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incremental stability is found for linear HCN chains at about the sixth monomer. The capability is not limited to clusters of identical molecules. it is just as easy to examine “soups” of any combination of the species for which electrical properties and the MMC c and d parameters have been obtained. Right now, this is helpful for anticipating spectroscopic features of mixed trimers and tetramers, but eventually this may provide potentials for the solvation of a complex in a droplet or for the effects of impurities on crystal structures. IV. Summary The wealth of information that has become available from spectroscopic study of small molecules, combined with a good number of high level theoretical studies, is making the prospect of general construction of realistic intermolecular potentials ever more promising. The current state of this art shows good agreement between values obtained with the most complete models and experimental values. Of course, the comparisons themselves remain limited by dynamical issues that are a special consequence of the flatness of these surfaces. In other words, we are not precisely sure how accurate the binary cluster surfaces are, as yet. Even so, we see certain essential predictions (e&, stabilities, qualitative forms of structures, existence of isomers, interconversion barriers) holding true for a healthy variety of small clusters. The basis for this capability is a rather strict use of old notions which all come down to intermolecular electrical interaction as a first ingredient. While the MMC scheme and other procedures in use are an alternative to ab initio calculation, it must be said that the detail now possible in electrical analysis (Le., extensive treatment of multipole polarization) has resulted from the tremendous advances
in ab initio electronic structure methods. They are the best source for electrical properties at this stage, and via the electrical properties, ab initio techniques serve as the basis for schemes such as MMC. An idea that we have advanced, that polarization is the primary electronic structure change upon weak interaction, is a significant point for generalizing any electrical interaction model from binary to large clusters. The preliminary studies with the MMC scheme of clusters with 3-10 submolecules look encouraging, but there is still a good deal of testing before we can be entirely sure that the main source of nonpairwise interactions is mutual polarization. The polarization model of (PM) of Stillinger and David9’ seems to have anticipated this view of the role of polarization. PM is an empirical approach that represents molecules by point charges and dipole polarizabilities, with modification of the electrostatics to account for charge reorganization. It is not reestricted to rigid species and has been developed for molecules92other than water, the first target. The latest step, then, is not a new thing to do but a surer, most extensive treatment of polarization. So, if it holds that polarization is the connecting element between binary and large cluster potentials, then the technology already at hand, with a few more refinements and embellishments, should be a powerful development for simulations and dynamics studies of a wide assortment of problems. Acknowledgment. This work was supported, in part, by a grant from the Physical and Theoretical Chemistry Program of the National Science Foundation (CHE-8721467). (91) Stillinger, F. H.; David, C. W. J . Chem. Phys. 1978, 69, 1473. (92) Turner, P. J.; David, C W. J . Chem. Phys. 1981, 74, 512.
ARTICLES Origin of the “Gauche Effect” in Substituted Ethanes and Ethenes Kenneth B. Wiberg,* Mark A. Murcko, Keith E. Laidig, Department of Chemistry, Yale Uniuersity, New Haven, Connecticut 0651 I
and Preston J. MacDougallt Department of Chemistry. McMaster University, Hamilton, Ontario, Canada L8S 4MI (Receiued: May 15, 1989; I n Final Form: May I . 1990)
I ,2-Disubstituted ethanes having strongly electronegative substituents prefer the gauche conformation over the usual trans form. This is not the result of an attractive interaction in the gauche form, but rather a destabilizing interaction in the trans rotamer. The interaction appears to be the result of the formation of bent bonds, and the lower stability of a twist bent bond over that of syn bent bond. The lower energy of cis-l,2-difluoroethene as compared to its trans form may be explained in the same way.
It is well established that, whereas butane prefers the anti conformation over gauche by 0.9 kcal/mol,’ 1,2-difluoroethane prefers the gauche over anti by about 0.6 kcal/moL2 This “gauche effect” has been of considerable interest,’ and a general trend toward increasing proportion of the gauche rotamer as the substituent electronegativity is increased has been noted.4 We have ‘Present address: Department of Chemistry, Texas A&M University, College Station, TX 77843.
