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Infrared intensities of water and water dimer

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Title:
Infrared intensities of water and water dimer
Creator:
Zilles, Barbara Ann ( Dissertant )
Person, Willis B. ( Thesis advisor )
Vala, Martin T. ( Reviewer )
Muga, M. Luis ( Reviewer )
Bailey, Thomas L. ( Reviewer )
Brooks, James K. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1980
Language:
English
Physical Description:
ix, 133 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Atoms ( jstor )
Coordinate systems ( jstor )
Dimers ( jstor )
Electrons ( jstor )
Matrices ( jstor )
Molecules ( jstor )
Monomers ( jstor )
Spectral bands ( jstor )
Symmetry ( jstor )
Tensors ( jstor )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Monomers ( lcsh )
Polymers and polymerization ( lcsh )
Water -- Analysis ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The atomic polar tensors for water monomer and the linear dimer have been calculated by the self-consistent field method using the 4-31G basis set. The atomic polar tensors have been used to obtain infrared absorption intensities for these two molecules. Atomic polar tensors were also determined from experimental intensities for water monomer. These experimental polar tensors were then added to the difference between the theoretical dimer and monomer polar tensors to obtain scaled polar tensors for the dimer. Intensities from the monomer polar tensors are compared with values determined from gas-phase measurements. Intensities from both scaled and unsealed 4-31G polar tensors for the dimer are compared with a simulated spectrum, which has been fitted to the experimental spectrum of water dimer isolated in the nitrogen matrix. In addition, the atomic polar tensors for the water monomer and dimer were analyzed using the charge, charge flux, overlap model. The major change from monomer to dimer was found in the charge flux tensor on the hydrogen-bonded hydrogen atom, in the diagonal element along the OH bond. The change in this element was thus shown to be the origin of the characteristic intensity enhancement accompanying hydrogen-bond formation. This result was interpreted in terms of the physical characteristics of the charge flux tensor. The interpretation was consistent with the vibronic charge transfer and dynamic polarization models for the intensity enhancement. Additional conclusions could be made, particularly concerning the effect of the hydrogen-bond on interatomic interactions in the water dimer.
Thesis:
Thesis (Ph. D.)--University of Florida, 1980.
Bibliography:
Includes bibliographic references (leaves 127-132).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Barbara Ann Zilles.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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INFRARED INTENSITIES OF WATER AND WATER DIMER


By

BARBARA ANN ZILLES















A DISSERTATION PRESENTED TO THE GRADUATE
COUNCIL OF THEC UNIVERSITY OF: RORIDA IN PARTIAL
FU~LFILLMIENT OF THIE REQUIREMENTS FOR THIE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA
1980



















ACKNOWLEDGMENTS


I would like to take this opportunity to thank Dr. Person, whose

dedication to helping people learn has benefited me a great deal. In

addition I would like to thank Jerry Rogiers and Roberto M~aia, as well

as past members of Dr. Person's research group, for many stimulating

discussions. Partial support from NSF Research Grants No. CHE-74-21471

and CHE-78-18940, and from the Division of Sponsored Research,

University of Florida, is also gratefully acknowledged.



















TABLE OF CONTENTS


Page

ii

v

vii

viii




1

1

2

5

5
12


20
20
22

25

25

26

29

33

50

58

67

67

71

77


ACKNOWLEDGMENT S------- --------------- --- ---- -------------------

LIST OF TABLES------------------------------------------

LIST OF FIGURES-----------------------------------------

ABSTRACT---- --------------------------------- -------------------

CRAPTER

1. INTRODUCTION--------------------------------------

1.1. Perspective-------------------------------------

1.2. Vibrational Properties of H-bonded Complexes-----------

1.3. Experimental Spectra of Water Dimer--------------------
1.3.1.Matrix Isolation Spectroscopy and Structure
of the Water Dimer-------------------------------
1.3.2.Review of Experimental Studies-------------------

1.4, Theoretical Calculation of Vibrational Properties of
the Wdater D imer-------------------------- --------------
1.4.1.Frequencies-------------------------------
1.4.2.Intensities-------------------------------

2. DESCRIPTION OF CALCULAT IONS---------------------------------

2.1. Outline-----------------------------------------

2.2. Normal Coordinate Analysis-----------------------------

2.3. Intensity Relations----------------------------------

2.4. Application to Intensities for Water and Water Dimer---

2.5. Calculation of Theoretical Polar Tensors---------------

2.6. Scaled Polar Tensors and Intensities-------------------

3. COMPARISON OF THEORETICAL INTENSITIES WITH EXPERIMENT-------

3.1. Simulation of Experimental and Theoretical Spectra-----

3.2. Water Monomer Intensities------------------------------

3.3, Water Dimer Intensities------------------------------











4. POLAR TENSOR ANALYS IS--------------------------------------- 93


4.1. Total Polar Tensors----------------------------------- 93

4.2. Quantum Mechanical Analysis---------------------------- 99
4.2.1.Discussion of Model------------------------------ 99
4.2.2.Results and Implications for the Intensity
Enhancement of the H-bond Band------------------- 106


APPEND IX----------------------------------------------- 119


REFERENCES------------------------------------------ 127


BIOGRAPHICAL SKETCH---------------------------------------- 133

















LIST OF TABLES


Table Page

1-1. Assignment of bands in the V3(H-B) and V1(HI-B) regions to
dimer and trimer by various workers----------------------- 14

2-1. Equilibrium position vectors for water monomer------------ 38

2-2. Definition of internal and symmetry coordinates for water
monomer------------------------------------------ 38

2-3. The B matrix for water monomer---------------------------- 39

2-4. Equilibrium position vectors for the linear water dimer--- 41

2-5. Definition of internal and symmetry coordinates for the
linear water dimer-------------------------------------- 41

2-6. The B matrix for water dimer------------------------------ 43

2-7. Experimental F and L matrices for water monomer in the gas-
phase--------------------------------------------- 45

2-8. Experimental F and L matrices for water monomer in the N2
matrix------------------------------------------- 45

2-9. Experimental F and L matrices for the linear water dimer
in the N2 matrix-------------------------------------- 46

2-10. Scaled 4-31G F and L matrices for the linear water dimer
in the N2 matrix-------------------------------------- 47

2-11. Experimental atomic polar tensors for water monomer in the
gas-phase---------------------------------------- 51

2-12. Transformation matrix from the molecular coordinate system
of the monomer to the bond system of the monomer Hi atom-- 62

2-13. Experimental polar tensor for the monomer H atom in the
bond system----------------------------------------- 62

2-14. Transformation matrix from the molecular coordinate system
of the dimer to the bond system of dimer H atoms---------- 65

2-15. Transformation matrices from the molecular coordinate sys-
tem of the dimer to the MSM of dimer 0 atoms-------------- 65










Table Page

3-1. Lorentzian parameters for fitted experimental spectrum---- 68

3-2. Theoretical and experimental intensities for gas-phase H20 73

3-3. Intensities calculated for H20 from experimental gas-phase
polar tensors using two different L matrices-------------- 74

3-4. Intensities calculated from ab initio 4-31G APT's for water
monomer and dimer-------------------------------------- 78

3-5. Intensities calculated from scaled dimer 4-31G APT's and
experimental monomer polar tensors------------------------ 79

3-6. Dimer intensities calculated from unsealed CNDO APT's at
different 0-0 distances--------------------------------- 87

3-7. Dimer intensities calculated from scaled CNDO APT's at
different 0-0 distances--------------------------------- 87

3-8. Dimer intensities calculated from scaled 4-31G APT's using
two different L matrices--------------------------------- 90

4-1. Ab initio 4-31G polar tensors for H atoms of the water
monomer and dimer, in the bond system--------------------- 94

4-2. Ab initio 4-31G polar tensors for O atoms of the water
monomer and dimer, in the molecular system of the dimer---- 97

4-3. Charge, charge flux, and overlap contributions to ab initio
4-31G APT's for 0 atoms of the water monomer and dimer, in
the molecular system of the dimer------------------------- 107

4-4. Charge, charge flux, and overlap contributions to ab initio
4-31G APT's for O atoms of the water monomer and dimer, in
the molecular system of the dimer------------------------- 108

4-5. Individual contributions to P ,Hb(charge flux), in the bond

system of Hb, for (H20)2 and HCN-HF----------------------- 116
















LIST OF FIGURES


Fig. Page

1-1. Spectrum fitted to experimental spectrum of water
isolated in the N2 matrix--------------------------------- 7

1-2. The linear water dimer------------------------------------ 9

1-3. The centrosymmetric cyclic dimer-------------------------- 11

2-1. Hlolecular coordinate system of water monomer-------------- 36

2-2. Molecular coordinate system of the linear water dimer----- 40

2-3. Bond coordinate system for atom H1 of the water monomer--- 60

2-4. Molecular coordinate system of the monomer for 05 of the
water dimer----------------------------------------- 64

3-1, Spectrum fitted to experimental spectrum of water
isolated in the N2 matrix--------------------------------- 69

3-2. Comparison of fitted experimental spectrum of H20 in the N2
matrix with spectra calculated using experimental gas-
phase polar tensors and two different L matrices---------- 76

3-3. Comparison of fitted experimental spectrum of water dimer
isolated in the N2 matrix with spectra calculated using
unsealed and scaled 4-31G polar tensors------------------- 81

3-4. Comparison of spectra of the water dimer calculated from
unsealed CNDO polar tensors at different 0-0 distances---- 85

3-5. Comparison of spectra of the water dimer calculated from
scaled CNDO polar tensors at different 0-0 distances------ 89

















Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in
Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy


INFRARED INTENSITIES OF WATER AND WATER DIMER

By

Barbara Ann Zilles

June 1980

Chairman: Willis B. Person
Major Department: Chemistry


The atomic polar tensors for water monomer and the linear dimer have

been calculated by the self-consistent field method using the 4-31G basis

set. The atomic polar tensors have been used to obtain infrared absorp-

tion intensities for these two molecules. Atomic polar tensors were also

determined from experimental intensities for water monomer. These

experimental polar tensors were then added to the difference between the

theoretical dimer and monomer polar tensors to obtain scaled polar ten-

sors for the dimer. Intensities from the monomer polar tensors are com-

pared with values determined from gas-phase measurements. Intensities

from both scaled and unsealed 4-31G polar tensors for the dimer are com-

pared with a simulated spectrum, which has been fitted to the experimental

spectrum of water dimer isolated in the nitrogen matrix.

In addition, the atomic polar tensors for the water monomer and dimer

were analyzed using the charge, charge flux, overlap model. The major

change from monomer to dimer was found in the charge flux tensor on the

viii









hydrogen-bonded hydrogen atom, in the diagonal element along the OH bond.

The change in this element was thus shown to be the origin of the

characteristic intensity enhancement accompanying hydrogen-bond

formation. This result was interpreted in terms of the physical

characteristics of the charge flux tensor. The interpretation was con-

sistent with the vibranic charge transfer and dynamic polarization models

for the intensity enhancement. Additional conclusions could be made,

particularly concerning the effect of the hydrogen-bond on interatomic

interactions in the water dimer.















CHAPTER 1

INTRODUCTION

1.1. Perspective


The study of infrared absorption intensities has recently assumed a

new prominence in molecular spectroscopy. Experimentally, the capability

for obtaining high resolution digital absorbance data--on a routine basis

over a wide frequency range--has contributed to this development. Thus,

for example, the difficulty of obtaining reliable absorption coefficients

for overlapped bands has been considerably reduced. Spectrum-fitting

programs are commonly available, which can adjust the intensity parameters

(along with the linewidths and band center frequencies) to minimize the

difference between the experimental and fitted spectrum. While the number

of adjustable parameters in a complex spectrum may be large, the experi-

mental data points are many times more numerous.

Simultaneously, the theoreticians are beginning to make significant

advances in the quantum mechanical calculation of infrared intensity

parameters. This progress has stimulated the development of new formal-

isms for interpreting the intensities. These formalisms have led to

unprecedented success in understanding molecular intensities and predic-

ting them from chemical and structural information about thle molecule

[1-171.

The water dimer has been another subject of intense experimental and

theoretical interest, during the past twelve years. Mlorokuma and Pederson

performed the first theoretical calculation for this dimer in 1968 [18].







Since that time numerous ab initio and empirical studies have been carried

out. There are two motivations for this interest. First, the water dimer

may be regarded as a prototype of liquid water. Direct application of the

results for water dimer to the study of liquid water may be severely

limited. Nonadditivity of pairwise interactions is generally held to be

important 119]. In any case, a thorough understanding of the properties

of water dimer must precede that of liquid water.

Secondly, the water dimer is a prominent example of a hydrogen-

bonded (H-bonded) complex. These weak complexes have stimulated the

imagination of chemists since the 1930's, particularly because of their

importance in biological systems [20-21]. The H-bond illustrates the type

of interaction found in the more general class of electron donor-acceptor

complexes. This interaction can be described as a transfer of electronic

charge from an electron-rich portion of the electron donor molecule to the

electron-acceptor molecule. In H-bonded complexes the interaction in the

electron acceptor is localized in the region of a hydrogen atom. A large

number of H-bonded complexes can be formed with the water molecule acting

as electron acceptor. The water molecule is also an important electron

donor. Thus the water dimer, in which the water molecule acts as both thle

electron donor and acceptor, plays a unique role in the study of

H-bonding.


1.2. Vibrational Properties of H-bonded Complexes


A~mongi thle properties of Hi-bonded complexes that have been studied,

the historical importance of vibrational spectroscopy is well-know~n [221.

Not only can the infrared frequencies and intensities for a particular

complex be compared with a wealth of related data for other complexes and

for the uncomplexed molecules, but these properties also reveal much about








the bonding characteristics of the molecules, and how they are changed by

the H-bond.

The vibration which is most sensitive to the effect of the H-bond is

commonly referred to as the AHI stretch--where RAH is the electron acceptor

and AH is involved in the H-bond. For example, when the electron acceptor

is water, AH is the OH bond. However, in this case, the notion of a

single H-bonded AH stretch is complicated by the fact that there are two

OH bonds in water. In the isolated H20 molecule, these OH bonds are

symmetrically equivalent and couple to give a symmetric and an antisymmet-

ric stretching vibration. Moreover, even though only one of the H atoms

forms the Hi-bond, this coupling is only moderately reduced in the complex.

This is because intramolecular forces are much stronger than the H-bond

interaction.

Because of this coupling there is more than one absorption band in

such complexes whose normal coordinate contains a substantial contribution

from the H-bonded OH stretch. Similarly, the normal coordinate for the

infrared band most sensitive to the H-bond contains contributions not

only from the Hi-bonded OH stretch, but also from the other, "free," OH

stretch, as well as a small contribution from the bend. We therefore

refer to this band as the "H-bond band." Furthermore, it should be

emphasized that, in theoretical calculations for any complex where water

acts as the electron acceptor, a complete normal coordinate analysis is

essential for obtainingS valid frequencies or intensities.

For all H-bonded complexes, the H-bond band undergoes a shift to

lower frequency and a dramatic increase in intensity relative to the

isolated electron acceptor molecule. The magnitude of the frequency

shift has long been regarded as a measure of the strength of the H-bond

120, chap. 3J. Due to the developments in the field of infrared







intensities mentioned above, however, the intensity enhancement is

becoming increasingly recognized as a far more sensitive indicator of the

effect of the H-bond [22]. Accordingly, we have undertaken a thorough

study of the intensity of this band in the water dimer (as well as that

of other bands).

The intensity enhancement is also less readily understood than the

frequency shift. The latter results from a weakening of the AH bond upon

H-bond formation. Two main factors contribute to this reduction in bond

strength. First, the approach of the electron donor molecule on the one

side of the H-bonded H atom polarizes the electron density in the AH

bond. Some of this density is thus forced to the nonbonding region of

the A atom, where it cannot compensate for the AH nuclear repulsion. The

plot of electron density versus position along the H-bond, given by

Schuster for the water dimer [19, p. 74], is an excellent illustration

of this electrostatic polarization effect. Secondly, an equilibrium

charge transfer effect occurs, whereby electron density from the electron

donor molecule is excited into an antibonding AH orbital.

Neither the electrostatic polarization nor the equilibrium charge

transfer effects, however, can begin to account for the overwhelming

magnitude of the intensity enhancement. The intensity in the H-bonded

complex is generally a factor of 10 or more times that in the isolated

molecule, whereas the relative frequency shift is on thef order of a few

percent for non-ionic binary complexes.

The traditional approach to this problem follows that used to

investigate the physical phenomena that contribute to the stabilization

energy of H-bonded complexes [23]. Qualitative arguments concerning the

origin of the intensity enhancement are basically in agreement [24-271.







So far, however, the large magnitude of this enhancement has not been

accounted for on a quantitative basis [23, and references cited therein].

Accordingly, we have approached this problem by applying a quantum

mechanical model for the investigation of intensities. This model has

previously been applied to various molecules, but not to molecular

complexes [4,10]. We have found that such application leads to a unique

quantitative formulation for the H1-bond intensity enhancement. This

formulation is not only consistent with the general trend of ideas on

the subject, but also allows certain distinctions to be made which

further clarify the concepts involved.


1.3. Experimental Spectra of Water Dimer


1.3.1. Matrix Isolation Spectroscopy and Structure of the Water Dimer


Until comparatively recently, the majority of experimental H-bond

studies were carried out in solution. In addition to ill-defined medium

effects, thermal effects at the temperatures associated with liquids

complicate the interpretation of the spectra. Accordingly, the technique

of matrix isolation spectroscopy has been employed extensively in the

last decade for the study of H-bonded species. This technique involves

the rapid condensation, at cryogenic temperatures, of a dilute mixture

of the species to be studied in an inert "matrix" gas 128]. The matrix

gases used most frequently are nitrogen (N2) and argon (A~r). Interactions

of the matrix with the solute are therefore expected to be minimal.

Several workers have observed the infrared spectra of water isolated


in N2 [29-34] and in AZr [34-40] matrices. We discuss these experimental
studies in more detail in the following section. Absorption due to

water monomer, dimer and trimer (or higher multimer) generally appears








in these spectra. The relative abundance of the different aggregate

species depends on the concentration of water in the gaseous mixture, as

well as on other factors which are probably constant for a given set of

experiments.

The bands due to each species can in principle be identified by

observing spectra at several concentrations and noting the concentration

dependence of the band absorbances. The peak absorbances of bands due

to monomer are expected to decrease monotonically with concentration,

while trimer band absorbances are expected to increase monotonically.

The absorbances of dimer bands are expected to show a maximum with respect

to concentration. In practice, however, many of the bands are overlapped

so that two neighboring bands due to different species can show the same

concentration dependence. Thiis difficulty has led to some differences

in the bands assigned by various workers to the dimer, as discussed in

section 1.3.2.

Tursi and Nixon have obtained a spectrum of water isolated in the Np

matrix [30,31), in whiich the dimer bands are quite prominent and well-

resolved from the monomer and trimer bands, as compared with spectra

reported by other workers. A simulated spectrum, which we have adjusted

to give the best fit to thlis experimental spectrum, is illustrated in

Fig. 1-1. The method we have used to calculate the simulated spectrum

is discussed in section 3.1. Thie bands assigned to water monomer, dimer

and trimer (or higher multimer) by Tursi and Nixon are indicated in

Fig. 1-1. All workers have assigned the monomer bands in the same

manner: band I to the antisymmetric stretch, ; band V to the symmetric

stretch, V ; and band XI to the bend, v .

Tursi and Nixon assigned six absorption bands to the dimer as

indicated in Fig. 1-1. W~ith the exception noted below, six bands would









F RE OiU E iNCT Y C 1;I- 1 )






IFIX


DlI~ (I V D

III X"

III M
M XII




Fi.11 pcrmfte oeprmntlsetu ie yTri[1 o
wate islae inteN arx h Hstecigrgo fo
354 to34 m h m edn eini rm18 o13 m
Badsdu t wte mnme (),dier(D ndtrme o hg rmulie
(T ae ndcte imr ad ssgnensmae y urian Nxo (0
in tem ftelna wtrdmraeasoidctd o nume l
idntf bad fo eea icsini h et







generally be expected for thle water dimer, since there are three funda-

mental absorptions for each of the water molecules in the complex. Tursi

and Nixon assigned the six bands in terms of the "linear" structure for

the water dimer illustrated in Fig. 1-2. Their assignments are generally

accepted, and we now discuss how the frequency pattern expected for this

structure is consistent with that in the observed spectrum. In this

discussion we use the simulated spectrum in Fig. 1-1 for reference.

The dimer structure shown in Fig. 1-2 is characterized by the linear

arrangement of the atoms involved in the H-bond: O H,, and Os. These

atoms and atom Hs of the electron acceptor (EA) define a plane of sym-

metry; atoms 11 and H1 of the electron donor (ED) are symmetrically

oriented with respect to this plane. Thus the OHi stretches in the ED

molecule are symmetrically coupled just as in the monomer. That is, the

antisymmetric and symmetric stretches both have a contribution of equal

magnitude from each individual OH stretch. We further note that neither

H1 nor H, is Hi-bonded. Accordingly the spectrum of the linear dimer

would be expected to have three bands for the ED molecule shifted

slightly from the monomer bands vz, u2 and v3. Three such bands--bands

II, VI and X--are evident in the spectrum shown in Fig. 1-1. These bands

were assigned by Tursi and Nixon to V (ED), VI(ED) and V,(ED), respec-

tively.

Figure 1-2 shows that the two OH bonds in the EA molecule are not

syrmnetrically equivalent. Moreover, the OsHI bond is H-bonded and th~us

has a loWer stretching force constant than the "free" Os's6 bond. The

two stretching vibrations expected for the EA molecule would therefore

contain unequal contributions from these two OH stretches, but would

retain their basic symmetric and antisymmetric character as discussed in

section 1.2. A greater contribution from the H-bonded Os a, stretch would















HH

H O

Oz


ED E

H,









Fig. 1-2. The linear water dimer. The electron donor (ED) and electron acceptor (EA)
components are indiicated. Atom numbers For the normal coordinate analysis
are also indicated.








be expected for the symm~etric stretch, V (EA), whiich would therefore be

shifted to considerably lower frequency from vz of the monomer. Band VII

in Fig. 1-1 is consistent with this expectation. This band was assigned

to V (EA) by Tursi and Nixon. The antisymmetric stretch, V3(EA), would

be expected to have a greater contribution from the "free" OsH6 stretch

and a moderate shift to lower frequency from v3 of the monomer. Band III

exhibits such a shift and was assigned to V3(EA) by Tursi and Nixon.

The bending vibration of the EA molecule in Fig. 1-2 perturbs the

linearity of the H-bond. A considerable shiift to higher frequency from

V2 of the monomer would thus be expected for this vibration. Accordingly,

band IX in Fig. 1-1 was assigned to V2(EA) by Tursi and Nixon.

Some of the bands in Fig. 1-1 have also been interpreted in terms

of the cyclic structure for the water dimer [29,34,36]. This structure

has two H-bonds and is characterized by the nonlinear arrangement of the

two 0 atoms and the H atom involved in each H-bond. The structure of the

centrosymmetric cyclic dimer is illustrated in Fig. 1-3. This structure

is unique in that it has a center of inversion and thus obeys the exclu-

sion rule--three vibrations are active only in the infrared, and three are

active only in the Raman spectrum. The three infrared active vibrations

are the out-of-phase combinations (v~ 9 and v3 ) of the corresponding

monomer vibrations of the two water molecules forming the dimer.

The frequency shifts expected for these vibrations, from vy, f2 and

V3 of the monomer, would follow qualitatively the pattern discussed for

the vibrations of the EA molecule of the linear dimer. However, the

frequency shifts for the cyclic dimer should be smaller because the

nonlinear H-bonds are not as strong as linear H-bonds 136,41].


























SOz





H,

H4



Fig. 1-3. Thie centrosymmetric cyclic water dimer. The center of inversion is at the
midpoint between 0l and 02.





12

1.3.2. Review of Experimental Studies


The first study of matrix isolated water was carried out in the N2

matrix by Van Thiel, Becker and Pimentel, using A prism monochromator [291.

These authors attributed the dimer bands to the centrasyrmmetric cyclic

structure because only three dimer bands could be observed in their low

resolution spectra. Bands II, VI and X in Fig. 1-1 were not resolved

from the neighboring monomer bands.

In all the subsequent studies, the experimental spectra of water

isolated in the N2 matrix were basically the same as those obtained by

Tursi and Nixon. Three groups of workers [33-351 did not assign certain

bands in their spectra in agreement with the assignment made by Tursi and

Nixon (the TN assignment). These discrepancies involve bands assigned to

dimer and trimer in two regions of the spectrum. One of these regions

includes bands III and IV in Fig. 1-1. We recall that band III corre-

sponds to V3(EA) of the linear dimer according to the TN assignment. Now

the frequency shift of this vibration from VS of the monomer is due to

the H-bond interaction. We therefore refer to this region as the v (H-B)

region.

The other region in which different assignments have been made by

various workers includes band VII, which corresponds to Vf(EA) according

to the TN assignment. He designate this region as the Vg(H-B) region.

Figure 1-1 shows that in the V (H-8) region a relatively sharp dimer band

is superimposed on a broad trimer band according to the TN assignment.

The same situation occurs in the V1(H-B) region. In many of thle spectra

obtained by various workers a broad shoulder appears on the low frequency

side of band VII in Figure 1-1. We refer to thant broad absorption as

band VIII.





13

Suppose the TN assignment is correct and that the broad bands are

due to trimer while the sharper bands are due to dimer. Then thle absor-

bances of the two broad bands should vary differently with respect to

concentration from those of the two sharp bands. However, the concentra-

tion dependence of the "true" band absorbances would probably not be

observed, because the broad trimer band acts as a baseline for the sharper

dimer band. Rather, any increase in the absorbance of the broad trimer

band would result in an apparent increase in the absorbance of the dimer

band, even though the "true" absorbance of the latter band does not

increase. Thus, reliable assignments of these bands cannot be made solely

on the basis of their concentration dependence.

Table 1-1 summarizes the discrepancies between the TN assignment and

the assignments made by other workers for bands III, IV, VII and VIII.

In two of the studies, more than four bands in the OH stretching region

were assigned to dimer. Since any given structure for the water dimer

can have a maximum of four bands in this region (two bands for each of

the component water molecules), alternate interpretations had to be made

for the extra bands.

Huong and Cornut obtained spectra of water isolated in both the N2

and the Ar matrix [341. As indicated in Table 1-1, these authors

attributed bands IV and VII to the dimer. As discussed in section 1.3.1.,

two stretching vibrations, vs and vl are expected for the centrosym-

metric cyclic dimer. Accordingly, Huang and Cornut assigned band IV in

the U (H-B) region to V, and band VII in the u (Hl-B) region to 9 .

They assigned bands II, III, VI and VII to the linear dimer according to

the TN assignment. Their overall assignment is therefore not consistent

with the relative frequency pattern expected for linear and nonlinear

H-bonds (see section 1.3.1.). That is, the assignment attributes larger








Table 1-1. Assignment of bands in the V3(H-B) and VI(H-B) regions to
dimer (D) and trimer (T) by various workers. Bands III, IV
and VII are indicated in Fig. 1-1. Band VIII appears on the
low frequency side of band VII in many spectra.




V (H-B) Region V,(H-B) Region Total Number of
Dimer Bands in OH
III IV VII VIII Stretching Region

Tursi and Nixona D T D T 4

Huong and Cornutb D D D D 6

Barlettae D T D D 8C

Luckd T T D D 4


Tursi and Nixon [30].

Huong and Cornut [34).
SBarletta [33}. Band VIII is resolved into four bands in this work.
Luck [36]. See also Mann, Neikes, Schmidt and Luck [351.






15

frequency shifts to the bent H-bonds of the cyclic dimer than to the

linear H-bond of the structure shown in Fig. 1-2.

Barletta obtained excellent high resolution spectra of water isolated

in the N2 matrix [33). In these spectra four bands were resolved in the

absorption corresponding to band VIII in the V (H-B) region. These four

extrar" bands were attributed to the dimer. Of the eight bands thus

assigned to the dimer, four were observed to be considerably weaker than

the others. The four stronger bands were then assigned to fundamentals

of the linear dimer: bands II, III and VII in Fig. 1-1 and one of the

four "extra" bands in the V (H-B) region. According to this assignment

two vibrations (correspabding to band VII and the"extra" band) have

substantially lower frequencies than vl of the monomer. Only one such

vibration is expected for the linear dimer as discussed in section 1.3.1.

Moreover, band VI in Fig. 1-1 was considered too weak to be a fundamental.

This band corresponds to V (ED) according to the TN assignment. It is

shown below, however, that on the basis of our calculations for the

linear water dimer the intensity of the VI(ED) fundamental should indeed

be small.

Mlann, Neikes, Schmidt and Luck observed the spectra of water isolated

in the Ar matrix [35, see also reference 36J. Their spectra were

basically the same as those observed for water isolated in the N2 matrix

by the various workers. Table 1-1 shows that these authors assigned both

bands in the VB(H-B) region to the trimer, while both bands in the VI(H-B)

region were assigned to the dimer. These authors interpreted the dimer

bands in terms of the cyclic structure.

We note that the cyclic dimer can be asymmietric. If one of the two

water molecules in the structure shown in Fig. 1-3 is rotated somewhat,

the center of symmetry is eliminated. For this structure two weak







stretching bands would be expected, corresponding to the infrared inactive

vibrations of the centrosymmetric structure, namely, the in-phase combina-

tions (vq and UI ) of thle corresponding monomer vibrations of the two

water molecules in the dimer. According to these considerations, M~ann,

Neikes, Schmidt and Luck assigned bands II, VI and VII in Fig. 1-1, as

well as band VIII in the VI(H-B) region, to the asymmetric cyclic dimer
-+ +
stretching vibrations, \) V3 vy and vl respectively. This assign-

ment is consistent with the relative intensity pattern expected for the

out-of-phase and in-phase vibrations. However, the frequency shifts, from

the corresponding monomer bands, resulting from this assignment do not

seem entirely satisfactory. That is, while frequency shifts of 103 and

97 cm- are attributed to V3 and vl respectively, a relatively small

shift of 14 cm' is attributed to V1 and a relatively large shift of

17 m is attributed to .

On the other hand, Luck [36] objected to the frequency shift of

V (EA), the "H-bond band," resulting from the TN assignment on the basis

that it was too small (85 cm ~) for a linear dimer. Several experimental

studies were cited, which showed that linear H-bonds were stronger than

nonlinear H-bonds. It was pointed out that the water polymer (small

crystallites or amorphous clusters of ice) had linear H-bonds, and its

H-bond band occurred at 3220 cm- (in Ar). Luck then stated, "We would

expect the absorption of the linear dimers in the region of the H-bond

band of large aggfregantes ('polymer band')" [36, p. 552].

This reasoningl does not take into account thle impiortance of coopern-

tive effects in ice, which has a stabilization energy of 11 kcal/mole

[42]. In conitra~st, Clementi hans calculated a stabilizaition energy of







5.6 kcal/mole for the linear water dimer,* using a very large basis set

and including correlation effects [431. Furthermore, we point to the

good agreement of the theoretical frequency predicted for the H-bond band

of the linear dimer (see Appendix) with that of band VII in Fig. 1-1.

The intensity we have calculated is too large. However, evidence will

be presented below that this discrepancy is due to an artifact of the

calculation rather than to the structure of the model we have chosen.

Tursi and Nixon also observed the spectrum of D2anHDinteN

matrix in addition to that of H20. They found a one-to-one correspon-

dence between the H20 bands discussed in section 1.3.1. and those due to

D20. Corresponding bands were also found for HDO monomer. For the

linear HDO dimer, there are two possible isotopes since the EA molecule

can be completed either through the H atom or through the D atom. Tursi

and Nixon observed four stretching bands whose frequency pattern was

consistent with that expected for a linear HDO dimer with the EA completed

through the D atom. Two additional bands expected for the other isotope

were not observed, nor were any dimer bands in the HDO bending region

observed .

Tursi and Nixon used the nine frequencies they observed for H20, D20

and HDO monomers to obtain a force field for water monamer. Similarly,

they obtained a force field for the linear water dimer using twelve

freqences fr (20)2 and (D20)2, and the four frequencies for (HDO)2

We have used these two force fields in our intensity calculations, and

thley are discussed further below.



*' Experimental studies give estimates of 5.2 kcal/mole [44) and
3.0 keal/mole [451. But the theoretical value is the most reliable and
probably close to the true value.







Recently, the infrared spectrum of the various isotopes of water

isolated in the N2 matrix was again studied by Fredin, Nelander and

Ribbegard [32]. They made assignments for all the isotopic dimer bands

in terms of the linear structure, in agreement with Tursi and Nixon.

