Transition State Dynamics and Relaxation Processes in Solutions: A

13034

J. Phys. Chem. 1996, 100, 13034-13049

Transition State Dynamics and Relaxation Processes in Solutions: A Frontier of Physical Chemistry Gregory A. Voth* and Robin M. Hochstrasser* Department of Chemistry, UniVersity of PennsylVania, Philadelphia, PennsylVania 19104 ReceiVed: January 26, 1996; In Final Form: April 23, 1996X

The theoretical framework for describing solvent effects on solution phase reactions is summarized. When possible, the results are related to experiments involving geometric isomerization, to photoinduced reactive processes that are initiated at the transition state, and to the bimolecular reactions between molecules that are at van der Waals separations in liquids. Key differences between solution and gas phase transition state dynamics are pointed out, and the relaxation processes that determine wavepacket propagation in the solution phase are discussed. Quantum activated rate processes are described theoretically, and some results on proton transfer are given.

I. Introduction Much of conventional chemistry deals with reactions that take place in the presence of solvents. Often the solvent is a principal player in determining the rate or the outcome of a reaction. On the other hand, the most fundamental work on reaction dynamics has dealt with isolated molecules, often with reactants prepared in specific quantum states and traveling with well-defined velocities. Years of experimental and theoretical research on reaction dynamics of isolated systems has resulted in a reasonable, predictive understanding of the features determining the rates and the products of chemical reactions involving reactants with relatively few atoms. However, it appears unlikely that the detailed characteristics of chemical reactions in condensed phases will be obtainable from the same basic theoretical and experimental procedures that proved so successful for small, isolated systems. Complex chemistry, such as is usually found for reaction dynamics in solutions, is unlikely to be understood at this reasonable, predictive level by means of a reductionist approach. Probably a different strategy is needed, one based on quantum or classical statistical properties of the solvent that can deal directly with the issues that prevail in liquids. The interactions between chemically reacting solutes and the solvent make it impossible to examine the chemical dynamics of bimolecular reactions by the traditional methods that were developed and so brilliantly used by the pioneers of gas phase reaction dynamics.1-3 Many of these approaches rely on the collisionless conditions found in low-pressure gases and molecular beams. They allow the measurement of the vibrational, rotational, and translational energy and angular distributions of the products of reactions. They have also enabled the study of reaction products arising from specific quantum states of the reactants. In solutions, a particular distribution of quantum states of a reactant or product that might be prepared in an experiment or by a reactive process generally will evolve into an uninformative thermal distribution on the picosecond time scale. The information regarding the total nascent translational energy is stored as thermal energy in the solution on the time scale of the collisions between molecules in the liquid and could in principle be measured later. Perhaps methods also will be invented to examine the nascent orientational distributions of reaction fragments which are relatable to the torques experienced by the nuclei near the transition state. However, the nascent X

Abstract published in AdVance ACS Abstracts, July 1, 1996.

S0022-3654(96)00317-6 CCC: $12.00

rotational distribution is also thermalized rapidly and also contributes to heating the solution. We will see that the vibrations relax much more slowly, and nonequilibrium states are readily studied in simple experiments. Thus, the sum of the translational and rotational energies, the total energy released in the reaction, and the vibrational energy or state distribution can be regarded as observables of solution phase reactions. Specific distributions of reactant states are subject to the same relaxation effects. Therefore, vibrational states of reactants could be prepared many picoseconds prior to the reactive event and their effects studied, as in the gas phase, but the memory of specific translational and rotational states of reactants would be obliterated within a few collisions. The diffusion of reactants toward one another is so slow compared with these relaxation time scales that the steady state concentration of nonequilibrium states in a conventional solution of reacting solutes is negligible. These factors must be accommodated if bimolecular reactive processes are to be studied in the solution phase. To gain proper understanding of the reactions of even small molecules in solutions, a wide range of structural and dynamical effects need to be characterized. For example, different solvent configurations may lead to different solute structures and hence to inhomogeneities in the ensemble of reactive trajectories. This spreads the distribution of equilibrium geometries in the solution and therefore changes the concept of the transition state to an average, solvent mediated property. For the situation in which the solute vibrational motions occur on comparable time scales with those of coupled solvent modes, a multidimensional reaction surface that explicitly incorporates a collective solvent mode or modes must be considered. Other systems might be in a limit in which the solvent can easily follow the solute motions or even that the solvent simply senses a vibrationally averaged structure. These are just extreme situations, and a full treatment needs to connect the nonadiabatic limit to that of an equilibrium solvent coordinate. We will see that the manner in which energy is partitioned and by which the synchrony of nuclear motions in the reactant state (coherence) is transferred into the products of a reaction will depend on these conditions. Further efforts along these lines will bring the study of solvated chemical reactions to the level of requiring accurate calculations of the solute-solvent intermolecular forces. These forces are typically treated in terms of average properties. However, impulsive reactions may produce specific states of the solvent shell. Experiments are needed to determine accurate, solvated © 1996 American Chemical Society

Dynamics and Relaxation Processes in Solutions potential energy surfaces of small molecules in solution. Finally, knowledge of the relaxation parameters and reorientational dynamics and the effect of solvent friction on them is needed to understand the relaxation of the vibrational coherence and populations of vibrational states. Although it is instructive to compare condensed phase experiments with those of isolated molecules, it is clear that the way in which solvated reactions are to be discussed and understood still requires a formidable effort from both experimentalists and theorists along new directions. The effects of solvent on chemical reaction rates have been extensively studied theoretically over the past few decades. The central framework for our current level of understanding of these effects is provided by the transition state theory4,5 (TST) developed in the 1930s, the Kramers theory6 of 1940, the Grote-Hynes7 and related theories8 of the 1980s and 1990s, and the Yamamoto reactive flux correlation function formalism9 as extended and further developed by a number of workers.10,11 Each of these theoretical efforts has, in turn, spawned an enormous amount of work in its own right. There are many good reviews of this body of literature, some of which are cited in refs 8 and 12-15. We will organize the various issues involving solvent effects in chemical reactions around some key theoretical and experimental concepts as they stand today. Even more importantly, we will attempt to point out the gaps in our understanding and prediction of these effects. From this discussion, it will become evident that, despite the large body of theoretical work in this field, there are significant questions which remain unanswered as well as a need for greater contact between theory and experiment. In fact, an account of reaction dynamics in solutions that intimately integrates quantum statistical mechanics and experimental measurements is not yet possible because the theory and experiment have barely begun to make significant contact. On one hand, this is a situation that makes the field an exciting new frontier of physical chemistry. But on the other, it forces us to deal with theory and experiment rather separately in this article, finding the few areas of mutual support whenever possible. Thus, we begin with a basic theoretical framework for chemical reactions in liquid phases followed by a selection of experiments that are presented in some detail in order to dramatize the complexities of real systems. These experiments can be related to some of the formulations of reaction dynamics, but frequently they require to be couched in terms of models based on perturbation theories of relaxation which in large part form the language of the experimentalists. We therefore spend some space developing these models. Then we discuss diffusion-controlled reactions, typical of most of the common reactive processes occurring in chemistry. Finally, we survey quantum effects in solution phase dynamics and present two specific examples involving proton and electron transfer. II. Theoretical Framework for Solvent Effects in Liquid Phase Chemical Reactions To begin, we consider a system which is at equilibrium and undergoing a forward and reverse chemical reaction. For simplicity, we will focus on an isomerization reaction, but the discussion also applies to other forms of unimolecular reactions as well as to bimolecular reactions which are not diffusion limited. The equilibrium of the reaction is characterized by the mole fractions xR and xP of reactants and products, respectively, and an equilibrium constant Keq. For gas phase reactions, it is commonplace to introduce the concept of the minimum-energy path along some reaction coordinate, particularly if one is interested in reaction rates for a microcanonical ensemble. In condensed phase chemical dynamics, however, this concept is

