Analysis of vibrational absorption intensities in benzene and methyl benzenes

85

Journal 01 Mcilecular Structure, 213 (1992) 8598 Elsevier Science Publishers B.V., Amsterdam

Analysis of vibrational absorption intensities in benzene and methyl benzenes Boris Galabov’, Sonya Ilieva”, Todor Gounev’ and Derek Steeleb BDepartment of Chemistry, University of Sofia, Sofia 1126 (Bulgaria) ‘Department of Chemistry, Royal Holloway and Bedford New College, University of London Egham, Surrey, TWZO OEX (UK) (Received 19 May 1992)

Abstract Vibrational absorption intensities in the gas phase for toluene-d,, toluene-d, and pxyelene-d,, have been determined. An optimized set of dipole-moment derivatives associated with aromatic and methyl group vibrations have been determined using an appropriate double-mode refinement process from the experimental infrared intensity data for benzene-d,, toluene-d,, and toluene-d,. The signs of intensity parameters are determined by 4-3lG ab initio MO calculations. The set of dipole derivatives is used in predicting the spectrum of p-xylene.

INTRODUCTION

Vibrational spectra provide two main parameters that can be used for experimental characterization of the geometrical and electronic structure of molecules: the frequencies and intensities of the vibrational bands. Analysis of the observed vibrational wavenumbers via normal coordinate analysis permits the evaluation of the parameters of the potential force field of molecules [l]. The set of force constants represents a quantitative expression of the forces binding the atoms in a molecule. At present, the valence force field is generally accepted as the most plausible representation of the intramolecular forces from a physical point of view. The analysis of the observed vibrational intensities is aimed at determining molecular parameters that would appropriately illustrate the electric charge distribution in a molecule and its dynamics with vibrational distortions. A number of different molecular parameters are currently employed in interpreting intensities: electro-optical parameters (EOPs) [2], atomic polar tensors Correspondence to: Professor B. Galabov, Department of Chemistry, University of Sofia, Sofia 1126, Bulgaria.

0022-2860/92/$05.00 0 1992 Elsevier Science Publishers

B.V. All rights reserved.

86

B. Galabov et al./J. Mol. Struct., 273 (1992) 8598

(APTs) [3], eff ec t’ive atomic charges and charge fluxes (ECCFs) [4], bond charge tensors (BCTs) [5], and bond polar parameters (BPPs) [6]. It should be emphasized that force constants and intensity parameters are interconnected through the matrix of vibrational eigenvectors. In general terms we may say that intensities of vibrational bands are determined by two factors: (i) the charge reorganizations accompanying vibrational distortions; (ii) the amplitudes and forms of vibrations. The first factor is represented by the set of intensity parameters, while the second is reflected in the eigenvector matrix L [l]. A satisfactory solution of the vibrational problem requires, therefore, that the force constants and the respective L matrices reproduce correctly both observed vibrational frequencies and intensities. Such tests for the quality of experimental force fields are not popular, and have been performed on a limited number of small symmetric molecules. The quality of ab initio MO calculations of vibrational spectra is currently assessed by simultaneous determinations of force constants and intensities. It seems only natural that the same procedure is followed, at least where possible, in analysing experimental vibrational spectra. In this paper, experimental vibrational absorption intensities for benzene, toluene-d, and toluene-d, are analysed using normal coordinate transformation matrices evaluated from the force field for methyl benzenes proposed by La Lau and Snyder [7]. The valence force field of these authors has been determined from frequency data for a number of methyl benzenes including some isotopic derivatives. Experimental gas-phase infrared intensities for toluene-d, and toluene-d, determined in the present work and literature data for benzene are used. A set of local infrared intensity parameters are evaluated using experimental intensities and theoretical estimates of the respective dipole moment derivatives are obtained from RHF 431G ab initio MO calculations. The refined set of parameters is then used in predicting the intensities of the infrared bands of p-xylene. EXPERIMENTAL

The integrated infrared intensities of toluene-d, and toluene-d, were determined on a Perkin-Elmer model 983G ratio recording spectrophotometer at different partial pressures of the gas samples. Spectra were measured with spectral resolution varying from 3.6cm-l at 3000cm-’ to 2.1 cm-l at 1200 cm-’ (mode 5, filter 4). The integration was carried out with a specially written program using the numerical representation of the spectra accumulated with the Perkin-Elmer 3600 data station. The absolute intensities were calculated from the slopes of the Beer’s law plots. Each plot was defined by 3-15 individual points reflecting the dependence ApZ/pZ,where A is the absolute infrared intensity, p is the partial pressure

