- Email: [email protected]
An ab initio SCF calculation of the polarizability tensor, polarizability derivatives and Raman scattering activities of the water-dimer molecule
Received 3 August 1983
The electric polarizability tensor, polarizability derivatives and the resulting Raman scattering activities and depolarization ratios of the water dime& calculated within the framework of ab initio SCF theory, are reported. While the calculated mean polarizability of the dimer is almost the same as for two isolated monomers; the Rolaritibihty anisotropy is considerably larger. The mean polarisability derivatives with respect to normal coordinates are quite small in the case of intermolecular modes and significant deviations from- the morkmeric values occur only for three intramolecular modes, all stretches. The Raman intensities parallel the behaviour of the mean pohnizability derivatives. Consequently the main effect of hydrogen bonding on the Raman intensities of water is a change in the intensities of-the stretching modes. In the case of (H20), the Raman intensity of the intramolecular hydrogen-bond stretch is about twice that of the symmetric stretch of the Ha0 monomer.
1. Introduction
_
The water dimer is a hydrogen-bonded molecule of unique importance_ In a recent paper [l], hereafter referred to as I, we -presented a brief survey of the experimental and theoretical work that centred on the water dimer and reported on an -ab initio SCF study of the dipole-moment derivatives and infrared intensities that were carried out for (H,O),,~ as well. -as several of its deuterated derivatives. As remarked in I, the direct experimental determination of the properties of the water dimer is fraught with difficulties and even such basic electronic properties as the dipole moment and polarizability have not been, as yet, measured. Consequently ab initio quantum-chemical methods have a uniquely important role to play in the study of systems such as the water dimer. This paper is a direct continuation of I and presents the-results of an ab init&SCF study-of the polarizability, polarizability derivatives and Raman intensities of the water dimer. No experimental determination of any of these properties has,.-as yet, -been reported_ More surl~%ing per-
haps is-the fact that definitive Raman intensities for the water monomer itself were only relatively recently published [2]. In our theoretical work on the monomer we found that its polarizability and Raman intensities were quite accurately predicted by SCF calculations using a basis set cf moderate size [3]. We expect comparable accuracy from our dimer calculations, hence we are confident that the salient features of the Raman intensities are reproduced by our calculations as well as the effects of the hydrogen bonding on the polarizability and polarizability-derivative tensors.
2. Theory and cbmputing
methods
The intensity of an “off-resonance” Raman band within the harmonic approximation is adequately accounted . for by -Placzek’s classical ground-state polarizability theory [4-71, according -to which the differential cross section for Raman scattering--at right angles to. the direction of .incident plane-polarized light is given by L
0301~fU04/84/$03XKl0 Elsevier Science Publishers B.V. (Nor+-Holland Physics Publishing Division)
70
D.J. Swanton et al. / Ab inirio SCF calculations on the water dimer
where C is a constant, gi is the degeneracy of the i th vibrational mode and the quantity g,[45( a:)’ + 7($)‘], that we will denote by Ai, is known as the scattering activity. Z’ is the derivative of the average polarizability Z: E = + ( T(_~_~ + a_,._, + a,, )
- a_v.,.)z+ (a_. - a,,)’
+ (a,, - a,,)’
+6[(a,r)z+(a,,)2+(~,,)2])
(3)
by replacing all components of the polarizability by their derivatives. The derivatives are taken with respect to the given (ith) normal coordinate. The degree of depolarization, pi, of the Raman scattered light is then given as pi = 1,./l, = 3( y’)‘/[45(
2)’
+ 4( y’)‘]
,
(4)
where the incident light propagating in the x direction is-plane polarized in the L direction and the scattered light is observed in the _v direction. As discussed by Bogaard and Haines IS], it is advantageous to start with a set of Cartesian polarizability derivatives (CPDs), since such a set provides a “minimal, complete and uniform parameterization of Raman intensities” and their transformation to normal coordinates is simple and straightforward. Such an approach has been adopted in this work also. The CPD tensor of the ith atom, apl.&)
= aa,,/%(
8.
~(~)=~(o)+C(a~(b)/a~~)t~g B
(2)
and ( y’)’ is the square of the polarizability-derivative anisotropy that is obtained from the square of the polarizability anisotropy, y’: Y’ = ~((a,,
perturbing static electric field. Expanding the total energy E (8’) as a Taylor series in an applied field
B,Y.~
= X,Y,Z
(5)
was evaluated numerically from a set of polarizability tensors that were obtained at the equilibrium and a set of distorted geometries, as described in I in connection with the calculation of the Cartesian dipole-moment-derivative tensors. The components of the polarizability tensor were evaluated within the framework of the SCF approximation, i.e., using the coupled perturbed Hartree-Fock (CPHF) method [g-11], that allows the analytical determination of the second derivative of the Hartree-Fock energy with respect to a
+ 4 1 c (a*E( cP)/W’M7) B 7
$E;
+qa3j,
(6)
a given polarizability as
component
aBr = - (aWWa8fiato,)
at+.
is identified
b_O,
(7)
the negative sign in eq. (7) being consistent with the equations for the dipole moment. p( 8), ~(8)
= L$ + Cap,gY
+ o(a’),
$,Y
= x*y,=,
-r (8)
and i.e..
