Quantitative prediction and interpretation of infrared spectra


Material Information

Quantitative prediction and interpretation of infrared spectra
Physical Description:
vi, 257 leaves : ill. ; 28 cm.
Chin, S ( Steven ), 1957-
Publication Date:


Subjects / Keywords:
Molecular spectra   ( lcsh )
Infrared spectra   ( lcsh )
Spectrum analysis   ( lcsh )
Infrared spectroscopy   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1984.
Includes bibliographical references (leaves 251-256).
Statement of Responsibility:
by Steven Chin.
General Note:
General Note:

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 000500659
notis - ACS0291
oclc - 12084561
System ID:

Full Text








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


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.

CHE81-01131 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.



ACKNOWLEDGEMENTS ................................... ii

ABSTRACT ........................ ............ ....... v


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


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

6.1 The Predicted Spectrum for the Bifurcated
Water-Water Complex (R = 2.89 Ao) ........ 162
6.2 Additional Calculations on the Bifurcated
Water-Water Complex ...................... 175

7 THE OM-H20-OM COMPLEX, M = 0, -1, -2 ........... 179
7.1 Intensity Theory for Charged Species ...... 180
7.2 The OM.H20-OM Complex .................... 184
7.3 Intensity Analysis for the OM.H20-OM
Complex ...... ............................ 186
7.4 Frequency Analysis for the 0'H20O-OM
Complexes ................................ 198
7.5 Model for the Infrared Spectra of Beta-
Alumina .............. .................... 206

8 IONIZED WATER, H20 ............................ 210

9.1 Comments on the Charge Flux, Charge, and
Overlap Tensors ..................... .... 226
9.2 Other Applications for These Calculations.. 234


A SUMMARY OF PROGRAMS ........................... 238



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



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 4-31G

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


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.


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

electro-optical 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 (12-16).

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


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 rare-gas matrices, reaction

intermediates observed by time-resolved 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 rare-gas 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 hard-to-isolate

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

(19-29), and theoretical (30-41) 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 donor-acceptor 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 H-bonded 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 RAH--B, where RAH is

an electron acceptor, B is an electron donor molecule, and

the hydrogen bond is the H--B interaction. When water is

both the electron acceptor and the donor, A-H is the O-H 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 O-H 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 0-H 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 O-H bonds is important

to consider in analyzing complexes involving water molecules.

For H-bonded 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 H-bonded systems are vital to our understanding of

these complexes.

The frequency shift is usually thought to originate in a

weakening of the A-H bond as the H-bond 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 A-H bond into the non-bonding region of

the A atom, where it no longer compensates for the A-H 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 A-H 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 enhancement-is well known experimentally,

but not as well understood. The electrostatic polarization

and charge-transfer 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

O-H 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)


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





-4 Q)


0 4.)




0 (d






-t4 E-4

o o Cm~
00 r
0 0 -4

Ln Ln r-4



of ko I
I I 0
q.v Ln i
x 0 v

0 0 0
C) a ri

0~i II

m, I,,
o a
-4 r4 -



0-4 0

r4 d4






.rq 0

r.- 4)
Q 4)

--4 (a

a) 4

-,4 fJ
0 r.

-4-4 14J


-H 4-4 r-4
44 0















0 r


0 40


-4 (d


4-) IM
w )

4-) 4

0) w

r-H 4-4 r-
0 0
0 0 wl
-4 4Jk















- I
\ /
\ /


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 beta-alumina complex has been calculated. Water,

as an impurity in beta-alumina, is important to the electrical

conductivity and mechanical strength properties of beta-

alumina (44). This in turn is important towards the use of

beta-alumina as solid electrolyte in the sodium-sulfur 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 beta-alumina.

Infrared spectroscopy is used as a tool to probe the kinetics

and thermodynamics of the reaction of water with beta-alumina.

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

beta-alumina. Bates and coworkers have recently determined

the crystal structure and vibrational spectrum of hydrated

beta-alumina (44). The water molecules are located in the

crystal so that they are strongly hydrogen-bonded to the

oxygen atoms of the aluminate ion (see Fig. 1.4).


0 0

0 0
0 0
o n

. 0U

0 u

m II

40 0

Sp H 11 3o
4 c 0 0 -y 0

I e
0 mH
00 II

0 m II 3 0
04 I(D
-4 -

r C O I
WE-' I

O aa


4, /

J /



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 beta-alumina 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 0-H bonds,

weakening of the 0-H 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 beta-alumina 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.


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 Eckart-Sayvetz conditions (11).

The Eckart-Sayvetz conditions provide explicit defini-

tions for the translations and rotations of a molecule. They

are given below in terms of mass-weighted principal cartesian

coordinates (11).

1 m

1 s
o a

2 Mo= k ma qya

p3 = Em q
M3 M a x
o a a

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 mass-weighted, molecule-fixed, symmetry-adapted, prin-

cipal cartesian coordinate system, hereafter denoted as the

S coordinates.