B. Galabov et ok/J. Mol. Struct., 273 (1992) 85-93

87

of the sample and I is the path length. Linearity of the optical/detector system was shown to be within 0.5% by use of a calibrated sector system (Stanford Research Systems, Inc.). The zero transmittance of the PerkinElmer 9830 was corrected for stray radiation using software driven correction as supplied by the manufacturer. Toluene-d, (Reidel-de Haen, purity > 99.7%) and toluene-d, (Aldrich, purity > 99.96%) were used as purchased. A sample of purified p-xylene-do , a gas chromatography standard, was used. Overlapping band areas with pronounced individual absorptions were separated with the aid of a spectral curve fitting program [a]. The integrated intensities for 12 absorption areas of toluene-d,, 16of toluene-d, and 7 ofp-xylene were determined. The absolute infrared intensities for benzene were taken from the literature [9]. CALCULATIONS

Normal coordinate analysis The valence force field for benzene and methyl benzenes developed by La Lau and Snyder [7] is used throughout this study. There are a number of more recent calculations on the force fields of benzene, toluene and other alkyl benzenes [N-12]. Particularly accurate appears to be the force field for benzene determined by Pulay and Fogarasi [lo] on the basis of combined experimental and ab initio MO data. The force field of La Lau and Snyder [7] is, however, common for the entire series of methyl benzenes. It can, therefore, serve as an appropriate basis for testing the simultaneous transferability of force constants and intensity parameters. The normal coordinate analysis is performed in the set of ordinary internal coordinates as used by La Lau and Snyder [7]. The intensity analysis is carried out in terms of non-redundant internal coordinates describing distortions of separate fragments of methyl benzenes. In most cases these represent appropriate linear combinations of ordinary internal coordinates. These sets of vibrational coordinates are defined in Tables 1 and 2 and in Fig. 1. The force field of La Lau and Snyder [7] reproduces satisfactorily the observed vibrational frequencies of benzene, toluene and various xylenes including the respective fully deuterated derivatives. The resulting L matrices which define the forms of the vibrations are employed in determining an optimized set of local infrared intensity parameters and in predicting intensities for p-xylene. Intensity analysis As mentioned, there are a number of alternative theoretical formulation

88

B. Galabov et al./J. Mol. Struct., 273 (1992) 81t98

TABLE 1 Internal coordinates for methyl benzenes Coordinate”

Description

AT,

Aromatic C-C stretching C-CH, stretching Aromatic C-H stretching C-CH, in-plane deformation C-H in plane deformation C-C-C ring deformation C-H stretching in CH, HCH deformation in CH, HCC deformation in CH, C-CH, out-of-plane deformation Aromatic C-H out-of-plane deformation Aromatic C-C torsion C-CH, torsion

AR ASi A#‘, A@ AC Ah AQi Ar, AUi AR AM APi AZ, At,

“The internal coordinates are shown in Fig. 1.

TABLE 2 Local symmetry coordinates for methyl benzenes” Aromatic vibrations S, = AT S, = As S,, = (A& - A+‘)/2”” So.,,= AP S, = As1 Methyl vibrations S, = AR k$ = (A@ - AW)/2i” ,,eP = AM S,, = (ArI + AF, + Ar,)/3l” S_, = (iAr, - Ar, - Ar,)/12*‘* S,, = (- Aa, - Au, + AaQ + BP, + A/?,+ A/3,)/6l” S,,, = (2Aa, - Aaz - Aa,)/121’2 Srnckl= (2AS, - APz - AP3)/121’2 Sasz= (Ars - Ar,)/21i2 S,,, = @x3 - Aa,)/2l” Smu = (Ah - A~J2”” S, = At, “Descriptions of the internal coordinates are given in Table 1.