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Figure 1. Potential of mean force. A schematic diagram of the potential of mean force along the reaction coordinate for an isomerizing solute in the gas phase (solid line) and in solution (dashed line). Note the modification of the barrier height, the well positions, and the reaction free energy due to the interaction with the solvent.

not as useful. In fact, a search for the minimum-energy path in a liquid phase reaction would lead one to the solid state! Instead, one considers a free energy path along the reaction coordinate q, and the dominant effect of a solvent in a condensed phase reaction is to change the nature of this path (i.e., its barriers and wells). To illustrate this point, the free energy function along the reaction coordinate of an isomerizing molecule in the gas phase is shown by the solid line in Figure 1. In the condensed phase, the free energy function will most likely be modified by the interaction with the solvent, as shown by the dashed line in Figure 1. (It should be noted that, in the spirit of TST, the definition of the optimal reaction coordinate should probably be redefined for the condensed phase reaction, but for now it will taken to be the same coordinate as in the gas phase.) As can be seen from Figure 1, the solvent can modify the barrier height for the reaction, the location of the barrier along q, and the reaction free energy. It may also introduce dynamical effects that are not apparent from the curve, and we note that a classical framework has been implicitly used heresthe generalization to the quantum regime will be discussed later. It is worth contemplating the fact that a free energy can be directly relevant to the rate of a dynamical process such as a chemical reaction. Since the free energy function arises from an ensemble average over configurations and most chemical rate constants are thermally averaged quantities, it is not so surprising that they are interdependent. However, a rigorous treatment of their relationships is an essential step in a theory of reaction dynamics. As it turns out, the free energy curve for a solution phase chemical reaction (cf. Figure 1) can be viewed, in effect, as a natural consequence of Onsager’s linear regression hypothesis as it is applied to condensed phase chemical reactions, along with some additional analysis and simplifications.10 In this spirit, if one imagines a small perturbation of the populations of reactants and products away from their equilibrium values, then the regression hypothesis states that the decay of these populations back to their equilibrium values will follow the same time-dependent behavior as the decay of correlations of spontaneous fluctuations of the reactant and product populations in the equilibrium system. In the condensed phase, it is this powerful principle that connects a macroscopic dynamical quantity such as a kinetic rate constant with microscopic equilibrium quantities such as a free energy function along a reaction pathway. In turn, the effect of the solvent can be largely understood in the equilibrium, or quasiequilibrium, context in terms of the modifications of the free energy curve as shown in Figure 1. As we shall see, the

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Voth and Hochstrasser

remaining solvent effects which are not included in the equilibrium picture may be defined as “dynamical”. The regression hypothesis, stated mathematically for the system just described, is given in the classical limit by

∆NR(t) ∆NR(0)

)

〈δNR(0) δNR(t)〉 〈δNR(0)2〉

(1)

h R(t) - 〈NR〉 is the time-dependent difference where ∆NR(t) ) N between the number of reactant molecules N h R(t) arising from the initial nonequilibrium (perturbed) distribution and the final equilibrium number of the reactants 〈NR〉. On the right-hand side of the equation, δNR(t) ) NR(t) - 〈NR〉 is the instantaneous fluctuation in the number of reactant molecules away from the equilibrium value in the equilibrium ensemble, and the notation 〈...〉 denotes an ensemble average over initial conditions. The solution to the macroscopic kinetic rate equations yields an expression from the left-hand side of eq 1, that is, ∆NR(t) ) ∆NR(0) exp(-t/τrxn), where τrxn-1 is the sum of the forward and reverse rate constants, kf and kr, respectively. The connection with the microscopic dynamics of the reactant molecule comes about on the right-hand side of eq 1. In particular, in the dilute solute limit, the reactant and product states of the isomerizing molecule can be determined by the functions hR[q(t)] ) 1 hP[q(t)] and hP[q(t)], respectively, where hP[q(t)] ≡ h[q* - q(t)] and h(x) is the Heaviside step function. The product state function abruptly switches from a value of 0 to 1 as the reaction coordinate trajectory q(t) passes through the barrier maximum at q* (cf. Figure 1). The important connection between the macroscopic (exponential) rate law and the decay of spontaneous fluctuations in the reactant populations, as specified by the function hR[q(t)] ) 1 - hP[q(t)] and in terms of the microscopic reaction coordinate q, is valid in a “coarse-grained” sense in time, i.e., after a period of molecular scale transients, usually of the order of a few tens of femtoseconds. From the theoretical point of view, the importance of the connection outlined above cannot be minimized for it provides a link between the macroscopic (experimentally observed) kinetic phenomena and the molecular scale dynamics of the reaction coordinate in the equilibrium ensemble. As it turns out, it requires further analysis of the linear regression expression in eq 1 to achieve a useful expression for the rate constant from both the computational and the conceptual point of view. Such an expression was first provided by Yamamoto,9 but others have extended, validated, and expounded upon his analysis in considerable detail.10,11 We follow most closely here the work of Chandler10 in this regard in order to demonstrate the places in which solvent effects can appear in the theory, and hence in the value of the thermal rate constant. The key insight is to differentiate both sides of the linear regression formula and then carefully analyze its likely behavior for systems having a barrier height of at least several times kBT. The resulting expression for the classical forward rate constant in terms of the so-called “reactive flux” time correlation function is given by9-11

kf ) xR-1〈h˙ P[q(0)]hP[q(tpl)]〉 ) xR-1〈q˘ (0)δ[q* - q(0)]hP[q(tpl)]〉

(2)

where xR is the equilibrium mole fraction of the reactant. The classical rate constant is obtained from eq 2 when the correlation function reaches a “plateau” value at the time t ) tpl after the molecular scale transients have ended.10 From the above expression, it is apparent that the classical rate constant can be calculated by averaging over trajectories initiated at the barrier top with the velocity Boltzmann distribution for the reaction

coordinate and an equilibrium distribution in all other degrees of freedom of the system, then weighting those trajectories by their initial velocity, and correlating this initial flux over the barrier with the product state population function hP[q(t)]. The time dependence of the correlation function is then computed until the plateau value is reached, at which point it becomes essentially constant, and the numerical value of the thermal rate constant can be evaluated. It is interesting to note that the time-dependent behavior of the correlation function during the molecular transient time has an important origin.10,11 This behavior is due to trajectories that recross the transition state, and hence it can be can be proven10 that the classical TST approximation to the rate constant is obtained from eq 2 in the t f 0+ limit, i.e.,

kfTST ) xR-1 lim 〈q˘ (0)δ[q* - q(0)]hP[q(t)]〉 1f0+

-1

) xR 〈h[q˘ (0)]q˘ (0)δ[q* - q(0)]〉

(3)

It is, of course, widely appreciated that classical TST provides the central framework for understanding thermal rate constants (see the article by Truhlar et al. in the present issue) and also for quantifying the dominant effects of the solvent in liquid phase chemical reactions (see below). In order to “divide and conquer” the theoretical issue of solvent effects in thermally activated rate dynamics, it is useful to rewrite the exact classical rate constant in eq 2 as8-14

kf ) κkfTST

(4)

where κ is the dynamical correction factor (or “transmission coefficient”) which is given by

κ)

〈q˘ (0)δ[q* - q(0)]hP[q(tpl)]〉 〈h[q˘ (0)]q˘ (0)δ[q* - q(0)]〉

) 〈hP[q(tpl)]〉+ - 〈hP[q(tpl)]〉-

(5)

