Quantum energy flow during molecular isomerization

12 December 1997

Chemical Physics Letters 280 Ž1997. 411–418

Quantum energy flow during molecular isomerization David M. Leitner, Peter G. Wolynes Department of Chemistry, UniÕersity of Illinois, Urbana, IL 61801, USA Received 25 September 1997

Abstract Intramolecular energy flow greatly influences molecular isomerizations, particularly when the energy barrier to reaction is low, as in catalytic and biochemical reactions. We discuss here a simple quantum mechanical theory that describes the extent and rate of vibrational energy flow in molecules, and apply it for the first time to predict rates of isomerization. We consider trans-cis photoisomerization of stilbene, which has been extensively studied experimentally. Vibrational flow in stilbene plays a crucial role in moderating isomerization; the rate both in supersonic jets and low pressure gases is well described by the theory treating quantum flow. q 1997 Elsevier Science B.V.

Since the early part of the century physical chemists have sought to predict the nature and rate of energy flow within a molecule w1,2x. The central motivation for this goal is that energy flow is required to activate molecules and can influence the rate of chemical reactions in gas and condensed phases w3–6x. Still, the task of treating intramolecular energy flow correctly using quantum mechanics has proven formidable. The most commonly adopted unimolecular reaction rate theories, in particular Rice-Ramsperger-Kassel-Marcus ŽRRKM. theory w7x, assume that intramolecular energy flow occurs extremely quickly, thereby justifying use of transition state theory ŽTST. by which the rate can be calculated using equilibrium statistical mechanics w7x. While this assumption appears to be justified a posteriori for a variety of unimolecular fragmentation reactions in gas phase w7x, intramolecular energy flow nevertheless affects the rate when the barrier to

reaction is rather low. Isomerization reactions, in particular, often overcome barriers of only a few kcalrmole, an energy regime where, as we shall see, quantum flow can be slow enough to modify reaction rates of even sizable molecules. Below we demonstrate the influence of intramolecular energy flow on the isomerization of trans-stilbene. Stilbene isomerization has provided an important test for rate theories due to its accessibility to a wide range of photochemical and spectroscopic experiments w8–12x. Numerous studies of stilbene isomerization exist for trans-stilbene and substituted stilbenes as isolated molecules in a supersonic jet, in dilute gases, and in condensed media w8–12x. Isomerization follows photoexcitation to the S 1 excited electronic state, proceeding over a barrier to a 908 twisted form, from which it rapidly Ž- 1 ps. decays to the ground state with about a 50% chance to cis ŽFig. 1..

0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 1 2 0 - 2

D.M. Leitner, P.G. Wolynesr Chemical Physics Letters 280 (1997) 411–418

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Fig. 1. Schematic plot of potential energy surfaces of stilbene. Isomerization from the trans configuration follows photoexcitation to the S1 excited electronic state. The rate to escape to the twisted configuration is k iso . Following rapid conversion Ž -1 ps. to S 0 , stilbene relaxes with about a 50% chance to cis.

The isomerization rate coefficient is often expressed as a correction to the microcanonical transition state theory Žor RRKM. estimate, k TST , by introducing the transmission coefficient, k , whereby the energy dependent reaction rate is k Ž E . s k Ž E . k TST Ž E . .

Ž 1.

The transmission coefficient can take into account effects such as tunneling and the nonadiabaticity of curve crossing when the reaction involves two weakly coupled electronic potential surfaces. It also accounts for the so-called ‘‘recrossing’’ effects which can arise from sluggish energy redistribution. If energy redistribution is slow the reaction process can be visualized as a random walk through vibrational state space to the transition state w13x. For adiabatic reactions, i.e., those proceeding on a single Born-Oppenheimer potential energy surface, several derivations w13–15x lead to

k Ž E. s

k IVR Ž E . k IVR Ž E . q n R Ž E .

.

Ž 2.

