5772
J . Phys. Chem. 1990, 94, 5112-5778
mental error over this temperature range. Rather, the small decrease in fP and 4pwith increasing temperature in solid solution must be associated with a concomitant increase in the rate of nonradiative decay. No direct evidence of the process involved is available, but the small effective activation energy of 0.46 kJ mol-’ (38 cm-’) likely rules out a mechanism which involves more effective T,-So coupling via the population of higher vibrational states in T,. We note with interest, however, that the measured
energy spacing, 38 cm-], is of the same magnitude as the zero-field splitting parameters, ID*I, for the triplet states of several aromatic thionesgJ4 (16-28 cm-I), suggesting that radiationless decay via an upper spin sublevel of the triplet might be responsible. Acknowledgment. We thank the Natural Sciences and Engineering Research Council of Canada for its continuing financial support.
An ab Initio Study of the Structures, Vibrational Spectra, and Energetics of the Homocyclic Sulfur Molecules S,, n = 4-8 David A. Dixon* and E. Wasserman Central Research and Development Department,? E. I. du Pont de Nemours and Company, Experimental Station, Wilmington, Delaware 19880-0328 (Received: May 30, 1989; In Final Form: December 13, 1989)
High-level ab initio calculations have been employed to determine the geometries of cyclic S,, n = 4-8. Gradient optimization was done at the SCF level with a polarized double-f basis set. Where available, agreement with experiment for the structures is better than has previously been available from ab initio calculations. Vibrational spectra were calculated for comparison to experiment. Energies relative to S,,have been obtained for n = 5-7. Comparison of energetic and spectroscopiccalculations to experiment indicates that S4 is not a ring. The reaction S, + S8 = 2S7 has a substantial entropy contribution to AG.
Introduction
The higher levels of ab initio molecular orbital calculations are increasingly of merit in providing information for the experimentalist seeking a more complete view of a system. As a growing new scientific area, the results from numerical simulations can supplement the interpretation of a variety of measurements.! The family of homocyclic sulfur molecules, S,, is a rich source of molecular variety with a single element. The experimental structures show a variation of -0.2 A, or -IO%, in S-S bond length, an unusually large range for nominal single bonds, and IOo in S-S-S bond angle.2 Due to the work of SteudeP and S ~ h m i d ta, ~number of sulfur rings have been well-characterized experimentally. On the other hand, previous theoretical studies of these rings with molecular mechanics! semiempiri~al,~ or ab initio6 approaches have not yet reproduced the structural and vibrational spectroscopic parameters for these systems to the accuracy of theoretical studies of first-row elements. We have previously employed’ numerical simulation techniques (molecular orbital theory) to study structural and spectroscopic features of pentathiepins in regions of the potential energy surface inaccessible to present experiments. We report here a high-level a b initio study of the geometries, energetics, and vibrational spectra of the sulfur rings S,, n = 4-8. The structures of n = 6-8 are well-established experimentally but are uncertain for n = 4 and 5 . This group includes many of the most interesting structural features present in elemental sulfur rings. The larger rings, n 2 9, investigated experimentally by Steudel and co-workers2 are not considered here. It is important to note that these calculations are not substitutes for more experimentally coupled approaches, e.g., force-field analyses. The ab initio computations of the force field are done in the harmonic approximation and do not include electron,correlation as they are at the S C F level. In the treatment presented here for vibrations, frequency scaling is used to compensate approximately for anharmonicity and correlation. When theoretical assignments of vibrational frequencies are at variance with experimental conclusions, improved agreement might be obtained
by the future incorporation of additional terms in the vibrational Hamiltonian. Alternatively, another assignment for the observed frequency might be considered. Since a complete theoretical treatment should correspond to correct experimental assignments, disagreements should focus attention on possible changes in one approach or the other. With ab initio calculations of second-row elements being more demanding than those for the first row, we hope that the treatments reported here represent a step toward a deeper understanding of these electron-rich species. Calculations
The calculations were done with the programs HONDO* on an IBM-3081 computer and GRADSCF’ on a CRAY-1A or CRAYXMP/24 supercomputer. We have previously shown that a polarized double-!: basis set is required to calculate accurately the S-S bond length in H2S2.6J0 All calculations were thus done with the valence double-l basis set of McLean and Chandler” augmented by a set of polarization functions (Cd = 0.6) on sulfur giving a (12s8pld)/[5s3pld] basis set. The number of basis functions ( 1 ) Schaeffer, H. F., Ill Science 1986, 231, 1100. (2) (a) Steudel, R. Top. Curr. Cbem. 1982,102, 149. (b) Steudel, R. In Studies in Inorganic Chemistry; Miller, A., Krebs, B., Eds.; Elsevier: Amsterdam, 1984; Vol. 3, p 5. (3) Schmidt, M. Angew. Cbem., Int. Ed. Engl. 1973, 12, 445. (4) Kao, J.; Allinger, N. L. Inorg. Cbem. 1977, 16, 35. (5) (a) Dewar, M. J. S.; McKec, M. L. J. Comput. Cbem. 1983.4, 84. (b) Dewar, M. J. S.; Reynolds, C. H. Ibid. 1986, 7, 140. (c) Baird, N. C . Ibid. 1984, 5, 35. (d) Jug, K.; Iffert, R. Ibid. 1987, 8, 1004. (6) (a) Laitinen, R. S.; Randolph, B.; Pakkanen, T. A. J. Comput. Chem. 1987,8,658. (b) Kao, J. Inorg. Cbem. 1977, 16, 2085. (c) Ibid. 3347. (d) Feng, W.L.; Novaro, 0. Ini. J. Quantum Chem. 1984, 26, 521. (7) Chenard, B. L.; Dixon. D. A.; Harlow, R. L.; Roe, D. C.; Fukunaga, T.J. Org. Cbem. 1987. 52, 24. (8) (a) Dupuis, M.; Rys, J.; King, H. F. J . Chem. Phys. 1976.65, 1 1 I . (b) King, H. F.; Dupuis, M.; Rys,J. National Resource for Computer Chemistry Software CataIoK University of California, Berkeley, CA, 1980 Vol. I , program QHOZ (HONDO). ( 9 ) GRADSCF is an ab initio gradient program system designed and written by A. Komornicki at Polyatomics Research. (IO) Dixon, D. A.; Zeroka, D.; Wendoloski, J. J.; Wasserman, Z. R. J. Phys. Chem. 1985,89, 5334. ( 1 1 ) McLean. A. D.; Chandler, G. S. J. Chem. Phys. 1980, 72. 5639
‘Contribution no. 4955.