the dipole/electric-field
E( 8) - E(0)
= - J$(
8p’)
interaction
- db’,
energy,
(9)
0
where E(0) is the unperturbed SCF energy and p” is the static dipole moment. Computational details of the unperturbed calculations are given in I. The CPHF calculations were performed using the quadratically convergent Hartree-Fock (QCSCF) program [12] that solves the first-order CPHF equations iteratively, working directly with the two-electron supermatrix in the atomic-orbital basis. The basis set in these calculations is the [%,4p,ld;4s,lp] contracted gaussian basis that was used in I. It is based on Dunning’s (9s,5p) primitive set [13] and is essentially the same basis as in our previous polarizability and Raman-intensities study of Hz0 [3]. The transformation of the calculated CPDs to normal coordinates has been carried out according to the .equation %3,/W
= C+W%.,~ 6
W%y,k,
(10)
where the A matrix defines the normal coordinates
D. J. -Skmton- et al. J Ab initio’S
Q in terms of the atomic
Cartesian coordinates
x=&Q.
X: (11)
The equilibrium
geometj
and nomal~coordinzites
were -obtained from a potentialenergy surface for which -the ~intermolecular potential is the SCF potential of Popkie et al. [14], while the intramolecular potential derives from the experimental quartic (monomeric) force field of Kuchitsu and Morino [15]. The above force field, as discussed in I in some detail, is expected to yield normal coordinates of reasonable quality, especially for the intramolecular modes. The method used to obtain the equilibrium geometry and the A matrix from the force field is fully described in I.
3. Results and discussions The coordinate system and the geometry of the water-dimer molecule is shown in fig. 1, while in table 1 full details of the equilibrium geometry are given in the form of atomic Cartesian coordinates. The calculated polarizability-tensor components are listed in table 2. In order to facilitate comparison with the monomer we introduce, as in dimer”, i.e. a I, the concept of a “non-interacting dimer whose wavefunction is a Wartree, i.e. nonantisymmetrized, product of the unperturbed monomer wavefunctions. Thus any one-electron property of the non-interacting dimer is just the sum of the monomer properties with the monomers assuming the (proton) donor and acceptor positions with reference to the dimeric coordinate system. The polarizability tensor of the non-inter-
Table 1 Equilibrium geometry of the water, dimer. used in ,ihe SCF polariaabiity and polarizability-derivative c&ulations a) Atom
x HI H2
- 1.09996 - 3.57626
-0.12595 0.11127 1.55292
02 H3 H4
2.75790 3.52307 3.52307
0.10823 - 0.69153 -0.69153
Table 2 SCF energies (Et,). dipole moments (eat) and polarizabilities (oie*E;t) for the H,O dimer and the non-interacting H,O dimer. (Coordinate system as given in fig. 1)
aYY Qzz alp
Fig. 1. Geometry of the waterdimer.
0.0 1.43740 - 1.43740
acting water dimer is also given in table 2. We note that LY,, shows the largest change on hydrogen bonding, an increase by = 9% while a,._” and a,, decrease by - 4%. The net change is very small in the average polarizability Z but quite considergble in the anisotropy. Evidently, the nearly spherical polarizability tensor of H,O, or of a non-interacting dimer, becomes quite asymmetric ,when a hydrogen bond forms. The difference in the individual components are accentuated when the polarizability tensors are expressed in their respective principal-axis systems; for the Hz0 dimer, in atomic units (a;e’E, ‘), we obtain 18.455, 15.296 and 16.041, to be compared with 17.147, 15.744 and 16.539 for the non-interacting dimer. The calculated CPDs are listed in table 3, while the mean polarizability derivatives and squared polarizability-derivative anisotropies with respect
PY axr
1 ACCEPTOR
0.0 0.0 0.0
a) See fig. 1 for coordinate system and structure of (H,O)s.