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examined the origin of the gauche effect. We have shown that the correct rotamer preference for the above compounds may be obtained via ab initio molecular orbital calculations provided a sufficiently large basis set is u ~ e d . ~The .~ Wiberg, K. B.;Murcko, M . A. J . Am. Chem. SOC.1988, 110, 8029. (2) Huber-Walchli, P.; Giinthard, Hs.H. Spectrochim. Acta, Parr A 1981, 37A, 285. ( 3 ) Wolfe, S . Acc. Chem. Res. 1972, 5 , 102. (4) Philips, L.; Wray. V. J . Chem. Soc., Chem. Commun. 1973, 90. (1)
0 1990 American Chemical Society
Gauche Effect in Substituted Ethanes and Ethenes TABLE I: Calculated Energies of Difluoroethane" basis set optimizn 6-3 lG(d) 6-31G(d) 6-3 1G(d,p) 6-31G(d,p) 6-31 IG(d,p) 6-3 1G(d) 6-3 1+G(d) 6-3 lG(d) 6-31++G(d,p) 6-3 1G(d,p) 6-31 lG(d) 6-31 lG(d) 6-3 1 1++G(d,p)* 6-31 lG(d) MP2/6-3 I 1++G(d,p) 6-31 lG(d) MP3/6-3I l++G(d,p) 6-3 1 1G(d) MP4/6-3 I 1++G(d,p) 6-31 lG(d)
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trans -216.922 12 -216.928 89 -217.003 61 -216.938 34 -276.944 90 -216.996 59 -211.01229 -217.131 39 -211.144 71 -277.180 33
gauche -276.921 40 -216.928 24 -211.003 32 -216.938 10 -216.944 12 -276.996 28 -277.01 2 61 -211.132 64 -277.145 13 -271.181 45
AE 0.45 0.41 0.18 0.15 0.1 1 0.19 -0.20 -0.78 -0.64 -0.lOC
dnauche)
3.03 3.18 3.18 3.18
#Total energies are given in hartrees ( 1 hartree = 627.5 kcal/mol), and energy differences are given in kcal/mol. *The 6d option was used with this basis set. The frozen core approximation was used in the post-Hartree-Fock calculations. When corrected for differences in zero-point energies AH becomes -0.76 k a l / m ~ l . ~ and ( H Z g 8- Ho),
TABLE Ik Bond Propertiesa** compound gauche-difluoroet hane anfi-difluoroethane cis-difluoroethene trans-difluoroethene
bond
rAB
rbp
Pc
XI
X2
A3
c
C-F c-c C-F
1.5028 1.3661 1.5112 1.3684 1.3093 1.3245 1.308 1 1.3309
1.5033 1.3661 1.5115 1.3684 1.3117 1.3246 1.3090 1.3310
0.2821 0.2329 0.2182 0.2301 0.3739 0.2584 0.3753 0.2542
-0.5888 -0.3929 -0.5995 -0.3802 -0.9166 -0.5530 -0.9206 -0.5283
-0.5640 -0.3838 -0.5355 -0.31 15 -0.5335 -0.5089 -0.5325 -0.4900
0.2915 1.3318 0.2991 1.3303 1.7014 1.1212 1.6833 1.6604
0.0440 0.0231 0.1192 0.0234 0.1182 0.0865 0.1286 0.0181
c-c C-F c-c C-F c-c
O r A B is the conventional bond length in A, rbpis the length along the bond path, p c is the charge density (e/au3) at the bond critical point, the A'S are the second derivatives of p with respect to the coordinate directions (e/au5), and t is the ellipticity (Al/X2-l). *The data are based on 63 1 1++G(d,p)//6-3 1 G(d,p) wave functions for the difluoroethanes and 6-31 1++G(d,p)//6-31+G(d) wave functions for the difluoroethenes.
