Variation of disulfide bond stretching frequencies with disulfide

May 16, 1974 - dihedral angle and CSS bond angle of dimethyl disulfide in the gas phase. It gives ... stretching frequencies, these relationships, tog...
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1848

Van Wart, Shipman, and Scheraga

Variation of Disulfide Bond Stretching Frequencies with Disulfide Dihedral Angle in

Harold E. Van WartFa Lester L. Shlpman,2band Harold A. Scheraga" Department of Chemistry, Cornell University, lthaca, New York 14850 (ReceivedMay 16, 1974)

The CND0/2 semiempirical molecular orbital method has been used to calculate the variation of the energy, equilibrium S-S bond length, and S-S stretching frequency of dimethyl disulfide as a function of the CS-SC dihedral angle. The effects of variation of the CSS bond angle on the energy and (I-S stretching frequency were also examined. The CND0/2 method gives the experimentally observed values for the CS-SC dihedral angle and CSS bond angle of dimethyl disulfide in the gas phase. It gives values for the S-S stretching frequency which are too large and values for the equilibrium S-S bond length which are roughly 10% too small. However, the CNDO/2 method gives trends in S-S bond length and S-S stretching frequency that are in reasonable agreement with the experimentally observed trends. Some implications for interpretation of the variation of disulfide stretching frequencies in terms of the conformational properties of the GS-SC group are discussed.

1. Introduction

yl group at either of two nonequivalent gauche positions or a t a trans (CC-SS dihedral angles of --f60 and 180', respectively) position with respect to the distal sulfur across Recently, se.i era1 R.aman spectral investigations of compounds containing the CSSC moiety have been r e p ~ r t e d . ~ - ~the C-S bond. The coexistence of rotational isomers due to These studies were motivated by the importance of the diinternal rotations about single bonds may result in the presence of more than one band in a vibrational spectrum sulfide bond (of cysline) in influencing protein structure and by the prominence of the S-S stretching band in the attributable to the same normal mode. For example, rotaRaman spectra of disulfide-containing molecules as comtional isomers have been shown to be responsible for the plex as proteins. ne goal af the aforementioned work3-6 appearance of multiple C-C1 stretching bands in chlorinatwas to investigate the possibility that the S-S and C-S ed alkanes7 and the appearance of multiple C-S stretching stretching frequencies might correlate in a systematic way bands in alkyl mercaptansa8Sugeta, et al.,4 have suggested with the conformational properties of the CSSC group. Unthat rotation about the C-S bond affects the S-S stretching fortunately, there is no general agreement, yet, on the exisfrequency in alkyl disulfides and cystine. tence or naturca of such correlations between stretching Van Wart, et a1.,6 studied the Raman spectra of a series frequencies and conformation. If systematic relationships of disulfide-containing compounds whose CS-SC dihedral can be found between the dihedral angles for rotation angles varied widely but whose conformation about the Cabout the S-S and IC-S bonds and the S-S and C-S S bond remained in the region of the gauche conformation. stretching frequencies, these relationships, together with An approximately linear variation of the S-S stretching measured S-S and C-S stretching frequencies, could be frequency with CS-SC dihedral angle was reported.6 used as a sensitive probe for the determination of conforIf the.results of the studies of Sugeta, et al.,4 and Van mation about t h ? S--Sbond in cystine in proteins. Wart, et a1.,6 are to be consistent with each other, the posiAssignments of S-S stretching frequencies in the literations of the S-S stretching bands for a CS-SC dihedral ture vary from a low value of about 485 cm-l for a strained angle of about 90' and a gauche conformation about both cyclic disulfide6 to an upper value of about 550 cm-l for alC-S bonds should have been the same in each investigaiphatic disulfides in which the carbons adjacent to the dition. Sugeta, et al.,4 reported a value of 509 em-l for this sulfide bond are secondary or tertiary ~arbons.~J'If the exconformation in diethyl disulfide while the analogous value periniental valucms are taken for the bond lengths and bond taken from the empirical frequency us. CS-SC dihedral angles, the conformalion of the CSSC group is uniquely deangle plot of Van Wart, et a1.,6is roughly 523 cm-l. The orscribed by the values of the dihedral angles for rotation igin of this discrepancy in the results of the two studies about the S-S and C-S bonds. Two recent proposal^^.^ may lie in the assignments of the bands since Sugeta, et suggest that the positicln of the S-S stretching band may be al.,4 reported another S-S stretching band at 524 cm-l correlated with these dihedral angles. which they assigned to the conformation having one tesminal group gauche and the other trans about the C--S bond. Sugeta, et ~ t l . , studied '~ the S-S stretching band in a series of alkyl disulfides. In these open-chain compounds, the If, on the other hand, their 524-cm-l band were assigned to CS-SC dihedral angle varies little from its equilibrium a structure with two gauche conformations, this discrepancy between the two studies would disappear. value of approximately 90'. More than one S-S stretching band was observed in &heRaman spectra of many of those This difference in interpretation of results of the two compounds in wl ich the alkyl substituent was larger than a studies indicates that a knowledge of the position and relative energy of minima in the potential for rotation about methyl group. This was attributed to the presence, in therthe S-S and C-S bonds is essential for the correct interpremodynamic equilibrium, of gauche and trans rotamers tation of the S-S and C-S stretching bands, It has been asabout the 6-S band. In diethyl disulfide, for example, rotation about either 6-S bond can position the terminal methsumed, for example, that the minima for rotations about The Journal of Physic81 Chemi'stry, Vol. 78.