Here, the symbol 〈...〉( denotes an averaging over the fluxweighted distribution10,11 for positive or negative initial velocity of the reaction coordinate. Therefore, the effect of the solvent in a thermal rate constant can appear both in the value of the TST rate constant and in the value of the dynamical correction factor. These effects will be described separately below. We note that the two quantities are not independent of each other in that they both depend on the choice of the reaction coordinate q. The “variational” choice of q amounts to finding a definition of that coordinate which causes the value of κ to be as close to unity as possible, i.e., to minimize the number of recrossing trajectories. It seems clear that a fruitful area of research for the future will be to theoretically define the “best” reaction coordinate in a liquid phase chemical reactionsone in which the solvent is explicitly taken into account. In charge transfer reactions, for example, a collective solvent polarization coordinate can be treated as being coupled to a solute coordinate (see, e.g., ref 16), but a more detailed and rigorous microscopic treatment of the full solution phase reaction coordinate is clearly desirable for the future (see, e.g., ref 17). Cluster studies may also provide insight into the participation of the first few solvation shells in the reaction dynamics (see, e.g. ref 18). Before describing the effects of solvent on thermal rate constants, it is worthwhile to first reconsider the above analysis in light of current experimental work on condensed phase dynamics and chemical reactions. The formalism outlined above, while exceptionally powerful in that it provides a link between microscopic dynamics and macroscopic chemical kinetics, is intended to help us calculate and analyze only thermal

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rate constants in equilibrium systems. The linear regression hypothesis provides the key line of attack for this problem. To the extent that the thermal rate constant is the quantity of interestsand many times it is the primary quantity of intereststhis theoretical approach would appear to be the best one. However, in many experiments, e.g., nonlinear optical experiments involving intense laser pulses and/or photoinitiated chemical reactions, the system may initially be far from equilibrium and the above theoretical analysis may not be appropriate. Furthermore, quantities such as vibrational or phase relaxation rates which are described later in more detail are often experimentally measured and are only indirectly related to the thermal rate constant. It would therefore appear that more theoretical effort will be required in the future to relate experimental measurements to the particularly microscopic dynamics in the liquid phase which influence the outcome of such measurements and, in turn, to the more “traditional” quantities such as the thermal rate constant. III. Activation Free Energy and Solvent Effects Having separated the “dynamical” from “equilibrium” (or, more accurately, quasiequilibrium) effects, one can readily discover the origin of the activation free energy and define the concept of the potential of mean force by analysis of the expression for the TST rate constant. The latter can be written as10

kfTST )

(2πmβ)-1/2

∫-∞dq e q*

-βVeq(q)

e-βVeq(q*)

(6)

where β ) 1/kBT and Veq(q) is the potential of mean force (PMF) along the reaction coordinate q. The latter quantity is allimportant for quantifying and understanding the effect of the solvent on the value of the thermal rate constant. It is defined as

Veq(q) ) -kBT ln{∫dq′ dx δ(q - q′) exp[-βV(q′,x)]} + constant (7) where x are all coordinates of the condensed phase system other than the reaction coordinate, and V(q,x) is the total potential energy function. The additive constant in eq 7 is irrelevant to the value of the thermal rate constant in eq 6 If the PMF is expanded quadratically around its minimum in the reactant state (cf. Figure 1), i.e., Veq(q) ≈ Veq(q0) + (1/2)mω02(q - q0)2, then for q* . 〈q0〉 eq 6 simplifies to8,10

ω0 -β∆Fcl (8) e 2π where the actiVation free energy of the system is defined as ∆ ) Veq(q*) - Veq(q0). The PMF can be decomposed as F* cl Veq(q) ) V(q) + Weq(q), where V(q) is the intrinsic contribution to the PMF from the solute potential energy function, and therefore, by definition, Weq(q) is the contribution arising from the solute-solvent coupling. From Figure 1, it is evident that the latter coupling is responsible for the solvent-induced change in the activation free energy, the reaction free energy, and the position of the reactant and product wells. Thus, within the context of TST one can conclude that the solvent enters into the picture in a “simple” way through the above-mentioned modifications of the reaction coordinate picture in Figure 1. The changes in the free energy barrier height must be viewed as the dominant solvent effect on the TST rate constant since these changes appear in an exponential. For example, an error in calculating the solvent contribution to the barrier of 1 kcal/ kfTST )

mol (not uncommon!) translates into an error of a factor of 4 in the rate constantsa factor which is often larger than any dynamical and/or quantum effects such as those described below. This is a sobering fact for the theorist, and it is therefore no accident that the accurate calculation of the solvent contribution to the activation free energy has become the major goal of many theoretical and computational chemists. To meet this goal in the future will require four things: (1) increasingly accurate representations of the solute potentials, usually from highly demanding ab initio electronic structure calculations; (2) accurate representations of both the solvent potentials and the solute-solvent couplings; (3) accurate computational methods to compute, with good statistics, the activation free energy in condensed phase systems; and (4) improved theoretical techniques, both analytical and computational, to identify the microscopic origin of the dominant contributions to the activation free energy and the relationship of these effects to experimental parameters such as pressure, temperature, solvent viscosity, polarity, etc. Each of these challenges has generated a significant number of theoretical papers over the past few decadesstoo many in fact to fairly cite them heresand many of these efforts have been major steps forward. There seems to be little dispute, however, that much work remains to be done in all of these areas. Indeed, one of the computational “grand challenges” facing theoretical chemistry over the coming decades will surely be the quantitatiVe prediction (better than a factor of 2) of chemical reaction rates in highly complex systems. This effort is central to the needs of industry and government in, for example, the environmental fate prediction of pollutants. IV. Dynamical Correction and Solvent Effects While the TST estimate of the thermal rate constant is usually a good approximation to the true rate constant and contains most of the dominant solvent effects, the dynamical corrections to the rate can be important as well. In the classical limit, these corrections arise from solvent-generated forces which cause recrossings of the transition state and thereby a violation of the TST assumption. Mathematically, recrossings are responsible for a value of the dynamical correction factor κ in eq 4 which drops below unity. A considerable theoretical effort has been underway over the past 50 years to develop a general theory for the dynamical correction factor (see, e.g., refs 8-15). One approach to the problem is a direct calculation of κ using molecular dynamics simulation and the reactive flux correlation function formalism.10,11,19 This approach obviously requires the numerically exact integration of Newton’s equations for the many-body potential energy surface and a good microscopic model of the condensed phase interactions. Another widely used approach8,12-15 has been to employ a model for the reaction coordinate dynamics around the barrier top, e.g., the generalized Langevin equation (GLE). The latter is given by

dVeq(q) t - ∫0 dt′ η(t-t′;q*) q˘ (t′) + δF(t) (9) dq where m is the effective mass of the reaction coordinate, η(t-t′;q*) is the friction kernel calculated with the reaction coordinate “clamped” at the barrier top, and δF(t) is the fluctuating force from all other degrees of freedom with the reaction coordinate so configured. The friction kernel and force fluctuations are related by the usual fluctuation-dissipation relation mq¨ (t) ) -

η(t;q*) ) β〈δF(0) δF(t)〉q*

(10)

In the limit of a very rapidly fluctuating force, the above equation can be approximated by the simpler Langevin equation

