748
J . Phys. Chem. 1991, 95, 748-751
Deuterium Isotope Effect on the Photoisomerization Rates of Stilbene: An RRKM Analysis Based on Computed Vibrational Frequencies Fabrizia Negri and Giorgio Orlandi* Dipartimento di Chimica “G. Ciamician”, Universita’ di Bologna, 401 26 Bologna, Italy (Received: April 23, 1990)
Microcanonical rate parameters k ( E ) for stilbene photoisomerization are obtained by RRKM calculation based on computed vibrational frequencies of reactant and transition state. Two alternative transition states are considered, one formed by the avoided crossing between the 1B and 2A energy curves and the other arising on the 1B energy curve; for the former, nonadiabatic effects are estimated by computing 1B-2A vibronic couplings. The thermal averaged rate parameters k(7“) are obtained from the computed k ( E ) . The comparison of the rate coefficients observed in supersonic beams and in the dense phases with the computed rates indicates that the photoisomerizationof stilbene can be well described in the framework of a statistical theory and supports the nonadiabatic mechanism associated with the transition state formed by the 1B-2A avoided crossing. Both the rate parameters observed in supersonic beams and in the dense phase can be explained by attributing a different degree of adiabaticity and a different barrier height for the reaction in the two environments.
Introduction The trans-cis photoisomerization of stilbene has for a long time been a test case for the theories of photoisomerization, and for this reason it has been studied extensively in a variety of different environment^.'-'^ In most solvents this process has been found to proceed by overcoming a barrier of about 1000-1400 cm-I, which, according to quantum mechanical calculations,Iel6 arises from the crossing between the potential energy curve of the state excited optically, lB, and the potential energy curve of 2A, which is the state with a minimum a t the twisted (or perpendicular) configuration. Recently, trans-stilbene (tS) has been studied under collision-free conditions in supersonic molecular beams by laser excitation.8-” By this technique the energy dependence of k ( E ) , which is the rate coefficient of the photoisomerization process, was measured, and a well-defined threshold for this process was found at an excess vibrational energy of ca. 1200-1300 cm-I. There have been several attempts to analyze theoretically these results: all of them were based on methods of statistical unimolecular rate theory, although in principle the theory of radiationless transitions could also be used. Khundkar, Marcus, and Zewail” applied the RRKM method, using the computed18 SI frequencies of tS for both the initial and the transition state. Assuming the reaction coordinate to correspond to a =400-cm-’ ( I ) Saltiel, J.; DAgostino, J.; Megarity, E. D.; Metts, L.; Neuberger, K. R.; Wrighton, M.; Zafirion, 0. C. Org. Photochem. 1973, 3, 1 . (2) Dyck, R. H.; McClure, D. S. J . Chem. Phys. 1962, 36, 2326. (3) Sumitani, M.; Nakashima, N.; Yoshihara, Y.; Nagakura, S. Chem. Phys. Lett. 1977, 51, 183. (4) Good. H. P.;Wild, U. P.; Fischer, E. H.; Resevitz, E. P.; Lippert, E. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 126. (5) Heisel, F.; Miehe, J. A.; Sipp, B. Chem. Phys. Len. 1979, 61, 115. (6) Greene, 9. I.; Hochstrasser, R. M.; Weisman, R. B. J . Chem. Phys. 1979, 71. 544. Greene, B. I.; Hochstrasser, R. M.; Weisman, R. B. Chem. Phys. 1980.48, 289. (7) Hochstrasser, R. M. Pure Appl. Chem. 1980, 52, 2683. (8) Syage, J. A.; Lambert, W. R.; Felker, P. M.; Zewail, A. H.; Hochstrasser, R. M. Chem. Phys. Letr. 1982, 88, 266. (9) Amirav, A.; Jortner, J. Chem. Phys. Letr. 1983, 95, 295. (IO) Majors, T. J.; Even, U.; Jortner, J. J . Chem. Phys. 1984, 81, 2330. (11) Syage, J . A.; Felker, P. M.; Zewail, A. H. J . Chem. Phys. 1984,81, 4706. (12) Kim, S. K.; Courtney, S. H.; Fleming, G. R. Chem. Phys. Letf. 1989, 159, 543. (13) Courtney, S. H.; Fleming, G. R. J . Chem. Phys. 1985, 83, 215. (14) Orlandi, G.; Siebrand, W. Chem. Phys. Lett. 1975, 30, 352. (15) Tavan, P.; Schulten, K. Chem. Phys. Lett. 1978, 56, 200. (16) Orlandi, G.; Palmieri, P.; Poggi, G. J . Am. Chem. SOC.1979, 101, 3492. (17) Khundkar, L. R.; Marcus, R. A.; Zewail, A. H. J . Phys. Chem. 1983, 87. 2473. (18) Warshel, A. J . Chem. Phys. 1975.62, 214.