B. Galabov et al.lJ. Mol. Struct., 273 (1992) 8Ei-98

89

Fig. 1. Definition of internal coordinates for methyl benzenes.

of infrared intensities [2-6] that can be used in analysing the measured gas-phase data. For small, symmetric molecules, where all individual intensities for the active modes can be determined, an explicit non-approximate solution of the inverse intensity problem is possible. For such systems, APTs describing charge fluctuations associated with atomic displacements can be derived [3]. If the interest is in characterizing polar properties of valence bonds, the BPP formulation can be applied [6]. Both of these approaches employ as intensity parameters derivatives of the molecular dipole moment with respect to atomic and bond displacement coordinates, respectively. The relationships between different theoretical formulations of intensities have been discussed in a number of studies [B-16]. For larger and less symmetric molecules, non-approximate solutions for the inverse intensity problem are not possible, mostly due to considerable band overlap in many parts of the spectra. In such cases approximate solutions are sought following a least-squares refinement of parameters. In the present study we chose to perform such calculations employing as intensity parameters derivatives of the molecular dipole moment with respect to group symmetry coordinates. Such an approach was first used by Snyder [17] in calculations of infrared intensities for a number of n-alkanes in a crystalline state.

90

B. Galabovet al./J. Mol. Struct., 273 (1992) 8598

To carry out these calculations, an optimization program for infrared intensity parameters in Fortran was developed. It is described in the following section. As in all least-squares refinement procedures, the reliability of the results obtained depends strongly on the ratio between available experimental data and the number of parameters to be determined. It is always preferable to use as extended a set of experimental data as possible. It is also quite important that these are determined with reasonable accuracy. In the case of vibrational intensities such a requirement poses severe problems, since accurate individual integrated intensities for correctly assigned vibrational bands can only be obtained for smaller molecules possessing some symmetry. In the refinement process of intensity parameters carried out in this study, we used experimental gas-phase integrated infrared intensities for the four active modes of benzene reported by Overend and Youngquist [9] and later confirmed by measurements of Kondo et al. [18]. The experimental absolute infrared band intensities for toluene-d, and toluene-d, determined in the present study are also used. The observed intensities for toluene-d,, and benzene are given in the last column of Table 3. The corresponding results for toluene-d, are presented in Table 4. Twelve separate intensity values for toluene-d, were obtained. Where possible, partly overlapped bands were separated using a computer separation procedure [8]. It can easily be seen from the data given in Table 4 that many of the closely situated bands cannot be separated in any reasonable way. Better conditions for separation of overlapped bands were found in the case of toluene-c&. Altogether, 16 intensities for individual bands or for closely situated overlapped bands were determined. In the present study, 20 non-zero intensity parameters were refined to fit 32 experimental intensities. The conditions for carrying out the refinement process were substantially improved by imposing constraints on the parameter values resulting from RHF 4-31G ab initio MO calculations [19] of dipole-moment derivatives. The respective theoretical dipole-moment derivatives were calculated using the finite differences method for small displacements from the theoretical equilibrium geometry along the respective set of vibrational coordinates (Table 2). The ab initio MO estimates for the respective i?p/iB, derivatives are shown in Table 3. All parameter values refer to group local coordinate systems defined in Fig. 2. As a general rule all signs for ap/iW, , as obtained from the quantum mechanical calculations, were retained. Parameters shown by the ab initio calculations to have very small values were set to zero. The program DIMODE The programe DIMODE

developed in the present study calculates nrpole

91

B. Galabov et al./J. Mol. Struct., 273 (1992) 8&98 TABLE 3 Experimental and calculated intensities for benzene and toluene-d,, A CdC (km mol-‘)

VC.lC

(cm-‘)

Benzene 3054.7 3054.7

A em (km mol-‘)

elu

35.54 35.54

71.08

68.70

1487.9 1487.9

elu

9.53 9.53

19.06

16.10

1032.0 1032.0

elu

3.62 3.62

7.24

8.48

74.12

74.12

84.60

4.03 19.16

23.19

26.45

0.49 3.40 31.85

35.74

38.18

676.8 Toluene-d, 3056.5 3054.9 3057.7 3055.0 3054.6

al

b, al al

b,

2950.8 2949.6

b,

12.94 5.50

18.44

18.52

2900.8

al

10.42

10.42

11.05

1614.3 1589.2

al

b,

0.61 0.95

1.56

7.26

1503.8

al

11.50

11.50

14.87

3.78 2.45 1.16

7.39

6.54

1461.2 1451.3 1443.3

b,

4 b, b,

1375.4 1326.9

al

b,

3.26 0.02

3.28

3.08

1081.2

b,

3.52

3.52

4.30

1038.4 1026.0

b,

1.43 2.30

3.73

2.67

al

0.72 34.48 18.88 0.01

54.09

40.96

0.08 6.00

6.08

6.24

793.3 734.7 698.0 616.5 526.1 462.7

al

b, b,

b, al

b,

92 TABLE

B. Galabov et al./J. Mol. Struct., 273 (1992) 8598 4

Calculated

and experimental

intensities

for toluene-d,

A ealc (km mol-‘) 2277.0 2275.6

al

$Zmol-‘)