energies previously obtained for 1,2-difluoroethane along with additional calculations to examine the effect of basis set size are shown in Table I. With a large basis set, the gauche form was calculated to have the lower energy even at the R H F level. Since the single-determinant S C F calculations neglect electron correlation, the gauche preference cannot be assumed to result only from attractive nonbonded interactions' because the latter are a result of electron correlation. Indeed, a comparison of the C-C-F bond angles in the anti (lO8.l') and gauche ( 1 10.3') rotamers indicated that there are repulsive interactions in the latter.' This conclusion was reinforced by the observation that the torsional angle in the gauche form is 74',* considerably larger than the normal 60'. The only conclusion that is in accord with these observations is that the anti rotamer is destabilized by electronegative substi tuen ts.' A possible origin of anti destabilization may be found in an examination of the bond angles in methyl fluoride. Here, the conventional H-C-F bond angle is 108.9',9 only slightly smaller than tetrahedral. One might, however, have expected the angle to be considerably smaller.I0 An electronegative atom such as fluorine would be expected to prefer a carbon orbital with high p character, thus avoiding the much more tightly bound s electrons. If this were the case, the C-H bonds would have increased s character, leading to larger H-C-H angles, and a smaller H-C-F angle. The conventional bond angle does not necessarily give a proper description of a bond because many bonds (if not most) are bent. The most commonly recognized bent bonds are found in small ring compounds such as cyclopropane and cyclobutane derivatives, where bent bonds can lead to improved bonding. They also will be found in cases where steric interactions are present because orbital following is relatively small in bending deformations."
The bond path as described by Bader12 is a much more useful description of a bond. It is the path of maximum electron density connecting a pair of bonded atoms. With symmetrically substituted atoms, such as the carbon of ethane, the bond path of necessity coincides with the conventional bond. But, with methyl fluoride, this is not the case, and bent C-H bonds are found. The angle between the C-H and C-F bond paths at the carbon is only 106.7', in good accord with the expectation based on the simple hybridization argument.I3 If we extend this to 1,2-difluoroethane, we would expect bent bonds for both the anti and gauche rotamers. However, bond bending in the anti rotamer would lead to the C-C bond orbitals being bent in opposite directions, whereas with the gauche rotamer, they would bend roughly in the same direction. The former should lead to decreased overlap and a poorer bond as has been discussed by Dixon and Gassman for trans-bicyclo[4.1 .O]hept-3-ene based on a valence bond approach.14
Top. Curr. Chem. 1977, 70, 1.
Calculations of the bond paths using the 6-31++G(d,p) wave functions showed that this expectation was realized. The dashed lines indicate the directions for the bond paths at the carbon nuclei. The angular deviations were as follows: anti, al = 0.66', a2 = 2.15', bond path angle = 105.12', conventional angle = 107.93'; gauche, a I = 0.4S0, a2 = 2.79', bond path angle = 106.80°, conventional angle = 109.96'. It is not possible a t this time to estimate the energetic consequences of these bent bonds, but it clearly provides a simple and attractive explanation for the increased preference for gauche rotamers as the electronegativity of the substituent is increased. Some additional information derived from the above analysis is given in Table I1 which lists the charge densities at the bond
(8) Collomon, J. H.; Hirota, E.; Kuchitsu, K.; Lafferty, W. J.; Maki, A. G.; Pote, C. S.Landolr-Bornstein;Springer . - Verlaa: - Berlin, 1916; New Series, Group 11, Vol. 7. (9) Clark, W. W.: DeLucia. F. C. J. Mol. Strucf. 1976. 32. 29. (IO) Bent, H. A. Chem. Rev. 1961, 61, 275. ( I I ) Nakatsuji, H. J. Am. Chem. Soc. 1974, 96, 24.
(12) Runtz, G.; Bader, R. F. W.; Messer, R. R. Can. J . Chem. 1977,55, 3040. (13) Wiberg, K. B.; Murcko, M. A. J . Mol. Struct. 1988, 169, 355. (14) Dixon, D. A.; Gassman, P. G. J. Am. Chem. Sot. 1988, 110, 2309.