The transformation from mass-weighted 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 mass-weighted principal

cartesian coordinates. The matrix U can be constructed via

a Gram-Schmidt 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

(E(xo+Ax,y +Ay) E(x +Ax,yo) E(xo,y +Ay) + E(xoyo)
xy AxAy

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(xo-Ax) 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 o-Ax)-P (x)
a = a 0 a o a 0 a 0
xx Ax 2Ax 2Ax
Pa(xo+Ax)-P (xo-Ax)

Here P (xo) represents the permanent equilibrium dipole moment

in the a = x, y, or z direction, Pa(XOfx) and P (x-Ax) 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 mass-weighted cartesian

coordinates K. The force constants in S coordinates, K, can

be easily transformed to mass-weighted cartesians using the

U matrix. Since this matrix is orthogonal its inverse is

simply equal to U In mass-weighted 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
to time, and G1 is actually the identity matrix, E, since

the potential energy is now expressed in mass-weighted

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


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


where di is the degeneracy of the ith mode, and (aP/aQi) is

given in units of e-u 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 mass-weighted 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
z5 -__-1-
P< =O o -- ..... 3.
xx Izz

0 qe x AP AP3N
M I I 7 3N
o xx yy

The matrix P consists of the dipole moment derivatives

in mass-weighted cartesian coordinate space. By removing the

mass dependencethe derivatives in cartesian coordinates are


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


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


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 (3N-6)

columns correspondsto the dipole derivatives with respect to

one of the (3N-6) 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

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, out-of-plane bending (wagging), and linear

bending-aresufficient 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 ... (3N-6)

Rk Xi where i = i, 2, 3 ... (3N) (27a)

or in matrix notation

R= B X (27b)

where B is a (3N-6) 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


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 M-1 B+ (29)

where M is a (3N by 3N) diagonal matrix of the reciprocal

masses. The "inverse" of B is then obtained by (47)

B-1 G = B-1 B M1 B+ (30)

B-1 = M-1 B+ G-1 (31)

A = M-1 B+ G-1 (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

counter-clockwise 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

3N-6, 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)

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
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
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

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
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

vibrations are harmonic so that *m and in in equation 39 are

harmonic oscillator wavefunctions, then equation 40 becomes


(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 e-u where e is the atomic
unit of charge (1 e = 1.602 x 10 19C) and u is the atomic

mass unit. The units are e-u 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-
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 (55,63-65).

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)

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(


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-
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 charge-charge flux-overlap (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 LCAO-MO method is as follows (41)

= EQAR Z z D < AlrA B>] (59)

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)

The term D B is the density matrix element which is defined as

D 2 c .c i(62)
AV B i=1 A B

and S is the overlap integral

S A B = (63)

For simplicity, equation 58 can be rewritten as

p = Z QARA + E F(A,B) (64)

where F(A,B) is now given by

F(A,B) = -Z Z D i/ (1)rA (1)dTr (65)

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 off-diagonal 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)

Px = E + ZERV-C ) 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


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 (XC-AXC)-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 charge-flux


PC(overlap) = P [ (charge) + PC(charge-flux)] (70a)

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)

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
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:

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)

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

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)

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
is termed the anisotropy 8(CF). It is expressed as

(C =) -= e F -(UF)) (80)

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


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)
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 charge-flux 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


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

+ 2 P(CF) iP(O) i (86)

where the sum extends over all nine elements of the respective


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.


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 beta-alumina 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/6-II computer of the Northeast Regional Data

Center (NERDC) operating with OS MV/SP-JES2 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 4-31G basis set was used for all molecules (71). The

integral threshold was set at 1.0 x 10-6, and the convergence

limit at 5.0 x 10-5. 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 8-inch floppy disk for use

in the normal coordinate analysis. This analysis was performed

on this laboratory's S-100 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 8-inch 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 15-inch high resolution graphics screen (480 x 512) detach-

able keyboard and light pen. In addition, there is available

128K of S-100 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

S-100 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

8-inch 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 8-inch

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 4-31G 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 4-31G 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 4-31G basis set), and then the predicted differences were


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 off-diagonal force

constants are equal to zero and do not have to be calculated.

However, all the off-diagonal 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 off-diagonal force constants. The A1 block is a

symmetric 2 by 2 block, and therefore there is one non-zero

off-diagonal force constant. To calculate this off-diagonal

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.



H1 3


H1 0.0 -0.7847 -0.4772

02 0.0 0.0 0.0611

0.0 0.7847


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-
times up to 200 cm- .) Secondly, the 4-31G 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


Table 3.2 Predicted wavenumbers and intensities for isolated
water compared to the experimental wavenumbers and intensities.

Wavenumbers v (cm-1)

Mode Calc.









Intensity A (km/mole)













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 4-31G

basis set are generally accurate enough to be useful in in-

terpreting infrared spectra (17). For this particular type

of calculation (SCF with 4-31G basis set), the frequencies

are expected to be within 10-20% 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


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

100-150 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
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


















a' N

4.4 a'
4.4 a'


a, r-
a' 0
fi o

~R i

N 1
CG 0

0 -a

o .,

r J
N "


a' 5
(N I



N 0
r- f

in N
a' -

= a,


o -
a' .