4

0.52 0.07 0.64 13.42 20.73 5.93 2.85

2091.6

al

5.28

5.28

3.98

1572.1

al

1538.9

b,

0.76 0.26

1.02

8.37

1389.4

aI

4.49

4.49

5.85

1328.3

b,

1.06

1.06

0.55

1276.5

b,

0.03

0.03

0.65

1171.6

al

0.43

0.43

1.18

1044.3

0.37 3.34

5.14

1035.0

b, b, b, b,

954.9

al

0.01

862.8

b,

0.40

855.1

al

0.00

0.54

0.52

840.9

b,

0.14

825.0

al

1.65

815.9 809.0

b, b,

2.60 0.14

4.39

4.55

755.8

b,

0.45

0.45

0.32

725.2 723.0

al

2.00

b,

0.12

2.12

0..68

604.3 594.4

b, b,

0.70 0.00

0.70

0.87

541.4

h

23.55

23.55

20.50

491.8

al

0.21

0.27

0.33

403.0

b,

1.90

7.90

6.64

b,

2274.7

al

2273.6

al

2272.3

b,

2203.6 2200.6

1043.3 1037.7

b,

1.39 0.37

44.16

44.17

1.20

B. Galabov et al./J. Mol. Strut.,

(a)

93

273 (1992) 8598

(b)

Fig. 2. Definition of Cartesian reference systems for (a) aromatic and (b) methyl vibrations.

Moment nxrivatives with respect to internal vibrational coordinates, employing a double mode (di-mode) refinement process. The program allows the determination of a common set of infrared intensity parameters for a series of structurally related molecules and their isotopomers. Simultaneous optimization of alternative sets of dipole derivatives for different conformers can also be performed. The minimization function represents the sum of squares of the differences between calculated and experimental integrated infrared intensities. For molecules with some symmetry, the analysis can be carried out either in terms of molecular symmetry coordinates or in terms of local group symmetry coordinates. Owing to band overlap, the number of experimental integrated infrared intensities for a medium-sized molecule is usually smaller than the number of intensity parameters to be determined. Intensity data for structurally related molecules or isotopomers are, therefore, needed in order to improve the conditions for the refinement process. In the minimization part of the program, non-gradient simplex optimization [20] and gradient procedures [21] are combined; this secures better convergence in the refinement process. These procedures alternate with each other automatically, depending on the function surface in the course of the minimization. Various restrictions on initial estimates of the dipole moment derivatives can be imposed. The input consists of: (i) the normal coordinate transformation matrices (L) of the molecules studied; (ii) the experimental integrated infrared intensities of individual bands or overall intensities of overlapped

94

B. Galabov et al.lJ. Mol. Struct., 273 (1992) 85-96

TABLE 5 Ab initio and optimized a~+/dS, (< = x, Y, z) values for methyl benzenes (D A-’ and D rad-‘) No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Typeb of s