energy
(PROTON
z
Y
- 2.90723
Pr
DONOR
.-
Coordinates (ae)
01
Property
(PROTON)
-71
-
a Y2
H,O dimer 152.060268 1.2171 0.1036 18.085 15.670 16.041 -1.020 16:599 8.195
Non-interacting H ,O dimer - - 152.052252 1.0072 0.1178 16.632 16.259 16.539 - 0.676 16.477 1484
72
D.J. Swanton et al. I* Ab initio SCF calculations on the water dialer
Table 3 Components of the Cartesian polarizabili~y-derivative tensor as,_b(i) (a,e zE,-* ) for the water dimer. The coordinate system and the molecular geometry are given in fig. 1 and table 1. (asym6(i) = k,,/&-,(i)) Atom
Axis
Polarizability component (fly)
(i)
(6)
xx
YY
ZZ
XY
x=
01
x Y z x .p z x .b
- 7.200 - 3.428 0 9.294 1.000 0 - 1.670 2.314
0.737 - 6.292 0 0.996 0.210 0 - 1.950 6.198
- 0.650 - 0.998 0 0.976 - 0.022 0 - 0.564 1.014
- 1.542 1.612 0 0.041 -1.404 0 1.278 - 2.300
0 0 -0.310 .O 0 0.253 0 0
0. 0 - 0.402 0 0 - 0.026 0 0
: x .r z I Y -_ x .r z
0 - 3.245 2.132 0 1.616 - 0.980 1.996 1.616 - 0.980 - 1.996
0 - 1.287 2.724 0 0.798 - 1.256 1.498 0.799 - 1.256 - 1.498
0 - 3.336 3.832 0 1.846 - 1.796 4.262 1.846 - 1.796 - 4.262
0 1.500 1.044 0 0.670 0.716 0.668 0.670 0.716 0.668
-0.366 0 0 - 2.860 1.303 - 0.699 1.606 - 1.303 0.699 1.606
0.459 0 0 2.744 - 0.655
Hl
HZ
02
H3
H4
-
Y=
-
1.129 - 1.375 0.655 - 1.129 - 1.375
with the resulting depolarization ratios and Raman scattering activities. The analogous sets of calculated values for the non-interacting H,O dimer and H,O monomer are also listed in table 3. Note that in the case of the non-interacting dimer the normal coordinates used are those of the (interacting) dimer. Thus. on going from the monomer to the non-interacting dimer we can note the changes that are brought about by defining the
similar, whether one compares the dimer with the non-interacting dimer or indeed with the monomer. The same holds for the acceptor antisymmetric stretch. For donor OH stretches, irrespective of whether they correlate with the monomeric symmetric or antisymmetric stretch, the effective decoupling of the stretches alone brings about substantial changes, in 5’ especially, with a concorn itant increase in the scattering activities. In the case of the donor Ol-Hl stretch, a further sub-
normal
stantial
to normal
coordinates
modes
are given
as those
in table
of the true
4 together
dimer,
ivithout
CPDs. Considering the dimer results first, we note that the average polarizability derivatives for the intermolecular modes are v$ry small, both in absolute terms and in cornpark% with the squared polarizability-derivative anisotropies. Consequently the resulting depolarization ratios are close to their limiting value of 0.750 and the scattering activities are all very low. Further, as far as the intermolecular modes are concerned there are only small differences between the dimer and the non-interacting dimer. The situation is rather different for the intramolecular modes. As we may have anticipated, the results for the bending modes are very changing
the
monomeric
change
is brought
about
by
the changes
in
the CPD tensor on dimer formation, as evidenced by the changes that are observed on going from the non-interacting dimer to the dimer proper. Finally, mode 10, although described as an acceptor symmetric stretch, contains sufficiently significant contributions from the donor stretches, the result of which is a decrease in IE’l from the monomeric value and consequently a decrease in the scattering activity by approximately a factor of two. For this particular mode the changes are almost entirely due to the changes in the normal coordinate. In summary, the significant changes in scattering activities occur for the donor stretches and the acceptor symmetric stretch, the Ol-Hl
A” A”
A’ A’
A”
A’ A’ A’
A
A”
A’
2 3
4
5
6
7 8 9
10
11
12 54.87
50.61
34.21
2.76 1.88 49.75
0.003
0.36 4.43
1.60 1.42
1.14
0.404
0.750
0.270
0.628 0.712 0.076
0.750
0.749 0.744
0.750 0.750
0.695 4.9
334
207
282
12.5 7.9 1240
0.01
i.5 18.2
6.5 5.8
48.48
0.374
0,750
51.56
0.0
- 2.082
0.254
0.480 0.636 0.034
1.50 2.05 18.03 38.81
0,750
0.26
0.728
0.02 0.748
0.750 0.750
1.88 1.99
1.03
0.743
0.11
PI
2.599
- 0.274 -0.180 5.837
0.0
0.006 0.014
0.0 0.0
- 0.009
W2
A, 0,4
312
211
337
8.1 9.2 970
1.1
4.2
0.06
I,7 8.1
0.0
0.0
4.751
- 0.241 - 0.241 4.751
51.59
51.59
26.87
1.78 1.78 26.87
(Y/l2
CT;
A,
2,
P,
E; (0’
HZ0 monomer
Non-irneracting Hz0 dimer
0.750
0.750
0.072
0.548 0.548 0.072
p,:
,’
211
,211
704
8.8, 8.8 704
A,
4 $
B $
F,
s ‘I” ,$
(8 a ‘3’ % 8, ,P. s, c: *%
P 2 R Y ,‘$ ./,g. L.
h s, * ‘K ,, e
depolarization ratios. p, and Raman ,scattering’
H,O dimer
in-plane H-bond 0.089 bend H-bond torsion 0.0 out-of-plane 0.0 H-bond bend H-bond stretch - 0.007 in-plane H-bond - 0.005 shear out-of-plane 0.0 H-bond shear acceptor bend -0.218 donor bend -0.094 donor Ol-Hl 6.276 stretch (symmetric) acceplor symmelric 2.325 stretch acceptor anli0.0 symmetric stretch donor Ol-H2 stretch 2.042 (aniisymmetric)
Description of mode 19
‘) See ref. [16] for a pictorial representation of the normal modes.