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dVeq(q) - ηˆ (0) q˘ (t) + δF(t) (11) dq where ηˆ (0) is the so-called zero-frequency or “static” friction, ηˆ (0) ) ∫∞0 dt η(t;q*). An example of this approach is given later where the dissipative process, vibrational energy relaxation, is caused by the fluctuating solvent forces along the molecular mode, and it is characterized by a frequency-dependent friction constant. The GLE can be “derived” by invoking the linear response approximation for the response of the solvent modes to the motion of the reaction coordinate. It should be noted, however, that the friction kernel will not in general be independent of the reaction coordinate motion,20 i.e., a nonlinear response, so the GLE may have a limited range of validity.21-23 Furthermore, even if it is valid, the strength of the friction might be so great that second and third terms on the right-hand side of eq 9 could dominate the dynamics much more so than the force generated by the PMF. It should also be noted that even though the friction in eq 9 is approximated to be dynamically independent of the value of the reaction coordinate, the equation may still be nonlinear, depending on the nature of the PMF. For nonquadratic forms of Veq(q), even the solution of the reactive dynamics from the model perspective of the GLE becomes a nontrivial exercise. Two central results have arisen from the GLE-based approach to the dynamical correction factor. The first is the Kramers theory of 1940,6 based on the simpler Langevin equation, while the second is the Grote-Hynes theory of 1980,7 based on the GLE. Both have been extensively discussed and reviewed in the literature.8,12-15 The important insight of the Kramers theory is that the transmission coefficient for an isomerization or metastable escape reaction undergoes a “turnover” as one increases the static friction from zero to large values. For weak damping (friction), the transmission coefficient is proportional to the friction, i.e., κ ∝ ηˆ (0), because the barrier recrossings are caused by slow energy diffusion (equilibration) in the reaction coordinate motion as it leaves the barrier region. For strong damping, however, the transmission coefficient is inversely proportional to the friction, i.e., κ ∝ 1/ηˆ (0), because the barrier crossings are caused by the diffusive spatial motion of the reaction coordinate in the barrier region. The detection of the Kramers turnover behavior has been the object of several experimental studies as is discussed later. The key insight provided by the Grote-Hynes theory is that the time dependence of the friction can be quite important and must be taken into account. In the overdamped regime, this is done so within the insightful and compact formula7 mq¨ (t) ) -

κGH )

λ‡0 ωb,eq2 ; λ‡0 ) ‡ ωb,eq λ + ηˆ (λ‡)/m 0

(12)

0

where ηˆ (z) is the Laplace transform of the friction kernel, i.e., ηˆ (z) ) ∫∞0 dt e-ztη(t), and ωb,eq is the magnitude of the unstable PMF barrier frequency. The derivation of this formula assumes a quadratic approximation to the barrier such that Veq(q) ≈ Veq(q*) - (1/2)mωb,eq2(q - q*)2. Research in the past few years has demonstrated that a multidimensional TST approach can also be used to calculate an even more accurate transmission coefficient than κGH for systems that can be described by the full GLE with a nonquadratic PMF. This approach has allowed for variational TST improvements24 of the Grote-Hynes theory in cases where the nonlinearity of the PMF is important and/or for systems which have general nonlinear couplings between reaction coordinate and the bath force fluctuations. The Kramers turnover problem has also been successfully treated within the context of the GLE

and the multidimensional TST picture.25 A multidimensional TST approach has recently been applied18 to a realistic model of an SN2 reaction and may prove to be a promising way to elaborate the explicit microscopic origins of solvent friction. We now consider some experimental results that pertain to these issues. V. Barrier Crossing Experiments and Their Relation to Theory The bulk of the experiments that have some obvious relationship to the theories of barrier crossing have involved geometric isomerizations of molecules. After considerable experimental effort over a period of 15 years some progress was made in understanding the essential characteristics of barrier crossing, but much remains to be learned. These studies have also provided opportunities to evaluate how gas phase dynamics principles need to be modified for solution phases. There have been experimental studies on many different types of molecules, but the present discussion will be confined to studies of the cis-trans isomerization of stilbene and the boat-chair conformational dynamics of cyclohexane. trans-Stilbene isomerizes to the cis form when excited with ultraviolet light pulses. There are reviews of the results of experiments on both trans- and cis- stilbene to which the reader is referred for a more complete description of all that has been done.26-31 The stilbene system would appear at first sight to have the essential ingredients needed to test the theory. The isomerization time constants for the trans isomer are in the range 30-150 ps in near ambient temperature liquids; the rates are viscosity, temperature, and pressure dependent. The reactive motion could be the classic one-dimensional coordinate involving rotation about the double bond, the rates are slow enough that the light pulse effectively prepares a thermal distribution on the trans side of the barrier, and the isomerization barrier height of the isolated molecule is about 6 times kBT at 300 K. The photochemical isomerization might be expected to be a thermally stimulated barrier crossing reaction with a onedimensional reaction coordinate and therefore an appealing candidate for detailed comparison with theoretical predictions. The earliest attempts fitted the viscosity dependence of the barrier crossing rate to the high-friction Kramers equation26,32a,33 by assuming that the frequency-dependent shear viscosity is proportional to the friction on the reactive motion in accordance with the Stokes-Einstein relationship. Reasonable fits were obtained only when the barrier frequency was chosen to be a few tens of cm-1, corresponding to what appeared to be unphysical flat topped barriers. An interesting observation was that the rate monotonically decreased with viscosity, even including a very low viscosity liquid such as butane.32a These results32a,34 showed that the Kramers turnover regime was not to be found in the range of frictions exhibited by normal liquids which were clearly in the high damping regime. Similar results were found for other molecules.32b Attention then turned to supercritical fluids where the friction could be continuously varied at relatively low densities by changing the pressure. The predicted maximum in the rate of isomerization was first seen in supercritical ethane35 and has since been fully characterized in a number of solvents.30,36,37 Although some of the qualitative features of the theory are seen in these experiments, there are major outstanding questions remaining in the interpretation of the observations. Before outlining what these are, the observations on isolated molecules need to be mentioned. The dependence of the barrier crossing rate on internal energy was first examined in low-pressure gases under thermal conditions,26,27,38 in what were the first picosecond laser studies of unimolecular reactions in the gas phase. Although a threshold, as expected from RRKM theory, was indicated, the need for

Dynamics and Relaxation Processes in Solutions experiments at lower temperature under beam conditions was compelling.38b The first molecular beam study, which appeared a few years later,39 showed the onset of barrier crossing very clearly and permitted accurate determinations of k(E), the microcanonical rate coefficients, free from convolution with the thermal distribution. The average of k(E) over the distribution of states above the barrier (at 1250 cm-1) should give the transition state rate of isomerization. These and other features were accurately determined in subsequent work,40 and the energy dependence of k(E) was fitted to RRKM-like expectations. There still has been no clear resolution of whether the reaction involves the nonadiabatic curve crossing predicted from quantum chemical calculations.41 The peak rate of isomerization, at the Kramers turnover, seen in these supercritical fluid experiments is in the range (50 ps)-1, which is about an order of magnitude faster than the transition state rates calculated from the function k(E) that was determined from the gas phase experiments. This discrepancy is important to understand since it must be the result of significant differences between the gas and solution phase conditions. One difference is that the degree of intramolecular vibrational energy redistribution (IVR) may not be the same in both cases: the IVR may be incomplete in the isolated molecule and more closely approach a distribution over all the modes of the system in the solution phase because of collision-induced relaxation.36c However, even in solutions the IVR is not completed on the time scale of the isomerism. Another difference is that solvent-solute interaction may modify the barrier to isomerization and allow a different reaction pathway.36a In addition, if there were a curve crossing as part of the isomerism process, the solvent interactions might be expected to modify the potential surface in the crossing region.42 With this backdrop we can discuss further the solution phase dynamics. The experiments on barrier crossing in stilbene involve a preparation stage in which the Franck-Condon modes are excited by a light pulse. We must examine whether the equilibrium conditions of the theory are achieved. Vibrational energy relaxation will ultimately lead to a thermal distribution over all the modes of the solute. However, equilibrium is achieved only after times long compared with {(1 + exp[-EV/ kBT])kV)}-1, where EV and kV are the vibrational energy and relaxation rates of vibrational states V. The vibrational relaxation of the higher frequency Franck-Condon modes initiates IVR processes. The overall cooling time of a molecule like stilbene in solution is in the range 11-14 ps.43 Populations of some of the Franck-Condon modes persist for tens of picoseconds,44 but this may not influence the reaction much, since there is experimental evidence that not only do these modes remain out of equilibrium with the bulk of the others but they may be largely spectator modes in isomerization.45 In the condensed phase the isomerization rate of trans-stilbene is independent of excitation energy, consistent with the important relaxation processes being fast. A challenging issue important in relating experiment and theory concerns the friction, η, that appears in theoretical expressions and its practical relationships to the viscosity, temperature, or pressure. When these macroscopic parameters are varied, the friction on a particular motion is modified, but so are other properties. For example, changing solvents as similar as paraffin hydrocarbons in order to vary the viscosity can change the potential surfaces, the way in which the solvent molecules pack around the solute and the volume available to the isomerizing molecule. There is direct evidence of solventinduced potential surface modification from the observation of different wavepacket dynamics on the excited surfaces of cisstilbene in different solvents.46 One measure of the friction on

