A study of the vibrational behavior of imidazole by the ab initio gradient method

A study of the vibrational behavior of imidazole by the ab initio gradient method

Journal of Molecular Structure (Theochem), 136 ( 1 9 8 6 ) 3 3 9 - - 3 5 0 Elsevier Science Publishers B.V., A m s t e r d a m - Printed in The Nether...

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Journal of Molecular Structure (Theochem), 136 ( 1 9 8 6 ) 3 3 9 - - 3 5 0 Elsevier Science Publishers B.V., A m s t e r d a m - Printed in The Netherlands




Department of Chemistry, The University of Texas at Austin, Austin, TX 78 712 (U.S.A.) (Received 12 August 1985 )


The complete harmonic vibrational force field of imidazole has been computed at the Hartree--Fock level using the 4-21 basis set of Gaussian orbitals. The harmonic constants were scaled by using scale factors previously derived by fitting the computed force field of benzene to the observed benzene vibrational spectrum. The resulting scaled force field was then used to predict the vibrational spectrum of imidazole. The N--H wagging and N--H stretching frequencies, which have no counterpart in benzene, were poorly predicted, but the mean deviation between experiment and prediction was only 9 cm -1 for the other in-plane (K) vibrations and 27 cm -1 for the out-of-plane (A") vibrations. In order to fit the N--H modes and to obtain a better fit for the out-of-plane vibrations, a new set of scale factors, some of which turned out to be identical with those from benzene, was derived by fitting the computed spectrum of imidazole to the observed spectrum. This new set of scale factors was used to predict the vibrational spectra of several deuterated forms o f imidazole (-1D, -3D, and -4D). The imidazole scale factors are presumably more accurate in scaling computed spectra of other 5-membered heterocyclic ring compounds. Dipole moment derivatives were also calculated and used to predict infrared intensities which are compared with experimental values. A few uncertainties in the experimental assignment of the imidazole spectrum can be clarified. INTRODUCTION

The application of vibrational spectroscopy to the study of biological problems has met with many successes, but detailed analysis of the spectra in t h e m a n n e r t h a t is d o n e f o r s m a l l e r m o l e c u l e s h a s b e e n h a m p e r e d b o t h b y d i f f i c u l t i e s in o b t a i n i n g a s p e c t r a o f t h e i s o l a t e d (gas p h a s e ) s p e c i e s a n d b y p r o b l e m s in a c h i e v i n g a f u l l i n t e r p r e t a t i o n o f t h e c o m p l i c a t e d s p e c t r a . T h e r e is r e a s o n t o b e l i e v e t h a t r e c e n t a d v a n c e s in c o m p u t a t i o n a l q u a n t u m c h e m i s t r y c a n h e l p w i t h t h e l a t t e r d i f f i c u l t y a n d p r o v i d e a s s i s t a n c e in t h e c o m p l e t e a n d r e l i a b l e a s s i g n m e n t o f o b s e r v e d s p e c t r a as w e l l as m a k i n g u s e f u l l y r e l i a b l e predictions of the spectra of unknown molecules. In a recent series of papers [1--5] we have computed ab initio the comp l e t e h a r m o n i c f o r c e f i e l d o f b e n z e n e a t t h e d o u b l e - z e t a H a r t r e e - - F o c k level, and used that force field to predict the infrared and Raman spectra of *On leave from Fudan University, Shanghai, People's Republic of China. 0166-1280/86/$03.50

© 1986 Elsevier Science Publishers B.V.

