QUANTITATIVE PREDICTION AND INTERPRETATION
OF INFRARED SPECTRA
By
STEVEN CHIN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1984
ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to
Dr. Person, without whom this work would not have been
possible. He has been everything I could possible hope
for in an advisor, and more. His patience, understanding
and guidance have been the primary inspiration for this
work.
I would also like to express my thanks to Drs. Tony
Ford and Krystyna Szczepaniak, with both of whom I was for
tunate to have many stimulating discussions.
I am also grateful to all the past and present members
of this research group for their friendship over the years.
Each of you, in your own special way have made an important
contribution to this work.
Partial financial support from NSF Research Grant No.
CHE8101131 is gratefully acknowledged, as is a supplemental
fellowship from the Graduate School at the University of
Florida. In addition, support of some of the computation
time from the College of Liberal Arts and Sciences at the
University of Florida, and from the Northeast Regional Data
Center is gratefully acknowledged.
Finally, I would like to thank Ms. Pam Victor for making
the extra effort that was needed to put this thesis in its
final typed form.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ................................... ii
ABSTRACT ........................ ............ ....... v
CHAPTER
1 INTRODUCTION ......... ..... ...... ......... ..... 1
2 THEORY .......................................... 18
2.1 Coordinate Systems ........................ 18
2.2 Normal Coordinate Analysis ............... 22
2.3 Infrared Intensity Analysis ............... 25
2.4 Transformation to Different Coordinate
Systems ................. ................. 29
2.4.1 Internal Force Constants .............. 30
2.4.2 Atomic Polar Tensors ................. 32
2.4.3 The S Coordinates ..................... 33
2.5 Experimental Infrared Intensities ......... 35
2.6 Theoretical Infrared Intensities .......... 36
2.7 Characteristic Invariants of Atomic
Polar Tensors ............................ 39
2.8 Quantum Mechanical Interpretation of Atomic
Polar Tensors: The Charge, Charge Flux,
Overlap Theory (CCFO) .......... .......... 44
3 THE WATER MONOMER ............................. 54
3.1 Calculation Procedure ..................... 55
3.2 The Predicted Spectra for the Water
Monomer ................................. 58
4 THE LINEAR WATER DIMER ......................... 72
4.1 The Predicted Spectrum for the Linear
Dimer ............................... ..... 75
4.2 Intensity Analysis for Linear Pair of
Water Molecules .......................... 83
4.3 Frequency Analysis for the Linear Pair
of Water Molecules ....................... 94
4.4 Further Calculations on Weaker and Stronger
Interacting Linear Pairs of Water
Molecules ................................ 102
Page
CHAPTER
5 THE CYCLIC WATER WATER COMPLEX ................. 122
5.1 Intensity Analysis for the Cyclic Complex.. 130
5.2 Frequency Analysis for the Cyclic Complex.. 139
5.3 Further Calculations on the Cyclic Complex
at Different Interaction Distances ....... 147
5.4 Comparisons Between Linear and Cyclic
Complexes .................. .............. 158
6 THE INPLANE BIFURCATED WATERWATER COMPLEX .... 161
6.1 The Predicted Spectrum for the Bifurcated
WaterWater Complex (R = 2.89 Ao) ........ 162
6.2 Additional Calculations on the Bifurcated
WaterWater Complex ...................... 175
7 THE OMH20OM COMPLEX, M = 0, 1, 2 ........... 179
7.1 Intensity Theory for Charged Species ...... 180
7.2 The OM.H20OM Complex .................... 184
7.3 Intensity Analysis for the OM.H20OM
Complex ...... ............................ 186
7.4 Frequency Analysis for the 0'H20OOM
Complexes ................................ 198
7.5 Model for the Infrared Spectra of Beta
Alumina .............. .................... 206
8 IONIZED WATER, H20 ............................ 210
9 GENERAL COMMENTS AND FINAL SUMMARY ............. 223
9.1 Comments on the Charge Flux, Charge, and
Overlap Tensors ..................... .... 226
9.2 Other Applications for These Calculations.. 234
APPENDIX
A SUMMARY OF PROGRAMS ........................... 238
B NORMAL COORDINATE TRANSFORMATION MATRICES ...... 245
C REDUNDANT INTERNAL COORDINATES ................. 247
REFERENCES ............................................. 251
BIOGRAPHICAL SKETCH ................................ 257
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
QUANTITATIVE PREDICTION AND INTERPRETATION
OF INFRARED SPECTRA
By
Steven Chin
April 1984
Chairman: Willis B. Person
Major Department: Chemistry
Ab initio quantum mechanical calculations at the self
consistent field level have been performed for interacting
water molecules to predict their infrared spectra. These
calculations have been made using Gaussian 76 and a 431G
basis set. The calculated ab initio force field has been
used to predict the fundamental absorption frequencies and
the corresponding normal coordinate transformation. This
normal coordinate transformation has then been used in con
junction with Atomic Polar Tensors (APTs) that were also
found in the ab initio calculation to predict the absolute
integrated intensities for these fundamental normal
vibrations.
The role that the chemical environment plays in the
predicted spectra for interacting water molecules has been
investigated by analyzing the predicted spectra for different
assumed orientations of the "complex" at various assumed.inter
molecular distances, and by varying the electron donating
properties of the electron donor. The former effect was in
vestigated by predicting spectra for water pairs in the linear,
cyclic, and bifurcated configurations, at varying intermolecular
distances. The latter effect was studied by calculating the
spectra for water completed with two oxygen atoms which have
varying charge (0M H20 0 where M = 0, 1, or 2) where
the oxygen atoms with the larger negative charge are stronger
electron donors. Finally, the spectrum for ionized water,
H20 is also predicted.
These calculations indicate that both the frequencies
and intensities of the vibrations for water interacting with
its environment are predicted to be strongly dependent upon
the nature of this environment. The predicted spectra for
these water molecules in different chemical environments are
then compared with the predicted spectrum for isolated mono
meric water, and also experimental spectra when available.
The vibrational changes from isolated water to the interact
ing water molecule in these different environments are then
analyzed and discussed to provide some insight into the effect
of intermolecular interactions on the vibrational spectrum of
the interacting molecule.
CHAPTER 1
INTRODUCTION
The infrared spectrum of a molecule is an extremely rich
and valuable source of information. For many years it has
been used in both the qualitative and quantitative analysis
of chemical compounds. The concept of group frequencies (1),
and its use in the identification of molecular compounds is
one of the most fundamental and fruitful applications of
infrared spectroscopy. As a result, most experimental spec
troscopists, and theoreticians working with them, tradi
tionally placed great emphasis on the locations of the
band centers, or absorption frequencies. In recent years,
however, there has also been an increased interest not only
on the fundamental infrared absorption frequencies, but also
on the absolute integrated intensities (2,3). Specifically,
there has been a great deal of interest in the parameteriza
tion and interpretation of infrared intensities. The use of
electrooptical intensity parameters (EOPs) (4,5) has been
pioneered by the Soviet workers (6), while the use of Atomic
Polar Tensors (APTs) was first introduced by Biarge, Herranz,
and Morcillo (7), and later reformulated by Person and Newton
(8) and used in conjunction with quantum mechanical calcula
tions of signs of (3P/3Q)s. Both methods have had good success
in the parameterization of infrared intensities (9). The APT
method is particularly useful in quantum mechanical calculations
and therefore has been adopted here.
The success of the quantum mechanical calculations of APTs
offers hope that in the future the prediction of infrared in
tensity parameters may be almost routine. Using these inten
sity parameters, the absolute integrated intensities can be
predicted, provided that the normal coordinate transformation
is well known. The normal coordinate transformation can be
easily obtained from the force field of the molecule by per
forming a normal coordinate analysis (10,11). Unfortunately,
the available experimental data are usually not sufficient to
determine completely an accurate vibrational force field. For
this reason quantum mechanical calculations of vibrational
force fields have become increasingly more popular (1216).
These calculations, coupled with the calculation of APTs, have
made possible the completely ab initio prediction of infrared
spectra (including both band frequencies and absolute
intensities).
These predictions are useful for several reasons. They
can be extremely helpful in assigning complicated spectra
(17), especially spectra for relatively large (> 10 atoms)
molecules. A second application of these types of calcula
tions concern chemical systems which cannot be isolated in
pure form and packaged in a bottle. These systems include
radicals and ions (18) frozen in raregas matrices, reaction
intermediates observed by timeresolved spectroscopy or
observed on catalytic surfaces, molecular fragments existing
in flames and plasma, small molecular compounds that are be
lieved to exist in space, and dimers or other intermolecular
complexes that are found in raregas matrices or supersonic
expansions. In recent years there has been an increased
interest in the application of vibrational spectroscopy
towards characterizing these unstable and hard to isolate
molecular species. However, this task is made much
more difficult by the absence of spectroscopic standards
and standard spectra. Therefore, any spectral determination
of the concentration of these transient species must rely on
a separate estimate of its intensities. The development of
ab initio calculations of these intensities appears to be
well suited for making these estimates. Additionally,
the spectral patterns predicted by these calculations (includ
ing both wavenumbers and intensities) can be qualitatively
used as a direct confirmation of the spectral assignment for
the species, especially in the absence of standard spectra.
The calculated intensities would be especially valuable in
the case that the observed wavenumbers of the unstable species
could not be compared with those of a structurally similar
stable molecule; or in the opposite situation where the ob
served wavenumbers matched equally well with more than one
structure. In situations such as these the correct assign
ment must match the predicted wavenumbers as well as the
predicted intensity pattern.
There are very encouraging developments in the ab initio
predictions of infrared spectra of unstable or hardtoisolate
molecular species. However, these methods must be further
tested if one is to be able to routinely predict reliable
spectra for these types of species. Therefore, we have used
the ab initio method to predict and interpret spectra of
some of these types of species.
The water dimer has been the subject of many experimental
(1929), and theoretical (3041) investigations. One reason
for this interest is that it is thought to be a prototype
model for liquid water, which is of paramount importance in
the life of biological systems. A second reason is that the
water dimer is a classic example of a donoracceptor complex,
in this case, specifically a hydrogen bonded complex. The
water dimer represents a unique example of a hydrogen bonded
system because one water molecule is the electron donor and
another is the electron acceptor.
The importance of vibrational spectroscopy in the study
of Hbonded systems is well documented (42,43). The infrared
spectra of such completed species exhibit characteristic
spectral changes when compared with that of the corresponding
isolated molecules and with the spectra of other complexes.
The comparisons among these spectra are valuable sources of
information concerning the properties and characteristics
of these systems.
The hydrogen bond interaction can be described as an
intermolecular interaction through a hydrogen atom. Typi
cally, the interaction is of the type RAHB, where RAH is
an electron acceptor, B is an electron donor molecule, and
the hydrogen bond is the HB interaction. When water is
both the electron acceptor and the donor, AH is the OH bond
of one water molecule, and the electron donor B is the oxygen
atom in the other water molecule. In the water dimer, this
interaction is slightly more complicated by the presence of
two OH bonds. In the isolated molecule, these two bonds are
symmetrically equivalent, and therefore are strongly coupled
to form two stretching vibrations, a symmetric stretch and an
asymmetric stretch. In the complex the symmetry is reduced in
relation to the isolated monomer, the 0H bonds are no longer
symmetric, and the coupling between these two bonds is reduced.
In general, the forces governing intramolecular coupling are
much stronger than those governing hydrogen bonding. Therefore,
this reduction in coupling between the two OH bonds is important
to consider in analyzing complexes involving water molecules.
For Hbonded complexes, the vibration most perturbed by
the interaction is the symmetric stretch. This band undergoes
a shift to lower energy, and an increase in intensity. The
similarities and differences between these two changes in the
spectra of Hbonded systems are vital to our understanding of
these complexes.
The frequency shift is usually thought to originate in a
weakening of the AH bond as the Hbond is formed. This in
turn is believed to be due primarily to two effects: the
polarization effect, where the donor molecular polarizes elec
tron density from the AH bond into the nonbonding region of
the A atom, where it no longer compensates for the AH nuclear
repulsion; and the electron transfer.effect, where the donor
molecule donates electron density into the antibonding
orbitals of the acceptor molecules, thereby weakening the bond.
Regardless, the net result is that the AH bond is weakened,
and so the frequency shifts towards lower energy. This effect
can be analyzed by calculating the internal force constants
for the complex.
The intensity enhancementis well known experimentally,
but not as well understood. The electrostatic polarization
and chargetransfer effects just mentioned cannot individually
account for the great enhancement in the intensity. Zilles
and Person (39) have recently applied a quantum mechanical
model of the APTs in order to interpret the intensity enhance
ment for the water dimer. They have found that the largest
single most important contribution to the intensity enhance
ment is from the increase in charge flows in the complex
compared to the monomer. This interpretation can be taken
one step further by attempting to compare the nature of these
charge flows in the linear dimer with those in other systems
involving water as the electron acceptor.
