736
-0.00204
-0.00261
-0.00595
1
0.00193
-C.00454
—C.00166
-0.0C232
-0.00424
2
0.00156
-0.0 05l8
-0.00 149.
-0.00209
-0.00316
3
0.00149
-0.00255
-0.00141
-0.00191
-0.00244
4
0.00157
-0•0 0 2 2o
-0.00136
-0.00178
-0.00194
5
0.00171
-0.00219
-0.00133
-0.00158
-0.00157
6
0.00189
-0.00219
-0. OiO 1 29
—0.00160
-0.00 129
7
0.00209
-0.00225
-0.OC126
-0.00154
-0.00 107
a
0.0 0231
-0.00232
- 0 . C 0 1 2 2
-0.00150
-0.00090
-i
0.00254
-C.00242
-C.00119
-0.00146
-0.00075
10
0.0 02 7 8
— 0.00253
-0.00116
-0.00144
-0.00063
11
0.00304
-0.00¿64
-0.00114
-0.0C142
-0.00052
i 2
C •0 Ojj1
-0.00277
-0.00 1 l 1
-0.0014 1
-0.00042
1 3
0 .0 0359
-0.00291
-0.00110
-0.00140
-0.00033
l 4
0 .00390
-0.0080o
-0.00109
— 0.0C140
-0.00023
15
0.0 04aa
-0.00322
-0.001C9
-C.00 140
-0.00014
1 6
0.00456
-0.00839
-0.00110
-0.00140
-0.00004
1 7
0.00492
-0.00359
-0.00 1 11
-0.00141
0.00005
18
0.00531
-0.00379
-0.00113
-0.0C142
O.OCOlo
1 9
0 .0 057a
-0.004 02
-0.00115
-0.00143
0.0 C 0 27
2C
0.00517
-0.00427
-0.00119
-0.00144
0.00038
21
0 . 0 0664
-0.00454
-0.00122
-0.0014o
0.00050
22
0.00715
-0.00483
—0.C0127
-0.00147
0.00063
¿3
0.00770
-0.0 051 8
-0.00132
-0.00149
0.00077
24
0.0 0829
—0.00350
-0.C 0l53
-0.00151
0.0C092

N
O
1
a
3
4
5
6
7
8
S
10
1 1
12
1 3
14
1 6
1 6
1 7
1 8
i 9
ao
ai
aa
23
¿4
343
TABLE 115
The Integrals for the Fluid
of Table 113
J( Ni 44 1 )
0.00105
-0.00051
-0.00116
-0 .00138
-0.0 0139
-0.00151
-0.00119
-0.00106
-0 .00093
-0.0 008 0
-0.0006 9
-0.C 0058
-0 .0 004 9
- 0.0 0040
-o .ooo3a
-0 ,0002o
-0 .00013
-0.00011
-0.00005
0.00000
0 .00005
0.00011
0.00015
o.oooao
0.00036
J(NÍ442J
0.00599
0.00187
-0.000 13
-0.00105
-o.ooi4a
-0.00151
-0.00146
-0.00135
-o .ooiai
-3.00107
-0.0009a
-C .00078
-0.00065
-0 .00053
-0.00041
-0.00031
-o.oooa i
-0.00011
-0.00002
C.00006
0.00014
o. cooaa
0.00039
0.000 36
0.00043
J(N;443)
0.00043
0.00004
-0.00035
-0.00045
-0.00057
— 0.00064
-0.00067
-0.000o8
-0.C 0067
—0.00066
-0.00064
-0.0 0063
-0.00063
-0.00061
-0.0C061
-0.00062
-0.00062
-0.00 064
— 0.C006Ó
-0.COO 70
-0.00074
-0.00080
-0.0 0Cd6
-0.000 95
-0.00105
J(N;444)
0.00385
0.00239
0.00163
0.00130
0.OOC98
0.000Q6
0.00080
0.00078
0.00C79
o.ocoa i
0.00084
o. c o o a 7
0.00093
0.00C96
0.C0103
0.00103
0.001 1 4
0.GO 121
0.OC133
0.00136
0.00144
0.00154
0.00164
0.00174
0.00136
j(n;600)
-0.00323
-0.0C385
-0.00400
-0.00393
-0.00377
-0.00353
-0.00340
-0.00335
-0.00313
-0.00303
-0.00296
-0.00391
-0.00289
-0.00288
-0.0028 9
-0.00292
-C.00295
-0.00301
-0.GO 303
-0.00315
-0.00524
-0.00335
-0.00345
-0.00 359
-0.00373