Moreover, they found that increasing the intensity of the irradiating

beam and/or decreasing the temperature of the sample led to the appearance

of the bands for linear (HDO)2 not observed by Tursi and Nixon--the dimer

bands in the HDO bending region and those corresponding to the EA

completed through the H atom. Thus, all the bands for (D20)2 and (HDO)2

have been observed at those frequencies which are expected for the linear

structure shown in Fig. 1-2. The consistency of these results with those

for (H20)2 strongly confirms the assignments given in section 1.3.1.

As mentioned above, Huong and Cornut [34] and Mann, Neikes, Schmidt

and Luck [351 have obtained spectra of water isolated in the Ar matrix

which are basically the same as those in the N2 matrix. Several workers

have obtained a totally different type of spectrum for water in the Ar

matrix 137-401. These workers have assigned the majority of bands in the

spectrum to vibration-ratation transitions of the monomer. The relative

absorbances of these bands show a complicated time and temperature

dependence. This behavior has been interpreted in terms of conversion

from ortho to para spin states of the hydrogen nuclei [37-39]. According

to this interpretation, the most recent results for the time and tempera-

ture dependence of the various bands indicate the following [391.

Unexpectedly rapid spin conversion occurs during the deposition process--

to an ortho/para ratio characteristic of a temperature roughly halfway

between room temperature and th~e temperature of the matrix. Thereafter,

the conversion process continues at a slower rate in matrices with







relatively high water concentrations. But in dilute matrices no conver-

sion occurs after deposition.

This interpretation is rather elaborate, and it should be pointed out

that other explanations may exist for the profuse spectrum of water in the

Ar matrix. Matrices giving rise to this type of spectrum have been

deposited at lower temperatures (4-100K) than those giving rise to the

simple spectrum (16-200K). At higher temperatures, the matrix is likely

to be annealed during deposition so that the orientation of the Ar

crystallites in the matrix is more uniform. It is possible that the

importance of nonequilibrium effects in the lower temperature matrix

results in a larger number of interactive sites, each having a different

energy and thus giving rise to different frequency shifts for the vibra-

tions. In addition, the possibility of N2 impurity in the Ar matrix

cannot be ignored. Mixed aggregates of H20 and N2 molecules could

account for some of the bands in the profuse spectrum.*

Within the context of attributing most of the bands to monomer

vibration-rotation transitions, Ayers and Pullin have assigned five bands

in the profuse spectrum of water in Ar to the linear water dimer [40].

In the frequency region corresponding to that expected for the V3(ED)

vibration, no band could be attributed to the dimer, all bands in this

region having been assigned to the monomer. The "absence" of this band

is somewhat disturbing since our calculations predict the intensity of

v3(ED) to be second only to that of thle H-bond band, V (EA). Possibly

one of the "rotatingi monomer" bands could be attributed to this vibration.



* In this regard, a number of bands have been induced in the spectrum of
hydrogen halides (RX) in the Ar matrix by doping with N2 impurity. These
bands have been attributed to various aggregates of the HX and N2 mole-
cules in the Ar matrix [46].








1.4. Theoretical Calculation of Vibrational
Properties of the Water Dimer


1.4.1. Frequencies


Matrix isolated spectra of the water dimer have provided new data

that are important for the study of H-bonding. Theoretical calculations

are an additional source of such data. Moreover, the development of

theoretical methods is now approaching a level, where they can be used as

a reliable guide for interpretation of the experimental spectra.

Curtiss and Pople have performed a set of quantum mechanical calcu-

lations with the 4-310 basis set in order to obtain force constants for

water monomer and the linear dimer [47]. The vibrational frequencies for

monomer and dimer were then calculated from each force field. These

frequencies agree reasonably well with the experimental ones observed by

TN, but they are all 10% too large. It is typical for Hartree-Fock

calculations to overestimate frequencies by approximately this amount

[19, p. 33].

The absolute differences between the calculated and experimental

frequencies (about 360 cm- in the stretching region and 160 cm-' in the

bending region) might seem too large for the calculations to be of any

use for the experimentalist. However, the factors in the calculation

that lead to overestimation of the force constants, and thus of the

frequencies, should be nearly constant for the monomer and dimer. One

method of comparing calculation with experiment, which minimizes the

effect of constant errors in the calculation, is to examine the calculated

and observed frequency shifits from monomer to dimer.

An alternate approach makes use of the fact that the physical

phenomena responsible for the frequency shifts (for example, changes in







bond strengths) appear directly as changes in the force constants. Thus,

in this approach one is concerned with the change in each force

constant, Af(th.), which is obtained from the theoretically calculated

monomer and dimer force constants, according to


af(th.) = f(th., dimer) f~th., monomer),


A set of scaled dimer force constants can then be calculated using the

experimental monomer force constant:


f(scaled, dimer) = f(exptl., monomer) + af(th.).


This set of scaled dimer force constants is then used in a standard

normal coordinate calculation to obtain dimer frequencies which can be

compared directly with experiment.

In this scaling procedure, errors in the theoretically calculated

monomer and dimer force constants compensate more effectively than they

do in the comparison of frequency shifts. This is because, in the normal

coordinate calculation of the frequencies the dimer force constants are

weighted differently, by the geometrical parameters and the masses, from

the monomer force constants. The force constant errors are thus also

weighted differently in the calculated frequencies.

This scaling procedure has been applied to the theoretical force

constants calculated by Curtiss and Pople (see Appendix). It has also

been applied to a set of force fields calculated by the CNDO method

for the linear water dimer. The resulting frequencies predicted for the

water dimer agree quite will with the experimental frequencies.








1.4.2. Intensities


Little work has been done in the area of theoretical intensity

calculations for the water dimer. In the work that has been done, only

the intensity of the H-bonded OH stretch has been calculated, and normal

modes have not been taken into account [48, and references cited therein].

Because theoretical intensities can provide valuable assistance in the

interpretation of experimental spectra, we have calculated a complete set

of intensities for the normal modes of the linear water dimer.

As discussed in section 2.4., the intensities are proportional to

the square of the derivative of the molecular dipole moment with respect

to the normal coordinates. These dipole derivatives, like the force

constants, reflect the chemical bonding characteristics of the molecule.

One of the goals of infrared intensity theory is to study how the dipole

derivatives change from one molecule to another and interpret these

changes in terms of the types of atoms and bonds present. Dipole deriva-

tives can also be expressed in terms of other forms of the vibrational

coordinates which are more useful for interpretation than the normal

coordinates.

The space-fixed cartesian representation for the dipole derivatives

has proven particularly advantageous for understanding the chemical

factors that determine the intensities 11-101. These dipole derivatives

are composed of a set of atomic polar tensors (APT's), one for each atom

in thle molecule (11. Studies have indicated that the values of the APT

elements for a given atom tend to be relatively independent of which

molecule the atom is in [4,7]. Remarkable success has been achieved in

predicting intensities for one molecule by transferring APT's from other

chemically related molecules, and then transforming these APT's to





23

dipole derivatives with respect to the normal coordinates of the first

molecule [5]. These results suggest that the APT's are fundamental

indicators of the chemical properties of atoms in molecules.

Accordingly, we have calculated APT's for the water monomer and

dimer using quantum mechanical methods in order to predict intensities

for these two molecules. As for the force constants, we expect that

the errors in the quantum mechanical calculation of the APT's are nearly

the same for the dimer complex as for the monomer. To the extent that

this is true, we might expect that the changes in the APT's from monomer

to dimer are well-represented by the calculation. With this idea in

mind, we have applied a scaling procedure, analogous to that described

above for the force constants, to obtain A~PT's for atoms in the dimer

from the known APT's for those atoms in the monomer.

First, the change in the APT of atom A from monomer to dimer,

aAP(th.), was obtained from the theoretical calculation by,


() aPA(th.) = PA(th., dimer) PA(th., monmer).


Then the experimental APT for atom A in the monomer was used to obtain

the scaled dimer APT:


(2) PA(scaled, dimer) = PA(exptl., monomer) + aPA(th.).


The APT's from both the ab initio quantum mechanical calculation, PA(th.,

dimer), and from th~e scaling procedure, P (scaled, dimer), w~ere used to

obtain intensities for the absorption bands of the water dimer. These

theoretical dimer APT's thus provided an independent source of data, which

we have used for comparison with experiment. The comparison will be

discussed in detail in chapter 3.






24

The APT's also contain more information thann do the intensities for

a relatively large, unsymmetrical molecule like the water dimer. That is,

the intensities can be obtained from the APT's, but the APT's can only be

calculated theoretically. Both the APT's and the intensities reflect

the redistribution of charge that takes place as the molecule vibrates.

In the APT's, this vibrational redistribution of charge is resolved into

contributions from each atom.

As mentioned previously, in the study of H-bonded complexes our

chief interest lies in the changes that take place relative to the uncom-

plexed molecule. W~e hlave thus compared the theoretical APT's which we

have calculated for each atom in the dimer with the corresponding APT in

the monomer. We have found that changes in the APT's from monomer to

dimer are restricted to a few critical elements. Moreover, the major

changes are restricted to those dimer atoms that are involved in the

H-bond. We have also used thle quantum mechanical definition of the

dipole moment to resolve each APT into charge, charge flux, and overlap

contributions. Again, we have found that changes from monomer to dimer

are restricted to a few critical elements. These results are presented

and discussed in chapter 4.















CHAPTER 2

DESCRIPTION OF CALCULATIONS

2.1. Outline


The essence of our method lies in the quantum mechanical calculations

and in the scaling procedure used for the dimer polar tensors, given in

Eqs. I and 2. However, we wish to predict the intensities for the

normal modes of vibration of the molecule, and thus the calculation of

intensities from polar tensors (APT's) is intimately bound up with the

normal coordinate analysis. The forms of the normal coordinates depend

on the force field and on the geometry and atomic masses of the molecule.

Our purpose in this chapter is to include enough details about our

calculations so that the reader can reproduce our results or obtain

analogous results, using a different force field or theoretical APT's

from a different basis set. In order to do this, we must refer in some

detail to the equations involved in the normal coordinate analysis; thus,

in section 2.2 we set down these equations. In section 2.3 we give the

equations relating the intensities and the APT's. Then in section 2.4

we describe how we have applied these equations and give all the data

we have used, except: for thle theoretical APT's. In section 2.5 we

describe the numerical and quantum mechanical methods we have used in

calculating the theoretical APT's. Finally, in section 2.6, we give our

procedure for calculating the scaled APT's and intensities for the dimer.

25








2.2. Normal Coordinate Analysis


As mentioned previously, there are several ways of representing the

vibrational coordinates of the molecule. The normal coordinates, Qi, are

characterized by the fact that each fundamental absorption band depends on

a single Q. within the harmonic oscillator approximation. Moreover, in

the normal coordinate representation, both the kinetic and potential

vibrational energy of thie molecule are a sum of independent contributions.

Each of these contributions also depends on a single Q.. We have used the

Wilson FG matrix method to obtain the normal coordinates. The most impor-

tant equations in this method are presented here. Detailed discussion of

these equations can be found elsewhere [49).

The internal valence coordinates, R., provide a useful representation

for the force constants. For an N-atomic molecule, th~e (3N-6)-dimensional

column vector of internal coordinates, R, is related to that of normal

coordinates, 9, by the matrix equation:


(3) R =LQ.


The normal coordinate transformation matrix, L, is determined by solving

the eigenvalue problem,


(4) GFL = LAZ.


The eigenvalues of this equation form the elements of the diagonal matrix

of frequency parameters, A, whiile the eigienvectors are given by the

corresponding columns of thle L matrix. The solution depends on the

inverse kinetic energy matrix, G, and the potential energy matrix, F.

The latter is composed of the force constants, given by





27


a V
F =
ij aR,aR.


where V is the vibrational potential energy of the molecule.

As discussed in the previous chapter, the space-fixed cartesian

coordinates constitute the most useful representation for the intensities.

The 3N-dimensional vector of these coordinates, X, is related to the

vector, R, by


R = X .


The [(3N 6) x 3N]-dimensional B matrix is

Eq. 4, according to


related to the G matrix in


G = BM 'Bi


where Bf indicates the transpose of B.

masses, M -, is given by


(8- 1
11 51


m,


The diagonal matrix of inverse





i= 1,2,3


i =4,5,6






i = 3N-2, 3N-1, 3N


for the N atoms of the molecule.

A third type of vibrational

eigenvalue problem of Eq. 4. In

the G and F matrices factor into

coordinate vector, S, is related


coordinates is used to simplify the

the symmetry coordinate representation,

different symmetry blocks. The symmetry

to R by








S = UR

S = EX


The symmetrized B) matrix, 6, is given by


8 = UB


The syrmmetrized F matrix, F, is then defined as


a2E
F..


F = UFUt


(11a)


(11b)


where U is an orthogonal transformation, that is Ut = U '. Similarly,

the symmetrized G matrix, G, is given by


G = 8 1 St3


G = UGUt


(12a)


(12b)


Finally, we define the symmetrized L matrix:


(13a) s = LQ

(13b) L = UL


It can be seen from Eqs. 11b, 12b and 13b that Eq. 4 is equivalent


GFL = LAZ


Each of the matrices in this equation is block diagonal. Thus, neither

the symmetry nor the normal coordinates in a given block can mix with

those in the other blocks. There is one block for each of the







irreducible representations of the vibrations in the symmertry group to

which the molecule belongs.

The units we have used for the various matrices can be discussed on

the basis that there are tw~o types of symmetry coordinates, S.. Those

involving bond stretches (r-type) have units of length. For those

involving angle deformations (6-type) we have used units of radians

(rad). There are, then, three types of force constants, depending on

whether Si and Sj in Eq. 11a are bothi r-type, both 6-type, or one of

each. The resulting force constants all have similar magnitudes, if the

energy in that equation is given in units of millidyne A (md A, which is

10-11 erg) and the length in R. The units of the force constant, Fij,

in the three cases for Si and S. cited above, are then md (A) L, md 8,

rad ', or md rad ', respectively. W~hen Sk is r-type, the kth row of 6 is

unitless, whereas it has units of rad (A) we kis6tp. o h

masses we have used atomic mass units (u). Thus, the kth row of L has

units of u when Sk is r-type and units of rad (A)- u when Sk is

6-type.

The G matrix in Eq. 14 is completely determined by the 6 and HI-

matrices, according to Eq. 12a. Thus, the solution to Eq. 14 depends on

the geometry, masses and force constants of the molecule. Thle eigenvector

matrix, L, describes the forms of the normal coordinates. Hence, L

enters into the calculation of the intensities.


2.3 Intensity Relations


There are a number of wanys of expressing infrared intensities.

Overend [50] has given thle relation between thle inteGrated molar absorp-

tion coefficient of the ith band, A. and thle derivative of the dipole
1'

moment vector, p, with respect to the _ith normal coordinate, Q *:




30


SN A H2 i a ap 2 1



Here, NA is Avogadro's number, and c is the speed of light in cm sec .
This equation gives A. in cm male' whn3/8.i n s Hwvr

for the latter we have used the atomic units electron (e) u .Thus, we

have used the equation



(15) Ai Ae2 (N9~2 q



where qe gives the electronic charge (qe = 4.803 x 10-10 esu), and A. is

still in cm male This equation assumes that the vibrational frequency

of each band is large enough so that all of the molecules are in the

ground vibrational state at the temperature of the measurement. It also

involves the assumptions of mechanical and electrical harmonicity, that

is, the dipole moment is expanded as a linear function of thie Q and the

harmonic oscillator wavefunctions are used for the vibrational Schrodinger

equation.

The remaining equations in this section have been treated previously

[1-3,6J. Eq. 15 may be expressed in terms of the PQ matrices defined by
Person and Newton [1],


(16) A 2N


where

ap. 3 = x,y,z
Qjk aQk' k = i,...,3N-6


is the matrix of dipole derivatives with respect to the normal





31

coordinates. The matrix of dipole derivatives with respect to the 3N

space-fixed cartesian coordinates, PX is related to the P~ matrix by


(17) Pq = PXA


The AL product matrix gives the transformation from the vector of normal

coordinates, Q, to the vector of space-fixed coordinates, X:


(18) X = AL

= ALQ.


The A and A matrices are related by


A =AUt


and are given by (3)



(19) A = M BftG-1




The AL product matrix has units of u b, and PX has units of e.

The PX matrix is comprised of N juxtaposed APT's for each of

the atoms of the molecule. If kA represents the cartesian coordinates of

atom A, the APT on atom A, PA, is given by

ap
(20) (PA
jk kA j ,k -x,y,z.


Then the _P matrix has the form


(21) PX = 1 P2 -- N)


From Eqs. 17 and 21 we can see that it is necessary to specify the

A and L matrices, in addition to the APT's, in order to determine the P





32

matrix and therefore the intensities. From Eqs. 19, 12a and 8, we see

that the A matrix is determined by the 8 matrix and the masses of the

molecule.

In order to obtain intensity predictions for the fundamental vibra-

tions of water and water dimer, we have transformed the theoretically

calculated APT's to the P_ matrix, according to Eqs. 17 and 21. We shall

also need the inverse transformation from th~e Pg to the PX matrix. ~This

is because we wish to use the experimental APT's for the water monomer

in our scaling procedure (see Eq. 2), and these must be obtained from

the experimental intensities. The desired inverse transformation is

given by


(22) P =P L 8B + D.


In this equation, the D matrix is the permanent dipole moment rotation

matrix resulting from the transformation from the molecule-fixed normal

coordinate representation to the space-fixed cartesian coordinate

representation.

Like the PX matrix, the D matrix is comprised of N juxtaposed

tensors. It has the form of Eq. 21 with PA replaced everywhere by DA


A

(23) DA t(~)


Each of the factors in Eq. 23 is a second order tensor of rantk 3. All

three elements of the diagonal tensor, mA are equal to the mass of atom

A; o is the permanent dipole moment vector of the molecule; and rAoi

the equilibrium position vector of atom A with respect to the center of

mass. The notation, ((v)) [represented by ( golf in Eq. 23, for example],

indicates the tensor formed from the vector, v, according to [6]






33

(24) (( )) = 0 vz y

-v 0 v
z x
v -v 0
Sy x


In Eq. 23, I- is the inverse of the moment of inertia tensor, I, which

is given by




(25) I = mA((r O)t(((r 0)



Thus, in order to determine the D matrix, we must know the masses

and position vectors of the atoms with respect to the center of mass of

the molecule, rA' and the permanent dipole moment, p. eavusd A

for the units of the dipole moment, u for the masses, and 1 for the

position vectors in Eqs. 23 and 25. The rotational tensor, D therefore

has units of e.


2.4. Application to Intensities for Water and Water Dimer


We may now apply these equations to the intensity calculations for

water monomer and dimer. For the monomer we are interested in evaluating

the theoretical APT's by comparing the resulting intensities with the

experimental gas-phase intensities and with those from other calculations.

Thus, we have used the A matrix and the experimental gas-phase L matrix

in Eqs. 16 and 17 to calculate the intensities from these APT's. The A

matrix was determined from the monomer 6 matrix and the atomic masses,

according to Eq. 19. The L matrix was obtained by solving Eq. 14, using

the experimental values of the force constants for the gas-phiase

monomer [511. As mentioned above, the G matrix in Eqs. 14 and 19 is

determined from the 8 matrix and the masses by Eq. 12a. Since we are





34

also interested in the observed intensities for the monomer in the N2

matrix,* we have calculated a set of intensities using the L matrix

derived from the force constants reported for the monomer in the

matrix 1301.

For the matrix-isolated dimer, we have calculated a set of intensi-

ties using the corresponding A and L matrices with the APT's obtained

directly from the theoretical calculations for the dimer. The L matrix

was derived from the force constants reported for the dimer in the N2

matrix (30], as well as the 6 matrix and masses, using Eq. 14. An

additional set of intensities was calculated from the scaled APT's using

the same A and L matrices. In order to investigate the effect of changing

the L matrix on the predicted intensities, we have also obtained an L

matrix using force constants derived from the theoretical calculations of

Curtiss and Pople [47]. Intensities were then obtained using this L

matrix and the scaled dimer APT's. To obtain the latter we have used the

experimental APT's for the monomer along with the monomer and dimer APT's

calculated theoretically, according to Eqs. I and 2.

The experimental monomer APT's were calculated from the experimental

intensities by first obtaining the Pq matrix elements according to Eq. 16,

and then by using this Pq matrix with the monomer 8, L and D matrices in

Eq. 22. Experimental intensities are not available for the monomer in

the matrix, so we have assumed they are about equal to those in the gas-

phase. To be consistent, we have used the monomer L matrix derived from

the gas-phase force constants to calculate the exper~imental monomer APT's.


Here we refer to the inert solid environment discussed in section 1.3.
The word "matrix," in this sense, should not be confused with the
mathematical term. The meaning of the wiord should be clear, from the
context in which it is used, in all cases.





35


From the above discussion, it is clear what data must be specified

in order for the reader to reproduce our results from the water mon~omer

and dimer APT's. For the monomer these include the 8 matrix, the masses,

and the experimental force constants, both for the gas-phase and for the

monomer in the N2 matrix. For the dimer we must give the B matrix, the

masses, and the experimental force constants for the dimer in the matrix,

as well as those derived from the theoretical calculation. We must also

specify the data from which the experimental monomer AiPT's were obtained.

For completeness we give both the data and the resulting APT's. We also

give the position vectors from which the two 8 matrices were obtained,

and the L matrix derived from each set of force constants we have used.

For the masses we have used mO = 16.0 and mH = 1.008 u throughout.

The 8 matrix defines the relationship between the space-fixed cartesian

and symmetry coordinates of the molecule, according to Eq. 9b. It is

obvious that a given 6 matrix applies to a particular ordered sequence

for both sets of coordinates. For the cartesian coordinates it is

sufficient to give the order of the atoms; the atomic x, y and z coordi-

nates are ordered consecutively.

In practice the B matrix is obtained first, thus defining the

internal coordinates according to Eq. 6. Then, in order for the symmetry

coordinates to have the properties described in the previous section,

they are obtained as the normalized results of the symmetry projection

operators on the internal coordinates. This determines the UI matrix

according to Eq. 9a, and the I matrix is thecn calculated from Eq. 10.

The coordinate system, atom numbering, and bond lengths and angles

for H20 are shown in Fig. 2-1. As indicated, the origin is centered on

the 0 atom. The equilibrium position vectors were determined from the

experimental structural parameters, rOH = 0.9572 A and 6 = 104.520 [52].
























































Fig. 2-1. Molecular coordina te sys tem of wa ter monomer.
Atom numbers, bond lengths and angle are indicated.








These vectors are given in Table 2-1. The definition of internal

coordinates, which wzas used in calculating the 83 matrix, is given on the

left side of Table 2-2. The H20 molecule has C2v symmetry. The reducible

representation spanned by the internal coordinates is given by


r, = 2a, + b2


The corresponding symmetry coordinates are given on the right side of

Table 2-2, and the resulting 8 matrix in Table 2-3.

The coordinate system for the linear water dimer is shown in

Fig. 2-2. The origin is on the 0 atom of the electron donor molecule

(atom 3). The atom numbering is given in Fig. 1-2. The equilibrium

position vectors we have used are given in Table 2-4. Large basis set

calculations for the water dimer have indicated that the bond distances

and angles of the two component water molecules do not differ appreciably

from their values in the monomer 153,54]. Accordingly, we have used the

experimental values of rOH and t3 given above for each of the water mole-

cules in the dimer. We have taken the intermalecular geometry from the

minimum energy structure calculated for the dimer by Hankins, Mloscow~itz

and Stillinger near the Hartree-Fack limit 153]. For the dimer structure

illustrated in Fig. 2-2 and discussed in section 1.3.1, the intermolecu-

lar geometrical parameters which determine the position vectors are the

0-0 distance and the angle Q, which the C2 axis of thle electron donor

makes with~ the~ negantive y axis as show~n in the figures. Trhe theroretical

values for these parameters are RO- = 3.00 Ai and O = 4100 1531.

The internal coordinates of water dimier are defined on thle left side

of Table 2-5 in terms of the bond lengths and angles in Fig. 2-2. N~ote

that we have only included the six intramolecular vibrational coordinates

and not the six additional intermolecular coordinates. This is because









Table 2-1. Equilibrium position vectors for water muo~noer. Coordinate
system and atom numbering are given in Fig. 2-1. Units
are K.


a
The prime indicates that values are not with respect
to the center of mass.


Table 2-2. Definition of internal and symmetry coordinates for water
monomer. Bond lengths and angles are indicated in Fig. 2-1.





39

Table 2-3. The 6 matrix for water monomer. Row 2 has units of rad A.
All other rows are unitless.


SynmetryCartesian Coordinate
Coordinate x1 y, z


0.0 0.559178 0.432805
0.0 0.639448 -0.826156
0.0 0.559178 0.432805




X2 Y2 Z2

0.0 -0.559178 0.432805
0.0 -0.639448 -0.826156
0.0 0.559178 -0.432805




x, y, z

0.0 0.0 -0.865611
0.0 0.0 1.652312
0.0 -1.118355 0.0


Sz

S2
S







S

S2



S



S


S3


















r2

8
'I


Fig. 2-2. Molecular coordinate system of the linear water dimer. Bond lengths and angles
are indicated.





































Bond lengths


Atom
No. : 1 2 3 4 5 6

xo 0.756950 -0.756950 0.0 0.0 0.0 0.0

ye -0.448812 -0.448812 0.0 2.042800 3.0 3.239987

zo 0.376598 0.376598 0.0 0.0 0.0 -0.926627


Internal Symmetry
Coordinates Coordinates Symmetry Units


R, = Orl S1 = R A A

R2 a2 S2 = (R1 + R2)//2 A' A

R, = Ar, S, = R, A' A

R, = ar, S, = Rg A' rad

Rs e Sg = R A rad

R, = AO2 S, = (RI R2)/2A


Table 2-4. Equilibrium position vectors for the linear water dimer.


Coordinate system is given in
in Fig. 1-2. Units are 1.


Fig. 2-2 and atom numbering in


Table 2-5. Definition of internal and


symmetry coordinates for the linear
and angles are indicated in


water dimer.
Fig. 2-2.





42

with one possible exception [40, and references cited therein], the six

low-frequency intermolecular absorptions of the water dimer have not yet

been observed. Thus, there are no reliable frequencies from which to

determine the force constants, and no experimental spectrum in the

literature with which to compare the intermolecular intensities. The

dimer molecule in Fig. 2-2 has Cs symmetry, and the reducible representa-

tion of the six internal coordinates is given by



TR = Sa' + a"

The corresponding symmetry coordinates are given on the right side of

Table 2-5. The 6 matrix obtained from these data appears in Table 2-6.

To calculate the 8 matrices in Tables 2-3 and 2-6, and to solve the

eigenvalue problem of Eq. 14, we used the normal coordinate programs WMA~iT

and CHARLY [551. The operation of these programs has been discussed in

detail elsewhere 156]. Program WMIAT calculates the 8 and G matrices and

outputs a transformation of the C matrix which facilitates the diagonali-

zation of the GF product. The input to WHAT includes the masses and

equilibrium position vectors of the atoms, the definition of the internal

coordinates, and the U matrix. We note that the 8 matrix must represent

a transformation to a true molecule-fixed coordinate system--that is, it

must be invariant to rigid translations and rotations of the whole male-

cule. Program WHALT insures this by using the "little s vector" technique

[491 to calculate the B matrix in Eq. 6 and then transforming

according to Eq. 10. The input to program CIIARLY includes the output from

WMIAT and the force constant matrix, F. The output from CHARLY includes the

frequencies (determined from the A matrix) and the eigenvector matrix, L.

In order to obtain the experimental gas-phase L matrix for H20, we used

the F matrix given by Cook, De Lucia and Helmingier- [51]. It was obtained

































SymmetryCartesian Coordinate
lordinate x4 y, zo x5 5 zs xs y, z


S, 0.0 0.0 0.0 0.0 -0.250717 0.968060 0.0 0.250717 -0.968060

S, 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

S, 0.0 -1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0

S, 0.0 0.0 1.044713 0.0 -1.011346 -1.306641 0.0 1.011346 0.261928

Ss 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
S, 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0


Table 2-6. The G matrix for water dimer, Rows 4 and 5 have
units of rad A O-. All other rows are unitless.


Symmetry
Coordinate


Cartesian Coordinate


0.0

-0.331548

0.0

0.632872

-0.331548


0.0

-0.331548

0.0

0.632872

0.331548


0.0

0.278202

0.0

-0.531043

-0.278202


0.0

0.0

0.0

0.0

-1.118355


0.0

0.663096

0.0

-1.265744

0.0


0.0

0.559178

0.0

0.639448

0.559178


0.0

0.278202

0.0

-0.531043

0.278202


0.0

-0.559178

0.0


-0.639448

0.559178


0.0

-0.556404

0.0

1.062085

0.0








by fitting the observed infrared frequencies, as well as the quartic

distortion coefficients from microwave data, for H20 and its deuterated

and tritiated isotopes. This F matrix is given in Table 2-7 along with

the resulting L matrix elements.

Thre experimental F matrix for water monomer in the N2 matrix was

taken from Tursi and Nixon [30]. It was obtained from a fit to the

frequencies of H120 and its deuterated isotopes in the matrix. The _F and

L matrices are given in Table 2-7. It can be seen that the stretch-bend

interaction force constant, F ,' has a different sign for H20 monomer in

the N2 matrix from that for H20 in the gas-phase. The effect of this

change on the intensities is discussed in th~e following chapter.

The experimental F matrix for water dimer in the N2 matrix was also

obtained from Tursi and Nixon [301, and was derived in thle same manner

as that for H20. It is given in Table 2-9 in terms of the symmetry

coordinates of Table 2-5, together with the corresponding L matrix.

Intermolecular coupling force constants were not included in the analysis

of Tursi and Nixon. The F and L matrix elements for the electron donor

(ED) molecule (see Fig. 1-2) appear in the upper part of Table 2-9, and

those for the electron acceptor (EA) molecule in the lower part.

The "theoretical" force constants wye have used for the water dimer

were derived from those calculated with the 4-31G basis set by Curtiss

and Pople [67], using th~e scaling procedure described in section 1.4.1.

The details of the derivation are given in the Appendix, as well as the

comparison of the resulting frequencies with experiment. The agreement

is quite satisfactory. This F matrix and the corresponding L matrix are

given in Table 2-10. W'e note that intermolecular coupling force

constants were included by Curtiss and Pople, and the F and L matrix

elements for these interactions are given in the middle part of the table.














FI 7.653 md A


Fl2 0.5124 md rad '


F22 0.6398 md rad-2


F,, 7.838 md A-1


L 1.013 u


L,2 -0.1117 ub


21, 0.0800 rad K U-4


L22 1.526 rad A- u-


L33 1.034 U2


Table 2-7. Experimental F and L matr-ices for water monomer in the
gas-phase. The F matrix is from Cook, DeLucia and
Helminger [51].


Table 2-8. Experimental F and L matrices for water mon~omer in the N2
matrix. The F matrix is from Tursi and Nixon [30).












Electron Donor


F2 7.3205 md 1

Fs -0.4088 md rad I

Fss 0.6927 md I rad 1
F 7.6545 md -1A


Electron Acceptor

F1 7.498 md 1- L .49

F1 -0.2200 md L- 0.79

F, -0.3350 md rad- L1 0.0979 u
F3 7.036 md -1 L3 -.42

F -0.3350 md rad- La .98

F, 0.7211 md 1 rad- L3~ 0.1057 u

LL -0.1082 rad I- u 2i

Lit -0.2824 rad 1-1 u-
a 1 -4
1.498 rad Au


Table 2-9. Experimental F and L matrices for the linear water dimer in
the N2 matrix. The F matrix is from Tursi and Nixon [30].


1.012 u

0.1198 u

-0.2664 rad X-u-

1.505 rad u-

1.034 u-l





47

Table 2-10. Scaled 4-31G F and L matrices for thle linear water dimer
in thle N2 matrix. The F matrix was obtained by scaling
the 4-31G F matrix of Curtiss and Pople [47), as
described in the Appendix.


Electron Donor

F2 7.428 md X L22 -1.018 u
Fss 0.6963 md 1 rad-2 55 152 adA u

F 7.804 md 1 103



Intermolecular

Fl2 -0.0170 md AI- L 0.08

F2 -0.0070 md A- LL -0.0287 u
F s 0.0027 md A rad- L2 .01

L,, 0.0504 u
Los ~ ~ 0.17ra
Ls 0.1137 rad A u




Electron Acceptor

F, 7.616 md A- Li .93

F, -0.182 md A Lig 0.4270 u

F, 7.126 md A- Ljl -0.4399 u

F 0.7330 md A rad Lzz 0.9265u

L44 1.524 rad A u







However, stretch-bend coupling force constants were neglected by

Curtiss and Pople for both the ED and EA molecules, while they were

included by Tursi and Nixon. The effect on the intensities of neglecting

these interactions is discussed in the following chapter. In calculating

the L matrix of Table 2-10 from Eq. 14, the cross-terms in the G matrix

corresponding to the stretch-bend interactions were constrained to zero.