0022-3654/90/2094-5712$02.50/0
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990 5773
Homocyclic Sulfur Molecules, S, n = 4-8 TABLE I: Calculated Geometry Parameters" for S, n = 4-8 r(S-S)b value
e(s-s-w
value
r(S-S-S-S)'
value
n=4 90 86
s-s-s s-s-s
n=5
1
1-2 2-3 1-1'
2.074 2.066 2.138
2
1'-1-2 1-2-3 2-3-2'
99.0 102.3 94.2
1'-1-2-3 1-2-3-2' 2-1'-1-2
37.4 60.7
95.8 101.6 102.5
1'-1'-2-3 1-2-3-2' 2'- 1'-1-2
-5 1.4 19.7 63.7
0.0
2
1-2 2-3 1-1'
2.060 2.114 2.070
1'-1-2 1-2-3 2-3-2'
s-s
2.070 (2.068)
s-s-s
n = 6d 102.6 (102.6)
s-s-s-s
73.8 (73.8)
n = 7c
1-2 2-3 3-4 1-1'
2.029 2.079 2.060 2.144
(2.050) (2.179)
1'-1-2 1-2-3 2-3-4 3-4-3'
s-s
2.060 (204.6-205.2)
s-s-s
(1.996)
(2.100)
109.5 105.5 103.0 106.4
(107.1) (105.3) (102.2) (106.1)
n = 8 (DU)f 107.2 (107.3-109.0)
1'-1-2-3 1-2-3-4 2-3-4-3' 2'-l'-l-2
s-s-s-s
8 1.4 (84.0) 104.6 (107.5) 76.8 (75.2) 0.0 (1.1) 99.7 (98.5)
"Experimental values in parentheses. *Bond distances in angstroms. cBond angles in degrees. "Reference 15. eReference 16. /Reference 17. ranged from 80 for S4 to 160 for S8. Geometries were gradient optimizedI2 with HONDO within the following symmetries: S4, D4* and D2d;S5, C, and C,; S6, Djd;S7, C,;s8, D4,+ All vibrational calculations used GRADSCF, which incorporates rapid methods for the analytic calculation of second derivatives.I3 Correlation energy corrections at the MP-2 levelI4were also calculated with GRADSCF.
Results Geometries. The optimized geometries of the sulfur rings are given in Table I together with the available experimental results for SS,l5S7,I6and S8.17 The total energies are given in Table 11. The S4 molecule is predicted to have a nonplanar structure of D2dsymmetry. The bond length of 2.1 12 8, is longer than the calculated 2.065 A of twisted, ground-state H2S: (experimental = 2.055 and more like the 2.1 18 8, calculated for cis-H2S2. The bond angle in S4 is calculated to be 86', significantly smaller than the S C F value of 97.3' computed for B(HSS) in cis-H2S2.I0 ( I 2) (a) Komornicki, A.; Ishida, K.; Morokuma, K.; Ditchfield, R.; Conrad, M. Chem. Phys. Lett. 1977, 45, 595. McIver, J. W., Jr.; Komornicki, A. Ibid. 1971. IO,303. (b) Pulay, P. In Applications of Electronic Structure Theory; Schaefer, H. F., 111, Ed.; Plenum: New York, 1977; p 153. ( I 3) (a) King, H. F.; Komornicki, A. In Geometrical Derivatives of Energv Surfaces and Molecular Properties; Jsrgenson, P.,Simons, J., Eds.; Reidel: Dordrecht, Netherlands, 1986; NATO AS1 Series C; Vol. 166, p 207. (b) King, H. F.; Komornicki, A. J . Chem. Phys. 1986,84, 5645. (c) McIver, J. W., Jr.; Komornicki, A. J . Am. Chem. SOC.1972, 94, 2625. (14) (a) Moller, C.; Plesset, M. S.Phys. Rev. 1934, 46, 618. (b) Pople, J. A.; Binkley, J. S.;Seeger, R. Int. J . Quantum Chem. Symp. 1976, IO, I . (15) Steidel, J.; Pickardt, J.; Steudel, R. Z . Naturforsch. 1978,336, 1554. (16) Steudel, R.; Reinhardt, R.; Schuster, F. Angew. Chem. 1977,89,756; Angew. Chem., Inr. Ed. Engl. 1977. 16, 715. (17) (a) Coppens, P.; Yang, Y. W.; Blessing, R. H.; Cooper, W. F.; Larsen, F. K. J . Am. Chem. SOC.1977, 99, 760. (b) Goldsmith, L. M.; Strause, C. E. Ibid. 1977, 99, 7580. (c) Watanabe, Y. Acta. Crystallogr. 1974,308, 1396. (18) Winnewisser, M.; Haase, J. Z . Naturforsch. 1958, 231, 56.
