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of C. B. Bryden are greatly appreciated.
References and Notes (1) Alley, E. 0.; Layton, B. R.; Minyard, J. P., Jr. J . Agric. Food Chem. 1974, 22, 729. (2) Davls, K. M. C. In (3)
A. R. Gregory and D. F. Williams
The Journal of Physical Chemjstty, Vol. 83, No. 20, 1979
“MolecularAssociation”;Foster, R., Ed.; Academic Press: New York, 1975; pp 151-213. Stothers, J. B. “Carbon-13 NMR Spectroscopy”;Academic Press:
New York, 1972. (4) Griffith, R. C.; Grant, D. M.; Roberts, J. D. J. Org. Chem. 1975, 40, 3726. (5) (a)Kosower, E. J. “An Introduction to Physical Organic Chemistry”; Wiiey: New York, 1968; Part 2. (b) Gordon, A. J.; Ford, R. A. “The Chemist’sCompanion”;Wiley-Interscience: New York, 1972. (6) Gutmann, V.; Wychera, E. Inorg. Nucl. Chem. Left. 1966, 2 , 257.
“CoordinationChemistry in Non-Aqueous Solutions”;Springer-Verlag: New York, 1988. (7) Wilson, N. K.; Zehr, R. D. J . Org. Chem., 1979, 44, 1278. (8) Alley, E. G.; Layton, B. R.; Minyard, J. P.,Jr. J . Org. Chem. 1976, 41, 462.
(9)
Wilson, N. K. J . Am. Chem. SOC. 1972, 9 4 , 2431. Scatchard, 0.
Ann. N . Y . Acad. Sci. 1949, 51, 660. (10) Benesi, H. A.; Hildebrand, J. H. J. Am. Chem. Soc. 1949, 71, 2703. (1 1) The Zvalue is defined as the transition energy in kcailmoi for the
(12)
longest wavelength absorption band (a charge-transfer band) for l-ethyl-4-(carbonyimethoxy)pyridinium iodide in the ~olvent.~ The donor number of the base B is defined as the negative enthalpy in kcal/moi of the reaction of B with the Lewis acid antimony pentachioride, B + SbCi, B-+SbC15.e Alley, E. G.; Layton, B. R.; Minyard, J. P., Jr. J. Agric. Food Chem., +
(13)
1974, 22, 442.
Stilbene. A Critical Commentary and Some New Calculations on Its Structure and Isomerization’ Allan R. Gregory” and Dlgby F. Williams Division of Chemistry, Natlonal Research Council of Canada, Ottawa, Canada K1A OR6 (Recelved January 15, 1979; Revised Manuscrlpt Received April 23, 1979) Publicatlon costs assisted by the Natlonal Research Council of Canada
The thermal and photochemical cis-trans isomerizations of both ethylene and stilbene are symmetry-forbidden and symmetry-allowedreactions, respectively. However, the nonequivalence of cis- and trans-stilbene,interactions between the ethylene and phenyl chromophores, and the availability of additional degrees of freedom to the stilbene molecule guarantee important differences between the ethylene and stilbene reactions. The relative strengths and weaknesses of previous theoretical studies of the properties and cis-trans isomerism of stilbene are discussed as a prelude to the presentation of the results of a new molecular orbital investigation. These results are in good agreement with experiment wherever contact is possible. Thus, the minimum energy geometries calculated for both isomers possess a C2 axis of symmetry, the phenyl groups of cis-stilbene are predicted to be rotated by -50’ about the C-Ph bonds, the trans isomer is calculated to be -5 kcal/mol more stable than the cis isomer, and the first absorption band is predicted to shift to the blue on passing from trans- to cis-stilbene. The trans triplet is predicted to correspond to an absolute energy minimum, the perpendicular triplet is predicted to be slightly less stable than the trans triplet, and the cis triplet is predicted to correspond to a saddle point. The section through the potential surface for rotation about the central bond in the triplet state that contains the planar reactant and product exhibits no other stationary points. To this extent, the present calculations support a recent suggestion that the observed barriers to isomerization in the triplet state are due to intersections between the ground and triplet state surfaces near the perpendicular configuration. However, they also hint at the possible existence of other barriers to isomerization that both near the perpendicular configurationand intrinsic to the triplet surface. These barriers lie along the coordinate for symmetrical rotation about the C-Ph bonds. In general, the C=C-C bond angle plays a crucial role in determining the shapes of the potential surfaces on the trans side of the perpendicular configuration.
I. Introduction The structure and cis-trans isomerization of stilbene, 1,2-diphenylethylene, have been investigated experimentally2-23 and theoretically21-44pB many times. Two photoisomerization pathways can be distinguished according to the multiplicity of the initially prepared state. The singlet pathway is relatively easy to study experimentally by direct optical excitation since several transitions from the ground state to excited singlet states have high oscillator strengths. Direct optical excitation of the triplet manifold is a spin-forbidden process and therefore more difficult. It was first achieved in the presence of oxygen a t high pressures3 and later in the presence of a heavy atom perturberas The initial attempts to detect the lowest triplet state after flash excitation were frustrated by its anomalously short lifetime a t room temperature.8 It was finally observed by triplet-triplet absorption following flash excitation in glassy solvents at liquid nitrogen
and argon temperature^.^ Photochemical studies with triplet sensitizers and quenchers have also provided information about the shape of the potential energy surface for rotation about the double bond in the lowest triplet state.7v8J0J1J4A detailed knowledge of the shapes of the potential energy surfaces for cis-trans isomerization in the lower-lying singlet and triplet states is of course essential for a full understanding of each reaction mechanism. The flash excitation and photochemical experiments indicate that (1) the trans triplet (3t)correponds to a minimum on the potential energy surface, (2) the cis triplet (3c)probably corresponds to a “maximum”, and (3) 3c can rotate to 3t. Lifetime studies suggest that the 3tspecies can rotate a t room temperature toward a twisted configuration (3p) with a much shorter lifetime due to the smaller energy gap between the triplet state and the ground state.1°J2J3 Measurements of the temperature dependence of photostationary cis-trans ratios lead to the
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conclusion that the energy of the 3p state is higher than More recently, that of the 3t state by -800 cal/m01.~J~ the temperature dependence of the initial rates of isomerization in the lowest triplet state has been studied in this laboratory by directly populating it optically with an argon laserF1 These experiments led to the conclusion that there are barriers of 2040 and 1310 cal/mol to 3t 3c and 3c 3t isomerization, respectively, at or near the 3p configuration. At almost the same time, Inoue et al. reported an experimental study of the sensitized cis-trans photoisomerization of c y c l ~ o c t e n e .Both ~ ~ groups rationalized their results in terms of a model in which the potential surfaces for isomerization of the ground and lowest triplet states intersect near the perpendicular configuration. However, the results for both molecules are also consistent with a triplet potential surface for rotation about the double bond in which two maxima intrinsic to the surface are separated by a minimum near the perpendicular configuration. For stilbene, this situation could conceivably arise from the mixing of a triplet ~ t a t e , , ~ p h , in which the excitation is localized in the benzene rings, with another of the same symmetry, 3e, localized in the ethylene fragment. The minimum near the perpendicular configuration could then be attributed to 3e and the maxima to 3ph-3e mixing, provided 3ph lies at lower energies in the cis and trans configurations and destabilizes as the molecule rotates about the double bond. This does not seem likely, but a triplet potential surface with the geometrical features in question has in fact been calculated for stilbene by Ting and McClure (TML31 The cis-trans isomerism of stilbene is a “thermally forbidden, photochemically allowed” reaction in the Woodward-Hoffmann sense.46Accordingly, the adequacy of different molecular orbital (MO) models for constructing potential energy surfaces for such reactions and for constructing wave functions for diradicals is reviewed briefly in section 11. The results of previous MO studies of the equilibrium conformations and cis-trans isomerism of stilbene and related molecules are evaluated in section I11 by comparing them with known experimental results. In section IV, we present the results of new calculations for stilbene. They raise the possibility that the observed barriers to isomerization in the triplet state are to be found along a hitherto unsuspected internal coordinate. Our conclusions are summarized in the final section.