If this had not been done, the interaction terms in the GF product matrix

would not be zero but would depend on the values of the stretch and bend

diagonal force constants. This somewhat arbitrary procedure was also

adopted by Curtiss and Pople. Due to the neglect of the stretch-bend

interactions by Curtiss and Pople, there are fewer F and L matrix

elements listed in Table 2-10 for the ED and EA molecules of the dimer.

These values are given in the upper and lower parts of the table, respec-

tively.

We now give the data used to obtain the experimental APT's, PA, for

the monomer. The experimental dipole derivatives with respect to

dimensionless normal coordinates, ap /aq have been given by Clough,

Beers, Klein and Rothiman for water monomer in the gas-phase [57, see also

reference 58]. Although the total integrated intensities give only the

magnitudes of the dipole derivatives (see Eq. 15), these authors chose

the signs so that the observed vibration-ratation effects on intensities

were reproduced in the fundamental bands of H20. These sign choices

agree with those obtained by a number of aib initic calculations (see

section 3.2).

The ap /aq. values given by these authors were converted to ap./SQj

to give the elements of the PQ matrix, using the equation [591


q. = 2E(cwj /h) 'Q








where h is Planck's constant in ergi sec, .j is the jth harmonic frequency

in em ~, and c is the speed of light in cm sec '. We used the harmonic

frequencies, wl = 3832.2, w2 = 1648.5, and wz = 3942.5 em ~, given by

Benedict, Gailar and Plyler [52], in thle conversion.

Each component of the dipole moment vector belongs to a different

irreducible representation of the C2v symmetry group. Hence there are

only three non-zero elements in the PQ matrix for H120: apz/~l aP z 2

and ap /aQ3' according to the axis system shown in Fig. 2-1. The values

of these elements derived from the data of Clough, Beers, Klein and

Rothman are 0.0479, -0.2344 and 0.2139 e u b, respectively.

This PQ matrix was used with the L matrix in Table 2-7 and the 6

matrix in Table 2-3 to calculate P L-IS, which was then added to the D

matrix to obtain P according to Eq. 22. The P L '6 matrix is composed

of three juxtaposed tensors (the vibrationall tensors"): one for each of

the atoms in the H20 molecule. These tensors are given in the upper

part of Table 2-11 for H1 and 03 of Fig. 2-1. For all the tensors given

in this table, the corresponding H2 tensor is the same as that for H1,

except that the off-diagonal elements of the H2 tenSer h8Ve the opposite

sign from that for the H1 tensor (according to the transformation shown

in Eq. 30 below).

The rotational tensors, DA, were calculated from Eq. 23 using the

effective dipole moment Ear the ground vibrational state, pzo = .81eA

taken from Clough, Beers, K~lein and Rothrman [571. Since y pxo a = 0,

the last row of ((p")) is zero (see Eq. 24). Hence, only thle x and y

components of the diagonal inverse moment of inertia tensor, I were

used in Eq. 23. These values were calculated using the equilibrium
position vectors with respect to the center of mass, roi q25 Te

values of the x and y coordinates, xAO and y 0, with respect to the








center of mass are the same as those given in Table 2-1 with respect to

an origin on the 0 atom as shown in Fig. 2-1. Because the center of mass

is displaced in the positive z direction away from the 0 atom, the values,

z0, 0 00658 1 o zo= 0.5~2031 3 were used in Eq. 25 rather


than the z~O' of Table 2-1. The resulting principal moments of inertia

are I =1.76993 A u and Iz = 0.614651 ~2u, while the corresponding
xx yy
elements of I- are just the reciprocals of these values.

The DA tensors obtained using Eq. 23 and the parameters given above

are shown in the middle part of Table 2-11. The total APT's, PA, were

obtained as the sum of the vibrational and rotational tensors and are

given in the lower part of the table. These experimental APT's for the

monomer were used in Eqs. 1 and 2 together with the monomer and dimer

APT's calculated theoretically to obtain scaled APT's for the dimer.

In the remainder of this chapter we describe the methods we have used for

obtaining the theoretical APT's and the scaled APT's for the dimer.


2.5. Calculation of Theoretical Polar Tensors


The theoretical APT's for water monomer and dimer were calculated

numerically from the theoretical values of the dipole moments for the

equilibrium configuration and for a set of displaced configurations for

each molecule. The calculated dipale moments were fitted to a linear

function of the cartesian displacement coordinates, aAj This treatment

is consistent with the assumption of electrical harmonicity used to derive

Eq. 15 and the linear relation between normal coordinates and the space-

fixed cartesian coordinates given by Eq. 18. Thus,







Table 2.11. Experimental atomic polar tensors for water monomner in the
gas phrase (PA). Vibrational and rotational tensors are
also given. Units are e, where le = 1.602 x 10-'9 coulombs.
Coordinate system is given in Fig. 2-1.


Atom H, 0




P L 8 0.0 0.1156 0.0895 0.0 -0.2312 0.0

0.0 -0.0624 0.1489 0.0 0.0 -0.2978


0.3295 0.0 0.0 0.6591 0.0 0.0

DA 0.0 0.1144 -0.1665 0.0 -0.2289 0.0

0.0 0.0 0.0 0.0 0.0 0.0


0.3295 0.0 0.0 0.6591 0.0 0.0

PA 0.0 0.2301 -0.0770 0.0 -0.4601 0.0

0.0 -0.0624 0.1489 0.0 0.0 -0.2978





52



(27) p = p +jA = x,y,z

A =1,...,N

A .

ij 1J



is the ith component of the total dipole moment of the molecule for the

displaced configuration, and po is the value of that component for the

equilibrium configuration.

The displaced configurations we have used involve the displacement

of a single atom, A, along the x, y or z direction of the molecular axis

system. Thus, for each displaced configuration, only one of the terms in

the double summation of Eq. 27 is non-zero. We have used, in general, two

configurations for each atom and each direction--one involving a displace-

ment in the positive x direction, for example, and one involving a

displacement of the same magnitude in the negative x direction. The

dipole moment components for the equilibrium and the two displaced

confgurtion--pa A) an p -ajA)--were used to determine the

first, second or third column of the APT on atom A when j^ was x, y or z,

respectively. Thus, Eq. 27 leads to


P (+njA oP Pi i A-
(28) (P i) =+
ij 2 AijA 2|8jAl


i, i = x,y,z


The condition thrat the two terms in brackets should be nearly equal was

used to check the calculations.

To determine the APT on one of the atoms in each of the molecules

H20 and (H20)2, we have used the relation [1],




53

(29) =0



We have also used the fact thlat the two H1 atoms in the monomer and in the

electron donor molecule of the dimer are symmetrically equivalent. Thus,

in both cases, we have obtained P 2 from P using the transformation [2]


(30) PH2 =TpHl


where T is the matrix for reflection across the xz symmetry plane for H20

(T, = T, =- 1, T, = -1, Ti = 0, i / j); and the yz symrmetry plane for

(H20)2 11, = -1, Ty, = T,, = 1, Ti = 0, i Z j).
The equilibrium configurations about which the displacements were

made for H20 and (H120)2 are given by the position vectors in Tables 2-1

and 2-4, respectively. As noted previously, thle H120 configuration was

derived from experimental measurements, while that of (H20)2 was derived

from the extended basis set calculation by Hankins, Moscowitz and

Stillinger [53]. Mlore recently, Dyke, Mack and Hluenter have obtained

structural parameters for water dimer produced by expanding the vapor

through a supersonic nozzle jet [60]. They used electric resonance

spectroscopy to determine microwave transition frequencies, and their

results indicated an equilibrium structure similar to that given in

Table 2-4 (with RO- = 3.00 1), except that RO- .81.1 n

a = 58('6)o. Although the experimental value of t appears to differ

significantly from the value of 400 which wef have used, the wavefunction

for thie wJt~er dimer is not expected to be very sensitive to thiis

parameter. A basis set similar to that used by Hannkins, H~oscowitz and

Stillinger has been used by Popkie, Kistenmacher and Clementi (61] to

calculate the bindingi energy of the dimer as a function of C:, withi all

other structural parameters held constant at the same values as those








consistent with Table 2-4. For values of O = 150, 300, 450, and 600,

binding energies of -4.49, -4.58, -4.58, and -4.39 kcal/mole, respec-

tively, were obtained. Thus, the calculated binding energy of the dimer

is not very sensitive to this parameter, and the calculated APT's would

be expected to be relatively independent of the value of $ chosen for the

equilibrium configuration.

We used two different quantum mechanical methods [62] to calculate

the dipole moments for each set of geometrical configurations: the ab

initio self-consistent field (SCF) method using a 4-31G basis set and

thie approximate SCF method of complete neglect of differential overlap

(CNDO).

The SCF method involves iteratively solving a set of simultaneous

linear equations, the Roothaan equations, that arise from the Schrodinger

equation when the ith molecular orbital, $ is expanded as a linear

combination of basis set orbitals, A~~, on each atom, that is,



(31) =




The Roothaan equations are solvedl for the cA whose coefficients are


integrals involving the basis set orbitals, QA


In the ab initio SCF method, the integrals are evaluated analytically

From the functional form of thre basis set. In the CNDO methodl many of

these integirals are neg~lected, and cominatiotins of others are parameter-

ized--th~at is, adjusted to fit ab iniitio SCF results for orbital energies

and eigenvectors of small molecules. Both the ab initio SCF method and

the approximations in the CNDO method have been described in reference 62.




55

Thle CNDO/2 param~eterization of the integrals was used in this work, and

this has been given in detail also in reference 62.

The CNDO calculations were performed using QCPE program CNINDO [63].

Essentially, the only input to this program is composed of the atomic

position vectors for each geometrical configuration. The 4-31G calcula-

tions were carried out using version 5 of the IBHLOL program by

Clementi and Hlehl [64]. This program is designed to accept "contracted"

(fixed linear combinations of) Gaussian functions as basis set orbitals.

The input to the program includes the exponents for the "primitive"

(uncontracted) Gaussian functions, the contraction coefficients, and the

coefficients for thle symmetrized orbitals, as well as the geometrical

configuration data.

The syrmmetrized orbitals we used were the same as the contracted

orbitals, except for those involving 1s orbitals on H1 and H2 of both H20

and (H20)2 (see Figs. 1-2 and 2-1). For these orbitals, the symmetrized

functions, Xls and Xls Were constructed, where


+ 1
Xls + Q1s(H ) + P1s (H )


for Al symmetry of H20 and A' symmetry of (H20)2; and



Xls- ~[ 1s(H "I) 1s 2,


for 8, symmetry of HO2 and A" symmetry of (H20) .NadN r h

corresponding normalization factors.

For the Gaussian exponents and contraction coefficients wre have used

those given by Ditchfield, Hebire and Pople [65J. Their 4-31G contraction

is the result of a systematic sequence of studies aimed at finding a

computationally efficient basis set while maintaining an optimal degree





56

of accuracy and flexibility. The basis set consists of a single con-

tracted Caussian orbital (CGO) for the core orbitals and two CGO's for

each valence orbital--an "inner" CGO, having relatively large exponents,

and an "outer" CCO with smaller exponents. The core CGO's consist of

four primitive Gaussians, the inner CCO's of three primitive Gaussians,

and the outer or diffuse CGO's of one Gaussian, each. We have used the

optimized exponents and coefficients given in Tables I and II of

reference 65, modified by the scaling parameters for the H20 molecule

recommended in Table III, according to Eq. 7 of that reference.

The output from the IBMOL program included the total energy and the

eigenvectors--that is, the coefficients c .in Eq. 31. The input to

IBMIOL and the eigenvector output were used in program POPULAN (66] to

calculate the dipole moments and the Mlulliken population analysis for

each configuration.

The analytic form for the expectation value of the dipole moment is

related to the Mullikenn grass atomic charge [67] on each of the atoms.

We have made use of this relation to analyze further the intensity

changes upon going from water monomer to dimer. Moreover, the dipole

moments calculated by the CNINDO program involve an approximation, in

addition to those used in calculating the eigenvectors, that is best

indicated by considering the expectation value of the dipole moment



in terms of the Hulliken gross atomic charges. For closed shell mole-

cules, such as water and water dimer,

is given in terms of the

molecular orbitals, Qi, by


N n-

(32)

= ZRA 2 g_ (i() 1i1)
A=1 i=1





57

In this equation, n is the number of orbitals (half the number of elec-

trons), RA is the position vector of nucleus A and rl that of electron 1.

ZA is the charge of nucleus A. Using Eq. 31 for the molecular orbital
and the definition of the density matrix element, DA B, which is



D AY= 2 ci .ci-~~


Eq. 32 becomes


(33)

= ZA tA -i u D A B MA B A


Here we have used the bracket notation for integrals:

A rj B (1)rl m (1)drl



The electronic position vector may be expressed as


(34) ~1 = A A '


where rA is the position vector of electron 1 with respect to nucleus A.

Substituting Eq. 34 into Eq. 33 and noting that RA comes out of the

integral and summations in the second term of Eq. 33, we have



(35)

=i rA A B DA BS A





B pA g A B


where S~~~ is the overlap integral:




58


S~AB =iA VB


The Mlulliken gross atomic population on atom A, N(A), is defined

as [67]


(36) N'(A) = L D S
B pA B BA


The gross atomic charge on atom A, QA,' is [67]


(37) QA = ZA N(A)


Substituting Eqs. 36 and 37 into Eq. 35 gives



(38)

= [QA A D A B WAR B A


In the CNINDO program the dipole moments are approximated by

neglecting terms for which B f A in the second term of Eq. 38. The only

remaining non-zero terms are those in which pA = 2sA and vA = 2pA. Thus,

this is the "sp polarization" term defined by Pople and Segal [681. The

gross atomic charges are also approximated in the CNINDO program, in that

SA in Eq. 36 is replaced by 6 A ,where 6 is the Kronecker delta.

The dipole moments for the appropriate configurations from programs

CNINDO and POPULAN~ were used in Eq. 28 to calculate the theoretical APlT

elements.


2.6 Scaled Polar Tensors and Intensities


The theoretically calculated APT's for the water monomer were

subtracted from those of the dimer in order to obtain the scaled APT's








for water dimer, according to Eqs. I and 2. Wde note that thle orientation

of the H20 monomer molecule with respect to the principal axis system in

Fig. 2-1 differs from the orientation of each of the component H20

molecules of the dimer with respect to the molecular coordinate system of

the dimer in Fig. 2-2. The values of the APT's in a given coordinate

system depend on the molecular orientation with respect to that coordinate

system. Thus, since we have calculated the monomer APT's in the coordi-

nate system of Fig. 2-1 and the dimer APT's in the coordinate system of

Fig. 2-2, we have transformed the APT's for atom A in the monomer and

dimer to a rotated axis system, before subtracting them. The axes of

this rotated coordinate system have the same orientation within respect to

the H20 molecule which contains atom A in the monomer as in the dimer.

In general, a transformed APT, (PA)', in a rotated axis system is

related to that in the original axis system, P by (3]


(39) (PA) RPAR


The transformation matrix, R, has elements, R.,, which are the cosines

between the ith rotated axis and the jth original axis, i,j = x,y,z.

The transformation matrix is orthogonal, that is, R- = Rf. For the

water monomer and dimer we have used two different rotated axis systems,

one for the H atoms and one for the 0 atoms.

The "bond system" is particularly useful for considering APT's for

H atoms. The bond system for H1 of the monomer is illustrated in

Fig. 2-3 and can be uniquely defined for each H atom, H., as follows.

The positive y axis is directed along the 01 bond containing H. from 0 to

Hi. The z axis is perpendicular to this y axis and lies in the plane of
the H20 moiety which contains H The positive z axis is directed

between the two OH bonds of that H20 maiety, as shown in Fig. 2-3. The
















































Fig. 2-3. Bond coordlinate system for atom Hz1 of the water monomer. The x axis is determined
by the right-handl rule.








x axis is perpendicular to the plane of the 1120 moiety which contains Hi'

and the positive x direction is determined by the right-hand rule. For

both the monomer and dimer, the APT for H1 in the band system of H, is

identical by symmetry with the APT for H2 in the bond system of H .

Thus, for the water monomer there is only one distinct APT for the H atom

in the bond system.

The theoretically calculated APT's for the monomer and dimer H atoms

in the bonzd system were used in Eq. 1. Similarly, the experimental APT

for the monomer H atom in the bond system was used in Eq. 2. The

theoretical and experimental APT's for the monomer H atom in the bond

sysem erecalulaed s (A)'in q.39, where PA was the APT for H1 in

the molecular coordinate system of the monomer shown in Fig. 2-1, and

R was the transformation matrix from that coordinate system to the bond

system of H~ (see Fig. 2-3). This transformation matrix is given in

Table 2-12. Table 2-13 shows the experimental APT for the monomer H atom

in the bond system, obtained by transforming the APT for H1 given in

Table 2-11. The theoretical APT's for H. (i = 1,4,6) of the dimer in

the bond system were also obtained as (P) in Eq. 39. where PA was the

theoretical APT for H. in the molecular coordinate system of the dimer in

Fig. 2-2, and R was the transformation matrix from that coordinate

system to the bond system of Hi. These transformation matrices are given

in Table 2-14 for i = 1,4,6.

For the 0 atoms a unique bond system cannot be defined since each

0 atom belongs to more than one bond. Aiccordingly,~ we transformed the

theoretical APT's for the two dimer 0 atoms, 0,, to a rotated axis system

whose orientation, with respect to the H20 moiety which contains O is

the same as the orientation illustrated in Fig. 2-1 for the monomer

molecule and axes. This rotated axis system is called the "molecular



















1 0 0

0 cos(37.740) cos(52.260)

0 cos(142.260) cos(37.740)


62

Table 2-12. Transformation matrix, RHI, from the molecular coordinate
system of the monomer, shown in Fig. 2-1, to the bond system
of the monomer H1 atom, shown in Fig. 2-3.


Table 2-13.


Experimental polar tensor for the monomer H atom in the bond
system. Units are e, where le = 1.602 x 10" ouoms


P 1 (exptl.,monomer)

0.3295 0.0
0.0 0.1322
0.0 -0.0495


0.0
-0.0641

0.2468








system of the monomer" (M~SMI) for the dimer 0 atoms and is illustrated in

Fig. 2-4 for Os'

The transformed theoretical APT's for the dimer 0 atoms were calcu-

lated as (PA)' in Eq. 39, where PA was the APT for 0. in the molecular

coordinate system of the dimer shown in Fig. 2-2, and R was the transfor-

mation from that system to the MSM of 0., i = 3,5. These transformation

matrices are given in Table 2-15 for 03 and Os of the dimer. The theoret-

ical APT for the monomer 0 atom in the coordinate system of Fig. 2-1 was

subtracted from each of these transformed API's for the dimer 0 atoms,

according to Eq. 1. The resulting changes in the theoretical APT's from

monomer to dimer, (apO)l, in thle MISM were then added to the experimental

APT for the monomer 0 atom given in Table 2-11, according to Eq. 2.

The resulting scaled APT's for the dimer 0 atoms were thus expressed

in the MISM. The same procedure was used for the theoretical monomer and

dimer APT's for the H atoms in the bond system and the experimental bond

system AlPT given in Table 2-13 for the monomer H atom. The scaled APT's

for the dimer 0 and H atoms, in their corresponding rotated axis systems

were then transformed back to the molecular coordinate system of the

dimer shown in Fig. 2-2. The inverse of the transformation given in

Eq. 39 was used, namely


(40) PA = Rir(PA ,R


where (P A)' repre~sents thle scaled AP~T for thec dimecr in th rotaited nxis

system; PA, the scaled AZPT in thle molecular system of thle dimer; and R

the appropriate transformation matrix given in Tables 2-14 and 2-15 for

each atom. The scaled APT for H2 in the molecular system of the dimer

was obtained from that for H according to Eq. 30. All of the scaled















H H

.H, O,

Oz



Z' 'H









Fig. 2-4. Molecular coordinate system of the monomer for Os of the water dimer. Thie x axis
is determinedl by the rigiht-hand rule.







Table 2-14. Transformation matrix, R1i, from the molecular coordinate
system of the dimer, sh~own in Fig. 2-2, to the bond system


of dimer H. atoms, 1


= 1,4,6.


0 cos(1300) cos(1400)
RHI cos(37.740) cos(117.960) cos(66.830)
cos(127.740) cos(127.290) cos(59.450)


1 0 0
RH 0 -1 0
0 0 -1


-1 0
RH 0 cos(75.480) cos(165.480)
0 cos(165.480) cos(104.520)




Table 2-15. Transformation matrices from the molecular coordinate
system of the dimer to the MSM of dimer 0. atoms, i = 3,5 (see
text and Fig. 2-4).



0 cos(500) cos(400)
R0z -1 0 0
0 cos(1400) cos(500)


1 0 0
Rs 0 cos(142.260) cos(52.260)
0 cos(127.740) cos(142.260)




66

APT's for thle dimer in that system were then juxtaposed to form the P

matrix, according to Eq. 21.

This scaled dimer P, matrix was multiplied by the AL product matrix

according to Eq. 17 to obtain the Pg matrix. This PQ matrix was then

used to obtain the intensities, according to Eq. 17. Similarly, the

(unscaled) theoretical APT's for the dimer, as well as those for the

monomer, were juxtaposed to form the corresponding PX matrices, which

were then used to obtain the intensities. These theoretical APT's are

presented and discussed in chapter 4 and the resulting intensities in

chapter 3.














CHAPTER 3

COMPARISON OF: THEORETICAL INTENSITIES WITH EXPERIMENT

3.1 Simulation of Experimental and Theoretical Spectra


In this chapter we compare the intensities calculated from the

theoretical APT's with experimental spectra for water monomer and dimer

in the N2 matrix. Experimental values have not been determined for the

intensities of either species in the matrix. However, as mentioned in

section 1.3.2, Tursi has given a spectrum of water isolated in N2, in

which the monomer and dimer bands were quite well resolved [31). We have

simulated this experimental spectrum using a sum of Lorentzian line-

functions. That is, the absorbance y. for each frequency point x. was

calculated as



(41) y (x ) = 3 -- - -- - ,j 1
(x-W.) + Aj2


where the sum was taken over thef three monomer bands, six dimer bands,

and two trimer bands in Tursi's spectrum. In Eq. 41, S. represents the

integrated absorbance of each band; a., the half bandwidth at half

maximum; and o., the frequency of the band center.

The absorbance values y (x ) depend primarily on the aj and Sj

parameters of those bands with peak Frequencies in the neighborhood of x..

We have adjusted all three parameters of each band so that thie simulated

spectrum could be superimposed on the experimental spectrum. The final

"fitted" parameters are given in Table 3-1. The "fitted experimental"







Table 3-1. Lorentzian parameters for fitted experimental spectrum.




S,b c ad
Species inch em cm cm


M 13.8 3725.7 2.8
D 3.99 3713.9 1.8

D 1.88 3697.5 2.1

T 3,46 3689.2 5.6

MI 3.71 3633.2 1.7

D 0.47 3626.1 1.0

D 3.08 3547.8 2.0

D 1.73 1618.6 1.0

T 1.79 1612.3 3.7

D 2.20 1600.3 1.0

M 7.85 1596.7 1.7


Monomer (MI), dimer (D), or trimer (TL).

Integrated band absorbance.
frequency of band center.
Half bandwidth at half maximum.






















CI


P1 I /-
r'l M


=115.00


l:'i I.1I!1


MI


Fig. 3-1. Spectrum fitted to experimental spectrum given by Tursi [311 for water isolated
in thre N2 matrix. Bands due to water monomer (H), dimer (D) and trimer or higher
miiltimer (T) are indicated.





70

spectrum calculated from these parameters according to Eq. 41 is show

in Fig. 3-1.

As described in chapter 2, we have calculated quantum mechanical

absorption coefficients, Aj, for each band of water monomer and dimer.

We have used these calculated absorption coefficients to obtain theoret-

ical spectra according to Eq. 41 for comparison with experiment. For

the experimental matrix spectrum, the amount of each absorbing species

in the path of the irradiating beam has not been determined, and the

absolute absorbance has not been given. Thus, we have calculated relative

theoretical spectra on the same scale as that used for the spectrum in

Fig. 3-1. This was done using relative integrated absorbances, S.(r)

with respect to a fitted band, band r. One band for each species was

chosen as the fitted band. The relative integrated absorbances for the

remaining bands were calculated using the fitted absorbance Sr from

Table 3-1, according to


S A.
(42) S (r) =-r


where Aj and Ar are the calculated absorption coefficients for the jth

and rth bands.

For purposes of comparison with the theoretical spectra, Eq. 41 w~as

used to calculate separate fitted experimental spectra for water monomer

and for the dimer, summing over the corresponding three and six bands,

respectively. In calculating the theoretical spectra, the fitted 3. and

o. parameters in Table 3-1 were used for all thec bands.







3.2 Water Mlonomer Intensities


The infrared intensities from the 4-31G polar tensors are compared

with those obtained from several other ab initio calculations in the

upper part of Table 3-2. The latter intensities were taken from works by

Pulay [691; Smith, Jorgensen and iOhrn [70]; and Krohn and Kern [71).

Pulay presented dipole moment functions for two different basis sets--

"Pulay I" and "Pulay II." The experimental L matrix in Table 2-7 was

used to obtain the intensities from our 4-31G calculation. We used the

same L matrix to calculate the intensities from the two dipole moment

functions given by Pulay and also from that given by Smith, Jorgensen

and iihrn. We report the intensities given by Krohn and Kern directly.

They included the effects of electrical anh~armonicity on the dipole

moment function, using terms up to fourth order in the normal coordinates.

They also used an experimental normal coordinate transformation, which

included the effects of mechanical anharmonicity through cubic and

quartic force constants.

The experimental intensities measured in the gas-phase by several

workers are given in the lower part of Table 3-2. The spread in the

measured values indicates that at least some of the discrepancy between

experimental and calculated values may be attributed to uncertainties in

the former. The experimental values from Clough, Beers, Klein and

Rothiman (underlined) are the most recent and probably the most reliable.

These intensities correspond to the PQ matrix elements which wie used to

obtain the experimental APT's, as discussed in section 2.4. Thle signs

reported for those P~ matrix elements agree with thle signs predicted by

nearly all the ab Iinitio calculations represented in the upper part of

Table 3-2. The only discrepancy is in the sign calculated from the





72

smaller basis set used by Pulay, Pulay I, for th~e aPZ/ 1 element. That

calculation predicted a negative sign for this element while the others

predicted a positive sign, in agreement with the experimental sign choice.

The calculated intensities in Table 3-2 are ordered according to the

quality of the basis set, as evidenced by the calculated energy minimum,

E There appears to be an overall tendency for the calculated inten-

sities of each band to increase as the basis set is improved. The upper

limits of the calculated intensities are in excess of the best experimen-

tal values, particularly for intensities Al and A2. This may be attrib-

uted in part to the error in the Hartree-Fock approximation. It can be

seen that the 4-31G intensities agree with the experimental values

within a factor of 2.3. This Icvel of agreement has been found for a

number of other 4-31G calculated intensities 14,72,73]. Hence, we expect

that the intensities for the water dimer can also be predicted from 4-31G

calculations within a factor of 2.5 or better.

Before discussing the dimer intensities, however, it is useful to

compare the experimental intensities of the monomer isolated in the N2

matrix with those from the gas-phase spectrum. Thus we have calculated

a spectrum using the experimental gas-phase intensities of H20 by the

method of section 3.1. Relative absorbances were obtained by using the

experimental Ai (underlined values) from Table 3-2 in Eq. 42. For

reasons given below, vs was chosen as the fitted band (that is, r = 3

in Eq. 42). The resulting spectrum (B) is compared with the fitted

experimental spectrum of water monomer in the N2 matrix (A) in Fig. 3-2.

These spectra indicate that the gas-phase intensities are quite different

from those in the matrix. Relative to A the gas-phase value of Al is

too small, while that of A2 is too large.






Table 3-2. Theoretical and experimental intensities for gas-phase 1120.
Units are km/mole (Eo in liartrees).



TheoreticalAAAE
Calculations 1, 2 8, E

Pulay In 1.91 66.6 6.03 -75.867
4-31Gb 1.07 90.1 19.7 -75.909
Pulay IIa 8.77 93.5 34.3 -76.035
Smith, et aZ.e 13.6 108.5 65.3 -76.041
Krobin and Kernd 15.4 99.0 64.7 -76.051


Experimental 2.24e 53.6e 44.6e -76.431E
Measurements 2.5 5.h 59.8i

2.26 548k 46.63
49.21 40-59m
v ** 59.3n 42.30

103.4P
71-894


a Pulay (69].
Present work.
c Smith, Jorgensen and Chrn [70).
Krohln and Kern [71].
e Clough, Beers, Klein and Rothman 157).
Quoted by J. Smith [74, p. 31].
SToth [75).
Goldstein [76].
Mlaclay [771.
3Flaud and Camy-Peyrct [78].
Ludw~ig, Iferriso and Abeytn [791.
Coldman~ and Oppenheim [80).
SJaffe antd Benedict 1811.
n Bn Arveh 182].
o Hirshfeld, JaFfe and Ross [83}.
K rakow and Hlealy 184).
SVon Rosenberg, Pratt and Bray [85],







One possible source of this discrepancy is a change in the mechanical

effects on the H20 molecule upon going from the gas-phase to the N2

matrix. According to the theory of infrared intensities, mechanical

effects influence the intensities through the L matrix while electrical

effects are reflected in the APT's. We have thus attempted to obtain an

indication of the mechanical effects on the H20 intensities as follows.

Intensities were calculated from L(N ) and PA~g). The PA(g) repre-

sent the experimental APT's determined from the gas-phase data as

described in section 2.4 (see Table 2-11), while L(N ) denotes the L

matrix determined from the force field given by Tursi and Nixon [301 for

H20 in the N2 matrix (see Table 2-8). These intensities are compared in
Table 3-3 with intensities calculated from L(g) and PA(g)--where L(g) is

the L matrix determined from the force field given by Cook, DeLucia and

Helingr [1) or 20 in the gas-phase (see Table 2-7). Since the PA~g

were determined using L(g), the latter intensities are just the experi-

mental gas-phase intensities.



Table 3-3. Intensities calculated for H20 from experimental
gas-phase polar tensors using two different L
matrices. Units are km/mole.




A, A2 3.

L(N2) 9.80 46.0 44.6

L(g) 2.24 53.6 44.6




Table 3-3 shows that the intensities obtained using L(N ) show a

substantial increase in Al and a decrease in A22 relative to the gas-phase

intensities. As pointed out in section 2,4, the major difference between






















Fig. 3-2. Comparison of fitted experimental spectrum of H20 in the N2
matrix (A) with spectra calculated using experimental gas-
phase APT's. Spectrum (B) was obtained using L(g) and
represents the experimental gas-phase intensities. Spectrum
(C) was calculated using L(N ). All frequencies and band
widths have been fitted to t~e experimental spectrum of H120
in the N2 matrix. For the calculated spectra (B and C),
the intensity of V3 also has been fitted. Band assignments
are indicated at the bottom of the spectrum (C).








the force fields corresponding to L(N ) and L(g), is that the values of

the stretch-bend interaction force constant, F12, have opposite signs

(compare Tables 2-7 and 2-8). This discrepancy probably accounts for

much of the difference between the two sets of intensities in Table 3-3.

We note that the same value of Aj was obtained using both force fields.

This is because v3 is the only vibration in the B2 symmetry group, and

thus the corresponding L matrix element (L 3) is independent of the force

constants.

We have calculated a spectrum from the intensities obtained using

L(N2) by the method of section 3.1. This spectrum (C) is compared with

that calculated from the gas-phase intensities (B) and with~ the fitted

experimental spectrum of H20 in the N2 matrix (A) in Fig. 3-2. For the

calculated spectra (B and C), Vs was chosen as the fitted band because

As does not depend on the force field. Figure 3-2 indicates that the

intensities calculated using L(N ) give substantially better agreement

with the experimental intensities in the N2 matrix than do the gas-phase

intensities. This result suggests that the mechanical changes in the

H20 molecule on going from the gas-phase to the N2 matrix environment

(if well-represented by the L matrices in Tables 2-7 and 2-8), may

account for much of the discrepancy between the H20 intensities in the

two environments.