TABLE 11: Total Energies for Homocvclic S. Molecules (in a d n SCF MP-2 4 (D4h) -1 589.962 561 -I 590.453 907 -1 590.464 0 16 4 (Du) -1589.969 149 5 (CS) -1987.507 948 -1988.141 120 5 (C,) -1987.507 848 -1988.140868 6 (Did) -2385.036 243 -2385.793 177 7 (CS) -2782.538914 -2783.428 639 7 (C,) -2782.538 951 -2783.428 603 8 (Du) -3181.073 445 -3 180.058 865
The inversion barrier via a square-planar transition state (vide infra) of D4* symmetry is 6.3 kcal/mol at the correlated MP-2 level and 4.1 kcal/mol at the SCF level. The bond length increases by 0.005 8, in the D4,,transition state. For S5, both a C, structure (half-boat) and a C, structure (twist-boat) were considered. The C,structure is a true minimum on the potential energy hypersurface, whereas the C, form is a transition state, Le., one negative direction of curvature. The two structures are almost isoenergetic with the C,structure being 0.06 kcal/mol more stable at the S C F level and 0.16 kcal/mol more stable at the correlated MP-2 level. The low energy required to reach the C, transition state suggests a very floppy (pseudorotating) molecule. The C,structure has four bonds near the length of a normal S-S bond (2.065 A from H2S2)6and one long bond of 2.138 A with a dihedral angle ~(2'-1'-1-2) of 0'. Thus the 1-1' bond should resemble the S-S bond in cis-H2S; but is 0.02 A longer in S5,consistent with the larger atomic radii of the S atoms as compared to the H atoms in H2S2. Except for 8(2-3-2'), the bond angles are near loo', consistent with the values found for QHSS) of 97.3' in cis-H2S2.I0The increased angles may be due to the increased size of the S atoms. The unique angle b'(2-3-2') is somewhat smaller than the others and B(HSS) in H2S2. The torsion angles are all smaller than the optimum value of 90' found for H2S2.6,10
5774 The Journal of Physical Chemistry, Vol. 94, No. 15, 1990
For the C2 structure for S5, there are two long bond lengths of 2.1 14 A associated with the smallest torsion angles of 20’. The three shorter bonds are similar to those in ground state (7 = 90) H2S2. There are two bond angles of 96’ and three near 102’. These values bracket the 97.3’ found for B(HSS) in H2S2. The torsion angles are larger in the C2 structure than in the C, and are thus closer to the optimum value found for H2Sz. Thus the strain due to torsion is lower in the C2 structure, balancing other sources of strain, and the C2 and C, structures have comparable energies. The D3, structure for s6 should be one with minimal strain. The bond length is 2.070 A compared to the experimental value of 2.068 A.I5 The calculation of the bond and torsional angles for S6also reproduces the experimental values. The torsion angle of 73.8’ approximates that of ground state H2S2and B(S-S-S) of 102.6 is larger than B(HSS) of 97.3’ in cis-H2S2. The allotrope S7 has low symmetry and large alternations of bond lengths around the ring. The molecule is very floppy and should easily pseudorotate, as has been suggested by Steudel et aI.l9 In the calculated C, structure, there is one torsion angle of 0’. and the associated bond is long (2.144A). The two adjacent bonds are quite short, 2.029 A. Two bonds at 2.060 8, are close to “normal” length (s6 and sg), with the 2-3 and 2’-3‘ bonds slightly elongated at 2.079 A. The theoretical values follow the trends found experimentally.I6 The experimental alternations are larger than the theoretical differences. For example r(l-1’) is observed to be 0.035 A longer than calculated and r( 1-2) and r( 1’-2’) are found to be 0.033 A shorter than calculated. These differences between theory and expeirment may be compared with the good agreement found for H2S2,S 6 , and S8 (vide infra). Determination of the structural parameters for S7could be complicated by low-energy vibrational modes. The S-S-S bond angles are in good agreement except for B( 1’-1-2) and B( I-1’-2’), which are calculated to be 2.4” larger than experiment. This discrepancy is consistent with the largest differences being found for r( 1-1’) and r( 1-2) (and r( 1’-2’))~ The calculated and observed torsion angles agree to within 3’. To obtain more understanding about the structure of S7,we examined the structures of the sulfanes H2S3 and H4S4in constrained geometries that mimic regions of S7. The structures studied are shown in Figure I , and various geometric parameters are given in Table 111. The geometries were gradient optimized with the previously employed DZ+D basis set for S and a polarized double-f basis set for H.20 Although the global minimum for H2S3 is the trans structure and that for H2S4 is skew, the fragments most appropriate for comparison with S7are the cis for H$3 and the cis-cis for H2S4. The cis structure for H2S3(a minimum on the potential energy surface) is higher in energy than the trans by only 0.25 kcal/mol at the SCF level and 0.19 kcal/mol at the MP-2 level. The cis-cis structure for H2S4, which is a transition state characterized by one negative direction of curvature, is 9.1 kcal/mol higher in energy than the skew at the SCF level and 8.2 kcal/mol at the MP-2 level. The S-S bond distance in cis-H2S3 is 2.063 8, as found for the 3-4 and 3’-4‘ bonds in S,. The cis-cis isomer of H2S4 has one long S-S bond of 2.1 53 A and two short of 2.033 A. These are similar to the values calculated for the 1-l’, 1-2, and 1’-2’s-S bonds in S7. The short bonds are essentially identical, whereas the long bond in H2S4is -0.01 A longer than the 1 - 1 ’ bond in S7. Thus, to a good approximation S7 can be considered as a cis-cis-S,R2 fragment bonded to a cis-S3R2 fragment. The bonds connecting the two fragments, 2-3 and 2’-3‘, are 0.01-0.02 A longer than an unstrained S-S bond. These results indicate that the significant increase in r( 1-1’) above a normal S-S bond is due to the 0’ torsion angle,21as was earlier concluded by Steudel from experimental correlations. The calculated structure of sg is of high symmetry (D4d), analogous to S6. There are different crystal structures for S8, (19) Steudel, R.; Schuster, F. J . Mol. Srruct. 1978, 44, 143. (20) Dunning, T. H., Jr.; Hay, P. J. In Methods of Electronic Structure Theory; Schaefer, H . F., 111, Ed.; Plenum: New York, 1977; p I . (21) See also: Laitinen, R. S.; Pakkanen, T. A. J . M o l . Struct. ( T H E O C H E M ) 1984, 108, 263. These authors report MINI-I ‘ ( S ) calculations on H2S4and obtain similar results.