frequency and the barrier (or threshold) energy to be 1250 cm-I, they obtained a rate coefficient k ( E ) significantly higher than that observed. A major reason for this discrepancy is the arbitrary assumption of identical frequencies in the transition state and in the reactant, but other effects (for example, an efficient reverse process and nonadiabatic effects) cannot be ruled out. Subsequently TroeI9 and Felker and ZewailZ0proposed two optimized RRKM analyses that are based on different mechanisms and make use of different molecular parameters. The model by T r d 9 assumes the process to be adiabatic and involves a suitable scaling of transition-state frequencies of the ethylene moiety. The model by Felker and ZewailZoemploys for the transition state the same frequencies as for the reactant molecule but it takes into account the nonadiabatic effects which are to be expected since the barrier for the reaction is due to an avoided crossing between two potential energy curves. The nonadiabatic transmission coefficient, introduced in the RRKM treatment, is chosen in such a way as to match the observed rates constant k ( E ) for transstilbene-do (tS-do) and trans-stilbene-d,, (tS-d,,). The two model^^^.^^ reproduce very well the photoisomerization rates observed in supersonic beams. As pointed out by Troe and Schroeder,2’ this success does not imply that the two models offer a realistic description of the process, nor that the conditions for the validity of RRKM treatment are met in stilbene. This point is underscored by a very recent study of photoisomerization in supersonic beams of stilbene-d2 (tS-d,) and stilbene-d,, (tS-d,,) isotopomers.22 Surprisingly, the photoisomerization rates k ( E ) of tS-d,, were found higher than the rates of tS-d,. These results could be accounted for by neither of the models. They were recently rationalized by a more elaborate analysis,23which allowed for partial intramolecular vibrational redistribution (IVR) but was still based on an empirical choice of transition-state frequencies. RRKM calculations can provide meaningful answers to the question of energy randomization and of the mechanism of photoisomerization only if the parameters used in the analysis are obtained from sources different from the fitting of the observed rate constants. One of these sources is obviously a quantum chemical calculation. In this article we report a new RRKM analysis of photoisomerization rates in tS-do, tS-d,, tS-d,,, and tS-d,, in which all the necessary vibrational frequencies, of both the initial and the transition state, are computed by a quantum (19) Troe, J. Chem. Phys. Lett. 1985, 114, 241. (20) Felker, P. M.; Zewail, A. H. J . Phys. Chem. 1985, 89, 5402. (21) Schroeder, J.; Troe, J. J . Phys. Chem. 1986, 90,4215. (22) Courtney, S. H.; Balk, M. W.; Philips, L. A.; Webb, S. P.; Yang, D.; Levy, D. H.; Fleming, G. R. J . Chem. Phys. 1988, 89, 6697. (23) Nordholm, S. Chem. Phys. 1989, 137, 109.