Furthermore, the reduction in coupling between the two
OH bonds usually creates some mixing between the symmetric
stretching vibrations and both the bending and assymetric
stretching vibrations. So, in effect, it is no longer strictly
a symmetric stretching vibration, but a mixture of all three
vibrations. Naturally, a consequence of this mixing is that the
normal vibrations in the complex are no longer the same vibra
tions as those in the isolated molecule. The intensities, as
well as the frequencies, are strongly dependent upon the form
of the normal vibrations. The effect of these changes on both
the frequencies and intensities are also important to
understanding the spectral changes present in the complex,
and these effects will be examined in detail.
The experimental studies of the infrared spectrum of the
water dimer are quite varied with respect to the interpretation
of the results. Most of these results have been previously
summarized (41 and references citedtherein). The most gener
ally accepted interpretation of the infrared spectrum is that
of Tursi and Nixon (19). They have assigned the six high fre
quency modes of the water dimer in a nitrogen matrix. Their
assignment is consistent with a linear structure (see Fig.
1.1) for the dimer, and is accepted by many as the correct
interpretation for the dimer. Other workers prefer the cyclic
(see Fig. 1.2) structure (20,21,24), while still otherspostu
late the existence of the bifurcated structure (see Fig. 1.3)
(26).
Most theoretical calculations support the linear struc
ture as the most stable geometry for the water dimer (31,34).
Zilles and Person (39) have used the experimental force field
of Tursi and Nixon (19) for this linear structure and quantum
mechanical APTs to predict and interpret the infrared intensi
ties of the linear dimer. To expand and further develop our
understanding of the infrared spectra of interacting molecules
we have used ab initio quantum mechanical calculations to pre
dict the vibrational force fields and APTs for not only the
linear configuration, but also the cyclic and the bifurcated
s.H
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structures. We will examine the spectra predicted for these
three different configurations and discuss the differences
among them, as well as the differences between our calcula
tions and those of Zilles and Person (39). In addition, we
have tried to simulate the extremum, namely complete electron
donation, by calculating the spectrum of H20 .
Finally, the infrared spectrum of a model for water iso
lated in a betaalumina complex has been calculated. Water,
as an impurity in betaalumina, is important to the electrical
conductivity and mechanical strength properties of beta
alumina (44). This in turn is important towards the use of
betaalumina as solid electrolyte in the sodiumsulfur battery.
Until recently, the nature of the hydration reaction of beta
alumina was not known. Bates and coworkers (45) have deter
mined that the water molecules diffuse into the conduction
layers and form strong electrostatic bonds with the mobile
cations, thereby affecting the properties of the betaalumina.
Infrared spectroscopy is used as a tool to probe the kinetics
and thermodynamics of the reaction of water with betaalumina.
Therefore, before attempting to use the infrared spectrum as
a probe for other properties, it is important to first under
stand the structure and vibrational properties of hydrated
betaalumina. Bates and coworkers have recently determined
the crystal structure and vibrational spectrum of hydrated
betaalumina (44). The water molecules are located in the
crystal so that they are strongly hydrogenbonded to the
oxygen atoms of the aluminate ion (see Fig. 1.4).
x
II
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Ok
0 0
0 0
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Consequently, this is a good system to use to expand our
studies on hydrogen bonded water molecules.
The quantum mechanical calculation of the spectrum of
hydrated betaalumina takes on importance for several reasons.
First, it offers the opportunity to verify (or dispute) the
experimental spectrum and the interpretation of this spectrum.
Secondly, it represents a different example of an intermolecular
interaction involving a water molecule. It offers the oppor
tunity to investigate the role that the electron donor plays
in intermolecular interactions. And lastly, it is a challenge
to our ability to simulate a complicated chemical system by
using a rather simple model.
A summary of the calculational procedure developed for
this work, along with a review of both normal coordinate
analysis and infrared intensity analysis,is given in Chapter
2. The results of these calculations on isolated water mono
mers and the linear, cyclic,and bifurcated forms of a pair
of interacting water molecules are presented in Chapters3,
4, 5, and 6 respectively. The importance of some of the
effects just discussed, such as coupling between 0H bonds,
weakening of the 0H bond, charges flows, and the mixing of
normal coordinates, are discussed with respect to each of the
systems calculated. The predicted spectrum for our model of
water isolated in a betaalumina complex is given in Chapter
7, and the predicted spectrum for H20 is presented in
Chapter 8. A final summary and suggestions for future cal
culations are presented in Chapter 9.
CHAPTER 2
THEORY
The theory of both normal coordinate analysis (10,11,46)
and infrared intensity analysis (2,3,41,47,48) is well known
and described fully elsewhere. We have developed a slightly
different approach to analyze simultaneously the normal coor
dinated and infrared intensities. This method will be dis
cussed in detail, including the similarities and differences
with the more traditional procedures.
2.1 Coordinate Systems
A polyatomic molecule of N atoms has 3N degrees of
freedom. Of these 3N degrees of freedom some are rigid body
motions (translations and rotations) and some are elastic
body motions (vibrations). The analysis of the molecular
vibrations is best done in a coordinate system which allows
the maximum separation of the overall motions into transla
tions, rotations, and vibrations. This problem has been dis
cussed in the past (11,49,50), and amounts to choosing a
coordinate system such that the interactions between vibra
tions and the translations and rotations are minimized. The
necessary coordinate system is a molecule fixed coordinate
system which translates and rotates along with the molecule.
These conditions require that there is no component of either
translation or rotation present in a vibration, are are known
as the EckartSayvetz conditions (11).
The EckartSayvetz conditions provide explicit defini
tions for the translations and rotations of a molecule. They
are given below in terms of massweighted principal cartesian
coordinates (11).
1 m
1 s
o a
2 Mo= k ma qya
1
p3 = Em q
M3 M a x
o a a
m
5 o o
4= I (Yzqx zxq ) (1)
a yy a a
m o a
p5 = E( ) (e q x q )
a yy a a
a zz a
Here pl, pl 2 p32 p3 4 p5, and p6 represent translation and
rotation in the x, y, and z directions, respectively, M is
the total mass of the molecule, m is the mass of atom
xa, y, and za are the principal cartesian coordinates for
atom a, qx qy and qz are the mass weighted principal
a a a
cartesian coordinates for atom a, and Ixx, Iyy, and Izz are
the moments of inertia along the principal cartesian axes,
or the principal moments of inertia.
The remaining vibrational coordinates should reflect these
constraints, and at the same time form a complete, nonredundant,
vector representation of the vibrational subspace. The calcu
lations can be further simplified by utilizing the symmetry of
the molecule and by requiring that all the coordinates be
orthogonal to each other. The resulting coordinate system
is a massweighted, moleculefixed, symmetryadapted, prin
cipal cartesian coordinate system, hereafter denoted as the
S coordinates.
The transformation from massweighted principal cartesian
coordinates q to 5 coordinates can be written in matrix
notation as
S = Uq (2a)
where U is an orthogonal transformation matrix and q is the
column vector representation of the massweighted principal
cartesian coordinates. The matrix U can be constructed via
a GramSchmidt orthogonalization (51) of the irreducible
representations of the cartesian coordinates to the Eckart
Sayvetz conditions. The linear combinations of the principal
cartesian coordinates which reduce as the irreducible repre
sentations of the point group of the molecule can be found
by following the prescription given in Cotton (52). These
linear combinations are then made orthogonal to the Eckart
Sayvetz conditions, giving the U matrix.
Since U is constructed as an orthogonal matrix, its in
verse is simply U+ where U+ represents the transpose of U.
Equation 2a can now be rewritten as
S= US (2b)
where all the symbols have their previously defined meanings.
Equation 2b can now be applied in the method of finite differ
ences (53) in order to calculate the force constants and dipole
derivatives in S coordinates. The values of S are incremented
by a specific amount, usually 0.01 u A, and equation 2b is
used to calculate the values at q of these displaced values
of T.
The Gaussian 76 ab initio molecular orbital program (54)
is then used to calculate the energies and dipole moments for
the equilibrium geometry and also at appropriate displacements
from equilibrium. The force constants are derived by assuming
a pure quadratic potential, and then calculating the second
derivatives of this potential function. Mathematically, these
constants can be calculated by using the following equations
(from (53))
E(x +Ax) + E(x Ax) 2E(x )
F = (3)
xx x2
Ax
(E(xo+Ax,y +Ay) E(x +Ax,yo) E(xo,y +Ay) + E(xoyo)
xy AxAy
(4)
where Fxx represents a diagonal force constant, Fxy an off
xx xy
diagonal force constant,E(xo) the energy at the equilibrium
configuration, E(x +Ax) and E(xoAx) the energies at the pos
itive and negative displacements of coordinate x from
equilibrium respectively, E(xo+Ax,y +Ay) the energy at the
simultaneous positive displacement of both coordinate x and
y, and Ax the numerical value of the displacement of coor
dinate x. The dipole derivatives are approximated by averag
ing the displacements at the positive and negative distortions,
to account in part for the nonlinearity of the dipole moment.
3a AP P (x +Ax)P (x ) P (x oAx)P (x)
a = a 0 a o a 0 a 0
xx Ax 2Ax 2Ax
(5)
Pa(xo+Ax)P (xoAx)
2Ax
Here P (xo) represents the permanent equilibrium dipole moment
in the a = x, y, or z direction, Pa(XOfx) and P (xAx) repre
sent these same components of the dipole moment at the positive
and negative displacements of coordinate x respectively, and
Ax is the displacement of coordinate x.
2.2 Normal Coordinate Analysis
After the force constants have been calculated, the next
step is to derive the vibrational frequencies and the nature
of the normal vibrations (or the normal coordinate
transformation). For a harmonic oscillator the solution of
Newton's equations of motion gives the secular equation (11),
IGF EAj = 0 (6)
where F is the potential energy matrix, G is the inverse kin
etic energy matrix, E is the identity matrix and A is the
eigenvalue matrix. The solution to this system of homogen
eous linear equations gives the eigenvalues, or normal
frequency parameters A, and the eigenvectors, or column vector
representations of the normal coordinates, also known as the
normal coordinate transformation matrix, L.
If the potential energy matrix is represented in mass
weighted cartesian coordinates, then the G matrix becomes
the identity matrix (11). Equation 6 can be rewritten as
K X = 1 A (7)
where K represents the matrix of force constants in mass
weighted cartesian coordinates, a is the eigenvector matrix
in these coordinates, and A is the diagonal eigenvalue matrix.
The values of A are related to the vibrational frequencies by
vi(cm ) = m) (2 c) or = 1303.1*" (8)
1 A 21
and the matrix z is the orthogonal normal coordinate trans
formation which transforms from normal coordinates to mass
weighted cartesian coordinates
q = & Q (9)
where Q is the vector of normal coordinates. Thus in mass
weighted cartesian coordinates solving the secular equation
reduces to finding the matrix Z which diagonalizes the
potential energy matrix, K.
9 K Z = A (10)
Therefore the force constants calculated in S coordinates must
be transformed to force constants in massweighted cartesian
coordinates K. The force constants in S coordinates, K, can
be easily transformed to massweighted cartesians using the
U matrix. Since this matrix is orthogonal its inverse is
simply equal to U In massweighted cartesians the poten
tial and kinetic energies are equal to
+ 1;
2T = q G q (11)
+
2V = q K q (12)
respectively, where q represents differentiation with respect
i
to time, and G1 is actually the identity matrix, E, since
the potential energy is now expressed in massweighted
cartesian coordinates.
In S coordinates the potential and kinetic energy can
be written as
7+ 1 7
2T = S G S (13)
2V = S KS (14)
where G is the inverse kinetic energy matrix in S coordin
ates and K is the potential energy matrix in S. Since
S = U q and S = q + (15)
equation 13 and 14 can be rewritten as
S+ + 1
2T = q U G U q (16)
2V = q U K U q
(17)
By comparing equations 12 and 17 the transformation from K
to K can be written as
K = U+KU (18)
The secular equation can now be solved in terms of mass
weighted cartesian coordinates using equation 10.
Here L+ is a matrix of dimension 3N by 3N, and each row
of this matrix represents one of the 3N normal modes. Six (or
five for linear molecules) of these modes are the translations
and rotations, and the remaining modes are the fundamental
vibrations. Each row of + consists of N (3 by 1) vectors
corresponding to the motion of each atom N in that particular
normal coordinate. By analyzing the corresponding vectors for
each atom we can determine what motions constitute that par
ticular normal vibration. This will be discussed in more
detail later. The matrix A is a diagonal one containing
the eigenvalues of the secular equation. Six of these eigen
values correspond to the translations and rotations of the
molecule, and are therefore of value zero. The remaining
values give the vibrational frequencies via equation 8.