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11. Molecular Orbital Models for Cis-Trans Isomerism The potential surfaces for cis-trans isomerization of stilbene might be expected to resemble either the ground state or an excited state potential surface for the isomerization of the parent molecule, ethylene. However, as we shall see, important differences arise from (i) the nonequivalence of cis- and trans-stilbene, (ii) interactions between the ethylene and phenyl chromophores, and (iii) the availability of additional degrees of freedom to the stilbene molecule. In molecular orbital terms, the thermal isomerization of both molecules are electronically forbidden reactionsbecause they involve either a real or an avoided energy crossing between an occupied orbital, $, and an unoccupied orbital, $‘, at or near the perpendicular configuration. A real crossing occurs along those reaction paths which maintain a molecular symmetry which results in the orbitals $ and $‘ belonging to different symmetry species of the molecular point group. Reaction paths of this type are available to both ethylene and stilbene. Configuration interaction (GI) between the closed shell singlets, lG2 and ‘V2,is likely to be important in the crossing region where
their energies are comparable. Since the transition state is expected to be found in just this region, estimates of its energy by single configuration (SC) MO methods can be in serious error. The difficulties and pitfalls that may be encountered when such methods are used as starting points for the calculation of “symmetry-forbidden”reaction paths have been discussed at length in the l i t e r a t ~ r e .SC ~~~~ closed shell SCF methods pose more problems than non-SCF methods because of their unequal treatment of the two orbitals $ and $’. Extensive GI will sometimes be required to counter the intrinsic bias of such method^.^^^^ In the crossing region, orbitals obtained from a SC open shell SCF calculation of the triplet 3$1y might be expected to provide a better starting point for constructing the singlet wave functions. However, the proper starting point within an SCF formalism for the computation of thermal reaction paths accompanied by a change in orbital occupancy is a double (or bi) configuration (DC) SCF methodS6O Recent ab initio calculations for ethylene,66 methylene,56and [2.2.2]pr0pellane~~ are consistent with these expectations. The crossing or avoided crossing of two orbitals $ and $’ containing two electrons requires the molecule to pass through a diradical region on the potential hypersurface. and In such a region, three singlet configurations, lq2, $V2, and a single triplet configuration, 3$$‘, are of comparable Since the molecule has very low symmetry at most points, $ and $’ generally belong to the same rather than to different symmetry species and all three singlet configurations can mix. Consequently, the diradical singlet state is generally a mixture of the three configurations l@, l$$’, and 1$’2. Singlet wave functions have been calculated for diradicals by finding the eigenvectors of a closed shell or open shell Hartree-Fock Hamiltonian and then performing a 3 X 3 CI calculation.60 Methods of this type, however, are not wholly satisfactory for the computation of potential surfaces when the motion of the nuclei can carry the molecule into a region of higher or lower spatial The proper starting point for the calculation of a wave function for the lowest energy diradical singlet state is a triple (or tri-) configuration (TC) SCF method. The situation is less complicated for diradical triplet states because 3$$’ is usually the only low-lying triplet configuration. In the absence of other configurations of comparable energy, GI is not so important. However, potential surfaces on which the molecule traverses different spatial symmetries still pose problems for SC open shell SCF methods.61@ One-electron methods such as the extended Huckel (EH) method65cannot distinguish between singlets and triplets. Since GI is relatively unimportant for the lowest triplet state, potential surfaces calculated for diradicals by these methods are likely to be better approximations to the triplet surface than to the singlet surface. An additional argument, based on the variation of the Coulomb integral J($,1y) and the exchange with geometry, favors the same concluintegral K($,$’) sionSB6
‘w,
111. Potential Surfaces for Stilbene Potential energy surfaces for the c-t isomerism of stilbene have been calculated in a number of semiempirical MO mode19.26,28,29,31,32,34,41 Most of these263132%4,41are restricted to the a electrons and therefore neglect the effect of hyperconjugation, which is expected to be greatest in the region of the perpendicular configuration. In all cases, a Czaxis of symmetry has been maintained during the course of the reaction. In m ~ s t ,only ~ one ~ ~ internal ~ ~ , ~ ~ ~ coordinate, the angle of rotation, w , about the central bond
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t' 18
/
obtained potential curves for rotation about the central bond in the ground and excited states that are quite asymmetric with respect to the perpendicular (w = 90') configuration. They obtained a minimum at w = O', an absolute minimum at 150', and maxima at 90 and 180" in the ground state curve and a minimum at 105' and maxima at 0 and 180' in the curve for the lowest triplet state. The calculated and observed barriers to thermal trans cis isomerization are -50 and 49 kcal/mol, respectively. Using the Coulomb repulsion integral approximation, TM obtained curves that are nearly symmetric with respect to the perpendicular configuration. Their ground state curve exhibits minima at w = 0 and 180°, rather than at 0 and 150", and a maximum at 90' corresponding to a barrier of -48 kcal/mol to thermal t c isomerization. The lowest triplet curve is nearly flat in the intervals 0 < w < 30 and 150 < w < 180' and exhibits a minimum at 90" and maxima at -60 and 120'. The cis isomer was incorrectly calculated to be more stable than the trans isomer in the ground state. Similar calculations for butadiene give the same erroneous result.68 Momicchioli et a1.,32,34 who used an intermediate approximation for Ec, obtained asymmetric potentials. Their ground state surface exhibits minima at (w,4) = (0,O) and ca. (175,50) and a maximum at ca. (90,0), while the lowest triplet surface exhibits a minimum at (0,O) and a maximum in the neighborhood of the cis configuration. These authors also reported that the location of the cis ground state minimum in (a,$)space and the height of the barrier to rotation through the (180,90) geometry are affected by the hardness of the core repulsion. The Pariser-Parr-Pople which BG and Momicchioli et a1.32,34 adopted, is a SC closed shell SCF method. It therefore provides an unsatisfactory starting point for describing the lowest-lying states, Le., the diradical states, in the region of the perpendicular configuration. The second important closed shell configuration in the diradical region was neglected entirely in the calculations by BG and in the first calculation^^^ by Momicchioli et al. The latter calculations predict a barrier of 103 kcal/mol to thermal t c isomerization. In their subsequent calculation^,^^ Momicchioli et al. took the second important closed shell configuration into account by constructing a DC wave function from the SC closed shell orbitals. They obtained an appreciable lowering of the thermal barrier to isomerization (-15 kcal/mol). In all the PPP calculations, the potential curve for the lowest triplet state was found to cross the ground state curve in the region of the perpendicular configuration. The importance of the lowest doubly excited closed shell configuration in the diradical region and the inherent weaknesses of constructing the diradical states from orbitals calculated in a SC closed shell SCF model were, however, both clearly recognized by TM who adopted the Pariser-Parr In this model, the wave functions are constructed from Huckel MO's. They found that the lowest triplet curve does not cross the lowest singlet curve, but only approaches it very closely in the perpendicular configuration. Their model, however, is deficient in other ways. Core repulsion and nonbonded interactions destabilize the perpendicular configuration to a lesser extent than the cis and trans configurations. An approximation such as the Coulomb repulsion integral approximation" which underestimates the core repulsion energy will therefore discriminate against the perpendicular configuration in favor of the cis and trans configurations. A model which neglects nonbonded interactions will also have this bias and one which has both these features will
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23
I
Figure 1. The numbering scheme and the axis convention that were used in the present work.