0022-3654/91/2095-0748$02.50/00 199 1 American Chemical Society
Photoisomerization Rates of Stilbene
The Journal of Physical Chemistry. Vol. 95, No. 2, 1991 149
chemical method based on the QCFF/PI This Hamiltonian provided a satisfactory assignment of vibrational frequencies in excited states of several aromatic molecules and in particular in the SIstate of tSZ6 In this way, not only the molecular parameters of tS-do but also the effects of partial and total deuteration are computed directly without the need of further arbitrary assumptions. Since two alternative hypotheses have been proposed for the mechanism of photoisomerization, namely an adiabatic and a nonadiabatic one, we performed two sets of calculations. The aim of this analysis is to ascertain whether the photoisomerization rate parameters k(B) of tS-do and of its isotopically substituted derivatives tS-dz, tS-dlo, and tS-d12can be reproduced by an RRKM treatment based on unbiased molecular parameters and whether these rates are better described by the adiabatic or the nonadiabatic mechanism.
Theoretical Background Vibrational Frequencies. Optimized geometries and vibrational frequencies are computed by the semiempirical QCFF/PI Hami l t ~ n i a nwhich ~ ~ . ~was ~ proved to be very effective in describing these properties in aromatic and conjugated compounds.2630 Ab initio methods, although in principle more satisfactory, are not capable at present of obtaining accurate force fields in excited states of large aromatic molecules such as stilbene. We employ an upgraded version of the original QCFF/PI program. The details of such upgrading have been described elsewhere30 and are here briefly summarized: (i) the energy gradient and Hessian are computed by taking into account the off-diagonal elements of the Fock matrix expressed in the molecular orbital (MO) basis; (ii) doubly excited configurations (DECs) are included in the configuration interaction (CI) treatment beside the usual singly excited configurations (SECs); (iii) force fields are computed by numerical differentiation of analytical gradient rather than by the analytical procedure adopted in the original program. Although still approximate, this procedure has the advantage of taking into account the contribution of electron density change to the energy gradient and to the Hessian and, in fact, was very successful in reproducing the SIfrequencies of tSZ6and the SIand Ti frequencies of polyenes.30 The CI matrix considered in these calculations includes 200 energy selected SECs and DECs built from the five highest occupied MOs and the five lowest virtual MOs. Vibrational frequencies are computed at the optimized SItrans geometry and at the optimized transition states. Transition States. According to most theoretical calculat i o n ~ , ' ~the - ' ~small barrier for the t r a n s 4 photoisomerization of stilbene is the result of the crossing between the potential energy curve of the state 1 B, labeled SIat the trans geometry, and the potential energy curve of the state 2A, which is higher than SI at the trans geometry and is labeled S2. The minimum of the latter curve is located at the twisted configuration (0 = 90') and arises from the avoided crossing with the 1A (So) energy curve. A number of theoretical and experimental studies have focused on the role and on the nature of the 2A electronic state that is involved in the crossing with lB.31-34 The 1B-2A energy curves crossing (24) Warshel, A.; Karplus, M. J . Am. Chem. Soc. 1972, 94, 5612. (25) Warshel, A.; Levitt, M. QCPE No. 247; Indiana University, 1974. (26) Negri, F.; Orlandi, G.; Zerbetto, F. J . Phys. Chem. 1989,93, 5124. (27) Warshel, A. In Modern Theoretical Chemistry; Segal, G . A,, Ed.; Plenum Press: New York, 1977; Vol. 7, Part A, p 133. (28) Lasaga, A. C.; Aerni, R. J.; Karplus, M. J . Chem. Phys. 1980, 73, 5230. Hemley, R. J.; Brooks, B. R.; Karplus, M. J . Chem. Phys. 1986,85, 6550. (29) Orlandi, G.; Zerbetto, F. Chem. Phys. Lerr. 1985, 120, 140. Negri, F.; Orlandi, G. J . Phys. Chem. 1989, 93. 4410. (30) Zerbetto, F.; Zgierski, M. 2.; Negri, F.;Orlandi, G. J . Chem. Phys. 1988, 89. 3681. Negri, F.; Orlandi, G.; Brouwer, A. M.; Langkilde, F. W.; Wilbrandt, R. J . Chem. Phys. 1989, 90, 5944. Negri, F.; Orlandi, G.; Zerbetto, F.; Zgierski, M. Z. J . Chem. Phys. 1989, 91, 6215. (31) Hohlneicher, G.; Dick, B. J . Phorochem. 1984, 27, 215. (32) Olbrich, G. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 209. (33) Fuke, K.; Sakamoto, S.;Veda, M.; Itoh, M. Chem. Phys. Lerr. 1980, 74, 546. (34) Stachelek, T. M.; Pazoha, T. A.; McClain, W. M.; Drucker, R. P. J . Chem. Phys. 1977, 66,4540.