3.3. Water Dimer Intensities


In this section we present and discuss the theoretical intensity

calculations for the water dimer. The dimer bands and intensities are

designated by the same notation used in section 1.3.1. That is, the

vibrations corresponding to the electron donor water molecule of the

dimer complex are labeled (ED) and those corresponding to the electron








acceptor are labeled (EA). Analogous numbering to that for the monomer

is used--that is, v, refers to the antisymmetric stretch, vl to the

symmetric stretch, and v2 to the bend.

W~e begin by comparing the intensities calculated from the theoretical

APT's for the dimer with corresponding intensities for the monomer. In

all cases we used the experimental L matrices for water monomer and

dimer isolated in the N2 matrix (determined from the force fields given

by Tursi and Nixon [301) to obtain intensities from the APT's. Table 3-4

shows the intensities obtained from the ab initio APT's calculated for

water monomer and dimer using the 4-31G basis set.



Table 3-4. Intensities calculated from ab in~ific 4-31G APT's for water
monomer and dimer. Units are km/mole.



A,(ED) A3 1E) A(ED) A (EA) A? (EA) A2(ED)

Monomer 19.7 19.7 10.1 10.1 81.2 81.2

Dimer 44.4 43.5 26.4 165.6 86.0 87.6




According to the calculation, the intensities of the four dimer

stretching vibrations are strongly enhanced relative to the corresponding

monomer intensities, while the bending intensities remain nearly the

same. A dramatic increase by a factor of nearly 17 is predicted for

V (EA), while the other three stretching intensities are predicted to

increase by factors ranging: from 2.2 to 2.6.

Table 3-3 shows that the arb initio APT's from thle 4-31G calculation

for the monomer, underestimate the stretching intensities by a factor of

2 and overestimate the intensity of the bend by a factor of 2. It might

be expected that the same errors would be found in the APT's calculated







for the dimer using the 4-31G basis set. In this event some improvement

in the predicted dimer intensities could be obtained by scaling these

theoretical APT's according to the procedure described in section 1.4.2.

We have thus used the theoretical monomer and dimer APT's, as well

as the experimental APT's for the monomer, in Eqs. 1 and 2 of that

section, to calculate scaled dimer APT's. The intensities obtained from

the scaled dimer APT's are compared in Table 3-5 with those obtained

from the experimental APT's for the monomer. Both sets of intensities

were calculated using the corresponding matrix isolated L matrix.



Table 3-5. Intensities calculated from scaled dimer 4-31G APT's and
experimental monomer polar tensors. Units are km/mole.



[A (ED)] [A (EA)] [A (ED)] [A (EA)] [A2(EA)] [A2(ED)I

Monomer 44.6 44.6 9.80 9.80 46.0 46.0

Dimer 79.2 74.6 26.0 182.2 49.9 50.9




Again, the intensities of the bending vibrations are predicted to be

about the same for the monomer and dimer. The predicted intensity of

V (EA) is enhanced by a factor of nearly 19, while those of the other

three stretchiing vibrations are 1.7 to 2.7 times greater than in the

monomer.

Calculated spectra were obtained as described in section 3.1 using

the intensities, A~i hown in Tables 3-4 and 3-5. These spectra are

compared in Fig. 3-3 with the fitted experimental dimer spectrum. For

the calculated spectra, v (EA) was chosen as the fitted band, since the

intensities of thie bending modes are expected to be least affected by

H-bond formation [56, see also Tables 3-4 and 3-5]. The other bend
























Fig. 3-3. Comparison of fitted experimental spectrum of water dimer
isolated in the N2 matrix (A) with spectra calculated
using unscaled (B) and scaled (C) 4-31G polar tensors.
For the calculated spectra (B and C), the intensity of
v2 (EA) has been fitted to the experimental spectrum of
water dimer in the N2 matrix. All frequencies and band-
widths have also been fitted. Band assignments are in-
dicated at the bottom of spectrum (C).

































n

111
7
L1
ill



n



ii
''






ri
,i




rj

CI"

LLJ
3

L~Jc:
L7~ 3
LL
O
r--



-1 _



i t--

rlJ IJ II O .C.U

E


"? !~








C:
'''



13

~1
iJ I II~
w






rlJ


W

rlI~
LL~3
.7
LL h
LL-: (~ 5
,T




ii wc


Ir
I
i-----
U. I11J U.:O I.OU


Ci

r r
I-
I--'
LL~



Li-
LL,
r-






u
rr

b 1 lu

RL; ~IJI' b IiI liE


0 00


0.50 Uuu


0.50 0a 00 0.50 1.00


7=-







V,(ED) strongly overlaps with the corresponding monomer band in the

experimental spectrum (see Fig. 3-1). The value of the absorbance used

to fit this band is therefore less certain.

Figure 3-3 shows that in the spectrum calculated using the ab initio

APT's (B), the intensities of the antisymmetric stretches are much

smaller than those in the fitted experimental spectrum (A). For the

spectrum obtained using scaled APT's (C), the overall agreement is better

except for V1(EA), whose intensity is predicted to be approximately twice

that in the fitted experimental spectrum. The normal coordinate for

V (EA), contains the largest contribution from the H-bonded OH stretch.

This vibration therefore corresponds to the "H-bond band" whose intensity

was discussed in section 1.2.

This overestimation of the B-bond band intensity is probably related

to the fact that the dimer stabilization energy calculated with the 4-31G

basis set is too large--8.2 kcal/mole [47), as compared to the most

accurate value of 5.6 kcal/mole (43). In general, small basis set

molecular orbital calculations have been found to predict this extra

stabilization, and this phenomenon has been investigated for a number

of H-bonded complexes [86-90; 19, pp. 63-65).

The overestimation of the stabilization energy (and presumably of the

H-bond band intensity) is a consequence of the small basis set which

inadequately represents the orbitals on each molecule of the complex. In

the molecular orbital method, the total energy of the complex is

optimized. The additional basis functions on one molecule are then

allowed to compensate for the limited number of basis functions on the

other molecule, in order to achieve optimization. Thus, intermolecular

orbital mixing occurs to a greater extent in small basis set calculations





83

than in those using large basis sets, and "basis set superposition error"

results.

An additional measure of the H-bond strength is given by the inter-

molecular separation, the 0-0 distance in the case of water dimer. A

stronger H-bond interaction results in a shorter 0-0 distance. The

minimum energy configuration calculated with the 4-31G basis set for the

water dimer occurs for an 0-0 distance of 2.83 A [471. This compares

with the experimental [60] and most accurate theoretical [431 value of

2.98 81.

Because both the H-bond band intensity and the 0-0 distance are

correlated with the dimer stabilization energy, and the latter is

affected by basis set superposition error in the calculations, we

decided to investigate the dimer intensities as a function of 0-0

distance.

Accordingly we have calculated an entire set of scaled A~PT's for the

water dimer at each of three different 0-0 distances, RO- = 3.0, 2.8

and 2.6 H1, using the relatively economical CNDO method. These calcula-

tions also allow us to see whether the scaling procedure which we have used

for the dimer APT's can override some of the shortcomings of the

approximate CNDO method.

Figure 3-4 shows the calculated spectra obtained using unsealed

APT's from the CNDO calculation. The intensities predicted for the

stretches are very small compared to the values calculated for the bends,

with the exception of A 1(EA). These results are similar to those

obtained with the ab inzitio 4-31G calculated APT's (spectrum (B) in

Fig. 3-3]. However, the situation is more exaggerated with the CNDO

calculation, and A l(EAZ) is predicted to be significant only for the
























Fig. 3-4(. Comparison of spectra of the water dimer calculated from
unsca;?ed CNDO polar tensors at different 0-0 distances.
Spectrum (A) corresponds to RO-0) =o 3.0 A; specru (B)
to RO-0 = 2.8 K: and spectrum ()t O0=26A h
intensity of v2 (EA) and the frequencies and bandwidths
have been fitted to the experimental spectrum of water
dimer in the N2 matrix. Band assignments are indicated
at the bottom of spectrum (C).












C e

5
-M_..
---,
o.Olj 0.50


h
v


,C



)- uJ -



U-J
I
W-1 0
CC,
u-


C
Frl
t-


C
C:


b=


Li~ b



II
--
---'
"ci.ou 0.:0


I -







U.00 U.50


HBORRNE


0.00
RBSCERNC ,


MSDRB ANC ~ iE





86

strongest H-bonded water dimer, that with RO- = 2.6 W. Table 3-6 shows

the corresponding values predicted for the intensities; they are indeed

unreasonable.

Figure 3-5 shows the calculated spectra obtained using scaled APT's

from the CNDO calculation. First wue note that the relative intensities

are a great deal better, thus indicating the value of scaling the APT's

for obtaining quantitative intensity estimates from approximate calcula-

tions. Secondly, Fig. 3-5 shows that the intensity predicted for V (EA)

is indeed a very sensitive function of the 0-0 distance, and in fact

doubles for every 0.2 A decrease in ROO

These results confirm that the intensity of this H-bond band Irela-

tive to A (EA)] is amazingly sensitive to RO-0 and, hence, the strength

of the H-bond. Although we have not quantitatively investigated the

effect of basis set superposition error on the 4-31G calculated intensity

of this band, we can infer qualitatively from these results that the

effect would be large. Accordingly, the intensity of the H-bond band in

the experimental spectrum obtained by Tursi and Nixon [see fitted

spectrum (A) in Fig. 3-3] is not inconsistent with the value that might

be predicted for the linear dimer from a large basis set calculation.

The numerical values of the intensities corresponding to the

calculated spectra in Fig. 3-5 are given in Table 3-7. In addition to

the dramatic change for the intensity of the H-bond band, vl(EAl), the

intensity of V,(EA) is also predicted to increase substantially as R
0-0
decreases. The increase in A (EA) can be noticed in the calculated

spectra shown in Fig. 3-5, particularly for RO- = 2.6 A. The sensitivity

of the V3(EA) vibration to the H-bond strength probably results from the








Table 3-6. Dimer intensities calculated from unscaledl CNDO APT's at
different 0-0 distances. Units are km/mole.




RO-0 A [A (ED)] [A (EA)] [A (ED)] [A (EA)] [A2(EA)] IA (ED)I


3.0 10.9 7.41 0.328 4.96 32.0 33.9

2.8 9.49 6.01 0.208 26.2 32.0 34.8

2.6 7.43 8.68 0.089 97.8 33.4 35.9










Table 3-7. Dimer intensities calculated from scaled CNDO APT's at
different 0-0 distances. Units are km/mole.




RO-0 R [A,(ED)] [A (EA)] [A (ED)] [A (EA)} (A2(EA)] [A2(ED)]

3.0 51.3 52.6 11.7 58.3 45.7 49.4

2.8 54.6 65.6 12.5 109.7 45.6 50.4

2.6 60.0 91.5 13.7 231.1 46.9 51.7




















Fig. 3-5. Comparison of spectra of the water dimer calculated from
scaled CNDO polar tensors at different 0-0 distances.
Spectrum (A) corresponds to RO= 3.0 K; spectrum (B)
to RO-0 = 2.8 81; and spectrum e), to RO-0 = 2.6 A. The
intensity of v2 (EA) and the frequencies and bandwidths
have been fitted to the experimental spectrum of water
dimer in the N2 matrix. Band assignments are indicated
at the bottom of spectrum (C).




















U. 0U. 0


1






r'


h
O

L I-
Ir: I ~r in



r


L' ~ m






'1
ii






w
rr
U


W._
iL~3
U_,
r--


C
C~
v
2
P
I;?


I_


L i



,,
;7
c.


U IJIJ O.:1


.00 0.50 1.00


0.00 0.50
RBS~bRNE ii


o .0 o
A B5 0i ~EMN ~iEE


.Ul 0. 0







intramolecular coupling discussed in section 1.2. Basically this coupling

is described by the L matrix which, in addition to the P, matrix (composed

of the APT's) determines the PQ matrix and, hence, the intensities (see

Eqs. 16 and 17).

Accordingly, we wish to consider the extent to which an alternate

choice for the L matrix can affect the calculated intensities. One such

alternate L matrix is that determined from the scaled 4-31G F matrix.

This F matrix was obtained from the ab initio 4-31G force fields calcu-

lated by Curtiss and Pople for the water monomer and linear dimer, using

the scaling procedure discussed in section 1.4.1 (see Appendix). Both

the scaled F matrix and the resulting L matrix have been given in

Table 2-10.

Table 3-8 shows the intensities calculated using the scaled 4-31G

L matrix. These intensities are compared in the table with those

obtained using thle experimental L matrix which was derived from the F

matrix given by Tursi and Nixon for the water dimer in the N2 matrix.

The latter L matrix has been used for all the intensity calculations

presented thus far and has been given in Table 2-9.



Table 3-8. Dimer intensities calculated from scaled 4-31G APT's using
two different L matrices. Units are km/mole.



[A (ED)] [A (EA)] [A (ED)] [A (EA)] [A2(EA)] [A2(ED)]

Scaled L 79.2 86.4 12.8 162.5 67.4 63.3

Exptl. L 79.2 74.6 26.0 182.2 49.9 50.9





Table 3-8 shows that a minor reduction is calculated for AI(EA) and

a minor increase for A,(EA) when the scaled 4-31G L matrix is used,





91

relative to the intensities calculated with the experimental L.

However, the major effect of the alternate L matrix on the intensities

in Table 3-8 is found in the intensities of the bending vibrations,

V2 (EA) and v2(ED), and of the syrmmetric stretch of the electron donor

molecule, V (ED).

As discussed in section 2.4, the major differences between the two

force fields are the neglect of intramolecular stretch-bend coupling

constants in the scaled 4-31G force field and the neglect of intermolec-

ular coupling constants in the experimental force field. Comparison of

Tables 2-9 and 2-10 shows that the L matrix elements for stretch~-bend

interaction included in the experimental force field are larger than

those for intermolecular interaction included in the scaled 4-31G force

field. This is also what we intuitively expect based on the relative

importance of intra- and intermolecular forces. We can infer from the

results in Table 3-8 that the effect of neglecting stretch-bend inter-

action in the force field results in an increase in the bending inten-

sities of the dimer and a decrease in the intensity A (EA).

In summary, it has been shown that the theoretical intensities

predicted for the water dimer are sensitive both to the H-bond strength

and the force field used in the calculation. Within these limitations,

we feel that the intensities calculated using scaled APT's from the

4-31G calculation [corresponding to spectrum (C) in Fig. 3-3] agree well

enough with the fitted experimental spectrum [spectrum (A) in Fig. 3-3]

to be wsorthy~ of some conisideration in~ thle interprettatin of thle experimental

dimer spectrum.

Various interpretations of the spectrum have been discussed in

section 1.3.2. We mentioned that the band assigned to V1(ED) by Tursi

and Nixon 130) was regarded by Barletta [33] as an overtone or




Full Text

PAGE 1

INFRARED INTENSITIES OF WATER AND WATER DIMER By BARBARA ANN ZILLES A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1980

PAGE 2

ACKNOWLEDGMENTS I would like to take this opportunity to thank Dr. Person, whose dedication to helping people learn has benefited me a great deal. In addition I would like to thank Jerry Rogers and Roberto Maia, as well as past members of Dr. Person's research group, for many stimulating discussions. Partial support from NSF Research Grants No. CHE-74-21471 and CHE-78-18940, and from the Division of Sponsored Research, University of Florida, is also gratefully acknowledged.

PAGE 3

TABLE OF CONTENTS Page 11 ACKNOWLEDGMENTS LIST OF TABLES v LIST OF FIGURES vii ABSTRACT v iii CHAPTER 1 . INTRODUCTION 1 1.1. Perspective 1 1.2. Vibrational Properties of H-bonded Complexes 2 1.3. Experimental Spectra of Water Dimer 5 1.3.1. Matrix Isolation Spectroscopy and Structure of the Water Dimer 5 1.3. 2. Review of Experimental Studies 12 1.4. Theoretical Calculation of Vibrational Properties of the Water Dimer 20 1.4.1. Frequencies 20 1.4.2. Intensities 22 2. DESCRIPTION OF CALCULATIONS 25 2.1. Outline 25 2.2. Normal Coordinate Analysis 26 2.3. Intensity Relations 29 2.4. Application to Intensities for Water and Water Dimer 33 2.5. Calculation of Theoretical Polar Tensors 50 2.6. Scaled Polar Tensors and Intensities 58 3. COMPARISON OF THEORETICAL INTENSITIES WITH EXPERIMENT 67 3.1. Simulation of Experimental and Theoretical Spectra 67 3.2. Water Monomer Intensities 71 3.3. Water Dimer Intensities 77

PAGE 4

4. POLAR TENSOR ANALYSIS 93 4.1. Total Polar Tensors 93 4.2. Quantum Mechanical Analysis 99 4. 2. 1. Discussion of Model 99 4. 2. 2 .Results and Implications for the Intensity Enhancement of the H-bond Band 106 APPENDIX 119 REFERENCES 127 BIOGRAPHICAL SKETCH 133

PAGE 5

LIST OF TABLES Table Page 1-1. Assignment of bands in the V (H-B) and Vj (H-B) regions to dimer and trimer by various workers 14 2-1. Equilibrium position vectors for water monomer 38 2-2. Definition of internal and symmetry coordinates for water monomer 38 2-3. The B_ matrix for water monomer 39 2-4. Equilibrium position vectors for the linear water dimer 41 2-5. Definition of internal and symmetry coordinates for the linear water dimer 41 2-6. The B_ matrix for water dimer 43 2-7. Experimental F_ and L_ matrices for water monomer in the gasphase 45 2-8. Experimental F_ and L_ matrices for water monomer in the N„ matrix 45 2-9. Experimental F_ and L_ matrices for the linear water dimer in the N„ matrix 46 2-10. Scaled 4-31G F and L_ matrices for the linear water dimer in the N„ matrix 47 2-11. Experimental atomic polar tensors for water monomer in the gas-phase 51 2-12. Transformation matrix from the molecular coordinate system of the monomer to the bond system of the monomer Hj atom — 62 2-13. Experimental polar tensor for the monomer H atom in the bond system 62 2-14, Transformation matrix from the molecular coordinate system of the dimer to the bond system of dimer H atoms 65 2-15. Transformation matrices from the molecular coordinate system of the dimer to the MSM of dimer atoms 65

PAGE 6

Table Page 3-1. Lorentzian parameters for fitted experimental spectrum 68 3-2. Theoretical and experimental intensities for gas-phase HO 73 3-3. Intensities calculated for HO from experimental gas-phase polar tensors using two different L_ matrices 74 3-4. Intensities calculated from ab initio 4-31G APT's for water monomer and dimer 78 3-5. Intensities calculated from scaled dimer 4-31G APT's and experimental monomer polar tensors 79 3-6. Dimer intensities calculated from unsealed CNDO APT's at different 0-0 distances 87 3-7. Dimer intensities calculated from scaled CNDO APT's at different 0-0 distances 87 3-8. Dimer intensities calculated from scaled 4-31G APT's using two different L matrices 90 4-1. Ab initio 4-31G polar tensors for H atoms of the water monomer and dimer, in the bond system 94 4-2. Ab initio 4-31G polar tensors for atoms of the water monomer and dimer, in the molecular system of the dimer 97 4-3. Charge, charge flux, and overlap contributions to ab initio 4-31G APT's for atoms of the water monomer and dimer, in the molecular system of the dimer 107 4-4. Charge, charge flux, and overlap contributions to ab initio 4-31G APT's for atoms of the water monomer and dimer, in the molecular system of the dimer 108 4-5. Individual contributions to P ^(charge flux), in the bond yy system of H , for (HO) and HCN-HF 116

PAGE 7

LIST OF FIGURES Fig. Page 1-1. Spectrum fitted to experimental spectrum of water isolated in the N_ matrix 7 1-2. The linear water dimer 9 1-3. The centrosymmetric cyclic dimer 11 2-1. Molecular coordinate system of water monomer 36 2-2. Molecular coordinate system of the linear water dimer 40 2-3. Bond coordinate system for atom H of the water monomer 60 2-4. Molecular coordinate system of the monomer for of the water dimer 64 3-1. Spectrum fitted to experimental spectrum of water isolated in the N matrix 69 3-2. Comparison of fitted experimental spectrum of HO in the N matrix with spectra calculated using experimental gasphase polar tensors and two different L_ matrices 76 3-3. Comparison of fitted experimental spectrum of water dimer isolated in the N„ matrix with spectra calculated using unsealed and scaled 4-31G polar tensors 81 3-4. Comparison of spectra of the water dimer calculated from unsealed CNDO polar tensors at different 0-0 distances 85 3-5. Comparison of spectra of the water dimer calculated from scaled CNDO polar tensors at different 0-0 distances 89

PAGE 8

Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INFRARED INTENSITIES OF WATER AND WATER DIMER By Barbara Ann Zilles June 1980 Chairman: Willis B. Person Major Department: Chemistry The atomic polar tensors for water monomer and the linear dimer have been calculated by the self-consistent field method using the 4-31G basis set. The atomic polar tensors have been used to obtain infrared absorption intensities for these two molecules. Atomic polar tensors were also determined from experimental intensities for water monomer. These experimental polar tensors were then added to the difference between the theoretical dimer and monomer polar tensors to obtain scaled polar tensors for the dimer. Intensities from the monomer polar tensors are compared with values determined from gas-phase measurements. Intensities from both scaled and unsealed 4-31G polar tensors for the dimer are compared with a simulated spectrum, which has been fitted to the experimental spectrum of water dimer isolated in the nitrogen matrix. In addition, the atomic polar tensors for the water monomer and dimer were analyzed using the charge, charge flux, overlap model. The major change from monomer to dimer was found in the charge flux tensor on the

PAGE 9

hydrogen-bonded hydrogen atom, in the diagonal element along the OH bond. The change in this element was thus shown to be the origin of the characteristic intensity enhancement accompanying hydrogen-bond formation. This result was interpreted in terms of the physical characteristics of the charge flux tensor. The interpretation was consistent with the vibronic charge transfer and dynamic polarization models for the intensity enhancement. Additional conclusions could be made, particularly concerning the effect of the hydrogen-bond on interatomic interactions in the water dimer.

PAGE 10

CHAPTER 1 INTRODUCTION 1 . 1. Perspective The study of infrared absorption intensities has recently assumed a new prominence in molecular spectroscopy. Experimentally, the capability for obtaining high resolution digital absorbance data — on a routine basis over a wide frequency range — has contributed to this development. Thus, for example, the difficulty of obtaining reliable absorption coefficients for overlapped bands has been considerably reduced. Spectrum-fitting programs are commonly available, which can adjust the intensity parameters (along with the linewidths and band center frequencies) to minimize the difference between the experimental and fitted spectrum. While the number of adjustable parameters in a complex spectrum may be large, the experimental data points are many times more numerous. Simultaneously, the theoreticians are beginning to make significant advances in the quantum mechanical calculation of infrared intensity parameters. This progress has stimulated the development of new formalisms for interpreting the intensities. These formalisms have led to unprecedented success in understanding molecular intensities and predicting them from chemical and structural information about the molecule [1-17]. The water dimer has been another subject of intense experimental and theoretical interest, during the past twelve years. Morokuma and Pederson performed the first theoretical calculation for this dimer in 1968 [18]. 1

PAGE 11

2 Since that time numerous ab initio and empirical studies have been carried out. There are two motivations for this interest. First, the water dimer may be regarded as a prototype of liquid water. Direct application of the results for water dimer to the study of liquid water may be severely limited. Nonadditivity of pairwise interactions is generally held to be important [19]. In any case, a thorough understanding of the properties of water dimer must precede that of liquid water. Secondly, the water dimer is a prominent example of a hydrogenbonded (H-bonded) complex. These weak complexes have stimulated the imagination of chemists since the 1930's, particularly because of their importance in biological systems [20-21]. The H-bond illustrates the type of interaction found in the more general class of electron donor-acceptor complexes. This interaction can be described as a transfer of electronic charge from an electron-rich portion of the electron donor molecule to the electron-acceptor molecule. In H-bonded complexes the interaction in the electron acceptor is localized in the region of a hydrogen atom. A large number of H-bonded complexes can be formed with the water molecule acting as electron acceptor. The water molecule is also an important electron donor. Thus the water dimer, in which the water molecule acts as both the electron donor and acceptor, plays a unique role in the study of H-bonding. 1.2. Vibrational Properties of H-bonded Complexes Among the properties of H-bonded complexes that have been studied, the historical importance of vibrational spectroscopy is well-known [22], Not only can the infrared frequencies and intensities for a particular complex be compared with a wealth of related data for other complexes and for the uncomplexed molecules; but these properties also reveal much about

PAGE 12

3 the bonding characteristics of the molecules, and how they are changed by the H-bond. The vibration which is most sensitive to the effect of the H-bond is commonly referred to as the AH stretch — where RAH is the electron acceptor and AH is involved in the H-bond. For example, when the electron acceptor is water, AH is the OH bond. However, in this case, the notion of a single H-bonded AH stretch is complicated by the fact that there are two OH bonds in water. In the isolated HO molecule, these OH bonds are symmetrically equivalent and couple to give a symmetric and an antisymmetric stretching vibration. Moreover, even though only one of the H atoms forms the H-bond, this coupling is only moderately reduced in the complex. This is because intramolecular forces are much stronger than the H-bond interaction. Because of this coupling there is more than one absorption band in such complexes whose normal coordinate contains a substantial contribution from the H-bonded OH stretch. Similarly, the normal coordinate for the infrared band most sensitive to the H-bond contains contributions not only from the H-bonded OH stretch, but also from the other, "free," OH stretch, as well as a small contribution from the bend. We therefore refer to this band as the "H-bond band." Furthermore, it should be emphasized that, in theoretical calculations for any complex where water acts as the electron acceptor, a complete normal coordinate analysis is essential for obtaining valid frequencies or intensities. For all H-bonded complexes, the H-bond band undergoes a shift to lower frequency and a dramatic increase in intensity relative to the isolated electron acceptor molecule. The magnitude of the frequency shift has long been regarded as a measure of the strength of the H-bond [20, chap. 3] . Due to the developments in the field of infrared

PAGE 13

4 intensities mentioned above, however, the intensity enhancement is becoming increasingly recognized as a far more sensitive indicator of the effect of the H-bond [22] . Accordingly, we have undertaken a thorough study of the intensity of this band in the water dimer (as well as that of other bands). The intensity enhancement is also less readily understood than the frequency shift. The latter results from a weakening of the AH bond upon H-bond formation. Two main factors contribute to this reduction in bond strength. First, the approach of the electron donor molecule on the one side of the H-bonded H atom polarizes the electron density in the AH bond. Some of this density is thus forced to the nonbonding region of the A atom, where it cannot compensate for the AH nuclear repulsion. The plot of electron density versus position along the H-bond, given by Schuster for the water dimer [19, p. 74], is an excellent illustration of this electrostatic polarization effect. Secondly, an equilibrium charge transfer effect occurs, whereby electron density from the electron donor molecule is excited into an antibonding AH orbital. Neither the electrostatic polarization nor the equilibrium charge transfer effects, however, can begin to account for the overwhelming magnitude of the intensity enhancement. The intensity in the H-bonded complex is generally a factor of 10 or more times that in the isolated molecule, whereas the relative frequency shift is on the order of a few percent for non-ionic binary complexes. The traditional approach to this problem follows that used to investigate the physical phenomena that contribute to the stabilization energy of H-bonded complexes [23] . Qualitative arguments concerning the origin of the intensity enhancement are basically in agreement [24-27].

PAGE 14

5 So far, however, the large magnitude of this enhancement has not been accounted for on a quantitative basis [23, and references cited therein]. Accordingly, we have approached this problem by applying a quantum mechanical model for the investigation of intensities. This model has previously been applied to various molecules, but not to molecular complexes [4,10]. We have found that such application leads to a unique quantitative formulation for the H-bond intensity enhancement. This formulation is not only consistent with the general trend of ideas on the subject, but also allows certain distinctions to be made which further clarify the concepts involved. 1.3. Experimental Spectra of Water Dimer 1.3.1. Matrix Isolation Spectroscopy and Structure of the Water Dimer Until comparatively recently, the majority of experimental H-bond studies were carried out in solution. In addition to ill-defined medium effects, thermal effects at the temperatures associated with liquids complicate the interpretation of the spectra. Accordingly, the technique of matrix isolation spectroscopy has been employed extensively in the last decade for the study of H-bonded species. This technique involves the rapid condensation, at cryogenic temperatures, of a dilute mixture of the species to be studied in an inert "matrix" gas [28] . The matrix gases used most frequently are nitrogen (N„) and argon (Ar) . Interactions of the matrix with the solute are therefore expected to be minimal. Several workers have observed the infrared spectra of water isolated in N^ [29-34] and in Ar [34-40] matrices. We discuss these experimental studies in more detail in the following section. Absorption due to water monomer, dimer and trimer (or higher multimer) generally appears

PAGE 15

6 in these spectra. The relative abundance of the different aggregate species depends on the concentration of water in the gaseous mixture, as well as on other factors which are probably constant for a given set of experiments. The bands due to each species can in principle be identified by observing spectra at several concentrations and noting the concentration dependence of the band absorbances. The peak absorbances of bands due to monomer are expected to decrease monotonically with concentration, while trimer band absorbances are expected to increase monotonically. The absorbances of dimer bands are expected to show a maximum with respect to concentration. In practice, however, many of the bands are overlapped so that two neighboring bands due to different species can show the same concentration dependence. This difficulty has led to some differences in the bands assigned by various workers to the dimer, as discussed in section 1.3.2. Tursi and Nixon have obtained a spectrum of water isolated in the N_ matrix [30,31], in which the dimer bands are quite prominent and wellresolved from the monomer and trimer bands, as compared with spectra reported by other workers. A simulated spectrum, which we have adjusted to give the best fit to this experimental spectrum, is illustrated in Fig. 1-1. The method we have used to calculate the simulated spectrum is discussed in section 3.1. The bands assigned to water monomer, dimer and trimer (or higher multimer) by Tursi and Nixon are indicated in Fig. 1-1. All workers have assigned the monomer bands in the same manner: band I to the antisymmetric stretch, \> 3 ; band V to the symmetric stretch, v.; and band XI to the bend, V . l '2 Tursi and Nixon assigned six absorption bands to the dimer as indicated in Fig. 1-1. With the exception noted below, six bands would

PAGE 16

C X [ V ,(ED)] 0.00 £ EH S is o o 3 o 6 i — i -H i— I JZ TD Cfl t-l Mi C E m c O 'H nj O H SCO w O) 00 y A [VjCEA) o > [V,(ED)] C E E >, n3 U CI T) HI cu a)

PAGE 17

a generally be expected for the water dimer, since there are three fundamental absorptions for each of the water molecules in the complex. Tursi and Nixon assigned the six bands in terms of the "linear" structure for the water dimer illustrated in Fig. 1-2. Their assignments are generally accepted, and we now discuss how the frequency pattern expected for this structure is consistent with that in the observed spectrum. In this discussion we use the simulated spectrum in Fig. 1-1 for reference. The dimer structure shown in Fig. 1-2 is characterized by the linear arrangement of the atoms involved in the H-bond : 0. , H^ , and 5 . These atoms and atom H of the electron acceptor (EA) define a plane of symmetry; atoms H and H 2 of the electron donor (ED) are symmetrically oriented with respect to this plane. Thus the OH stretches in the ED molecule are symmetrically coupled just as in the monomer. That is, the antisymmetric and symmetric stretches both have a contribution of equal magnitude from each individual OH stretch. We further note that neither Hj nor H 2 is H-bonded. Accordingly the spectrum of the linear dimer would be expected to have three bands for the ED molecule shifted slightly from the monomer bands Vj , V £ and v 3 . Three such bands — bands II, VI and X — are evident in the spectrum shown in Fig. 1-1. These bands were assigned by Tursi and Nixon to v 3 (ED) , V (ED) and V, (ED) , respectively. Figure 1-2 shows that the two OH bonds in the EA molecule are not symmetrically equivalent. Moreover, the H u bond is H-bonded and thus has a lower stretching force constant than the "free" 5 H g bond. The two stretching vibrations expected for the EA molecule would therefore contain unequal contributions from these two OH stretches, but would retain their basic symmetric and antisymmetric character as discussed in section 1.2. A greater contribution from the H-bonded OH stretch would

PAGE 18

s> < LU <

PAGE 19

10 be expected for the symmetric stretch, V (EA) , which would therefore be shifted to considerably lower frequency from V, of the monomer. Band VII in Fig. 1-1 is consistent with this expectation. This band was assigned to Vj(EA) by Tursi and Nixon. The antisymmetric stretch, V 3 (EA), would be expected to have a greater contribution from the "free" S H 6 stretch and a moderate shift to lower frequency from V 3 of the monomer. Band III exhibits such a shift and was assigned to V 3 (EA) by Tursi and Nixon. The bending vibration of the EA molecule in Fig. 1-2 perturbs the linearity of the H-bond. A considerable shift to higher frequency from V 2 of the monomer would thus be expected for this vibration. Accordingly, band IX in Fig. 1-1 was assigned to V 2 (EA) by Tursi and Nixon. Some of the bands in Fig. 1-1 have also been interpreted in terms of the cyclic structure for the water dimer [29,34,36]. This structure has two H-bonds and is characterized by the nonlinear arrangement of the two atoms and the H atom involved in each H-bond. The structure of the centrosymmetric cyclic dimer is illustrated in Fig. 1-3. This structure is unique in that it has a center of inversion and thus obeys the exclusion rule — three vibrations are active only in the infrared, and three are active only in the Raman spectrum. The three infrared active vibrations are the out-of-phase combinations (Vj , V 2 ~ and V 3 ~) of the corresponding monomer vibrations of the two water molecules forming the dimer. The frequency shifts expected for these vibrations, from V, , V and 1 ' 2 V 3 of the monomer, would follow qualitatively the pattern discussed for the vibrations of the EA molecule of the linear dimer. However, the frequency shifts for the cyclic dimer should be smaller because the nonlinear H-bonds are not as strong as linear H-bonds [36,41],

PAGE 20

11 u -a >, c U K e 3 W Hi o J2

PAGE 21

12 1.3.2. Review of Experimental Studies The first study of matrix isolated water was carried out in the N ? matrix by Van Thiel, Becker and Pimentel, using a prism monochromator [29] These authors attributed the dimer bands to the centrosymmetric cyclic structure because only three dimer bands could be observed in their low resolution spectra. Bands II, VI and X in Fig. 1-1 were not resolved from the neighboring monomer bands. In all the subsequent studies, the experimental spectra of water isolated in the Nmatrix were basically the same as those obtained by Tursi and Nixon. Three groups of workers [33-35] did not assign certain bands in their spectra in agreement with the assignment made by Tursi and Nixon (the TN assignment) . These discrepancies involve bands assigned to dimer and trimer in two regions of the spectrum. One of these regions includes bands III and IV in Fig. 1-1. We recall that band III corresponds to V 3 (EA) of the linear dimer according to the TN assignment. Now the frequency shift of this vibration from v 3 of the monomer is due to the H-bond interaction. We therefore refer to this region as the V (H-B) region. The other region in which different assignments have been made by various workers includes band VII, which corresponds to V (EA) according to the TN assignment. We designate this region as the V (H-B) region. Figure 1-1 shows that in the V 3 (H-B) region a relatively sharp dimer band is superimposed on a broad trimer band according to the TN assignment. The same situation occurs in the V (H-B) region. In many of the spectra obtained by various workers a broad shoulder appears on the low frequency side of band VII in Figure 1-1. We refer to that broad absorption as band VIII.

