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Anharmonic vibrational wave functions, infrared band intensities, and dipole moments of water
JOURNAL
OF MOLECULAR
SPECTROSCOPY
(1983)
lt&234-244
Anharmonic Vibrational Wave Functions, Infrared Band Intensities, and Dipole Moments of Water YUMIKO
YAMAOKA
Faculty of Pharmaceutical Sciences, Kobe-Gakuin
University, Nishi-ku, Kobe, Japan
AND KATSUNOSUKEMACHIDA Faculty of Pharmaceutical Sciences, Kyoto University, Sakyo-ku, Kyoto, Japan
A new and simple method to expand the anharmonic vibrational wave functions with respect to the harmonic oscillator wave functions is proposed. The coefficients of the expansion are given as matrix elements of the S function of the contact transformation in the perturbation theory and the explicit expressions of these coefficients are given within the approximation to the second order in X. As an example of the expansion, the wave functions of water molecules were calculated and applied to the calculation of infrared band intensities and average values of dipole moments in several states. INTRODUCTION
Determination of the anharmonic potential functions for molecular vibrations cannot be accomplished without precise assignments of fundamental and summation frequencies in infrared and near-infrared spectra. In attempting to assign a band to a specific vibrational level of a polyatomic molecule, however, we often encounter complicated anharmonic resonances. If a molecule involves three resonances of the Fermi type ViN uj + Uk, uj N ul + u,, and uk N up + up, we should take account of 14 levels resonating in either the first or the second order in the vicinity of 2Vi. In the region around 3u,, the corresponding resonance polyad contains 30 levels. In such cases it is almost impossible to assign the observed bands correctly without recourse to some preliminary knowledge on the band intensities. Theoretical prediction of the intensities of fundamental and summation bands incorporating both the electrical and the mechanical anharmonicity is thus highly desirable. In addition to the frequencies and the band intensities of vibrational levels, the r, structure, the dipole moment of the vibrationally excited states, and many other physical properties have been known to bc perturbed by the anharmonicity of the potential. It is of fundamental importance to determine the vibrational wave functions under the anharmonic potential as precisely as possible for evaluating the expectation values of these properties. In this report a new and simple method for expanding the vibrational wave functions of polyatomic molecules with respect to the harmonic oscillator wave functions is 0022-2852/83 Copyright All
rights
0
$3.00
1983 by Academic
of reproduction
Press.
in any form
Inc. reserved.
234
VIBRATIONAL
WAVE
FUNCTIONS
OF Hz0
235
proposed. The expansion coefficients are expressed in terms of the matrix elements
of the S function of the contact transformation in the perturbation theory. Transition moments of a number of vibrational levels of water are calculated as an example of application of the present method. PROCEDURE
perturbation theory of molecular vibrations developed by Nielsen et al. (Iis chosen so as to eliminate the cubic and quartic terms in the transformed Hamiltonian H’ = THT-‘. (1) In the
j), the contact transformation
Since the ma&ix elements of the transformed Hamiltonian on the basis of the harmonic wave functions & are given by (#W’I~,)
= (CT-%,)*IH
IV-‘&))
= L,
(2)
the wave function $,, of the primitive Hamiltonian H may well be approximated by T-‘4, and is expanded as k, = T-I& = C amn& (3) m where the coefficient a,,,,, is given by amn = ($&IT-‘I&). If we restrict the expansion to the linear terms of X, the transformation is given by T-’ = T;’ = exp(-iXS,) N 1 - AS,
(4) operator T-’ (5)
where TI is the operator which eliminates the cubic terms in the vibrational Hamiltonian and S, is the corresponding S function. Then the expansion coefficients umn in Eq. (3) are given by using the matrix elements of the S function as Grin = &WI- ~~(~~lS,I~,).
(6)
As S1 contains only cubic terms of q and p, the second term in Eq. (6) is zero unless the summation of the absolute values of the differences in the vibrational quantum number between the states m and n is equal to 1 or 3. This condition restricts the number of harmonic wave functions C#J,which contribute to the perturbed wave function $, and facilitates the subsequent calculation. The explicit expressions of SI and its matrix elements are given in Appendix 1. The second contact transformation makes the approximation correct to the quadratic terms of X, where the transformation operator T-’ is expressed by T--l = T;’ T;’ N 1 - ihS, - X2($/2 + is,).
(7)
In addition to the matrix elements of S, and S2, those of ST are required to calculate the coefficients a,,,” in this approximation. They are easily obtained by using the relation
(4w:b”)
= F (4~lS,l~k)(dJX IS&“).
The matrix elements of SZ are also given in Appendix 1.
236
YAMAOKA AND MACHIDA
For polyatomic molecules the usual perturbation method often breaks down by the accidental degeneracy of the energy levels. When such a resonance occurs, the resonance denominator becomes nearly equal to zero and the transformed Hamiltonian gives an abnormally large perturbation energy. Dropping the corresponding denominator we get a modified transformation operator T’ which leaves the off-diagonal matrix elements between the resonating levels unchanged from those of the unperturbed Hamiltonian (I, 6). After transformation by the modified operator the energy matrix is diagonalized numerically to give the resonance-perturbed wave function by using the eigenvectors. The matrix elements of a given operator f which depends on the nuclear coordinates are calculated by (IclZlfl$,) = c &+#Q IfldJa,,. (9) if The time required to perform this calculation can be saved greatly by utilizing the sparseness of the matrices of ai, and (titlfl@,). To calculate the infrared band intensities, it is useful to apply the effective chargecharge flux model (7, 8) to higher order. The dipole moment P is defined as /.l = 2
eiXi
(10)
where ei and xi are the electric charge and the position vector, respectively, of the atom i. Differentiating Eq. (10) with respect to the dimensionless normal coordinates, we have
and
de. a2Rj c c-__x._ i , aRj aqd% 1
(12)
The effective charges and their derivatives with respect to the internal coordinates were taken as adjustable parameters in the following calculation. DISCUSSION
There have been two well-known methods of approximation which may be complementary to each other in studying the vibrational wave functions, i.e., the variational and the perturbation methods. By applying the variational method Suzuki proposed the direct diagonalization of the vibration-rotation Hamiltonian based on the harmonic oscillators to calculate the vibrational anharmonicity (9). Other basis sets such as Morse oscillators have also been used in the variational method to obtain the vibrational energy levels (10). However, the application of the variational method to large molecules is seldom seen because it requires the diagonalization of an enormous matrix.