340 normal and isotopically substituted benzenes. A small set of scale factors was then derived to correct the directly computed force field elements to obtain the best fit to the observed benzene spectra. The method, resulting in a scaled quantum-mechanical (SQM) force field, has been described fully in a paper dealing with application of the procedure to somewhat simpler molecules [6]. Having obtained scale factors for benzene, which measure the error in the computation of the force constants along various types of vibrational coordinates, the transferability of those scale factors to related molecules was tested. The force fields of pyridine [2, 5], naphthalene [3], and aniline [4] were calculated ab initio and corrected, without any fitting, by the scale factors derived for benzene. The resulting SQM force fields for these substances were then compared with known observed spectra and the predictive power of the technique was verified by noting that the mean deviation between predicted and observed fundamental frequencies fell in the range between 5 and 10 cm -1 in every case. This suggests that the spectra of u n k n o w n molecules can be predicted a priori [7] if the spectra of some related compound is known. The assumption is only that residual errors in the calculation from neglect of electron correlation and finite basis set are similar in the two molecules for a given type of vibration. The method is valuable for assisting in spectral assignments even when some experimental data are available so t h a t a completely a priori prediction is not needed. The calculations already referred to [1--6] as well as similar calculations on cubane [8], triprismane [9], CH3POF2 [10], and other molecules currently in progress have shown that missing or disputed fundamental vibrational assignments can be made by use of the SQM force field that fits the transitions whose identity is known with certainty. For such spectral assignment, the computed vibrational intensities are extremely helpful as well as the predicted frequencies. There are, of course, other research groups engaged in the computation of vibrational frequencies [11]. The unusual feature of the approach used in this work is the evaluation of a small number (but not one) of scale factors that are characteristic of a given vibrational type and realization of the prime importance of expanding the potential energy surface around the true minimum energy point if the scale factors are to be transferable. Other precautions which are routinely observed have been pointed out earlier [6]. With these precautions, we have been able to achieve accuracy in the predictions that is limited only by the isolated harmonic oscillator approximation that underlies all of this work. Imidazole is a compound of fundamental biological importance as a c o m p o n e n t of nucleic acid and related substances. There is an extensive literature on its vibrational spectra in condensed phases [12--20], including considerable interest in crystal lattice vibrations as well as intramolecular vibrations. Due mainly to the influence of hydrogen bonding in the solid state and the high boiling point (256°C) of imidazole, it is difficult to obtain a good vibrational spectrum of the free molecule. Fortunately,

341 P e r c h a r d and c o - w o r k e r s [12, 13] have s u c c e e d e d in o b t a i n i n g v a p o r phase vibrational d a t a and King [23] has r e p o r t e d t h e results f r o m infrared spectros c o p y using the l o w - t e m p e r a t u r e m a t r i x isolation t e c h n i q u e . These are the o n l y e x p e r i m e n t s in which t h e interesting N--H vibrations are n o t o b s c u r e d b y i n t e r m o l e c u l a r e f f e c t s in c o n d e n s e d phases. METHOD AND RESULTS Ab initio calculations were carried o u t w i t h i n t h e M O - - L C A O - - S C F a p p r o x i m a t i o n using t h e H a r t r e e - - F o c k gradient p r o g r a m T E X A S [21] using the standard 4-21 a t o m i c basis set [22] which is similar t o the 4 - 3 1 G Gaussian basis set o f Ditchfield e t al. [23] b u t is less t i m e - c o n s u m i n g and gives results o f at least equal accuracy. B o t h t h e c o m p u t a t i o n a l t e c h n i q u e s [22] and the m e t h o d s used in o b t a i n i n g the scaled q u a n t u m m e c h a n i c a l (SQM) f o r c e field [6] have been described previously. It is i m p o r t a n t f o r the accurate t r a n s f e r o f vibrational scale factors to c h o o s e the t r u e equilibrium g e o m e t r y as the p o i n t at which the t h e o r e t i c a l f o r c e field is evaluated [ 6 ] . In m a n y cases this can best be a p p r o x i m a t e d b y using the c o m p u t e d m i n i m u m - e n e r g y g e o m e t r y with suitable small corrections f o r c o n s i s t e n t errors arising f r o m neglect o f e l e c t r o n c o r r e l a t i o n and TABLE1 Reference geometry of imidazolea R(1, R(2, R(3, R(4, R(5, R(1, R(2, R(4, R(5,

2)= 1.363 3)-- 1.312 4) = 1.381 5)= 1.362 1)= 1.376 6) = 0.991 7 ) : 1.077 8)= 1.070 9) : 1.071

L(5, 1, L(1, 2, L(2, 3, L(3, 4, L(4, 5, L(5, 1, L(1, 2, L(5, 4, L ( I , 5,

2) 3) 4) 5) i)

= 106.9 = 112.0 = 104.9 = 110.7 = 105.5 6 ) = 126.9 7) = 122.5 8) = 127.9 9) = 121.9

aDistances in A, angles in degrees.