PAGE 22

13 Suppose the TN assignment is correct and that the broad bands are due to trimer while the sharper bands are due to dimer. Then the absorbances of the two broad bands should vary differently with respect to concentration from those of the two sharp bands. However, the concentration dependence of the "true" band absorbances would probably not be observed, because the broad trimer band acts as a baseline for the sharper dimer band. Rather, any increase in the absorbance of the broad trimer band would result in an apparent increase in the absorbance of the dimer band, even though the "true" absorbance of the latter band does not increase. Thus, reliable assignments of these bands cannot be made solely on the basis of their concentration dependence. Table 1-1 summarizes the discrepancies between the TN assignment and the assignments made by other workers for bands III, IV, VII and VIII. In two of the studies, more than four bands in the OH stretching region were assigned to dimer. Since any given structure for the water dimer can have a maximum of four bands in this region (two bands for each of the component water molecules) , alternate interpretations had to be made for the extra bands. Huong and Cornut obtained spectra of water isolated in both the N„ and the Ar matrix [34]. As indicated in Table 1-1, these authors attributed bands IV and VII to the dimer. As discussed in section 1.3.1., two stretching vibrations, V and Vj , are expected for the centrosymmetric cyclic dimer. Accordingly, Huong and Cornut assigned band IV in the v (H-B) region to v 3 and band VII in the Vj (H-B) region to Vj . They assigned bands II, III, VI and VII to the linear dimer according to the TN assignment. Their overall assignment is therefore not consistent with the relative frequency pattern expected for linear and nonlinear H-bonds (see section 1.3.1.). That is, the assignment attributes larger

PAGE 23

u Table 1-1. Assignment of bands in the V (H-B) and Vj(H-B) regions to dimer (D) and trimer (T) by various workers. Bands III, IV and VII are indicated in Fig. 1-1. Band VIII appears on the low frequency side of band VII in many spectra. V 3 (H-B) Region Vj (H-B) Region III IV VII VIII Total Number of Dimer Bands in OH Stretching Region Tursi and Nixon Huong and Cornut Barletta C Luck D

PAGE 24

15 frequency shifts to the bent H-bonds of the cyclic dimer than to the linear H-bond of the structure shown in Fig. 1-2. Barletta obtained excellent high resolution spectra of water isolated in the N matrix [33] . In these spectra four bands were resolved in the absorption corresponding to band VIII in the V (H-B) region. These four "extra" bands were attributed to the dimer. Of the eight bands thus assigned to the dimer, four were observed to be considerably weaker than the others. The four stronger bands were then assigned to fundamentals of the linear dimer: bands II, III and VII in Fig. 1-1 and one of the four "extra" bands in the Vj (H-B) region. According to this assignment two vibrations (corresponding to band VII and the"extra" band) have substantially lower frequencies than V of the monomer. Only one such vibration is expected for the linear dimer as discussed in section 1.3.1. Moreover, band VI in Fig. 1-1 was considered too weak to be a fundamental. This band corresponds to v x (ED) according to the TN assignment. It is shown below, however, that on the basis of our calculations for the linear water dimer the intensity of the V X (ED) fundamental should indeed be small. Mann, Neikes, Schmidt and Luck observed the spectra of water isolated in the Ar matrix [35, see also reference 36]. Their spectra were basically the same as those observed for water isolated in the N matrix by the various workers. Table 1-1 shows that these authors assigned both bands in the V 3 (H-B) region to the trimer, while both bands in the V (H-B) region were assigned to the dimer. These authors interpreted the dimer bands in terms of the cyclic structure. We note that the cyclic dimer can be asymmetric. If one of the two water molecules in the structure shown in Fig. 1-3 is rotated somewhat, the center of symmetry is eliminated. For this structure two weak

PAGE 25

16 stretching bands would be expected, corresponding to the infrared inactive vibrations of the centrosymmetric structure, namely, the in-phase combinations (v and V ) of the corresponding monomer vibrations of the two water molecules in the dimer. According to these considerations, Mann, Neikes, Schmidt and Luck assigned bands II, VI and VII in Fig. 1-1, as well as band VIII in the V (H-B) region, to the asymmetric cyclic dimer + + stretching vibrations, V 3 , V 3 , Vj and Vj , respectively. This assignment is consistent with the relative intensity pattern expected for the out-of-phase and in-phase vibrations. However, the frequency shifts, from the corresponding monomer bands, resulting from this assignment do not seem entirely satisfactory. That is, while frequency shifts of 103 and 97 cm * are attributed to V 3 and V , respectively, a relatively small shift of 14 cm is attributed to V , and a relatively large shift of 3 Jo 171 cm is attributed to V . On the other hand, Luck [36] objected to the frequency shift of VjCEA), the "H-bond band," resulting from the TN assignment on the basis that it was too small (85 cm ) for a linear dimer. Several experimental studies were cited, which showed that linear H-bonds were stronger than nonlinear H-bonds. It was pointed out that the water polymer (small crystallites or amorphous clusters of ice) had linear H-bonds, and its H-bond band occurred at 3220 cm 1 (in Ar) . Luck then stated, "We would expect the absorption of the linear dimers in the region of the H-bond band of large aggregates ('polymer baud')" [36, p. 552]. This reasoning, does not take into account the importance of cooperative effects in ice, which has a stabilization energy of 11 kcal/mole [42]. In contrast, Clementi has calculated a stabilization energy of

PAGE 26

17 5.6 kcal/mole for the linear water dimer,* using a very large basis set and including correlation effects [43]. Furthermore, we point to the good agreement of the theoretical frequency predicted for the H-bond band of the linear dimer (see Appendix) with that of band VII in Fig. 1-1. The intensity we have calculated is too large. However, evidence will be presented below that this discrepancy is due to an artifact of the calculation rather than to the structure of the model we have chosen. Tursi and Nixon also observed the spectrum of D~0 and HDO in the N~ matrix in addition to that of H„0. They found a one-to-one correspondence between the HO bands discussed in section 1.3.1. and those due to DO. Corresponding bands were also found for HDO monomer. For the linear HDO dimer, there are two possible isotopes since the EA molecule can be complexed either through the H atom or through the D atom. Tursi and Nixon observed four stretching bands whose frequency pattern was consistent with that expected for a linear HDO dimer with the EA complexed through the D atom. Two additional bands expected for the other isotope were not observed, nor were any dimer bands in the HDO bending region observed. Tursi and Nixon used the nine frequencies they observed for H„0, D„0 and HDO monomers to obtain a force field for water monomer. Similarly, they obtained a force field for the linear water dimer using twelve frequencies for (HO) and (DO), and the four frequencies for (HDO) . We have used these two force fields in our intensity calculations, and thev are discussed furttier below. * Experimental studies give estimates of 5.2 kcal/mole [44] and 3.0 kcal/mole [45]. But the theoretical value is the most reliable and probably close to the true value.

PAGE 27

18 Recently, the infrared spectrum of the various isotopes of water isolated in the N matrix was again studied by Fredin, Nelander and Ribbegard [32] . They made assignments for all the isotopic dimer bands in terms of the linear structure, in agreement with Tursi and Nixon. Moreover, they found that increasing the intensity of the irradiating beam and/or decreasing the temperature of the sample led to the appearance of the bands for linear (HDO)„ not observed by Tursi and Nixon — the dimer bands in the HDO bending region and those corresponding to the EA complexed through the H atom. Thus, all the bands for (D„0)„ and (HDO). have been observed at those frequencies which are expected for the linear structure shown in Fig. 1-2. The consistency of these results with those for (H„0) 2 strongly confirms the assignments given in section 1.3.1. As mentioned above, Huong and Cornut [34] and Mann, Neikes, Schmidt and Luck [35] have obtained spectra of water isolated in the Ar matrix which are basically the same as those in the N matrix. Several workers have obtained a totally different type of spectrum for water in the Ar matrix [37-40] . These workers have assigned the majority of bands in the spectrum to vibration-rotation transitions of the monomer. The relative absorbances of these bands show a complicated time and temperature dependence. This behavior has been interpreted in terms of conversion from ortho to para spin states of the hydrogen nuclei [37-39] . According to this interpretation, the most recent results for the time and temperature dependence of the various bands indicate the following [39]. Unexpectedly rapid spin conversion occurs during the deposition process — to an ortho/para ratio characteristic of a temperature roughly halfway between room temperature and the temperature of the matrix. Thereafter, the conversion process continues at a slower rate in matrices with

PAGE 28

19 relatively high water concentrations. But in dilute matrices no conversion occurs after deposition. This interpretation is rather elaborate, and it should be pointed out that other explanations may exist for the profuse spectrum of water in the Ar matrix. Matrices giving rise to this type of spectrum have been deposited at lower temperatures (4-10°K) than those giving rise to the simple spectrum (16-20°K). At higher temperatures, the matrix is likely to be annealed during deposition so that the orientation of the Ar crystallites in the matrix is more uniform. It is possible that the importance of nonequilibrium effects in the lower temperature matrix results in a larger number of interactive sites, each having a different energy and thus giving rise to different frequency shifts for the vibrations. In addition, the possibility of N impurity in the Ar matrix cannot be ignored. Mixed aggregates of H„0 and N„ molecules could account for some of the bands in the profuse spectrum.* Within the context of attributing most of the bands to monomer vibration-rotation transitions, Ayers and Pullin have assigned five bands in the profuse spectrum of water in Ar to the linear water dimer [40] . In the frequency region corresponding to that expected for the V (ED) vibration, no band could be attributed to the dimer, all bands in this region having been assigned to the monomer. The "absence" of this band is somewhat disturbing since our calculations predict the intensity of V 3 (ED) to be second only to that of the H-bond band, v (EA) . Possibly one of the "rotating monomer" bands could be attributed to this vibration. * In this regard, a number of bands have been induced in the spectrum of hydrogen halides (MX) in the Ar matrix by doping with N. impurity. These bands have been attributed to various aggregates of the HX and N_ molecules in the Ar matrix [46] .

PAGE 29

20 1.4. Theoretical Calculation of Vibrational Properties of the Water Dimer 1.4.1. Frequencies Matrix isolated spectra of the water dimer have provided new data that are important for the study of H-bonding. Theoretical calculations are an additional source of such data. Moreover, the development of theoretical methods is now approaching a level, where they can be used as a reliable guide for interpretation of the experimental spectra. Curtiss and Pople have performed a set of quantum mechanical calculations with the 4-31G basis set in order to obtain force constants for water monomer and the linear dimer [47] . The vibrational frequencies for monomer and dimer were then calculated from each force field. These frequencies agree reasonably well with the experimental ones observed by TN, but they are all 10% too large. It is typical for Hartree-Fock calculations to overestimate frequencies by approximately this amount [19, p. 33]. The absolute differences between the calculated and experimental frequencies (about 360 cm l in the stretching region and 160 cm" 1 in the bending region) might seem too large for the calculations to be of any use for the experimentalist. However, the factors in the calculation that lead to overestimation of the force constants, and thus of the frequencies, should be nearly constant for the monomer and dimer. One method of comparing calculation with experiment, which minimizes the effect of constant errors in the calculation, is to examine the calculated and observed frequency shifts from monomer to dimer. An alternate approach makes use of the fact that the physical phenomena responsible for the frequency shifts (for example, changes in

PAGE 30

21 bond strengths) appear directly as changes in the force constants. Thus, in this approach one is concerned with the change in each force constant, Af(th.), which is obtained from the theoretically calculated monomer and dimer force constants, according to Af(th.) = f(th., dimer) f(th., monomer). A set of scaled dimer force constants can then be calculated using the experimental monomer force constant: f (scaled, dimer) = f(exptl., monomer) + Af(th.). This set of scaled dimer force constants is then used in a standard normal coordinate calculation to obtain dimer frequencies which can be compared directly with experiment. In this scaling procedure, errors in the theoretically calculated monomer and dimer force constants compensate more effectively than they do in the comparison of frequency shifts. This is because, in the normal coordinate calculation of the frequencies the dimer force constants are weighted differently, by the geometrical parameters and the masses, from the monomer force constants. The force constant errors are thus also weighted differently in the calculated frequencies. This scaling procedure has been applied to the theoretical force constants calculated by Curtiss and Pople (see Appendix) . It has also been applied to a set of force fields calculated by the CNDO method for the linear water dimer. The resulting frequencies predicted for the water dimer agree quite will with the experimental frequencies.

PAGE 31

22 1.4.2. Intensities Little work has been done in the area of theoretical intensity calculations for the water dimer. In the work that has been done, only the intensity of the H-bonded OH stretch has been calculated, and normal modes have not been taken into account [48, and references cited therein] , Because theoretical intensities can provide valuable assistance in the interpretation of experimental spectra, we have calculated a complete set of intensities for the normal modes of the linear water dimer. As discussed in section 2.4. , the intensities are proportional to the square of the derivative of the molecular dipole moment with respect to the normal coordinates. These dipole derivatives, like the force constants, reflect the chemical bonding characteristics of the molecule. One of the goals of infrared intensity theory is to study how the dipole derivatives change from one molecule to another and interpret these changes in terms of the types of atoms and bonds present. Dipole derivatives can also be expressed in terms of other forms of the vibrational coordinates which are more useful for interpretation than the normal coordinates. The space-fixed cartesian representation for the dipole derivatives has proven particularly advantageous for understanding the chemical factors that determine the intensities [1-10]. These dipole derivatives are composed of a set of atomic polar tensors (APT's), one for each atom in the molecule [1] . Studies have indicated that the values of the APT elements for a given atom tend to be relatively independent of which molecule the atom is in [4,7]. Remarkable success has been achieved in predicting intensities for one molecule by transferring APT's from other chemically related molecules, and then transforming these APT's to

PAGE 32

23 dipole derivatives with respect to the normal coordinates of the first molecule [5]. These results suggest that the APT's are fundamental indicators of the chemical properties of atoms in molecules. Accordingly, we have calculated APT's for the water monomer and dimer using quantum mechanical methods in order to predict intensities for these two molecules. As for the force constants, we expect that the errors in the quantum mechanical calculation of the APT's are nearly the same for the dimer complex as for the monomer. To the extent that this is true, we might expect that the changes in the APT's from monomer to dimer are well-represented by the calculation. With this idea in mind, we have applied a scaling procedure, analogous to that described above for the force constants, to obtain APT's for atoms in the dimer from the known APT's for those atoms in the monomer. First, the change in the APT of atom A from monomer to dimer, AP (th.), was obtained from the theoretical calculation by, (1) AP A (th.) = P A (th., dimer) P (th., monmer) . Then the experimental APT for atom A in the monomer was used to obtain the scaled dimer APT: (2) P (scaled, dimer) = P (exptl. , monomer) + AP (th.). The APT's from both the ab initio quantum mechanical calculation, P (th., dimer), and from the scaling procedure, P (scaled, dimer), were used to obtain intensities for the absorption bands of the water dimer. These theoretical dimer APT's thus provided an independent source of data, which we have used for comparison with experiment. The comparison will be discussed in detail in chapter 3.

PAGE 33

24 The APT's also contain more information than do the intensities for a relatively large, unsymmetrical molecule like the water dimer. That is, the intensities can be obtained from the APT's, but the APT's can only be calculated theoretically. Both the APT's and the intensities reflect the redistribution of charge that takes place as the molecule vibrates. In the APT's, this vibrational redistribution of charge is resolved into contributions from each atom. As mentioned previously, in the study of H-bonded complexes our chief interest lies in the changes that take place relative to the uncomplexed molecule. We have thus compared the theoretical APT's which we have calculated for each atom in the dimer with the corresponding APT in the monomer. We have found that changes in the APT's from monomer to dimer are restricted to a few critical elements. Moreover, the major changes are restricted to those dimer atoms that are involved in the H-bond. We have also used the quantum mechanical definition of the dipole moment to resolve each APT into charge, charge flux, and overlap contributions. Again, we have found that changes from monomer to dimer are restricted to a few critical elements. These results are presented and discussed in chapter 4.

PAGE 34

CHAPTER 2 DESCRIPTION OF CALCULATIONS 2.1. Outline The essence of our method lies in the quantum mechanical calculations and in the scaling procedure used for the dimer polar tensors, given in Eqs. 1 and 2. However, we wish to predict the intensities for the normal modes of vibration of the molecule, and thus the calculation of intensities from polar tensors (APT's) is intimately bound up with the normal coordinate analysis. The forms of the normal coordinates depend on the force field and on the geometry and atomic masses of the molecule. Our purpose in this chapter is to include enough details about our calculations so that the reader can reproduce our results or obtain analogous results, using a different force field or theoretical APT's from a different basis set. In order to do this, we must refer in some detail to the equations involved in the normal coordinate analysis; thus, in section 2.2 we set down these equations. In section 2.3 we give the equations relating the intensities and the APT's. Then in section 2.4 we describe how we have applied these equations and give all the data we have used, except for the theoretical APT's. In section 2.5 we describe the numerical and quantum mechanical methods we have used in calculating the theoretical APT's. Finally, in section 2.6, we give our procedure for calculating the scaled APT's and intensities for the dimer. 25

PAGE 35

26 2.2. Normal Coordinate Analysis As mentioned previously, there are several ways of representing the vibrational coordinates of the molecule. The normal coordinates, Q., are 1 characterized by the fact that each fundamental absorption band depends on a single Q. within the harmonic oscillator approximation. Moreover, in the normal coordinate representation, both the kinetic and potential vibrational energy of the molecule are a sum of independent contributions. Each of these contributions also depends on a single Q.. We have used the Wilson FG matrix method to obtain the normal coordinates. The most important equations in this method are presented here. Detailed discussion of these equations can be found elsewhere [49] . The internal valence coordinates, R., provide a useful representation i for the force constants. For an N-atomic molecule, the (3N-6)-dimensional column vector of internal coordinates, R, is related to that of normal coordinates, Q, by the matrix equation: (3) R = LQ. The normal coordinate transformation matrix, L, is determined by solving the eigenvalue problem, (4) GFL = LA. The eigenvalues of this equation form the elements of the diagonal matrix of frequency parameters. A, while the eigenvectors are given by the corresponding columns of the L matrix. The solution depends on the inverse kinetic energy matrix, G, and the potential energy matrix, F. The latter is composed of the force constants, given by

PAGE 36

27 < 5 > F ij = 9R^R7 1 J where V is the vibrational potential energy of the molecule. As discussed in the previous chapter, the space-fixed cartesian coordinates constitute the most useful representation for the intensities, The 3N-dimensional vector of these coordinates, X, is related to the vector, R, by (6) R = BX . The [(3N 6) x 3N] -dimensional B matrix is related to the G matrix in Eq. 4, according to (7) G = BM _1 Bt where Bt indicates the transpose of B. The diagonal matrix of inverse masses, M , is given by (8) CM" 1 ).. = , i = 1,2,3 li m. , i = A, 5, 6 , i = 3N-2, 3N-1, 3N for the N atoms of the molecule. A third type of vibrational coordinates is used to simplify the eigenvalue problem of Eq. 4. In the symmetry coordinate representation, the G and F matrices factor into different symmetry blocks. The symmetry coordinate vector, S, is related to R by

PAGE 37

28 (9a) S = UR (9b) S = SX The symmetrized B matrix, 8, is given by (10) 8 = UB The symmetrized F matrix, F, is then defined as (Ha) r :, ' F ' ij 3S.9S. i J (lib) F = UfUt , where U is an orthogonal transformation, that is Ut = if 1 . Similarly, the symmetrized G matrix, G, is given by (12a) G = BM _1 8t (12b) G = UGUt Finally, we define the symmetrized L matrix: (13a) S = LQ (13b) L = UL It can be seen from Eqs. lib, 12b and 13b that Eq. 4 is equivalent to (14) GFL = LA Each of the matrices in this equation is block diagonal. Thus, neither the symmetry nor the normal coordinates in a given block can mix with those in the other blocks. There is one block for each of the

PAGE 38

29 irreducible representations of the vibrations in the symmetry group to which the molecule belongs. The units we have used for the various matrices can be discussed on the basis that there are two types of symmetry coordinates, S.. Those involving bond stretches (r-type) have units of length. For those involving angle deformations (8-type) we have used units of radians (rad). There are, then, three types of force constants, depending on whether S. and S. in Eq. 11a are both r-type, both 0-type, or one of each. The resulting force constants all have similar magnitudes, if the o o energy in that equation is given in units of millidyne A (md A, which is 10 erg) and the length in A. The units of the force constant, F.., in the three cases for S. and S. cited above, are then md (A) L , md X rad , or md rad , respectively. When S, is r-type, the kth row of S is O — i unitless, whereas it has units of rad (A) when S, is 9-type. For the masses we have used atomic mass units (u) . Thus, the kth row of L has ! 2 ° -i -h units of u when S. is r-tvpe and units of rad (A) u , when S, is k k 8-type. The G matrix in Eq. 14 is completely determined by the B and M 1 matrices, according to Eq. 12a. Thus, the solution to Eq. 14 depends on the geometry, masses and force constants of the molecule. The eigenvector matrix, L, describes the forms of the normal coordinates. Hence, L enters into the calculation of the intensities. 2.3 Int ensity Relations There are a number of ways of expressing infrared intensities. Overend [50] has given the relation between the integrated molar absorption coefficient of the ith band, A., and the derivative of the dipole l moment vector, p, with respect to the ith normal coordinate, Q :

PAGE 39

30 A. = N TT A

PAGE 40

31 coordinates. The matrix of dipole derivatives with respect to the 3N space-fixed cartesian coordinates, P is related to the P matrix by (17) ? -q = ? -x^ The AL product matrix gives the transformation from the vector of normal coordinates, Q, to the vector of space-fixed coordinates, X: (18) X = ALQ = ALQ . The A and A matrices are related by A = AUt and are given by [3] (19) A = M _1 BtG _1 A = M _1 BfG _1 . The AL product matrix has units of u~^, and P, has units of e. — -X The P^ matrix is comprised of N juxtaposed APT's for each of A the atoms of the molecule. If k represents the cartesian coordinates of atom A, the APT on atom A, P , is given by A 9p i (20) (P A ) — 1 , A J K A j,k=x,y,z Then the P matrix has the form (21) P x = (P 1 P 2 --P N ) . From Eqs. 17 and 21 we can see that it is necessary to specify the A and t matrices, in addition to the APT's, in order to determine the P -Q

PAGE 41

32 matrix and therefore the intensities. From Eqs. 19, 12a and 8, we see that the A matrix is determined by the 8 matrix and the masses of the molecule. In order to obtain intensity predictions for the fundamental vibrations of water and water dimer, we have transformed the theoretically calculated API's to the P matrix, according to Eqs. 17 and 21. We shall also need the inverse transformation from the P to the P matrix. This Q -x is because we wish to use the experimental APT's for the water monomer in our scaling procedure (see Eq. 2), and these must be obtained from the experimental intensities. The desired inverse transformation is given by (22) p = ? n L~ l B + D . -a -gIn this equation, the D matrix is the permanent dipole moment rotation matrix resulting from the transformation from the molecule-fixed normal coordinate representation to the space-fixed cartesian coordinate representation. Like the P_ x matrix, the D matrix is comprised of N juxtaposed tensors. It has the form of Eq. 21 with P replaced everywhere by D A . The rotational tensor, D , for atom A is given by (23) D A = ((p°)) l" 1 ((? A °))m A . Each of the factors in Eq . 23 is a second order tensor of rank 3. All three elements of the diagonal tensor, m , are equal to the mass of atom A; p is the permanent dipole moment vector of the molecule; and r ° is A the equilibrium position vector of atom A with respect to the center of mass. The notation, ((v)) [represented by ((p )) in Eq. 23, for example], indicates the tensor formed from the vector, v, according to [6]

PAGE 42

33 (24) ((v)) = V -v z y v -v " y x In Eq. 23, I is the inverse of the moment of inertia tensor, I, which is given by (25) I = m A ((r A ))+((r A 0)) A Thus, in order to determine the D matrix, we must know the masses and position vectors of the atoms with respect to the center of mass of the molecule, r, , and the permanent dipole moment, p . We have used e A A for the units of the dipole moment, u for the masses, and A for the position vectors in Eqs. 23 and 25. The rotational tensor, D , therefore has units of e. 2.4. Application to Intensities for Water and Water Dimer We may now apply these equations to the intensity calculations for water monomer and dimer. For the monomer we are interested in evaluating the theoretical API's by comparing the resulting intensities with the experimental gas-phase intensities and with those from other calculations. Thus, we have used the A matrix and the experimental gas-phase L matrix in Eqs. 16 and 17 to calculate the intensities from these APT's. The A matrix was determined from the monomer S matrix and the atomic masses, according to Eq . 19. The L matrix was obtained by solving Eq. 14, using the experimental values of the force constants for the gas-phase monomer [51]. As mentioned above, the G matrix in Eqs. 14 and 19 is determined from the 8 matrix and the masses by Eq. 12a. Since we are

PAGE 43

34 also interested in the observed intensities for the monomer in the N matrix,* we have calculated a set of intensities using the L matrix derived from the force constants reported for the monomer in the matrix [30] . For the matrix-isolated dimer, we have calculated a set of intensities using the corresponding A and L matrices with the APT's obtained directly from the theoretical calculations for the dimer. The L matrix was derived from the force constants reported for the dimer in the N„ matrix [30] , as well as the B matrix and masses, using Eq. 14. An additional set of intensities was calculated from the scaled APT's using the same A and L matrices. In order to investigate the effect of changing the L matrix on the predicted intensities, we have also obtained an L matrix using force constants derived from the theoretical calculations of Curtiss and Pople [47] . Intensities were then obtained using this L matrix and the scaled dimer APT's. To obtain the latter we have used the experimental APT's for the monomer along with the monomer and dimer APT's calculated theoretically, according to Eqs. 1 and 2. The experimental monomer APT's were calculated from the experimental intensities by first obtaining the P matrix elements according to Eq. 16, and then by using this P matrix with the monomer 8, L and D matrices in Eq. 22. Experimental intensities are not available for the monomer in the matrix, so we have assumed they are about equal to those in the gasphase. To be consistent, we have used the monomer L matrix derived from the gas-phase force constants to calculate the experimental monomer APT's. * Here we refer to the inert solid environment discussed in section 1.3. The word "matrix," in this sense, should not be confused with the mathematical term. The meaning of the word should be clear, from the context in which it is used, in all cases.