J. Phys. Chem., Vol. 100, No. 31, 1996 13039 the isomerization motion, which involves substantial angular motion of the two halves of the molecule, can be obtained from rotational diffusion rates which are well-known to have nonlinear relationships with solvent viscosity.47,48 By using the deviations from linearity found in rotational diffusion experiments, an effective friction for isomerization experiments can be obtained.47,48 The characterization of friction is not always clear in experiments that vary the pressure since rotational diffusion times are usually linear with pressure,49 but the constant of proportionality depends on solvent and is not fully understood.29 These deviations are likely due to specific solvent effects involving variations in excluded volume,50 coupling of internal and external motions leading to reaction paths in which the friction is reduced by overall motion of the whole molecule,31,45 and multidimensional effects involving internal modes providing higher barriers but lower friction paths.51 Even if the reaction coordinate for the stilbene isomerization were onedimensional in the isolated molecule, it will be more complex in solutions. This arises because the nuclear motion through the reaction coordinate in the presence of friction generates torques that tend to rotate the whole molecule in space.43b,d In the case of stilbene this rotation angle is about 25° and allows the reactant and product structures to occupy approximately the same region of space in the liquid. In effect, the reactive trajectory follows a minimum-friction path, forcing the molecule to rotate in order that it sweeps through the least volume of solution. This coupling between internal and overall motion makes it even more difficult to find a practical definition of the effective friction on the reaction coordinate. Clearly, the stilbene isomerization is not the simple paradigm that can be readily used to test the theory. Many interrelated factors need to be incorporated to obtain the full picture. The complexity of the actual situation is evident from the recent proposal52 that the pressure and temperature dependence of isomerization could be represented by a combination of standard unimolecular rate theory and a Kramers one-dimensional model with allowance for a decrease in barrier height with increasing solvent density and a temperature-dependent barrier frequency. The boat-chair isomerization of cyclohexane53 exhibits surprising behavior given the above discussion on the stilbene dynamics, and yet it is considered to be brought into relation with theory.54,55 In this example, the rate of isomerism increases with increasing viscosity in liquid CS2 at ambient temperature.53 Such a result suggests that the dynamics is in the inertial or energy diffusion regime of the Kramers theory. This is quite a different conclusion than reached for stilbene-like systems where at normal liquid state densities the rate of isomerization decreases with increasing viscosity, indicating that the dynamics is in the strong damping or spatial diffusion regime. It is expected that the Kramers maximum in the rate will shift to higher friction as the number of internal modes coupled to the reaction coordinate increases.56 The IVR process and the vibrational relaxation on the reactive coordinate have quite different effects on the barrier crossing dynamics. Vibrational relaxation causes the system to shift back and forth along the reaction coordinate, whereas IVR equilibrates the reactive energy with other modes of the system. The isomerism barrier in cyclohexane is about 4200 cm-1, at which energy IVR in other molecules of comparable size is often extremely fast. However, the molecular dynamics simulations of the cyclohexane isomerism55 clearly show periodic behavior in the reactive trajectories near the top of the barrier, indicating that the system is indeed in the inertial regime at the liquid density of CS2: The reactive mode is effectively decoupled from the bath of internal vibrational excitations! It remains to be seen whether there is a significant group of molecules in this category.

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It is clear that a next step in establishing a full mechanistic picture of barrier crossing in stilbene and related molecules will require excellent quantum mechanical calculations of the reactive surface, simulations of the free energy surfaces under a variety of conditions, and molecular dynamics calculations. Quantum chemical calculations including solvent already form an active field of research54 that will hopefully be extended. Some of these difficulties will be removed when molecules “simpler” than stilbene and other aromatics are employed in isomerization studies, perhaps with atomic or diatomic solvents. This will become possible with the new and more flexible laser technology that is becoming available at the present time. At present, the theory is often for more idealized systems than the laboratory samples at the disposal of experimentalists. However, it is hoped that this is also will change by theory addressing more complex conditions. An ultimate goal of this research, both theoretical and experimental, must be to understand fully those chemical reactions occurring in the practical situations of chemical and biochemical synthesis. VI. Transition State Dynamics Initiated by Photoexcitation of Triatomic Molecules in Solution We now turn to a discussion of experimental results for reactions in solutions involving small molecules. In such cases there is some expectation of defining the forces responsible for the reactive motions. The collinear bimolecular reaction

A + BA f AB + A has a transition state structure [A-B-A]‡. Such a structure must correspond to an excited electronic-vibrational state of the linear triatomic molecule ABA. Therefore, optical excitation of ABA at the appropriate wavelength will yield [A-B-A]‡ in a form that is compressed along the totally symmetric stretch coordinate as a result of the Franck-Condon principle, as shown in Figure 2. The same forms of spectroscopy that traditionally are used to examine the structure and dynamics of excited states can thereby provide comparable information about these transition states. This notion has been widely employed in the study of reactions in the gas phase.57 It is an ideal experimental configuration for solution reactions as well because the effects of even ultrafast relaxation processes on the nuclear motion near the transition state are made directly accessible with the system being prepared in a well-defined metastable state. In one approach excitation can be carried out with femtosecond pulses and methods of ultrafast spectroscopy used to examine the subsequent events on time scales shorter than the vibrational periods. Another approach to obtain information about the nuclear dynamics at the Franck-Condon state is to examine the resonance Raman scattering using excitation wavelengths near the relevant optical transition.58 In fact, any type of scattering or multiphoton process that is resonantly enhanced by the state of interest will provide some information about the transition state dynamics. These Raman methods have also been used to study reactions that are much more complicated than the collinear triatomic example, such as in the photodissociation of polyatomic molecules65 and in isomerization reactions.60 In the solution phase only two collinear type reactions have been studied with short time resolution. These are the dissociations of I3 into I2- and I61 and of HgI2 into IHg and I.62 The latter reaction has also been studied in detail in the gas phase using femtosecond excitation pulses63 so the opportunity arises for detailed comparisons of gas and solution phase processes near the transition state. The I3- reaction requires solvent participation at this point. Both reactions produce the diatomic fragments in less than a vibrational period and with a

Figure 2. Impulsive generation of HgI from HgI2. On the left are shown some data from ref 62 on the absorption of HgI generated impulsively from the photolysis of HgI2 at close to t ) 0. The delayed probe signal oscillates at some effective frequency that depends on which part of the well is probed (right-hand side). The oscillations decay in amplitude due to dephasing of the wavepacket. Thus, the experiment exposes information regarding the reactive motion (0-150 fs), the range of vibrational frequencies in the well, and the relaxation dynamics of the wavepacket. A cartoon of the specific motions including the reactive motion is shown in Figure 5.

relatively narrow distribution of bond lengths. The time resolution used in these experiments is shorter than the vibrational period so the energy eigenstates are not seen separately. With these conditions it is more convenient to discuss the dynamics in terms of eigenstates of the displacement of the nuclei from their equilibrium positions. Each of these is of course a superposition of energy eigenstates. The ensemble of molecules thus prepared has a coherent polarizability, oscillating in response to the in-phase vibrational motion of all the molecules: Clearly the reactions generate products that are described by a wavepacket, constituted by a definite set of energy states and their associated phases. It has been learned that to understand a solution phase reaction, information is needed about the distribution of equilibrium geometries of reactant in the solution; the solvated reaction coordinate, which involves modes of both the solute and the solvent; the way in which energy is partitioned and coherence is transferred into reaction products; and the nature of the solvent-induced relaxation of the coherence and the populations. These properties are strongly coupled, and they present exciting challenges to theory and experiment. We will choose the example of the photodissociation of HgI2 in ethanol to picture transition state dynamics in a solution phase reaction. VII. Theories of Relaxation All the observables of systems reacting impulsively, such as I3- or HgI2, can be determined in principle from the density matrix of the wavepacket. A key property is the probability density that a product molecule will have displacement x, at time t, which when given in terms of the reduced density matrix F, as