A’
Symmetry
1
Mode (4
Table 4 Mean polarizability gradients E; (u,eZE[‘~-‘/z ) squared polarizability-gradient anisotropies (u,‘)~ (aie4E{‘p-‘) activities A, (1O-34C4 NW2kg-‘) for the (H,O), molecule and the non-interacting H,O dimer.
D.J. Swamon et al. / db initio SCF calculations on the water dimer
stretch (symmetric) increasing in intensity by nearly a factor of two in comparison with the monomer. Large as these changes are, they are nowhere near as spectacular as the changes in infrared intensities that are discussed in I. The results for the HZ0 dimer (tabIe 4) may be compared with the analogous results for the deuterated dimers DOD-0D2, DOD-OH, and HOH-ODz that are given in table 5. As for (H20)?, the intermolecular scattering activities are very low for these deuterated species. The intermolecular activities, especially in the case of the stretches, show large variations, depending on the nature of the deuteration. This may greatly aid the assignment of observed Raman bands, should they become experimentally accessible. Note that the force-constant matrix is given in I, hence the normal coordinates can be readily calculated for any other isotopically substituted water-dimer molecule. consequently any interested worker could generate Raman intensity data for such dimers.
4. Summary
and conclusions
In this paper we presented and discussed the polarizability, polarizability derivatives and the resulting Raman scattering activities and depolarization ratios that had been calculated for the waterdimer molecule. As none of the above properties have been observed experimentally, the quality of the results cannot be assessed by comparison with experiments. Nevertheless, since our previous calculations on the Hz0 molecule were shown to possess reasonable predictive capability, we expect the same to be true in the case of the water dimer. As noted, the effect of hydrogen bonding on the polarizability tensor is principally on the anisotropy, rather than on the average polarizability. Consequently, the n -cn polarizability derivatives for the intermolecular vibrational modes are quite small, resulting in low values for the Raman scattering activities that are comparable with those for the intramolecular bonding modes. The intramolecular stretches, as in monomeric water, are the most Raman active, and also, most affected by hydrogen bonding. Thus for (H,O),, intensification is predicted for the donor stretches, by ap-
.. 02.
Swantpn et ~1. -;- Ab initio SCF cakulatio~~ on rhe water dimer
75
_
proximately a factor of two. for the donor H-bond stretch, whiIe the intensity ‘of the acceptor symmet-ric stretch drops to- about half the monomeric value. The differences bet&en the dimer and monomer molecules have b-&enanalyz& using a non-interacting H,O-ciimer model that allows a separation of the effects of hydrogen bonding into those that are due to changes in the normal coordinates and those due to changes in the cartksian polarizability-derivative tensor. The calculations were extended to a few deuterated water-dimer molecules and we noted that, especially in the case of the intramolecular stretctilg modes, deuteration produced large changes in the Raman scattering activities_ According to the calculated results. the Ram& band corresponding to intramolecular OH stretch, i.e. mode 9, is roughly twice as intense as the most intense monomeric band, i.e. that due to the symmetric stretch. WhiIe such a change is certainly very profound it is nowhere near as spectacular as for the corresponding infrared intensities where the analogous change is by = 2 orders of magnitude [l]_ While the current study of the water dimer furthers our understanding of hydrogen bonding and its effect on molecular properties, we hope that our calculated Raman scattering activities will aid further experimental work on this system.
RefererGs (11 D.J. Swanton, G.B. Bacskay and N.S. Hush, Chem. ‘Phys. 82 (1983) 303. 121 W.F. Murphy, Mol. Phys. 33 (1977).1701;
36 (1978) 727. -[3] LG. John. G.B.. Bacskay and N.S. Hush, Chem. Phys. 38 (1979) 319; 51 (1980) 49. [4] G. Placzek, in: Handbuch der Radiolo& VoL 6, ed_ E Marx (Akademische Verlagsgesellschaft, Leipzig, 1934) p_ 205. [5] L-A. Woodward, in: Raman spectroscopy, ed. H.A. Szymanski (Plenum Press, New York, 1967) ch. 1. [6] R.E Hester, in: Raman sptitroscopy, ed. H.A. Szymanski (Plenum Press. New York. 1967) ch. 4. (71 -RE_ Hester. in: Molecular spectroscopy, Vol. 2 (Chem. Sot. London, 1974) ch. 6. __ PI M-P. Bcgaard and R. Haines. Mol. Phys. 41 (1980) 1281. 191 A. Dalgamo. Advan. Phys. 11 (1962) 281. 1101 R-M. Stevens, R.M. Pitzer and W.N. Lipscomb. J. Chem. Phys. 38 (1963) 550. 1111 P-W. Langhoff, M. Karplus and R.P. Hirst. J. Chem. Phys. 44 (1966) 505. 1121 G.B. Bacskay. Chem. Phys. 61 (1982) 385. t131 T-H. Dunning Jr.. J. Chem. Phys. 53 (1970) 2823. 1141 H. Popkie, H. Kistenmacher and EL Clementi. J. Chem. Phys. 59 (1973) 1325. K. Kuchitsu and Y. Merino, Bull. Chem. Sot. Japan 38 1151 (1965) 814. WI D.F. Coker. J.R. Reimers and R.O. Watts, Australian J. Phys. 35 (1982) 623.