No. 18, 1974

Disulfide E3orld Stretehmg Frequencies

the C-S bona are at gauche and trans positions, but this has not been verified either experimentally or theoretically. Experimentally, it is known that the energy minimum for rotation about the S-S bond is f85" in dimethyl disulfideg,l0(similar to the case of hydrogen persulfidell). There are, as yet, no experimental values for the cis and trans (CS-SC dihedral angles of 0 and 180°,respectively) rotational barriers in dimethyl disulfide (see references and iscussion Jn ref I:! and 13). Because of the paucity of experimental data on the variation of the energy for rotation about the S-S and C-S bonds in the CSSC moiety, we found it useful to investigate this problem with the aid of the CNDOI2 semiempirical molecular cirbital method. This method was chosen because of its success in predicting reasonable barriers and shapes for the potential energy curve for rotation about the S-S bond in W2S2.14The purpose of the work presented in this paper is to elucidate the effects of rotation about the S-S bond on the energv, equilibrium S-S bond length, and S-S stretching frequency in the model compound dimethyl disulfide. In particular, it is desirable to determine whether the CNU0/2 method can account for the experimental6 trend in S-S stretching frequency with CS-SC dihedral angle. Further, such calculations provide information for values of the CS-SC dihedral angle not yet encountered experimentally i n model compounds (especially CS-SC dihedral angles in ',he range of 100-180'). Although it is expected that C-S szretching frequencies are sensitive to torsion about the 8-43b o d , this problem will not be considered here. The effects OF r~otationabout C-S bonds in the CSSC moiety on the S-d s r e t c ~ i n gfrequency will be reported in a subsequent paper.

imethyl Disulfide There are two experimental gas-phase structures for dimethyl disulfide. a microwave geometry by Sutter, et aL,9 and an electron diffraction geometry by Beagley and McAloon.lc These two geometries are in close agreement and their structural parameters are given in Table I for comparison. Far the calculations reported in this paper, all bond lengths and bond angles that are not varied have been fixed a t the microwave geometry, unless specified otherwise. In all calculations, the methyl hydrogens were staggered wit) respect to the S-S bond. Note that CS-SC dihedral angles of 0 to 180' are equivalent to angles of 0" to -180' becarusr of the C2 symmetry of the molecule. 11, CNDQ/2

CNDO/Z is a semiempirical SCF-LCAO-MO theory developed by Pople and coworkers.15J6 In the present study, standard CNIl0/2 parameterization was used17 and all calculations were done in double precision. The d orbitals on the sulfurs we*eincluded in the basis set. The SCF procedure was considered to have converged when the electronic energy changed by less than eV between one iteration and the next. Figure I shows the variation of the energy of dimethyl disulfide with S-S distance at a fixed CS-SC dihedral angle of YO'. No1,e that the energy curve has a distinctly anharmonic shape. The shape of the energy curve a t various other values 0,' the CS-SC dihedral angle in the range of 0-180" was investigated and found to be similar in shape to that shown In Figure 1. The energy was calculated at 0.1-A