(Figure l),has been varied, and furthermore, except for Momicchioli et a1.,32134authors reporting n-electron calculations of the surfaces have also neglected nonbonded interactions. However, even a cursory examination of steric interactions in the planar cis configuration (w = 180') leads to the conclusion that this configuration is very unfavorable. (To emphasize just how unrealistic this configuration is, we note that rigid rotation about the central bond results in H-16 lying closer to C-10 than to C-3 and H-22 lying closer to C-3 than to C-lo!) Hence which omit such interactions from consideration cannot be expected to give realistic potential surfaces in the neighborhood of the cis isomer. The steric strain in the planar cis configuration can of course be relieved by rotating the phenyl groups about the phenyl-ethylene "single" bonds. This was recognized by Momicchioli et al.,32i34who calculated energies as a function of both w and the angle 4 for symmetrical twisting (& = & = 4, Figure 1, C2 axis conserved) about the "single" bonds exocyclic to the phenyl rings. Their model predicts (i) that the ground state of trans-stilbene is planar, but that the potential surface in the 4 direction is rather flat, (ii) that the ground state of the cis isomer is nonplanar with -50°, and (iii) that w is strongly coupled to 4 on the cis side of the perpendicular configuration. These results are in accord with those obtained from similar 7-electron calculations of the geometries and electronic spectra of cisand t r u n s - ~ t i l b e n e . ~ ~ ~ ~ ~ The molecular structures of both isomers have been determined in the gas phase recently by electron diffraction.15J6 Both were found to be nonplanar and to possess C2 symmetry. The phenyl groups were found to be rotated -30 and 43' about the C-Ph bonds in the trans and cis isomers, respectively. On the other hand, X-ray studies indicate that the trans isomer is almost planar in the crystalline statea2It seems reasonable to conclude on the basis of the crystallographic and theoretical evidence that the barrier to rotation of the trans isomer through the planar geometry is small. Such a conclusion, however, is difficult to reconcile with the gas phase electron diffraction results which suggest that the phenyl groups are not rotating excessively even at 280 "C." The method used to calculate the core repulsion energy, Ec, also exerts an important influence on the shape of the potential surfaces. The point change approximation and the Coulomb repulsion integral approximation6' provide upper and lower bounds, respectively, to Ec. Using the point charge approximation,BorreU. and Greenwood (BG)26
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Structure and Isomerization of Stilbene
therefore underestimate the energies of the trans and cis configurations relative to the perpendicular configuration to a large degree, perhaps even to the extent of predicting a triplet surface with a minimum near the perpendicular configuration separated by two maxima. The equilibrium conformations, electronic spectra, and cis-trans isomerization pathways of stilbene have also been s t ~ d i e d ~ "by~ semiempirical ~ ~ ~ ~ s ~ ~all-valence electron MO methods. An ab initio MO study of conformationhas been reported very recently42 and other ab initio studies are in p r o g r e s ~ .All ~ ~the * ~ published calculations employ the SC closed shell SCF model. Ljunggren and Wettermark (LW)29used Santry's version70 of the CND0/2 modeP which does not give rotationally invariant results.72 This model predicts for the ground state of the trans isomer that there is a small barrier to rotation of the phenyl groups through the planar configuration and that the energy is almost constant in the interval 50 5 4 I90". Similarly, the energy of the ground state of the cis isomer was found to be essentially constant in the interval 30 I4 I90". There are hints in the potential curves of very shallow minima at 4 = 60-70". The cis isomer was also calculated to be 1.6 kcal/mol more stable than the trans isomer. Knop and Knop30used the standard CND0/2 mode171to investigate the conformation of cis-stilbene. This model predicts that the energy of the molecule decreases monotonically and rapidly with increasing 4 (175"). Bally et al.39found that the MINDO/3 predicts that the energy of the trans isomer behaves in the same way, but that it falls off more slowly. There is a significant qualitative discrepancy between these results and those obtained from the n-electron models for symmetric out-of-plane twisting of the phenyl rings. Momicchioli et ale's n-electron model predicts that the 4 = 90" conformation is an energy maximum for both isomers.32 In this conformation, the n-electron systems of the phenyl rings are completely decoupled at the oneelectron level from that of the ethylene fragment. The energy barriers to rotation through the 4 = 90" conformation were calculated to be -22 and 6 kcal/mol in the trans and cis isomers, respectively. The CND0/2 and MIND0/3 models predict that this conformation is either an energy minimum or a very flat energy maximum. As already noted, the values of obtained by gas phase electron diffraction for the trans and cis isomers are -30 and 43", respectively. The electron diffraction results15J6 also indicate that the phenyl groups in both isomers are not rotating extensively. The CND0/2 and MIND0/3 models thus fail to predict the properties of both isomers correctly whereas the n-electron models only have difficulty in coping with those of the trans isomer. It is now well documented that the CNDO/2 and IND074models underestimate steric repulsion and thus permit nonbonded atoms to approach too closely.75 Clearly, such methods are likely to give distorted potential surfaces in the neighborhood of the planar cis and trans configurations and to underestimate the barriers to rotation through these geometries. In light of the electron diffraction results for stilbene,16J6it is also clear that the CNDOI2 and MIND0/3 models underestimate conjugative destabilization (the increase in the "a"-electron energy) and/or overestimate hyperconjugative stabilization (the 6--7 mixing), which accompany out-of-plane twisting of the phenyl rings. The standard CNDO/2 model also fails to predict the conformations and barriers to rotation of other conjugated molecule^.^^ Experimentally, the most stable conformation of 1,3-butadiene, a molecule structurally related to stilbene, is the planar trans form ( 4 =
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2655
0') and the barrier to rotation about the central CC bond through the conformation with orthogonal vinyl groups (4 = 90") is 7.2 k ~ a l / m a l .The ~ ~ standard CND0/2 model predicts that a gauche form with 6 c.! 130' is the most stable conformation and that the barrier measured from Ab initio calculations the trans form is -1 kcal/m01.