VlBRATIONAL
WAVE
FUNCTIONS
OF Hz0
237
On the other hand, the second-order perturbation theory developed by Nielsen has been successfully used for the calculation of vibrational frequencies of molecules containing more than four atoms (6, 1 I). Explicit expressions for the first-order wave functions have also been given in this method (12, 13). An advantage of the variation method over the perturbation method has been the flexibility of the former to a higher-order approximation than the first. Even in the perturbation method, however, it is rather easy to obtain the second-order wave functions if we use the matrix elements of the S function, The difference between the variational method and the perturbation method lies in the way of truncation. The former truncates the matrix elements with higher quantum numbers, while the latter truncates the higher-order force constants. Our method corresponds to an analytical diagonalization of the infinite matrix of the Hamiltonian containing quadratic through quartic force constants, so it might be regarded as a hybrid of the two types of approximation. The merit of this method is that we need to handle only the wave functions related to the experimental data available in case by case. The higher-order expressions of infrared band intensities of polyatomic molecules based on the perturbation method were first given by Secroun et al. (14) for the case of bent triatomic molecules. The general expressions for band intensities and average values of physical observables were established by Overend and his co-workers (IS17). Their method consists of the contact transformation of the operator of any physical quantities expanded in terms of the dimensionless normal coordinates and the term by term calculation of the matrix elements of the transformed quantities. Meanwhile, no explicit expressions of band intensities or other physical quantities are required in our method because all the computations are reduced to the matrix multiplication in the form of Eq. (9). It is not necessary to use different formulae for different combinations of vibrational quanta: see Appendix 1. Even the energy levels and the rotational constants can be calculated without using the equations derived by Nielsen. The obtained wave functions are also applicable to knowing the actual form of molecular vibrations as cross-sectional plots of wave function amplitudes (IO). EXAMPLE
As an example of the method in the preceding section, the wave functions of several states of water were calculated within the approximation to the first order in X. The structural parameters were taken to be r(OH) = 0.9576 A and L(HOH) = 104.29” according to Benedict et al. (18) (Fig. 1). The quadratic through the quartic force constants of water were taken from Set III by Smith and Overend (19). It has been known that vi resonates with 2~ through the Fermi-Dennison resonance and 2v, with 2~ through the Darling-Dennison resonance. The resonance denominators ofthe Fermi-Dennison resonance vi-2~ were omitted from the expansion coefficients and the energy matrix related to these resonances were diagonalized numerically to obtain the energy levels and the wave functions. All computations were carried out on a NEAC System 300 computer at Kobe-Gakuin University. Some of the obtained wave functions are shown in Table I. For the calculations of band intensities of this molecule, it is necessary to determine the atomic charges and their derivatives with respect to the internal coordinates. The
238
YAMAOKA AND MACHIDA
FIG. 1. The definition of molecular axes and internal coordinates of water molecule.
atomic charges were estimated from the equilibrium dipole moment of Hz0 (20) with a slight modification. The first derivatives were so chosen as to reproduce the observed dipole moment derivatives with respect to dimensionless normal coordinates (20). The second derivatives were estimated by the method of least squares with reference to the same set of the observed intensities as used in the variational method by Carney et al. (IO). Two sets in Table II reproducing the observed band intensities equally well were obtained. In Table III, the derivatives of the dipole moment with respect to the dimensionless normal coordinates calculated from two sets of parameters are compared with the values derived from the recently observed data by Flaud and Camy-Peyret (21-23) and those determined from the ab initio MO calculation by
TABLE I Coefficients of Vibrational Wave Functions for Lower Levels of Hz0 (umn X 1000)
3
010
000
0
000 100 010 200 110 020 002 300 210 120 030 102 012 400 310 220 130 040 202 112 022 320 230 140
-11 -
1000
-158 11 -23 3 5 -11 -38 9 -
050
-
122 032
l
1000 5 -145 -39 14 -23 4 9 -21 -38 13 -
Involved
ozo* 11 71 39 -24 2 997 -8 7 -132 -115 20 -3 0 0 -1 -4 1 -22 5 12 -33 -38 16
100* 157 997 -3 -341 35 -71 -109 -0 9 8 -1 -45 5 7 -11 -53 9 2 -0 -1 2 3 -1
in a resonance
001 101 011 201 111 021 003 301 211 121 031 103 013 401 311 221 131 041 203 113 023 321 231 141 051 123 033
polyad
001
011
1000 -321 87 -23 3 5 -11 -65 16 -
-87 1000 5 -308 68 24 -23 4 9 -21 -65 23 -
-
of "1 = 2v2
021*
101'
30
320
94
996
-68 -54 10 996 -18 7 -294 16 35 -4 0 1 -1 -9 1 -23 5 12 -33 -65 28
6 -570 111 -94 -188 -1 28 -2 -3 -45 5 7 -11 -92 15 2 -0 -1 3 6 -3
239
VIBRATIONAL WAVE FUNCTIONS OF Hz0 TABLE II Effective Atomic Charges and Their Flux of Water Molecule” set I
Parameter eH
set
II
unit
1.5859
10-10
es"
deH/dr
-1.0384
10-10
eSU/A
deH/dr'
-0.2397
10-lo
eSU/A
de,/d$
0.4146
d2eH/dr2
1.9
10-10
esu/rad
0.3
10-10
em/A'
d2eH/drdr'
0.7
-
10-10
eSU/A2
d2eH!dr' 2
0.6
-
10-10
W/A2
-1.5
0.3
10-10 esu/A.rad
d2eH/dr'd$
0.6
-0.8
10-10
esu/A.rad
d2e,/de2
0.5
0.2
10-10
esu/rad2
d2eH/drd$
=For descriptions relations
were
e. = -2eH,
of coordinates
see Fig.