2.3 Infrared Intensity Analysis
The absolute infrared intensity of the ith vibrational
mode of a molecular is proportional to the square of the
dipole moment derivative with respect to motion along the
ith normal coordinate. In units of km/mole this becomes (2)
A. (km/mole) = 974.9 d.i (aP/aQ.) 2
(19)
where di is the degeneracy of the ith mode, and (aP/aQi) is
given in units of eu where e is the atomic unit of charge,
equal to 1.602 x 1019C. If the value of the dipole moment
derivative with respect to the normal coordinate is known,
then the intensity for that mode can be easily calculated
with equation 19. The problem of infrared intensity analysis
then reduces to evaluating and interpreting this derivative.
As stated previously, the method of finite differences
can be used to approximate the dipole moment derivatives. By
using equation 5, the dipole moment derivatives in S coordin
ates can be calculated. Since the normal coordinate transfor
mation is given in terms of massweighted cartesians, then the
dipole moment derivatives must be transformed to this same
coordinate system in order to utilize this normal coordinate
transformation to calculate the intensities. This transfor
mation is written as
P = P U (20)
where P is the 3x3N matrix of the dipole derivatives in mass
weighted cartesians and PS is the corresponding 3x3N matrix
of dipole derivatives in S. The first six (or five for linear
molecules) columns represent the dipole derivatives with re
spect to the three translational and three rotational motions
respectively. These values can be calculated from the charge
of the molecule (qe), the mass of molecule (M ), the principal
moments of inertia (Ixx, yy, and Izz) and the vector compon
ents of the permanent dipole moment Px' Py, and Pz"
These relations are
given by King (55) and have recently been reformulated by
Rogers and Hillman (56), and are summarized below in columns
one through six. The remaining columns of this matrix, PS,
are calculated using equation 5, and represent the dipole
derivatives with respect to the vibrational ? coordinates.
The three rowsof this matrix represent differentiation of
one of each of the three vector components of the dipole
moment; either P P' or Pz
e z x x
q P P AP APx
M 0 0 0 O.....
yy zz
o i
o I I 7 3N
S P PO AP AP
z5 __1
P< =O o  ..... 3.
xx Izz
0 qe x AP AP3N
M I I 7 3N
o xx yy
(21)
The matrix P consists of the dipole moment derivatives
in massweighted cartesian coordinate space. By removing the
mass dependencethe derivatives in cartesian coordinates are
obtained
P = P M (22)
x 
where Px is composed of the juxtaposition of the Atomic Polar
Tensors (APTs) (57) for the N atoms in the molecule, and M
is a diagonal matrix represented by the triplet sets of the
square root of the masses of the atoms in the molecule. The
Px matrix has the general form (3 x 3N)
x x x x x x
xA A A N N
aP aP aP ap ap ap
XA .................. a a
z z z z z z
A A A 3N N N
(23)
The APTs will be discussed in more detail later.
The transformation from P to PQ, and in essence the abso
lute integrated intensities, can be accomplished with the
matrix
PQ P I (24)
where P is the matrix of dipole moment derivatives in normal
coordinate space. This matrix is also 3 by 3N, with the first
five or six columns corresponding to the dipole derivatives
with respect to the translations and rotations. The values of
these derivatives are all zero since there is no component of
vibration in a pure translation or rotation, and hence no in
tensity or frequency for these modes. Each of the next (3N6)
columns correspondsto the dipole derivatives with respect to
one of the (3N6) vibrational normal modes. The derivative of
one of the three vector components of the dipole moment Px,
Py, and Pz are again each responsible for one of the three rows
of this matrix. This matrix has the following representation
/0 0 0 0 0 0 x x
BQ7 3QN
Q 3Q7 aQ3 (25)
3 7 3N
aP aP
S0 0 0 0 0 0
aQ 7 * *** 3Q 3
The absolute integrated intensity for the ith mode is
simply proportional to the sum of the squares of the vector
components of the dipole moment derivatives for the ith
column. The actual value of the absolute intensity in km/mole
is found using equation 19, where the following relationship
is now true
2 7 2 2 2
p = X + + ap (26)
2.4 Transformation to Different
Coordinate Systems
The S coordinates are very convenient for the calculations
of frequencies and intensities. However, the detailed analyses
of the frequencies and intensities are best accomplished in two
different coordinate systems. The frequency parameters are
most easily analyzed in terms of force constants in internal
coordinates and the intensity parameters will be discussed
in terms of dipole derivatives with respect to space fixed
cartesian coordinates, or APTs. Therefore, the frequency and
intensity parameters in S coordinates must be transformed to
the corresponding parameters in internal and cartesian coor
dinates respectively.
2.4.1 Internal Force Constants
As chemists, we have a desire to interpret the absorption
frequencies in terms of molecular structure. Therefore, the
necessary frequency parameters should be related to the molecu
lar geometry. The most useful frequency parameters are thus
the force constants in internal coordinates (10,11).
The internal displacement coordinates are changes in
bond lengths and bond angles of a molecule. For most appli
cations five coordinates bond stretching, angle bending,
torsional motion, outofplane bending (wagging), and linear
bendingaresufficient to describe completely all of the vi
brational motions (10,11). For more unusual situations other
specific internal coordinates can be defined by following the
prescription of Califano (11). More detailed information on
internal coordinates can be found in references 10 and 11.
The internal coordinates are related to the cartesian
coordinates by
( Rk k = 1, 2 ... (3N6)
Rk Xi where i = i, 2, 3 ... (3N) (27a)
or in matrix notation
R= B X (27b)
where B is a (3N6) by 3N matrix with typical element defined
as (OR /Xi) and R is the column vector representation of
internal coordinates. The B matrix is a function solely of
the molecular geometry, and formulas for the calculation of
its matrix elements for the five most common type of internal
31
coordinates have been derived (11). The A matrix transforms
the internal coordinates back to cartesian coordinates
X = AR (28)
where A is apparently the inverse of B. However, since the
B matrix is not a square matrix, but rather a rectangular
one, it does not have an inverse, and the A matrix is calcu
lated via the G matrix (58).
In internal coordinates
G = B M1 B+ (29)
i
where M is a (3N by 3N) diagonal matrix of the reciprocal
masses. The "inverse" of B is then obtained by (47)
B1 G = B1 B M1 B+ (30)
B1 = M1 B+ G1 (31)
A = M1 B+ G1 (32)
At this point is it appropriate to mention that A in equation
32 can be calculated only if the G matrix is a square non
singular matrix. This in turn means that there cannot be
any redundancies defined in the B matrix elements. This is
often a problem if one wishes to fully utilize the symmetry
of the molecule. The methods of dealing with this problem
will be discussed later; suffice it to say there that the
problem exists and must be resolved.
The relationship between the force constants in mass
weighted cartesian coordinates K and in internal coordinates
Fi is now given by
F. = A K A (33)
where A is the transpose of the A matrix, and F. is the force
constant matrix in internal coordinates.
2.4.2 Atomic Polar Tensors
The APT has been defined in the first three columns of
equation 23 above, and equation 20 calculates the APTs in
principal cartesian coordinates. However, usually the prin
cipal cartesian coordinate system is not ideally oriented
for the interpretation of infrared intensities. This arises
in part because the actual values of the polar tensor elements
depend upon the orientation of the axis system used. Therefore,
the bond coordinate system is usually the best system forana
lyzingthe APTs. The transformation of the APT from one coor
dinate system to any other system is done by rotating the axis
system. As the axis system is rotated, the values of the polar
tensors change according to the rotation performed. The new
values of these tensors can be calculated by performing the
appropriate tensor rotations on these tensors. The matrices
for the rotation of the axis system by the angle e about
either the X, Y, and Z axes respectively are (from (59))
1 0 0 cose 0 sine cose sine 0
R = 0 cose sine R = 0 1 0 R = sine cose 0 (34)
0 sine cos \sine cose 0 0 1
where the sign of the angle 6 is determined by viewing along
the rotation axis from the positive direction. If the old
axis system must be rotated in a clockwise direction to
transform to the new coordinate system, then the angle is
positive, and if the old axis system must be rotated in a
counterclockwise direction to transform to the new coordin
ate system, then the angle is negative. The rotation matrices
defined in equation 34 are the ones usually used for the rotation
of a cartesian axis system (59). The transformation for tensors
is given by
+
T = R T R = x, y, or z (35)
where T is in thenew coordinate system, and T is the tensor in
the old coordinate system.
2.4.3 The S Coordinates
We have just seen the advantages of interpreting the pre
dicted frequencies in terms of internal force constantsand the
predicted intensities in terms of APTs. Why, you may ask,
have we chosen to perform the actual calculations in S
coordinates? This question is discussed below.
The transformation from cartesian coordinates to S coordin
ates is performed by the orthogonal matrix U (see equation 2a).
The orthogonality of U makes the reverse transformation matrix,
from S coordinates to q coordinates, simply the transpose of
U, U+ (see equation 2b). In internal coordinates the analo
gous equation to equation 2a is equation 27. The reverse
transformation is then performed by equation 28. However,
since the B matrix in equation 27 is not square, the A matrix
in equation 28 cannot be obtained by simple inversion of B,
but must be obtained using equation 32. Equation 32 requires
that all of the redundancies which are often necessary to com
pletely describe the vibrational degrees of freedom using only
the previously defined five internal coordinates be removed.
The method of finite differences requires the transformation
given by either equation 2b or equation 28. In internal coor
dinates equation 28 is often not easily obtained because the
redundancy cannot be removed in any unique manner. Therefore,
we avoid the problem with redundant coordinates by using
equation 2b, or the S coordinates.
The cartesiah coordinates used in the definition of the
APT also avoid the redundancy problem. However, these coor
dinates do not utilize the symmetry of the molecule, nor do
they explicitly separate the vibrational degrees of freedom
from the translations and rotations. This means that in
cartesian coordinates there are 3N coordinates rather than
3N6, and the symmetry of the molecule cannot be used to
simplify the problem. This greatly increases the computa
tional effort necessary to predict the infrared spectrum.
Therefore, since the S coordinates explicitly separate
the vibrations from the translations and rotations, and
utilize the symmetry of the molecules, they are
preferred to cartesian coordinates.
2.5 Experimental Infrared Intensities
The experimental absolute infrared band intensity is
given by (60)
A (km/mole) = (10 b ln(Io/I) dv (36)
band
where Ai is the absolute integrated molar absorption coeffi
cient or intensity for the ith fundamental mode in k/mole,
c is the sample concentration in moles/liter, a is the path
length in cm, Io is the intensity of the incident light in
the absence of the sample, I is the intensity of the trans
mitted light after passing through the sample, v. is the
i
wavenumber of the band in cm and the integration is per
formed over the entire band region. The factor of 100 con
verts Ai into the units of km/mole. In these units a very
strong band would have an intensity of around 1000 km/mole,
while a very weak band would have a value of 0.10 km/mole (57).
The alternate definition of the experimental integrated
absolute band intensity uses integration over Inv. instead
1
of ri (60). This quantity is given as the symbol i and is
defined as
r (cm2/mole) = (c) In(I /I) d(lnv.) (37)
1 band
where all the symbols have their previously defined meanings
and units. The definition of ri is preferred for many purposes
(61,62), but most spectrometers record spectra linear in wave
numbers, so integration over v. is much easier to measure
1
experimentally. As a result, Ai is the more commonly reported
value. The difference between A. and r. is not too severe,
1 1
however, the two quantities being related through
A. = F. v. (38)
1 1 1
1
Here v. is the band center in cm Ai is the intensity
in km/mole and ri measures the intensity in cm /mole.
The quantities Ai measure the total vibrational strength
for a transition from the ground vibrational state to an
excited vibrational state (v. = 0 => v. = 1) (60). Theo
retically, the most important quantity related to this
strength is the transition dipole moment mn(i) (from (57))
imn) = m P n d T (39)
where the transition occurs from state m to state n, P is the
dipole moment operator, tm is the wavefunction for vibrationally
excited state m, and 9n is the wavefunction for the initial
state n.
2.6 Theoretical Infrared Intensities
The absolute intensity of an infrared absorption band is
related to the theoretical quantity u n(i) by (60)
873N v, 2
A. = av ) (i) d. (40)
1 3hm I1
where N is Avagadro's number, v. is the wavenumber of the
av 1
absorption band, h is Planck's constant, c is the speed of
light, and d. is the degeneracy of the ith vibration. If the
1
vibrations are harmonic so that *m and in in equation 39 are
harmonic oscillator wavefunctions, then equation 40 becomes
(60)
(N wrd. /, 2
Ai(km/mole) = 32 i (41)
where (3P/aQi) is the dipole moment charge with respect to
motion along the ith normal coordinate. If the proper con
stants are substituted into equation 41, then that equation
can be simplified (57) to
A (km/mole) = 974.9 I(3P/3Qi) 12 di (42)
_
where (3P/aQi) has units of eu where e is the atomic
19
unit of charge (1 e = 1.602 x 10 19C) and u is the atomic
mass unit. The units are eu for (aP/aQi) when the value
of P is expressed in e A (where 1 e A = 4.8 Debye) and the
normal coordinates Q are in units of A u
Equation 42 indicates that the intensity for the ith
mode depends on the form of the ith normal coordinate. Un
fortunately, the normal coordinates are strongly dependent
upon the molecule in question. They usually involve motion
of all the atoms in the molecule simultaneously, and the
amplitudes of motion depend also on the masses of the atoms.