Ab initio values” x

Optimized values 2

Y

Sr S, S,, SOOP Sd

- 0.0640 - 0.6800 0 0 _

0.0370 0 0.5270 0

2

- 0.0900 0 0 0.8256 0 0.1789 0 0 0 0 0 _

0.8265 0 0 0 0 0 0 - 0.6294 - 0.2642 - 0.0304

SOOP S*, Sas1 S,, Sddl Srockl Sss2 ZZkZ S,

0

0.0000

x

Y

- 0.1000

0.0600 0 0.2344 0 - 0.0025 0 0.2000 0 0 0

_ .0.3517

0 0 0 0.5538 0.1000 0.2000 0.0003

0 0.2053 - 0.1400 0 0 0 0

0

- 0.4910

0

0 0 - 0.1152 - 0.5000 0 0 0.5868 0 0.2475 0 0 0 0 0 0

0.7530 _ 0 0 1.0600 0 - 0.7389 0 0.3861 - 0.2834 0 0 0

-

z 0.0000 0 0

0.5642 0 0 0 0.6500 0

“1D = 3.33564 x 10e30coulomb metre; 1A = 10-l-m. bThe values for ap/aS, Nos. l-8 refer to internal coordinates sit T, , $,, 4; and Q1 (Fig. 1) and to the Cartesian system shown in Fig. 2(a). The remaining ap/aS, values (Nos. 9-17) refer to the Cartesian system defined in Fig. 2(b). ‘Obtained from RHF 431G ab initio MO calculations.

bands; (iii) an array containing the direction cosines of local group and molecular Cartesian reference systems: (iv) a topological array defining the correspondence between the set of parameters to be optimized and the elements of the matrix Ps containing the dipole moment derivatives with respect to internal (symmetry) coordinates; (v) initial values for the parameters. The best starting values for the intensity parameters are provided by appropriate ab initio MO calculations. DISCUSSION OF RESULTS

Infrared

intensity parameters

Under the conditions described in the previous section, a set of twenty infrared intensity parameters for benzene and methyl benzenes were evaluated. The optimized quantities are shown in Table 5. As mentioned, except for the signs which are constrained by the ab initio calculations, parameter values were allowed to freely change in the refinement process.

95

B. Galabov et al./J. Mol. Struct., 273 (1992) 8%98 TABLE

6

Calculated

and experimental

intensities

for p-xylene

A de (km mol-I)

b 1” b 2u b 2u b 3” b 1”

11.76 35.35

14.78

1375.1

b 1” b 2u b 3” b 2u b I”

1282.0

b,

1.02

1202.6

b I”

0.00

1113.8

b 2u

2.47

1037.1

b 3” b I”

3.16

974.8

a,

0.00

964.5

b 2”

0.00

807.7

b 3”

32.37

732.0

b 1”

1.57

491.2

b 3”

15.69

3057.0 3054.6 2950.8 2949.6 2900.8 1521.6 1455.1 1451.2 1405.3

1025.7

25.87

~A,,,, (kmmol-‘)

A ew

(km molb’)

104.64

141.40

28.33

46.40

1.02

_

10.84 20.82

0.68 4.90 1.50 6.47

2.20

_ 2.47

6.21

5.36

4.90

_ 32.37

28.35 _

15.69

13.01

The comparison of ab initio and optimized values shows, however, very satisfactory correlation between the two sets. The calculated infrared intensities for benzene, toluene-d,, and toluene-d, as obtained from the refined set of intensity parameters and the L matrices associated with the field of La Lau and Snyder [7], are presented in Tables 3 and 4. In general terms, the correspondence between calculated and observed intensities is quite satisfactory. A serious problem was encountered with the intensities of the aromatic bands near 1600 cm-l for toluene-d,, and 1550 cm-l for toluene-d,. Under the sign constraints imposed by the ab initio MO calculations, the integrated intensities of these bands are substantially underestimated. This difficulty can be partly overcome by allowing the parameter i3p/&S,,,, to reach a relatively large negative value instead of positive as predicted by theory. We preferred, however, to retain all the signs obtained from MO calculations. In any case, such a change in ap/Z$,,, strongly affects the band

96

B. Galabov et al./J. Mol. Struct., 273 (1992) 85-98

3200

2800

1000

1400 WAVENUMBER

600

(cm-‘)

Fig. 3. (A) computed and (B) experimental infrared spectra of p-xylene in the gas phase (10 torr, 12 cm cell).

intensities at 1504cm-l in toluene-d,, and 1389cm-l in toluene-d,. The respective calculated values become much smaller and deviate considerably from experiment. There is basically no satisfactory compromise in this respect. A possibility always exists that the problem is associated with the structure of the respective L-matrices. We may state that the L-matrix elements do not satisfactorily reproduce the intensities of the aromatic bands in toluene near 1600 cm-’ and 1500 cm-l. These discrepancies may arise from the fact that the force field of La Lau and Snyder [7] has been determined from observed frequencies of samples in the solid state. Nevertheless, an acceptable fit of the observed vibrational intensities appears to be an important criterion for the accuracy of the vibrational eigenvectors. C-H stretching frequencies and intensities have often been used as sensitive indicators for changes in electronic structure and the effects of the surroundings on bond properties [22-241. From the experimental data for the stretching bands, the average intensity for the aromatic C-H bond in benzene is 11.4 km mol-l, while in toluene it is increased to 12.9 km