344
TABLE 116
The
, 620
Integrals J
66° ,
- J for
the Fluid
11 LI
of Table 113
N
J(N;620)
J(NÍ640)
J(Ni 660)
C
-G.00111
0.00076
-0.00898
1
-0.00140
0.00 063
-0.00636
2
-0.00 142
0.00057
-0.00490
3
-0.00132
0,0005o
-0.00406
4
-0.00119
0.0 0055
-0.00355
5
-0.00106
0.0C057
-0.00323
6
-0.00093
0i00053
-0.00302
7
-0.00081
0i00060
-0.00288
a
-0.00072
0.00061
-0 .00277
9
-0.00063
0.00063
-0.00270
10
-0.00056
0.00065
-0 .00265
11
-0.00050
0.00068
-0.00261
1 2
-0.00043
0.0007 1
-0.00258
1 3
-0.00040
0*00073
— 0.0 0257
1 4
-0.00036
0.00077
-0.00255
1 5
- 0.00033
0.00060
-0.00256
1 ó
-0. 00030
0.00084
-0.00253
17
-0.00027
C.00089
-0.00257
i a
-0.00C25
0.00093
-0.0 02 59
1 9
-0.00023
0.00099
-0 .00262
20
-0.00021
0.00104
-0.00265
21
-0.00020
0*00110
-0.00268
22
-0.00018
0.00117
-0.00272
23
-0.00016
0.00124
-0 .00277
24
-0.00015
0.00132
-0.00283

345
TABLE 117
The Integrals for a Lennard-Jones
—2 n n ^
plus Anisotropic Overlap Fluid, pa = .85,
kT/e = 1.287, 5 =• 0.30
N
j { n ; o o o )
Jln;200 )
J(N;220)
J(N;221 )
J(N ;222)
0
• • • •
-0.0 762 0
0.09534
0.03353
0.00746
1
• • • 0
-0.07097
0.09093
0.C2797
0.00732
2
• • • »
-0.C7531
0.08823
0.02366
0.00741
3
• * • 4
-0.08490
0.08739
0.02030
0.00753
4
1 .2 040 9
-0.09553
0.08319
0.01767
0.00779
5
0 .78573
-0.10652
0.09041
0.01558
0.00803
6
0.62071
-0.11743
0.09382
0.01391
0.00828
7
0.5 3933
-0.12621
0.09829
0.01256
0.00857
e
0 .49194
-0.13395
0.10374
0.01146
0.00888
9.
0 .46219
-0. 1 4982
0.11013
0.01056
0.00924
1 0
0.44318
-0.16103
0. 1 1747
0.00982
0.00964
11
0.43154
-0.17275
0.12578
0.00920
0.01009
12
0.42545
-C.185 1 9
0.13514
0.00868
0.C10 60
1 3
0 «4236 7
-0. 19855
.0.14 564
0.00825
0.01118
14
0.4 262 0
-0.21301
0.15737
0.00790
0.01 183
1 5
0.43206
— 0.22680
0.17049
0.00760
0.01255
16
0.44128
-0.2461 1
0.16513
0.00735
0.01337
1 /
0.45381
-0.26520
0.20149
0. C07 1 6
0.Cl*29
10
0.46966
"0.28632
0.21977
0.00700
0.01532
1 9
0.48895
-0.30974
0.24021
0.00688
0.01647
20
0.51137
-C.33578
0.26309
0.00679
0.01775
21
0.53867
-0.36401
O.28872
0.00673
0.Cl 920
22
0.56967
-0.39722
0.31748
0.00670
0 .C2081
23
0.60526
-0.43347
0.34976
0.00659
0.02262
2 4
0.64592
-0.47407
O•38605
0.00672
0.02464

N
O
1
2
3
4
5
6
7
8
9
I 0
1 1
12
1 3
1 4
1 5
16
17
18
19
20
21
22
2 3
24
346
TABLE 118
t n t400 440
The Integrals J - J
-n-
-n-
of Table 117
for the Fluid
J(n;400 )
0.02677
0.02933
0.0 3359
0.05803
0.04270
0.0 4737
0.05212
0.05705
0.06224
0.06777
0.07376
0.08028
0.08744
0.0 9584
0.10409
0.1 l 382
0.12465
0.15674
0 .1502 7
0 . 16 d4 2
0.18242
0.20151
0.22300
0.24720
0.27450
J(N Í 420)
-0.C3674
-C.03283
-0.08166
-0.03262
-0.03445
-0.03702
-0.040 15
-0.04378
-0.04787
-0.05246
-0•0d7o6
-0.06329
— 0.06964
-0.07673
-0.08464
-0.09349
-0.1 0340
-0.1 1451
-0.12699
-0.14102
-0. 1538 1
-0.1 7461
-0.19470
-0.21740
-0.24303
J(N;4211
-0.00511
-0.00366
-0.00235
-0.00242
-0.00221
-0.00212
-0.00210
-0.00213
-0.00218
-0.00224
-0.00232
-0.00240
-0.002d0
-0.C0260
-0.00271
-0.00283
-0.00296
-0.00311
-0.00328
-0.0034 7
-0.00368
-0.GO 392
-0.00420
-0.00451
-0.00466
J(Ni 4221
0,0021 1
0.00064
-0.00031
-0.00095
-0.00142
-C.00178
-0.00210
-0.C3239
-0.C0267
-0.00297
-0.00327
-0.00360
-0.00396
-0.00436
-0.00478
-0.00525
-0.00577
-C.00634
-0.00697
-0.0G7O7
-0.00844
-0.00929
-0.01024
-0.01126
-0.01245
J(N J440)
0.01794
0.01 570
0.01680
0.01767
C.0190 1
0.02069
0.02266
0.02491
0.02744
0.03028
0.05348
0.03708
0.04113
0.04569
0.05083<