The interpretation and parameterization of infrared intensi
ties in terms of these quantities are not easily accomplished
because of the dependence of the form of the normal coordin
ates on the molecule in question. This dependence makes it
difficult to compare these derivatives (8P/3Q) among different
molecules. Instead, it appears to be more useful to attempt
to use the fact that the atoms in a molecule are connected by
bonds, and that these atoms (in similar chemical environments)
should exhibit similar properties within different molecules.
The most natural coordinate system to use to try to inter
pret the absolute intensities are the cartesian coordinates.
These coordinates are uniquely defined for every atom in the
molecule, and are not dependent upon the geometry of the
molecule. They can be oriented in the same direction in any
molecule, thereby offering the possibility of directly compar
ing parameters among different molecules. This is necessary
if we hope to be able to interpret intensities in terms of
chemical structure. The transformation from normal coordin
ates to cartesian coordinates can be derived using the chain
rule of differentiation
aP iP i
 i a (43)
ax Q ax.
SE L (44)
aX.j aQ i ij
where (3P/aQi) are derivatives of the dipole moment with re
spect to normal coordinates Qi, (3P/aQj) are these same deriva
tives with respect to cartesian coordinates X., and (aQ./aX.)
is the derivative of the normal coordinate with respect to a
cartesian coordinate. This last derivative is one of the
elements in the normal coordinate transformation matrix. This
matrix is given the representation L and represents the
transformation from cartesian coordinate to normal coordinates
= L+ X (45)
The concept of the dipole derivatives in cartesian coor
dinates, or the Atomic Polar Tensor (APT), was introduced in
1961 by Biarge, Herranz, and Morcillo (7), and later reformu
lated by Person and Newton (8). The matrix of APTs for any
particular molecule (termed P ) is 3 by 3N and consists of
the juxtaposition of N 3 by 3 second rank tensors, one for
each atom in the molecule. Each element of the Nth APT is
defined as (aP /ax.), where P is the ath component (either
x, y or z) of the total dipole moment, and x. is the direc
3
tion of the displacement of the jth coordinate (also either
x, y, or z). More detailed discussions on the formalism of
the APT are available in the literature, (2,3,7,8,41,47,48).
However, it should be remembered that the APT is simply a
more easily interpretable representation of the dipole
derivatives in normal coordinates.
2.7 Characteristic Invariants of
Atomic Polar Tensors
The actual values of the elements of the APT depend,
of course, on the orientation of the axis system used. For
this reason it is imperative that the coordinate system being
used always be reported along with the APT in order to properly
interpret the APT. We begin our discussion of how to interpret
the APT by starting with those properties which are invariant
to the axis system utilized.
The mechanics of molecular vibrations impose two important
conditions on the APTs. These conditions, and also the other
invariant properties of the APT, will be briefly discussed here.
A more thorough discussion is available elsewhere (55,6365).
The translational symmetry of the dipole moment requires
that the APTs for a molecule sum to a diagonal tensor; each
element in this tensor is equal to the total charge in the
molecule. From (55)
S[V () I+ = q E (46)
N e
where the summation runs over all the atoms in a molecule,
N is the cartesian gradient operator (a/ax, a/ay, a/az),
which when applied to a vector quantity (P, the dipole moment)
gives a 3 by 3 tensor. E is the identity matrix whose only
nonzero elements are ones along the diagonal. For a neutral
molecule this reduces to the requirement that the elements
of the APTs sum to zero, which is commonly expressed as the
null condition
: [VN(P) ] = o E (47)
N
A second important condition is an intensity sum rule
first derived by Crawford (63). This sum rule is most
easily derived following (55) through an analogy with the
molecular kinetic energy T,
2T = (T) M (T)+ (48a)
3X X
ax axr
1 aT 2 aT ( 2 + aT ( 2
2T = + l + (48)
N MN x yN zN
where k, y and 2 represent differentiation with respect to
time. The moment in cartesian space (aT/aX) can be trans
formed to moment in normal coordinate space using the chain
rule of differentiation.
( + (aT + ( (49)
where w represents rotational motion, and X translational
motion. In these coordinate the kinetic energy takes on
the form
2TT 1 T 12 1 aT (50)
i 6i a \ Raa a0 oJ
The relationship between equation 49 and the corresponding
equation for the dipole derivatives is obvious
aP\ + 1^\I^Q\ + /a p Xo)
+ ] +(51)
aX aQ ax a x ax aX
Their equivalence is complete because in the harmonic oscil
lator approximation aQ/Xa = aQ/ax and aw/ak = 3u/3X. There
fore an expression analogous to equations 48b and 50 for the
kinetic energy can be written for the dipole derivatives
1 aP (2 P aP P 2 1 aP 2 1 2
N + + + E + E +X
N MN N i aa a Xo(
(52)
The above equation can be simplified by summation over the
three components of the dipole moment Px, Py' Pz, giving
the intensity sum rule
2 (2%)2 2 2
+ [a = Za + + (53)
or, by using equation 42 and multiplying both sides of equa
tion 53 by the appropriate constants
av 2 (54)
SA. + = av X (54)
i 3c2 N
where XN defines the "effective charge" according to King
2 1 P a 2 + AP \2 + (LP 21 1 P 2 +P N (55)
N2 2 2 (55)
N 3 axN j azN 3
The term Q defines the translational and rotational correc
tions in equation 54
Na, 2 2 2 2 2 + 2 (56)
=av x y y z + z x e (56)
0 = + + 56)
3c zz xx Iy y
where the first three terms in the brackets of equation 56
represent the contributions to the dipole moment derivatives
due to rotation of the axis system, and the last term is the
correction due to translation. This last term arises only
when the permanent total charge qe on the molecule is not
equal to zero.
For any molecule the LHS of equation 53 is invariant to
a similarity transformation. Therefore the quantities on the
RHS of this equation must also be invariant to similarity
transform. This is equivalent to stating that the sum of
the square of the elements of the APT for atom N is invar
iant to axis rotation.
The importance of the effective charge as defined in
equation 55 is twofold. First, it is invariantto similarity
transformation, and therefore is not dependent upon the axis
system defined. Ideally, these quantities should then be
transferable from an atom in one molecule to the same atom
in another molecule (as long as this atom is in similar chem
ical environments in both molecules). Secondly,' the effective
charge is important because from equation 54 we can now see
that the effective charge for an atom (normalized by the re
cripocal of the mass for that atom) measures the contribution
of that atom to the total intensity sum for the molecule.
A second quantity useful in characterizing APTs is the
mean dipole moment pN. From (55)
N 1 N 1
p = Trace (P ) = 1 (57)
3 x 3 a 3N azNy
The trace of any tensor is invariant to a similarity transform,
so the mean dipole moment is also invariant.
The last invariant property of the APT is related to the
two previously defined invariant properties according to
2 9 2 N
N= (XN p ) (58)
where 8 is defined as the anisotropy. It measures the devi
N
ation of the APT from the constant diagonal tensor, p E (55).
2.8 Quantum Mechanical Interpretation
of Atomic Polar Tensors: The Charge,
Charge Flux, Overlap Theory (CCFO)
The chargecharge fluxoverlap (CCFO) model was first
introduced by King (55,66,67) and is related to the equilib
rium charge, charge flux concepts of Decius (68). The reader
is referred to the original references, and also some more
recent work illustrating the application of this model in
interpreting infrared intensities (13,39,41,56,69) for a more
detailed description of this model. Here only a brief outline
of the more important concepts of this model will be discussed.
The exact quantum mechanical definition of the dipole
moment within the LCAOMO method is as follows (41)
= EQAR Z z D < AlrA B>] (59)
A B p AB VB A
where pA represents the orbitals on atom A, vB represents the
orbitals on atom B, rA is the position vector of electron 1
with respect to nucleus A, DWAVB is the density matrix element
between orbital VA on atom A and orbital vB on atom A, and
QA is the gross atomic charge on atom A. The electronic pos
ition vector may be defined as specifying the position of an
electron, and may be expressed as
rl = R + rA (60)
where RA is the position vector of nucleus A and rl is the
electron position vector.
QA is the gross atomic charge on atom A and is given by
QA = Z z E D S (61)
AB A A VB v AB AVB
The term D B is the density matrix element which is defined as
F"AVB
n
D 2 c .c i(62)
AV B i=1 A B
and S is the overlap integral
lIAVB
S A B = (63)
For simplicity, equation 58 can be rewritten as
p = Z QARA + E F(A,B) (64)
A A,B
where F(A,B) is now given by
F(A,B) = Z Z D i/ (1)rA (1)dTr (65)
IA' B A B A B
The first term in equation 64 is obtained from the net charges
located at the nuclear positions and can be assigned to a
single atom. The second term is essentially a hybridization
term which measures the contribution due to the displacement
of charge away from the center of the nuclear position. This
effect is proportional to the offdiagonal density matrix
element between orbitals centered on atoms A and B and is
therefore intrinsically associated with a pair of atoms (41).
The APT for atom A is derived by applying the gradient
operator (with respect to the cartesian coordinates of atom
A) to the dipole moment vector of the molecule
Px = APT(C) = [VC(6)6 (66)
Application of this equation to the quantum mechanical defini
tion of the dipole moment gives (Equation 64)
P xC = [VC (QA A)]+ C Z[E(F(A,B))]+ (67a)
A AB
Px = E + ZERVC ) A E Z[C (F(A,B))]+ (67b)
x A C AB
= PC(C) + PC(CF) + pC(0) (67c)
Thus the APT can be broken down into three components, which
are represented by the three terms on the RHS of equation 67b.
These three components are the charge, charge flux, and over
lap tensors respectively. The physical meaning for each of
these tensors will be briefly outlined below.
The charge tensor is a diagonal tensor, each element
being composed of the net atomic charge on the atom in
question
QA = A NA (68)
where QA is the net atomic charge, NA is the Mulliken popula
tion (70), and ZA is the number of electrons in the neutral
atom. The charge contribution to the APT is static, and in
volves only one atom A. The charge flux tensor is defined
(41) as
+
o o
+ N (XCAXC)N (XC+AX )
pC(CF) = RA (QA)] = R A C C NA( C) C (69)
A A 2XC (6
where NA(X0 + AXc) is the Mulliken population for atom A after
displacement of coordinate X for atom C along the positive
direction, NA(X AXc) is the Mulliken population of this
same atom after displacement along this same coordinate in
the negative direction, RA is the vector describing the equi
librium cartesian coordinates of atom A, AX is the numerical
value of the displacement of coordinate XC, and the summation
runs over all atoms in the molecule. The vector RA is then
multiplied by transpose of the vector quantity given in paren
thesis in equation 69 to give the charge flux tensor for atom C.
These quantities measure the contributions from each atom
to the dipole moment change due to the redistribution of charge
among all the atoms as one particular atom is displaced along
one of its cartesian coordinates. In effect, these terms arise
as a result of the changes in the atomic charges as one atom
moves in either the x, y, or z direction. The contribution to
the dipole moment change is obtained by "normalizing" the
changes in the atomic charges by the respective atomic
coordinates. This contribution can be considered as a "dia
tomic contribution" (41) because the quantities in equation
69 involve a pair of atoms A and C.
The last component of the APT is the overlap tensor.
This is a "triatomic contribution" (41) involving atoms A,
B and C. Physically, the overlap tensor corresponds to the
changes in hybridization between atoms A and B as atom C is
displaced along one of its cartesian coordinates. The effects
on the dipole moment derivatives come about due to the changes
in the displacement of charge from the nuclear centers. This
last tensor is usually obtained from the difference of the
total APT and the sum of the charge, and chargeflux
contributions.
PC(overlap) = P [ (charge) + PC(chargeflux)] (70a)
x
PC(O) = P C pC() PC(CF) (70b)
It can be shown by analogy with the total APT that each
of the three component tensors also has invariant properties.
For the charge tensor this is already obvious. The charge ten
sor is a constant diagonal tensor, therefore the trace and the
sum of the squares of the diagonal elements are also invariant.