P(x,t) ) 〈x|F(t)|x〉 ) ∑φ* m(x) φn(x) Fmn(t)

(13)

this formulation permits the propagation of the wavepacket P(x,t)

Dynamics and Relaxation Processes in Solutions

J. Phys. Chem., Vol. 100, No. 31, 1996 13041

based on knowledge of the energy eigenstates. This would be straightforward if the dynamical system were, for example, a harmonic or a Morse oscillator. The density matrix is in principle known exactly in the gas phase for a known potential surface but is more complicated in general because it requires information about the coupling of the solute molecule to the solvent. Even in a case as simple as the collinear reactions discussed above, the density matrix will not be known a priori throughout the motion. The initial reactive motion is controlled by a surface that involves all three atoms. After a time comparable with half of a vibrational period one of the atoms can be ejected beyond the interaction distance, and the relevant energy eigenfunctions φm(x) may become similar to those known for the gas phase diatomic fragment. In the early stages, when the eigenfunctions are not at all known, it is usually more efficient to propagate the wavepacket by direct numerical solution of the Schrodinger equation64 on an approximate potential surface. Second-order relaxation theory,65 an example of which is Redfield theory,66 can be used to calculate the reduced density matrix elements needed in eq 13, through the relaxation equations:

F˘mn(t) ) -iωmnFmn(t) + ∑ΓmnpqFpq(t)

(14)

pq

where the ωmn ) ωm - ωn are the Bohr frequencies of the solute. The so-called secular relaxation parameters Γmnpq are directly related to correlation functions of the solvent bath. When m ) n and p ) q ) m, they are the population state-to-state rate constants that determine the T1 relaxation of the oscillator. When m * n and p * q, they represent the coherence loss (m ) p, n ) q), corresponding to the T2 relaxation of each pair of levels, and coherence transfer (m * p, n * q), corresponding to the medium-induced transfer of coherence from one level pair to another. Much of the essential physics of coherence and population relaxation can be seen from the results of a harmonic oscillator linearly coupled to a harmonic bath. In that case there is a general operator form for eq 14 that yields the master equations for populations and coherences. In the case that there is population relaxation characterized by a time T1 and quadratic pure dephasing, T′2, the operator master equation has the secular form65

F˘(t) ) -iω0[ηˆ ,F] 1 1 nj + [F,ηˆ ]+ + njF - (nj + 1)bFb† - njb†Fb T1 2 1 [nˆ ,[nˆ ,F]] (15) T′2

{(

}

)

where nˆ ()b†b) is the number operator for the oscillator, nj is the Boson occupation number, and ω0 is the oscillator frequency including any shift due to coupling to the bath. The relaxation rates are

1/T1 )

2π p

∑i |µi|2δ(pω0 - pωi)

(16a)

8π (16b) nj∑|µi|2δ(pωi) p i where ui is the coupling of the system to the ith bath oscillator with frequency ωi. The evolution of the populations (eq 17) is obtained by taking an average of F in the state |n〉; the coherence master equation involving all the Bohr frequencies (eq 18) is obtained by projecting over 〈n| and |n + m〉; and a general equation for 〈b〉(t), from which can be calculated the relations for the position and momentum, can be obtained by recognizing 1/T′2 )

that d/dt[〈b〉(t)] is Tr{F˘(t)b} and that x ) (p/2mω0)1/2(b† + b) and px ) (pmω0/2)1/2(b† - b):

F˘nn ) -

1 {[n + (2n + 1)nj]Fnn T1 (nj + 1)(n + 1)Fn+1,n+1 - nnjFn-1,n-1} (17)

F˘m,m+n ) iω0nFm,m+n -

1 × T1

[{(nj + 12)(2m + n) + nj}F

m,m+n

- njxm(m + n)Fm-1,m+n-1 -

]

(nj + 1)xn(n + m + 1)Fm+1,m+n+1 -

(

)

1 1 d 〈b〉(t) 〈b〉(t) ) - iω0 + + dt T1 T′2

n2 (18) T′2 (19)

These results show that the damping of the oscillator’s position and momentum does not depend on the initial conditions of excitationsin other words, they do not relax on the time scales of the state-to-state rates that normally scale with quantum number. Therefore, it is an exact result for harmonic systems in this limit that the coherence loss in wavepacket dynamics is not explicitly determined by the dynamical parameters of each level pair. VIII. Applications of Relaxation Theories to Real Systems The range of validity of the foregoing approach in modeling the dynamics of real systems remains to be fully demonstrated experimentally. It is only exact for a harmonic oscillator linearly coupled to a bath of harmonic oscillators. Only in the case of HgI have experimental results for wavepacket motion been fitted to eqs 17 and 18 as far as we know. In that case the wavepacket oscillations were shown to persist for much longer than the stateto-state population relaxation times (see Figure 2). In fact, the coherence retention time for HgI was approximately given by the 2-3 ps, T1 relaxation time for the energy in the oscillator, as expected from eq 18. This coherence retention appears to occur also in the case of I2- by our inspection of the published data. The loss of the coherence by the energy relaxation has been simulated for hot harmonic oscillators67 using eq 14 with the parameters (16) obtained from classical simulations. Wavepacket oscillations observed in experiments are useful in determining the character of the molecular states involved in reactive and stationary processes. The oscillations can persist for long times in the condensed phase, even though the memory of the free propagation involving all the Bohr frequencies of the potential, such as occurs in the gas phase and given by the first term in eq 14, is lost very rapidly.62 This suggests that a simple classical picture will often be useful in describing wavepacket motion. Such a picture for HgI is given below. Together, eqs 13 and 14 provide a simple model for the motion of an impulsively prepared wavepacket on a surface consisting of levels approximately equally spaced at a frequency of ωm,m+1. The appropriateness of eqs 13, 17, and 18 can be tested by experiment62c and simulations. A classical simulation can be employed to evaluate the symmetric parts of the correlation functions that are needed to define the relaxation matrix, thus yielding a semiclassical description of the wavepacket motion.68 In a first analysis for a molecule weakly coupled to the surrounding solvent, the potential function could be chosen as the potential of mean force (eq 7). For eq 14 to be valid, the correlation time of the solvent response must be short compared with the relaxation times. If this were not true, the relaxation would be reversible and the system should not be described as a solute molecule weakly coupled to a bath. In the case of