>
g W
t rsns
e
cis
Figure 1. Qualitative potential energy curves of the ground So ( I A ) and the excited SI(IB)and S2(2A) electronic states as a function of the ethylene torsional coordinate 8. The curve crossings are shown as avoided (-)
and as allowed (- - -). The barrier corresponds to A ts.
>
F C
Y
trans
e
ci s
Figure 2. Qualitative potential energy curves of So (1A) and S,(IB) electronic states along 8, according to ref 35. The barrier corresponds to B ts.
is allowed when the molecule retains the C, symmetry axis but becomes avoided when the molecule is distorted along non-totally symmetric normal coordinates of b type. A qualitative picture
750 The Journal of Physical Chemistry, Vol. 95, No. 2, 1991
of the So, SI,and Sz potential energy curves is shown in Figure 1.
In a recent study based on MNDO-CI calculations35 relative minima on the 1B energy curve were found a t the trans and the twisted geometries that are separated by a barrier that does not result from a crossing between energy curves. Qualitative So and SI potential energy curves according to the calculation of ref 35 are sketched in Figure 2. Since the MNDO Hamiltonian, like all NDO Hamiltonians, tends to underestimate the energy of twisted configurations (see, for example, ref 36), this result should be taken with caution as it may reflect merely the bias of the method. In spite of this, for completeness, we shall take into account also the possibility that photoisomerization of stilbene takes place entirely on the 1B energy curve. On the basis of the foregoing we consider the two following alternatives: (a) The transition state lies at the crossing between the 1B and 2A energy curves and is labeled A transition state (A ts). It is determined by calculating the potential energy curve of 1B along 8, using the QCFF/PI Hamiltonian and relaxing all the other coordinates. As the molecule approaches the 1B-2A curve intersection, it loses its symmetry properties and the crossing becomes avoided (see Figure 1). The maximum of the S,curve formed at the crossing is taken as the A ts. If the 1B-2A crossing is strongly avoided, the photoisomerization process is adiabatic, and it can be handled by the standard RRKM theory. If it is weakly avoided the process is nonadiabatic: in this case the rate coefficients can be evaluated by the modified RRKM expression given by Felker and ZewaiLZ0 (b) The transition state corresponds to the barrier located on the 1B energy curve, somewhere between the two minima found at the trans and the twisted geometry, as suggested in ref 35. This transition state is labeled B (B ts). This geometry is determined as the maximum of the 1B potential energy curve along 8, upon optimization of all the other coordinates. In order to neglect the state 2A, the QCFF/PI calculations were performed with the only inclusion of SECs in the CI treatment. The barrier on 1 B is found very close to the twisted geometry, at the angle 8 = 86.7', which is shifted by some 25O with respect to the geometry found by the MNDO/CI c a l c ~ l a t i o n .Since ~ ~ the nature of the B state is not very sensitive to changes of the angle 8, we believe that the force field and the vibrational frequencies computed at the B ts defined above are reliable. The RRKM calculations employing these frequencies yield an alternative description of the photoisomerization rates in the context of the adiabatic mechanism. RRKM Calculation of Rate Coefficients. The k ( E ) rate coefficient of stilbene photoisomerization for the adiabatic mechanism is computed by the usual RRKM expression3'
k ( E ) = SE-"p+(Ev+) dE,+/hp(E) = W(E- E O ) / [ h p ( E ) ] 0
(1)
where Eo is the height of the energy barrier, E is the excess vibrational energy, p ( E ) is the density of vibrational states in the reactant state at the energy E, p+(Ev+)is the density of vibrational states in the transition state at the excess energy Ev+,N+(E - Eo) is the number of vibrational states of the transition state with an energy less or equal to E - Eo,and h is Planck's constant. The k ( E ) rate coefficient for the nonadiabatic mechanism is evaluated according to the modified RRKM expressionz0 k ( E ) = ~ E - E o ( -l P)p+(Ev+)dE,+/hp(E)
(2)
where p+, EV+,p , and E have the same meaning as in eq 1 and P is given by
(35) Troe, J.; Weitzcl, K. M. J . Chcm. Phys. 1988, 88, 7030. (36) Weitzcl, K. M.;BHssler, H. J. Chem. Phys. 1986, 84, 1590. (37) Marcus, R. A. J . Chcm. Phys. 1952, 20, 359.