1.
The following
assumed:
deo/dr
= -d+dr
d2e0,'dr2 = -d2eH/dr2
- deH/dr',
deo/d#
- d2eH,'drv2, d2eO/drdr'
d2e0,'drd+ = -d2eH/drd$
= -2dea/d$, = -2d2eH/drdr',
-d2eH,'dr'd$ and d2e,/d$2
= -2d2e,/dQ2.
Ermler and Krohn (24). In spite of the disagreement in signs of the second derivatives with the ab initio method, Set II reproduces the observed derivatives very well. By using the parameters tabulated in Table II, the infrared band intensities of H20, D20, and HDO were calculated. The results are compared with the observed intensities TABLE III Calculated Derivatives of Dipole Moment of HZ0 with Respect to Dimensionless Normal Coordinates” Axis z
Y
ab initio'
"e
-1.8587
-1.8587
al
0.0216
0.0216
0.06052
"2
-0.1617
-0.1617
-0.22275
-1.997490
511
-0.0098
0.0052
512
0.0049
-0.0065
0.00828
822
-0.0051
0.0137
-0.00758
833
-0.0052
0.0028
0.0026
a3
-0.0951
-0.0951
-0.1196731
0.0003
-0.0098
0.0070180
'23
-0.0447
0.0296
-0.0262108
The
in Debye.
signs
Ref.
(20).
Ref.
(24)
Obsd.c
Obsd.d
-1.8473 0.0216 -0.161
0.0217 -0.17946
-0.0017
813
aUnits
b
This work Set I set II
Index
-0.00843 0.013
-0.0950
-0.09714
0.0267
Description of indices are the same. as in Ref.(Z4). of the normal coordinates were taken in the Same way as
'Ref.(20).
dRef.(21)-(23).
2.28
4.41
0.0068
0.0039
0.617
0.973
0.0003
011
120
021
200
101
002
(IO).
0.059
2.67
0.640
0.065
0.0042
4.82
0.113
42.6
2.01
0.351
52.2
set II
aRef.
0.108
110
42.6
100
001
0.344
52.2
010
I
020
set
v,v*LJ3
HZ0 ~~__
0.032
4.50
0.318
0.340
0.021
5.45
0.110
48.2
2.18
0.396
63.8
Obsd.
I
0.064
1.32
0.256
0.027
I
0.663
0.078
0.343
0.036
0.028
2.35
0.359
26.3
10.4
0.094
46.2
set
HDO
(IO).
0.00078
0.892
0.183
0.066
0.0040
5.69
0.039
20.6
8.60
0.320
47.9
VX.a
in Ref.
0.0017
1.98
0.052
26.4
1.79
0.171
28.3
set II
et ~2. cited
0.0058
0.385
0.211
0.0009
0.0022
1.70
0.048
26.4
1.96
0.105
28.3
set
Dz0
b R. A. McClatchey
0.0090
1.94
0.379
0.146
0.0069
14.2
0.115
43.0
11.6
0.473
90.7
Var.a
b
__~_~..
Calculated Infrared Band Intensities of H,O, D20, and HDO (in km.
TABLE IV
1.85
0.020
0.638
0.035
0.043
1.99
0.889
26.3
10.0
0.565
46.2
set II
mole-')
1.14
0.166
0.564
0.0099
0.016
6.59
2.03
27.8
13.8
0.302
74.8
Var.a
30.1
13.5
54.7
Obsd.
_~_ b
VIBRATIONAL
WAVE
FUNCTIONS
241
OF Hz0
and those calculated by the variational method (10) in Table IV. The average values of dipole moment in the ground states of HzO, DzO, and HDO and in some vibrationally excited states of Hz0 were also calculated and are compared with the observed and calculated values (Table V). In this calculation, Set II reproduces not only the observed isotope effect of the ground state dipole moment (20) very well but also the dipole moment in the u2 state observed by Kuze et al. (25). The difference between the dipole moments of the v2 state and the ground state of H2”0 observed by Johns and McKellar (26), 0.03 10, is consistent with our result of H2’*0, 0.0318. A slight disagreement with the values by Riley et al. (13) might be due to their elimination of unobserved derivatives of the dipole moment. The success of Set II in elucidating the infrared intensities and the dipole moments simultaneously may be taken as proving the excellence of the wave functions calculated from the present method. 1
APPENDIX
The explicit formula of Si has been given by Amat et al. (3) and the vibrational part of S, is expressed in the form of summation over the combination of three normal modes as {S abc&dbPc + ~~b~(~a‘?d)c Sl(vib) = 2 + P&&b) abc @
+ %f(%GPa
+ PdMc) + S%(WLPb
+
Pbqc%))
=
c
%-JbC).
(A-1)
nbc (nsbac)
Because the matrix elements of pJh and go are identical except for their signs, the matrix elements of each term in Eq. (A-l) are reduced to a simple expression:
In Eq. (A-2), a:, the creation or the annihilation operator for the ith vibrational quantum, is used to simplify the equation. The following are some examples of the wave functions in terms of at and a;: TABLE V Average Values of Dipole Moment in the Ground and Vibrationally
Tsotope HP
Axis
z
Calcd.
state
set
I
Excited States (in Debye) Obsd.
Pet II
Rileva
0 00
-1.8761
-1.8552
-1.8455
100
-1.8964
-1.8606
-1.8552
010
-1.8631
-1.8234
-1.8108
001
-1.9036
-1.8746
-1.8667
Clouqhb -1.8546
-1.827
D2O
2
000
-1.8719
-1.8566
-1.855R
HDO
a
000
-1.7480
-1.7306
-1.7318
b
0
0.6683
0.6567
aRef.
(13).
0
0
0.6739 b
Ref.
(20).
'Ref.
(25).