The electric polarizability tensor, polarizability derivatives and the resulting Raman scattering activities and depolarization ratios of the water dime& calculated within the framework of ab initio SCF theory, are reported. While the calculated mean polarizability of the dimer is almost the same as for two isolated monomers; the Rolaritibihty anisotropy is considerably larger. The mean polarisability derivatives with respect to normal coordinates are quite small in the case of intermolecular modes and significant deviations from- the morkmeric values occur only for three intramolecular modes, all stretches. The Raman intensities parallel the behaviour of the mean pohnizability derivatives. Consequently the main effect of hydrogen bonding on the Raman intensities of water is a change in the intensities of-the stretching modes. In the case of (H20), the Raman intensity of the intramolecular hydrogen-bond stretch is about twice that of the symmetric stretch of the Ha0 monomer.
1. Introduction
_
The water dimer is a hydrogen-bonded molecule of unique importance_ In a recent paper [l], hereafter referred to as I, we -presented a brief survey of the experimental and theoretical work that centred on the water dimer and reported on an -ab initio SCF study of the dipole-moment derivatives and infrared intensities that were carried out for (H,O),,~ as well. -as several of its deuterated derivatives. As remarked in I, the direct experimental determination of the properties of the water dimer is fraught with difficulties and even such basic electronic properties as the dipole moment and polarizability have not been, as yet, measured. Consequently ab initio quantum-chemical methods have a uniquely important role to play in the study of systems such as the water dimer. This paper is a direct continuation of I and presents the-results of an ab init&SCF study-of the polarizability, polarizability derivatives and Raman intensities of the water dimer. No experimental determination of any of these properties has,.-as yet, -been reported_ More surl~%ing per-
haps is-the fact that definitive Raman intensities for the water monomer itself were only relatively recently published [2]. In our theoretical work on the monomer we found that its polarizability and Raman intensities were quite accurately predicted by SCF calculations using a basis set cf moderate size [3]. We expect comparable accuracy from our dimer calculations, hence we are confident that the salient features of the Raman intensities are reproduced by our calculations as well as the effects of the hydrogen bonding on the polarizability and polarizability-derivative tensors.
2. Theory and cbmputing
methods
The intensity of an “off-resonance” Raman band within the harmonic approximation is adequately accounted . for by -Placzek’s classical ground-state polarizability theory [4-71, according -to which the differential cross section for Raman scattering--at right angles to. the direction of .incident plane-polarized light is given by L
0301~fU04/84/$03XKl0 Elsevier Science Publishers B.V. (Nor+-Holland Physics Publishing Division)
70
D.J. Swanton et al. / Ab inirio SCF calculations on the water dimer
where C is a constant, gi is the degeneracy of the i th vibrational mode and the quantity g,[45( a:)’ + 7($)‘], that we will denote by Ai, is known as the scattering activity. Z’ is the derivative of the average polarizability Z: E = + ( T(_~_~ + a_,._, + a,, )
- a_v.,.)z+ (a_. - a,,)’
+ (a,, - a,,)’
+6[(a,r)z+(a,,)2+(~,,)2])
(3)
by replacing all components of the polarizability by their derivatives. The derivatives are taken with respect to the given (ith) normal coordinate. The degree of depolarization, pi, of the Raman scattered light is then given as pi = 1,./l, = 3( y’)‘/[45(
2)’
+ 4( y’)‘]
,
(4)
where the incident light propagating in the x direction is-plane polarized in the L direction and the scattered light is observed in the _v direction. As discussed by Bogaard and Haines IS], it is advantageous to start with a set of Cartesian polarizability derivatives (CPDs), since such a set provides a “minimal, complete and uniform parameterization of Raman intensities” and their transformation to normal coordinates is simple and straightforward. Such an approach has been adopted in this work also. The CPD tensor of the ith atom, apl.&)
= aa,,/%(
8.
~(~)=~(o)+C(a~(b)/a~~)t~g B
(2)
and ( y’)’ is the square of the polarizability-derivative anisotropy that is obtained from the square of the polarizability anisotropy, y’: Y’ = ~((a,,
perturbing static electric field. Expanding the total energy E (8’) as a Taylor series in an applied field
B,Y.~
= X,Y,Z
(5)
was evaluated numerically from a set of polarizability tensors that were obtained at the equilibrium and a set of distorted geometries, as described in I in connection with the calculation of the Cartesian dipole-moment-derivative tensors. The components of the polarizability tensor were evaluated within the framework of the SCF approximation, i.e., using the coupled perturbed Hartree-Fock (CPHF) method [g-11], that allows the analytical determination of the second derivative of the Hartree-Fock energy with respect to a
+ 4 1 c (a*E( cP)/W’M7) B 7
$E;
+qa3j,
(6)
a given polarizability as
component
aBr = - (aWWa8fiato,)
at+.
is identified
b_O,
(7)
the negative sign in eq. (7) being consistent with the equations for the dipole moment. p( 8), ~(8)
= L$ + Cap,gY
+ o(a’),
$,Y
= x*y,=,
-r (8)
and i.e..