1849

-d

I 2.o

1.5

2.5

Figure 1. Variation of the energy of dimethyl disulfide as a function, of S-S bond length for the experimental bond lengths and bond angles but with a CS-SC dihedral angle of 90'. The energy is normalized to zero at the minimum. TABLE I: Calculated and Experimental Values of Bond Lengths and Bond Angles for Dimethyl Disulfide

-

Microwave values

4

Electron diffraen valuelo

CNDO/Z value ~

Rs-s, Rc-s, A Rc-n, A CSS bond angle, deg SCH bond angle, deg CS-SC dihedral angle, deg

2.038 1.810 1.097 102.8

2.022 (3)a 1.806 (2)

108.9

106.5

1.090 (7) 104.1 (3)

~~

1.842

103.2

(I.O) 83 .7,* 85 .Ob

a Numbers in parentheses are the reported errors i n the last significant figure. These values correspond to the minima of the solid and dashed curves, respectively, of Figure 2.

*

intervals in S-S bond length except near the minimum where the intervals were reduced to 0.01 A. The curve was drawn through 18 calculated values of the energy. Note that the CNDO/2 value for the equilibrium S-S bond distance is 1.84 A for a CS-SC dihedral angle of 90", roughly 10% shorter than the experimental values of 2.049 and 2.0210 A. The variation of the energy with the dihedral angle for rotation about the S-S bond is shown in Figure 2 for the experimental S-S distance (solid curve) and for the optimized CND0/2 distances (dashed curve). The solid curve was calculated by keeping all bond lengths and bond angles at their experimental values and by varying the CS-SC dihedral angle in 15" increments. To obtain energies from which the dashed curve in Figure 2 was constructed, a value for the CS-SC dihedral angle was first selected. Then at least four values of the energy were calculated by varying the S-S bond length, Rs-s, in 0.01-A steps close to the minimum. Four energies were then used to obtain the energy a t the minimum, Eo, the equilibrium S-S bond length, Rs-seq, The Journal of Physical Chemisiry, Vol. 78, No. 18, 1974

Q 85

Van Wart, Shipman, and Scheraga

1.92

- 2.10

I .9c

- 2.00

1.88

- 2.06

1.86

- 2.04

1.84

$202

J E ? Iyv)

a

1.82 FS-- SC Dihedral Angle (degrees)

Figure 2. Variation of the energy of dimethyl disulfide as a function of CS-SC dihedral angle, for the experimental bond lengths and bond angles (solid curve, Rs-seq = 2.04 A); the dashed curve was obtained similarly, except that the optimized CNDO/2 values for the S-S bond length (eq 1) were used for the calculation. The energy is normalized to zero at the! minimum of each curve.

Figure 3. Dependence of the equilibrium S-S bond length on the CS-SC dihedral angle. The curve corresponds to the data from which the dashed curve of Figure 2 was computed. V refers to ex-

the disulfide stretching force constant, Ks-s, and the anharmonicitg constant, Ka, shown in the equation

Rs-seq.

E = E,

-t

(1/2)K'g-s(Rs-g - RS-sep)' + (1/6)Ka&,-s - R S - S ~ ) ~(l)

Values of E o , RS-S@'~~ and Ks-s obtained by fitting four energies at 0.02-A intervals agreed well with the values from 0.01-A intervals. The dashed curve in Figure 2 is the variation of E( with the CS-SC dihedral angle (calculated in 10' i n t e n d s from 0 to 180°).18 The value of the CS-SC dihedral angle predicted by the CNDQ/2 method is 83.7* for the solid curve in Figure 2 and 85.0' for the dashed curve. Either of these values is in excellent agrelement with the experimental values of 84.79 and 83.9°.10 'The value of Rs-seq at 85.0' is also 1.84 A. The calculated values for the cis and trans barriers for rotation about the S--S bond (from the minima at f83.7' to those CS-SC dihedral angles corresponding to the cis and trans conformations) are 1' 7.7 and 10.8 kcal/mol, respectively (from the solid curve), No experimental values for the cis and trans barriers are available for comparison. The variation of the equilibrium S-S bond length, Rs-seq, with CS-SC ISihedral angle is shown in Figure 3. These values were obtained from eq 1. The CND0/2 method gives a monotonic increase in the equilibrium S-S bond length in going from a CS-SC dihedral angle of 90° to either 0 or 180'. A similar variation in S-S bond length with Cs-SC dihedral anglth has also been observed experiment a l w 9 The triangles shown in Figure 3 represent values of the S-S bond length (determined by X-ray or neutron diffraction s t ~ d i e s l on ~ - compounds ~~ exhibiting a variety of CS-SC dihedral angles. The solid squares in Fig.ure 3 are the experimental values for the S-S bond length of dimethyl d i ~ u l f i d elo. ~The reasons for the scatter of the experimental points have bren discussed by Jones, et ~ 1 1 . ~(The 9 have only the CSSC fragment in comThe Journal of Pkysicai Chemistry, Vo/. 78, No. 18. 1974