N(C 2 1 z[pN(C) 2ii 2 (71)
1
1NC) C = TN (71)
p (C) = Trace (Charge Tensor) = P C = (72)
where P (C).. is a diagonal term of the charge tensor for atom
11
iI
N, given by Q N The trace of the charge tensor is termed the
mean atomic charge, while the sum of the squares of the ele
ments in the charge tensor is called the effective atomic
charge. The anisotropy of the charge tensor is by defini
tion always equal to zero:
N
0(C) = 0 (73)
The charge flux tensor has been defined in equation 69. Using
this definition and the chain rule of differentiation, the
following equation can be written
Q 3X Q + aX a + ax (74)
where all the symbols have their previously defined meanings
and where the last two terms go to zero because atomic charges
are not dependent upon either translation or rotation. By
comparing this with equations 51 and 52, the relationship
given below can be written
1 QA 2 + QA 2 QA 2 I QA 2
E +  = i (75)
N MN N NN
where it is obvious that the RHS of equation 75 is invariant
to similarity transform, so therefore the LHS of this equa
tion must also be invariant to similarity transform. If both
sides of equation 75 are premultiplied by the quantity RA2
(xA +y +zA ), then the following equation results
1 2 2 2 //QA 2 /QA\2 [QA 2
E (x +y +z ) 7 +
N MN A A A xN YN 'zN
S(x 2+y +z (76)
i a1i) A A A
The quantity (xA2 +yA2 +zA2 ) is the square of the distance of
atom A from the origin, and this distance is invariant to
rotation of the axis system (similarity transform). Therefore,
a product on the LHS of equation 76 is invariant to a similar
ity transform since both quantities on the RHS of this equa
tion are invariant. If the LHS of equation 76 is expanded,
the following result is obtained
1 2 aA)2 2 QA 2 2 A 2 2 DA\ 2
LHS = M( (axi xA j) + x A2 (A + y
N M axN AyN a zN A
2 aQ 2 2 aq\2 2 A2 2 aQA 2 2(aQA\2
YA yN A IA +A AxN A NA N zA
(77)
If we now sum the RHS of this equation over all atoms A, and
add the resulting terms, we obtain the squares of the nine ele
ments of the charge'flux tensor for the Nth atom. Since we are
summing invariant quantities, then the total sum is also invar
iant to axis rotation. Therefore, we conclude that the sum of
the squares of the elements of the charge flux tensor is invar
iant to axis rotation. This invariant quality is analogous to
the effective charge for the APT, and is termed the effective
charge of the charge flux tensor, X (CF).
S(CF)2 2 ,= EA Q ++ (78)
A TA N\ ( / \ N
Furthermore, as is true for any tensor, the trace of the
charge flux tensor for the N atom is invariant to axis
rotation. As a result, the trace of the charge flux tensor
becomes the second invariant property of that tensor.
N N (79)
N 1 1 ] N
p(CF) = 3 [Trace (Charge Flux) = P(CF) (79)
N
where P(CF)) i is a diagonal element of the charge flux tensor
for the Nth atom. The third invariant property of the charge
flux tensor is derived from the previous two invariants and
N
is termed the anisotropy 8(CF). It is expressed as
(C =) = e F (UF)) (80)
N NC NCF) 8
A similar argument can also be made for the overlap
tensor. Each term of the overlap tensor is defined in equa
tion 67c, where F(A,B) is given in equation 65. An equation
similar to equation 73 can be written for the overlap terms
aF(A,B) F(A,B) 3Q (A,B) a\ + (A,B) o
(81)
where as in the case of the charge flux tensor, the last two
quantities in equation 81 are equal to zero. This makes the
correspondence between equation 74 and equation 81 exact, and
the corresponding argument leads to the following result:
S1 3 AB)2 + (AB)\2+ F (82)
Here the LHS of this equation is equivalent to the sum of the
squares of the overlap tensor. Since the RHS of this equation
is dependent only on the form of the normal coordinate and is
independent of the axis system used, then the LHS also becomes
invariant to axis rotation. The sum of the squares of the
overlap tensor is then invariant to axis rotation and is
termed the effective overlap, defined here in the LHS of
equation 82. Again, as is true for any tensor, the trace
over the overlap tensor becomes the second invariant property
of that tensor. The trace of the overlap tensor, described
in an analogous way to the traces of the previous tensors,
is termed the mean overlap, and is defined as
N(0) = [Trace (overlap)] = Z(P ()) (83)
i
whereP (0)o)i is a diagonal element of the overlap tensor.
Finally, the anisotropy of the overlap tensor is also invar
iant to axis rotation and is defined as
(BNO 2 (( 2 (2) (84)
The three component tensors of the APT, namely the charge
charge flux,and overlap tensors, each has its own invariants
analogous to the total APT. The sum of the squares of all of
the elements for each individual tensor is invariant to axis
rotation. For the total APT this quantity is the "effective
charge," for the charge tensor the "effective atomic charge,"
the chargeflux tensor gives the "effective charge flux," and
the overlap tensor yields the "effective overlap." The rela
tionship between the last three terms and the "effective
charge" as defined by King and Mast (65) is given below
N2 N._.2
N N._ 2 N._ 2 N. 2
X = X (C) + X (CF) + X (0) + Residuals
(85)
where the residuals come about because the sum of the squares
do not equal the square of the sum. These residuals are de
fined as follows:
2 N N N N
Residuals = z Z2 P(C)i.P(CF)ij + 2 P(C)ijP(0)
i j
N N
+ 2 P(CF) iP(O) i (86)
where the sum extends over all nine elements of the respective
tensors.
The trace of the total APT has been previously defined as
the mean dipole moment in equation 57. The trace of the charge,
the charge flux, and the overlap tensors are defined as the
mean atomic charge, the mean charge flux, and the mean overlap
and are defined in equations 72, 79, and 82 respectively. The
anisotropy for each tensor derives from the two previous invar
N N
iants for each tensor, X and p and is defined for the
charge, charge flux, and overlap tensors in equation 73, 80,
and 84 respectively. The significance of each of these
terms will be discussed later.
CHAPTER 3
THE WATER MONOMER
The frequencies and intensities were calculated for water
interacting with a number of different chemical environments.
The different environments were chosen to simulate different
experimental conditions. Among those investigated were water
interacting with another water molecule in several different
geometries. These orientations include the linear, the cyclic,
and the bifurcated pair of interacting water molecules, as
shown in Figures 1.1, 1.2, and 1.3 respectively. In addition,
the spectrum of water isolated in a betaalumina complex was
modelled by performing calculations on H20 interacting with O
1 2
atoms, 0 atoms, and 0 atoms. Finally, the spectrum of
ionized water, H20 was also calculated. The results calcu
lated for these species were then compared with those results
calculated for isolated water. The changes in the calculated
spectral parameters relative to the isolated molecule were
dependent on both the geometrical orientation and the chemical
environment of the water molecule. Therefore, calculations
were done first on the isolated water molecule, and then these
results were compared with the results obtained for a water
molecule that was allowed to interact with its environment.
3.1 Calculation Procedure
For water and all other molecules the force constants and
dipole derivatives in T coordinates were calculated on the
Amdahl 470 V/6II computer of the Northeast Regional Data
Center (NERDC) operating with OS MV/SPJES2 Release 2 and
using the Gaussian 76 system of ab initio molecular orbital
programs (54). This program was modified to allow the calcu
lation of the dipole moments of ions. This modification was
first suggested by Rogers and Hillman (56), and permits the
program to calculate the dipole moment for charged species in
the same manner as done in the case of neutral molecules. The
standard 431G basis set was used for all molecules (71). The
integral threshold was set at 1.0 x 106, and the convergence
limit at 5.0 x 105. These values are standard for use with
Gaussian 76. There has been some work on the dependence of
the calculated results on the values used for these limits
(72), but this problem was not considered here. The molecular
geometry was input using principal cartesian coordinates, and
the density matrix for the converged set of wavefunctions was
stored on the IBM 3380 disk pack at the NERDC, to be used as the
initial guess wavefunction for the calculations at the dis
placed geometries. This procedure reduces the amount of com
puter time needed for the ab initio calculations on the
displaced geometries. Typically, a calculation on the
equilibrium geometry would take 20 iterations before conver
gence was met, while a calculation on a displacement geometry
would only take three or four iterations if the density matrix
for the equilibrium geometry was used as the initial guess.
Two programs, ORTHO and PUNCH, were written for the Amdahl
470 to set up the appropriate disk data sets at the NERDC
for input to Gaussian 76. ORTHO writes the U transformation
matrix to a 3380 disk pack. PUNCH uses this matrix as input
and performs the matrix algebra described by equation 2b.
This results in the calculation of the cartesian coordinates
of the displaced geometries. These coordinates are then
written to an IBM 3380 disk pack in the proper format for
input to Gaussian 76. Finally, Gaussian 76 is used to cal
culate the energies and dipole moments of the equilibrium
geometry and the displaced geometries. These results are
then used in conjunction with.equations 3, 4 and 5 to give
the force constants and dipole derivatives in S coordinates.
These two programswere designed to complement each
other, each program creating an IBM 3380 disk data set for
use in the next program. This eliminates some of the errors
that can occur if one has to type all the data into the com
puter by hand (either using cards or a remote terminal).
After the ab initio calculations were completed, the cal
culated force constants K, and the calculated dipole deriva
tives P were transferred to an 8inch floppy disk for use
in the normal coordinate analysis. This analysis was performed
on this laboratory's S100 bus microcomputer. This system in
cludes the CompuPro/Godbout 8085/8088 CPU integrated to oper
ate with CP/M 2.2 (73) and CP/M 86. Two Shugart SA 801R disk
drives for use with 8inch floppy disks are used for data and
program storage. Other peripherals include a Televideo 910
terminal, Centronics 739 dot matrix printer, Hayes Stack
Smartmodem 1200 (for connecting to the NERDC) and a Micro
angelo Graphics Subsystem. The graphics subsystem includes
a 15inch high resolution graphics screen (480 x 512) detach
able keyboard and light pen. In addition, there is available
128K of S100 static memory, of which 64K is accessible by
the 8 bit 8085 CPU. Software which is currently used operates
with the 8 bit processor (8085) and includes Microsoft FORTRAN
80, Microsoft BASIC 80, Microsoft BASIC COMPILER, and Micro
soft muMATH/muSIMP 80 (74). All of the normal coordinate pro
grams used here (see Appendix A for a brief description of
each program) operates with the 8 bit 8085 CPU and 64K of
S100 static memory. A whole series of programs was written
to perform the normal coordinate analysis. These programs
were designed to be run in sequence; each program creates
8inch floppy disk data sets for use in the following programs.
The proper sequence of programs necessary to perform a complete
normal coordinate analysis is given in Appendix A, along with
the necessary data input for each program (either from an 8inch
disk data set or by the operator via the Televideo terminal).
All of the programs were written in either Microsoft FORTRAN
(74) or Microsoft BASIC (74). The library subroutines neces
sary to run the main programs are also listed in Appendix A.
3.2 The Predicted Spectra for
the Water Monomer
A bond length of 0.951 AO and a bond angle of 111.20 was
used for the calculation on the isolated water molecule. The
numbering of the atoms and the orientation of the principal
cartesian axis system used in these calculations is shown in
Table 3.1. This geometry corresponds to the absolute minimum
on the potential curve (theoretical geometry) for the ab initio
SCF calculation of isolated water with the 431G basis set by
Lathan, Hehre, Curtiss and Pople (75). There is some debate as
to whether it is best to use the theoretically or experimentally
determined geometry when calculating the spectroscopic proper
ties of molecules. However, for some of the systems investi
gated here there are little or no experimental data available.
As a result, in order to be consistent, the theoretical geom
etries (calculated at the SCF level with the 431G basis set)
were used whenever possible. When this geometry was not avail
able, other available information (either from other calcula
tions at different levels or from experimental work) was used
to formulate a "best guess" at a reasonable geometry. This
was done rather than searching the potential surface for the
absolute minimum (i.e., optimizing the geometry) because the
main interest is in spectral changes of the interacting species,
and not absolute numbers. Therefore, calculations for the iso
lated species (in this case water) and then for the interacting
species were performed at the same level (ab initio level with
the 431G basis set), and then the predicted differences were
examined.
The principal cartesian coordinates for the theoretical
geometry of the isolated water molecule (R = 0.951 AO, 6 =
111.20) are given in Table 3.1.
In order to do a full calculation of the frequencies and
intensities, all the diagonal force constants, all the off
diagonal force constants, and all the dipole derivatives
must be calculated according to equations 3, 4, and 5,
respectively. Since the force constant matrix in S coordinates
is block factored by symmetry, some of the offdiagonal force
constants are equal to zero and do not have to be calculated.
However, all the offdiagonal force constants within the same
symmetry block must be calculated.