the dipole/electric-field
E( 8) - E(0)
= - J$(
8p’)
interaction
- db’,
energy,
(9)
0
where E(0) is the unperturbed SCF energy and p” is the static dipole moment. Computational details of the unperturbed calculations are given in I. The CPHF calculations were performed using the quadratically convergent Hartree-Fock (QCSCF) program [12] that solves the first-order CPHF equations iteratively, working directly with the two-electron supermatrix in the atomic-orbital basis. The basis set in these calculations is the [%,4p,ld;4s,lp] contracted gaussian basis that was used in I. It is based on Dunning’s (9s,5p) primitive set [13] and is essentially the same basis as in our previous polarizability and Raman-intensities study of Hz0 [3]. The transformation of the calculated CPDs to normal coordinates has been carried out according to the .equation %3,/W
= C+W%.,~ 6
W%y,k,
(10)
where the A matrix defines the normal coordinates
D. J. -Skmton- et al. J Ab initio’S
Q in terms of the atomic
Cartesian coordinates
x=&Q.
X: (11)
The equilibrium
geometj
and nomal~coordinzites
were -obtained from a potentialenergy surface for which -the ~intermolecular potential is the SCF potential of Popkie et al. [14], while the intramolecular potential derives from the experimental quartic (monomeric) force field of Kuchitsu and Morino [15]. The above force field, as discussed in I in some detail, is expected to yield normal coordinates of reasonable quality, especially for the intramolecular modes. The method used to obtain the equilibrium geometry and the A matrix from the force field is fully described in I.
3. Results and discussions The coordinate system and the geometry of the water-dimer molecule is shown in fig. 1, while in table 1 full details of the equilibrium geometry are given in the form of atomic Cartesian coordinates. The calculated polarizability-tensor components are listed in table 2. In order to facilitate comparison with the monomer we introduce, as in dimer”, i.e. a I, the concept of a “non-interacting dimer whose wavefunction is a Wartree, i.e. nonantisymmetrized, product of the unperturbed monomer wavefunctions. Thus any one-electron property of the non-interacting dimer is just the sum of the monomer properties with the monomers assuming the (proton) donor and acceptor positions with reference to the dimeric coordinate system. The polarizability tensor of the non-inter-
Table 1 Equilibrium geometry of the water, dimer. used in ,ihe SCF polariaabiity and polarizability-derivative c&ulations a) Atom
x HI H2
- 1.09996 - 3.57626
-0.12595 0.11127 1.55292
02 H3 H4
2.75790 3.52307 3.52307
0.10823 - 0.69153 -0.69153
Table 2 SCF energies (Et,). dipole moments (eat) and polarizabilities (oie*E;t) for the H,O dimer and the non-interacting H,O dimer. (Coordinate system as given in fig. 1)
aYY Qzz alp
Fig. 1. Geometry of the waterdimer.
0.0 1.43740 - 1.43740
acting water dimer is also given in table 2. We note that LY,, shows the largest change on hydrogen bonding, an increase by = 9% while a,._” and a,, decrease by - 4%. The net change is very small in the average polarizability Z but quite considergble in the anisotropy. Evidently, the nearly spherical polarizability tensor of H,O, or of a non-interacting dimer, becomes quite asymmetric ,when a hydrogen bond forms. The difference in the individual components are accentuated when the polarizability tensors are expressed in their respective principal-axis systems; for the Hz0 dimer, in atomic units (a;e’E, ‘), we obtain 18.455, 15.296 and 16.041, to be compared with 17.147, 15.744 and 16.539 for the non-interacting dimer. The calculated CPDs are listed in table 3, while the mean polarizability derivatives and squared polarizability-derivative anisotropies with respect
PY axr
1 ACCEPTOR
0.0 0.0 0.0
a) See fig. 1 for coordinate system and structure of (H,O)s.