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ground state HgI the PMF, the gas phase potential, and the potential estimated from the wavepacket dynamics as described below were all quite similar.62,69 The appropriateness of the foregoing equations to a particular dynamical problem depends on the how the separation of the Hamiltonian of the system into the “solute” and “bath” is made. In the simplest possible separation, the quantum numbers m, n, etc. correspond solely to the states of the solute molecule, such as I2- or HgI in the foregoing examples, and the bath is just the solvent. The bath, whatever its constitution, is presumed to be at thermal equilibrium and essentially unperturbed by the relaxation of the solute.65 When there is strong coupling or when there is there is a significant perturbation on the solvent by the relaxing solute, this theory is inapplicable. For example, when HgI is highly excited, the population relaxation rate is probably fast enough that the nearby solvent molecules are heated up substantially. This will result in a change in the solvent solute coupling, a time dependence in the effective bath temperature, and changes in the net rate of energy deposition from the solute into the solvent. In principle, this difficulty could be met by incorporating some of the modes of the solvent into the description of the solute. For example, there may be some solvent molecules that are relatively strongly bonded to the dissolved molecule such as those in the first solvent shell. An obvious example is the hydrogen bonding of water to ions which appears to influence greatly their relaxation times.70 This group of solvent molecules might reasonably be considered part of the reactive system, and the bath would then be the remaining solvent molecules. The new separation into system and bath will more likely satisfy the conditions for the validity of eq 14 or 18, but now m, n, etc. incorporate states of a solute-solvent complex, and IVR in a “supermolecule” is a better description for the relaxation processes. In any event, a much improved knowledge of the mode coupling and hence of the energy flow between solvent and solute modes will be needed. IX. More on Energy and Phase Relaxation and Wavepacket Dynamics in Solutions: Comparison of Theory and Experiment The transfer of vibrational energy from the solute directly into motions of the solvent (for example, as described by eq 9) is an important form of friction that influences the motion along reaction coordinates. There are as yet very few measurements of such vibrational relaxation rate constants in solutions at normal temperatures, so the theoretical predictions of them are not yet well tested. Of course, there are many measurements of vibrational relaxation times of the higher frequency modes of polyatomic molecules in liquids, a subject that has been frequently reviewed.71-73 However in the majority of these larger molecule cases the relaxation process deposits energy into both solvent modes and other internal modes of the solute. Only in very few examples have the pathways of relaxation or the energy partitioning actually been determined. The motion of the relevant nuclear coordinate can be modeled by a GLE. Then the time-dependent friction along the coordinate is proportional to the autocorrelation function of the random part of the fluctuating force. The frequency-dependent friction ηˆ (ω) can be calculated from classical molecular dynamics simulations, and the T1 time can be obtained as follows:74

1/T1 ) [tan(βhω/2)/(βhω/2)]ηˆ (ω)

(20)

The prefactor in the square brackets is nearly unity for frequencies less than 1/β. This approach was used to calculate T1 for CH3Cl treated as a diatomic,75,76 I2,77 I2-,77 and HgI69 in a variety of solvents. In the latter three cases the agreement with experiment is surprisingly good. There are no experimental

Figure 3. Calculated spectrum of the forces exerted along the internuclear axis of a diatomic molecule (HgI). The figure shows the results of molecular dynamics simulations of HgI in ethanol. The frequency-dependent friction on the bond motion, ηˆ (ω), is the Fourier transform of the force-force autocorrelation function. At the 130 cm-1 frequency of HgI the friction is dominated by the Lennard-Jones interactions with the charges (Coulombic friction) playing only a minor role. At higher frequencies the role of specific modes of the solvent becomes evident. The value of ηˆ (130 cm-1)/kBT is 2 ps, in good agreement with measurements of T1 (see ref 69).

tests yet of eq 3 or 6 for higher frequency vibrations of diatomics. The simulations used to calculate η(ω) also contain a wealth of structural indications concerning the specific solvent motions that are responsible for the relaxation.69,75-77 Detailed theoretical indications or predictions of this type are needed to aid in the design of experiments to further test the theory. A molecular dynamics simulation aimed at obtaining ηˆ (ω) was recently carried out for HgI 69 in ethanol (see Figure 3). The calculated energy relaxation time of ca. 2 ps agrees well with the experimental value,62 in support of the validity of linear response theory. Somewhat surprisingly, the excess vibrational energy appears to be dissipated mainly through the short-range Lennard-Jones interactions with solvent oxygen atoms which are held close to the Hg by long-range Coulombic interactions. The coupling to the solvent is strong as a result of the polarity of the system. However, the solvent nuclear response η(t) is a few times faster than the vibrational period, indicating that the HgI-ethanol system is close to the adiabatic regime. The importance of Lennard-Jones forces was also noted for the I2relaxation.77 There were numerous attempts to compute the parameters of vibrational relaxation from gas phase collision models in particular the isolated binary collision (IBC) model.78 Experimental results for the vibrational relaxation of I2 in Xe79 have indicated that the relaxation is controlled by binary collisions. On the other hand, MD simulations80 for CH3Cl in a polar molecular solvent manifested significant cross-correlations between the Coulombic and Lennard-Jones force fluctuations. As demonstrated above (eqs 17-19), when the population relaxation is very fast, which is the case in most of the examples given, the wavepacket dynamics is also controlled by T1 relaxation and not by state-to-state relaxation. When T1 is much smaller than the vibrational period, it is easy to see that stateto-state relaxation will not destroy the coherence since the

Dynamics and Relaxation Processes in Solutions

Figure 4. Wavepacket motion in a harmonic well with damping. The figure shows cartoons of how wavepackets propagate in the presence of population (T1) and pure dephasing (T′2) relaxation. The pictures represent quantum predictions for a harmonic oscillator linearly (AC) and quadraticallly (D) coupled to a bath of harmonic oscillators. (A) The case of no damping. (B) The motion in the limit where the vibrational frequency is slow compared with the T1 damping rate (T′2 ) ∞). The wavepacket narrows as the relaxation occurs. In the presence of T1 relaxation, with 1/T1 comparable with the vibrational frequency, the wavepacket swings down the surface, re-forming each half-cycle to form a wavepacket only slightly altered in width. (D) The motion in the presence of pure dephasing (T′2) when the energy is such that n2/T′2 is fast compared with the vibrational period. The fluctuations in energy due to the changes in the surface curvature occurring more rapidly than the position can adjust cause a spreading of the wavepacket as the molecule vibrates. Each of these descriptions is quasiclassical in nature and is readily deduced from eq 19.

energy changes without there being much disturbance of the distribution of position eigenstates. This is a situation where a simple classical model becomes preferable because information about the eigenstates themselves is not so useful in visualizing the evolution of the spatial location and shape of a wavepacket. The system behaves much like a classical ball rolling on an anharmonic surface with friction. In the case of HgI the initial impulsive reaction generates excitations with a mean vibrational quantum number in the range of 15. At this energy the state

J. Phys. Chem., Vol. 100, No. 31, 1996 13043 to-state relaxation time of 200 fs is not much faster than the vibrational period. Thus, the wavepacket can be envisaged (Figure 4) as undergoing a swinging motion as it loses energy and flops from one side of the surface to another while only gradually spreading into a broader distribution of internuclear separations. If the T1 time were extremely short compared with the vibrational period, which would occur at sufficiently high quantum numbers, the wavepacket would slide down the repulsive wall of the surface. Only when it reaches regions where the relaxation becomes slower than the period will the wavepacket move over to the attractive wall of the potential. These classical pictures are consistent with the quantum results of eqs 17-19: they incorporate T1 relaxation and coherence transfer, the latter being directly concerned with the spread of internuclear separations that arises from relaxation. A typical reactive process is that undergone by HgI2 (see Figure 5). The HgI2 is excited by femtosecond pulses into a region where the bonds are compressed relative to HgI along the totally symmetric mode and vertically excited along the asymmetric stretch coordinate. Motion on the excited state surface begins with trajectories that are combinations of these stretches. The bending motions have a period of about 2 ps and are too slow to displace much during the reactive motion (Figure 2). At the beginning of the motion the situation is probably similar to that in the gas phase,63 but the relaxation processes cause energy exchange with the medium on a time scale faster than the vibrational periods. In the experiment, the reactive region and the HgI are probed by measuring the transient absorption at various optical frequencies. The first peak in the signal appears when the HgI first reaches the attractive portion of its potential surface. This takes a time of 0.45 of one HgI stretching period (i.e., 120 fs), which is determined by the repulsive forces on the reactive potential. As the wavepacket proceeds along the exit channel, the attractive side of the ground state of HgI is probed on each successive cycle. The properties of the wavepacket, such as period and relaxation time of the oscillations, are dependent on the frequency of the probe pulses. The information obtained from probing different regions of the ground and excited state potential surfaces yields much needed information on these

Figure 5. Reactive processes during photodissociation of HgI2. A LEPS surface for the stretching coordinates of IHgI is shown. The excitation generates a narrow distribution of configurations around the transition state but compressed along the totally symmetric stretch. The asymmetric stretch is also excited by the light pulse but is not displaced on average. The packet spreads asymmetrically during the reactive motion resulting in HgI and IHg. The solvent stops the translation of IHg and I after a few vibrational periods during which time the wavepacket relaxes in the IHg ground state well. The open ellipses signify the regions that are probed in the experiments presented in Figure 2.