H8 \

H cs\i c2\


H6 Fig. 1. Labeling of atoms in imidazole. The molecule is completely planar.


use of a finite basis set [6, 22]. For imidazole, however, there is an extremely thorough microwave spectroscopic study [24] leading to a substitution structure of apparent high reliability. We have made very small corrections to the rs bond lengths, probably insignificant except for C--H and N--H distances, to correct for the expected small differences between the rs structure and the true equilibrium structure. The resulting molecular geometry, which we have used as the reference state, is shown in Table 1. The internal coordinates used, obtained according to our earlier recommendations [22], are given in Table 2. While we believe t h a t it makes little difference whether the second derivatives of energy with respect to coordinate displacements are computed analytically or numerically from the first derivatives at geometries displaced along the internal coordinates, in this case we have used the latter procedure. Finite displacements of the internal coordinates (Table 2) that correspond to displacements of 0.02 A or 2 ° in individual valence coordinates were used, with both positive and negative displacements being made to minimize the effect of cubic anharmonicity [6]. Due to the molecular Cs s y m m e t r y of imidazole, ab initio gradient calculations were needed at only 37 geometries (including the reference geometry) to determine the complete theoretical harmonic vibrational force field consisting of 231 independent elements, given in parentheses in Tables 3 and 4. The directly computed force field elements were next scaled according to the scheme

--nb'~scaled-- CiFi the°r

a n d .~,.s.caled z~ = ( CiCj) 1/2 .b-~~t.heor


with the set of six scale constants, Q, taken as those optimized for benzene [1]. The resulting scaled force field was used to calculate the vibrational fundamentals, as shown in Table 5. For this transfer of scale factors, it was assumed that the difference between N and C was insignificant as well as the difference between a 5-membered and a 6-membered ring. It can be TABLE 2 I n t e r n a l c o o r d i n a t e s for i m i d a z o l e a ql = R(2, 7);q~ = R(1, 6);q~ = R ( 5 , 9 ) ; q 4 = R ( 4 , 8 ) ; q s = R ( 1 , 2);

q6 = R(5, 1 ) ; q v = R(4, 5);q8 = R(3, 4);q9 = R(2, 3); q,0 = L(7, 2, 3 ) - - L(7, 2, 1);q,~ = L(6, 1, 2 ) - - L(6, 1, 5); q~2 = L(9, 5, 1 ) - - / _ ( 9 , 5, 4);q13 : L(8, 4, 5 ) - - L(8, 4, 3); q l ~ : a ~ + a(a 2 + a s ) + b ( % + a 4 ) ; q ~ s = ( a - b ) ( a : - a 5 ) + ( 1 - - a ) ( a 3 - a 4); qx6 : 7 out-of-(3, 1, 2)plane; qx7 : 6 out-of-(2, 5, 1)plane; ql8 : 9 out-of-(1, 4, 5)plane; q . : 8 out-of-(5, 3, 4 ) p l a n e ; qm = b ( % + r s ) + a(T~+ r 4 ) + r3;q2~=(a--b)(r4--r2)+ ( 1 - - a ) ( r 5 - - ~ ) . aa ffi cos 144°; b = cos 72°; a I = L(1, 2, 3); a2 = L(5, 1, 2); a 3 = L(4, 5, 1 ) ; a 4 = L(3, 4, 5); a s = L(2, 3, 4); r I = dihedral angle (3, 2, 1, 5) = dL(3, 2, 1, 5); r~ = dL(2, 1, 5, 4); r 3 = d/_(1, 5, 4, 3); r 4 = dL(5, 4, 3, 2); T5 = dL(4, 3, 2, 1).




0 0. 0


~~i ~

•. ~ ~ ~


~ ~ ~ ,


~ ® ~ o. ~ ~ o. ~ ' ~ . o . ~



o . ~ . ~



344 TABLE 4 SQM and (directly computed) out-of-plane force fields of imidazole a Internal Coordinates









(0.53) 17 18 19 20 21

---0.12 (--0.03) 0.02

0.14 (0.27) --0.02



0.01 (0.01) 0.00 (0.01) 0.18 (0.25)

0.02 (0.03) 0.07 (0.12) --0.09 (--0.14)


(0.48) --0.04 (--0.07) --0.17 (--0.23) 0.05 (0.07)


(0.55) 0.19 (0.27) 0.04 (0.05)

0.52 (0.66) --0.03 (--0.04)



aUnits are such that FijAqiAq j is in mdyn A with distances in A and angles in radians. For meaning of SQM force field, see text. Directly computed values (without scaling) are in parentheses.