~~9~~ generally give results that are in much better agreement with e ~ p e r i r n e n t .Martin ~~ et aL30 have attributed the failure of the CNDO model to give realistic potential surfaces for conjugated molecules to the assumption of spherical orbitals in the calculation of the two-center two-electron integrals. This hypothesis is supported by the improved results obtained for 1,3-butadiene in an NDDO model which utilizes the standard CNDOIB parameterization insofar as possible.81 In the NDDO model, the anisotropy of the two-electron two-center integrals that are not neglected entirely is retained. However, the barrier to thermal t c isomerization is still underestimated by -50%. Furthermore, the explanation given by Martin et cannot be the only reason for the failure of the CND0/2 model because a PCILOS2calculation, which also utilized the approximations and parameterization of the standard CND0/2 model, gave results in qualitative agreement with experiment as well as higher, but still unsatisfactory estimates of the rotational barrie r ~ , ~ ~ The NDDO model mentioned above also gives results for stilbeneMthat are in better agreement with experiment and the n-electron models. The potential curve for symmetric out-of-plane twisting of the trans isomer exhibits a minimum at -20", a small barrier of -0.3 kcal/mol to rotation through the planar geometry, and a barrier of -4.5 kcal/mol to rotation through the 4 = 90' geometry. The curve for the cis isomer exhibits a minimum at -38' and a barrier of -3.8 kcal/mol to rotation through the 4 = 90" geometry. Like Santry's CND0/2 model, the NDDO model incorrectly predicts the cis form to be more stable than the trans form. Although these results and the CNDO/2 result for 1,3-butadiene may simply reflect the general tendency of NDO methods to favor more crowded s t r u c t ~ r e s , ~it ~should J ~ be borne in mind that only a limited number of internal coordinates were varied in the semiempirical studies of stilbene and that a full geometrical optimization might order the stabilities of the isomers correctly. This situation stands in sharp contrast to that which holds at the ab initio level. The results of an ab initio calculation,42which is also based on semirigid geometries taken from X-ray diffraction measurements, are in good overall agreement with experiment. is calculated to be 19 and 52" for the trans and cis isomers, respectively, and the trans isomer is calculated to be the more stable isomer. The calculated barriers to rotation of the trans form through the planar and 4 = 90" conformations are -0.4 and 9 kcal/mol, respectively. The difference of -13 kcal/mol in the barriers to rotation through the 90" conformation given by the ab initio model and the aelectron model of Momicchioli et al.32is a rough measure of the extent to which hyperconjugation stabilizes that conformation. LW also used Santry's CNDO/2 SC closed shell SCF mode170to calculate a potential curve for the rigid rotation of stilbene about the central CC bond in the ground state. They obtained minima at w = 0 and MOO, a maximum at 90°, and a barrier of 129 kcal/mol to thermal t c isomerization. The inherent deficiencies of such a model for calculating a reaction path that proceeds through a diradical region have already been described. In addition,
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however, LW held 4 fixed at 90’ throughout the rotation and thus denied the perpendicular configuration the stabilization that would ordinarily be expected to accrue from delocalization of the “a)’-electronsover the two benzyl fragments. Pedersen et a1.28used the standard CNDO/2 SC closed shell SCF mode171to calculate a potential curve for the rigid rotation of stilbene about the central bond with 4 set equal to Oo and obtained a barrier of -150 c isomerization. kcal/mol to thermal t At the one-electron level in the n-electron approximation, rotation about the central bond decouples the 7 electrons of the two benzylic fragments and completely destroys the central n bond. The length of the central bond, Rcc, in the ground state of stilbene is therefore expected to increase to a maximum at w = go”, just as it does in the ground state of ethylene. The increase for ethylene has been calculated by ab initio methods to be -0.13-0.14 A.65 The variation of Rcc with w is also likely to be important for stilbene, but it has been neglected in all but one investigation. LW optimized Rcc in the perpendicular configuration and, on comparing the result with the value assumed for Rcc throughout the rest of their study, concluded that the stretching of the double bond is negligible. However, it is unlikley that this conclusion is reliable since the central “double” bond is reduced formally to a single bond by the rotation in question. Pedersen et al.2sused the standard CND0/2 open shell unrestricted Hartree-Fock modeP to calculate a potential curve for the rigid rotation of stilbene in its lowest triplet state about the central bond. The curve they obtained has the following properties: (i) it crosses the ground state near the perpendicular configuration, (ii) it exhibib an absolute minimum at -90’) a secondary minimum at Oo, and a maximum at -15O, and (iii) 3p is -30 kcal/mol more stable than 3t. The latter two properties are inconsistent with the properties of the profile deduced from experiment (vide supra). The maximum at -15’ in the CNDO/2 curve is also unsatisfactory for theoretical reasons. A similar maximum was found in a curve calculated in an INDO MO model for the rotation of ethylene about the C-C bond in the lowest triplet statess4 The triplet wave function was constructed from the IND074SC closed shell SCF orbitals. The latter maximum has no counterpart in the curves calculated for the same process in ab initio MO models.M@Both maxima may owe their existence in part to the neglect of overlap by NDO models which results in the energies of both the n and n* orbitals associated with the (central) CC bond being underestimated to an increasing degree as w Oo. The situation for stilbene is aggravated by the tendency, due in part to the neglect of overlap, of CNDOI2 and INDO models to underestimate nonbonded overlap r e p u l s i ~ n s . ~ ~ J ~ ~ Bruni et al.%have calculated potential energy curves for the c-t isomerism of a related system, styrene (i.e., phenylethylene), in a CIPSIs7PCILOs2model. On finding that u-T mixing introduces absolute minima at w = 90’ in the curves for S1 and T1,they concluded that c-r mixing plays a primary role in photochemical rearrangements involving rotation about double bonds. Bendazzoli et a1.@have also investigated the isomerism of styrene. Invoking the virtual orbital approximation, they constructed SC and CI wave functions for the low-lying states from the MO’s obtained from an ab initio STO-3GmSC closed shell SCF calculation for the ground state. They investigated the importance of c--n coupling by including a limited number of 6-7 excitations in their CI calculations. The conclusion of Bruni et alas6has been criticized on the basis of their result^.^^^^^ Such criticism, however, overlooks the equi-
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vocal nature of the ab initio model itself and the fact that the SC SCF wave function, which forms the starting point for the ab initio study, already includes g-n mixing at the one-electron level (i.e., hyperconjugation), which Bruni et a1.86 introduce into their wave functions separately. Nevertheless, since Bruni et al. also adopt the standard CND0/2 approximations and parameterization, it does seem likely that their calculation overestimates the stabilities of the S1 and T1states of styrene in the perpendicular configuration. There is a growing consensus that the neglect of overlap has energetic consequences that cannot be i g n ~ r e d . ~ ~ ~ ~ The well-known EH model has its limitations, but it does not suffer from this or other major weaknesses of the SC SCF CNDO and INDO models. As a full-overlap model, it predicts correctly that the antibonding combination of two interacting orbitals is destabilized more than the bonding combination is stabilized and it is able to cope with nonbonded overlap repulsions in a qualitatively correct fashi0n.7~As a non-SCF one-electron model, it does not discriminate between the two orbitals that characterize diradical geometries, but treats them instead on an equal footing. It is also free of the ambiguity that sometimes arises from the existence of multiple solutions for SC SCF models.44-54~63”~96 In addition, the EH potential curve for the rotation of 1,3-butadieneabout the central CC bonds1 is in qualitative agreement with e ~ p e r i m e n t .However, ~~ the EH model also underestimates the barrier to thermal t c is~merization~~ and, compared to ab initio appears to overestimatethe barrier to rotation through the cis configuration. Despite this and its well-known deficiencies and limitations, it nevertheless appears that the EH model is potentially capable of providing reliable insights into the structure and isomerization of stilbene.