(5) Wiberg, K. B.; Murcko, M. A. J. Phys. Chem. 1987, 91, 3616. (6) For related calculations, see: Dixon, D. A.; Smart, B. E. J. Phys. Chem. 1988. 2729. - . - ., 92. -, (7) Epiotis, N. D.; Cherry, W. R.; Shaik, S.; Yates, R. L.; Bernardi, F.
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The Journal of Physical Chemistry, Vol. 94, No. 18, 1990
TABLE 111: Calculated Energies of 1,2-Difluoroetbenes" basis set optimizn 6-31G(d) 6-3 1G(d) 6-3 1G(d) 6-3 IG(d, ) 6-31G(d) 6-3 I I G((p) 6-3 1+G(d) 6-31 +G(d) 6-31+G(d) 6-31 ++G(d,p) 6-31+G(d) 6-31 l++G(d, ) 6-31+G(d) 6-3 1 1 ++G(d(p) MP2/6-31 I++G(d,p) 6-31+G(d) MP3/6-31 I++G(d,p) 6-31 +G(d) MP4/6-31 I++G(d,p) 6-31 +G(d)
Wiberg et al.
trans -275.721 74 -275.725 76 -275.79960 -275.735 76 -275.739 83 -275.807 56 -275.820 15 -276.504 22 -276.50997 -276.547 50
cis -275.721 31 -275.725 35 -275.799 59 -275.736 67 -275.740 73 -275.808 90 -275.821 53 -276.505 66 -276.5 1 1 02 -276.548 76
AE 0.27 0.26 0.01 -0.57 -0.56 -0.84 -0.87b -0.90 -0.66 -0.79C.d
dcis) 2.68 2.80 2.85 2.85 2.85 2.8 1
'Total ener ies are given in hartrees and energr differences are in kcal/mol. *The use of a still larger basis set has been reported to increase the R H F energ {ifferences b about 0.1 kcal/mol. CWhen corrected for differences in zero-point energies and (H298- H,,), AH becomes -0.66 kcal mol. JThe calculatec! vibrational frequencies (6-31G(d), unscaled) were as follows: trans, 3456.8, 3446.8, 1954.7, 1428.5, 1427.9, 1280.0, 1264.3, 1044.7, 947.7, 603.1, 378.1, 346.3 cm-';cis, 3465.1, 3438.5, 1971.3, 1546.1, 1405.5, 1244.2. 1125.0, 1014.7, 900.6, 837.7, 550.7, 252.8 cm-'.
critical points (pc, the points having a minimum value of p along the bond paths) and the lengths along the bond paths. The bond path lengths for the C-C bonds were significantly longer than the conventional lengths as expected for bent bonds. As is often found, pc for a given type of bond was related to the bond length. The table also includes the second derivatives of p at the critical point (X,,X2, A,) and the ellipticity (e = Xl/hz - I ) . It is interesting to note that the ellipticity of the C-C bond in anti-difluoroethane is significantly larger than that for gauche. It should be noted that the importance of bond bending in determining properties was probably first discussed by Bartell in considering the structure of c y c l o b ~ t a n e . ~It~is known that the methylene groups of cyclobutane tilt inwards as the ring is puckered, although simple steric arguments would have suggested the opposite motion in order to minimize cross-ring nonbonded repulsion. However, if one invokes a bent bond description and assumes that the methylene group will maintain local C,, symmetry with respect to the bond path vectors at the carbon, it is easily seen that the methylene groups will prefer to be tilted inwards.
Another case in which these considerations may apply is 1,2difluoroethene where the cis isomer is more stable than trans by 1.0 f 0.1 kcal/mol.I6 Dixon et a1.I' have shown that this energy difference may be reproduced via a b initio calculations using large basis sets. The results of some additional calculations using the standard Pople basis setsI8 are shown in Table 111. Again, diffuse functions were particularly effective in reproducing the correct energy difference by using these basis sets. The bond path directions were calculated for both rotamers giving the result: trans, al = 1.12O. a2= 3.27O, bond path angle = 117.94O,conventional angle = 120.10°; cis, a , = 1.04', a2 = 5.85", bond path angle = 117.81 O, conventional angle = 122.62'. The twist-bent C=C bond in the trans isomer would be expected to be destabilized more than the syn-bent bond in the cis isomer.