seen that this assumption worked reasonably well except for the vibrations involving the N--H group. In order to obtain the best possible force field for imidazole, we have added three new scale factors for the N--H stretch, N--H deformation, and N--H wagging modes. These, together with the six scale factors previously adopted for benzene were reoptimized by fitting the computed spectrum of imidazole to the observed spectrum. The resulting nine scale factors are given in Table 6. It can be seen that the heavy atom stretch scale factors remain unchanged from the benzene values and that the largest difference is between the optimized N--H values and the crude assumption that N--H is identical with C--H. The main entries in Tables 3 and 4 give the force fields scaled with the new constants. We believe these to be the best available harmonic force fields for imidazole. The spectra predicted by these force fields are given in Table 5. Derivatives of the dipole m o m e n t s have been evaluated and used to compute the infrared absorption intensities by methods previously described [22--25]. The resulting values as shown in Table 5 where they are compared with experimental data reported by Perchard and co-workers [12, 13]. As a test of the scale factors optimized on the normal imidazole spectrum and a confirmation of the accuracy of assignments, we have also calculated the vibrational spectra of three deuterated forms of imidazole. The results are given in Table 7. No vapor phase isotopic spectra are available, so in Table 7 we make a comparison with the spectra of solid imidazole observed by Perchard and co-workers [12, 13]. In the solid state the N--H vibrations are badly disturbed by intermolecular hydrogen bonding so t h e y should not,








~-~ ~-~

~.~ ~

¢0 o~ o~ ~





scale factors

C--H stretch C--H deformation C--H wagging C--C stretch C--N stretch Ring deformation Ring torsion N--H stretch N--H deformation N--H wagging

Benzene a

4-Pyrone b

Aliphatic c

Maleirnide d

Pyrrole e

Imidazole f

0.863 0.797 0.739 0 911 i 0.808 0.768

0.866 0.839 0.781 0.938

0.866 0.800 0.730 g

0.835 0.785 h0.697

0.853 0.786 0.697 0.930~ 0.877 J 0.777 0.755 0.819 0.787 0.451

0.863 0.811 0.674

i i

0.803 0.760

0.878 0.793 0.731 0.844 0.790 0.510

0.911 0.777 0.788 0.824 0.777 0.494

aReference 1. bCs~sz~tr,P.; Cs~sz~r, A. ; Somogyi, A. ; Dinya, Z. ; Holly, S.; G~l, M., Boggs, J. E. unpublished results. CReference 6. dCs~sz~r, P.; Cs~sz~r, A.; Hars~nyi, L.; Boggs, J. E. unpublished results, exie, Y.; Fan, K. ; Boggs, J. E., unpublished results, fThis work. g0.920 for C--C and 0.866 for CfC. h0.903 for C--C and 0.869 for C=C. iFor transfer of benzene scale factors to imidazole, C--N and N--H were taken as the same as C--C and C--H. and do n o t, agree with the frequencies c o m p u t e d for the isolated molecule. F o r these motions and a few others, we also show in Table 7 a comparison between the f r e q u e n c y shifts on deuteration, as predicted and as observed. It can be n o ted that these shifts are well r e p r o d u c e d even though the absolute solid state frequencies do n o t give direct information on the intramolecular force field. DISCUSSION

The fundamental vibrational frequencies of the normal isotopic species o f imidazole, calculated f r o m the force field (1) w i t h o u t scaling, (2) scaled with the benzene scale factors, and (3) scaled with factors optimized for imidazole, are r e p o r t e d in Table 5 along with a variety of experimental measurements. The spectra f r om bulk phases show clearly the large effect of hydrogen bonding between molecules, particularly in the striking shifts observed in th e N--H wagging and N--H stretching motions. Only the vapor phase spectra o f Pechard and co-workers [12, 13] and to some e x t e n t the matrix spectra o f King [18] are free of this perturbation. Our analysis is based primarily on the vapor phase results, although the assignments and frequencies observed by King are mostly consistent with these. The w ork in the condensed phases is mainly of use in considering questions of dubious assignments o f fundamentals. First, as expected, the spectral calculations with the unscaled force field are grossly in error with an average deviation of 143 cm -1 . Absolute calculations with high accuracy would require use of a basis set at least triple zeta in the valence shell with added polarization functions and a t horough treatm e n t o f electron correlation [26, 2 7 ] , a level of c o m p u t a t i o n which cannot be c o n t e m p l a t e d with existing computers for a molecule this large.






g r¢3

N r..