PAGE 44

35 From the above discussion, it is clear what data must be specified in order for the reader to reproduce our results from the water monomer and dimer APT's. For the monomer these include the 8 matrix, the masses, and the experimental force constants, both for the gas-phase and for the monomer in the N matrix. For the dimer we must give the 8 matrix, the masses, and the experimental force constants for the dimer in the matrix, as well as those derived from the theoretical calculation. We must also specify the data from which the experimental monomer APT's were obtained. For completeness we give both the data and the resulting APT's. We also give the position vectors from which the two 8 matrices were obtained, and the L matrix derived from each set of force constants we have used. For the masses we have used m = 16.0 and m^ = 1.008 u throughout. The 8 matrix defines the relationship between the space-fixed cartesian and symmetry coordinates of the molecule, according to Eq. 9b. It is obvious that a given 8 matrix applies to a particular ordered sequence for both sets of coordinates. For the cartesian coordinates it is sufficient to give the order of the atoms; the atomic x, y and z coordinates are ordered consecutively. In practice the B matrix is obtained first, thus defining the internal coordinates according to Eq. 6. Then, in order for the symmetry coordinates to have the properties described in the previous section, they are obtained as the normalized results of the symmetry projection operators on the internal coordinates. This determines the U matrix according to Eq . 9a, and the 8 matrix is then calculated from Eq . 10. The coordinate system, atom numbering, and bond lengths and angles for H-0 are shown in Fig. 2-1. As indicated, the origin is centered on the atom. The equilibrium position vectors were determined from the expe rimental structural parameters, r = 0.9572 A and 9 = 104.52° [52] OH

PAGE 45

36 Fig. 2-1. Molecular coordinate system of water monomer. A torn numbers, bond lengths and angle are indicated.

PAGE 46

37 These vectors are Riven in Table 2-1. The definiton of internal coordinates, which was used in calculating the B matrix, is given on the left side of Table 2-2. The H_0 molecule has C~ symmetry. The reducible I 2v representation spanned by the internal coordinates is given by r = 2a. + b R i 2 The corresponding symmetry coordinates are given on the right side of Table 2-2, and the resulting 8 matrix in Table 2-3. The coordinate system for the linear water dimer is shown in Fig. 2-2. The origin is on the atom of the electron donor molecule (atom 3). The atom numbering is given in Fig. 1-2. The equilibrium position vectors we have used are given in Table 2-4. Large basis set calculations for the water dimer have indicated that the bond distances and angles of the two component water molecules do not differ appreciably from their values in the monomer [53,54]. Accordingly, we have used the experimental values of r and 8 given above for each of the water molecules in the dimer. We have taken the intermolecular geometry from the minimum energy structure calculated for the dimer by Hankins, Moscowitz and Stillinger near the Hartree-Fock limit [53] . For the dimer structure illustrated in Fig. 2-2 and discussed in section 1.3.1, the intermolecular geometrical parameters which determine the position vectors are the 0-0 distance and the angle
PAGE 47

38 Table 2-1. Equilibrium position vectors for water monomer. Coordinate system and atom numbering are given in Fig. 2-1. Units are A. Atom No. :

PAGE 48

39 Table 2-3. The 8 matrix for water monomer. Row 2 has units of rad A l All other rows are unitless. Symmetry Coordinate Cartesian Coordinate 'i 0.0 0.0 0.0 0.559178 0.639448 0.559178 0.432805 -0.826156 0.432805 0.0 0.0 0.0 -0.559178 -0.639448 0.559178 0.432805 -0.826156 -0.432805 0.0 0.0 0.0 y. o.o 0.0 -1.118355 -0.865611 1.652312 0.0

PAGE 49

40 N -*-

PAGE 50

41 Table 2-4. Equilibrium position vectors for the linear water dimer. Coordinate system is given in Fig. 2-2 and atom numbering in in Fig. 1-2. Units are A. Atom

PAGE 51

42 with one possible exception [40, and references cited therein] , the six low-frequency intermolecular absorptions of the water dimer have not yet been observed. Thus, there are no reliable frequencies from which to determine the force constants, and no experimental spectrum in the literature with which to compare the intermolecular intensities. The dimer molecule in Fig. 2-2 has C symmetry, and the reducible representation of the six internal coordinates is given by F R 5a' + a" . The corresponding symmetry coordinates are given on the right side of Table 2-5. The 8 matrix obtained from these data appears in Table 2-6. To calculate the 8 matrices in Tables 2-3 and 2-6, and to solve the eigenvalue problem of Eq . 14, we used the normal coordinate programs WMAT and CHARLY [55] . The operation of these programs has been discussed in detail elsewhere [56] . Program WMAT calculates the 8 and G matrices and outputs a transformation of the G matrix which facilitates the diagonalization of the GF product. The input to WMAT includes the masses and equilibrium position vectors of the atoms, the definition of the internal coordinates, and the U matrix. We note that the 8 matrix must represent a transformation to a true molecule-fixed coordinate system — that is, it must be invariant to rigid translations and rotations of the whole molecule. Program WMAT insures this by using the "little £ vector" technique [49] to calculate the B matrix in Eq . 6 and then transforming according to Eq. 10. The input to program CHARLY includes the output from WMAT and the force constant matrix, F. The output from CHARLY includes the frequencies (determined from the A matrix) and the eigenvector matrix, L. In order to obtain the experimental gas-phase L matrix for H„0, we used the F matrix given by Cook, De Lucia and Helminger [51]. It was obtained

PAGE 52

X

PAGE 53

44 by fitting the observed infrared frequencies, as well as the quartic distortion coefficients from microwave data, for H„0 and its deuterated and tritiated isotopes. This F matrix is given in Table 2-7 along with the resulting L matrix elements. The experimental F matrix for water monomer in the N matrix was taken from Tursi and Nixon [30] . It was obtained from a fit to the frequencies of HO and its deuterated isotopes in the matrix. The F and L matrices are given in Table 2-7. It can be seen that the stretch-bend interaction force constant, F , has a different sign for H„0 monomer in the N matrix from that for H„0 in the gas-phase. The effect of this change on the intensities is discussed in the following chapter. The experimental F matrix for water dimer in the N„ matrix was also obtained from Tursi and Nixon [30] , and was derived in the same manner as that for H„0. It is given in Table 2-9 in terms of the symmetry coordinates of Table 2-5, together with the corresponding L matrix. Intermolecular coupling force constants were not included in the analysis of Tursi and Nixon. The F and L matrix elements for the electron donor (ED) molecule (see Fig. 1-2) appear in the upper part of Table 2-9, and those for the electron acceptor (EA) molecule in the lower part. The "theoretical" force constants we have used for the water dimer were derived from those calculated with the 4-31G basis set by Curtiss and Pople [47], using the scaling procedure described in section 1.4.1. The details of the derivation are given in the Appendix, as well as the comparison of the resulting frequencies witli experiment. The agreement is quite satisfactory. This F matrix and the corresponding L matrix are given in Table 2-10. We note that intermolecular coupling force constants were included by Curtiss and Pople, and the F and L matrix elements for these interactions are given in the middle part of the table.

PAGE 54

4 5 Table 2-7. Experimental F and L matrices for water monomer in the gas-phase. The F matrix is from Cook, DeLucia and Helminger [51] . 7.653 md A * 0.5124 md rad 0.6398 md A rad" 7.838 md A ' 1.013 u -0.1117 u 1.034 u Table 2-8. Experimental F and L matrices for water monomer in the N, matrix. The F matrix is from Tursi and Nixon [30] . F 12 F 22 F 3 3

PAGE 55

46 Table 2-9. Experimental F and L matrices for the linear water dimer in the N matrix. The F matrix is from Tursi and Nixon [30] . Electron Donor -0.4088 md rad 0.6927 md A rad 7.6545 md A _1 1.012 u 2 0.1198 u 2 -0.2664 rad A _1 u~^ 1.505 rad A ' u ' 1.034 u * Electron Acceptor 7.498 md A ' -0.2200 md X _1 -0.3350 md rad" 7.036 md X" 1 -0.3350 md rad" 0.7211 md K rad" 0.9492 u 0.3794 u~ 0.0979 u~' -0.4026 u" 0.9387 u~' 0.1057 u~ ; o-l 0.1082 rad A u 1.498 rad A

PAGE 56

47 Table 2-10. Scaled 4-31G F and L matrices for the linear water dimer in the N2 matrix. The F matrix was obtained by scaling the 4-31G F matrix of Curtiss and Pople [47] , as described in the Appendix. Electron Donor 0.6963 md A rad 2 7.804 md X" 1 -1.018 u * 1.524 rad A~ 1.034 u"" 2 Intermolecular -0.0170 md A -0.0070 md A _1 0.0027 md A rad" -0.0088 u * -0.0287 u"" 1 0.0418 u~^ 0.0504 u~' S 0.1137 rad A" Electron Acceptor 7.616 md A -0.182 md A 7.126 md A _I 0.7330 md A rad" L n

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43 However, stretch-bend coupling force constants were neglected by Curtiss and Pople for both the ED and EA molecules, while they were included by Tursi and Nixon. The effect on the intensities of neglecting these interactions is discussed in the following chapter. In calculating the L matrix of Table 2-10 from Eq . 14, the cross-terms in the G matrix corresponding to the stretch-bend interactions were constrained to zero. If this had not been done, the interaction terms in the GF product matrix would not be zero but would depend on the values of the stretch and bend diagonal force constants. This somewhat arbitrary procedure was also adopted by Curtiss and Pople. Due to the neglect of the stretch-bend interactions by Curtiss and Pople, there are fewer F and L matrix elements listed in Table 2-10 for the ED and EA molecules of the dimer. These values are given in the upper and lower parts of the table, respectively. A We now give the data used to obtain the experimental APT's, P , for the monomer. The experimental dipole derivatives with respect to dimensionless normal coordinates, 3p./3q., have been given by Clough, Beers, Klein and Rothman for water monomer in the gas-phase [57, see also reference 58] . Although the total integrated intensities give only the magnitudes of the dipole derivatives (see Eq. 15), these authors chose the signs so that the observed vibration-rotation effects on intensities were reproduced in the fundamental bands of H ? 0. These sign choices agree with those obtained by a number of ab initio calculations (see section 3.2). The 3p./3q. values given bv these authors were converted to 3p./3Q 1 J i J to give the elements of the P matrix, using the equation [59] q. = 2ir(cu)./h) 2 Q.

PAGE 58

49 where h is Planck's constant in erg sec, CO, is the jth harmonic frequency in cm , and c is the speed of light in cm sec *. We used the harmonic frequencies, u) 1 = 3832.2, U), = 1648.5, and oj 3 = 3942.5 cm l , given by Benedict, Gailar and Plyler [52], in the conversion. Each component of the dipole moment vector belongs to a different irreducible representation of the C„ symmetry group. Hence there are only three non-zero elements in the P matrix for H„0: 9p /9Q 15 9p /9C> and 9p /9Q 3 , according to the axis system shown in Fig. 2-1. The values of these elements derived from the data of Clough, Beers, Klein and Rothman are 0.0479, -0.2344 and 0.2139 e u 2 , respectively. This P matrix was used with the L matrix in Table 2-7 and the 8 matrix in Table 2-3 to calculate PL 1 B, which was then added to the D matrix to obtain P according to Eq . 22. The PL *S matrix is composed of three juxtaposed tensors (the "vibrational tensors"): one for each of the atoms in the H„0 molecule. These tensors are given in the upper part of Table 2-11 for H and of Fig. 2-1. For all the tensors given in this table, the corresponding H tensor is the same as that for Hj , except that the off-diagonal elements of the H tensor have the opposite sign from that for the H tensor (according to the transformation shown in Eq. 30 below) . The rotational tensors, D , were calculated from Eq. 23 using the effective dipole moment for the ground vibrational state, p ° = 0.3861 e A, z taken from Clough, Beers, Klein and Rothman [57]. Since p ° = p ° = 0, x y the last row of ( (p_J ) is zero (see Eq . 24). Hence, only the x and y components of the diagonal inverse moment of inertia tensor, I , were used in Eq. 23. These values were calculated using the equilibrium position vectors with respect to the center of mass, r °, in Eq . 25. The values of the x and y coordinates, x and y , with respect to the

PAGE 59

50 center of mass are the same as those given in Table 2-1 with respect to an origin on the atom as shown in Fig. 2-1. Because the center of mass is displaced in the positive z direction away from the atom, the values, z ° = -0.0655689 A, z ° = z ° = 0.520313 £, were used in Eq . 25 rather 3 M l H 2 than the z ' of Table 2-1. The resulting principal moments of inertia are I = 1.76993 A 2 u and I = 0.614651 A 2 u, while the corresponding XX yy ' b elements of I are just the reciprocals of these values. The D tensors obtained using Eq. 23 and the parameters given above are shown in the middle part of Table 2-11. The total APT's, P A , were obtained as the sum of the vibrational and rotational tensors and are given in the lower part of the table. These experimental APT's for the monomer were used in Eqs. 1 and 2 together with the monomer and dimer APT's calculated theoretically to obtain scaled APT's for the dimer. In the remainder of this chapter we describe the methods we have used for obtaining the theoretical APT's and the scaled APT's for the dimer. 2.5. Calculation of Theoretical Polar Tensors The theoretical APT's for water monomer and dimer were calculated numerically from the theoretical values of the dipole moments for the equilibrium configuration and for a set of displaced configurations for each molecule. The calculated dipole moments were fitted to a linear function of the cartesian displacement coordinates, Aj . This treatment is consistent with the assumption of electrical harmonicity used to derive Eq . 15 and the linear relation between normal coordinates and the spacefixed cartesian coordinates given by Eq . 18. Thus,

PAGE 60

51 Table 2.11. Experimental atomic polar tensors for water monomer in the gas phase (P' ) . Vibrational and rotational tensors are also given. Units are e, where le = 1.602 x 10 -19 coulombs, Coordinate system is given in Fig. 2-1. Atom

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52 (27) 9 J t A J Aj . .A i,j = x,y,z A = 1,... ,N A j ij Aj .A .A, where (P ) . . is an element of the APT on atom A (see Eq. 20). Here p. is is the ith component of the total dipole moment of the molecule for the displaced configuration, and p.° is the value of that component for the equilibrium configuration. The displaced configurations we have used involve the displacement of a single atom, A, along the x, y or z direction of the molecular axis system. Thus, for each displaced configuration, only one of the terms in the double summation of Eq. 27 is non-zero. We have used, in general, two configurations for each atom and each direction--one involving a displacement in the positive x direction, for example, and one involving a displacement of the same magnitude in the negative x direction. The dipole moment components for the equilibrium and the two displaced configurations — p. , p . (+Aj ), and p.(-Aj ) — were used to determine the first, second or third column of the APT on atom A when j was x, y or z, respectively. Thus, Eq. 27 leads to (28) (P A ). ij P,(+Aj A ) P, 2|Aj .A, P,° " P,(-Aj A ) 2 Aj .A, .A i, .1 x , y , z The condition that the two terms in brackets should be nearly equal was used to check the calculations. To determine the APT on one of the atoms in each of the molecules H^O and (H„0)„, we have used the relation [1],

PAGE 62

I 53 (29) / P A = We have also used the fact that the two H atoms in the monomer and in the electron donor molecule of the dimer are symmetrically equivalent. Thus, in both cases, we have obtained P 2 from P l using the transformation [2] (30) P H 2 = TP H iTt where T is the matrix for reflection across the xz symmetry plane for H„0 ( T i i = T 33 = 1> T 22 = ~*' T i-i = °> i ^ J)? and tne Y z symmetry plane for (H 2 0) 2 (T u = -1, T 22 = T 33 = 1, T.. = 0, i t j). The equilibrium configurations about which the displacements were made for Yi^O and (HO) are given by the position vectors in Tables 2-1 and 2-4, respectively. As noted previously, the H„0 configuration was derived from experimental measurements, while that of (HO) was derived from the extended basis set calculation by Hankins, Moscowitz and Stillinger [53] . More recently, Dyke, Mack and Muenter have obtained structural parameters for water dimer produced by expanding the vapor through a supersonic nozzle jet [60] . They used electric resonance spectroscopy to determine microwave transition frequencies, and their results indicated an equilibrium structure similar to that given in Table 2-4 (with R Q _ o = 3.00 A), except that R = 2.98(±0.01) X and $ = 58 (±6)°. Although the experimental value of f; appears to differ significantly from the value of 40° which we have used, the wave function for the water dimer is not expected to be very sensitive to this parameter. A basis set similar to that used by Hankins, Moscowitz and Stillinger has been used by Popkie, Kistenmacher and Clementi [61] to calculate the binding energy of the dimer as a function of $, with all other structural parameters held constant at the same values as those

PAGE 63

54 consistent with Table 2-4. For values of = 15°, 30°, 45°, and 60°, binding energies of -4.49, -4.58, -4.58, and -4.39 kcal/mole, respectively, were obtained. Thus, the calculated binding energy of the dimer is not very sensitive to this parameter, and the calculated APT's would be expected to be relatively independent of the value of chosen for the equilibrium configuration. We used two different quantum mechanical methods [62] to calculate the dipole moments for each set of geometrical configurations: the ab initio self-consistent field (SCF) method using a 4-31G basis set and the approximate SCF method of complete neglect of differential overlap (CNDO) . The SCF method involves iteratively solving a set of simultaneous linear equations, the Roothaan equations, that arise from the Schrodinger equation when the ith molecular orbital, \b . , is expanded as a linear i combination of basis set orbitals, $ , on each atom, that is, M A ' , /_ ) c . <5> 1 *T~ *— U . 1 (31) ty. = / / c .4 — * — U 1 u A y A M A M A The Roothaan equations are solved for the c . , whose coefficients are integrals involving the basis set orbitals, <5> In the ab initio SCF method, the integrals are evaluated analytically from the functional form of the basis set. In the CNDO method many of these integrals are neglected, and combinations of others are parameterized — that is, adjusted to fit ab initio SCF results for orbital energies and eigenvectors of small molecules. Both the ab initio SCF method and the approximations in the CNDO method have been described in reference 62.

PAGE 64

55 The CNDO/2 parameterization of the integrals was used in this work, and this has been given in detail also in reference 62. The CNDO calculations were performed using QCPE program CNINDO [63] . Essentially, the only input to this program is composed of the atomic position vectors for each geometrical configuration. The 4-31G calculations were carried out using version 5 of the IBMOL program by Clementi and Mehl [64] . This program is designed to accept "contracted" (fixed linear combinations of) Gaussian functions as basis set orbitals. The input to the program includes the exponents for the "primitive" (uncontracted) Gaussian functions, the contraction coefficients, and the coefficients for the symmetrized orbitals, as well as the geometrical configuration data. The symmetrized orbitals we used were the same as the contracted orbitals, except for those involving Is orbitals on H, and H of both H 1 2 2 and (H 2 0) 2 (see Figs. 1-2 and 2-1). For these orbitals, the symmetrized Is A ls functions, x, and x, » were constructed, where X ls + = TT^is^) + # lB lH 2 )] N for A x symmetry of HO and A' symmetry of (HO); and X ls -=^ ls ( Hl ) -* ls (H a )] for B 2 symmetry of H„0 and A" symmetry of (H ? 0) N + and N~ are the corresponding normalization factors. For the Gaussian exponents and contraction coefficients we have used those given by Ditchfield, Hehre and Pople [65]. Their 4-31G contraction is the result of a systematic sequence of studies aimed at finding a computationally efficient basis set while maintaining an optimal degree

PAGE 65

56 of accuracy and flexibility. The basis set consists of a single contracted Gaussian orbital (CGO) for the core orbitals and two CGO's for each valence orbital — an "inner" CGO, having relatively large exponents, and an "outer" CGO with smaller exponents. The core CGO's consist of four primitive Gaussians, the inner CGO's of three primitive Gaussians, and the outer or diffuse CGO's of one Gaussian, each. We have used the optimized exponents and coefficients given in Tables I and II of reference 65, modified by the scaling parameters for the H„0 molecule recommended in Table III, according to Eq. 7 of that reference. The output from the IBMOL program included the total energy and the eigenvectors — that is, the coefficients c . in Eq . 31. The input to IBMOL and the eigenvector output were used in program POPULAN [66] to calculate the dipole moments and the Mulliken population analysis for each configuration. The analytic form for the expectation value of the dipole moment is related to the Mulliken gross atomic charge [67] on each of the atoms. We have made use of this relation to analyze further the intensity changes upon going from water monomer to dimer. Moreover, the dipole moments calculated by the CNINDO program involve an approximation, in addition to those used in calculating the eigenvectors, that is best indicated by considering the expectation value of the dipole moment

in terms of the Mulliken gross atomic charges. For closed shell molecules, such as water and water dimer,

is given in terms of the molecular orbitals, \p , , by -v Y -> Y r -+ : P> = L Z A R A " 2 fJ , l' i (l)r 1 -i..(l)dx 1 (32) A=l " " i=l

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57 In this equation, n is the number of orbitals (half the number of electrons), R is the position vector of nucleus A and r that of electron 1. Z is the charge of nucleus A. Using Eq. 31 for the molecular orbital and the definition of the density matrix element, D , which is Vb n D = 2 / c .c . VB 1-1 V V ' Eq. 32 becomes (33)

= I [Z A R A ffi D <" A l ? l'VJ A B y A v B M A B Here we have used the bracket notation for integrals: %i^iiv = i v i)? iV 1)dT i • A B The electronic position vector may be expressed as (34) ^ = \ + r A , ->where r is the position vector of electron 1 with respect to nucleus A. Substituting Eq. 34 into Eq. 33 and noting that R comes out of the integral and summations in the second term of Eq . 33, we have (35) <> = Z. [R A [Z A -Z^Z.D S ] ~r k A L A *— l — *— y,v„ U,v„ A B U, V D A B A B A B ZZZd v ] By A v B AB where S is the overlap integral W A V B

PAGE 67

58 S = The Mulliken gross atomic population on atom A, N(A) , is defined as [67] -SZ, (36) N(A) = Z_/Z. D S The gross atomic charge on atom A, Q , is [67; (37) Q A = Z A N(A) . Substituting Eqs. 36 and 37 into Eq. 35 gives Z_ [Q A R A -Z_Zz_ D „ ] J ~ LVl A"A J —' l — "y v„ '""A 1 "A 1 A B y A V B A B In the CNINDO program the dipole moments are approximated by neglecting terms for which B f A in the second term of Eq. 38. The only remaining non-zero terms are those in which p, = 2s. and v = 2p . Thus. A A A K A ' this is the "sp polarization" term defined by Pople and Segal [68] . The gross atomic charges are also approximated in the CNINDO program, in that S in Eq. 36 is replaced by 6 , where 6 is the Kronecker delta. M A B TB The dipole moments for the appropriate configurations from programs CNINDO and POPULAN were used in Eq . 28 to calculate the theoretical APT elements. 2.6 Scaled Polar Tensors and Intensities The theoretically calculated APT's for the water monomer were subtracted from those of the dimer in order to obtain the scaled APT's

PAGE 68

59 for water dimer, according to Eqs. 1 and 2. We note that the orientation of the H„0 monomer molecule with respect to the principal axis system in Fig. 2-1 differs from the orientation of each of the component H„0 molecules of the dimer with respect to the molecular coordinate system of the dimer in Fig. 2-2. The values of the APT's in a given coordinate system depend on the molecular orientation with respect to that coordinate system. Thus, since we have calculated the monomer APT's in the coordinate system of Fig. 2-1 and the dimer APT's in the coordinate system of Fig. 2-2, we have transformed the APT's for atom A in the monomer and dimer to a rotated axis system, before subtracting them. The axes of this rotated coordinate system have the same orientation with respect to the H„0 molecule which contains atom A in the monomer as in the dimer. A In general, a transformed APT, (P ) ' , in a rotated axis system is related to that in the original axis system, P , by [3] (39) (P A ) ' = RP A Rt The transformation matrix, R, has elements, R.., which are the cosines between the ith rotated axis and the j th original axis, i,j = x,y,z. The transformation matrix is orthogonal, that is, R = Rt. For the water monomer and dimer we have used two different rotated axis systems, one for the H atoms and one for the atoms. The "bond system" is particularly useful for considering APT's for H atoms. The bond system for H of the monomer is illustrated in Fig. 2-3 and can be uniquely defined for each H atom, H., as follows. The positive y axis is directed along the OH bond containing H. from to H.. The z axis is perpendicular to this y axis and lies in the plane of the H.O moiety which contains H.. The positive z axis is directed 2 l v between the two OH bonds of that H„0 moiety, as shown in Fig. 2-3. The

PAGE 69

60 r

PAGE 70

61 x axis is perpendicular to the plane of the II moiety which contains H , 2 i and the positive x direction is determined by the right-hand rule. For both the monomer and dimer, the APT for H in the bond system of H. is identical by symmetry with the APT for H in the bond system of H . Thus, for the water monomer there is only one distinct APT for the H atom in the bond system. The theoretically calculated APT's for the monomer and dimer H atoms in the bond system were used in Eq . 1. Similarly, the experimental APT for the monomer H atom in the bond system was used in Eq. 2. The theoretical and experimental APT's for the monomer H atom in the bond A A system were calculated as (P ) ' in Eq. 39, where P was the APT for Hj in the molecular coordinate system of the monomer shown in Fig. 2-1, and R was the transformation matrix from that coordinate system to the bond system of h^ (see Fig. 2-3). This transformation matrix is given in Table 2-12. Table 2-13 shows the experimental APT for the monomer H atom in the bond system, obtained by transforming the APT for Hj given in Table 2-11. The theoretical APT's for H. (i = 1,4,6) of the dimer in the bond system were also obtained as (P ) ' in Eq . 39. where P A was the theoretical APT for H. in the molecular coordinate system of the dimer in Fig. 2-2, and R was the transformation matrix from that coordinate system to the bond system of H.. These transformation matrices are given in Table 2-14 for i = 1,4,6. For the atoms a unique bond system cannot be defined since each atom belongs to more than one bond. Accordingly, we transformed the theoretical APT's for the two dimer atoms, 0., to a rotated axis system i J whose orientation, with respect to the H„0 moiety which contains , is the same as the orientation illustrated in Fig. 2-1 for the monomer molecule and axes. This rotated axis system is called the "molecular

PAGE 71

62 Table 2-12. Transformation matrix, R l , from the molecular coordinate system of the monomer, shown in Fig. 2-1, to the bond system of the monomer H atom, shown in Fig. 2-3. R H ! cos(37.74°) cos(142.26°) cos(52.26°) cos(37.74°) Table 2-13. Experimental polar tensor for the monomer H atom in the bond system. Units are e, where le = 1.602 x 10" coulombs . u P J (exptl. , monomer) 0.3295 0.0 0.0 0.0 0.1322 -0.0495 0.0 -0.0641 0.2468

PAGE 72

63 system of the monomer" (MSM) for the dimer atoms and is illustrated in Fig. 2-4 for . The transformed theoretical APT's for the dimer atoms were calcuA A lated as (P )' in Eq . 39, where P was the APT for 0. in the molecular coordinate system of the dimer shown in Fig. 2-2, and R was the transformation from that system to the MSM of 0., i = 3,5. These transformation matrices are given in Table 2-15 for and of the dimer. The theoretical APT for the monomer atom in the coordinate system of Fig. 2-1 was subtracted from each of these transformed APT's for the dimer atoms, according to Eq. 1. The resulting changes in the theoretical APT's from monomer to dimer, (AP )', in the MSM were then added to the experimental APT for the monomer atom given in Table 2-11, according to Eq. 2. The resulting scaled APT's for the dimer atoms were thus expressed in the MSM. The same procedure was used for the theoretical monomer and dimer APT's for the H atoms in the bond system and the experimental bond system APT given in Table 2-13 for the monomer H atom. The scaled APT's for the dimer and H atoms, in their corresponding rotated axis systems were then transformed back to the molecular coordinate system of the dimer shown in Fig. 2-2. The inverse of the transformation given in Eq. 39 was used, namely (40) P A = Rt(P A ) 'R where (P ) ' represents the scaled APT for the dimer in the rotated axis A system; P , the scaled APT in the molecular system of the dimer; and R the appropriate transformation matrix given in Tables 2-14 and 2-15 for each atom. The scaled APT for H 2 in the molecular system of the dimer was obtained from that for H , according to Eq . 30. All of the scaled

PAGE 73

64 U

PAGE 74

65 Table 2-14. Transformation matrix, R 1 , from the molecular coordinate system of the dimer, shown in Fig. 2-2, to the bond system of dimer H. atoms, i = 1,4,6.

PAGE 75

66 APT's for the dimer in that system were then juxtaposed to form the P — X matrix, according to Eq. 21. This scaled dimer P matrix was multiplied by the AL product matrix according to Eq. 17 to obtain the P matrix. This P matrix was then used to obtain the intensities, according to Eq . 17. Similarly, the (unsealed) theoretical APT's for the dimer, as well as those for the monomer, were juxtaposed to form the corresponding P matrices, which were then used to obtain the intensities. These theoretical APT's are presented and discussed in chapter A and the resulting intensities in chapter 3.

PAGE 76

CHAPTER 3 COMPARISON OF THEORETICAL INTENSITIES WITH EXPERIMENT 3.1 Simulation of Experimental and Theoretical Spectra In this chapter we compare the intensities calculated from the theoretical APT's with experimental spectra for water monomer and dimer in the N„ matrix. Experimental values have not been determined for the intensities of either species in the matrix. However, as mentioned in section 1.3.2, Tursi has given a spectrum of water isolated in N , in which the monomer and dimer bands were quite well resolved [31] . We have simulated this experimental spectrum using a sum of Lorentzian linefunctions. That is, the absorbance y. for each frequency point x. was calculated as S.A. (4i) y.OO = i Y ^ . j = i,n, 11 V 3 (x.W .) 2 + Aj 2 i J where the sum was taken over the three monomer bands, six dimer bands, and two trimer bands in Tursi's spectrum. In Eq. 41, S. represents the integrated absorbance of each band; A., the half bandwidth at half maximum; and co., the frequency of the band center. 1 The absorbance values y.(x.) depend primarily on the A. and S. i i 2 1 parameters of those bands with peak frequencies in the neighborhood of x. We have adjusted all three parameters of each band so that the simulated spectrum could be superimposed on the experimental spectrum. The final "fitted" parameters are given in Table 3-1. The "fitted experimental" 67

PAGE 77

68 Table 3-1. Lorentzian parameters for fitted experimental spectri

PAGE 78

69 a) E :"J A U.i'fj o. c .o h 5 S ft 6 H N C E 1.00 >

PAGE 79

70 spectrum calculated from these parameters according to Eq . 41 is shown in Fig. 3-1. As described in chapter 2, we have calculated quantum mechanical absorption coefficients, A for each band of water monomer and dimer. We have used these calculated absorption coefficients to obtain theoretical spectra according to Eq. 41 for comparison with experiment. For the experimental matrix spectrum, the amount of each absorbing species in the path of the irradiating beam has not been determined, and the absolute absorbance has not been given. Thus, we have calculated relative theoretical spectra on the same scale as that used for the spectrum in Fig. 3-1. This was done using relative integrated absorbances, S . (r) with respect to a fitted band, band r. One band for each species was chosen as the fitted band. The relative integrated absorbances for the remaining bands were calculated using the fitted absorbance S from r Table 3-1, according to (42) s.(r) = S A. r J J A^ where A and A are the calculated absorption coefficients for the ith J r j and rth bands. For purposes of comparison with the theoretical spectra, Eq. 41 was used to calculate separate fitted experimental spectra for water monomer and for the dimer, summing over the corresponding three and six bands, respectively. In calculating the theoretical spectra, the fitted A and a) parameters in Table 3-1 were used for all the bands.

PAGE 80

71 3.2 Water Monomer Intensities The infrared intensities from the 4-31G polar tensors are compared with those obtained from several other ab initio calculations in the upper part of Table 3-2. The latter intensities were taken from works by Pulay [69]; Smith, Jorgensen and Ohrn [70]; and Krohn and Kern [71]. Pulay presented dipole moment functions for two different basis sets — "Pulay I" and "Pulay II." The experimental L matrix in Table 2-7 was used to obtain the intensities from our 4-31G calculation. We used the same L matrix to calculate the intensities from the two dipole moment functions given by Pulay and also from that given by Smith, Jorgensen and Ohrn. We report the intensities given by Krohn and Kern directly. They included the effects of electrical anharmonicity on the dipole moment function, using terms up to fourth order in the normal coordinates. They also used an experimental normal coordinate transformation, which included the effects of mechanical anharmonicity through cubic and quartic force constants. The experimental intensities measured in the gas-phase by several workers are given in the lower part of Table 3-2. The spread in the measured values indicates that at least some of the discrepancy between experimental and calculated values may be attributed to uncertainties in the former. The experimental values from Clough, Beers, Klein and Rothman (underlined) are the most recent and probably the most reliable. These intensities correspond to the P matrix elements which we used to obtain the experimental APT's, as discussed in section 2.4. The signs reported for those P_ n matrix elements ap,ree with the signs predicted by nearly all the ab initio calculations represented in the upper part of Table 3-2. The only discrepancy is in the sign calculated from the

PAGE 81

72 smaller basis set used by Pulay, Pulay I, for the 9P /6Q element. That calculation predicted a negative sign for this element while the others predicted a positive sign, in agreement with the experimental sign choice. The calculated intensities in Table 3-2 are ordered according to the quality of the basis set, as evidenced by the calculated energy minimum, E Q . There appears to be an overall tendency for the calculated intensities of each band to increase as the basis set is improved. The upper limits of the calculated intensities are in excess of the best experimental values, particularly for intensities A x and A 2 . This may be attributed in part to the error in the Hartree-Fock approximation. It can be seen that the 4-31G intensities agree with the experimental values within a factor of 2.3. This level of agreement has been found for a number of other 4-31G calculated intensities [4,72,73]. Hence, we expect that the intensities for the water dimer can also be predicted from 4-31G calculations within a factor of 2.5 or better. Before discussing the dimer intensities, however, it is useful to compare the experimental intensities of the monomer isolated in the N„ matrix with those from the gas-phase spectrum. Thus we have calculated a spectrum using the experimental gas-phase intensities of HO by the method of section 3.1. Relative absorbances were obtained by using the experimental A. (underlined values) from Table 3-2 in Eq. 42. For reasons given below, V 3 was chosen as the fitted band (that is, r = 3 in Eq. 42). The resulting spectrum (B) is compared with the fitted experimental spectrum of water monomer in the N 9 matrix (A) in Fig. 3-2. These spectra indicate that the gas-phase intensities are quite different from those in the matrix. Relative to A , the gas-phase value of A is too small, while that of A is too large. 2

PAGE 82

73 Table 3-2. Theoretical and experimental intensities for gas-phase H„0. Units are km/mole (E in Hartrees) . Theoretical

PAGE 83

74 One possible source of this discrepancy is a change in the mechanical effects on the U molecule upon going from the gas-phase to the N matrix. According to the theory of infrared intensities, mechanical effects influence the intensities through the L matrix while electrical effects are reflected in the APT's. We have thus attempted to obtain an indication of the mechanical effects on the H„0 intensities as follows. A A Intensities were calculated from L(N ) and P (g) . The P (g) represent the experimental APT's determined from the gas-phase data as described in section 2.4 (see Table 2-11), while L(N ) denotes the L matrix determined from the force field given by Tursi and Nixon [30] for H^O in the N. matrix (see Table 2-8) . These intensities are compared in Table 3-3 with intensities calculated from L(g) and P (g) — where L(g) is the L matrix determined from the force field given by Cook, DeLucia and Helminger [51] for HO in the gas-phase (see Table 2-7). Since the P (g) were determined using L(g), the latter intensities are just the experimental gas-phase intensities. Table 3-3. Intensities calculated for HO from experimental gas-phase polar tensors using two different L matrices. Units are km/mole.