energy
(PROTON
z
Y
- 2.90723
Pr
DONOR
.-
Coordinates (ae)
01
Property
(PROTON)
-71
-
a Y2
H,O dimer 152.060268 1.2171 0.1036 18.085 15.670 16.041 -1.020 16:599 8.195
Non-interacting H ,O dimer - - 152.052252 1.0072 0.1178 16.632 16.259 16.539 - 0.676 16.477 1484
72
D.J. Swanton et al. I* Ab initio SCF calculations on the water dialer
Table 3 Components of the Cartesian polarizabili~y-derivative tensor as,_b(i) (a,e zE,-* ) for the water dimer. The coordinate system and the molecular geometry are given in fig. 1 and table 1. (asym6(i) = k,,/&-,(i)) Atom
Axis
Polarizability component (fly)
(i)
(6)
xx
YY
ZZ
XY
x=
01
x Y z x .p z x .b
- 7.200 - 3.428 0 9.294 1.000 0 - 1.670 2.314
0.737 - 6.292 0 0.996 0.210 0 - 1.950 6.198
- 0.650 - 0.998 0 0.976 - 0.022 0 - 0.564 1.014
- 1.542 1.612 0 0.041 -1.404 0 1.278 - 2.300
0 0 -0.310 .O 0 0.253 0 0
0. 0 - 0.402 0 0 - 0.026 0 0
: x .r z I Y -_ x .r z
0 - 3.245 2.132 0 1.616 - 0.980 1.996 1.616 - 0.980 - 1.996
0 - 1.287 2.724 0 0.798 - 1.256 1.498 0.799 - 1.256 - 1.498
0 - 3.336 3.832 0 1.846 - 1.796 4.262 1.846 - 1.796 - 4.262
0 1.500 1.044 0 0.670 0.716 0.668 0.670 0.716 0.668
-0.366 0 0 - 2.860 1.303 - 0.699 1.606 - 1.303 0.699 1.606
0.459 0 0 2.744 - 0.655
Hl
HZ
02
H3
H4
-
Y=
-
1.129 - 1.375 0.655 - 1.129 - 1.375
with the resulting depolarization ratios and Raman scattering activities. The analogous sets of calculated values for the non-interacting H,O dimer and H,O monomer are also listed in table 3. Note that in the case of the non-interacting dimer the normal coordinates used are those of the (interacting) dimer. Thus. on going from the monomer to the non-interacting dimer we can note the changes that are brought about by defining the
similar, whether one compares the dimer with the non-interacting dimer or indeed with the monomer. The same holds for the acceptor antisymmetric stretch. For donor OH stretches, irrespective of whether they correlate with the monomeric symmetric or antisymmetric stretch, the effective decoupling of the stretches alone brings about substantial changes, in 5’ especially, with a concorn itant increase in the scattering activities. In the case of the donor Ol-Hl stretch, a further sub-
normal
stantial
to normal
coordinates
modes
are given
as those
in table
of the true
4 together
dimer,
ivithout
CPDs. Considering the dimer results first, we note that the average polarizability derivatives for the intermolecular modes are v$ry small, both in absolute terms and in cornpark% with the squared polarizability-derivative anisotropies. Consequently the resulting depolarization ratios are close to their limiting value of 0.750 and the scattering activities are all very low. Further, as far as the intermolecular modes are concerned there are only small differences between the dimer and the non-interacting dimer. The situation is rather different for the intramolecular modes. As we may have anticipated, the results for the bending modes are very changing
the
monomeric
change
is brought
about
by
the changes
in
the CPD tensor on dimer formation, as evidenced by the changes that are observed on going from the non-interacting dimer to the dimer proper. Finally, mode 10, although described as an acceptor symmetric stretch, contains sufficiently significant contributions from the donor stretches, the result of which is a decrease in IE’l from the monomeric value and consequently a decrease in the scattering activity by approximately a factor of two. For this particular mode the changes are almost entirely due to the changes in the normal coordinate. In summary, the significant changes in scattering activities occur for the donor stretches and the acceptor symmetric stretch, the Ol-Hl
A” A”
A’ A’
A”
A’ A’ A’
A
A”
A’
2 3
4
5
6
7 8 9
10
11
12 54.87
50.61
34.21
2.76 1.88 49.75
0.003
0.36 4.43
1.60 1.42
1.14
0.404
0.750
0.270
0.628 0.712 0.076
0.750
0.749 0.744
0.750 0.750
0.695 4.9
334
207
282
12.5 7.9 1240
0.01
i.5 18.2
6.5 5.8
48.48
0.374
0,750
51.56
0.0
- 2.082
0.254
0.480 0.636 0.034
1.50 2.05 18.03 38.81
0,750
0.26
0.728
0.02 0.748
0.750 0.750
1.88 1.99
1.03
0.743
0.11
PI
2.599
- 0.274 -0.180 5.837
0.0
0.006 0.014
0.0 0.0
- 0.009
W2
A, 0,4
312
211
337
8.1 9.2 970
1.1
4.2
0.06
I,7 8.1
0.0
0.0
4.751
- 0.241 - 0.241 4.751
51.59
51.59
26.87
1.78 1.78 26.87
(Y/l2
CT;
A,
2,
P,
E; (0’
HZ0 monomer
Non-irneracting Hz0 dimer
0.750
0.750
0.072
0.548 0.548 0.072
p,:
,’
211
,211
704
8.8, 8.8 704
A,
4 $
B $
F,
s ‘I” ,$
(8 a ‘3’ % 8, ,P. s, c: *%
P 2 R Y ,‘$ ./,g. L.
h s, * ‘K ,, e
depolarization ratios. p, and Raman ,scattering’
H,O dimer
in-plane H-bond 0.089 bend H-bond torsion 0.0 out-of-plane 0.0 H-bond bend H-bond stretch - 0.007 in-plane H-bond - 0.005 shear out-of-plane 0.0 H-bond shear acceptor bend -0.218 donor bend -0.094 donor Ol-Hl 6.276 stretch (symmetric) acceplor symmelric 2.325 stretch acceptor anli0.0 symmetric stretch donor Ol-H2 stretch 2.042 (aniisymmetric)
Description of mode 19
‘) See ref. [16] for a pictorial representation of the normal modes.