K"zeC
242
YAMAOKA
Iu~u&2&p&
AND MACHIDA
= lu, + 1 Ub+ 1 U, + 1)
l&z~U~U,U,) = Iv, + 2 U, + 1) ~u,+u,-u,U,u,) = ItI, u, - 1) and Ja,u,u,-u,)
= /u’, - 3).
The symbol V&, used for the u-dependent factor in our previous paper (6) corresponds to the quartic combination lUtUfU~UrViUjUkUj)*
The sign before each wi corresponds to the sign of a:, as in (u,ubu,~-iS(ubc)l&b+u;u,u~u,) The S function conveniently as &(vib) =
2
katx {(ue 2’ W, + @b - w,
of the second contact transformation [ c*
abed
=
{$jki(PiPjPti/
+
q/PiPjPk)
+
l)(“b;
‘)(;)r2.
operator
&jki(qtqjqkPI
T2 is rewritten
+
Pfliqjqk))]
iwzj
(nsbsc-sd) =
8(ubcd).
C
abed (n
(A-4)
The matrix elements of each term in Eq. (A-4) are classified into two cases according to whether the sum of the absolute values of the differences in quantum numbers over all the normal modes is 4 or 2. (i) The case 2 IAUiI = 4: i
1 (U,t)bUPdl-i9(UbCd)lU~U~UefU~t),Ubl),Ud) = fw, + t&bf w,+ wd
X
1
(1 + 6i, + a,,)( 1 + bh + 61,)
(ii) The case 2 i
1AUi ) = 2:
(U~Ub~PdI~~~~qdI~~~~~~~~~~“bu~ud).
(A-5)
VIBRATIONAL
WAVE FUNCTIONS
-C{F(&+w,+:pfW m x
c +u
OF Hz0
m &J
243
c
(1 + 26,,)( 1 + 6brn+ &,)( 1 + 2&b + 26,, - 6&&J’
1 1 1 1 + kabmkacm + + + 8 ( w,,, + w, f wb ‘+,, - w, T wb w,,, - 0, * w, w, + w, i w, 1
+
-
%I x
(1
+
x
+
1 w,
+
w,
+
wb
hm
+
Jbm)(l
+
+
6bc +
26,b
(1
+
‘t,,
‘&I
+
r +
wb &m)(1
1 wm +
+ &b
+
26,‘. + 2&b&,)-’
I
+
w,
f
w,
&zc -
}I
w,
-
wa +
w, >
&zb&zc)
(2),2)b2),1q~qbqc~~~a,‘~,~b~,).(A-6)
The case (ii) is a special case of the case (i) with the wave function of the type la~&&&,t$,u~). In Eqs. (A-5) and (A-6), fk, is defined to be 1 when the operators ak+and a: are of the same type with respect to the double sign, and -1 otherwise. The summation with * denotes the sum over all the combinations of i, j, k, and 1 taken to be a, b, c, and d under the restriction of the indicated inequality. The rotational contribution &Y~ has already been given by Amat et al. (3).
APPENDIX
L(HOH) at, a;
:
kl
$ Q4n h h H
Ho,H,,
**-
H' k abc
k abed P x Vi,
Vi +
etc. W,
Vi,
ZVi,
2: NOMENCLATURE
Bond angle between two OH bonds. The creation and the annihilation operators. Coefficient in the expansion of the wave function. The equilibrium rotational constant. Kronecker’s delta. Electric charge of the ith atom. An operator of a molecular property. A flag defined in Appendix 1. Bending coordinate of water. Harmonic wave function of state m. Plan&s constant. Plan&s constant divided by 27r. Hamiltonian. Expanded Hamiltonian. Transformed Hamiltonian. Cubic force constant in dimensionless normal coordinates. Quartic force constant in dimensionless normal coordinates. Dipole moment. A ordering parameter of smallness. Vibrational states or vibrational frequencies. Harmonic frequency of ath normal mode.
244
YAMAOKA AND MACHIDA
Vibrational wave function of state m. Momenta conjugate to qs. Dimensionless normal coordinate. OH stretching coordinates of water. Molecular structure in the ground state. Bond length of OH bond. Internal coordinate. S functions corresponds to the transformation operator T, T,,T2, . . . . Coefficient in the expression of S, . Coefficient in the expression of S2. Contact transformation operators. Quantum number of the ith normal mode. The rotational contribution to the vibrational Hamiltonian. Coriolis coupling constant. RECEIVED:
November
12, 1982 REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
H. H. NIELSEN, in “Handbuch der Physik” (S. Fliigge, Ed.), Vol. 37, Springer-Verlag, Berlin, 1959. M. GOLDSMITH, G. AMAT, AND H. H. NIELSEN, J. Chem. Phys. 24, 1178-l 182 (1956). G. AMAT, M. GOLDSMITH, AND H. H. NIELSEN, J. Chem. Phys. 27, 838-844 (1957). G. AMAT AND H. H. NIELSEN, J. Chem. Phys. 27, 845-850 (1957). G. AMAT AND H. H. NIELSEN, J. Chem. Phys. 29,665-672 (1958). K. MACHIDA AND Y. TANAKA, J. Chem. Phys. 61, 5040-5049 (1974). J. C. DECIUS,J. Mol. Spectrosc. 57, 348-362 (1975). M. AKIYAMA, J. Mol. Spectrosc. 84, 49-56 (1980). I. SUZUKI, J. Mol. Spectrosc. 80, 12-22 (1980). G. D. CARNEY, L. L. SPRANDEL, AND C. W. KERN, Adv. Chem. Phys. 37, 305-379 (1978). A. C. JEANNOTTE II, C. MARCOTT, AND J. OVEREND, J. Chem. Phys. 68,2076-2080 (1978). M. TOYAMA, T. OKA, AND Y. MORINO, J. Mol. Spectrosc. 13, 1931213 (1964). G. RILEY, W. T. RAYNES, AND P. W. FOWLER, Mol. Phys. 38, 877-892 (1979). C. SECROUN,A. BARBE, AND P. JOUVE,J. Mol. Spectrosc. 45, l-9 (1973). S. J. YAO AND J. OVEREND, Spectrochim. Acta Part A 32, 1059-1065 (1976). M. L. LAB~DA AND J. OVEREND, Spectrochim. Acta Part A 32, 1033-1042 (1976). I. SUZUKI, K. SCANLON, AND J. OVEREND, Spectrochim. Acta Part A 35, 1341-1346 (1979). W. S. BENEDICT,N. GAILAR, AND E. K. PLYLER, J. Chem. Phys. 24, 1139-1165 (1956). D. F. SMITH, JR., AND J. OVEREND, Spectrochim. Acta Part A 23,471-483 (1972). S. A. CLOUGH, Y. BEERS,G. P. KLEIN, AND L. S. ROTHMAN, J. Chem. Phys. 59,2254-2259 (1973). J. M. FLAUD AND C. CAMY-PEYRET, J. Mol. Spectrosc. 55, 278-310 (1975). C. CAMY-PEYRET AND J. M. FLAUD, Mol. Phys. 32, 523-537 (1976). J. M. FLAUD, C. CAMY-PEYRET, J. Y. MANDIN, AND G. GUELACHVILI,Mol. Phys. 34,4 13-426 (1977). W. C. ERMLER AND B. J. KROHN, J. Chem. Phys. 67, 1360-1374 (1977). H. KUZE, T. AMANO, AND T. SHIMIZU, J. Chem. Phys. 75,4869-4872 (198 1). J. W. C. JOHNSAND A. R. W. MCKELLAR, Cannd. J. Phys. 56,737-743 (1978).