Here k IVR is the intramolecular vibrational redistribution ŽIVR. rate, and n R is the intrinsic reaction rate when the reactant is poised in the transition region, and should be taken as the reaction coordinate vibrational frequency. The usual transition state rate is k TST Ž E . s n RŽ E . P ) Ž E ., where P ) is the microcanonical probability at energy E that the reactant is in the transition region ŽFig. 2.. From Eq. Ž2. it is clear that it is the relation between k IVR and n R that determines the extent to which energy flow

influences the reaction rate, rather than, as may be thought, the relation between k IVR and k TST . Since n R ) k TST , k IVR can affect the rate even when k IVR ) k TST and the reaction course is simple exponential. Finding the transmission coefficient has been impeded by the difficulty of calculating k IVR for even an isolated polyatomic molecule. When the molecule is in contact with others, k IVR also contains contributions both from the quantum flow rate of the q isolated reactant, k IVR , and from collisions of the reactant with its environment. The standard and simplest procedure to treat collisions assumes that they result in microcanonical equilibrium at each energy, so that the vibrational flow rate in Eq. Ž2. is k IVR s q k IVR q nc , where nc is the collision rate. In large molecules with low isomerization barriers, few vibrational modes need be excited at energies sufficient to allow reaction. Quantum effects on intramolecular energy flow are thus especially important. Detailed quantum mechanical computations of energy flow, however, remain extremely challenging despite the enormous progress made in recent years w16x. Often classical mechanics, which has the practical advantage of relatively straightforward implementation for larger molecules, is adopted to treat energy flow, but this is inappropriate if the molecule is cold. In trans-stilbene, only a few of the 72 modes are excited near the barrier energy, and it is thus a quantum mechanical object. Recent classical trajectory simulations of stilbene isomerization, while at the same time strongly suggesting the importance of slow and restricted vibrational redistribution in the isomerization, also demonstrate the difficulty of extracting realistic flow rates from a classical model of a system so near its zero-point energy w17x. We have recently developed a statistical prescription for determining rates of quantum energy flow in a molecule w18x. Much like the RRKM theory, this theory requires knowing only some coarse-grained information about a molecule’s energy levels. We refer to the theory as Local Random Matrix Theory ŽLRMT., since it actually describes the extent and rate of vibrational flow for an ensemble of systems, each a very close description of a particular one, for example trans-stilbene. What distinguishes LRMT from random matrix theories applied earlier to interpret complex spectra w19,20x is that LRMT embodies the local nature of flow in the vibrational space of a

D.M. Leitner, P.G. Wolynesr Chemical Physics Letters 280 (1997) 411–418

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Fig. 2. Reaction dynamics of a polyatomic molecule in vibrational quantum number space. The state of the system at any time is specified in a basis indicating the number of quanta in each vibrational mode of the molecule, Ž n R ,n1 ,n 2 ,... .; n R is the number of quanta in the reactive mode, n1 the number of quanta in the first bath mode, etc. Motion in an isolated molecule is confined to an energy shell of total energy E. States lie on a lattice in quantum number space; for simplicity only states on Žpurple. and some below Žblack. the energy shell are shown. When enough quanta, n C , occupy the reactive mode, isomerization can occur. The microcanonical transition state theory, or RRKM, rate is given by n R , the reactive mode frequency, times the equilibrium fraction of states on the energy shell with sufficient excitation in the reactive mode. The RRKM theory assumes transport to the activated mode is fast. At low excitation energies, flow is slow and a typical reactive trajectory resembles a quantum mechanical random walk among states on the energy shell; one such path is illustrated. According to local random matrix theory ŽLRMT., the residence time in a given state depends on its anharmonic coupling VQ to states with Q different excitations, and the density of states to hop to, rQ ; VQ and rQ for Q s 2 are indicated, but other couplings exist and are accounted for. If the couplings are small the overall reaction rate depends on the time the molecule takes to find the transition states Žcubes., rather than the intrinsic rate n R .

molecule ŽFig. 2.. Molecular spectroscopists have long appreciated that vibrational redistribution occurs via pathways typically involving the exchange of only few quanta at a time, thereby giving rise to a picture of local flow in the vibrational quantum number space w21x. LRMT introduces selection rules for energy transfer in the vibrational space yielding the sequential structure for vibrational flow deduced by spectroscopists. Vibrational flow rates fluctuate

from state to state, but the average rate predicted by LRMT reads q k IVR s

2p "

Ý ²< VQ < 2 :rQ Ž E . ,

Ž 3.