CS- SC Dihedral Angle (degrees)

perimental X-ray or neutron diffraction data compiled by Jones, et ab,'' and to additional data from ref 20-22; S refers to the microThe left- and right-hand ordinates wave data reported in Table refer to the theoretical and experimental values, respectively, of

mon.) Since the CNDOI2 method does not give the correct experimental value for the equilibrium S-S bond length, the experimental points have been plotted near the theoretical curve by shifting the scale (see scale on right-hand ordinate). Note that the theoretical trend for the variation of S-S bond length with CS-SC dihedral angle for the model compound dimethyl disulfide agrees with the experimental trend within the precision indicated by the scatter of the experimental points. The S-S stretching frequency (in wave numbers) is plotted in Figure 4 as a function of the CS-SC dihedral angle. The frequencies were calculated using the equation (21 where c is the speed of light, p is the reduced mass for the S-S stretching motion, and Ks-s is the S-S stretching force constant whose calculation was described earlier (see eq 1). Curve 3 was calculated using the experimental value9 of 102.8' for the CSS bond angle. Since the CSS bond angle is known to vary experimentally (especially in going from open-chain compounds to five- and six-membered disulfide-containing rings), some representative calculations were carried out for other CSS bond angles and these results are also plotted in Figure 4. For comparison, the observed experimental trend6 in S-S stretching frequency with CS-SC dihedral angle is shown as an insert in Figure 4. The CNDO/2 method gives values for the S-S stretching frequency that are too high by about a factor of 2. The approximately linear variation in calculated stretching frequency with CS-SC dihedral angle in the 20-80° range is similar to the experimental trend. However, the per cent variation in the calculated curve is about 3 times that in the experimental curve. Although it is unlikely that the ws,s = [l/(2nc)lE&.,/F)i'2

Disulfide Bond ~ t ~ e ~Frequencies ( ~ ~ ~ n g I

1851 I

I

*.I51

CSS Bond Angle (degrees)

I

0

30

I

I

120 150 CS-SC Dihedral Angle (degrees) 60

90

I

180

Figure 4. Variation of the, S-S stretching frequency of dimethyl disulfide with the CS-SC dihedral angle calculated for CSS bond angles of (1) 1 0 7 O , (2) 105O, (3) 102.8', (4) lolo, and (5) 99'. The insert shows the experimental' variation of S-S stretching frequency with CS-SC dihedral angle for a number of constrained compounds.

CND0/2 method yields the correct absolute energy of dimethyl disulfide, it describes the variation of the energy of the molecule with CS-SC dihedral angle adequately. Furthermore, although these calculations do not give the correct absolute value of the equilibrium S-S bond length, they do lead to a Variation in S-S bond length with CS-SC dihedral angle that is similar to the experimentally observed behavior. The variation in the S-S stretching frequency with CS-SC dihedral angle (in the 40-140° range) shown in Figure 4 is a natural consequence of the trends in energy and bond length.23 Although the calculated values of the S-S &retching frequencies are too large, more confidence may 'be placed in the calculated trend. For example, 8berhammer24recently used the CNDOI2 method to investigate the influence of various substituents on bond lengths and bond angles in various sulfur compounds. He found that, although the CND0/2 method sometimes gave quantitatively incorrect results, in all cases it gave the correct qualitative description of the influence of different s ~ ~ s t i t u e non t sthese bond lengths and bond angles, Consider tho e 5ection of curve 3 in Figure 4 drawn as a dashed line at small values of the CS-SC dihedral angle. In this range a9 c o ~ ~ f o r ~ a tof ~ odimethyl ns disulfide, the two methyl groups are quite close together. Since the equilibrium S-S bond length predicted by the GND0/2 method i s smaller than the experimental value, the methyl groups are even closer than they would be in reality. It is likely that the real molecule relieves the high strain energies (which would be less than the calculated energies because of the larger e x ~ e K ~ S-S ~ e bond ~ ~ alength) ~ encountered in this region by a combination of CSS bond angle bending and methyl group rotation. Neither of these degrees of freedom was allowed in the calculations (because of the high cost of computes time), This is a high-energy region of conformational space avoided by the real molecule in the absence of the constraint of ring closure. Hence, we attach no physical significance to the dashed section of the curve.