B. Galabov et al./J. Mol. Struct., 273 (1992) 8&98

97

mall’. The intensity per methyl C-H bond is found to be 9.8 km mallI. These differences are probably significant and reflect the changing intrinsic properties of the repective bonds in these molecular groupings. It was not possible to separate the overlapping bands in the C-H stretching region for p-xylene. The overall intensity per C-H bond is, however, markedly increased. That is why the predicted values for this band, as discussed further in the text, are too low.

Prediction of vibrational intensities in p-xybne The set of refined intensity parameters together with the transferable force constants for methyl benzenes as developed by La Lau and Snyder [7] were simultaneously used in predicting the vibrational spectrum of pxylene. In Table 6, the calculated intensities are compared with the experimental values determined in the present study. The simulated and observed spectra of p-xylene are shown in Fig. 3. In general terms, the agreement between calculated and observed spectra is satisfactory, although some differences are detected. As mentioned, the overall intensity in the C-H stretching region is underestimated by about 26%. The total calculated intensity in the 1521-1375cm-’ interval is also substantially lower than the experimental values. In view of the preceding discussion, these differences may be attributed to the changing properties of the C-H bonds in this molecule. There is a weak band at 732 cm-l in the simulated spectrum, not found in the experimental one. Such a band is, however, observed in the solid-state spectrum [7]. These differences between spectra in different phases affect, very possibly, the quality of intensity predictions in other parts of the spectrum as well. However, in spite of these problems the principal intensity features of the infrared spectrum of p-xylene are correctly predicted. REFERENCES 1 2 3 4 5 6 7 8 9 10 11

E.B. Wilson, Jr., J.C. Decius and P.S. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955. L.A. Gribov, Intensity Theory for Infrared Spectra of Polyatomic Molecules, Consultant Bureau, New York, 1964. J. Biarge, J. Herranz and J. Morcillo, An. Quim. Ser. A, 57 (1961) 81. J.C. Decius, J. Mol. Spectrosc., 57 (1975) 348. A.J. van Straten and W.M.A. Smit, J. Mol. Spectrosc., 62 (1976) 297. B. Galabov, J. Chem. Phys., 74 (1981) 1599. C. La Lau and R.G. Snyder, Spectrochim. Acta, Part A, 27 (1971) 2073. L. Antonov and S. Stoyanov, unpublished results. J. Overend and M.J. Youngquist, in M. Davis (Ed.), Infrared Spectroscopy and Molecular Structure, Elsevier, Amsterdam, 1963, p. 368. P. Pulay and G. Fogarasi, J. Chem. Phys., 74 (1981) 3999. J.A. Draeger, Spectrochim. Acta, 41 (1985) 607.

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B. Galabov et al./J. Mol. Struct., 273 (1992) 8&96 M. Tasumi, T. Urano and M. Nakata, J. Mol. Struct., 146 (1986) 383. M. Gussoni and S. Abbate, J. Chem. Phys., 65 (1976) 34. A. Rupprecht, J. Mol. Spectrosc., 89 (1981) 356. S. Ilieva, T. Dudev and B. Galabov, Croat. Chem. Acta, 63 (1990) 143. T. Dudev and B. Galabov, Spectrochim. Acta, Part A, 48 (1992) 1153. R.G. Snyder, J. Chem. Phys., 42 (1965) 1744. T. Nakanaga, S. Kondo and S. Saeki, J. Chem. Phys., 70 (1979) 2471. R. Ditchfield, W.J. Hehre and J.A. Pople, J. Chem. Phys., 54 (1971) 724. J.A. Nelder and R. Mead, Comput. J., 7 (1965) 308. B.D. Bunday, Basic Optimization Methods, Edward Arnold, London, 1984. D.C. McKean, J. Mol. Struct., 113 (1984) 251. K.M. Gough and W.F. Murphy, J. Chem. Phys., 87 (1987) 1509. A. Kindness, D.C. McKean and D. Stewart, J. Mol. Struct., 224 (1990) 363.