Q

where Q indicates a distance in vibrational quantum number space; ²< VQ < 2 : is the mean square coupling to states a distance Q away; rQ Ž E . is the density of

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D.M. Leitner, P.G. Wolynesr Chemical Physics Letters 280 (1997) 411–418

such states; and " is Planck’s constant over 2p . The couplings arise from the anharmonicity of the molecule and are analogous to the terms that determine thermal conductivity in solids. rQ Ž E . depends in general on VQ , reflecting a broadening of the distribution of levels at Q due to coupling. Determining precise values of the parameters VQ and rQ requires a detailed quantum chemical analysis of the energy surface of the reactant molecule. Nevertheless, with only modest information about the vibrational frequencies of the reactant, we can reasonably estimate rQ w18x. A recent breakthrough by Gruebele and coworkers w16,22x also allows us to estimate VQ from the spectroscopic data, as discussed below. Using Eq. Ž3., we recently calculated flow rates for several organic molecules w18x, giving results that compare well with those found in very large scale computational studies of the quantum dynamics. At sufficiently low vibrational energy, relaxation gives way to localization of vibrational excitation, i.e., a near-separable normal or normalrlocal mode picture obtains. No irreversible energy flow occurs. At the point where Eq. Ž3. is no longer valid, LRMT predicts a sharp but continuous localization transition in large molecules w18,23x. The location of the transition, like flow rates, depends on the local quantities VQ and rQ . Below this transition LRMT predicts the extent of vibrational redistribution in a now only limited portion of the energetically accessible vibrational space w18x. Here we apply LRMT to predict isomerization rates of trans-stilbene. A difficulty with theoretical analyses of any photochemical process originates from uncertainties in the potential energy surfaces of the excited states. Several potential surfaces for stilbene isomerization have been proposed w17,24,25x, each leading via an RRKM analysis to somewhat different isomerization rates. Nevertheless, reasonable semiquantitative predictions emerge from a number of them. In particular, the semiempirical potential surface of Negri and Orlandi w25x highlights the importance of nonadiabatic effects on the rate, as proposed earlier by Felker and Zewail w26x. Early theoretical estimates for isomerization rates of isolated stilbene found a discrepancy of nearly an order of magnitude between RRKM and the lower experimental rates w27x. Subsequent reassessments of possible reaction barriers, reaction coordinate frequencies,

and revised potential surfaces have reduced this difference some w17x. Nevertheless, RRKM theory predicts a saturation with increasing pressure of the rate for stilbene isomerization in the gas phase in a pressure regime where collision rates are higher than isomerization rates, whereas the experimental rate under these conditions continues to rise w12,28–30x. This strongly suggests the influence of limited internal energy flow, as pointed out earlier w12,15,29x. Quantitative discrepancies between RRKM theory and the observed rates in isolated molecules can be substantially reduced by introducing nonadiabatic corrections to the RRKM calculation, which arise due to the weak electronic coupling between S 1 and S 2 surfaces w25x. However, when the nonadiabatic correction dominates, the transmission coefficient cannot increase with pressure except when there are resonances tuned in by collisions w6x. In the absence of slow intramolecular energy flow, barring such special resonances, friction conspiring with nonadiabatic effects cannot explain the observed increase in isomerization rates with increasing pressure. Nordholm w15x argued that the isomerization rate data can be explained if limited vibrational flow is accounted for. He pointed out that separate measurements of vibrational redistribution rates by Zewail and coworkers w31x would strongly suggest that vibrational flow could influence the isomerization. From the available data, Nordholm proposed a simple empirical formula for flow rates that explained some trends in the experimental kinetics. LRMT predicts rather different vibrational flow rates than the empirical formula at energies outside the bounds of the then available data, but all evidence suggests that a consistent picture appears only when vibrational flow rates are explicitly calculated in predicting isomerization rates. We have used in our calculations the semiempirical potential energy surface of Negri and Orlandi w25x, which is a development of an earlier surface calculated by Warshel w24x. The Negri-Orlandi surface provides frequencies for the reactant and activated complex, and suggestive parameters for nonadiabatic corrections. Anharmonicities coupling the vibrational modes and giving rise to flow are not provided by their calculation. We estimate the coupling terms by adopting the potential energy factorization approach of Gruebele and coworkers w16,22x.