OF MOLECULAR
SPECTROSCOPY
(1983)
lt&234-244
Anharmonic Vibrational Wave Functions, Infrared Band Intensities, and Dipole Moments of Water YUMIKO
YAMAOKA
Faculty of Pharmaceutical Sciences, Kobe-Gakuin
University, Nishi-ku, Kobe, Japan
AND KATSUNOSUKEMACHIDA Faculty of Pharmaceutical Sciences, Kyoto University, Sakyo-ku, Kyoto, Japan
A new and simple method to expand the anharmonic vibrational wave functions with respect to the harmonic oscillator wave functions is proposed. The coefficients of the expansion are given as matrix elements of the S function of the contact transformation in the perturbation theory and the explicit expressions of these coefficients are given within the approximation to the second order in X. As an example of the expansion, the wave functions of water molecules were calculated and applied to the calculation of infrared band intensities and average values of dipole moments in several states. INTRODUCTION
Determination of the anharmonic potential functions for molecular vibrations cannot be accomplished without precise assignments of fundamental and summation frequencies in infrared and near-infrared spectra. In attempting to assign a band to a specific vibrational level of a polyatomic molecule, however, we often encounter complicated anharmonic resonances. If a molecule involves three resonances of the Fermi type ViN uj + Uk, uj N ul + u,, and uk N up + up, we should take account of 14 levels resonating in either the first or the second order in the vicinity of 2Vi. In the region around 3u,, the corresponding resonance polyad contains 30 levels. In such cases it is almost impossible to assign the observed bands correctly without recourse to some preliminary knowledge on the band intensities. Theoretical prediction of the intensities of fundamental and summation bands incorporating both the electrical and the mechanical anharmonicity is thus highly desirable. In addition to the frequencies and the band intensities of vibrational levels, the r, structure, the dipole moment of the vibrationally excited states, and many other physical properties have been known to bc perturbed by the anharmonicity of the potential. It is of fundamental importance to determine the vibrational wave functions under the anharmonic potential as precisely as possible for evaluating the expectation values of these properties. In this report a new and simple method for expanding the vibrational wave functions of polyatomic molecules with respect to the harmonic oscillator wave functions is 0022-2852/83 Copyright All
rights
0
$3.00
1983 by Academic
of reproduction
Press.
in any form
Inc. reserved.
234
VIBRATIONAL
WAVE
FUNCTIONS
OF Hz0
235
proposed. The expansion coefficients are expressed in terms of the matrix elements
of the S function of the contact transformation in the perturbation theory. Transition moments of a number of vibrational levels of water are calculated as an example of application of the present method. PROCEDURE
perturbation theory of molecular vibrations developed by Nielsen et al. (Iis chosen so as to eliminate the cubic and quartic terms in the transformed Hamiltonian H’ = THT-‘. (1) In the
j), the contact transformation
Since the ma&ix elements of the transformed Hamiltonian on the basis of the harmonic wave functions & are given by (#W’I~,)
= (CT-%,)*IH
IV-‘&))
= L,
(2)
the wave function $,, of the primitive Hamiltonian H may well be approximated by T-‘4, and is expanded as k, = T-I& = C amn& (3) m where the coefficient a,,,,, is given by amn = ($&IT-‘I&). If we restrict the expansion to the linear terms of X, the transformation is given by T-’ = T;’ = exp(-iXS,) N 1 - AS,
(4) operator T-’ (5)
where TI is the operator which eliminates the cubic terms in the vibrational Hamiltonian and S, is the corresponding S function. Then the expansion coefficients umn in Eq. (3) are given by using the matrix elements of the S function as Grin = &WI- ~~(~~lS,I~,).
(6)
As S1 contains only cubic terms of q and p, the second term in Eq. (6) is zero unless the summation of the absolute values of the differences in the vibrational quantum number between the states m and n is equal to 1 or 3. This condition restricts the number of harmonic wave functions C#J,which contribute to the perturbed wave function $, and facilitates the subsequent calculation. The explicit expressions of SI and its matrix elements are given in Appendix 1. The second contact transformation makes the approximation correct to the quadratic terms of X, where the transformation operator T-’ is expressed by T--l = T;’ T;’ N 1 - ihS, - X2($/2 + is,).
(7)
In addition to the matrix elements of S, and S2, those of ST are required to calculate the coefficients a,,,” in this approximation. They are easily obtained by using the relation
(4w:b”)
= F (4~lS,l~k)(dJX IS&“).