Figure 5. Variation of the energy of dimethyl disulfide as a function of CSS bond angie, for the experimental values of all other bond angles and bond lengths, at a CS-SC dihedral angle of 90'. The energy is normalized to zero at the minimum. An interesting feature of curve 3 in Figure 4 is the local minimum at a CS-SC dihedral angle of 140-150O. Values of the anharmonicity constant, K,, calculated for points in this region are quite normal compared to K , values for other regions above 30" (a typical value for K,/Ks-s in the region of 30-180' being -5.0 A-1). Furthermore, there are no close atomic contacts associated with this region. Thus one might expect to observe CS-SC dihedral angles in this region. Unfortunately, there are no experimental frequency determinations for this region of CS-SC dihedral angles which could support or contradict the existence of the calculated minimum. In the 90-180' range of CS-SC dihedral angles, the S-S stretching frequency is highest at 90". The effect of small changes in the CSS bond angle on the S-S stretching fr-equency is shown in Figure 4. There is no appreciable change in the shape of the curves from one CSS bond angle to another although the curves are shifted to somewhat higher or lower frequencies. Hence, the trends observed in the S-S stretching frequency do not seem to be significantly influenced by small changes in the CSS bond angle. The calculated potential energy curve for CSS bond angle bending is shown in Figure 5- The calculated equilibrium bond angle is 103.2", in excellent agreement with the experimental values of 102A9and 104.1°.10

V. Discussion Effects of Rotation about the S-S Bond. Several approximate molecular orbital studies, in addition to the present one, have been carried out on dimethyl disulfide.l2,13,25-27 These studies have been concerned primarily with the electronic absorption spectra of disulfides, although the variation of the energy for rotation about the S-S bond was also considered by Boyd,12 Perahia and Pullman,26 and Yamabe, et al.,27 Only the present study has dealt with the effects of rotation about the S-S bond on the equilibrium S-S bond length and S-S stretching frequency. A comparison of some of the results i s given in Table 11. The CND0/2 method provides the best value for the equilibrium CS-SC dihedral angle. The computed barrier heights show quite a spread, with the values obtained by the CNDOj2 method being on the high side. The results of all the approximate molecular orbital studies indicate that rotation about the S-S bond is hindered and that the equilibrium CS-SC diThe Journalof Physical Chemistry, Voi. 78. No. 18, 1974

4 85

Van Wart, Shipman, and Scheraga

TABLE 11: Comparison of Results for Approximate i ~ ~ for ~ Dimethyl Disulfides Molecular O ~ b Theories GSSC dihedral angle, deg

Cie barrh, kcal/ moi

-__-_

Trans barrier, kcal/ mol

Methodb

Ref

_ l l l l -

-90 -90

100 83.745.0

7.0 45.9 2.'9 17 .'7

2.2 14.5 1.3

10.8

EHT ZDO-SCF PCILO CNDO/2

12 26 25

This study

These results refer to a conformation of the molecule in which the methyl hydrogens are staggered w:!th respect to the S-S bond. Abbreviations used stand for extended Wickel theory (EHT), zero differential overlap-selfconsistent field (ZDB-SCF), perturbative configuration interaction using localized orbitals (PCILO), and complete neglect of differential overlap (CND0/2).