D.M. Leitner, P.G. Wolynesr Chemical Physics Letters 280 (1997) 411–418

Assuming that terms higher than second order in the potential are factorizable, as demonstrated by Madsen et al. w22x, we calculate cubic anharmonic terms as the geometric mean of the anharmonic constants deduced for each of up to three modes that are coupled. To estimate each of these constants, we assume that the anharmonicity scales with mode frequency in stilbene as it does in other organic molecules. We have scaled the anharmonic constants with the frequencies as was done for propyne, where the factorization scheme was found to lead to spectra agreeing closely with experiment w16x. The factorization approach provides cubic anharmonic terms ranging from order 0.1 cmy1 to 10 cmy1 , which is very reasonable. Gruebele and coworkers w16,22x have also argued that higher order coupling terms, which give rise to VQ , Q ) 3, decrease exponentially with order in a way that is nearly universal among organic molecules, allowing us to estimate these terms for stilbene. Also required to calculate VQ is the number of quanta in each of the 72 modes, which is obtained from the same direct counting procedure w7x as is used to calculate the RRKM rate coefficient, k TST Ž E .. The density of states rQ is also determined as in previous applications of LRMT w18x, taking into account here the C2 h symmetry of trans-stilbene w25x. The results of our LRMT calculations of quantum

Fig. 3. Energy dependence of the quantum vibrational flow rate in stilbene. LRMT predictions are given by the curve; the dashed line shows how the vibrational flow rate would vary if it were to depend on the total density of states of stilbene, rather than the local density as is actually the case. Results of direct experimental measurements for IVR by Zewail and coworkers Žfilled circles. w11,31x, and Jean and coworkers Žopen circle. w32x are also shown.

415

Fig. 4. Predictions and experimental results for the rate of stilbene photoisomerization. The uppermost, dashed curve is the RRKM prediction. The middle curve Žlong dashes. gives the prediction for the isomerization rate including effects of finite quantum energy flow in stilbene. The lowest curve includes also a nonadiabatic correction. Points are the experimental results of Zewail and coworkers w11x. The jump in predicted rates at about 3100 cmy1 is due to 12 modes with about this frequency. At higher energies, predicted rates parallel those of the nonadiabatic calculations of Negri and Orlandi w25x, which do not account for energy flow.

vibrational flow rates in trans-stilbene are summarized in Fig. 3. The transition to vibrational flow is predicted to lie near 1250 cmy1 , close to the range 1170–1230 cmy1 where Felker et al. w31x observed the onset of extensive redistribution. Other available experimentally determined vibrational flow rates w11,32x are also seen to be in agreement with our estimates. Eq. Ž3. indicates that the flow rate varies with the local density of states, rather than the total reactant state density, as has been argued to be the case in molecules for some time w23,33x. Part of the validity of RRKM theory rests on the assumption that flow becomes extremely fast with increasing energy. The high total state density of most polyatomics is invoked as justification for this assumpq tion. The rise in k IVR with energy is clearly far slower than expected based on the total density of states of stilbene, as Fig. 3 indicates. Fig. 4 shows the isomerization rate of stilbene predicted using k TST and k of Eq. Ž1.. We have taken the barrier height to be E0 s 1100 cmy1 , which is consistent with all available experimental data. Isomerization rates of isolated stilbene determined by Hochstrasser, Zewail, Fleming, Levy and coworkers w9,11,30x were observed to be finite in all experiments probing energies above 1100 cmy1 . This