Dixon and Wasserman
H
H T
C HIS3
H
I
S
1’s-
,s-s
0
H
S I
H
Tr
Skew H2S4
Figure 1. Structures of S2 and S , sulfanes. C = cis, T = trans
dependent on the crystal al10trope.l~ Here we compare the structure for c&g, where the crystal site does not have Du symm e t r ~ . The ’ ~ ~range of experimental bond distances is 2.046-2.052 8, and for bond angles 107.3-109.0°. The calculated bond length is -0.01 8, longer than the experimental values, consistent with our previous work on H2S2.6*10 The calculated bond angles are 1 ’smaller than the experiment, and the calculated torsion angle is about 1’ larger. Vibrations. The calculated molecular vibrations and the infrared intensities are given in Table IV together with the experimental values where available. We also report scaled frequencies employing a factor of 0.9. Such scaling compensates for neglect of correlation and the use of harmonic frequencies, whereas the experimental values include anharmonic effects.22 We first discuss S6, S7, and S8 where comparison to experiment is possible. For s6, there is overall good agreement between the scaled and the experimental except for the a,, and eg modes. The a,, mode is forbidden in the IR and Raman regions and difficult to observe. We calculate a scaled value of 466 cm-I as compared to the experimental assignment of 390 cm-I associated with a very weak band.24 We also calculate a value of -470 cm-l for the eBstretch, which is 20 cm-l higher than the assigned value of 450 cm-1.20,21The value of 180 cm-I assigned for the lowest e, mode2, is 21 cm-I higher than our scaled value. The potential energy surface for S7 is complicated by a lowenergy pseudorotation mode. At the SCF level, the C, structure had an imaginary frequency of 27i cm-I. The structure was distorted along this mode and reoptimized. The new C, structure had the smaller imaginary frequency of 21i cm-I and was only 0.02 kcal/mol lower in energy than the earlier C,structure at the SCF level. However, at the MP-2 level the C, structure was slightly lower (0.02 kcal/mol) than the C,structure. The similarities in energy from the two calculations suggest that S7 has a “free” pseudorotation mode (a”) with a frequency 530 cm-l. Such a pseudorotation mode has been previously suggested by Steudel and SchusterI9 on the basis of thermodynamic data for S,. Subsequently, Steudel et aLZ5observed broadening of some of the S-S stretches in solution by Raman spectroscopy, providing further experimental evidence for pseudorotation.
-
(22) Dixon, D. A. J . Phys. Chem. 1988, 92, 86. (23) Berkowitz, J.; Chupka, W. A,; Bromels, E.; Belford, R. L. J . Chem. Phys. 1967, 47,4320. . (24) (a) Nimon, L. A.; Neff, V . D.; Cantley, R. E.; Buttlar, R. 0.J . Mol. Spectrose. 1967.22, 105. (b) Nimon, L. A.; Neff, V. D. Ibid. 1968,26, 175. (25) Steudel, R.; Papavassilliou, M.; Jensen, D.; Seppelt. K . Z. Nuturforsch. 1988, 43b. 245.
Homocyclic Sulfur Molecules, S,, n = 4-8
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990 5775
TABLE 111: Geometry Parameters for S3and S, Sulfanes” molecule r(S-H) r(S-SH) r(S-S) B(HSS) C-H S 1.332 2.063 98.5 I-H~S:~ c,c-H~S~ c,t-H2S, I,C-H~S~ t,t-H$4 skew-H2S;
1.332 1.333 1.331 1.334 1.333 1.333
2.063 2.033 2.035 2.050 2.05 1 2.062
2.153 2.155 2.129 2.125 2.063
98.5 99.2 99.4 99.0 99.0 98.1
e(SSS)
r(HSSS)
106.5 106.5 109.0 109.0 100.6 100.6 106.4
81.9 87.9 86.8 86.6 89.5 89.4 86.5
r(SSSS)
0.0 0.3 180.0 180.0 85.9
AE(SCF)*
AE(MP-2)b
no. i VC
0.24
0.19
0.0 9.1
0.0
0 0
8.3 5.1 5.0 0.0
1 1
8.2 7.4 6.0 6.0
1
0.0
0
I
(I Bond distances in angstroms. Bond angles in degrees. See Figure 1 for labels. Energy in kcal/mol relative to most stable conformer. CNumber of imaginary frequencies. dE(SCF) = -1 193.683512 au; E(MP-2) = -1 194.070941 au. CE(SCF)= -1591.191 576 au; E(MP-2) = -1591.705342 au.