T- 0
I '
N r
0; 0 .0O


*N-, 0 5

o a
* .0

(N C 4

a,~~ 0 O
N a, a'

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
N '
* *

7' a
Nl N* -,

K -
.4 (4

*0 jy


.a (

0 00
-Nf o o
00 -
o'p ?' ? ^
"-._i A g

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 (12-16), 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 10-19 C.
The coordinate system used is defined below.



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







P 02 =

p H3 =













= .4754 e
= .4268 e

= .2187 e2

= .1292 e2
= .1071 e

= .0995 e2


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

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 O-H bond stretching

coordinates, and the H-O-H 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 10-19 C). The coordinate system is defined in
Table 3.4.

P(C) = 0


P(CF) = 0






0 0

.4017 0

0 .4017

0 0

.0550 .0370

.1040 -.0450

0 0

-.1827 -.1790

-.1990 -.1307


P(C) = 0


P(CF) =

P(0) =




0 0

-.8034 0

0 -.8034

0 0

-.1090 0

0 .0910

0 0

.3644 0

0 .2604

X022 = .6455 e2

X02 = .0067 e2

X02 = .0755 e2

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 off-diagonal terms for H1.

.1614 e2

.0017 e



H 2

= .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 md-AO, then the

units of the stretching force constants are md-AO, the bending

force constants are given in md-Ao/rad2, and the stretch-bend

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.


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 4-31G

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

(0--0) 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 0--0 dis-

tance of 2.974 A. This value is considerably closer to the

experimental value than is the value calculated with the 4-31G

basis set. This suggests that the 4-31G basis set overestimates

the strength of the hydrogen bond, but if the deficiencies of

the basis set are eliminated then more accurate results are


Table 4.1 Principal cartesian coordinates for the linear water
dimer at the theoretical equilibrium geometry (AO). Atom
numbering as in Figure 1.1.


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



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 O-H

bond length involved in the hydrogen bond is predicted to be

0.958 A0. The remaining three 0-H bonds are predicted to be

all the same, and 0.950 A0. The H-O-H 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 O-H bond length (H4-05) 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 3N-6 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 off-diagonal 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 non-zero off-diagonal force constants. In the A"

block there are an addition 3+2+1 or 6 off-diagonal 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)



. o
*4 'n -C I 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>

3 ~'3
Ia _, a >
(1 D w- a 34 a p cD o

) 3- 3w

2 E
as .-

- 1 I 3
O s-S S -

U) 0 I0
(U 4-1 1 3' 3

4 M-1 --
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
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 4-31G 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 4-31G 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 cm-1, and 3872 cm-1 (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.
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 cm-1 is mostly electron donor bending

coupled out-of-phase 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 cm-1 and the one at 1714 cm-1

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 out-of-plane (where reference is

made to the plane of the molecule) bending motions. The vibra-

tion predicted at 835 cm-2 is mainly the 03-H4-5 out-of-plane

(of the electron acceptor) bend (hydrogen bond bend) along with
some electron donor twisting motion. The mode at 749 cm cor-

responds to in-plane electron acceptor twisting in phase with a

little out-of-plane (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
cm is the out-of-phase combination of these same electron

acceptor twisting and electron donor wagging motions just
described. The mode at 313 cm can be characterized as pre-

dominately electron donor twisting motion mixed with electron

acceptor hydrogen (free hydrogen) wagging motion. At 210 cm

the vibration corresponds to the out-of-plane (of electron

acceptor) twisting of the electron acceptor mixed

with some in plane (of electron donor) electron donor bending.
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
[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
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


P = -.0406


P H2 = .0406



P03 0

Px = 0


P 5=


-.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



















0 0 .482

.3645 .0763 0

.0066 .1000 0







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.


.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
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

Table 4.4 Calculated charge tensors for the theoretical linear
dimer compared to those of isolated water (e) (le= 1.602 x
10-19 C). Atom numbering and coordinate system as given in
Figure 1.1.

Linear Dimer


P(C) =

P(C) =























0 0

237 0

0 .4237

0 0

,237 0

0 .4237







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 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


H2 -.1293
P(CF) =


P(CF) =

P(CF) =

P(CF) =





P(CF) =

.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/























































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 10-19 C). Atom numbering and coor-
dinate system are given in Figure 1.1.

P(6) = -.1699


P(O) .1699


P(0) = 0


P(0) = 0


P(0) = 0


P(0) = 0



































0 o

























































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
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





O) p

MO 0

4-) --I

r e
m 1

0 4
41 a)


4l 4



) (n

'-I -r

(U -H

0 (0
i Q)

H 0


S -4


-4 0 m m" Ln
1-1 14 r N L- m
-4 r-4 14 t -4 -4

% .D L co -
oo -4 *-r ,-'4
-l mf CN -4 r-4

0 H4

06 X




0 m
a, ^-
1r-1: *v

'C *

o: p,.
2 *3

-r cu

i-l B s^

o "
1 '5
C g= C

- 12

N (B


Ln co

m m
Ln CM m

r-l m


LA )



.* r-4
-4 1-4

-4 N
'C' i-4

T r-4

o oo
m-4 r-4

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
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 tensors--the PH4 term, the PH1 and the PH2 term--and
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 20-30
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
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