The water molecule has C2v symmetry. The 9 degrees of
freedom reduce to 3A1 + A3 + 2B1 + 3B2, with the three vibra
tional coordinates reducing as 2A1 + B2. According to equa
tion 3, two displacements are needed for each calculation of
the three diagonal force constants, one in the positive dir
ection and one in the negative direction. This is a total of
six displacements that are needed to calculate the three diag
onal force constants. The B2 symmetry block is 1 by 1, and
contains no offdiagonal force constants. The A1 block is a
symmetric 2 by 2 block, and therefore there is one nonzero
offdiagonal force constant. To calculate this offdiagonal
force constant using equation 4, one additional displacement
is needed, the simultaneous positive displacement of both
coordinates. This makes a total of seven displacement geometry
and one equilibrium geometry calculations that are necessary
Table 3.1 Principal cartesian coordinates for the isolated
water monomer at the theoretical geometry (A). Atom number
ing is defined below.
z
0
2
H1 3
X Y Z
H1 0.0 0.7847 0.4772
02 0.0 0.0 0.0611
0.0 0.7847
0.4772
for the full calculation of the frequencies and intensities
for the isolated water molecule.
The calculated frequencies and intensities for the water
monomer at its theoretical geometry are summarized in Table
3.2, where the experimental values (76,77) are also listed
for comparison. As is usually the case with ab initio cal
culations, the predicted frequencies and intensities do not
agree exactly with the experimental values (17,78). This is
not too surprising considering the assumptions made and the
errors inherent in these types of calculations. First, we
have assumed that the molecule is a pure harmonic oscillator,
and that the potential curve can be well represented by a
parabola. In reality, the potential curve is anharmonic,
and the experimentally observed parameters reflect this
anharmonicity. We have chosen to ignore this anharmonicity
in our calculations. Strictly speaking this is not correct;
therefore, the comparison between our calculated harmonic
frequencies and the experimentally observed anharmonic fre
quencies cannot be expected to be in complete agreement.
(The effect of anharmonicity can be quite significant, some
l
times up to 200 cm .) Secondly, the 431G basis set employed
in these calculations is not large enough to correctly describe
the electronic wavefunction. The inadequacies of this basis
set (such as the improper behavior of the gaussian function
at small distances, the absence of polarization functions and
the inflexibility in the orbital exponents) are significant
enough that the electronic wavefunction does not describe
62
Table 3.2 Predicted wavenumbers and intensities for isolated
water compared to the experimental wavenumbers and intensities.
Wavenumbers v (cm1)
Mode Calc.
Harmonic
3983
1676
4104
3832
1648
3942
Exp.
Intensity A (km/mole)
Calc.
Exp.
Observed
3657
1595
3756
3.90
125
54
2.24
54
45
aReference 58
bReference 41
correctly the electronic behavior of this molecule, leading
to additional errors in these calculations. Finally, the
effect of neglecting electron correlation in these calcula
tions is not entirely known. However, from some preliminary
results by Bartlett (79), it appears that more sophisticated
calculations may lead to much better agreement with experi
mental values. He has calculated the APTs for the isolated
water molecule using "many body methods." The agreement be
tween his calculated results and the elements of the experi
mental APTs given by Zilles (41) is within 0.05 e. This re
sults suggests that the disagreement between the experimental
frequencies and intensities and our calculated values is a
result of the assumptions made and the methods used, and that
this disagreement can be reduced by using more advanced theo
retical methods. However, as a first approximation, the re
sults one can obtain using ab initio methods with the 431G
basis set are generally accurate enough to be useful in in
terpreting infrared spectra (17). For this particular type
of calculation (SCF with 431G basis set), the frequencies
are expected to be within 1020% of the experimental ones (71),
and the intensities are predicted to be within a factor of 2
of the experimental values (80). The wavenumbers and intensi
ties listed in Table 3.2 predicted for the water molecule with
the theoretical geometrical parameters do indeed reflect these
limits.
Calculations of the frequencies and intensities were also
done for the water monomer with varying bond angle and bond
lengths. Calculations were performed by varying the bond length
while fixing the bond angle, by varying the bond angle while
fixing the bond length, and by varying both the bond angle and
bond length simultaneously. In addition, the frequencies and
intensities were calculated for the experimentally determines
geometry, where the bond length is 0.9572 AO and the bond angle
is 104.50 (76,77). The calculated wavenumbers and intensities
for these various geometries are summarized in Table 3.3.
The data in Table 3.3 indicates that the values of the
calculated frequencies are fairly sensitive to the assumed
geometry. The calculation at the experimental equilibrium
geometry predicts wavenumbers in much better agreement with
the observed values than does the calculation at the theoreti
cal geometry. Changes as small as 0.006 A in the assumed bond
length for these calculations are responsible for changes of
100150 cm in the predicted frequencies for the bond stretches.
The bending frequency is less sensitive to changes in the
assumed molecular parameters, but is nevertheless still
affected. Surprisingly, a change of slightly less than 70 in
the assumed bond angle results in drastic changes of nearly
i
300 cm for the two stretching modes. However, while the
results on the frequencies indicate that close attention must
be paid to the assumed geometry to predict reliable absolute
numerical values, the predicted intensity values are much more
independent of the assumed molecular structure. The intensities
are predicted to be nearly the same for each of the normal modes
for all the different assumed structures. The largest
4
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4.4 a'
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a
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00
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r f
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a' 
= a,
aa
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.fl
a' .
T 0
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N r
0; 0 .0O
0
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o a
* .0
(N C 4
a,~~ 0 O
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a' 0 
N N ~
N11 0
o N N O 0
ON a
N1 N 
O 0 N 0 a
N a'a' N N N
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difference is the change by a factor of 2 in the predicted
intensity of the asymmetric stretching mode as the asummed
bond angle is change by 70. However, it is obvious that in
general the predicted frequencies are very sensitive to
changes in the assumed geometries, while the calculated
intensities are very much less sensitive to these same
changes in geometry.
At first glance, the strong dependence of the wavenumbers
on the assumed geometry might seem very discouraging concern
ing the possibility of calculating absolute absorption fre
quencies because the final results are strongly dependent
upon the initial assumed geometry. This problem has been
examined in the past (1216), but is not of interest here.
In this thesis we shall be interested primarily in the
changes that come about as a result of the interaction of
the water molecule with its environment. Hence, the internal
geometry of the water molecule can be chosen arbitrarily, and
the intermolecular distance and orientation between the water
molecule and the interacting species varied. In this way the
resulting changes in the predicted spectrum can be attributed
to the interaction and not a consequence of a geometry change.
By fixing arbitrarily the internal geometries of the water
molecules and varying only the intermolecular orientation or
distances, we can then compare the predicted wavenumbers and
intensities to observe their dependence only on the varying
intermolecular parameters. Any dependence on the initial
assumed geometry is eliminated because the internal geometry
Table 3.4 Predicted APTs and invariants for isolated water
at the theoretical geometry, in e, where le= 1.602 x 1019 C.
The coordinate system used is defined below.
z
2
H H
1 "3
I \2 2
.482 0 0 X2 = .1292 e2
H 2 2
P 0 .274 .142 = .1071 e
0 .095 .226 82 = .0995 e2
.960
0
0
.482
0
0
P 02 =
Px
p H3 =
x
0
.548
0
0
.274
.095
0
0
.452
0
+.142
.226
= .4754 e
2
= .4268 e
= .2187 e2
= .1292 e2
2
= .1071 e
= .0995 e2
v
is eliminated because the internal geometries of all the water
molecules are assumed to be the same.
The APTs (2) for the three atoms of the water molecule
are presented in Table 3.4. The results presented here are
for the principal cartesian coordinate system, but the coor
dinate system can be easily rotated to obtain APTs in any
other coordinate system using equation 35. The invariants
N
of the total APTs, XN, p and 8N, are presented also. The
breakdown of the APTs into their three component CCFO tensors
is given in Table 3.5. The overlap tensor is calculated by
equation 70b as the difference between the total APT and the
sum of the charge and charge flux tensors. All of the effec
tive invariants for the constituent CCFO tensors are given
in Table 3.5 also. These invariant properties will be dis
cussed later.
The internal force constants for the water monomer with
the theoretical geometry were calculated from the force con
stants in S coordinates using equations 18 and 33. The inter
nal coordinates were defined as the two OH bond stretching
coordinates, and the HOH angle bending coordinate. The B
matrix elements were calculated using the little s vectors
as described by Califano (11). The resulting B matrix was
then used to calculate the A matrix using equation 32. The
resultant force constants calculated with this A matrix are
given in Table 3.6, along with experimental values for
comparison. The B matrix elements used in these calculations
are not weighted. This means that the units of the stretching
Table 3.5 Predicted CCFO tensors and their effective invariants
for the isolated water molecule at the theoretical geometry, in
e (le= 1.602 x 1019 C). The coordinate system is defined in
Table 3.4.
/.4017
Hl
P(C) = 0
0
0
H1
P(CF) = 0
0
H1
P(0)
0803
0
0
0 0
.4017 0
0 .4017
0 0
.0550 .0370
.1040 .0450
0 0
.1827 .1790
.1990 .1307
.8034
P(C) = 0
0
02
P(CF) =
02
P(0) =
.1606
0
0
0 0
.8034 0
0 .8034
0 0
.1090 0
0 .0910
0 0
.3644 0
0 .2604
X022 = .6455 e2
C
22
X02 = .0067 e2
CF
2
X02 = .0755 e2
0
The invariants for H2 are identical to those for H1. The com
ponent tensors for H2 are obtained simply by switching the
signs on the offdiagonal terms for H1.
.1614 e2
.0017 e
XH12
C
H2
CF
H 2
O
= .0190 e2
Table 3.6 Predicted internal force constants for the water
monomer at the theoretical geometry compared to the experi
mental values. Units are md/AO where the weighting factor d,
where d = 1 AO, is used to weight the force constants involv
ing bending coordinates (see text and also reference 11).
Internal Force Constants (md/A)
Calculated Experimentala
R1 = R2 9.141 7.750
de3 0.714 0.668
R1R2 0.065 0.068
RI 3 0.300 0.270
aReferences 58; fit to observed frequencies.
coordinates are in AO, and the units of the bending coordin
ates are rad (11). As a result, the force constants calcu
lated using the A matrix derived from this B matrix are also
not weighted. If the energy is expressed in mdAO, then the
units of the stretching force constants are mdAO, the bending
force constants are given in mdAo/rad2, and the stretchbend
interaction constants are given by md/rad. In order to compare
these calculated force constants with other values in the lit
erature, care must be taken that the same weighting scheme is
used in both instances (48). In the remainder of this thesis
the weighting factor (11) d, where d = 1 AO will be used to
convert all force constants to units of md/AO without changing
the numerical values of these constants. The normal coordinate
transformation matrices obtained in these calculations, and all
subsequent calculations, are summarized in Appendix B.
CHAPTER 4
THE LINEAR WATER DIMER
The geometry for the linear dimer was taken from Curtiss
and Pople (30), and is the minimum energy structure for
(H20)2 obtained by optimizing all geometrical parameters with
respect to the total energy of the complex using the 431G
basis set. This geometry will now be referred to as the
theoretical equilibrium geometry for the linear dimer. The
principal cartesian coordinates are given in Table 4.1. This
molecule has Cs symmetry, having only a single symmetry element,
the plane of symmetry. The calculated heavy atom distance
(00) is 2.832 AO, slightly shorter than the distance deter
mined experimentally by Dyke, Mack and Muenter (81) of 2.98 A.
The discrepancies between the calculated value and the experi
mental value is due to the inadequacies of the basis set and
the absence of electron correlation. Calculations performed
with a larger basis set (including polarization functions) but
without electron correlation predict a value for the 00 dis
tance of 2.974 A. This value is considerably closer to the
experimental value than is the value calculated with the 431G
basis set. This suggests that the 431G basis set overestimates
the strength of the hydrogen bond, but if the deficiencies of
the basis set are eliminated then more accurate results are
predicted.
Table 4.1 Principal cartesian coordinates for the linear water
dimer at the theoretical equilibrium geometry (AO). Atom
numbering as in Figure 1.1.
X Y Z
H1 .7876 .2207 1.8360
H2 .7876 .2207 1.8360
03 0.0 .0397 1.3729
H4 0.0 .0271 .4999
05 0.0 .0613 1.4573
H6 0.0
6
.8109
1.8337
The calculation performed by Curtiss and Pople (30) pre
dict the hydrogen bond to be linear within 0.050, with an
optimized energy of 151.83038 hartrees. In addition, the OH
bond length involved in the hydrogen bond is predicted to be
0.958 A0. The remaining three 0H bonds are predicted to be
all the same, and 0.950 A0. The HOH angles are calculated
to be 111.30 in the electron acceptor, and 112.00 in the elec
tron donor. A calculation by Lathan, Hehre, Curtiss and Pople
(75) with the same basis set gives a bond length of 0.951 Ao
and a bond angle of 111.20 for the isolated water molecule,
with a total energy of 75.90864 hartrees. Therefore, the
binding energy of the dimer (the energy of the dimer minus
twice the energy of the monomer) is calculated to be 8.2
kcal/mole. The internal geometries of the individual monomer
units in the dimer are predicted to undergo slight changes
upon complex formation. The bonded OH bond length (H405) is
predicted to elongate by 0.007 A0, and the two internal bond
angles increase slightly. While these changes appear to be
very small, it is still surprising that the internal geometries
of the individual monomer units are predicted to change. Most
workers have assumed that the internal geometries of the two
monomer units are unchanged from the values of the isolated
molecule (81). Later in this chapter it will be shown that
this geometry in the monomer units is very important towards
understanding the spectral changes predicted for the dimer
relative to the isolated monomer.