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D.M. Leitner, P.G. Wolynesr Chemical Physics Letters 280 (1997) 411–418

barrier energy is also consistent with activation energies for stilbene in various solvents such as ethane w28x, which provides an estimate for E0 in the range 975–1255 cmy1 when accounting for differences in the vibrational energy content of the reactant and transition state complex. Calculation of k using Eq. Ž2. also requires the barrier-crossing frequency. We use n R s 284 cmy1 , and include a modest correction to account for increased energy of translation in the reactive mode with greater molecular energy. The precise value of n R is in fact not so important when q k IVR is small, since variation of k with n R is largely offset by variation of k TST with n R . Our choice for n R is close to values suggested in Refs. w17,25,26x. We observe in Fig. 4 that predicted rates corrected for finite k IVR are more accurate than RRKM estimates. Nonadiabatic effects can also play a role in moderating the isomerization rate. Predicted and experimental rates of isolated trans-stilbene isomerization can be brought into agreement w25,26x by introducing reasonable nonadiabatic parameters into the Landau-Zener estimate PLZ for the diabatic curvecrossing probability, where the adiabatic limit is reached when PLZ s 1. We can also account for nonadiabatic effects in terms of P LZ . Straub and Berne w34x derived a simple expression for the transmission coefficient of Eq. Ž1. that contains both nonadiabatic and adiabatic corrections, ky1 s ky1 ad q Ž1 y PLZ .r2 P LZ , where P LZ is the average curvecrossing probability assuming all curve crossings are independent. If k IVR is very small, the adiabatic correction due to slow vibrational relaxation clearly dominates, whereas nonadiabatic effects dominate with faster flow rates. Using the nonadiabatic parameters of Ref. w25x, we observe in Fig. 4 that effects of nonadiabaticity become more important above about 2000 cmy1 . While nonadiabatic effects alone cannot easily explain the observed rise in isomerization rate with pressure, a rate theory accounting for slow vibrational flow does explain this trend. We have computed the thermal isomerization rate, k ŽT ., at collision frequencies corresponding to the experimentally studied pressure range of 1 to around 100 atm for stilbene in methane vapor. Results for both adiabatic and nonadiabatic rates are plotted in Fig. 5, where for the latter we again use nonadiabatic parameters

Fig. 5. Variation of isomerization rates with pressure for stilbene in methane buffer gas. Grey curves correspond to an assumed collision rate of 0.025 psy1 ratm, and black curves to a collision rate of 0.2 psy1 ratm Žsee text.. The solid curves account for both nonadiabatic effects and quantum energy flow, and the dashed curves only for the latter. Points are experimental results of Fleming and coworkers, for which possible errors are also indicated w12x.

suggested in Ref. w25x. The grey curves correspond to the hard-sphere collision rate of 0.025 psy1 ratm estimated in Refs. w28–30x. We see there only little rise in the rates predicted below about 10 atm, then a clearer rise until saturation at a value near 7 nsy1 in the nonadiabatic calculation and about 15 nsy1 in the adiabatic. Also shown are the measured rates w12x. The observed rise in isomerization rate is more rapid at lower pressures than predicted, but is similar at higher pressures. The experimental isomerization rate saturates about a factor of 4 higher than predicted by our nonadiabatic calculation, and about a factor of 2 higher than by the adiabatic one. Possible measurement errors w12x could reduce this discrepancy somewhat, as could uncertainties in the parameters of the RRKM calculation. The most apparent difference between the experimental data and our calculations is the predicted slow rise in rate at lower pressures. Negri and Orlandi w25x argued for a larger collision diameter than used in the original hard-sphere estimate for the collision rate, which would increase the effective collision rate, which varies as the square of the collision diameter, and therefore k IVR . We illustrate this possibility with the black curves, where a rate of 0.2 psy1 ratm is assumed. The experimental data are now quite well described accounting for both k IVR and nonadiabatic effects, at least semiquantitatively.

