INFRARED INTENSITIES OF WATER AND WATER DIMER
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
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
ACKNOWLEDGMENT S------- --------------- --- ---- -------------------
LIST OF TABLES------------------------------------------
LIST OF FIGURES-----------------------------------------
ABSTRACT---- --------------------------------- -------------------
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-------------------------- --------------
2. DESCRIPTION OF CALCULAT IONS---------------------------------
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
BIOGRAPHICAL SKETCH---------------------------------------- 133
LIST OF TABLES
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
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-
2-8. Experimental F and L matrices for water monomer in the N2
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
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
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
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
Barbara Ann Zilles
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
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.
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
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 .
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
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 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
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 . 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 . 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
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
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
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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-
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
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].
Fig. 1-3. Thie centrosymmetric cyclic water dimer. The center of inversion is at the
midpoint between 0l and 02.
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)
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
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 .
Huong and Cornut [34).
SBarletta [33}. Band VIII is resolved into four bands in this work.
Luck . See also Mann, Neikes, Schmidt and Luck [351.
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
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  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
. 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
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 . 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  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 .
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 .
1.4. Theoretical Calculation of Vibrational
Properties of the Water Dimer
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 . 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.
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
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
dipole derivatives with respect to the normal coordinates of the first
molecule . 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.
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.
DESCRIPTION OF CALCULATIONS
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.
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
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
The diagonal matrix of inverse
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
F = UFUt
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
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
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  has given thle relation between thle inteGrated molar absorp-
tion coefficient of the ith band, A. and thle derivative of the dipole
moment vector, p, with respect to the _ith normal coordinate, Q *:
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
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 ,
(16) A 2N
ap. 3 = x,y,z
Qjk aQk' k = i,...,3N-6
is the matrix of dipole derivatives with respect to the normal
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
The A and A matrices are related by
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
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
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
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
(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
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
(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 
(24) (( )) = 0 vz y
-v 0 v
v -v 0
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
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
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 . 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.
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 .
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
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.
Table 2-3. The 6 matrix for water monomer. Row 2 has units of rad A.
All other rows are unitless.
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
Fig. 2-2. Molecular coordinate system of the linear water dimer. Bond lengths and angles
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
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
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- . It was obtained
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.
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 . 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 , 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
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).
F2 7.3205 md 1
Fs -0.4088 md rad I
Fss 0.6927 md I rad 1
F 7.6545 md -1A
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 .
-0.2664 rad X-u-
1.505 rad u-
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.
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
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
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-
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
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 , 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
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
(27) p = p +jA = x,y,z
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
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 ,
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 
(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 . Mlore recently, Dyke, Mack and Hluenter have obtained
structural parameters for water dimer produced by expanding the vapor
through a supersonic nozzle jet . 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
We used two different quantum mechanical methods  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
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,
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.
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 .
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 . 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
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
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
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
The analytic form for the expectation value of the dipole moment is
related to the Mullikenn grass atomic charge  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
= ZRA 2 g_ (i() 1i1)
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
= 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
=i rA A B DA BS A
B pA g A B
where S~~~ is the overlap integral:
S~AB =iA VB
The Mlulliken gross atomic population on atom A, N(A), is defined
(36) N'(A) = L D S
B pA B BA
The gross atomic charge on atom A, QA,' is 
(37) QA = ZA N(A)
Substituting Eqs. 36 and 37 into Eq. 35 gives
= [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
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)
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.
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)
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
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
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
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)
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
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
(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.
P1 I /-
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.
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
(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 ; 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
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).
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
v ** 59.3n 42.30
a Pulay (69].
c Smith, Jorgensen and Chrn [70).
Krohln and Kern .
e Clough, Beers, Klein and Rothman 157).
Quoted by J. Smith [74, p. 31].
3Flaud and Camy-Peyrct .
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 ,
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
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
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
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).
rlJ IJ II O .C.U
iJ I II~
LL-: (~ 5
U. I11J U.:O I.OU
b 1 lu
RL; ~IJI' b IiI liE
0.50 0a 00 0.50 1.00
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
than in those using large basis sets, and "basis set superposition error"
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  and most accurate theoretical [431 value of
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
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).
)- uJ -
MSDRB ANC ~ iE
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
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
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
Ir: I ~r in
L' ~ m
U IJIJ O.:1
.00 0.50 1.00
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 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,
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
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  as an overtone or