The irreducible representations of all 3N degrees of free
dom for this molecule are 11A' + 7A". The 3N6 or 12 vibrations
reduce as 8A' + 4A". For each of the 12 diagonal force con
stants two calculations at displaced geometries are necessary.
This gives a total of 24 ab initio calculations that are needed
to calculate all the diagonal force constants. For each non
zero offdiagonal force constant an additional ab initio cal
culation is necessary. In the A' symmetry block there are
7+6+5+4+3+2+1 or 28 additional calculations necessary to de
rive all nonzero offdiagonal force constants. In the A"
block there are an addition 3+2+1 or 6 offdiagonal force
constants that must be calculated. This bring the number of
ab initio calculations necessary to calculate the complete
vibrational force field to 24+28+6 or 58 displaced geometries
plus one equilibrium calculation, or a total of 59 calculations.
4.1 The Predicted Spectrum for
the Linear Dimer
The predicted frequencies and intensities for the theoret
ical equilibrium geometry for the linear configuration of the
water dimer are summarized in Table 4.2. The corresponding
frequencies and intensities for the isolated water molecule
are listed for comparison adjacent to those for the correspond
ing vibration of the linear dimer. In addition, the frequen
cies and relative intensities observed experimentally by Tursi
and Nixon (19) are also listed here for comparison. The pattern
predicted by our ab initio calculation is in good agreement with
the experimental spectrum assigned to the linear structure by
Tursi and Nixon (19). Behrens and Luck (27) argue that the in
terpretation of the spectrum of (H20)2 by Tursi and Nixon (19)
76
"I
. o
*4 'n C I 0
0
to44 0
4Jr EQ
0" N 0 ^ C ^
33 ,
0J >
0 3% 0 :
H 3
(l) (l r* i ? S : 2 ='
VP r~
>c 0>
ro
3 ~'3
at33
Ia _, a >
(1 D w a 34 a p cD o
) 3 3w
2 E
as .
 1 I 3
O sS S 
U) 0 I0
(U 41 1 3' 3
4 M1 
30
I % A I 0 s 0
, Nd "1 4
3' "3 .0
'31 e 'fl 0 4% .1 3 "3
r 3 2
HTU) g3
a3'
o + b o c c 3 3 "
U)C. r\ y 31^ ^*
U U 3 o%' 0
'3 0 03
(  g
U3 '3 
CO  = j 3S
 43 S0 .
 r ' 
C + 3; >. a3
31  I. Z) =
.4) 3 I 3 .
(0 P N "3 2 I
3 3: E
'0 q ln v rr = S : =^
X) ( l O* OY il
uI "3:a
*H E Jig l *? 3U 5 g = S^ ^s iS
2" z *3 1 2 5 1 i ; ^< : ^ 1
H ~ 3 a 3, 3 El S _33 ir t ^5 ' ^ i
j) 1 j I i l i^ a J ^ i; 0 
(N^a ? 'ii n i^^ t 1^ 1 r, 1t 1
3S9 1 ;1IUl
'O ' % ;5 J, . ^ = ^ ^
5r '^ z 'r "^ ^ u x
a,* i o 3
is not conclusive evidence for the linear structure. Their
arguments are based on the fact that all six of the bands
assigned to the linear structure do not follow the same con
centration dependence, as they should if they all belonged to
the same species. However, reliable assignments of these bands
cannot be made solely on the basis of their concentration
dependence because there are often overlapping and broad bands
corresponding to higher aggregates which serve to hinder reli
able measurements of relative intensities. The calculation of
Curtiss and Pople (30) along with other work (19,26,81) points
to the linear structure as the most energetically favorable one,
and using this structure in our calculations we have been able
to reproduce reasonably well the experimental spectrum of Tursi
and Nixon (19) (see Table 4.2).
While it is probably true that these ab initio 431G cal
culations are still not accurate enough to give reliable abso
lute values, it is encouraging that the correct patterns are
reproduced, including the spectral changes that occur upon
complex formation. The nature of these spectral changes is
of primary interest, and not the absolute values predicted
for the frequencies and intensities. Since these ab initio
calculations with the 431G basis set correctly reproduce the
observed spectral changes, we believe that these calculations
are valuable in understanding these changes. These changes
in turn are important to understand because their nature might
offer some valuable insights towards understanding intermolec
ular interactions.
The assignment of the vibrations as predominantly the
electron donor symmetric stretch, or electron acceptor bend
ing, for example, were made with the aid of a program which
was written to picture a vibrating molecule. The program,
VIBRATE, accepts the equilibrium cartesian coordinates for
each atom in the molecule and then draws a ball and stick
model of that molecule. This portion of the program has been
adapted for our microcomputer from the PLUTO program of Mother
well (82). The cartesian coordinates corresponding to the
displacements of an appropriate increment in both the positive
and negative direction along the normal coordinates are also
input from a disk data set into the program. Atoms (or circles)
are then turned on and off at these displaced cartesian coor
dinates for any desired normal mode. These circles are turned
on and off at both the positive and negative displacements,
giving the illusion that the atoms in the molecule are actu
ally vibrating along that particular normal coordinate. By
observing the motions of all the atoms in the molecule for any
particular normal vibration, the characteristic motions of
that normal mode can be accurately described. These vibrations
can be frozen on the graphics screen, then sentto the printer
for hard copy. This program was written in BASIC and uses the
Microangelo high resolution graphics screen for the display.
The low frequency modes were also assigned using this program.
This type of assignment has certain advantages over the Poten
tial Energy Distribution (PED) (58) descriptions of normal
vibrations because they allow one to actually visualize the
vibration. The PEDs can be misleading because they do not
describe the actual normal vibration, but only the contribu
tions from a given set of internal coordinates that make up
that particular normal vibration.
At this point let us just briefly comment on the charac
teristics of the 12 vibrational modes for the theoretical
linear configuration of two interacting water molecules. The
i i
highest frequency modes predicted at 4126 cm 4076 cm
3992 cm1, and 3872 cm1 (Table 4.2) are all stretching modes
and are isolated almost entirely in only one molecule. These
vibrations are the asymmetric stretch of the electron donor,
the asymmetric stretch of the electron acceptor, the symmetric
stretch of the electron donor, and the symmetric stretch of the
electron acceptor, respectively. The intermolecular interaction
does not couple or perturb the form of these high frequency
stretching modes at all, and in fact they remain localized in
one molecule. These normal modes behave as if two separate
molecules existed without any interaction. The two modes
occurring at 1800 cm and 1714 cm correspond to bending
vibration of the water units. However, these vibrations are
no longer localized, but are coupled between both molecules.
i
For instance, the vibration at 1800 cm1 is predominantly
the electron acceptor bending motion, coupled in phase (i.e.,
both molecules open their angles simultaneously) with some
of the bending mode of the electron donating molecule. The
vibration at 1714 cm1 is mostly electron donor bending
coupled outofphase with some electron acceptor bending.
In this lower wavenumber region the vibrations are no longer
localized on the individualized water molecules, but rather
become vibrations of the entire complex. While it is possible
to classify the vibration at 1800 cm1 and the one at 1714 cm1
as electron acceptor and electron donor bending, respectively,
strictly speaking this is not correct due to the presence of
intermolecular coupling which perturbs and mixes these
vibrations. These bending vibrations are not only coupled
with each other, but they are also coupled to some of the low
frequency intermolecular bending and twisting motions.
The low frequency (or intermolecular) modes are generally
characterized by twisting and outofplane (where reference is
made to the plane of the molecule) bending motions. The vibra
tion predicted at 835 cm2 is mainly the 03H45 outofplane
(of the electron acceptor) bend (hydrogen bond bend) along with
i
some electron donor twisting motion. The mode at 749 cm cor
responds to inplane electron acceptor twisting in phase with a
little outofplane (of electron donor) electron donor wagging
(electron acceptor twists upwards at the same time the hydrogen
atoms of the electron donor wag upwards). The vibration at 208
1
cm is the outofphase combination of these same electron
acceptor twisting and electron donor wagging motions just
i
described. The mode at 313 cm can be characterized as pre
dominately electron donor twisting motion mixed with electron
1
acceptor hydrogen (free hydrogen) wagging motion. At 210 cm
the vibration corresponds to the outofplane (of electron
acceptor) twisting of the electron acceptor mixed
with some in plane (of electron donor) electron donor bending.
i
Finally, the vibration at 103 cm is the hydrogen bond strech
ing mode.
The low frequency modes are presented here for complete
ness, but will not be discussed at any length. There is very
little experimental evidence about them available from studies
in this region of the spectrum. The uncertainties in the ab
initio calculations are magnified in this region because the
force constants are extremely small. These frequencies are
therefore expected to be much more sensitive to the assumed
geometry, and the confidence that one can place on the pre
dicted values of these absolute frequencies is certainly
limited. For some configurations some of the intermolecular
modes were predicted to have imaginary frequencies. This re
sult is a consequence of performing the calculations at dis
placements that are not about the absolute minimum of the
potential surface. It is not expected that these imaginary
frequencies affect greatly the frequencies of interest, namely
the high frequencies intramolecular modes, due to the large
energy separation. However, it is certainly interesting to
note here that the predicted intensities for the low fre
quency modes are in general equal to or greater than those
for the intermolecular vibrations. This might offer some
hope to experimentalists interested in investigating the
bands in this region. However, until now, the difficulties
(such as the presence of internal rotation, and a low signal
to noise ratio) in this region of the spectrum have made
the study of these bands extremely difficult. Even so, the
intensities of these modes merit a few comments.
Four of the six low frequency modes are predicted to have
a large intensity; greater in fact, than for the high frequency
modes. As a result of comparing the motions involved in the
high intensity low frequency modes, it was observed that these
four vibrations all involved a considerable contribution from
the atoms in the electron acceptor, while the remaining two
low intensity modes were mainly motion of the electron donor.
This suggests that perhaps most of the intensity for the com
plex originates in the electron acceptor.
In general, the data in Table 4.2 indicates that the
intramolecular vibrations of the electron donor are essen
tially not affected by the interaction with another water
molecule. However, the electron acceptor experiences con
siderable change, especially in the symmetric stretch
l
[v,(EA)]. This band is predicted to shift 110 cm to lower
frequency and to increase in intensity by a factor of almost
72, from 3.90 km/mole to 279 km/mole. The remaining modes
for this complex are not predicted to change very much from
the corresponding vibrations in the isolated water molecule.
Both asymmetric stretching vibrations are predicted to in
crease by slightly more than a factor of 2 in intensity, but
this is certainly less than the factor of 72 predicted for
the symmetric stretch of the electron acceptor.
4.2 Intensity Analysis for Linear
Pair of Water Molecules
The change in intensity predicted for the symmetric stretch
of the electron acceptor is extreme, from a relatively weak
band in the monomer to a strong band in the dimer. A detailed
ab initio calculation and analysis of the linear dimer has
recently been published by Zilles and Person (39). However,
our equilibrium calculation here utilizes a slightly different
approach so that our results are slightly different from those
reported previously (39). For instance, we have used an un
scaled ab initio force field and unsealed ab initio polar
tensors, while Zilles and Person (39) used the experimental
force field of Tursi and Nixon (19) with scaled APTs. Sec
ondly, while our equilibrium calculation was performed at
the theoretical geometry, Zilles and Person (39) chose to
use the experimental geometry. While the use of an experi
mental force field, scaled APTs and the experimental geometry
usually gives better agreement with experimental results, our
calculation is based on purely theoretical methods and there
fore gives an indication of the quality of predictions that
can be expected using these methods. Finally, we have ex
tended the work of Zilles and Person (39) by actually quan
tifying the predicted changes in the polar tensors to
changes in intensities. In addition, we have investigated
the dependence of the predicted spectrum on both intermol
ecular orientation and distance by performing similar calcu
lations at different geometries and intermolecular distances.
The results of our equilibrium calculation are very sim
ilar to the results reported by Zilles and Person (39). How
ever, there are some slight differences in the two approaches
so that caution must be used in making a different numerical
comparison between the two results. Zilles and Person (39)
found that the only significant change in the APTs of the
dimer relative to the isolated monomer was in the P term
yy
of the bonded hydrogen. Therefore they attributed the large
predicted intensity increase (in the dimer relative to the
monomer) in the symmetric stretching mode of the electron
acceptor as due to this change in this term of this polar
tensor. When the CCFO analysis was performed on the APTs
by Zilles and Person (39), they found most of the increase
in the APTs for the dimer was due to an increase in charge
flux as the dimer was formed. Our results are similar, and
are briefly summarized below.