The agreement between theory and e ~ p e r i m e n t l for ~ * ~S7~ is quite reasonable except for the assignment of three stretching bands. The calculations for the four a’ stretches range from 509 to 410 cm-I. whereas the experimental values are 518-360 cm-I. The calculations of the lowest two a’ stretching frequencies give 460 and 410 cm-’ as compared to the assigned values of 400 and 360 cm-l. On the basis of our calculations, we would consider the observed band at 400 cm-’ as the fourth a‘ stretch and the assigned a” transition at 460 cm-’ as a possible superposition of a’ and a” modes. A similar situation would occur with the lowest energy a” stretching mode assigned at 396 cm-I. On the basis of the calculations, this transition could be derived from the lowest a’ stretch. The calculations indicate that the lowest energy a” stretch should be -461 cm-I. The observed transition near 480 cm-l could be a superposition of the second a’ and a” modes. Both are predicted to have low intensities and would be difficult to differentiate in the infrared region. Thus the major differences in the calculated and experimental assignments are for the lowest a’ and a” frequencies. Such a difference is also found in s8 (vide infra), suggesting that there could be larger correlation corrections to these lower energy stretching frequencies. The magnitudes of the differences in the calculated and observed S-S bond lengths for the shortest (1-2) and longest (1-1’) bonds is -0.035 A. For the highest stretching frequencies this difference leads to a correction of IO cm-’ in scaled calculated frequencies. It would be surprising if the difference was significantly larger for the lowest stretching frequency. Thus these calculations cannot account for the observed transition at -360 cm-I. However the potential energy surface for S7 is very flat due to the presence of the low-energy pseudorotation mode. The second derivatives were calculated analytically and determine the specific curvature at a single point on the potential energy surface. Because the pseudorotation mode in S7involves substantial changes in S-S bond lengths around the ring, there could be strong couplings of the stretches. Our calculations, of course, do not include such potentially important couplings. Much larger anharmonic corrections could result, and our calculations were in the harmonic approximation. One other difference is that calculations give a value of 186 cm-’ for the a” transition experimentally assigned at 239 cm-I. The calculated vibrational spectrum for s8 is in good agreement with e~periment~’-~l except for the b, and e, stretching modes. The calculated values agree with the assigned values for the infrared-active stretch of e, symmetry and with the assignment of the Raman-active a, and e2 stretching modes, with calculation of the latter 15 cm-I too high. Vibrational spectra of oriented single crystals of orthorhombic sulfur29show three bands near 470 cm-I, consistent with the calculated results for the a,, e,, and e2 bands. The computed Raman-active e, stretch is essentially degenerate with the ez stretch at 489 cm-I in contrast to the
-
-
(26) Gardner, M.; Rogstad, A. J. Chem. Soc., Dalton Trans. 1973,599. (27) (a) Scott, D. W.; McCullogh, J. P.J. Mol. Spectrosc. 1961,6, 372. (b) Scott, D. W.; McCullogh, J. P.; Kruse, F. H.J. Mol. Spectrosc. 1964. 13, 313. (28) Steudel, R.; Maude, H . 4 . Z.Naturforsch. 1978, 33a, 951. (29) Gautier, G.;DeBeau, M. Specrrochim. Acta 1974, 30A, 1193. (30) Gautier, G.;DeBeau, M. Spectrochim. Acra 1976, 32A, 1007. (31) (a) Guthrie, G.B., Jr.; Scott, D. W.; Waddington, G. J. Am. Chem. SOC.1954. 76, 1488. (b) Bernstein, H. J.; Powling, J. J. Chem. Phys. 1950, 18, 1018.
experimental assignment of the e, stretch 30 cm-I lower than the e2 mode. The original experimental results” for the e3 stretch have been confirmed by measurements of the vibrational spectra for single crystals of both orthorhombicB and monoclinic P3O sulfur that show Raman bands near 440 cm-I. This suggests that of a larger correlation correction (a scale factor of 0.82) is needed for the e, stretch as compared to those for the a,, e,, and ez stretches. The b, stretch is forbidden in the IR and Raman regions. The frequency is calculated at 487 cm-l to be essentially degenerate with the e2 and e, stretches rather than the assigned value of 424 cm-I. Although a weak Raman band has been observedz9 near 420 cm-’ in single crystals of orthorhombic sulfur, the assignment of the b, stretch has usually been used on force-field calculations that yielded frequencies3, ranging from 532 cm-l to the value of 424 cm-’ given in Table IV. In fact, our calculations predict all of the stretches to be of comparable energy (within 25 cm-I). The b2 transition is calculated to be 256 cm-l versus experiment at 239 cm-I. At least five bands below 100 cm-l have been observed. We assign the -60-cm-’ transition as the e2 torsion rather than the band in the solid near 80 cm-1.z7+28 Scott et al.27bsuggest a value of 56 cm-’ for this mode in the vapor. The second derivative calculations for S4 predict stretches at 504,478, and 454 cm-] and the inversion mode at 170 cm-I. For comparison, the imaginary inversion frequecy is 127i cm-l in the D4htransition state. The infrared spectrum of a species attributed to S4 has been reported in a matrix.3z The reported peak positions are 681,660,636,483, 320, and 270 cm-I. Comparison with our values shows that only the bands at 483 and 320 cm-’ could be attributed to S4 in a ring and the former would be forbidden in the infrared. The authors attribute the observed spectrum to an open structure for S4. Comparison of the calculated energetics of cyclic S4 to the experimental thermodynamic data (vide infra) are consistent with this conclusion. The ground state of S, is calculated to be a half-chair of C, symmetry as all frequencies were real (positive directions of curvature). The C2 structure (analogous to a twist-boat) is a transition state characterized by one negative direction of curvature (imaginary frequency) of 32i cm-I, whose absolute value is similar in magnitude to that of the pseudorotation mode in the C, geometry. The C, structure has two stretches near 500 cm-l, two near 450 cm-I, and one near 430 cm-’, corresponding to localized motion about bonds of different length. The lowest energy mode at -35 cm-’ involves pseudorotation so that cyclic S,, like S7, is probably a fluxional molecule. The stretches in the C2structure are similar to those found for the C, structure, although the values are more evenly spread acrms the range 430-500 cm-I. The lowest energy stretching mode calculated for Ss (C,) is 429 cm-l as compared to 4 IO cm-l for S7. These modes are clearly associated with the 1-1’ bonds in Ssand S,. The small frequency difference is consistent with the slightly longer 1-1’ bond calculated as 2.144 8, in S7 versus 2.138 8, in S,. Energies and Thermodynamic Functions. The thermodynamic properties of S, were calculated in the harmonic oscillator, rigid-rotor approximation using standard formulas of statistical mechanics3, with the scaled frequencies and computed geometries. (32) Meyer, B.; Stroyer-Hansen, T. J. Phys. Chem. 1972, 76, 3968. (33) McQuarrie, D. Srarisrical Mechanics; Harper and Row: New York, 1976.