gl~ e~





~.~ <~ g

d Z


348 The frequency predictions using the benzene scale factors are indicative of the best absolute predictions that could have been made for the imidazole spectrum if no experimental measurements had ever been made. It can be seen in Table 5 that the major errors are for the N--H wag and the N--H stretch. The error in the calculation of these is obviously significantly different from the error in calculating the corresponding C--H motions in benzene so that a different scale factor is required. With the omission of these two frequencies, the mean deviation between the predicted frequencies (using scale factors from benzene) and the gas-phase experimental values, is 9 cm -1 for the in-plane (A') vibrations and 27 cm -1 for the out-of-plane (A") modes. The results for the in-plane frequencies are on the order of accuracy we have come to expect [1--6], and are n o t much in excess of what might be expected as the difference between the computed harmonic frequencies and the experimental anharmonic onces. Any effects of interaction between modes are also n o t considered in our isolated harmonic oscillator approximation. The 27 cm -~ average deviation for the out-of-plane motions shows that the transfer of the corresponding scale factors from benzene was less accurate, although the results have an accuracy that would be useful in the original assignment of the spectrum or in later correction of misassignments. The out-of-plane error arises mostly in the C--H wagging motions, suggesting that the error in the quantum-mechanical calculation of these is also significantly different in this 5-membered heterocyclic ring than it is in the 6membered benzene ring. We would like to obtain the best possible force field for imidazole to assist in the prediction of other vibrational properties and we would also like to determine optimal scale factors for this ring for potential transfer to more complex biologically significant molecules. We have, therefore, reoptimized four of the six benzene scale factors and added 3 additional ones to take specific account of the N--H motions in imidazole. Any changes in the benzene C - H stretch or heavy atom stretch scale factors produced completely insignificant improvements in the fit so they were left unaltered. The scale factors obtained are given in Table 6 and the resulting c o m p u t e d spectrum in Table 5. The resulting mean deviation between experiment and computation is 6.0 cm -~ (6.9 cm -1 for the in-plane vibrations and 3.7 cm -1 for the out-of-plane ones). Comparison of the scale factors optimized for different molecules (Table 6) reveals a high degree of uniformity. However, the small systematic differences that can be found are important for consideration of the transferability of these factors from known molecules to u n k n o w n ones. Most obvious is the observation that the scale factor for the out-of-plane C--H motion (C--H wag) is consistently lower by an average of 8% in the 5-membered rings (imidazole, pyrrole, and maleimide) than it is in the 6-membered rings or in the aliphatic compounds. To a lesser extent, an average of 3%, the ring deformation scale factors are also lower in the 5-membered rings. It is clear that more data of this sort is badly needed in order to evaluate the limita-

349 tions on accuracy that may be expected in the prediction of u n k n o w n vibrational spectra by the SQM method. A number of comments must now be made regarding a few of the spectral assignments shown in Table 5. The N--H in-plane deformation mode, v12, was not assigned in the vapor spectrum. Although infrared absorption intensities calculated at the level used here cannot have quantitative accuracy, they are expected to be approximately correct, and it is seen t h a t v12 is predicted to be one of the weaker bands in the spectrum. In the matrix spectrum, King [18] assigned va in the region which has a medium-strength absorption in the gas phase corresponding to a ring deformation motion. To us, it seems more reasonable to adopt the solution Raman value [15] of 1160 cm -1 for this fundamental, although the computed value is intermediate between the two possibilities. At least part of the deviation between the experimental and predicted values must arise from solvent interaction perturbing the solution Raman frequency. The v~ C - H stretch is predicted to be the weakest fundamental in the entire spectrum. In the gas phase it was assumed to be coincident with v19, but it seems more reasonable to believe that it should be near the Raman frequency of 3060 cm -1 but too weak to be distinguished as a fundamental in the absorption experiment. We have, therefore, adopted the 3060 cm -1 frequency for comparison with the computed value. It should be remembered that the three A' C--H stretches are probably perturbed by some degree of Fermi resonance. The assignments of v6 and vs were originally reversed in Perchard's analysis [12, 13]. This was corrected by King [18] who gave the assignments shown in Table 5. King's reassignment is very strongly supported both by the frequencies and relative intensities from the present calculations. We suspect that the frequency of 668 cm -~ given by Perchard and coworkers [12, 13] may be a misprint. Their value is copied by King in a later paper, but King also shows a gas-phase spectrum (his Figure 3) in which the band in question is clearly at 658 cm -1 rather than at 668 cm -~ . We have chosen the revised figure for comparison in Table 5, but the difference is too small to distinguish with certainty from our work. The scale factors for imidazole were optimized using data from experimental work on the normal isotopic species only. The predictions for several deuterated forms shown in Table 7, then, can give an indication of the accuracy with which these scale factors fit a wider variety of transitions than were used in obtaining them. Unfortunately, the isotopic experimental information is known only from experiments in condensed phases. Either solid phase absorption or Raman so the C--H motions, and especially the N--H motions, are expected to be distorted from the pure molecular values. With the exception of vibrational modes involving these motions, the agreement shown in Table 7 appears quite satisfactory. For the C--H and N--H motions where the differences between prediction and experiment seem excessive, we have calculated the frequency shifts on deuteration and have