A’
Symmetry
1
Mode (4
Table 4 Mean polarizability gradients E; (u,eZE[‘~-‘/z ) squared polarizability-gradient anisotropies (u,‘)~ (aie4E{‘p-‘) activities A, (1O-34C4 NW2kg-‘) for the (H,O), molecule and the non-interacting H,O dimer.
D.J. Swamon et al. / db initio SCF calculations on the water dimer
stretch (symmetric) increasing in intensity by nearly a factor of two in comparison with the monomer. Large as these changes are, they are nowhere near as spectacular as the changes in infrared intensities that are discussed in I. The results for the HZ0 dimer (tabIe 4) may be compared with the analogous results for the deuterated dimers DOD-0D2, DOD-OH, and HOH-ODz that are given in table 5. As for (H20)?, the intermolecular scattering activities are very low for these deuterated species. The intermolecular activities, especially in the case of the stretches, show large variations, depending on the nature of the deuteration. This may greatly aid the assignment of observed Raman bands, should they become experimentally accessible. Note that the force-constant matrix is given in I, hence the normal coordinates can be readily calculated for any other isotopically substituted water-dimer molecule. consequently any interested worker could generate Raman intensity data for such dimers.
4. Summary
and conclusions
In this paper we presented and discussed the polarizability, polarizability derivatives and the resulting Raman scattering activities and depolarization ratios that had been calculated for the waterdimer molecule. As none of the above properties have been observed experimentally, the quality of the results cannot be assessed by comparison with experiments. Nevertheless, since our previous calculations on the Hz0 molecule were shown to possess reasonable predictive capability, we expect the same to be true in the case of the water dimer. As noted, the effect of hydrogen bonding on the polarizability tensor is principally on the anisotropy, rather than on the average polarizability. Consequently, the n -cn polarizability derivatives for the intermolecular vibrational modes are quite small, resulting in low values for the Raman scattering activities that are comparable with those for the intramolecular bonding modes. The intramolecular stretches, as in monomeric water, are the most Raman active, and also, most affected by hydrogen bonding. Thus for (H,O),, intensification is predicted for the donor stretches, by ap-
.. 02.
Swantpn et ~1. -;- Ab initio SCF cakulatio~~ on rhe water dimer
75
_
proximately a factor of two. for the donor H-bond stretch, whiIe the intensity ‘of the acceptor symmet-ric stretch drops to- about half the monomeric value. The differences bet&en the dimer and monomer molecules have b-&enanalyz& using a non-interacting H,O-ciimer model that allows a separation of the effects of hydrogen bonding into those that are due to changes in the normal coordinates and those due to changes in the cartksian polarizability-derivative tensor. The calculations were extended to a few deuterated water-dimer molecules and we noted that, especially in the case of the intramolecular stretctilg modes, deuteration produced large changes in the Raman scattering activities_ According to the calculated results. the Ram& band corresponding to intramolecular OH stretch, i.e. mode 9, is roughly twice as intense as the most intense monomeric band, i.e. that due to the symmetric stretch. WhiIe such a change is certainly very profound it is nowhere near as spectacular as for the corresponding infrared intensities where the analogous change is by = 2 orders of magnitude [l]_ While the current study of the water dimer furthers our understanding of hydrogen bonding and its effect on molecular properties, we hope that our calculated Raman scattering activities will aid further experimental work on this system.
RefererGs (11 D.J. Swanton, G.B. Bacskay and N.S. Hush, Chem. ‘Phys. 82 (1983) 303. 121 W.F. Murphy, Mol. Phys. 33 (1977).1701;
36 (1978) 727. -[3] LG. John. G.B.. Bacskay and N.S. Hush, Chem. Phys. 38 (1979) 319; 51 (1980) 49. [4] G. Placzek, in: Handbuch der Radiolo& VoL 6, ed_ E Marx (Akademische Verlagsgesellschaft, Leipzig, 1934) p_ 205. [5] L-A. Woodward, in: Raman spectroscopy, ed. H.A. Szymanski (Plenum Press, New York, 1967) ch. 1. [6] R.E Hester, in: Raman sptitroscopy, ed. H.A. Szymanski (Plenum Press. New York. 1967) ch. 4. (71 -RE_ Hester. in: Molecular spectroscopy, Vol. 2 (Chem. Sot. London, 1974) ch. 6. __ PI M-P. Bcgaard and R. Haines. Mol. Phys. 41 (1980) 1281. 191 A. Dalgamo. Advan. Phys. 11 (1962) 281. 1101 R-M. Stevens, R.M. Pitzer and W.N. Lipscomb. J. Chem. Phys. 38 (1963) 550. 1111 P-W. Langhoff, M. Karplus and R.P. Hirst. J. Chem. Phys. 44 (1966) 505. 1121 G.B. Bacskay. Chem. Phys. 61 (1982) 385. t131 T-H. Dunning Jr.. J. Chem. Phys. 53 (1970) 2823. 1141 H. Popkie, H. Kistenmacher and EL Clementi. J. Chem. Phys. 59 (1973) 1325. K. Kuchitsu and Y. Merino, Bull. Chem. Sot. Japan 38 1151 (1965) 814. WI D.F. Coker. J.R. Reimers and R.O. Watts, Australian J. Phys. 35 (1982) 623.