-__
._ -
.
,
Figure 1. Difference between the charge density maps for rruns-1,2difluoroethene (upper) using the 6-3 l+G(d) and 6-3 1G(d) basis sets and for ns-I,2-difluoroethene (lower) using the same basis sets. The solid contours indicate regions in which charge density has been moved via the addition of diffuse functions to the basis set
Is it possible to obtain information on the nature of the difference between conformers in another way? We have explored (15) Bartell, L. S.; Andersen, B. J. Chem. Soc., Chem. Commun. 1973, 786. (16) Craig, N. C.; Piper, L. G.;Wheeler, V. L. J . Phys. Chem. 1971, 75, 1453. (17) Dixon, D. A.; Smart, B. E.;Fukunaga, T. Chem. Phys. L e r r . 1986, 125, 447. (18) Hchre, W. J.; Radom, L.;Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986.
the use of charge density difference maps. It is seen from Table I1 that, whereas the 6-31G(d) basis set incorrectly gives the trans form the lower energy, the 6-3 1+G(d) basis set gave a satisfactory energy difference. In order to determine the role of the diffuse functions in reproducing the greater stability of the cis form, the mo wave functions were obtained by using both the 6-31G(d) and 6-31+G(d) basis sets at the 6-31+G(d) geometry. The charge density distributions were calculated for both sets by using a 101 X 101 grid, and the difference was obtained. The difference maps for the two isomers are shown in Figure 1. It is readily apparent
J . Phys. Chem. 1990, 94, 6959-6962
that the principal role of the diffuse functions was to move charge into the bonding regions between atoms, and for the C-F and C-H bonds, the added charge density lies along the bond axes. However, with the C-C bonds, the added charge density was displaced from the line of centers between the carbons. In the cis case, the density was pushed upwards toward the region between the fluorines, and in the trans case, it was moved in a sigmoid fashion. This corresponds exactly to the directions for the bond paths. The hypothesis that bond bending is responsible for the destabilization of the trans conformers is in accord with the observation that the preference for gauche or cis isomers decreases as the electronegativity of the substituent decreases. The degree of bond bending is directly related to the electronegativity of the substituent.'* As the electronegativity is decreased, bond bending decreases, and the normal steric interactions in the cis or gauche
6959
rotamers become predominant. Thus, with n-butane or 2-butene, the trans conformers are preferred. Calculations. The ab initio calculations were carried out using GAUSSIAN 8619 and standard basis sets.'* The bond paths were calculated by using the PROAIMS package.20 Acknowledgment. This investigation was supported by a grant from the Office of Basic Energy Sciences, Department of Energy. (19) GAUSSIAN86; Frisch, M. J.; Binkley, J. S.;Schlegel, H. B.; Raghavachari, K.; Melius, C. F.; Martin, R. L.; Stewart, J. J. P.; Bobrowicz, F. W.; Rohlfing, C. M.; Kahn, L. R.; DeFrees, D. J.; Seeger, R.; Whiteside, R. A,; Fox, D. J.; Fleuder, E.M.; Pople, J. A. Carnegie-Mellon Quantum Chemistry Publishing Unit: Pittsburgh, PA, 1984. (20) Biegler-Konig, F. W.; Bader, R. F. W.; Tang, T. H. J. Compur.
Chem. 1982, 3, 317.