The matrix elements of SZ are also given in Appendix 1.
236
YAMAOKA AND MACHIDA
For polyatomic molecules the usual perturbation method often breaks down by the accidental degeneracy of the energy levels. When such a resonance occurs, the resonance denominator becomes nearly equal to zero and the transformed Hamiltonian gives an abnormally large perturbation energy. Dropping the corresponding denominator we get a modified transformation operator T’ which leaves the off-diagonal matrix elements between the resonating levels unchanged from those of the unperturbed Hamiltonian (I, 6). After transformation by the modified operator the energy matrix is diagonalized numerically to give the resonance-perturbed wave function by using the eigenvectors. The matrix elements of a given operator f which depends on the nuclear coordinates are calculated by (IclZlfl$,) = c &+#Q IfldJa,,. (9) if The time required to perform this calculation can be saved greatly by utilizing the sparseness of the matrices of ai, and (titlfl@,). To calculate the infrared band intensities, it is useful to apply the effective chargecharge flux model (7, 8) to higher order. The dipole moment P is defined as /.l = 2
eiXi
(10)
where ei and xi are the electric charge and the position vector, respectively, of the atom i. Differentiating Eq. (10) with respect to the dimensionless normal coordinates, we have
and
de. a2Rj c c-__x._ i , aRj aqd% 1
(12)
The effective charges and their derivatives with respect to the internal coordinates were taken as adjustable parameters in the following calculation. DISCUSSION
There have been two well-known methods of approximation which may be complementary to each other in studying the vibrational wave functions, i.e., the variational and the perturbation methods. By applying the variational method Suzuki proposed the direct diagonalization of the vibration-rotation Hamiltonian based on the harmonic oscillators to calculate the vibrational anharmonicity (9). Other basis sets such as Morse oscillators have also been used in the variational method to obtain the vibrational energy levels (10). However, the application of the variational method to large molecules is seldom seen because it requires the diagonalization of an enormous matrix.
VlBRATIONAL
WAVE
FUNCTIONS
OF Hz0
237
On the other hand, the second-order perturbation theory developed by Nielsen has been successfully used for the calculation of vibrational frequencies of molecules containing more than four atoms (6, 1 I). Explicit expressions for the first-order wave functions have also been given in this method (12, 13). An advantage of the variation method over the perturbation method has been the flexibility of the former to a higher-order approximation than the first. Even in the perturbation method, however, it is rather easy to obtain the second-order wave functions if we use the matrix elements of the S function, The difference between the variational method and the perturbation method lies in the way of truncation. The former truncates the matrix elements with higher quantum numbers, while the latter truncates the higher-order force constants. Our method corresponds to an analytical diagonalization of the infinite matrix of the Hamiltonian containing quadratic through quartic force constants, so it might be regarded as a hybrid of the two types of approximation. The merit of this method is that we need to handle only the wave functions related to the experimental data available in case by case. The higher-order expressions of infrared band intensities of polyatomic molecules based on the perturbation method were first given by Secroun et al. (14) for the case of bent triatomic molecules. The general expressions for band intensities and average values of physical observables were established by Overend and his co-workers (IS17). Their method consists of the contact transformation of the operator of any physical quantities expanded in terms of the dimensionless normal coordinates and the term by term calculation of the matrix elements of the transformed quantities. Meanwhile, no explicit expressions of band intensities or other physical quantities are required in our method because all the computations are reduced to the matrix multiplication in the form of Eq. (9). It is not necessary to use different formulae for different combinations of vibrational quanta: see Appendix 1. Even the energy levels and the rotational constants can be calculated without using the equations derived by Nielsen. The obtained wave functions are also applicable to knowing the actual form of molecular vibrations as cross-sectional plots of wave function amplitudes (IO). EXAMPLE
As an example of the method in the preceding section, the wave functions of several states of water were calculated within the approximation to the first order in X. The structural parameters were taken to be r(OH) = 0.9576 A and L(HOH) = 104.29” according to Benedict et al. (18) (Fig. 1). The quadratic through the quartic force constants of water were taken from Set III by Smith and Overend (19). It has been known that vi resonates with 2~ through the Fermi-Dennison resonance and 2v, with 2~ through the Darling-Dennison resonance. The resonance denominators ofthe Fermi-Dennison resonance vi-2~ were omitted from the expansion coefficients and the energy matrix related to these resonances were diagonalized numerically to obtain the energy levels and the wave functions. All computations were carried out on a NEAC System 300 computer at Kobe-Gakuin University. Some of the obtained wave functions are shown in Table I. For the calculations of band intensities of this molecule, it is necessary to determine the atomic charges and their derivatives with respect to the internal coordinates. The
238
YAMAOKA AND MACHIDA
FIG. 1. The definition of molecular axes and internal coordinates of water molecule.