In order to attempt to understand the large predicted
intensity increase for the symmetric stretching mode of the
electron acceptor, the intensity parameters were transformed
to APTs. The APTs for the theoretical linear configuration
in the coordinate system given in Figure 1.1, are given in
Table 4.3, along with the properly rotated APTs for the theo
retical water monomer. First, it must be remembered that the
intensities are related to the normal coordinates via equation
24. Since the normal coordinates are mass weighted motions of
all the atoms in a molecule, the heavier atoms will have smaller
amplitudes and therefore contribute less to the normal coordinates.
Table 4.3 Calculated APTsfor theoretical linear dimer alQog
with corresponding APTs for monomer (e) (le= 1.602 x 10" C).
Atom numbering and coordinate system as given in Figure 1.1.
Linear Dimer
Monomera
.3284
Hi
P = .0406
x
.0331
.3284
P H2 = .0406
.0331
.6055
P03 0
x
0
.4827
H4
Px = 0
0
1.0010
P 5=
0
.0777 .0596
.3231 .1079
.0968 .4388
+.0777 .0596
.3231 .1079
.0968 .4388
0 0
.7884 .2154
.1633 .8416
0 0
.5892 .0688
.0803 .3700
0 0
.8115 .0078
.0433 .5055
.274
.081
.040
S.274
.081
.040
.548
0
0
.482
0
0
.964
0
0
.4670
0
0
0 0 .482
.3645 .0763 0
.0066 .1000 0
0
.369
.003
0
.044
.131
aThe APTs for the atoms in the monomer are for coordinate sys
tems rotated from Table 1 separately for each atom to corres
pond to the coordinate system of the dimer in Figure 1.1.
H6
P
x
.121 .074
.295 .113
.113 .412
.121 0.074
.295 .113
.113 .412
0 0
.591 .228
.228 .825
0 0
.148 .089
.042 .351
0 0
.517 .047
.047 .483


Therefore, when analyzing infrared intensities the lighter
atoms are of greater importance than the heavier atoms.
While bearing this fact in mind, the results in Table 4.3
show reasonably large changes in the APTs of both the oxygen
and bonded hydrogen atoms of the electron acceptor. The P
yy
terms for both of these atoms change by a substantial amount.
The P term for the oxygen atom decreases by 0.308 e, while
the P term for the bonded hydrogen increases by 0.441 e.
Since the lighter atoms play a more important role in the cal
culation of infrared intensities, the following discussion will
emphasize the bonded hydrogen atom when trying to explain the
large predicted intensity increase in the symmetric stretching
mode of the electron acceptor for the dimer.
The polar tensors were further analyzed to determine
which, if any, of the three component tensors was mainly re
sponsible for the change in the total APT. The charge,
charge flux, overlap (CCFO) analysis was performed for all
of the atoms in the linear pair of water molecules. The
resultant charge, charge flux and overlap tensors are listed
for the linear pair and the corresponding water monomer in
Tables 4.4, 4.5, and 4.6, respectively.
The three tables, Tables 4.4, 4.5, and 4.6 indicate that
the charge and overlap tensors for all of the atoms are rela
tively constant in going from the monomer to the linear pair.
There are slight changes, but nothing significant. However,
the charge flux tensors show most of the changes reflected in
the total APTs. For the P term of the bonded hydrogen, the
YY
Table 4.4 Calculated charge tensors for the theoretical linear
dimer compared to those of isolated water (e) (le= 1.602 x
1019 C). Atom numbering and coordinate system as given in
Figure 1.1.
Linear Dimer
Monomer
.4237
H1
0
P(C) =
P(C) =
0
03
P(C)
H4
P(C)
05
P(C)
H6
P(C)
.8184
0
0
.4632
0
0
.8734
0
0
.3812
0
0
.4
.4
.8
.4
.8
.3
0 0
237 0
0 .4237
0 0
,237 0
0 .4237
.4017
0
0
.4017
0
0
0 0 .8034
184 0 0
0 .8184 0
0 0
632 0
0 .4632
0 0
1734 0
0 .8734
0 0
812 0
0 .3812
0
.4017
0
0
.4107
0
0
0
.4017
0
0
.4017
0 0
.8034 0
0 .8034
Table 4.5 Calculated charge flux tensors for the theoretical
linear dimer compared to the corresponding sensors for iso
lated water monomers (e) (le= 1.602 x 10 C). Atom number
ing and coordinate system as given in Figure 1.1.
Linear Dimer
.0957
H1
.1293
P(CF)
.0663
.0957
H2 .1293
P(CF) =
.0663
.1769
P(CF) =
0
.0268
H4
P(CF) =
0
05
(CF)
P(CF) =
+
.0184
0
0
.0061
H6
P(CF) =
0
.0661 .0370
.0470 .0151
.0087 .0132
.0661 .0370
.0470 .0151
.0087 .0132
0 0
.0760 .0320
.0218 .0337
0 0
.4495 .0276
.0352 .0905
0 0
.2354 .0436
.0919 .0499
0 0
.0442 .0141
.0960 .0334/
S053
.089
.054
.053
.089
.054
.105
0
0
0
0
0
0
0
0
0
0
0
\
Monomer
+.034
.033
.020
.034
.033
.020
0
.066
.040
0
.089
.012
0
.043
.091
0
.046
.104
.012
.020
.012
.021
.020
.012
0
.040
.025
0
.051
.081
0
.091
.029
0
.040
.054
Table 4.6 Calculated overlap tensors for the theoretical linear
dimer compared to the corresponding tensors for isolated water
monomers (e) (le= 1.602 x 1019 C). Atom numbering and coor
dinate system are given in Figure 1.1.
.1910
H
P(6) = .1699
.0994
.1910
H2
P(O) .1699
.0994
.3898
03
P(0) = 0
0
.0463
H
"4
P(0) = 0
0
.1461
05
P(0) = 0
0
.0919
H6
P(0) = 0
0
.1438
.0536
.0881
.1438
.0536
.0881
0
.1060
.1851
0
.3236
.0451
0
.2972
.0541
0
.0275
.0904
.0966
.0928
.0283
.0966
.0928
.0283
0
.1834
.0569
0
.0412
.0031
0
.0486
.3180
0 o
.1026
.3144
.181
.170
+.094
.181
.170
.094
.360
0
0
.065
0
0
.161
0
0
.065
0
0
.155
.074
.093
+.155
.074
.093
0
.146
.188
0
.343
.054
0
.329
.044
0
.013
.101
.095
.093
.022
.095
.093
.022
0
.188
.047
0
.038
.030
0
.044
.291
0
.084
.325
increase in the total APT is predominately due to the increase
in the charge flux tensor, and not a result of a change in the
charge or overlap tensor. Quantitatively, the increase in the
charge flux tensor is responsible for 0.360 e, or 82% of the
change in the Pyy term of the total APT for this atom.
A program was written, PXPQ (see Appendix A), to determine
exactly what percentage of the total intensity increase from
monomer to dimer was due to the difference in the normal coor
dinates (or due to the mechanical effect), and what amount
was a result of the change in charge, charge flux and overlap
in the bonded hydrogen. This program uses a normal coordinate
transformation matrix and a set of polar tensors as input. It
allows the operator to adjust any of the elements in any of the
polar tensors for any of the atoms as he wishes. The intensi
ties using the input normal coordinate transformation matrix
and these adjusted polar tensors are then recalculated. If
we use the rotated monomer polar tensors as the initial input
APTs, and the calculated normal coordinate transformation for
the linear dimer, the calculated intensities will show the
effects due to the changing normal coordinates (mechanical
effect), but not the electronic effects from the interaction.
If we then add only the increase in charge flux for the P
yy
term of bonded hydrogen and recalculate the intensities,
these intensities will reflect only the normal coordinate
effect and the effect of the increase in charge flux for the
bonded hydrogen atom. The results of both of these calcula
tions are given in columns four and five of Table 4.7. The
I
0
CE
m
(U)
0I
0
4.
O) p
MO 0
4) I
r e
m 1
0 4
41 a)
.4
H
4l 4
00
40
ro
(U1
) (n
'I r
C,
(U H
rl
0 (0
i Q)
H 0
UE4
0
fi
S 4
0
4 0 m m" Ln
11 14 r N L m
4 r4 14 t 4 4
II
% .D L co 
oo 4 *r ,'4
l mf CN 4 r4
14
0 H4
06 X
aox
cc
'C
a)
0I
82~
0 m
P
a, ^
1r1: *v
'C *
o: p,.
2
2 *3
r cu
<
il B s^
o "
1 '5
C
g= C
a)
 12
OOCf
N (B
2'ci
Ln co
m m
Ln CM m
rl m
Ln CN
LA )
cI
n
.* r4
4 14
4 N
'C' i4
T r4
o oo
m4 r4
l 
normal coordinate effect is minimal, as evidenced by only a
slight increase in the intensities of the symmetric stretching
modes, from 3.90 km/mole to 4.57 km/mole for the electron
donor and to 11.6 km/mole for the electron acceptor. This is
to be expected based on the fact that the stretching vibrations
of the dimer were already seen not to couple or become per
turbed much by the interaction. When the charge flux increase
for the P term of the bonded hydrogen is added to the monomer
APTs, the predicted intensities for the electron acceptor ex
hibit considerable changes. The asymmetric stretch increases
from 47 km/mole to 123 km/mole, and the symmetric stretch in
creases from 11.6 km/mole to 235 km/mole. The predicted in
tensities for this pair of vibrations in the complex are 110
km/mole and 279 km/mole respectively. By adding only the
increase in charge flux for the Pyy term of the bonded hydro
gen of the electron acceptor (H4) all of the predicted inten
sity increase for the asymmetric stretch of the electron
acceptor for the linear pair can be reproduced. By comparing
columns five and nine of Table 4.7, it is obvious that prac
tically all of the predicted intensity increase for the linear
dimer is a direct consequence of the increase in charge
flux in the P term of the bonded hydrogen. The effect
YY
of both the change in charge and overlap tensors on this
intensity increase can be seen by successively adding the
differences from dimer to monomer in the Pyy term of the charge
and overlap tensors to those APTs used for column five. It is
seen in columns six and seven that these contributions are
indeed minimal; containing only 6% and 7% of the total pre
dicted intensity increase respectively.
The only band of the linear dimer which is not adequately
reproduced in column seven is the asymmetric stretch of the
electron donor. It is still a factor of 2 too low compared to
the predicted intensities in column nine. If we refer back to
Table 4.3 and look again at the hydrogen atoms for the electron
donor, there is an increase of 0.054 e in the P term of both
hydrogen atoms in the dimer relative to the monomer. This
change is relatively small, but is still large enough to be
responsible for the increase in intensity for the asymmetric
stretch of the electron donor. From Table 4.5, 0.0427 e or
78% of this change is due to an increase in charge flux. The
terms for both hydrogens were increased by this amount, and
the resulting predicted intensities are given in column eight
of Table 4.7. By adding the difference in charge flux from
dimer to monomer to the Pxx term, the predicted intensity for
the asymmetric stretching mode of the electron donor rises
from 55 km/mole to 83 km/mole. The value of 83 km/mole is
in closer agreement with the predicted value for the dimer
of 110 km/mole. Therefore, by considering only the increase
in charge flux in the dimer relative to the monomer, almost
76% of the intensity of the symmetric stretch for the electron
donor can be accounted for. Therefore, we conclude that the
intensities for the six intramolecular vibrations of the dimer
are well reproduced by adjusting only the three terms in the
polar tensorsthe PH4 term, the PH1 and the PH2 termand
yy xx xx
that most of the change in these terms as the dimer is formed
is due to a change in charge flux and not to changes in
charge or overlap.
4.3 Frequency Analysis for the Linear
Pair of Water Molecules
The predicted frequencies for the six intramolecular
vibrations of the linear dimer (see Table 4.2) are in good
agreement with the predicted values for the corresponding
vibration in the isolated monomer, differing by only 2030
1
cm However, three of these vibrations are predicted to
occur at wavenumbers considerably different from the corres
ponding vibration in the monomer. The symmetric stretching
vibration of the electron acceptor is predicted to decrease
l
110 cm in the dimer relative to the monomer. The two bend
ing vibrations in the dimer are predicted to increase by 124
1 i
cm and 39 cm In order to understand these changes the
calculated frequency parameters or force constants were analyzed.
The frequency parameters in 3 coordinates were transformed
to internal force constants using equations 18 and 33. The
internal coordinate definitions are given in Table 4.8, and
the resultant B and A matrices were calculated with the pro
grams BMAT and AMATRIX respectively (see Appendix A). The
weighting factor d, where d = 1 Ao has been applied to all
the calculated force constants and therefore the units of