hedraP angle Cor open-chain disulfides is approximately f90". It is desirable to understand, in structural terms, the nature of the interactions which give rise to the features of the energy us. CS-Sc' dihedral angle curve. The problem of partitioning the total energy from a molecular orbital calculation intio simple, structurally meaningful components (e.g., lone pair-lone pair repulsions, etc.) is a current topic of developmental research in quantum chemistry. Since the energy-partitioning techniques have not yet been developed to the point that there i s general agreement on the components giving rise to the rotational barrier in a molecule as simple as ethane, we have not attempted to perform an energy-partitioning analysis on the much more complicated CSSC moiety. K t is of interest that the correct equilibrium CS-SC dihedral angle can be obtained by the approximate molticular orbital calculations without inclusion of dispersion er,ergies. Upon rotation about the S-S bond from a CS-SC dihedral angle of about 90', the disulfide bond weakens. This weakening manifests rtself as an increase in the equilibrium 53-23 distance and B dm-ease in the S-S stretching frequency. These trenois are found both in the present theoretical results and in X-ray and neutron diffraction results (in the ease of the equilibrium S-S bond length) and Raman spectroscopic reslults (in the case of the s-S stretching frequency). It would appear that these experimental trends are the (direct or inclirect) result of bond weakening under torsion. If these experimental trends were not due primarily to interactions within the GSSC group, itself, the computations reported here for the simple model compound dimethyl disulfide would not have been expected to reproduce them. Finally, the present cdculated results cover a wider range of CS-SC! dihedral angles than has been observed experimentally and therefore should be useful in interpreting the results OS future experiments designed to investigate more of the range of possible CS-SC dihedral angles. Implications for Przderpretmg S-S Stretching Frequencies frorn Proteins and Other Molecules. In the absence of structural constraints such as ring closure, the equilibrium CS-SC dihedral angle would be expected to lie in the 70 to 110' (or -70 to -4 ioa) range. This corresponds to an energy change of less than I kcal/mol from the preferred conformations of about $90'. (See Figure 2, solid line.) This expectation is supported by the multitude of CS-SC dihedral angles in the 70 to l l O o (and -70 to - 1 l O O ) range reported in X-ray and neutron diffraction studies on a variety of disulfides.19 Jones, et a1.,l9 have also found that this trmd is upheld for cystine residues in proteins. As seen in h e Journal oi Physictai Chemistry, Vo!. 78, No. 18. 1974

Figure 4,the S-S stretching frequency oes not change appreciably throughout the 70' to 110' range of CS-SC dihedral angles. An implication is that variations in the S-S stretching frequency for protein molecules may reflect changes predominantly in the CC-SS dihedral angles and CSS bond angles for the cystine residue. This emphasizes the need for more detailed information about the potential function for rotations about the C-S bonds. There is, on the other hand, a class of antibiotic molecules whose nucleus is the epidith~apiperaz~n~dione group. The CS-SC dihedral angle in this group is very low (about 10-15°).2s The interpretation of disulfide stretching frequencies for this group of molecules must be based, in part, on the weakening of the CS-SC bond shown in the present study. Other biologically important molecules may have CS-SC dihedral angles in the middle range of 30 to 60' (-30 to -60') or 120 to 160' (-120 to -160O). One might expect cyclic oligopeptides with cystine S-S bonds across the ring to have high enough strain energies to permit disulfide torsional angles in this range. Lysine-vasopressin, oxytocin, and malformin-A may be examples of such molecules. In these cases, the effects of 6%-SC bond weakening must also be considered. This study has provided evidence that CS-SC bond weakening is associated with a lowering of the disulfide stretching frequency and that this bond weakening is a function of the CS-SC! dihedral angle. The effects of rotation about the C-S bonds, as mentioned earlier, are under investigation and should provide other relevant information about the dependence of disulfide stretching frequencies on conformat~~n.