D.M. Leitner, P.G. Wolynesr Chemical Physics Letters 280 (1997) 411–418

Better quantitative calculations await more accurate quantum chemical calculations of the energy surface of stilbene in isolation and stilbene interacting with its environment. While general trends in stilbene isomerization rates are explained satisfactorily accounting for quantum flow, there are possible effects not addressed by the present theory. Troe and coworkers w35x have argued that the higher rates at higher pressures are due to a direct reduction of the reaction barrier with pressure. This effect, however, should be small for stilbene in methane over the pressure range in which the isomerization rate rises an order of magnitude w12x. In any event an RRKM calculation corrected for intramolecular quantum flow could easily accomodate such an effect. Balk and Fleming proposed a plausible explanation for the low pressure data, invoking the possibility that the torsional motion of stilbene that leads to isomerization could have a disproportionately high probability of being directly excited by collisions with methane w29x. Such an effect is excluded within the standard RRKM formalism. The effect of such mode specificity would be similar to that of increasing the collision rate at low pressure, which we have seen can account for the variation of the isomerization rate observed experimentally w12x. The ongoing study of stilbene photoisomerization under many conditions has revealed it to be both rich and evidently quite complicated. While a variety of effects possibly particular to stilbene await explanation, especially in the liquid phase experiments, explicit treatment of quantum energy flow described here overcomes the difficulties thus far encountered in explaining experimental rates in isolated molecules and gases. LRMT predictions for vibrational flow rates in stilbene compare well with those directly measured in the laboratory, which themselves suggest the importance of intramolecular energy flow during stilbene isomerization. Further measurements of k IVR in stilbene would of course improve our understanding of its role in the reaction. We expect the picture developed here for stilbene, in which the reaction can be described starting from a Kassel-like picture where vibrations are nearly good quantum numbers, and the rate depends on the transitions between these states, pertains to a great many rearrangements of relatively cold reactants.

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Note added: Since we completed this manuscript, we became aware of recent calculations by Gershinsky and Pollak wJ. Chem. Phys. 107 Ž1997. 812x of t-stilbene isomerization rates in gas and liquid phases. These authors adopt the potential surface constructed by Vachev et al. wJ. Phys. Chem. 99 Ž1995. 5247x. Using this surface, they calculate RRKM values for the rate that agree reasonably well with rates measured in the jet experiments, in contrast to other calculations using both older w24,25x and newer w17x stilbene surfaces. They attribute this to the much higher number of low frequency modes contained in the Vachev et al. surface compared to the others. Bolton and Nordholm, however, who specifically aimed at constructing a more realistic surface than Vachev et al.’s, found RRKM rates using their own surface to be quite close to calculations using Negri and Orlandi’s w17x. Normal mode frequencies of the Bolton-Nordholm surface compare well with experiment, as their analysis reveals w17x. The number of low frequency modes of their surface is consistent with the number for Warshel’s w24x and Negri and Orlandi’s w25x, but is far lower than that for Vachev et al.’s. Our own calculation of the ergodicity transition using the surface of Vachev et al. yields a threshold of f 670 cmy1 , much below the observed transition at f 1200 cmy1 . With the Vachev et al. surface the energy flow rate predicted by our theory would be sufficient to allow the RRKM assumption to be valid. On the basis of these considerations, we believe that the consistency of Gershinsky and Pollak’s results with RRKM and rapid energy flow depends on features of their potential surface less reliable than those of other available surfaces.

Acknowledgements The authors thank Professors M. Gruebele, R.M. Hochstrasser, J.M. Troe and A.H. Zewail for helpful discussions and critical reading of the manuscript. DML is grateful for the hospitality of the Department of Physics, Bilkent University, Ankara, Turkey, where part of this work was done. This work was supported by NSF Grant CHE 95-30680.

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