PAGE 84

Fig. 3-2. Comparison of fitted experimental spectrum of H„0 in the N„ matrix (A) with spectra calculated using experimental gasphase APT's. Spectrum (B) was obtained using L(g) and represents the experimental gas-phase intensities. Spectrum (C) was calculated using L(M ). All frequencies and band widths have been fitted to the experimental spectrum of HO in the N matrix. For the calculated spectra (B and C) , the intensity of V also has been fitted. Band assignments are indicated at the bottom of the spectrum (C) .

PAGE 85

76 (A) J U £ IN L. I i . ^/ J6.CC (B) : pe-;ue: "*> Vi ri (C)

PAGE 86

77 the force fields corresponding to ^(N„) and 1(g) , is that the values of the stretch-bend interaction force constant, F , have opposite signs (compare Tables 2-7 and 2-8) . This discrepancy probably accounts for much of the difference between the two sets of intensities in Table 3-3. We note that the same value of A was obtained using both force fields. This is because v 3 is the only vibration in the B symmetry group, and thus the corresponding L matrix element (L ) is independent of the force constants. We have calculated a spectrum from the intensities obtained using L(N„) by the method of section 3.1. This spectrum (C) is compared with that calculated from the gas-phase intensities (B) and with the fitted experimental spectrum of H„0 in the N matrix (A) in Fig. 3-2. For the calculated spectra (B and C) , V was chosen as the fitted band because A 3 does not depend on the force field. Figure 3-2 indicates that the intensities calculated using i-(N„) give substantially better agreement with the experimental intensities in the N„ matrix than do the gas-phase intensities. This result suggests that the mechanical changes in the H„0 molecule on going from the gas-phase to the N_ matrix environment (if well-represented by the L matrices in Tables 2-7 and 2-8) , may account for much of the discrepancy between the H„0 intensities in the two environments. 3.3. Water Dimer Intensities In this section we present and discuss the theoretical intensity calculations for the water dimer. The dimer bands and intensities are designated by the same notation used in section 1.3.1. That is, the vibrations corresponding to the electron donor water molecule of the dimer complex are labeled (ED) and those corresponding to the electron

PAGE 87

78 acceptor are labeled (EA). Analogous numbering to that for the monomer is used — that is, V 3 refers to the antisymmetric stretch, V, to the symmetric stretch, and V to the bend. We begin by comparing the intensities calculated from the theoretical APT's for the dimer with corresponding intensities for the monomer. In all cases we used the experimental L matrices for water monomer and dimer isolated in the N matrix (determined from the force fields given by Tursi and Nixon [30]) to obtain intensities from the APT's. Table 3-4 shows the intensities obtained from the ab initio APT's calculated for water monomer and dimer using the 4-31G basis set. Table 3-4. Intensities calculated from ab initio 4-31G APT's for water monomer and dimer. Units are km/mole.

PAGE 88

79 for the dimer using the 4-31G basis set. In this event some improvement in the predicted dimer intensities could be obtained by scaling these theoretical API's according to the procedure described in section 1.4.2. We have thus used the theoretical monomer and dimer APT's, as well as the experimental APT's for the monomer, in Eqs. 1 and 2 of that section, to calculate scaled dimer APT's. The intensities obtained from the scaled dimer APT's are compared in Table 3-5 with those obtained from the experimental APT's for the monomer. Both sets of intensities were calculated using the corresponding matrix isolated L matrix. Table 3-5. Intensities calculated from scaled dimer 4-31G APT's and experimental monomer polar tensors. Units are km/mole. [A 3 (ED)] [A 3 (EA)] [Aj(ED)] [Aj(EA)] [A 2 (EA) ] [A 2 (ED) ] Monomer Dimer 44.6 79.2 44.6 74.6 26.0 9.80 182.2 46.0 49.9 46.0 50.9 Again, the intensities of the bending vibrations are predicted to be about the same for the monomer and dimer. The predicted intensity of Vj (EA) is enhanced by a factor of nearly 19, while those of the other three stretching vibrations are 1.7 to 2.7 times greater than in the monomer. Calculated spectra were obtained as described in section 3.1 using the intensities, A., shown in Tables 3-4 and 3-5. These spectra are compared in Fig. 3-3 with the fitted experimental dimer spectrum. For the calculated spectra, V 2 (EA) was chosen as the fitted band, since the intensities of the bending modes are expected to be least affected by H-bond formation [56, see also Tables 3-4 and 3-5]. The other bend

PAGE 89

Fig. 3-3. Comparison of fitted experimental spectrum of water dimer isolated in the N2 matrix (A) with spectra calculated using unsealed (B) and scaled (C) 4-31G polar tensors. For the calculated spectra (B and C) , the intensity of V 2 (EA) has heen fitted to the experimental spectrum of water dimer in the N2 matrix. All frequencies and bandwidths have also been fitted. Band assignments are indicated at the bottom of spectrum (C) .

PAGE 90

81 0.00 0.50 "ij.oo o.5o Id. go 0.50 1.00 a> 'ij.uO 50 HB30F.E;HI:ce b.OO 0.50 HBSuF.BUNCE O.UU 0.50 nBSOF-EiflNCE 1.00

PAGE 91

V (ED) strongly overlaps with the corresponding monomer band in the experimental spectrum (see Fig. 3-1). The value of the absorbance used to fit this band is therefore less certain. Figure 3-3 shows that in the spectrum calculated using the ab initio APT's (B) , the intensities of the antisymmetric stretches are much smaller than those in the fitted experimental spectrum (A). For the spectrum obtained using scaled APT's (C) , the overall agreement is better except for V (EA) , whose intensity is predicted to be approximately twice that in the fitted experimental spectrum. The normal coordinate for V,(EA), contains the largest contribution from the H-bonded OH stretch. This vibration therefore corresponds to the "H-bond band" whose intensity was discussed in section 1.2. This overestimation of the H-bond band intensity is probably related to the fact that the dimer stabilization energy calculated with the 4-31G basis set is too large — 8.2 kcal/mole [47], as compared to the most accurate value of 5.6 kcal/mole [43]. In general, small basis set molecular orbital calculations have been found to predict this extra stabilization, and this phenomenon has been investigated for a number of H-bonded complexes [86-90; 19, pp. 63-65]. The overestimation of the stabilization energy (and presumably of the H-bond band intensity) is a consequence of the small basis set which inadequately represents the orbitals on each molecule of the complex. In the molecular orbital method, the total energy of the complex is optimized. The additional basis functions on one molecule are then allowed to compensate for the limited number of basis functions on the other molecule, in order to achieve optimization. Thus, intermolecular orbital mixing occurs to a greater extent in small basis set calculations

PAGE 92

83 than in those using large basis sets, and "basis set superposition error" results. An additional measure of the H-bond strength is given by the intermolecular separation, the 0-0 distance in the case of water dimer. A stronger H-bond interaction results in a shorter 0-0 distance. The minimum energy configuration calculated with the 4-31G basis set for the o water dimer occurs for an 0-0 distance of 2.83 A [47]. This compares with the experimental [60] and most accurate theoretical [43] value of 2.98 A. Because both the H-bond band intensity and the 0-0 distance are correlated with the dimer stabilization energy, and the latter is affected by basis set superposition error in the calculations, we decided to investigate the dimer intensities as a function of 0-0 distance. Accordingly we have calculated an entire set of scaled APT's for the water dimer at each of three different 0-0 distances, R =3.0, 2.8 and 2.6 A, using the relatively economical CND0 method. These calculations also allow us to see whether the scaling procedure which we have used for the dimer APT's can override some of the shortcomings of the approximate CND0 method. Figure 3-4 shows the calculated spectra obtained using unsealed APT's from the CND0 calculation. The intensities predicted for the stretches are very small compared to the values calculated for the bends, with the exception of A l (FA) . These results are similar to those obtained with the ab initio 4-31G calculated APT's [spectrum (B) in Fig. 3-3]. However, the situation is more exaggerated with the CND0 calculation, and Aj (EA) is predicted to be significant only for the

PAGE 93

Fig. 3-4. Comparison of spectra of the water dimer calculated from unsealed CNDO polar tensors at different 0-0 distances. Spectrum (A) corresponds to Rq_q = 3.0 A; spectrum (B) to R _o = 2.8 A; and spectrum (C) to Rq-o = 2.6 A. The intensity of V 2 (EA) and the frequencies and bandwidths have been fitted to the experimental spectrum of water dimer in the N2 matrix. Band assignments are indicated at the bottom of spectrum (C) .

PAGE 94

85 0.00 0.50 SI o.oo 0.50 0.00 0.50 o LU, u.OO IfiSfiE 0.50 D H hi L t LU l UJq g. > 00 0.50 HtJ50RB'flNCE U.jjO 0.50 rtB50h;clflNCE

PAGE 95

86 strongest H-bonded water dimer, that with R = 2.6 X. Table 3-6 shows the corresponding values predicted for the intensities; they are indeed unreasonable. Figure 3-5 shows the calculated spectra obtained using sealed APT's from the CNDO calculation. First we note that the relative intensities are a great deal better, thus indicating the value of scaling the APT's for obtaining quantitative intensity estimates from approximate calculations. Secondly, Fig. 3-5 shows that the intensity predicted for V (EA) is indeed a very sensitive function of the 0-0 distance, and in fact o doubles for every 0.2 A decrease in R . These results confirm that the intensity of this H-bond band [relative to A 2 (EA)] is amazingly sensitive to R n _ n and, hence, the strength of the H-bond. Although we have not quantitatively investigated the effect of basis set superposition error on the 4-31G calculated intensity of this band, we can infer qualitatively from these results that the effect would be large. Accordingly, the intensity of the H-bond band in the experimental spectrum obtained by Tursi and Nixon [see fitted spectrum (A) in Fig. 3-3] is not inconsistent with the value that might be predicted for the linear dimer from a large basis set calculation. The numerical values of the intensities corresponding to the calculated spectra in Fig. 3-5 are given in Table 3-7. In addition to the dramatic change for the intensity of the H-bond band, V (EA) , the intensity of V 3 (EA) is also predicted to increase substantially as R decreases. The increase in A 3 (EA) can be noticed in the calculated spectra shown in Fig. 3-5, particularly for R„_„ = 2.6 A. The sensitivity of the V (EA) vibration to the H-bond strength probably results from the

PAGE 96

8 7 Table 3-6. Dimer intensities calculated from unsealed CNDO APT's at different 0-0 distances. Units are km/mole. Vo 1

PAGE 97

Fig. 3-5. Comparison of spectra of the water dimer calculated from scaled CNDO polar tensors at different 0-0 distances. Spectrum (A) corresponds to R n _ n = 3.0 A; spectrum (B) to Rq_q = 2.8 A; and spectrum (C) , to R()-0 = 2.6 A. The intensity of v 2 (EA) and the frequencies and bandwidths have been fitted to the experimental spectrum of water dimer in the N2 matrix. Band assignments are indicated at the bottom of spectrum (C) .

PAGE 98

89 LI.UU 0.5L u.oo H' 0.50 0. GO 0.50 00 S T, > 00 0.50 RBSCREJRNC e •J , 0.00 0.50 RB'SORE.'RNCE 0.0 . 5 hBSORBRNCE loo

PAGE 99

90 intramolecular coupling discussed in section 1.2. Basically this coupling is described by the L matrix which, in addition to the P matrix (composed of the APT's) determines the P matrix and, hence, the intensities (see Eqs. 16 and 17). Accordingly, we wish to consider the extent to which an alternate choice for the L matrix can affect the calculated intensities. One such alternate L matrix is that determined from the scaled 4-31G F matrix. This F matrix was obtained from the ab initio 4-31G force fields calculated by Curtiss and Pople for the water monomer and linear dimer, using the scaling procedure discussed in section 1.4.1 (see Appendix). Both the scaled F matrix and the resulting L matrix have been given in Table 2-10. Table 3-8 shows the intensities calculated using the scaled 4-31G L matrix. These intensities are compared in the table with those obtained using the experimental L matrix which was derived from the F matrix given by Tursi and Nixon for the water dimer in the N„ matrix. The latter L matrix has been used for all the intensity calculations presented thus far and has been given in Table 2-9. Table 3-8. Dimer intensities calculated from scaled 4-31G APT's usin^ two different L matrices. Units are km/mole.

PAGE 100

91 relative to the intensities calculated with the experimental L. However, the major effect of the alternate L matrix on the intensities in Table 3-8 is found in the intensities of the bending vibrations, V 2 (EA) and V 2 (ED) , and of the symmetric stretch of the electron donor molecule, V, (ED) . As discussed in section 2.4, the major differences between the two force fields are the neglect of intramolecular stretch-bend coupling constants in the scaled 4-31G force field and the neglect of intermolecular coupling constants in the experimental force field. Comparison of Tables 2-9 and 2-10 shows that the L matrix elements for stretch-bend interaction included in the experimental force field are larger than those for intermolecular interaction included in the scaled 4-31G force field. This is also what we intuitively expect based on the relative importance of intraand intermolecular forces. We can infer from the results in Table 3-8 that the effect of neglecting stretch-bend interaction in the force field results in an increase in the bending intensities of the dimer and a decrease in the intensity Aj (EA) . In summary, it has been shown that the theoretical intensities predicted for the water dimer are sensitive both to the H-bond strength and the force field used in the calculation. Within these limitations, we feel that the intensities calculated using scaled APT's from the 4-31G calculation [corresponding to spectrum (C) in Fig. 3-3] agree well enough with the fitted experimental spectrum [spectrum (A) in Fig. 3-3] to be worthy of some consideration in the interpretation of the experimental dimer spectrum. Various interpretations of the spectrum have been discussed in section 1.3.2. We mentioned that the band assigned to V (ED) by Tursi and Nixon [30] was regarded by Barletta [33] as an overtone or

PAGE 101

92 combination band on the basis that it was too weak to be a fundamental vibration. However, our calculated intensity for the v (ED) fundamental shows that it should be weak. It was also mentioned that Luck [36] assigned the band at 3698 cm 1 to the trimer. However, the relative intensity we have predicted for V 3 (EA), to which this band was assigned by Tursi and Nixon, agrees quite well with that in the fitted experimental spectrum. Moreover, Luck reassigned the bands, assigned to V (ED) , Vj(ED), and V 2 (ED) by Tursi and Nixon, to the cyclic dimer. We feel that the agreement of our intensity predictions for these three bands with the fitted experimental spectrum is more than fortuitous. Finally, according to the interpretation by Ayers and Pullin [40] of the profuse spectrum of water in Ar, an assignment was made in which all bands in the frequency region corresponding to V (ED) were attributed to vibration-rotation transitions of the monomer. Yet the intensity of this vibration is predicted to be larger than that of any other vibration with the exception of the H-bond band, V, (EA) . Taking into account all the considerations discussed in this section, the predicted intensity pattern assuming the linear dimer structure agrees satisfactorily with the fitted experimental spectrum. These predictions thus lend support to the assignment made by Tursi and Nixon.

PAGE 102

CHAPTER 4 POLAR TENSOR ANALYSIS 4.1. Total Polar Tensors The dipole derivatives used to calculate the intensities give us information about the dynamics of the electronic charge distributions as the molecule vibrates. The APT's are particularly useful, since they describe the redistribution of charge in the molecule as each atom in the molecule moves. The elements of the APT's are also independent of the atomic masses [1]. Thus, we are interested in investigating the changes in the APT's that are predicted to occur upon going from water monomer to dimer, according to the 4-31G calculations. As discussed in section 2.6 in connection with scaling the APT's, the monomer and dimer APT's must be evaluated in the same coordinate system in order for any comparison to be meaningful. We have used the bond system described in section 2.6 for comparing the H atom APT's (see Fig. 2-3). This axis system is uniquely suited for examining the APT's and how they are affected by dimer formation, in terms of the bonding characteristics of the molecule. The 4-31G calculated APT's in the bond system for the three nonequivalent H atoms of the dimer (Hj, h\ and H g in Fig. 1-2) are compared in Table 4-1 witti that for the monomer H atom. As discussed in section 2.4, the dimer APT's in the bond system are the same for Hj and H 2 , and both bond system APT's for the H atoms in the monomer are also the same. The 4-31G calculated APT for 93

PAGE 103

9 4 Table 4-1. Ab initio 4-31G polar tensors for H atoms of die water monomer (M) and dimer (D) , in the bond system. Units are e, where le = 1.602 x 10~ 19 coulombs. P H (M)

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95 the monomer H atom in Table 4-1 can be compared with the corresponding experimental APT in Table 2-13. Table 4-1 shows that the values calculated for most of the APT elements of the dimer do not change significantly from their corresponding values in the monomer. The major changes are predicted to occur in the P elements (second row, second column) for each H atom in the dimer. This element is the derivative of the dipole moment component along the OH bond with respect to displacement along that bond. The fact that changes from monomer to dimer in these APT's are restricted to the P yy elements shows that the effect of the H-bond interaction on the H atom APT is really quite simple and acts directly through eacli bond. The 4-31G calculation indicates an increase of 0.06 electrons (e) for the P t element of the electron donor atom, H , from the corresponding value in the monomer. For the free atom, H , of the electron acceptor molecule, a decrease in the P element of 0.04 e is predicted to occur upon yy v complex formation. The magnitude of these changes is dependent to some extent on the 4-31G basis set. However, the increase predicted in the P element for Hj and the decrease in this element for H fi are probably significant. The most dramatic change in the APT's calculated for the dimer H atoms occurs in the P element of the APT on H, , the H bonded H atom. yy k This element is rather small in the monomer, but shows an amazing increase of 0.32 e upon H-bond formation. Again, the exact magnitude of the change predicted for this element would depend on the basis set used in the calculation, but the fact that it is large and positive probably has general validity.

PAGE 105

96 The change predicted for the P element (third row, second column) of the H^ atom APT should also be mentioned. In the monomer this element — the derivative of the dipole moment component perpendicular to the bond (in-plane) with respect to displacement along the bond — is calculated to be a very small negative number. It decreases by 0.04 e in the dimer thus becoming substantial, but this change is overshadowed by the large change predicted for P . Thus, the derivative of the yy dipole moment vector with respect to displacement of H^ along the OH bond is essentially directed along that bond. Similar systematic changes might be expected for the atom APT's upon going from monomer to dimer. However, these changes depend on the coordinate system in which the APT's are evaluated. When we applied the scaling procedure, we transformed the APT's for the dimer atoms from the molecular coordinate system of the dimer (MSD) to that of the monomer (MSM) , as discussed in section 2.6. However, in this system (MSM) no systematic changes could be observed in the monomer and dimer APT's. Accordingly, we investigated these APT's in the MSD. In order to do this we had to transform the monomer atom APT's from the MSM to the MSD. The MSD is shown in Fig. 2-4 and is oriented differently with respect to each HO moiety in the dimer. Thus the transformation from the MSM to the MSD for is not the same as the corresponding transformation for 5 . We therefore obtained two transformed monomer APT's, one corresponding to each dimer atom. Each transformed monomer APT, P 1 (MSD), was obtained from the monomer APT in the MSM, P°(MSM), according to: P°i(MSD) = (R°i)tP°(MSM) R°i i = 3,5,

PAGE 106

97 Table 4-2. Ah initio 4-31G polar tensors for atoms of the water monomer (M) and dimer(D), in the molecular system of the dimer. Units are e, where le = 1.602 x 10~ 19 coulombs. P° 3 (M)

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98 The transformation matrices, R i have been given in Table 2-15. The transformed APT's for the monomer atom are compared in Table 4-2 with the corresponding dimer APT's, both calculated using the 4-31G basis set. Table 4-2 shows that the APT's calculated for both atoms are almost exactly the same in the monomer and dimer, except for the P element. yy In the MSD this element is the derivative of the dipole component along the linear H-bond with respect to displacement along the H-bond. In the unperturbed monomer, the corresponding element is predicted to be large and negative. In the H-bonded complex P is even more negative, yy decreasing by 0.15 e for 3 of the electron donor and by 0.21 e for 5 of the electron acceptor. Again, the exact magnitudes of these changes depend on the basis set used for the calculation, but the large decrease in the P element upon going from water monomer to dimer is significant. The fact that the remaining elements are almost exactly the same in the monomer and dimer is probably also significant. It is evident that the predicted changes in the APT's resulting from the H-bond interaction in the water dimer are orientationally welldefined. The largest effects are in the diagonal terms along the appropriate bond for each atom. For the H atoms the appropriate bond is the OH bond to which each H atom belongs. For the atoms it is the linear H-bond of the dimer molecule. By far the dominant changes are predicted for the atoms involved in the H-bond: 3 , H^ and 5 . As mentioned previously, we are particularly interested in the intensitv of the H-bond band. Onlv the chances in the APT's for H and affect this intensity. The relative contribution to the intensity from each APT element is related to the appropriate AL matrix element (Eqs. 16 and 17). The AL matrix elements for the H-bond band, corresponding to

PAGE 108

99 the P elements of H and , have opposite sign. Thus the large changes from monomer to dimer, shown in Tables 4-1 and 4-2 for these two P elements, reinforce each other in their effect upon the intensity of the H-bond band — since they also have opposite sign. However, the magnitude of the AL matrix element for H is much lark ger than that for , because the A matrix element is inversely proportional to the mass of the atom (see Eq. 19). Therefore the H-bond band intensity is dominated by the large change in the P element of H . yy * The prediction of strong orientational effects in the total APT's indicates that the influence of the H-bond on these quantities is localized in certain atomic displacements. This can be understood in terms of the linear arrangement of atoms -In the H-bond. But the chemical basis for the large magnitude of these effects is not obvious. We investigate this problem below by examining the chemical and physical factors that contribute to each APT element. 4 .2. Quantum Mechanical A n alys i s 4.2.1. Discussion of Mo del We have analyzed the water monomer and dimer APT's according to the charge, charge flux, overlap (CCFO) model. The term "charge flux" comes from Decius who developed the equilibrium charge-charge flux (ECCF) model for analyzing the physical factors that determine the intensities [11]. Decius and Mast applied this model to the intensities of a number of molecules [12], however it differs quite basically from the CCFO model used here. The latter model is essentially the same as that presented by King and Mast and applied to CH F and CF, [9,10]. The terminology and

PAGE 109

100 interpretation used by Kinp, and Mast are somewhat different from ours, and these discrepancies, as well as the more substantial ones between the CCFO and ECCF models, are discussed below. The CCFO model uses the exact quantum mechanical definition of the dipole moment within the LCAO-MO method, given by Eq. 38. That equation can be rewritten as (A3) = Zj Va + Aj ^ (A ' B) • A AA A,B where (44) F(A ,B) = / D f $ (l)r A $ (l)dT. <— i U V I u A V 1 II \) M A RJ m A R V V B AB The first term in Eq . 43 gives the contribution from the scalar charges, , that can be assigned to a single atom. These are the Mulliken gross atonic charges defined in Eqs. 36 and 37. The second term in Eq. 43 gives the contribution that is intrinsically associated with a pair of atoms. Such a contribution arises from the vector properties of the dipole moment. In the ECCF model, on the other hand, a set of effective charges, X,., is defined such that A (45) A A Thus the effective charge, C , , contains a contribution from the Q. in A A Eq. 43 and a contribution that is related to the F(A,B) in that equation. C Now, the APT on atom C, P , is given by the transpose of the gradient of the dipole moment vector with respect to the displacement of

PAGE 110

101 atom C, (46) P C = (7 J) 1 Thus, application of Eq. 46 to Eq. 43 gives (47a) P C = 2L [V r (Q A R.)]+ /_, [V F(A,B)]+ A L A A A,B C (47b) Z C =Q C E+ ^ R A (V C Q A )+ 2., [V C F(A,B)]A " * A , B c c c c (47c) P_ = _P (charge) + P (charge flux) + P (overlap) Similarly, application of Eq. 46 to Eq . 45 gives (48a) P C = ^ [V C (C A R A )]" A £ + Z, VVa)' (48b) P C -« C E+ Z, V V cV + A (48c) P C = P C (chargc)' + P C (charge flux) In Eq. 47b and 48b, E is the identity tensor and we have used the fact that 7(xy) = x7y + yVx.* Equation 47 gives the CCFO model for the AFT's, * See note p. 1C3.

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102 while the ECCF model is given by Eq. 48. * It should be pointed out that the overlap tensor, given by the third term in Eq. 47b is not neglected in Eq. 48. The latter equation is entirely equivalent to Eq . 47 for the C C total APT. However, the P (charge) and P (charge flux)' in Eq . 48c for the ECCF model are not the same as the corresponding unprimed quantities in Eq . 47c for the CCFO model. Rather both terms in the r ECCF model contain a contribution from P (overlap) in Eq. 47c. This is because of the fundamental difference in the definition of the charges, Q A an d C A The Q can be assigned to a single atom, according to the quantum mechanical expression for the APT (and the dipole moment), while the Z cannot. Comparing the two models, we see that there are only two terms for each APT in the ECCF model as compared to three for the CCFO model. As a result the parameters in the ECCF model can be determined directly from experiment for many molecules (though not for the water dimer). These parameters can also be obtained from theoretically calculated APT's. The CCFO parameters can be calculated directly from the quantum mechanical wavefunction. They cannot be obtained from experimental intensities. The smaller number of parameters in the ECCF model suggests that the numerical values obtained for each parameter may be more reliable. However, numerical significance was not a problem in our application of the CCFO model to the changes that take place from water monomer to dimer. * The ECCF model was actually presented by Decius for the dipole derivatives in the symmetry coordinate representation [11]. Decius and Mast have shown how the constraints of charge conservation, rotational invariance, and symmetry are applied for various types of molecules [12] Using these equations, the relationship between the symmetry coordinate ECCF model and its APT analog, described here, is straightforward.

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103 We have found that certain terms change by a large amount , while in all other terms the changes are very small. Moreover, the separation of the APT into the three terms of the CCFO model is quantum mechanically exact. Thus the physical significance of the three terms is welldefined. The general significance of the three terms has been discussed by King and Mast [9,10]. We wish to elaborate on this discussion by pointing out what the three terms have in common and the distinguishing characteristics of each term. The third term is also treated differently by King and Mast, with a resulting difference in interpretation. Each term in Eq. 47b depends on an "electrical" quantity involving charge and a "mechanical" quantity involving distance. In the charge tensor, given by the first term of Eq. 47b, the mechanical quantity is implicitly contained in the E tensor.* In both the charge and charge flux tensors, the electrical and mechanical quantities are separable into independent factors. However, in the overlap tensor, given by the third term of Eq. 47b, the electrical quantity involves the D ,, , $ , A B A and $ in Eq. 44. According to the latter equation, the electrical B and mechanical quantities in this tensor are not separable. This distinction between the charge and charge flux tensors, on the one hand, and the overlap tensor on the other, can be stated more specifically as follows. The charge and charge flux tensors depend on the electron density determined from the electronic wavefunction. But the dependence of the overlap tensor on the electronic wavefunction cannot be expressed in terms of the electron density function. * The E tensor arises from the fact that V„R, = E 5, r where 6 ._ is the ,„,— ,, C A — AC AC Kronecker delta.

PAGE 113

104 Each term in Eq . 47b also involves the gradient, and this is a reflection of dynamic effects in the APT. The charge tensor involves dynamic effects in an implicit manner only.* This tensor depends explicitly on just the static electrical factor. In the charge flux tensor the dynamic factor is electrical, while the static factor is mechanical. Both the electrical and mechanical factors are dynamic in the overlap tensor. The important distinction is that the charge flux and overlap tensors involve dynamic electrical effects, while the charge tensor does not. Finally, the first, second and third terms in Eq . 47b for the CCFO model represent progressively higher orders of interatomic interaction. The contribution of the charge tensor to the total APT on atom C, given by the first term in Eq. 47b, depends only on that atom. The charge flux tensor on atom C, given by the second term in Eq. 47b, includes interactions among all the atoms of the molecule, because it involves a sum of contributions from these atoms. Each term in this sum depends in general on a pair of atoms, A and C. The terms in this sum are thus independent diatomic contributions to the total APT. The overlap tensor, given by the third term in Eq . 47b, involves the highest order of interatomic interactions. This tensor is a double sum over pairs of atoms, each term in the sum depending in general on 3 atoms, A, B and C. When A = B in Eq . 44, F(A,B) represents the contribution of the sp (or spd) hybridization effects to the dipole moment. Thus F(A,A) represents diatomic effects, even though it depends on a single atom, because these hybridization effects are determined by the * See note p. 103.

PAGE 114

105 atom(s) bonded to atom A. When A / B, F(A,B) represents the dipole moment overlap effects that cannot be expressed in terms of the electron density, and is also diatomic. The general contribution to the overlap tensor, V C F(A,B), is therefore a tviatomic contribution to the total APT. The explicit triatomic dependence of the overlap tensor was eliminated in the model of King and Mast. They approximated the atomic orbitals on atom A by expanding them in terms of the orbitals on atom B. Actually this approximation was applied to Eq. 43 for the dipole moment. However, only the second term in that equation is intrinsically diatomic. Thus when the gradient operator is applied to Eq . 43, the charge and charge flux tensors, resulting from the first term in that equation, are exactly the same as those obtained by King and Mast. They obtained a different functional form for the third term in Eq. 47b — our overlap tensor. They interpreted their third term as an effect due to "interference phenomena resulting from the superposition of wavefunctions" [10, p. 9]. However, our calculation of the three terms in the CCFO model, for the water and water dimer APT's, is numerically equivalent to the three terms calculated by King for CH F and CF, [10]. This is because the third term in both models can be obtained by subtraction of the charge and charge flux tensors from the total APT on the left side of Eq . 47c. We have also calculated our overlap tensor directly, using the functional form of F(A,B) given in Eq . 44 and the output from the dipole moment n overlap population analysis option of program POPULAN. This P (overlap) agrees, to at least two decimal places, with that obtained by difference Q from the total P , for all the atoms of the water monomer and dimer.

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106 In summary, the three tensors in the CCFO model have distinguishing physical characteristics. These distinctive properties do not apply C C to the P (charge)' and P (charge flux)' tensors in the ECCF model of Decius. This is because each of the tensors in the ECCF model contains Q a contribution from P (overlap) in the CCFO model, according to the discussion following Eq. 48. Accordingly, P (charge)' in the ECCF model C r has the properties of P (overlap), as well as those of P (charge) in the C CCFO model. Similarly, P (charge flux)' also has the properties of Q P^ (overlap). Thus no general distinctions can be made in the physical significance of the two ECCF tensors. 4.2.2. Results and Implications for the Intensity Enhancement of the H-bond Band The charge, charge flux and overlap tensors calculated with the 4-31G basis set, for each of the independent H atoms of the water dimer are compared with those for water monomer in Table 4-3. These tensors refer to the bond system for each H atom, as discussed in section 4.1. These data are really quite remarkable in that, among all the tensor elements presented, there is only one which shows a significant change from water monomer to dimer. This element is P (charge flux) of H , yy ° i» ' the H-bonded H atom, which shows a large increase of 0.27 e. A similar situation is encountered in the CCFO analysis for the atoms. The dimer tensors on 3 and 5 , in the molecular coordinate system of the dimer (MSD) , are compared in Table 4-4 with the corresponding transformations of the monomer tensors (see discussion in section 4.1.). Both atoms show a significant change, but again, only for the p (charge flux) element. This element decreases by 0.14 e for and by 0.12 e for .

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107 Table 4-3. Charge, charge flux, and overlap contributions to ab initio 4-31G APT's for H atoms of the water monomer (N) and dimer (D) in the bond system. Units are e, where le = 1.602 X 10 -19 coulombs.

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108 Table 4-4. Charge, charge flux, and overlap contributions to ah initio 4-3 10 APT's for atoms of the water monomer (M) and dimer (D) in the molecular system of the dimer. Units are e, where le = 1.602 x 10~ 19 coulombs.