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IV. Extended Huckel Calculations For the reasons set out above, we have used the EH modeP5 to investigate the equilibrium conformations of stilbene and to calculate potential curves for the cis-trans isomerization of stilbene in its lowest states. Bond lengths and bond angles were held fixed at the values obtained for the trans isomer by electron diffraction in the gas phase (see Table 111,ref 16). Solution of the characteristic EH eigenproblem yields the MO’s and their energies, ti. We took the total energy of the ground state, E, to be 2Cici where the sum is over all (doubly) occupied orbitals. The energies of the lowest singly and doubly excited states are E + At and E + 2Ae, respectively, where At = ~ L U M O‘HOMO and the designations HOMO and LUMO refer to the lowest energy closed shell Configuration. The parameters were set at the values used by Hoffmann et al.97 Points on the potential curves for various torsional modes were calculated at loo intervals in the torsional coordinate. Figure 1 shows the atom number scheme and the axis convention that were adopted. In Figure 2 the high energy occupied and low energy unoccupied T MO’s of planar stilbene are constructed schematically in the Huckel model from the n MO’s of ethylene and the appropriate symmetry combinations of the HO and LU n MO’s of benzene. Since the local symmetry of the phenyl rings is ‘Tzu)’, the n MO’s of the ethylene fragment interact with the “a”, but not the “b” benzene combinations. The latter have nodes at C-2 and C-9. The figure implies that the HO and LU MO’s of stilbene are delocalized over the whole molecule. The former contains a significant contribution from the ethylenic n MO mixed in an antibonding way while the latter contains a significant contribution from the ethylenic n* MO mixed in a bonding way. The ethylenic compo-
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2657
Structure and Isomerization of Stilbene
TABLE I: Computed Orbital Energies for Planar trans-Stilbene symmetry no,
energy,eV
41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26
2b
C
C,
character
- 3.96 -4.53 -6.60 -7.91 -8.33 -8.33 - 9.37
b a b a a b b
n n n n n n n
-12.03 -12.75 -12.80 - 12.80 - 12.96 -13.16 - 13.28 -13.41 - 13.70
a
n
a a b
U
Czh
I/
n n n
b a b b a
U U U 71
-1193 -1194 3 a ( i )
:,:p -1195
W W
z
1
-1196
c;=c;:; Figure 2. Schematic construction of the high-energy occupied and low-energy unoccupied a MO’s of planar stilbene in the Huckel model from the a MOs of ethylene and the appropriate symmetry combinations of the HO and LU a MO’s of benzene.
-1200 -1201
nents of these two MO’s account for the high oscillator strength of the transition from the ground state to the lowest excited singlet state. They are also responsible for the crossing (avoided crossing) of the HO and LU MO’s that occurs when the molecule is rotated about the central bond and a Cz axis of symmetry is (is not) maintained during the rotation. The energies and symmetries of the higher energy of;cupied and lower energy unoccupied MO’s calculated in the EH model for the planar trans isomer are given in Table I. The second highest occupied orbital is predicted to be a u orbital. This is not in agreement with the order given by the CNDOIS2 model which predicts that the highest occupied u orbital lies below five higher energy occupied a orbitals.98 The EH model also predicts that u orbitals lie among the higher energy occupied T MO’s of n a ~ h t h a l e n e but , ~ ~this is not supported by ab initio calculations.lW Evidently, the EH model does not discriminate properly between u-u and T-a type interactions. This is to be expected since it makes no concession to local a-electron screening in conjugated systems.lol Potential energy curves for the symmetrical rotation = 42, Cz axis conserved) about the “single” CC bonds are plotted for the ground state, first singly excited state, and first doubly excited state of trans-stilbene (w = Oo) in Figure 3a(i). The ground state is predicted to be planar, but the minimum is very shallow. In agreement with the previous PPP, NDDO, and ab initio calculations, the EH model predicts that the 4 = 90” conformation is an energy maximum. The energy barrier to rotation through this conformation is calculated to be 7.1 kcal/mol. The strong preference for the planar geometry and the higher barriers to rotation through the 4 = 90° conformation in the two
-1202
--I -9I 5? 5
-12.5
”
Ib
W EO
30
50
70
90 +O
IO
30
50
70
90 +O
Flgure 3. Potential energy curves for the symmetrical rotation (4 = 4 2, C2,axis conserved) about the “slngle” CC bonds for the ground state, first singly excited state, and first doubly excited state of trans (w = 0, Figure 3a(i)) and cis (w = 180, Figure 3b(i)) stilbene. The energies of the HO and LU MO’s of the trans and cis forms are plotted in Figure 3a(ii) and 3b(ii), respectively.
excited states are due to the fact that the H W and LU MO’s are antibonding and bonding, respectively, between C-1 and C-2 and between C-8 and C-9. The energies of these two orbitals are plotted as a function of 4 in Figure Sa($. Potential curves for the same torsional mode are plotted for the same states of cis-stilbene (w = M O O ) in Figure 3b(i). The ground state minimum occurs at -50° and a barrier of -2.2 kcal/mol is calculated for rotation through the 4 = 90” conformation. As expected, the minima in the curves for the two excited states are shifted to smaller $ and the barriers to rotation through the q!~ = 90” conformation are higher in these states. The energies of the HO and LU MO’sare plotted in Figure 3b(ii). Potential curves for rotation about one “single” CC bond (& = w = 0) are plotted for the same states of transstilbene (w = 0) in Figure 4a(i). The three curves exhibit minima at 0” and become progressively harder with ex-
2658
A. R. Gregory and D. F. Wiillams
The Journal of Physical Chemisrry, Vol. 83, No. 20, 1979
-I I93 40(i)
4bIi)
-I 197 -I 198
- I I98
.,
bl
-I 199
t -1200
I
t
I
-I 201
\
i
/
-12oot-
t
/
b2 oo
02 0 0 ::
10
I
I
I
I
1
1
1
30
50
70
90
110
130
150
I I
170
wo c
-12.5l
I
,b
J
I
30
I
50
I
I
7C
Figure 4. Potential energy curves for rotation about one “single” CC bond (4* = 0) for the ground state, first singly excited state, and first doubly excited state of trans (Figure 4a(i)) and cis (Figure 4b(i)) stilbene. The energies of the HO and LU MO’s of the trans and cis forms are plotted in Figure 4a(ii) and 4b(ii), respectively.