5776 The Journal of Physical Chemistry, Vol. 94, No. IS, 1990
TABLE IV:
Vibrational Frequencies and Infrared Intensities for
Homoevclic S, II = 4-8
all big bzl bl"
e" a1 bl b2
e
542 576 346 141i 537
n=4 488 518 31 I 127i 483
560 189 53 I 369 505
n = 4 (Dw) 504 170 478 332 454
(D4h)
0.0 0.0 9.3 0.0
0.0 0.0 0.0 0.6 0.2
n = 5 (C,) a'
a"
aa
0.3 5.2 1.2 1.5 0.3 0.2 2.1 0.2 2.3
553 498 477 333 25 1 549 503 326 37
498 448 429 300 226 494 453 293 33
562 514 482 326 253 533 496 336 351
n = 5 (C,) 506 463 434 293 228 480 446 302 32i
528 288 518 360 536 237 520 177
475 259 466 324 482 21 3 468 159
565 531 513 456 312 257 I96 144 57 1 539 319 207 I79 0-20
509 478 462 410 28 1 23 I 176 130 514 485 46 1 287 186 161 0-20
522 239 541 284 521 213 543 I73 66 544 270
470 215 487 256 467 192 489 156 59 490 243
s6 ( D 3 d )
Sl
a'
a"
512
a, bl b,
el e2
e3
(C,)
Ss ( D d
0.1 1.6 1 .o
0.2 0.4 0.5 6.5 1.5
0 0 0 4.5 0 0 1.9 2.2
478: 471' 264, 262 390 312, 313 450, 448 201, 202 462, 463 I80
6.1 0.3 I .2 0.2 3.5
518/5178 480, 483 400, 402 360, 362 270, 268 239, 239 195, 184 155, 145 527, 510 460, 477 396, 397 285, 288 239, 242 185, 193 147, 155
1.o
0.6 0.5 0.1 0.3 0.0 0.6 1.8 0.0 0.0
Dixon and Wasserman from experimental spectroscopic data by Berkowitz et al.23and with measured experimental values of 84.6 cal/mol.K at 298.16 K and 93.1 cal/mobK at 400 K.23334The values are given in Table V . The values in Table V demonstrate that the calculations can reproduce the experimental measurements. The calculated thermodynamic properties for S7are sensitive to the pseudorotation frequency. In Table V, we report properties obtained with choices of 20 and 30 cm-I. The results for SoTare higher than those19 that employ a larger pseudorotation frequency of 147 cm-I, for which, SoT= 95.27 cal/mol.K at 298.15 K and I 12.8 I cal/mol.K at 500 K. This value of the pseudorotation frequency was taken from a spectroscopic assignment from the solid. The authors19 note that free pseudorotation would raise SoTby about 2 cal/mol.K at room temperature. Our calculated value of 117.6 cal/mol.K at 512 K with uPR = 30 cm-I can be compared to the experimental value of 116.9 f 3.8 cal/mol.K reported by Detry et al.35 This experimental value was obtained from mass spectrometric studies employing an electrochemical Knudsen cell. Rau et al.36reanalyzed the data of Detry et al.35 using estimated heat capacities and obtained SoT= 97.41 cal/ mo1.K at 298.15 K. For Ss, the values given are in reasonable agreement with those of Steude12?' considering that our smallest frequency is slightly lower. Because structural and spectroscopic data on S4 and S5 were not available, earlier workers had to estimate SoT.Our calculated value for SoTfor S5of 83.91 cal/mol.K is significantly higher than the earlier values 77.2 cal/mol.K28 and 73.74 ~ a l / m o l . K . ~The ~ estimated35SoTof Ss of 94.3 f 4.0 cal/mol.K at 592 K is lower than our calculated value of 99.98 cal/mol.K. Because s6 has three more degrees of vibrational freedom, it would be expected to have a higher entropy than that for S5. However, the entropies of Ss and s6 are calculated to be quite similar at room temperature. One of the additional modes for s6 is a stretch that is not substantially populated at 298 K. The two remaining modes at 177 cm-l (e,) in s6 are much higher in energy than the lowest energy (pseudorotation) mode in S5 at 33 cm-I. As a consequence, there is a significant contribution from this low mode in S5, and the entropies of s5 and s6 are comparable. At higher temperatures where the higher energy vibrational modes in s6 contribute, the entropy for s6 is larger, as expected. Our value of 84.35 cal/mol.K for the entropy of the ring structure of S, at 61 5 K is in good agreement with the estimated value of 83.9 f 2.5 cal/mol.K reported to Detry et al.35 The estimated ambient temperature values of 68.8 cal/mol-K2*and 74.22 ~ a l / m o l . Kbracket ~~ our value of 71.39 cal/mol.K at 298.15 K. However, we note that on the basis of the enthalpy data reported below, it is unlikely that S4 is a ring. Experimentally, the heats of formation of S2 and s8 are accurately and the energies of other S, species relative to these values have been obtained. The heats of formation of various s, molecules can be related to mfo (s8)by using eq 1-4. 7/gs8 3/qs8
0.0 0.0 0.0 7.3 0.4 4.4 0.0 0.0 0.0 0.0 0.0
475,h 475' 218, 217 424 239, 243 475,471 193, 184 415, 475 163, 152 83, 85 444,435 241, 248
a Frequencies in cm-l; calc = calculated; obs = observed. bCalculated frequency scaled by 0.9. 'Infrared intensity in km/mol. dReference 23. eReference 24. /Reference 19. ERefercnce 26. *Reference 28. 'Reference 27.
Entropies and heat capacities for S6, S7,and Ss have been computed. The values for s6 are in agreement with those determined
-
s7
(1)
s6
(3)
(4) These values corrected to 298.15 K are given in Table VI. Using a value of 24.3 kcal/mol for AH? we report the AHfO's for the various S, species in Table VII. As shown in Table VI, only reaction 4 shows a largc correlation correction (AE(SCF) - AhE(MP-2)). Reactions 1 and 3 show (34) Berkowitz, J.; Marquart, J . R. J . Chem. Phys. 1963, 39, 275. (35) Detry, D.; Drowart, J.; Goldfinger, P.; Keller, H.; Rickert, H. 2.Phys. Chem. (Munich) 1967, 55, 314. (36) Rau, H.; Kutty, T. R. N.; Guedes de Carvalho, J . R. F. J . Chem. Thermodynam. 1973, 5. 833. (37) Guthrie, G.E.,Jr.: Scott, D. W.; Waddington, G. J . Am. Chem. Soc. 1954, 76, 1488.