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109 It was established in section 4.1, that the dominant changes from water monomer to dimer in the total polar tensors occur in the P yy element on the , H and atoms. We now see that in each case the 3 4 5 changes are sharply limited to the charge flux tensor in the CCFO analysis. The charge flux accounts for 93% of the change in the total P element yy for 3 , 84% for H u , and 58% for 5 . Table 4-4 shows that the gross atomic charge on decreases by 0.06 e upon dimer formation. This change therefore accounts for 29% of the total P change for . However, it is also evident that the charge on y y 5 this atom is very large in both the monomer and dimer, and it changes by only 7% of the monomer value upon dimer formation. This change is not significant. Thus the major changes in the dimer APT's can be attributed unequivocably to the charge flux tensor. This result is of great consequence for understanding the characteristic intensity enhancement of the H-bond band. This enhancement has been discussed frequently in the literature. However, attention has been focused on the general nature of the H-bond interaction. The intensity enhancement has been regarded only as an additional property to be investigated and correlated with other characteristic properties of H-bonded complexes. The traditional controversy associated with the intensity enhancement therefore involves the relative importance of charge transfer and polarization effects [23, and references cited therein]. These two effects contribute to the second order energy of molecular complexes, obtained by the method of perturbation theory [91]. Formalisms have been developed [92] for resolving these contributions in whole complex MO calculations, such as that presented here. However, there are

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110 discrepancies in the definition of the two energy contributions, between the MO formalisms and perturbation theory [92]. In both methods, the distinction between charge-transfer and polarization effects cannot be made unambiguously, except at large intermolecular distances [19, p. 45]. Investigation of the intensity enhancement in terms of these effects has been limited primarily to empirical methods [93-100]. Accordingly efforts to account for the magnitude of the intensity enhancement, and the various contributions to it, have met with only qualitative success [23, and references cited therein]. Our approach to this problem differs significantly from previous approaches. We regard the intensity enhancement as an unusual phenomenon in the general theory of infrared intensities. Thus we have applied a model for investigation of intensities to an H-bonded complex. We have found that the magnitude of the enhancement in the water dimer can be quantitatively attributed to the increase in the charge flux tensor on H, , from that in the monomer. We interpret this result in terms of the distinguishing characteristics of the charge, charge flux and overlap tensors, discussed in the previous section. In doing this, we would like to address the possible implications of this result for other H-bonded complexes besides the water dimer. We restrict our attention in this regard, to the relatively weak interaction involved in non-ionic binary complexes. First we consider the fact that there is a negligible change in Q _P (charge) on each atom in the water dimer from that in the monomer. There is evidence that the charge tensors remain largely unaffected by the H-bond interaction, for H-bonded complexes in general. Del Bene has calculated the Mulliken gross atomic charges — and therefore the charge

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Ill tensors—with the ST0-3G basis set for 41 H-bonded dimer complexes [101103]. On the whole, these charges are nearly the same as the corresponding charges in the isolated electron acceptor molecules (monomers), which were also calculated. As discussed in section 4.1, the H-bond band intensity is dominated by P b where H, is the H-bonded H atom (H yy D v •» in the water dimer). Thus, we have calculated the difference between the monomer and dimer charges on H,, for the complexes reported by Del Bene. For all 41 complexes the charge differences are less than 0.05 e. Moreover the relative charge differences, AQ„ (rel.) — that is the b ratio of the charge difference to the monomer charge — is 27% or less for the 41 complexes. This change is insignificant, especially considering that AQ H (rel.) decreases with increasing basis set size.* For the water b dimer, for example, AQ H (rel.) is 19% with the ST0-3G basis set [101], as b compared to 13% with 4-31G and 5% with the large basis set used by Hankins . Moscowitz and Stillinger [53], The Aq„ (rel.) obtained from the studies of Del Bene can be used to estimate the contribution of the charge tensor to the intensity enhancement for the general H-bonded complex. Assuming that the H-bond band H intensity is completely determined by P b ,** the relative increase in u p b (charge) could only enhance the intensity by a maximum factor of 1.6. Since the intensity of the H-bond band in the dimer is generally larger * The absolute charge differences are not sensitive to the basis set. See references [104, p. 4994] and [19, p. 73]. Compare also the 6-31G* charge differences calculated by Dill, Allen, Topp and Pople [105] with those calculated by Kollman and Allen [104,106] and by Del Bene [102]. However, the monomer charges increase with basis set size so that AQjj (rel.) decreases, b ** Actually P b accounts for 87% of the intensity in our water dimer calculation. The remaining intensity is due to a sum of several small terras from other contributions.

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112 by a factor of 10 or 20 than the corresponding monomer band, the contribution from the charge tensor is not significant. Nevertheless the characteristics of the charge tensor are useful in interpreting the intensity enhancement, because they tell us what is not involved in this phenomenon. As discussed in the previous section, one characteristic of the charge tensor on H, is that it depends only on the H, atom. It is generally recognized that the intensity enhancement cannot be accounted for by the effects due only to R, . The charge tensor is also characterized by a static electrical factor. Both the dynamic and mechanical effects are only implicitly involved in the charge tensor. The importance of the dynamic electrical effect for the intensity enhancement, and its explicit dependence on the vibrational coordinate, has been emphasized by Pimentel and McClellan [24], Mulliken and Person [25], Zgierski [26] and Witkowski [27]. Pimentel and McClellan presented a pictorial model for the intensification which showed a polarization of charge induced by the displacement of li . Mulliken and Person explained the enhancement in terms of vibronic coupling between the "dative" and "no-bond" electronic wavefunctions. The dative, or charge-transfer wavef unction represents an excited electronic state, in which charge is donated from the nonbonding orbital on the electron donor molecule to an antibonding orbital on the electron acceptor molecule. The two states are vibronically coupled, in that their mixing coefficients in the total wavefunction are a function of the vibrational coordinate. Zgierski and Witkowski have also attributed the intensity enhancement to vibronic coupling. However, they considered an excited electronic state involving the nonbonding orbitals on both the electron donor and acceptor molecules.

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113 These three models may differ with regard to the nature of the H-bond, but they agree in their treatment of the intensity enhancement. According to these models the important factor in this enhancement is the variation of the electronic wavefunction at a frequency corresponding to that of the H-bond band. In other words, the intensification arises from the increase in the dynamic electrical effect associated with the charge flux tensor, as described in the previous section. However, the overlap tensor also involves a dynamic electrical effect. If the dominant change from monomer to dimer had been found in TJ p b (overlap) — or if both the charge flux and overlap tensors had shown substantial changes — this would also be consistent with the models described above. But P b (overlap) is the same as the corresponding U monomer value, while P b (charge flux) increases by a large amount. This result pinpoints the origin of the intensity enhancement more precisely, according to the distinctions between the charge flux and overlap tensors made in section 4.2.1. This enhancement is due to the variation of the electron density, which can be resolved into independent diatomic contributions. That part of the variation of the dipole moment, which cannot be expressed in terms of electron density, and which depends on hybridization and other triatomic effects, does not change from water monomer to dimer. Therefore it does not contribute to the intensity enhancement. The explicit dependence of the intensity enhancement on the electron density, as such, appears to be contained in the polarization model of Pimentel and McClellan. However, this explicit dependence should be emphasized. Moreover, the determining role played by the charge flux as opposed to the overlap tensor in the intensity enhancement allows us to understand the effect of interatomic interactions on this enhancement.

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114 In considering interatomic interactions in H-bonded complexes, we can distinguish between intramolecular and intermolecular effects. Intramolecular interactions involve atoms in the same water molecule in the water dimer complex, for example, while intermolecular interactions involve one atom in the electron donor molecule and one in the electron acceptor molecule. In the atomic polar tensors of H-bonded complexes, interatomic interactions are dominated by intramolecular effects. These effects in the isolated monomers are perturbed by the intermolecular interactions due to complex formation. The overlap tensor involves the contribution of higher order interatomic effects to the total polar tensor, as discussed in section 4.2.1. Thus the overlap tensor might be expected to contain only those intramolecular effects on the intensities that are not affected by the H-bond interaction. Accordingly, it might be assumed that the overlap tensor does not contribute to the intensity enhancement, in the general H-bonded complex. This assumption should be tested by calculating charge, charge flux and overlap tensors for other H-bonded complexes and their corresponding monomers. This calculation for the water monomer and dimer has shown that the interatomic interactions, which are perturbed by the effect of the H-bond, are isolated in the charge flux tensor. We can gain some insight into the nature of these interatomic interactions by investigating the individual contributions to this tensor. Another reason for considering these contributions is that the polarization model of Pimentel and McClellan was more specific than we have stated thus far. They actually attributed the intensity enhancement to a polarization of the base (electron donor molecule), induced by the displacement of H, .

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115 We examine onlv the contributions to the yy element on H since b the intensity of the H-bond band is determined primarily by this element. The corresponding charge flux contribution from atom A is given by y A (9Q A /3y H ), according to Eq . 47b. We note that y the equilibrium b " y coordinate of atom A, depends on the origin of the coordinate system. We have used the center of mass as the only unique origin of the molecule. IT The contributions to P b (charge flux) are given in Table 4-5 for water monomer and dimer. We have also included the corresponding contributions for the HCN-HF dimer for comparison. Curtiss and Pople reported the (9 /3y ) values obtained from their 4-31G calculation for A H b this dimer and for the HF monomer [107]. We have used their equilibrium geometrical parameters to determine the y values and obtain the contributions in Table 4-5. As indicated in the table, the contributions from electron donor (ED) atoms in the dimer should be compared with zero in the monomer. That is, the charges in the isolated ED molecule will not change when H, in the electron acceptor (EA) is displaced. The contribution from R is compared with the corresponding contribution from the H atom in the monomer which is displaced. It is interesting that the change in this monatomic contribution, for both complexes, actually detracts from the increase from monomer to dimer which is found for the total charge flux element — and which enhances the intensity in the dimer. The contributions from the remaining atoms in Table 4-5 give the charge flux on one atom with respect to displacement of another. In this sense, these contributions represent interatomic effects in the total atomic polar tensor. The contributions from 3 and N of the ED molecules

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116 u Table 4-5. Individual contributions to P b (charge flux), in tiie bond system of H b , for (H^OK and HCN-HF. Partial sums over the atoms of the ED and EA molecules are given. Units are e. y A (9Q A /9yn b ) for (H 2 0) 2

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117 also detract from the total increase in the charge flux element from monomer to dimer. The major increase occurs on 5 and F of the EA molecules. This increase gives the perturbation of the intramolecular interaction of and F with K, due to the intermolecular effect of complex formation. Smaller but substantial contributions come from the ED atoms not involved in the H-bond: Hj and H 2 of the water dimer, and H" and C of the HCN-HF dimer. These contributions represent purely intermolecular interactions. The individual contributions in Table 4-5 are subject to limited numerical accuracy. However, the sums over the atoms of the ED molecule and over those of the EA molecule are probably reliable, at least qualitatively. Table 4-5 shows that for the water dimer the total contribuTT tion to P 7 b (charge flux) from the EA is larger than that from the ED, but both are substantial. For the HCN-HF dimer, on the other hand, the total contribution from the EA is slightly smaller than that from the ED. But again, both are substantial. The 4-31G calculations for these two dimers thus indicate that the polarization of charge, which is responsible for the intensity enhancement, cannot be localized on the ED molecule. Rather, the perturbation of the intramolecular polarization effects by the formation of the H-bond must be considered in addition to the intermolecular effects. In all our discussion of the charge, charge flux, and overlap tensors, we have restricted our attention to those on H . The atomic polar tensor on this atom plays a determining role in the intensity enhancement, and this is our main interest. However, Table 4-4 shows that similar changes take place in the contributions to the atom polar

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118 tensors. That is, the charge and overlap tensors do not change significantly from monomer to dimer; but the charge flux tensors change by a large amount. We have also calculated the individual contributions to the charge flux tensor on $ . These results parallel those in Table 4-5 for the charge flux tensor on H, . b In other words, the results of the CCFO analysis for the atoms are similar in detail to those for H . Therefore, the important factors in the intensity enhancement also determine the effect of the H-bond on the overall vibrational redistribution of charge in the water dimer. These important factors include the dynamic electrical effect, which can be expressed in terms of the electron density function. With regard to interatomic interactions, diatomic effects are more important than monatomic or triatomic effects. And finally, both intermolecular effects and the perturbation of intramolecular effects are important for the diatomic interactions. It is hoped that these results provide some insight into the nature of the H-bond.

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APPENDIX We have determined a set of scaled force fields for the linear water diraer in order to obtain frequencies for comparison with experiment. As discussed in section 1.4.1, the scaling procedure we have used depends on theoretically calculated force constants for the water monomer and linear dimer. We have applied this scaling procedure to the 4-31G force fields calculated by Curtiss and Pople [47], as well as force fields we have calculated by the method of CNDO, for these two molecules. Curtiss and Pople obtained the linear dimer force field with respect to the theoretical 4-31G o equilibrium geometry, for which R = 2.832 A. The CNDO force fields were calculated with respect to three different equilibrium geometries with R = 3.0 A, 2.8 A and 2.6 A. The theoretical monomer and dimer force constants from the 4-31G calculation of Curtiss and Pople are given in Table A-l, while those from the CNDO calculations appear in Table A-2. The dimer force constants are given in terms of the symmetry coordinates defined in Table 2-5. These dimer symmetry coordinates correspond to differently numbered symmetry coordinates in the monomer or to monomer internal coordinates, both of which were defined in Table 2-2. Each monomer internal (F .) or symmetrv (F .) force constant is written on the same line in Tables A-l and A-2 as the dimer force constant to which it corresponds. The corresponding monomer and dimer theoretical force constants were subtracted to give the change 119

PAGE 129

120 Table A-l. A.b initio 4-31G force constants for water monomer and linear dimer calculated by Curtiss and Pople [47]. Units

PAGE 130

121 Table A-2. Force constants for water monomer and linear dimer calculated by the CNDO method. Units

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in each theoretical force constant, Af (th.): k (A-l) Af (th.) = f.(th., dimer) f, (th., monomer), k k k ' ' The Af (th.) were then added to the experimental force constant, f (exptl., monomer), with respect to the same monomer internal or symmetry coordinates to give the scaled dimer force constant, f (scaled, k dimer) : (A-2) f (scaled, dimer) = f, (exptl., monomer) + Af, (th.). k k k We are mainly interested in obtaining frequencies for comparison with the experimental spectrum of the water dimer obtained by Tursi and Nixon in the N matrix [30], Accordingly we have used the f (exptl., monomer) which they determined by fitting the observed monomer frequencies in the N matrix. These force constants have been given in Table 2-8, with respect to the monomer symmetry coordinates. The relevant experimental force constants with respect to the monomer internal coordinates are F M = 7.516 md A , F12 = -0.181 md A , and Fj 3 = -0.2986 md rad" 1 . The scaled F matrix derived from the 4-31G force fields in Table A-l, according to equations (A-l) and (A-2), has been given in Table 2-10. Table A-3 shows the scaled F matrices derived from the CND0 force fields in Table A-2. Each of these scaled F_ matrices was used (together with the linear dimer G matrix determined from the 8 matrix in Table 2-6) to set up the eigenvalue problem, Eq . 14. The equation was solved for the eigenvector matrix, J., and the diagonal frequency parameter matrix, A. The L matrix for the scaled 4-31G force field was used to obtain a set of diner intensities from the scaled 4-31G calculated APT's, as discussed in section 3.3. This L matrix has been given in Table 2-10.

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123 Table A-3. Scaled CNDO force constants for water monomer and linear dimer. r 2 2

PAGE 133

124 Here we present the results for the _A matrices. The fundamental absorption frequencies, oj. in cm , are related to the A matrix elements by A = (4tt 2 cV 2 /N a ) x 10~ 5 ii 1 A for the units we have used, where c and N, have been defined in A connection with Eq. 15. The predicted frequencies from the scaled forced fields are compared in Table A-4 with a number of experimentally observed values. The oj. in the table correspond to the dimer normal coordinate numbering, while the V, in brackets give the analogous monomer normal coordinate numbers for the vibrations of the electron acceptor (EA) and the electron donor (ED) molecules of the dimer (see Fig. 1-2) . The errors for each predicted frequency are given in parentheses with respect to the frequencies observed by Tursi and Nixon. For the 4-31G calculation the errors in the bending vibrations, V 2 (ED) and V 2 (EA), are quite large. However, this is primarily due to the neglect of stretch-bend interaction (see discussion in sections 2.4 and 3.3), rather than to the errors in the calculated force constants. The remaining frequencies agree quite well considering that the force field is harmonic while the observed frequencies are not. Some indication of the effect of this discrepancy is given by the "best fit" to the observed frequencies of Tursi and Nixon, shown in parentheses in Table A-4. For the CNDO calculations it can be seen that the frequencies most sensitive to the 0-0 distance, R n _ n » are the electron acceptor stretches, Vi(EA) and V 3 (EA). The intensities of these bands were also found to be most sensitive to R , as discussed in section 3.3.

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125

PAGE 135

126 Although these frequency calculations are approximate, they lend support to the assignment of the matrix-isolated spectra in terms of the linear water dimer. In particular, the frequency of the H-bond band, Vj (EA) , is not at variance with the predicted frequencies.

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REFERENCES 1. W. B. Person and J. H. Newton, J. Chem. Phys . 61, 1040 (1974). 2. J. H. Newton and W. B. Person, J. Chem. Phys. 64, 3036 (1976). 3. J. H. Newton, Ph.D. Dissertation, University of Florida (1974). 4. W. B. Person and J. D. Rogers, European Congress on Molecular Spectroscopy, Frankfurt, West Germany (September, 1979). J. D. Rogers and W. B. Person, Symposium on Molecular Spectroscopy, Columbus, Ohio (June, 1979); American Chemical Society National Meeting, Washington, D.C. (September, 1979). 5. J. H. Newton and W. B. Person, J. Chem. Phys. 68, 2799 (1978); J. Phys. Chem. 82^ 226 (1978). J. H. Newton, R. A. Levine and W, B. Person, J. Chem. Phys. 67 , 3282 (1977). B. J. Krohn, W. B. Person and J. Overend , J. Chem. Phys. 65s 969 (1976); 67, 5091 (1977). W. B. Person and J. Overend, J. Chem. Phys. j36, 1442 (1977). 6. J. F. Biarge, J. Herranz and J. Morcillo, An. R. Soc. Esp. Fis. Ouim. A57 , 81 (1961). 7. J. Morcillo, L. J. Zamorano and J. M. V. Heredia, Spectrochim. Acta 22, 1969 (1966). M. De la Lastra, A. Medina and J. Morcillo, An. R. Soc. Esp. Fis. Quim. B63, 1079 (1967) . J. Morcillo, J. F. Biarge, J. M. V. Heredia and A. Medina, J. Mol. Struct. 3, 77 (1969). 8. W. T. King, G. B. Mast and P. P. Blanchette, J. Chem. Phys. 56, 4440 (1972). G. B. Mast and W. T. King, J. Phys. Chem. 80, 2004 (1976). 9. W. T. King and G. B. Mast, J. Phys. Chem. 80, 2521 (1976). 10. W. T. King in "Vibrational Intensities," W. B. Person and G. Zerbi, Eds. (Elsevier, Amsterdam, to be published, 1980), chap. 6. 11. J. C. Decius, J. Mol. Spectrosc. 5_7, 348 (1975). 12. J. C. Decius and G. B. Mast, J. Mol. Spectrosc. 7£, 294 (1978). G. B. Mast and J. C. Decius, J. Mol. Spectrosc. 79, 158 (1980). 127

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128 13. M. Gussoni and S. Abbate, J. Chem. Phys. 65, 3439 (1976); J. Mol. Spectrosc. 62, 53 (1976). S. Abbate and M. Gussoni, J. Mol. Spectrosc. 66, 1 (1977). S. Abbate, M. Gussoni G. Masetti and G. Zerbi, J. Chem. Phys. 67^ 1519 (1977). 14. A. J. van Straten and W. M. A. Smit , J. Mol. Spectrosc. 5_6, 484 (1975); 62, 297 (1976); 65, 202 (1977). 15. L. S. Mayants and B. S. Averbukh, J. Mol. Spectrosc. 22_, 197 (1967). 16. S. K. Samvelyan, V. T. Aleksenyan and B. V. Lokshin,J. Mol. Spectrosc. 48, 47 (1973). 17. L. M. Sverdlov, Opt. Spek. 39, 268 (1975). L. M. Sverdlov, M. A. Kovner and E. P. Krainov, "Vibrational Spectra of Polyatomic Molecules" (Wiley, New York, 1974). 18. K. Morokuma and L. Pederson, J. Chem. Phys. 48, 3275 (1968). 19. P. Schuster in "The Hydrogen Bond," P. Schuster, G. Zundel and C. Sandorfy, Eds. (North-Holland Publishing, New York, 1976), Vol. 1, chap. 2. 20. G. C. Pimentel and A. L. McClellan, "The Hydrogen Bond" (Freeman, San Francisco, 1960). Many books have been published on the subject of H-bonding. This one covers the earlier work quite thoroughly, and the following reference covers the more recent work. 21. P. Schuster, G. Zundel and C. Sandorfy, Eds., "The Hydrogen Bond," 3 Vols. (North-Holland Publishing, New York, 1976). 22. D. Hadzi and S. Bratos, ibid.. Vol. 2, chap. 12. 23. H. Ratajczak, W. J. Orville-Thomas and C. N. R. Rao, Chem. Phys. 17, 197 (1976). 24. G. C. Pimentel and A. L. McClellan in "The Hydrogen Bond" (Freeman, San Francisco, 1960), p. 248. 25. R. S. Mulliken and W. B. Person, Ann. Rev. Phys. Chem. 13, 107 (1962) R. S. Mulliken and W. B. Person, "Molecular Complexes" (Wiley, New York, 1969). 26. M. Z. Zgierski, Acta Phys. Polon. A5_L . 877 (1977). 27. A. Witkowski, Mol. Phys. 29^ 1441 (1975). 28. H. E. Hallam in "The Hydrogen Bond," P. Schuster, G. Zundel and C. Sandorfy, Eds. (North-Holland Publishing, New York, 1976), Vol. 3, chap. 22. H. E. Hallam, Ed. "Vibrational Spectroscopy of Trapped Species" (Wiley, New York, 1973).

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129 29. M. Van Thiel, E. D. Becker and G. C. Pimentel, J. Chem. Phys . 27 486 (1957). — 30. A. J. Tursi and E. R. Nixon, J. Chem. Phys. _52, 1521 (1970). 31. A. J. Tursi, Ph.D. Dissertation, University of Pennsylvania (1970). 32. L. Fredin, B. Nelander and G. Ribbegard, J. Chem. Phys. 66, 4065, 4073 (1977). — 33. R. E. Barletta, Ph.D. Dissertation, Brown University (1977). 34. P. V. Huong and J.-C. Cornut , J. Chim. Phys. 72, 534 (1975). 35. B. Mann, T. Neikes, E. Schmidt and W. A. P. Luck, Ber. Bunsenges. Phys. Chem. 78., 1236 (1974). 36. W. A. P. Luck in "The Hydrogen Bond." P.Schuster, G. Zundel and C. Sandorfy, Eds. (North-Holland Publishing, New York, 1976), Vol. 2, chap. 11. 37. R. L. Redington and D. E. Milligan, J. Chem. Phys. 37, 2162 (1962); 39, 1276 (1963). 38. H. P. Hopkins, R. F. Curl, Jr. and K. S. Pitzer, J. Chem. Phys. 48, 2959 (1968). 39. G. P. Avers and A. D. E. Pullin, Chem, Phys. Lett. _29, 609 (1974); Ber. Bunsenges. Phys. Chem. 82, 62 (1978). 40. G. P. Avers and A. D. E. Pullin, Spectrochim. Acta 32A. 1629, 1641, 1689, 1695 (1976). 41. R. C. Kerns and L. C. Allen, J. Am. Chem. Soc. JJ30, 6587 (1978). 42. E. Whalley in "The Hydrogen Bond," P. Schuster, G. Zundel and C. Sandorfy, Eds. (North-Holland Publishing, New York, 1976), Vol. 3, chap. 29 (see section 29.3). 43. 0. Matsuoka, E. Clementi and M. Yoshimine, J. Chem. Phys. 64, 1351 (1976). 44. H. A. Gebbie, W. J. Burroughs, J. Chamberlin, J. E. Harris and H. G. Jones, Nature 221, 143 (1969). 45. R. W. Bolander, J. L. Kassner and J. T. Zung, J. Chem. Phys. 50, 4402 (1969). 46. D. Maillard, A. Schriver, J. P. Perchard , C. Girardet and D. Robert, J. Chem. Phys. 67, 3917 (1977). 47. L. A. Curtiss and J. A. Pople, J. Mol. Spectrosc. 55, 1 (1975).

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130 48. R. Janoschek in "The Hydrogen Bond," P. Scuster, G. Zundel and C. Sandorfy, Eds. (North-Holland Publishing, New York, 1976), Vol. 1, chap. 3. 49. E. B. Wilson, Jr., J. C. Decius and P. C. Cross, "Molecular Vibrations" (McGraw-Hill, New York, 1955). 50. J. Overend in "Infrared Spectroscopy and Molecular Structure," M. Davies, Ed. (Elsevier, Amsterdam, 1963), chap. 10. 51. R. L. Cook, F. C. De Lucia and P. Helminger, J. Mol. Spectrosc. 53, 62 (1974). 52. W. S. Benedict, N. Gailar and E. K. Plyler, J. Chem. Phvs . 24, 1139 (1956). 53. D. Hankins , J. W. Moscowitz and F. H. Stillinger, J. Chem. Phys . 53, 4544 (1970). 54. G. H. F. Diercksen, Theor. Chim. Acta 21, 335 (1971). 55. J. Overend, "Program Manual of the Molecular Spectroscopy Laboratory," University of Minnesota, Vol. 2 (August, 1966). We are grateful to J. Overend of the University of Minnesota for making these programs available to us. 56. G. E. Sanchez, Ph.D. Dissertation, University of Florida (1971). 57. S. A. Clough, Y. Beers, G. P. Klein and L. S. Rothman, J. Chem. Phys. 5_9, 2254 (1973). 58. R. A. McClatchey, W. S. Benedict, S.A. Clough, D. E. Burch, R. F. Calfee, K. Fox, L. S. Rothman and J. S. Garing, Air Force Cambridge Research Laboratories Report TR-73-0096 (January, 1973). J. S. Garing and R. A. McClatchey, Appl. Opt. 1_2, 2545 (1973). 59. G. Herzberg, "Infrared and Raman Spectra of Polyatomic Molecules" (Van Nostrand Reinhold, New York, 1945), p. 377. 60. T. R. Dyke, K. M. Mack and J. S. Muenter, J. Chem. Phys, 66, 498 (1977) 61. H.Popkie, H.Kistenmacher and E.Clementi, J. Chem. Phys. _5_9, 1325 (1973), 62. J. A. Pople and D. L. Beveridge, "Approximate Molecular Orbital Theory" (McGraw-Hill, New York, 1970). 63. T. A. Dobosh, Program QCTE 141, from Quantum Chemistry Program Exchange, Indiana University, Bloomington , Indiana. 64. E. Clementi and J. W. Mehl, IBM Research Laboratory Reports RJ 853 (May, 1971) RJ 883 (June, 1971) and RJ 889 (July, 1971). We are grateful to J. R. Sabin of the University of Florida Quantum Theory Project for making this program available to us.

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131 65. R. Ditchfield, W. J. Hehre and J. A. Pople, J. Chem. Phys..54, 724 (1971). 66. R. N. Kortzeborn, IBM Research Laboratory Report RJ 577 (June, 1969). We are grateful to J. R. Sabin of the University of Florida Quantum Theory Project for making this program available to us. 67. R. S. Mulliken, J. Chem. Phys . 23, 1833, 1841, 2338, 2343 (1955); 36, 326 (1962). 68. J. A. Pople and G. A. Segal, J. Chem. Phys. 43, S136 (1965). 69. P. Pulay, Mol. Phys. 18, 473 (1970). 70. J. A. Smith, P.Jorgensen and Y. Ohrn, J. Chem. Phys. 62, 1285(1975). 71. B. J. Krohn and C. W. Kern, J. Chem. Phys. 69, 5310 (1978). 72. C. E. Blom and C. Altona, Mol. Phys. 34, 177 (1977). C. E. Blom, C. Altona and A. Oksam, Mol. Phys. 34, 557 (1977). 73. P. Pulay, C. Fogarasi, F. Pang and J. E. Boggs , J. Am. Chem. Soc. 101 , 2550 (1979). 74. J. Smith, M.S. Thesis, University of Florida (1973). 75. R. A. Toth, J. Quant. Spectrosc. Radiat. Transfer 1_3, 1127 (1973). 76. R. Goldstein, J. Quant. Spectrosc. Radiat. Transfer U_, 343 (1964). 77. G. J. Maclay, J. Chem. Phys. 43, 185 (1965). 78. J. M. Flaud and C. Camy-Peyret, J. Mol. Spectrosc. 55, 278 (1975). 79. C. B. Ludwig, C. C. Ferriso and C. N. Abeyta, J. Quant. Spectrosc. Radiat. Transfer 5, 281 (1965). 80. A. Goldman and U. P. Oppenheim, Appl. Opt. _5» 107 3 (1966). 81. J. H. Jaffe and V. S. Benedict, J. Quant. Spectrosc. Radiat. Transfer 3, 87 (1963). 82. Y. Ben Aryeh , J. Quant. Spectrosc. Radiat. Transfer J_, 211 (1967). 83. M. A. Hirshfeld, J. H. Jnffe and G. Ross, J. Quant. Spectrosc. Radiat. Transfer 6, 311 (1966). 84. B. Krakow and A. R. Healy, J. Opt. Soc. Am. 5_9, 1490 (1969). 85. C. W. von Rosenberg, N. H. Pratt and K. N. C. Bray, J. Quant. Spectrosc. Radiat. Transfer, 10, 1155 (1970). W. Kolos, Theor. Chim. Acta 51, 219 (1979)

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132 87. A. Meunier, B.Levy and G. Berthier, Theor. Chim. Acta 29., 49 (1973) 88. A. Johansson, P. Kollman and S. Rothenberg, Theor. Chim. Acta 29^ 167 (1973). 89. S. Iwata and K. Morokuma, J. Am. Chem. Soc. 95, 7563 (1973). 90. D. Kocjan, J. Roller and A. Azman, J. Mol. Struct. 34, 145 (1976). 91. J. N. Murrell, M. Randic and D. R. Williams, Proc. Roy. Soc. A284 , 566(1965). 92. K. Morokuma, J. Chem. Phys . _5_5, 1236 (1971). K. Kitaura and K. Morokuma, Int. J. Quantum Chem. 10, 325 (1976). 93. H. Tsubomura, J. Chem. Phys. 24, 927 (1956). 94. P. C. McKinney and G. M. Barrow, J. Chem. Phys. 31, 294 (1959). 95. K. Szczepaniak and A. Tramer, J. Phys. Chem. Tl, 3035 (1967). 96. S. Bratos, Adv. Quantum Chem. 2, 209 (1967). 97. M. Dreyfus and A. Pullman, Theor. Chim. Acta 1_8, 388 (1970). 98. P. A. Kollman and L. C. Allen, Chem. Rev. 72, 283 (1972). 99. W. B. Person in "Spectroscopy and Structure of Molecular Complexes," J. Yarwood, Ed. (Plenum, New York, 1973), chap. 1. 100. H. Ratajczak and W. J. Orville-Thomas , J. Mol. Struct. 1_9, 237 (1973) 101. J. E. Del Bene, J. Chem. Phys. 5_5, 4633 (1971). 102. J. E. Del Bene J. Am. Chem. Soc. 9_5, 5460 (1973). 103. J. E. Del Bene, J. Chem. Phvs . 56, 4923 (1972); 5]_, 1899 (1972); 5J3, 3139 (1973); 62, 1314, 1961 (1975); 60, 4666 (1975); Chem. Phys. 15, 463 (1976). J. E. Del Bene and F. T. Marchese, J. Chem. Phys. 58, 926 (1973). J. E. Del Bene and W. L. Kochenour, J. Am. Chem. Soc. 98^, 2041(1976). J. E. Del Bene and I. Cohen, J. Am. Chem. Soc. 100, 5285 (1978). 104. P. A. Kollman and L. C. Allen, J. Am. Chem. Soc. 92, 4991 (1971). 105. J. D. Dill, L. C. Allen, W. C. Topp and J. A. Pople, J. Am. Chem. Soc. 97, 7220 (1975). 106. P. A. Kollman and L. C. Allen, J. Chem. Phvs. 52, 5085 (1970). 107. L. A. Curtiss and J. A. Pople, J. Mol. Spectrosc. 48, 413 (1973).

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BIOGRAPHICAL SKETCH Barbara Ann Zilles was born in Arcadia, Florida. She received the bachelor's degree in chemistry from Mount St. Mary's College in Los Angeles, California and the master's degree in teaching from the University of Florida. She has been with Old Dominion University in Norfolk, Virginia, as research assistant professor and with NASA Langley Research Center in Hampton, Virginia, as research contractor. She is receiving the Ph. D. degree in physical chemistry from the University of Florida. 133

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Willis B. Person, Chairman Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Martin T. Vala Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. 7 "1 / / / 1 h / C c'e M. Luis Muga Professor of Chemistrv I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Thomas L. Bailey Professor of Physic's

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I certify that I have read this studv and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jnaies K. Brooks Professor of Mathematics This dissertation was submitted to the Graduate Faculty of the Department of Chemistry in the College of Liberal Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June, 1980 Dean, Graduate School

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UNIVERSITY OF FLORIDA 3 1262 08553 3833


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