citation. Once again the minimum for the ground state is very shallow. The dl = 90” conformation is a maximum on each curve, but the barriers to rotation through this conformation are, in keeping with twisting about one “single” CC bond instead of both, much smaller than for symmetrical rotation (& = d2, C2 axis conserved). The barrier in the ground state is -3.3 kcal/mol. The energies of the HO and LU MO’s are plotted as a function of 4 in Figure 4a(ii). Potential curves for the same torsional mode are plotted for the same states of cis-stilbene (w = 180’) in Figure 4b(i). Compared to the curves for symmetrical rotation (Figure 3b(i)),the minima are displaced to higher &. This is a consequence of the additional steric repulsion arising from the coplanarity of the ethylene fragment and the second phenyl ring. In contrast to the situation which holds for the trans isomer, the c#q = 90’ conformation is a minimum on the potential curves for the ground and singly excited states and a relatively low maximum on the potential curve for the doubly excited state. The energies of the HO and LU MO’s are plotted as a function of 41 in Figure 4b(ii). The minimum energy geometries calculated for the ground states of both isomers possess a C2 axis of symmetry. The trans isomer is calculated to be -5 kcal/mol more stable than the cis isomer. The blue shift observed for the first absorption band on passing from trans- t o cis-stilbene is predicted. Steric hindrance forces more out-of-plane rotation of the phenyl groups in the cis form than in the trans form: the result is a larger HOMOLUMO energy gap in the cis form (see Figure 3). These EH results are different from those reported briefly by Momicchioli et al. in a footnote to their first paper.32Their EH calculations predict that trans-stilbene should be quasi-perpendicular (60 < 4 < 90’) and that there is a barrier of -8 kcal/mol to rotation through the planar conformation. The blue shift in the first absorption band of cis-stilbene relative to trans-stilbene cannot be explained now simply in terms of the HOMO-LUMO energy
Figure 5. Potential energy curves for symmetrical rotation (4 = q5 = 0, C2axis conserved) about the central bond for the ground state, first singly excited state, and first doubly excited state of stilbene. The energies of the most stable ground (4, = q5 N 50) and singly excited (4 = 4 40) state conformations calculated for cis (w = 180) stilbene are also shown.
gap. These differences are probably due in part to the use of different structural data. As indicated above, we chose to use the new data obtained recently from gas phase measurements.lsJ6 The C=C-C angle plays a particularly important role in determining the shape of the potential curves for rotation about the “single” C-C bonds. With slightly different bond lengths (rC4 = 1.33, rCw = 1.39, rc4 = 1.45, and rC-H = 1.10 8)and the C=C-C angle set equal to 120” instead of 127.7’, we found that the difference in energy between the planar and the 4 = 90’ conformations is calculated to be only -0.5 kcal/mol. LW’s CNDO/2 results also indicate a crucial role for the C=C-C angle. Momicchioli et al. set this angle equal to 120’ in their PPP calculation^.^^^^^ Potential energy curves for symmetrical rotation (41= Cp2 = 0, C2 axis conserved) of stilbene about the central bond are plotted for the ground state, first singly excited state, and first doubly excited state in Figure 5. This figure has the form characteristic of SC correlation diagrams for symmetry-forbiddenreactions.‘@*47 In agreement the with the PPP calculations of Momicchioli et a1.32*34 potential curves are quite asymmetric with respect to the perpendicular (w = 90’) configuration. As has already been indicated, the EH potential curve for the first singly excited state is more correctly identified in the diradical region with the lowest triplet state than with the first singly excited singlet state. This curve shows no evidence of the two maxima which appear at -60 and 120° in the potential curve calculated for the lowest triplet state by TM. It exhibits a minimum at 0’ and a region of rapidly increasing energy as w 180’ and the H-16-H-22 separation diminishes. It shares these features with the curve calculated by Momicchioli et al.34 On the other hand, the energy difference between the trans (w = 0”) and perpendicular (w = 90’) configurations is calculated by the EH model to be -0.7 kcal/mol and by Momicchioli et al.’s PPP model to be -14.3 kcal/mol. Thus only the EH model, which allows for hyperconjugation, correctly predicts the minimum to be very shallow. The location of the EH minimum is sensitive to the values chosen for the other geometrical parameters. Thus in calculations with r w = 1.33, r c q = 1.39, rC4 = 1.44, and rcH = 1.09 A and LC=C-C = 120°, we found that the minimum is
-
Structure and Isomerization of Stilbene
displaced to w N 70' and that a flat maximum associated with a barrier to rotation of -3.6 kcal/mol is obtained at w N loo. Since Figures 3(a)(i) and 5 imply that the lowest singly excited state is planar, it follows from the nodal properties of the HO and LU MO's that its minimum energy geometry will be one with a longer C-1-C-8 bond and shorter C-1-C-2 and C-8-C-9 bonds. a-electron calculation^^^^^^ support this conclusion. The triplet curve for the rotation of stilbene about the central bond is thus quite different from that for ethylene. It is generally accepted that the 3p species corresponds to an absolute minimum on the ethylene surface and that there is a sizeable barrier to rotation through the planar configuration.55~*5Joz The EH curve has this shape. With rcc = 1.339 and rCH = 1.086 A and LC=C-H = 121.2', we obtained a 30.4 kcal/mol barrier to rotation through the planar configuration. The ground state curve shown in Figure 5 is in good agreement with the curve calculated by Momicchioli et a1.32.34Both curves exhibit a minimum near 130" on the cis side of the perpendicular configuration. The energies of the most stable ground and triplet state conformations of cis stilbene (w = 180') calculated in the EH model are also shown. The 3c species is predicted to lie at a higher energy than the 3p species. The PPP results of Momicchioli et al.34suggest that 3c can rotate freely to 3t. Since this conclusion might be expected to hold in the EH model, the EH results at this stage appear to favor the first of the two explanations put forward in the Indroduction to account for cis-trans isomerization in the triplet state. However, Sumitani et alaz3have reasoned on the basis of unpublished ab initio results for ethylene and 1,3,5-hexatriene and the PCILO calculations of Bruni et a1.86for styrene that the triplet state surface does not cross the ground state surface near w = 90'. This conclusion is not supported by the results of earlier ab initio calculations for ethylene55,85and the results of most calculations for stilbene to date. The energies of the five highest occupied and five lowest unoccupied orbitals in planar trans-stilbene are plotted as a function of w in Figure 6. The HO and LU MO's have symmetries a, and b,, respectively, in C% symmetry, which reduce to a and b, respectively, in C2 symmetry. As shown above, the HO and LU MO's contain significant contributions from the a(a) and a*(b) orbitals, respectively, of the ethylene fragment. The two stilbene orbitals therefore cross at w = 90' as shown in the figure. That this crossing must take place can perhaps be best seen as follows. Let 2p1 and 2p8denote the orbitals at C-1 and C-8, respectively, from which the a and a* orbitals of the ethylene fragment are formed. Suppose that the right-hand half of the molecule is rotated about the central bond. For w = 0, the z axis is the C2 axis of symmetry. For w > 0, the C2 axis of symmetry lies in the y-z plane and makes an angle l/@ with the x-z plane. For w = 180°, the C2 axis of symmetry coincides with the y axis. As the rotation takes place, the 2p1 orbital changes from 2pZ1(w = 0') into 2pyl (w = 90') and finally into -2pz1 (w = 180'). At the same time, the symmetry orbital which transforms as the a species changes from a to a* while the symmetry orbital which transforms as b changes from a* to a. Therefore, the two orbitals a and b must cross. Alternatively, 2p1 and 2p8 can be resolved into their 2ps and 2p, components and the two halves of the molecule can be rotated in opposite directions through angle l / p about the central bond. This motion has the advantage that the z axis remains the C2 axis of symmetry and the ay,ay*,T,,and a,* orbitals associated with the ethylene
The Journal of Physical Chemistry, Vol. 83, No. 20, 7979 2859
I
I
-9.01
IO
30 50 70 90 110 I 3 0 150 clp
Figure 8. Variation of the energies of the five HO and five LU MO's of planar trans stilbene with symmetrical rotation (4, = 4 = 0, C2 axis conserved) about the central bond. The numbers and symmetries of the orbitals are given on the left- and right-hand sides, respectively, of the figure.