Homocyclic Sulfur Molecules, S, n = 4-8
The Journal of Physical Chemistry, Vol. 94, No. 15, 1990 5777
TABLE V: Calculated Thermodynamic Properties of S, as a Function of Temperature
T,K
W T ) - WO)'
CPb
SO;
Sd (Dw) 273.15 298. I5 300.00 400.00
500.00 600.00 615.00 700.00 800.00 900.00
1000.00 273.15 298.15 300.00 400.00 500.00 592.00 600.00 700.00 800.00 900.00
l000.00
3.20 3.61 3.64 5.38 7.20 9.08 9.37 10.99 12.92 14.85 16.80
16.22 16.68 16.71 17.92 18.57 18.94 18.98 19.18 19.34 19.45 19.52
69.88 7 1.33 71.43 76.42 80.50 83.92 84.35 86.86 89.43 91.71 93.77
21.13 21.74 21.78 23.34 24.17 24.62 24.65 24.95 25.15 25.29 25.39
82.03 83.91 84.04 90.55 95.85 99.98 100.31 104.13 107.48 110.45 113.12
25.83 26.60 26.65 28.64 29.69 29.78 30.30 30.68 30.93 31.11 31.24
82.12 84.42 84.58 92.55 99.07 99.78 104.54 109.24 113.36 117.01 120.30
S7 (CJ,uPR = 30 cm-Id 5.65 30.95 6.44 31.83 3 1.89 6.49 9.81 34.19 13.29 35.35 13.71 35.45 16.86 36.04 20.49 36.48 24.15 36.77 27.84 36.97 31.55 37.12
96.49 99.24 99.44 108.96 116.72 117.56 123.23 128.82 133.71 138.06 141.96
s5 (C')
4.03 4.57 4.61 6.87 9.25 1 1S O 1 1.70 14.18 16.68 19.2 I 21.74
s6
273.15 298.15 300.00 400.00 500.00 5 12.00 600.00 700.00 800.00 900.00 1000.00 273.15 298.15 300.00 400.00 500.00 512.00 600.00 700.00 800.00 900.00 1000.00
4.64 5.30 5.35 8.12 11.05 11.40 14.05 17.10 20.18 23.28 26.40
(D3d)
S, (CI), vPI = 20 273.15 298.15 300.00 400.00 500.00 5 12.00 600.00 700.00 800.00 900.00
I000.00
5.66 6.45 6.51 9.82 13.30 13.73 16.87 20.50 24.17 27.85 31.56
273.15 298.15 300.00 400.00 500.00 530.00 600.00 700.00 800.00 900.00 1000.00
6.46 7.37 7.43 11.27 15.30 16.54 19.44 23.64 27.88 32.15 36.44
30.95 31.83 31.89 34.15 35.35 35.45 36.04 36.48 36.77 36.97 37.12
97.30 100.05 100.24 109.76 117.53 118.37 124.04 129.63 134.52 138.86 142.77
35.83 36.84 36.91 39.53 40.92 41.21 41.73 42.23 42.57 42.80 42.97
98.72 101.91 102.13 113.15 122.14 124.53 129.68 136.15 141.81 146.84 151.36
Ss ( D d
aCorrections to enthalpy as function of temperature in kcal/mol. "eat capacity in cal/mol.K. Entropy in cal/mol-K. dupR = pseudorotation frequency.
correlation corrections of < I kcal/mol, while reaction 2 has a correlation correction of 2.5 kcal/mol. The agreement between theory and e ~ p e r i m e n t ~for ~ J reactions ~ 1 and 2 is within I kcal/mol. For reaction 3, the calculated value for AH is 1.4 kcal/mol outside the error limit of the experimental value and somewhat too large. For reaction 4 there is a clear discrepancy of -25 kcal/mol between theory and experiment, suggesting that S4 is not a ring. The agreement between the calculated and experimental AHq's for S7and s6 is within 1 kcal/mol. The AH? calculated for S5 is 4-7 kcal/mol larger than the experimental values. The discrepancy is sufficiently small given the experimental uncertainties that we can conclude that S5 may be a ring structure. The -23 kcal/mol difference between calculated and experimental AHyO's for cyclic S4 indicates that the ground state for S4 is not a ring and may be a linear or branched open chain. This energy difference is consistent with the strain energy found for other four-membered rings.38 The equilibration between s6, S7,and S8has been studied in methanol cyclohexane and acetonitrile solutions.3g To compare with the observed K = 2 X at 298 K for (5), we have calculated the various terms needed to obtain AG for reaction 5. The
-
(5) electronic component, AEelcc,is 5.9 kcal/mol. The zero-point correction of -0.3 kcal/mol is almost cancelled by the temperature correction to AH of 0.2 kcal/mol, giving AH (298 K) = 5.8 kcal/mol. The value of AS is 12.15 cal/mol.K at 298 K, giving an overall value AG (298 K) = 2.2 kcal/mol. Then K = 2.4 X IO-* at 298 K. This agreement with the experimental value provides further evidence of the ability to treat these S,, systems theoretically; also, there is a large entropy component in AG for reaction 5 at ambient temperatures. Comparison to Other Theoretical Work. There have been a variety of theoretical studies on the polysulfur systems discussed here. The structure of S4has been examined at a number of levels. Kao6boptimized several structures for S4with the STO-3G and 4-31G basis sets. The geometries of the S4ring determined with the STO-3G basis set are in reasonable agreement with our calculated results, whereas the bond distances are too long with 4-31G. Kao finds the global minimum at the STO-3G level to be an open-chain triplet, but the UHF procedure used to calculate the energy is biased in favor of the triplet. The 4-31G calculations predict the ring structure to be >20 kcal/mol less stable than the open-chain triplet. Although the UHF calculation favors triplets, this result is consistent with our analysis of the strain energy in S,. Feng and Novaro,6d on the other hand, find the open chain for S4to be less stable than ring S, at the SCF level with a 4-31G basis set, apparently calculated with an R H F wave function for the singlet. Semiempirical (MNDO) calculations4*yield a bond distance for D2d S4 that is 0.15 8, shorter then the -2.11 8, calculated at the ab initio level. The SINDOl semiempirical method, which includes d orbitals on S, leads to an improved bond distancegdthat is still 0.06 8, short. MM2 calculations3 on cyclic S, predict S4 to be planar, in contrast to our butterfly with a bond distance 0.03 8, shorter. The MM2 method does predict a significant amount of strain energy in cyclic S,. For cyclic SS, Kao6c found the two structures studied here to be of comparable energy at the STO-3G level. Our calculated geometries are in good agreement with his. Kao's results show that the triplet (UHF) chain and singlet (RHF) cyclic structures should be of comparable energy. The MM2 results3 for cyclic S, also predict the ready pseudorotation but not our calculated alternation in bond lengths. studied the geometries of S6,S7,and S8 rings Laitinen et with pseudopotentials and a minimum basis set for the valence space. Their bond distances are -0.20 A longer than experiment. (38) Greenberg, A.; Liebman, A. Strained Organic Molecules; Academic Press: New York 1978. (39) Tebbe, F. N.; Wasserman, E.; Peet, W. G.; Vatvars, A,; Hayman, A. C. J . Am. Chem. SOC.1982, 104, 4971.