Acknowledgment. We are indebted to Dr. T. Koetzle for providing the experimental diffraction data plotted in Figure 3, prior to publication. References and Notes (1) This work was supported by research grants from the National Institute of General Medical Sciences of the National Institutes of Health, U. S. Public Health Service (GM-14312), and from the PdationaO Science Foundation (GB-28469x3). (2) (a) NIH Predoctoral Trainee, 1970-1974. (b) NlH Postdoctoral Fellow, 1972-1974. (3) R. C. Lord and N. Yu. J. Mol. Bo/., 50, 509 (1970). (4) H. Sugeta, A. Go, and T. Miyazawa, Chem. felt., 83 (1972). (5) E. J. Bastian and R. B. Martin, J. Phys. Chem., 77, 1129 (1973). (6) H. E. Van Wart, A. Lewis, H. A. Scheraga, and F. D. Saeva, Proc. Nat. Acad. Sci. U. S.,70, 2619 (1973). (7) J. J. Shipman, V. L. Folt, and S. Krimm, Speetrochim. Acta, 18, 1603 (1962). (8) S. K. Nandy, D. K. Mukherjee, S. B. Roy, and G . S. Kastha, J. Phys. Chem., 77, 469 (1973). (9) B. Sutter, HI. Dreizler, and H. D. Rudolph, Z. Nalurforsch. A, 20, 1676 (1965). (10) 8. Beagley and K. T. McAloon, Trans. Faraday Soc., 67, 3216 (1971). (11) G. Winnewisser, N. Winnewisser, and W. Gordy, J. Chem. Phys., 49, 3465 (1958). (12) D. 8.Boyd, J. Amer. Chem. Soc., 94, 8799 (1912). (13) D. B. Boyd, Theor. Chim. Acta, 30, 137 (1973). (14) I. H. Hillier, V. R. Saunders, and J. F. Wyatt, Trans. Faraday Soc., 66, 2665 ('1970). (15) D. P. Santry and G. A. Segal, J. Chem. Phys., 47, 158 (1967). (16) J. A. Pople and D. L. Beveridge, "Approximate Molecular Orbital Theory," McGraw-Hill, New York, N. Y., 1970. (17) We are indebted to Professor R. Hoffmann for making the CNDO/2 computer program available to us. (18) An irregularity was encountered when calculating the energy (for the dashed curve of Figure 2 ) at a CS-SC torsionai angle of 0'. It was found that the energy jumped discontinuously about 60 kcallmol when the S-S distance was decreased from 1.99 to 1.98 d. This behavior arose because the Huckel approximation. which is used to generate an initial density matrix for the CNDOl2 interation, yielded a had starting guess for this matrix for S-S internuclear distances below 1.99 A. Starting from this bad guess, the CNDOR method consequently converged on an excited state. The problem was solved by reading in directly the Huckel-generated initial density matrix from the previous calculation at

Method of

Caltxilating Electrostatic Energy

1853

1.99 A for iise as a starting guess for calculations at 1.98A and shorter S-S internucleitr distances. (19)D. D. Jones, I. Bernal, hil. N. Frey, and T . F. Koetzle, Acta Crysta//ogr., Sect. E, 30!,1220 (19711). (20)G. H. Wahl, J. Bordner, D. N. Harp, and J. G. Gleason, J. Chem. Soc., them. Conimun., 985 (1972). (21)0. Foss and 0. Tjomsiand, Acta Chem. Scand., 12,181.0 (1958). (22) 0.Foss, K. dohnsen, and T. Reistad, Acta Chem. Scand., 18, 2345 (1964). (23)For a stretching potentia! of the form

Hence WS-S is determined by €o/(Rs-seq)*, as long as MN does not change with CS-SC dihedral angle. The constancy Qf MN (in the range of 40-160' in CS-SC dihedral angle) is verified by the fact that a plot of - [ ( K S - S ( R ~ - S ~ ~(which ) ~ ~ /is€equal ~ ] to MM vs. CS-SC dihedral angle is found to be approximately horizontal in the 40-160' range. Hence, the variation in WS-S with CS-SC dlhedral angle, shown in Figure 4, is a direct consequence of the trends in €0 and F7s-s- shown in Figures 2 and 3. (24)H. Oberhammer, Theor. Chim. Acta, 26, 79 (1972). (25)S. D. Thompson, D. G. Carroll, F. Watson, M. O'Donneil, and S. P. McGlynn J. Chem. fhys., 45, 1367 (1966). (26)D. Perahia and B. Buiiman, Biochem. Biophys. Res. Commun., 43, 65

(1971). (27) H. Yamabe, H. Kato, and T. Yonezawa, Bull. Chem. SOC.Jap., 44,604