atomic charges were estimated from the equilibrium dipole moment of Hz0 (20) with a slight modification. The first derivatives were so chosen as to reproduce the observed dipole moment derivatives with respect to dimensionless normal coordinates (20). The second derivatives were estimated by the method of least squares with reference to the same set of the observed intensities as used in the variational method by Carney et al. (IO). Two sets in Table II reproducing the observed band intensities equally well were obtained. In Table III, the derivatives of the dipole moment with respect to the dimensionless normal coordinates calculated from two sets of parameters are compared with the values derived from the recently observed data by Flaud and Camy-Peyret (21-23) and those determined from the ab initio MO calculation by
TABLE I Coefficients of Vibrational Wave Functions for Lower Levels of Hz0 (umn X 1000)
3
010
000
0
000 100 010 200 110 020 002 300 210 120 030 102 012 400 310 220 130 040 202 112 022 320 230 140
-11 -
1000
-158 11 -23 3 5 -11 -38 9 -
050
-
122 032
l
1000 5 -145 -39 14 -23 4 9 -21 -38 13 -
Involved
ozo* 11 71 39 -24 2 997 -8 7 -132 -115 20 -3 0 0 -1 -4 1 -22 5 12 -33 -38 16
100* 157 997 -3 -341 35 -71 -109 -0 9 8 -1 -45 5 7 -11 -53 9 2 -0 -1 2 3 -1
in a resonance
001 101 011 201 111 021 003 301 211 121 031 103 013 401 311 221 131 041 203 113 023 321 231 141 051 123 033
polyad
001
011
1000 -321 87 -23 3 5 -11 -65 16 -
-87 1000 5 -308 68 24 -23 4 9 -21 -65 23 -
-
of "1 = 2v2
021*
101'
30
320
94
996
-68 -54 10 996 -18 7 -294 16 35 -4 0 1 -1 -9 1 -23 5 12 -33 -65 28
6 -570 111 -94 -188 -1 28 -2 -3 -45 5 7 -11 -92 15 2 -0 -1 3 6 -3
239
VIBRATIONAL WAVE FUNCTIONS OF Hz0 TABLE II Effective Atomic Charges and Their Flux of Water Molecule” set I
Parameter eH
set
II
unit
1.5859
10-10
es"
deH/dr
-1.0384
10-10
eSU/A
deH/dr'
-0.2397
10-lo
eSU/A
de,/d$
0.4146
d2eH/dr2
1.9
10-10
esu/rad
0.3
10-10
em/A'
d2eH/drdr'
0.7
-
10-10
eSU/A2
d2eH!dr' 2
0.6
-
10-10
W/A2
-1.5
0.3
10-10 esu/A.rad
d2eH/dr'd$
0.6
-0.8
10-10
esu/A.rad
d2e,/de2
0.5
0.2
10-10
esu/rad2
d2eH/drd$
=For descriptions relations
were
e. = -2eH,
of coordinates
see Fig.
1.
The following
assumed:
deo/dr
= -d+dr
d2e0,'dr2 = -d2eH/dr2
- deH/dr',
deo/d#
- d2eH,'drv2, d2eO/drdr'
d2e0,'drd+ = -d2eH/drd$
= -2dea/d$, = -2d2eH/drdr',
-d2eH,'dr'd$ and d2e,/d$2
= -2d2e,/dQ2.
Ermler and Krohn (24). In spite of the disagreement in signs of the second derivatives with the ab initio method, Set II reproduces the observed derivatives very well. By using the parameters tabulated in Table II, the infrared band intensities of H20, D20, and HDO were calculated. The results are compared with the observed intensities TABLE III Calculated Derivatives of Dipole Moment of HZ0 with Respect to Dimensionless Normal Coordinates” Axis z
Y
ab initio'
"e
-1.8587
-1.8587
al
0.0216
0.0216
0.06052
"2
-0.1617
-0.1617
-0.22275
-1.997490
511
-0.0098
0.0052
512
0.0049
-0.0065
0.00828
822
-0.0051
0.0137
-0.00758
833
-0.0052
0.0028
0.0026
a3
-0.0951
-0.0951
-0.1196731
0.0003
-0.0098
0.0070180
'23
-0.0447
0.0296
-0.0262108
The
in Debye.
signs
Ref.
(20).
Ref.
(24)
Obsd.c
Obsd.d
-1.8473 0.0216 -0.161
0.0217 -0.17946
-0.0017
813
aUnits
b
This work Set I set II
Index
-0.00843 0.013
-0.0950
-0.09714
0.0267
Description of indices are the same. as in Ref.(Z4). of the normal coordinates were taken in the Same way as
'Ref.(20).
dRef.(21)-(23).
2.28
4.41
0.0068
0.0039
0.617
0.973
0.0003
011
120
021
200
101
002
(IO).
0.059
2.67
0.640
0.065
0.0042
4.82
0.113
42.6
2.01
0.351
52.2
set II
aRef.
0.108
110
42.6
100
001
0.344
52.2
010
I
020
set
v,v*LJ3
HZ0 ~~__
0.032
4.50
0.318
0.340
0.021
5.45
0.110
48.2
2.18
0.396
63.8
Obsd.
I
0.064
1.32
0.256
0.027
I
0.663
0.078
0.343
0.036
0.028
2.35
0.359
26.3
10.4
0.094
46.2
set
HDO
(IO).
0.00078
0.892
0.183
0.066
0.0040
5.69
0.039
20.6
8.60
0.320
47.9
VX.a
in Ref.
0.0017
1.98
0.052
26.4
1.79
0.171
28.3
set II
et ~2. cited
0.0058
0.385
0.211
0.0009
0.0022
1.70
0.048
26.4
1.96
0.105
28.3
set
Dz0
b R. A. McClatchey
0.0090
1.94
0.379
0.146
0.0069
14.2
0.115
43.0
11.6
0.473
90.7
Var.a
b
__~_~..
Calculated Infrared Band Intensities of H,O, D20, and HDO (in km.
TABLE IV
1.85
0.020
0.638
0.035
0.043
1.99
0.889
26.3
10.0
0.565
46.2
set II
mole-')
1.14
0.166
0.564
0.0099
0.016
6.59
2.03
27.8
13.8
0.302
74.8
Var.a
30.1
13.5
54.7
Obsd.
_~_ b
VIBRATIONAL
WAVE
FUNCTIONS
241
OF Hz0
and those calculated by the variational method (10) in Table IV. The average values of dipole moment in the ground states of HzO, DzO, and HDO and in some vibrationally excited states of Hz0 were also calculated and are compared with the observed and calculated values (Table V). In this calculation, Set II reproduces not only the observed isotope effect of the ground state dipole moment (20) very well but also the dipole moment in the u2 state observed by Kuze et al. (25). The difference between the dipole moments of the v2 state and the ground state of H2”0 observed by Johns and McKellar (26), 0.03 10, is consistent with our result of H2’*0, 0.0318. A slight disagreement with the values by Riley et al. (13) might be due to their elimination of unobserved derivatives of the dipole moment. The success of Set II in elucidating the infrared intensities and the dipole moments simultaneously may be taken as proving the excellence of the wave functions calculated from the present method. 1
APPENDIX
The explicit formula of Si has been given by Amat et al. (3) and the vibrational part of S, is expressed in the form of summation over the combination of three normal modes as {S abc&dbPc + ~~b~(~a‘?d)c Sl(vib) = 2 + P&&b) abc @
+ %f(%GPa
+ PdMc) + S%(WLPb
+
Pbqc%))
=
c
%-JbC).