Negri and Orlandi with = Az/IFI - Fz1 = ~ ~ / I -FFz1 I
where y is the nonadiabatic parameter, Vis the coupling between the diabatic states 1B and 2A, A = 2V is the energy gap between the resulting adiabatic curves a t the avoided crossing, F Iand Fz are the slopes of diabatic energy curves at the same geometry, and p is the reduced mass for the reaction (Le., torsional) coordinate. As pointed out above, the vibrational frequencies of the initial and the transition state, necessary to evaluate eqs 1 and 2 for the four isotopic species, are obtained by the QCFF/PI Hamiltonian.24,25The number and the density of states are computed by the Bunker-Hase program,38 which employs a direct-count procedure with 18 frequency groupings. The Eo barrier is taken as a fitting parameter. Only one barrier height parameter suffices for all the four isotopic species, since the zero-point energy difference between reactant and transition state for each isotopic species is obtained from the computed vibrational frequencies. The barrier height for the isotopic species x (x = tS-d,, tS-d,,, tS-d12) can be written as the sum Eo(X) = EO(d0)
+ EYX)
(4)
E'(x) being the correction to the barrier due to the zero-point
energy difference between the species x and the tS-do. In the evaluation of the density of states the following approximations are introduced: the rotations are neglected since the moments of inertia of the reactant and of the transition state are likely to be very similar; furthermore the anharmonicity effects are ignored, although they may be important especially for the reaction coordinate and at the A ts,which corresponds to a weakly avoided crossing. The y parameter appearing in eq 3 is treated as an empirical parameter to be determined by minimizing the difference between the computed and the observed k(E). However, to check whether the value obtained empirically is physically acceptable, we evaluate independently the y parameters by computing the 1B-2A vibronic couplings. Calculation of the Nonadiabatic Parameter y. The nonadiabatic parameter y, defined in eq 3, can be estimated from quantum chemical calculations of the energy curve slopes F,,Fz and of vibronic coupling Vbetween the states 1 B and 2A. For IF,- Fzl we adopt the estimate of Felker and ZewailMof 15 000 cm-I/rad, based on quantum mechanical calculations; this estimate is identical for the four isotopic species considered. The V coupling is the vectorial sum of the adiabatic vibronic couplings between 1B and 2A induced by all the b vibrations. The coupling V, due to ith b vibration can be written as
V, = ( ~ V ' " ~ ( * ( ~ B ) I ( ~ H / ~ Q ~ ) O )QilAw'") I*(~A) (5) where Qi is one of the b-type coupling modes, AUU)is the total vibrational wave function of the jth electronic state, v and w represent the two sets of vibrational quantum numbers which differ by 1 quantum in the inducing mode Qi, and q(1B) and q(2A) are the electronic wave functions of the states 1B and 2A, respectively. This integral can be factored as a product of a vibrational and an electronic factor through the Condon approximation:
Vi = ( * ( ~ B ) I ( ~ H / ~ Q ~ ) o ) * (Av'1)lQilAw'2)) (~A))
(6)
The main contributions to the vibrational integral in eq 6 are due to the inducing mode and to the totally symmetric modes. Keeping only these contributions and assuming that most of the modes are in their ground states for relatively small (