also shown these in Table 7. The isotopic shifts are seen to be calculated with somewhat greater accuracy than are the absolute frequencies for these bands in the isotopic species, but in some cases the differences remain substantial. It should not be surprising that such large differences remain, for example, in the N--H wagging and N--H stretching motions, when it is recalled that this hydrogen atom is involved in strong hydrogen bonding in the solid state from which the experimental values were obtained. The computations shown in Table 7 should perhaps be regarded as more useful when used as predictions to assist in the assignments from future gas phase experiments on these deuterated forms. REFERENCES 1 P. Pulay, G. Fogarasi and J. E. Boggs, J. Chem. Phys., 74 (1981) 3999. 2G. Pongor, P. Pulay, G. Fogarasi and J. E. Boggs, J. Am. Chem. Soc., 106 (1984) 2765. 3 H. Sellers, P. Pulay and J. E. Boggs, J. Am. Chem. Soc., in press. 4 Z. Niu, K. M. Dunn and J. E. Boggs, Mol. Phys., 55 (1985) 421. 5 G. Pongor, G. Fogarasi, J. E. Boggs and P. Pulay, J. Mol. Spectrosc., in press. 6 P. Pulay, G. Fogarasi, G. Pongor, J. E. Boggs and A. Vargha, J. Am. Chem. Soc., 105 (1983) 7037. 7 Even though no experimental data are needed for the molecule of interest, we speak of a priori rather than ab initio prediction since there is a transfer of information from a related molecule. 8 K. M. Dunn, P. Pulay, C. Van Alsenoy and J. E. Boggs, J. Mol. Spectrosc., 103 (1984) 268. 9 Y. Dai, K. M. Dunn and J. E. Boggs, J. Mol. Struct. (Theochem), 109 (1984) 127. 10 S. Von Carlowitz, W. Zeil, P. Pulay and J. E. Boggs, J. Mol. Struct. (Theochem), 87 (1982) 113. 11 See, for example, the recent review by G. Fogarasi and P. Pulay in B. S. Rabinovitch (Ed.), Ann. Rev. Phys. Chem., Vol. 35. Annual Reviews, Inc., Palo Alto, CA, 1984, p. 191. 12 C. Perchard, A.-M. Bellocq and A. Novak, J. Chim. Phys., 62 (1965) 1344. 13 A. N. Bellocq, C. Perchard, A. Novak and M. L. Josien, J. Chim. Phys., 62 (1965) 1334. 14 N. D. deCordes and J. L. Walter, Spectrochim. Acta, A24 (1968) 237. 15 L. Colombo, P. Bleckmann, B. Schrader, R. Schneider and Th. Plesser, J. Chem. Phys., 61 (1974) 3270. 16 M. Majoube, Proc. Int. Conf. Raman Spectrosc., 6th., 2 (1978) 76. 17 H. Wolff and H. MSller, Spectrochim. Acta, 32A (1976) 581. 18 S. T. King, J. Phys. Chem., 74 (1970) 2133. 19M. Majoube, J. Mol. Struct., 61 (1980) 129. 20M. Majoube and G. Vergoten, J. Chem. Phys., 76 (1982) 2838. 21 P. Pulay, Theor. Chim. Acta, 50 (1979) 299. 22 P. Pulay, G. Fogarasi, F. Pang and J. E. Boggs, J. Am. Chem. Soc., 101 (1979) 2550. 23 R. Ditchfield, W. J. Hehre and J. A. Pople, J. Chem. Phys., 54 (1971) 724. 24 D. Christen, J. H. Griffiths and J. Sheridan, Z. Naturforsch., 36A (1981) 1378. 25 H. Sellers, J. E. Boggs, A. V. Nemukhin and J. Alml~f, J. Mol. Struct. (Theochem), 85 (1981) 195. 26P. Pulay, J.-G. Lee and J. E. Boggs, J. Chem. Phys., 79 (1983) 3382. 27 J. E. Boggs, J. Mol. Struct., 130 (1985) 155.