(A-1)
nbc (nsbac)
Because the matrix elements of pJh and go are identical except for their signs, the matrix elements of each term in Eq. (A-l) are reduced to a simple expression:
In Eq. (A-2), a:, the creation or the annihilation operator for the ith vibrational quantum, is used to simplify the equation. The following are some examples of the wave functions in terms of at and a;: TABLE V Average Values of Dipole Moment in the Ground and Vibrationally
Tsotope HP
Axis
z
Calcd.
state
set
I
Excited States (in Debye) Obsd.
Pet II
Rileva
0 00
-1.8761
-1.8552
-1.8455
100
-1.8964
-1.8606
-1.8552
010
-1.8631
-1.8234
-1.8108
001
-1.9036
-1.8746
-1.8667
Clouqhb -1.8546
-1.827
D2O
2
000
-1.8719
-1.8566
-1.855R
HDO
a
000
-1.7480
-1.7306
-1.7318
b
0
0.6683
0.6567
aRef.
(13).
0
0
0.6739 b
Ref.
(20).
'Ref.
(25).
K"zeC
242
YAMAOKA
Iu~u&2&p&
AND MACHIDA
= lu, + 1 Ub+ 1 U, + 1)
l&z~U~U,U,) = Iv, + 2 U, + 1) ~u,+u,-u,U,u,) = ItI, u, - 1) and Ja,u,u,-u,)
= /u’, - 3).
The symbol V&, used for the u-dependent factor in our previous paper (6) corresponds to the quartic combination lUtUfU~UrViUjUkUj)*
The sign before each wi corresponds to the sign of a:, as in (u,ubu,~-iS(ubc)l&b+u;u,u~u,) The S function conveniently as &(vib) =
2
katx {(ue 2’ W, + @b - w,
of the second contact transformation [ c*
abed
=
{$jki(PiPjPti/
+
q/PiPjPk)
+
l)(“b;
‘)(;)r2.
operator
&jki(qtqjqkPI
T2 is rewritten
+
Pfliqjqk))]
iwzj
(nsbsc-sd) =
8(ubcd).
C
abed (n
(A-4)
The matrix elements of each term in Eq. (A-4) are classified into two cases according to whether the sum of the absolute values of the differences in quantum numbers over all the normal modes is 4 or 2. (i) The case 2 IAUiI = 4: i
1 (U,t)bUPdl-i9(UbCd)lU~U~UefU~t),Ubl),Ud) = fw, + t&bf w,+ wd
X
1
(1 + 6i, + a,,)( 1 + bh + 61,)
(ii) The case 2 i
1AUi ) = 2:
(U~Ub~PdI~~~~qdI~~~~~~~~~~“bu~ud).
(A-5)
VIBRATIONAL
WAVE FUNCTIONS
-C{F(&+w,+:pfW m x
c +u
OF Hz0
m &J
243
c
(1 + 26,,)( 1 + 6brn+ &,)( 1 + 2&b + 26,, - 6&&J’
1 1 1 1 + kabmkacm + + + 8 ( w,,, + w, f wb ‘+,, - w, T wb w,,, - 0, * w, w, + w, i w, 1
+
-
%I x
(1
+
x
+
1 w,
+
w,
+
wb
hm
+
Jbm)(l
+
+
6bc +
26,b
(1
+
‘t,,
‘&I
+
r +
wb &m)(1
1 wm +
+ &b
+
26,‘. + 2&b&,)-’
I
+
w,
f
w,
&zc -
}I
w,
-
wa +
w, >
&zb&zc)
(2),2)b2),1q~qbqc~~~a,‘~,~b~,).(A-6)
The case (ii) is a special case of the case (i) with the wave function of the type la~&&&,t$,u~). In Eqs. (A-5) and (A-6), fk, is defined to be 1 when the operators ak+and a: are of the same type with respect to the double sign, and -1 otherwise. The summation with * denotes the sum over all the combinations of i, j, k, and 1 taken to be a, b, c, and d under the restriction of the indicated inequality. The rotational contribution &Y~ has already been given by Amat et al. (3).
APPENDIX
L(HOH) at, a;
:
kl
$ Q4n h h H
Ho,H,,
**-
H' k abc
k abed P x Vi,
Vi +
etc. W,
Vi,
ZVi,
2: NOMENCLATURE
Bond angle between two OH bonds. The creation and the annihilation operators. Coefficient in the expansion of the wave function. The equilibrium rotational constant. Kronecker’s delta. Electric charge of the ith atom. An operator of a molecular property. A flag defined in Appendix 1. Bending coordinate of water. Harmonic wave function of state m. Plan&s constant. Plan&s constant divided by 27r. Hamiltonian. Expanded Hamiltonian. Transformed Hamiltonian. Cubic force constant in dimensionless normal coordinates. Quartic force constant in dimensionless normal coordinates. Dipole moment. A ordering parameter of smallness. Vibrational states or vibrational frequencies. Harmonic frequency of ath normal mode.
244
YAMAOKA AND MACHIDA
Vibrational wave function of state m. Momenta conjugate to qs. Dimensionless normal coordinate. OH stretching coordinates of water. Molecular structure in the ground state. Bond length of OH bond. Internal coordinate. S functions corresponds to the transformation operator T, T,,T2, . . . . Coefficient in the expression of S, . Coefficient in the expression of S2. Contact transformation operators. Quantum number of the ith normal mode. The rotational contribution to the vibrational Hamiltonian. Coriolis coupling constant. RECEIVED:
November
12, 1982 REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
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