Vibrational Energy Migration along Hydrocarbon Chains: A Model Study Alessandro Lami* and Giovanni Villani* Istituto di Chimica Quantistica ed Energetica Molecolare del CNR. Via Risorgimento 35, 56100 Pisa, Italy (Received: June 20, 1989; In Final Form: December 7, 1989)
The decay of an anharmonic bond coupled to a chain of eight harmonic bonds is studied numerically by use of a Lanczos procedure. The calculations show that the excitation moves coherently along the chain for at least 500 fs. The above results are used to comment on the interpretation of the chemical activations experiments on 1-decene by Rabinovitch and co-workers and to confirm (qualitatively) their hypothesis of sequential motion of the excitation along the chain.
Introduction The study of the way by which a vibrationally hot molecule redistributes its energy among all the modes is a main step toward a full comprehension of the factors that govern selectivity in chemical reactions. What one needs for investigating intramolecular vibrational energy transfer and randomization is a way for depositing energy in a certain part of the molecule and for monitoring the energy spreading into the molecule as a whole. This has been attempted by several researchers, employing a variety of different experimental techniques.' A very promising method consists in using a chemical activation mechanism for the energy deposition and in looking at the reaction products to verify if randomization has taken place before the reaction.24 The present paper is a theoretical-numerical investigation of the vibrational relaxation problem, stimulated by a very funda- . mental question posed by the experimental findings by Rabinovitch and c+workers,4 In their study of the decay of the radical obtained by addition of H to the 1-olefins, they discussed two alternative hypotheses for the spreading of the vibrational excitation (about 45 kcal/mol), which is initially deposited in the terminal part of the chain C,-C-C--C=C
+ H'
-
C,-C-C-C-C-H I
(the bar indicates vibrational excitation), The first hypothesis involves sequential motion of the excitation along the bonds in the chain, leading to the rupture between C3 ( I ) See, e.&; Gordon, R. J. Comments At. Mol. Phys. 1988, 31, 123. Bondybey, V. E. Annu. Rev. Phys. Chem. 1984, 35, 591. (2) Rymbrand, J. C.; Rabinovitch, B. S. J . Chem. Phys. 1971,54, 2275. (3) Trenwith, A. 8.; Rabinovitch, B. S.J . Phys. Chem. 1982, 86, 3447. (4) Trenwith, A. B.; Oswald, D. A.; Rabinovitch. B. S.; Flowers, M. C. J .
Phys. Chem. 1987, 91, 4398.
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and C4 and to the formation of propene. Since the bond rupture occurs before energy randomization, the reaction rate should be insensitive to pressure. Alternatively one may suppose that there is a rapid, nonsequential energy spreading, before the propene formation. The rate constant is expected, in this case, to be quite sensitive to pressure variations. The experimental results4 support the sequential decay hypothesis. As pointed out by Reinhardt and DuneczkyS the previous studies on model chains6*'cannot be used to discuss appropriately the problem of sequentiality in the bond excitation since they are based on the coupling of a local mode (simulating the initially excited part) to a set of delocalized normal modes for the rest of the chain. What one finds from such calculations is simply that the normal modes receive first a vibrational quantum, then a second one, etc., i.e., that there is sequentiality in the sense of the flow of quanta from the local mode to the bath of normal modes. The present paper sheds some light on the problem of energy migration. We focus on the I-decene and simulate the radical I by an excited anharmoniG CH3-CH2quantum oscillator coupled to eight CH2-CH2 harmonic bonds. The computational scheme is still based on a transformation to the normal modes for the chain, and so we are able to confirm sequentiality from the point of view of the flow of quanta into the normal modes. However, our analysis proceeds further by projecting the results on the basis of localized bond oscillators, and the numerical results clearly show that, on a subpicosecond time scale, the excitation moves sequentially along the chain, in (5) Reinhardt, W. P.; Duneczky, C. J. Chem. Soc., Faraday Tram. 2 1988, 84, 1511. ( 6 ) Hutchinson, J. S.; Reinhardt, W. P.; Hynes, J. T. J. Chem. Phys. 1983, 79. 4247. '(7) Hutchinson, J. S.; Marshall, K. T. J . Chem. Soc., Faraday Tram. 2 1988, 84, 1535.
0 1990 American Chemical Society