5778 The Journal of Physical Chemistry, Vol. 94, No. 15, 1990
Dixon and Wasserman
TABLE VI: Energy Components for Calculating AH (298 K) for Reactions 1-4" reactn
AE(SCF)'
AE(MP-2)C
1
7.9 5.0 18.1 37.8
6.7 1.5 18.7 45.6
2 3 4
4AH( r ) 0.0 -0.2 0.0 -0. I
APEd -0.2 -0. I -0.5 -0.6
AH(SCF)
AH(MP-2)
AH(expt)c
AH(expt)'
7.7 4.7 17.6 37.1
6.5 7.2 18.2 44.9
5.77 f 0.31 6.26 f 0.33 12.8 f 3.4 20.2 f 3.1
5.7 6.2 14.3 20.5
Energies in kcal/mol. 'Reaction energy, S C F level. Reaction energy, MP-2 level. dZero-point energy correction. 'Reference 35. /Reference 34
TABLE VII: Calculated and Experimental AHf' (298 K) for S, (kcal/mol) n
AHfo(calc)
AHfo(expt)"
8 7 6 5 4
24.3 27.8 25.4 33.4 57.1
24.3 27.2 24.4 26.1 34.8
Reference 36.
AHlo(expt)b 26.0 f 28.1 f 24.8 f 29.1 f 33.5 f
2.7 2.7 2.7 2.5 2.2
Reference 35.
They also do not find as large an alternation in bond lengths for S7as calculated here or determined experimentally. Laitinen and Pakkanen*' also calculated the structure of H2S4with the MINI-I basis set. They examined the S-S and S-SH bond distances at three torsion angles. Their earlier conclusions are similar to ours on the effect of variation of the torsion angle on the bond lengths and on H2S4as a model for the geometry of c clic S,. However, their bond lengths are longer by 0.06-0.08 . It is somewhat surprising considering the good agreement between the STO-3G results for S4 and S5 and our results that there is such a large difference in the bond lengths in Laitinen and Pakkanen's work. MNDO calculations" on S6 and Ss yield bond distances that are short by more than 0.1 A. Use of the SINDOl methoda leads to improved bond distances, in good agreement with experiment. The MM2 results3 for S6 and Ss are also in agreement with the experimental results. However, for S, MM2 again does not reproduce the bond alternation. After this work was completed, Hohl et aL40 published their work on the structures of S, systems using a density functional method combined with molecular dynamics and simulated annealing techniques in order to calculate the molecular geometries. The calculated structures for Ss-Ss are in qualitative agreement with our values. For S5,the density functional results show that
K
(40) Hohl, D.; Jones, R. 0.; Car, R.; Parrinello, M. J . Chem. Phys. 1988, 89, 6823.
both the C, and C, structures are minima, in contrast to our result that the C, structure is a transition state. The density functional bond distances are longer than our values, by 0.09 A for the 1-1' bond in the C2 structure. For S6, the density functional bond distance is also longer than the experimental value by 0.04 A, whereas for SBthe difference is 0.06 A. As expected from the longer bond distance, the density functional frequencies for Ss are below the experimental values. For S7,the density functional method gives a highly fluxional structure with approximate C, symmetry. Most of the bond distances are too long when compared with experiment, although the shortest bonds are in good agreement with our results. However, the long 1-1' bond is 0.07 A larger than the experimental value. For S4 the density functional method predicts that the structure is two loosely coupled S2 molecules in a D, geometry. This is in contrast to the work cited above, which suggested that some chain structure is likely. Conclusions
We have shown that the structure, spectra, and energies of S, rings can be calculated with a polarized, double-t basis set. The bond lengths are correlated with the torsion angle about that bond as originally suggested by SteudelzJ9 from experimental studies. The rings S5 and S7 are expected to be quite fluxional. The calculations give scaled values for the vibrational spectra that are in good agreement with the experimental values except for the lowest energy stretches, where larger scale factors may be necessary. This suggests that these molecules will provide a useful testing ground for studying correlation, anharmonic, and dynamic effects in the ab initio calculation of vibrational spectra. An internally consistent set of thermodynamic properties was calculated and agrees well with experiment and properties primarily calculated from experimental structures and force fields. The calculated enthalpies confirm the experimental values for S5-S7 and indicate that S4 is not a ring. Registry No. S4, 19269-85-3; Ss, 12597-10-3; S,, 13798-23-7; S7r 2 1459-04- I ; Sa, 10544-50-0.