fragment remain symmetry orbitals throughout the rotation. In this case, however, the cis isomer is formed in the x-z plane instead of the y-z plane. In C2 symmetry, azand ay*both transform as a species and a,, and a,* both transform as b species. Hence, as w increases from zero, the a orbital, at first dominated by a,, slowly mixes in ay*. At w = 90°, both a, and ay*contribute equally to the a orbital. As w increases from go', the as* contribution becomes larger and larger until at 180' the a, contribution to the a orbital is completely mixed out. Similarly, as w increases from zero, the b orbital, at first dominated by a,*, slowly mixes in ay At o = 90°, both contribute equally to the b orbital. As w increases from go", the ay contribution becomes larger and larger until at 180' the az* contribution is completely mixed out. Hence the ethylenic contribution to the a orbital changes from a, to ay*while the ethylenic contribution to the b orbital changes from a,* to ay. The a and b orbitals must therefore cross. A view based on energy considerations alone leads to the conclusion that the curves for the energies of the HO and LU MO's and for the total SC energies of the ground and doubly excited states exhibit cusps at w = 90°. Such an interpretation, however, is associated with a discontinuous change at 90' in the orbital occupancies which define the ground and doubly excited states.49@When a SC closed shell SCF model is used, there is the additional complication that the occupied and virtual orbitals and the total SC energies of the ground and doubly excited states are forced apart artificially (see Figure 2, ref 26 and Figures 3 and 4, ref 34). There is then a danger in the energy picture of mistaking the cusps in the energy curves for stationary points. In a first approximation, the "a" electrons of stilbene are more accurately perceived in the perpendicular configuration (w = 90°, 8, = O2 = 0') as two weakly coupled
A. R. Gregory and D. F, Williams 70(1)
-I
7b(l)
f
of -27 kcal/mol between the $ = 0 and 90’ conformations. However, these curves also exhibit a small secondary minimum at -70’. The energy of the triplet in this conformation is somewhat higher than it is in the ground state cis and trans equilibrium conformations, but this might not be the case following optimization of the “single” and central CC bond lengths. Such a possibility, which indicates that the multidimensional character of the system cannot be neglected, also suggests an alternative explanation for isomerism in the triplet state. If, after the initial preparation of the vibrationally excited triplet, the bond length changes in question and rotation about the central bond were to precede rotation about the “single” bonds, the system might be faced with surmounting a small barrier, b, intrinsic to the triplet surface itself, before achieving the preferred value of 4(O0)in the perpendicular (w = 90’) configuration and crossing to the ground surface. We can then identify b with the observed barrier to c t isomerization and b plus the energy difference between 3t and 3p with the observed barrier to t c isomerization. The barrier b itself will depend on the direction of isomerization due to coupling between the two torsional modes as o 90’. The EH results suggest that this mechanism is more likely to hold for cis trans isomerization because of the smaller difference between rp in the cis ground state and the value of $ corresponding to the maximum separating the two minima in the triplet curve shown in Figure 7a(i). The gas phase electron diffraction data15J6 for the two forms also point to this conclusion, but not as strongly. In any case, the present mechanism is more attractive than the mechanism proposed by Benson and Williams (BW).21 The latter requires the system to intersystem cross not just once, but twice, and the first of these crossings is not very favorable. Potential curves for the rotation of one “single” CC bond (& = 0) in the perpendicular (w = 90’) geometry are plotted for the same states in Figure 7b(i). The q+ = 90’ conformation is a maximum on each curve and the barrier to rotation through this conformation increases with excitation. As expected, these barriers are smaller than those for symmetrical rotation ($1= $z,Cz axis conserved). The behavior of the potential curves is determined by the behavior of the energies of the HO and LU MO’s (Figure 7b(ii)). One orbital remains essentially constant in energy while the energy of the other one increases.
-
Flgure 7. a(i) Potential curves for the Symmetrical rotation ( 4 , = 4 2r C2axis conserved) about the “single” CC bonds for the ground state, first singly excited state, and first doubly excited state of perpendicular (w = 90’) stilbene. a(ii) Variation of the energies of the HO and LU MO’s for the same torsional mode. b(i) Potential curves for rotation about one “single” CC bond (4 = 0)for the ground state, first singly excited state, and first doubly excited state of perpendicular (w = 90’) stllbene. b(ii) Variation of the energies of the HO and LU MO’s for the same torsional mode.
benzyl a-electron systems. The higher lying occupied and lower lying unoccupied orbitals of stilbene in zero order are simply plus and minus combinations of the a MO’s of the two benzyl system. The benzyl ?r MOs themselves can be constructed in a qualitative way by examining the interactions between the a MO’s of the benzene ring and the 2pa orbital associated with the exocyclic carbon atom. In the Huckel model, the energies of the benzene a MO’s are symmetrically disposed about the energy of the exocyclic orbital. As a consequence,the HOMO of the benzyl radical is nonbonding between the exocyclic carbon and the adjacent carbon atom in the ring. In the EH model, the inclusion of overlap shifts all the benzene orbitals to higher energies. The energies of these orbitals are no longer symmetrically disposed about that of the exocyclic orbital. On the basis of energy considerations alone, one would predict that the HOMO of the benzyl radical should be weakly antibonding between the exocyclic carbon and the ring, Our EH calculations for stilbene indicate, however, that this is not the case and that the interaction in question is weakly bonding. We have investigated rotations about the single bonds in the perpendicular configuration (w = 90’). Potential curves for the symmetrical rotation (dl = &, Cz axis conserved)about the “single” CC bonds are plotted for the ground state, first singly excited state, and first doubly state in Figure 7a(i) and the energies of the HO and LU MO’s are plotted in Figure 7a(ii). At first, the C-1-C-2 and C-8-C-9 “a”-type interactions are weakly bonding in both orbitals ($ = 0), but, as $ increases, they diminish in strength, change sign, and finally become weakly antibonding ($ = 90’). The two orbitals do not remain degenerate: the b orbital lies at a slightly lower energy throughout most of the rotation. The shift of these orbitals to higher energies is reflected in the potential curves for the electronic states, all of which show an energy difference
-
-
-
V. Conclusions In this paper, we have assessed the relative strengths and weaknesses of previous calculations on stilbene. These considerations underline recent warnings that the scope and limitations of a MO model should be investigated carefully before it is used in a predictive c a p a ~ i t y . ~ ~ ~ ~ ~ J Bearing this in mind, we have made a case for an EH investigation of stilbene and carried out calculations in this model by using gas phase structural data that became available only r e ~ e n t l y Wherever . ~ ~ ~ ~ ~contact is possible, the results are in good agreement with experiment. Our conclusions may be summarized as follows: (1) The EH model predicts that the phenyl groups of cis-stilbene in the gas phase are rotated ~ 5 0 about ’ the C-Ph bonds, but, in common with many other models, incorrectly predicts that the ground state of trans-stilbene in the gas phase is planar. However, X-ray studies indicate that the trans isomer is almost planar in the crystalline states2Moreover, the trans minimum is calculated to be very shallow and those models which do predict a realistic nonplanar geometry for the trans form also predict a very small barrier to rotation through the planar conformation.
Structure and Isomerization of Stilbene
Although these results imply that rotation about the C-Ph bonds is soft, this is not easily reconciled with the conclusion from a gas phase electron diffraction study that the phenyl groups are not rotating exce~sively.~~ The shape of the potential curve for symmetrical out-of-plane rotation is very sensitive to the C=C-C bond angle. (2) The EH model used in conjunction with a semirigid geometry model correctly predicts the trans isomer to be the more stable one. All-valence electron NDO models do not. (3) The EH model predicts the blue shift observed for the first absorption band on passing from trans- to cisstilbene. (4) The EH potential surface for rotation about the central bond in the lowest triplet state is remarkably consistent with experiment. In agreement with the PPP results of Momicchioli et al.,34the planar trans configuration is found to correspond to an absolute minimum on the triplet surface. Their PPP model fails, however, to predict a small energy difference between the 3t and 3p species. The r$ = 0” sections calculated by both models show no evidence of the two maxima which appear at -60 and 120” in the r$ = 0” section calculated by TM in a PP model. These maxima are considered to be artifacts of TM’s model and are attributed to the neglect of nonbonded interactions and to the use of an approximation which underestimates the core repulsion energy. To this extent, the EH model supports BW’s suggestion that intersections between the ground and triplet state surfaces are responsible for the observed barriers to isomerization in the triplet state. The PPP model of Momicchioli et al. not only provides this support, but also predicts that such crossings actually occur.34 (5) The EH results, however, also hint at the possible existence of other barriers to isomerization on the triplet surface near the perpendicular configuration. These barriers, unlike those of BW, are intrinsic to the triplet surface and, unlike those of TM, lie in the r$ direction rather than in the w direction. Present data, both experimental and theoretical, do not discriminate between the two possible “triplet” energy profiles. This matter can be settled theoretically only by computing the energies of the ground and triplet states accurately as a function of a minimum of four internal coordinates. These are w , $I, the central CC bond length, and the length of the “single” CC bonds.
Acknowledgment. We thank J-P. Laplante for computational assistance and W. Siebrand and other members of NATO Project No. 1190 for discussions and support. References and Notes (1) Issued as NRCC No. 17678. (2) J. M. Robertson and I. Woodward, Proc. R. SOC.London, Ser. A , 162, 568 (1937); C. J. Finder, M. G. Newton, and N. L. Allinger, Acta Ctystallogr.,Sect. B, 30, 411 (1974); J. Bernstein, ibid., 31, 1268 (1975). (3) D. F. Evans, J . Chem. Soc., 1351 (1957); A. Bylima and 2. R. Garbowski, Trans. faraday Soc., 65, 458 (1969). (4) H. Stegemeyer, J. Phys. Chem., 88,2555 (1962), and references cited therein. (5) R. H. Dyck and D. S. McClure, J. Chem. Phys.,38, 2326 (1962). (6) S. Malkin and E. Fischer, J . Phys. Chem., 68, 1153 (1964). (7) G. S. Hammond, J. Saltiil, A. A. Lamola, N. J. Turro, J. S. Bradshaw, D. 0. Cowan, R. C. Counsell, V. Vogt, and C. Dalton, J. Am. Chem. Soc., 88, 3197 (1964). (8) W. G. Herckstroeter and G. S. Hammond, J . Am. Chem. Soc., 88, 4769 (1966). (9) G. Heinrich, H. Blume, and D. Schulte-Frohlinde, Tetrahedron Lett ., 4693 (1967); W. G. Herkstroeter and D. S. McClure, J. Am. Chem. Soc., 90, 4522 (1968). (10) J. Saltiel, J . Am. Chem. Soc., 90, 6394 (1968). (1 1) J. Saltiel, J. T. D’Agostino, E. D. Megartty, L. Metts, K. R. Neuberger, M. Wrighton, and 0. C. Zafiriou, Org. Photochem,, 3, 1 (1971).
The Journal of Physical Chemistry, Vol. 83, No. 20, 1979 2661
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(1969). (101) N. 0.Lipari and C. B. Duke, J. Chem. phys., 63, 1748,1768 (1975); K. KrogMespersen and M. A. Ratner, Theor. Chlm. Acta, 47, 283 (1978). (102) A. J. Merer and R. S. Mulilken, Chem. Rev., 69, 639 (1969). (103) D. B. Boyd, J . Phys. Chem., 82, 1407 (1978). (104) NOTEACCEDIN PROOF:D. A. Luippold (Chem. Phys. Lett., 35, 131 (1975))has calculated potential curves for the rigid rotation of the Soand TI states of stilbene about the central bond in INDO SC SCF models and obtained results very similar to those of Pedersen et aL2* Luippold's TI curve, however, does not exhibit a secondary minimum at . ' 0 Luippoid also calculated the Socurve in an INDO GVB model and thus allowed for the second important closed shell MO conflguration. Use of the GVB wave function results in a substantial lowerlng of the thermal barrlers to isomerization, but not to such an extent that the crossing of the So and T, curves disappears.
Meaning of Diffusion-Controlled Association Rate Constants in Enzymology H. Nakatanl and H. B. Dunford" Department of Chemistty, University of Alberta, Edmonton, Alberta, Canada T6G 2G2 (Received March 9, 1979) Pubtication costs assisted by the University of Alberta
In the first stage of enzyme-ligand interactions diffusion-controlledassociation occurs which may be the most common and simplest elementary process. The rate constant for the diffusion-controlled association can be calculated by using suitable theoretical equations. The deviation of experimental values from results calculated by using the Smoluchowski equation suggests the existence of a restricted target area on the enzyme surface. Recently, a crevice or capture-windowmodel has been proposed which appears more realistic in terms of known enzyme structures. A simple alternative capture window model is proposed. The viscosity dependence of an and dissociation apparent association rate constant is interpreted in terms of diffusion-controlled formation (kw) (k1) of an encounter complex followed by a nondiffusion-controlled reaction (kJ. For the reaction of p nitroperbenzoic acid with horseradish peroxidase, h d i f f = 1.3 X lo8 M-l s-l and k-l/kz = 2.2 in water at 25 "C. Introduction The development of fast reaction techniques in solution has facilitated study of detailed reaction mechanisms for elementary processes on enzyme-ligand systems. At the first stage of enzyme-ligand interactions, a diffusioncontrolled bimolecular association process always occurs. The selectivity and recognition between enzyme and ligand during diffusion-controlled association seems an interesting and important subject for the understanding of enzyme function and mechanism. Although there are many data on bimolecular association rate constants on enzymeligand systems which are believed to be diffusion controlled,l few systems have been proven experimentally to 0022-3654/79/2083-2662$0 1 .OO/O
be diffusion controlled. However, two enzymes which operate with exceedingly fast rates are carbonic anhydrase and superoxide dismutase. For the former enzyme it was once felt that the rate was so fast that it must be facilitated by surface diffusion.2 However, this was based on the assumption that carbonic acid, not HC03-, acted as a s u b ~ t r a t e . ~Rate , ~ constants for both of the superoxide dismutase reactions5 E-Cu(I1) + 0 2 E-Cu(I) + 0 2
-+
E-Cu(I)
+ 02-
are estimated at 2
X
2H+
E-Cu(I1)
lo9 M-' s-l,
0 1979 American Chemical Society
+ HzO2