13044 J. Phys. Chem., Vol. 100, No. 31, 1996 energy surfaces in the solution. Similar considerations apply to the I3- photodissociation.61 The situation is expected to be different for higher frequency modes and those undergoing slower vibrational relaxation. In such cases the pure dephasing or adiabatic interactions with the solvent are more likely to dominate the dissipation of the wavepacket.81,82 Pure dephasing, characterized by T′2, is caused by fluctuations of the energies of level pairs. Interactions with the bath that are quadratic in the oscillator coordinate are needed to cause fluctuations in the potential energy function curvature for a harmonic oscillator. In an anharmonic oscillator the bath induced fluctuations of the potential and hence of the energy spacings require, at the lowest order, interactions that are both linear and quadratic in the oscillator displacement. A simple calculation62 shows that the importance of the pure dephasing terms in a Morse oscillator grows more rapidly with increasing quantum number than for a harmonic oscillator. In contrast to the population relaxation, pure dephasing can obliterate the wavepacket during the vibrational motion because the fluctuations in the energy are directly translated into a spread of positions by the nuclear motion (see Figure 4). X. Solvent Response and the Reaction Coordinate Currently, ultrafast experiments are capable of probing systems before there is much opportunity for solvent motion to occur. The energies of the initial and final states involved in the spectroscopic probe process are both fluctuating in response to the solvent dynamics. Therefore, one can expect that these dynamics will be sensed in optical experiments. For a very short time interval the solvent molecules are effectively immobilized, and the system responds as if it were inhomogeneous (i.e., it will exhibit photon echoes). Over much longer time intervals many systems will behave as if they are homogeneous. This situation was clearly recognized by Kubo 40 years ago.83 What has happened recently in this field is that the behavior through these limiting regimes has begun to be examined experimentally, and the theoretical description has begun to be refined by the development of more detailed models of the dynamics.84 Such models attempt to associate general theoretical descriptions of the physical behavior with the fewest possible number of parameters. One example is the multimode Brownian oscillator model which can model the short time dynamics of electronic spectra,84-87 including the dynamical Stokes shift between absorption and emission.84 The goal of this model is to allow for energy exchange between the solute and coupled oscillators that in turn are coupled to the bath of solvent molecules. Another approach is to use experimental solvent spectral densities to fit solute dynamical data.87b A challenge presented by such models is to associate specific solvent modes and solvent-solute interactions with the oscillators that are used to fit experimental data and to calculate these parameters from dynamical theories. The knowledge of how the solvent reacts to sudden changes of conformation or charge distribution in a solute molecule is of great importance in reaction dynamics. For example, it tells us whether the solvent can follow the reactant motion adiabatically so that equilibrium conditions can be assumed at each point on the reactive trajectory or whether nonequilibrium dynamics must be considered. It also tells us about the changing forces exerted on the solute once it becomes excited. An experiment that focuses directly on this question is the time dependence of the Stokes shift of the fluorescence spectrum. When a molecule is electronically excited, there can be a change in dipole moment, which makes the equilibrium energy of the excited state different from that of the Franck-Condon states reached by light absorption. Therefore, the electronic transition energy and hence

Voth and Hochstrasser the fluorescence spectrum shift in response to changes in the solvent configuration. The search for a full explanation of polar solvation dynamics and its influence on reaction dynamics has been intense during the past few years.88-94 Generally the energy shift has two time scales, one that is very fast (ca. 50300 fs) and another much slower (ca. picoseconds in most liquids at 300 K), generally smaller amplitude component. The separation of these time scales and the relative amplitudes of the fast and slower parts of the response are solvent-dependent. The picture that emerges from simulations,91,95 which first predicted the results, and later experiments96 is that the fast component corresponds to the energy changes in the solventsolute interaction associated with small-amplitude inertial motions of the solvent molecules. Furthermore, simulations have indicated that the fast dynamics associated with the response to charge separation in water are dominated by molecules in the first solvation shell.95b The multiple time scales of the solvent response imply that the friction on reaction coordinates that involve changes in the charge distribution of the solute molecule is frequency-dependent. The high-frequency part of this friction corresponds to the solvent inertial movements which are certainly fast compared with some but not all reactive processes. In the case of the vibrational motion of a molecule whose charge distribution changes with nuclear displacement, the adiabatic limit will not be achieved even in water when the frequency exceeds ca. 650 cm-1. Of course, when the vibrational period becomes very small compared with the solvent response time, the solvent will sense just the average solute charge distribution, but as always it is the intermediate case that is the most challenging! A few relaxation experiments do appear to be in the nonadiabatic regime. The concept of charge flow accelerating the vibrational relaxation rate has been discussed in relation to the dissociation and recombination of I2- 97 and the results compared with measurements of the vibrational relaxation dynamics.98 In that case the charge shift from [I-I]- to I + Iis strongly coupled to solvent, and the free energy path obtained from simulations involves a solvent coordinate. Furthermore, the actual reactive motion does not coincide exactly with the free energy pathway because the solvent cannot follow the rapid changes in solute coordinate. In the case of HgI the dipole moment changes by about 1 D as the molecule vibrates through one period and the solvent responds on about the same time scale. The impulsive excitation creates it in a compressed form. The changes in HgI charge distribution are coupled to the solvent charges by Coulomb forces. Clearly, the energy in the HgI mode must be changing as the molecule extends from its compressed starting position. The amount of this change depends on the magnitude of the coupling, but if it is large enough, the reactive potential surface should obviously incorporate the solvent coordinates as well as those of HgI2. XI. Probing the Basic Dynamical Parameters of the Reactions of Separated Reactants in Solutions Still another class of reactions that are ubiquitous in solutions are those that are diffusion-controlled. Here we are interested in the reactive part of such reactions that occurs only when the combining molecules are close together. The study of bimolecular reaction dynamics in solutions has required the creation of very special experimental conditions. We already discussed the stratagem of creating a transition state of the A + BA reaction by direct excitation of ABA and gave I3- and HgI2 as examples. Another approach is to create the two reactants as neighboring molecules in a solution by a fast photoreaction. For example, one of the reactants can be produced in a particular distribution of states by photolysis of a precursor. The precursor

Dynamics and Relaxation Processes in Solutions is dissolved in a solvent that consists of the other reactant. In this way a reactive radical can be suddenly created within a van der Waals separation of the molecules with which it is to react. The subsequent reaction is controlled by the forces between the reactants and the solvent influence on the dynamics. Two reactions have been reported by this technique. One involved Cl atoms from the photolysis of Cl2 in liquid chloroform, reacting with the HCCl3 molecules to produce HCl by hydrogen abstraction.99 In the other, CN radicals, generated from the photolysis of ICN in chloroform, abstracted a hydrogen to yield HCN and ClCN.100 Transient infrared spectroscopy was employed to detect the rate of HCl or HCN formation and the vibrational state distributions in these products. Some essential features of the dynamics of the CN radical abstracting H from the C-H bond of CHCl3 were examined recently using classical simulations.101 The results of the experiments are not as expected on the basis of the behavior of the isolated molecules. For example, in solution the time constant for HCN formation is 300 ps and temperature-insensitive, and the nascent vibrational state distribution is almost thermal. On the contrary, in the gas phase the reaction of thermal CN radicals with chloroform occurs on every collision, and a highly nonthermal (inverted) vibrational distribution in the C-H mode is generated. The changed vibrational state distribution immediately tells us that the potential energy surface determining the forces on the atoms near the transition state is significantly modified by the solvent: The transition state region in the solution appears to be closer to the HCN product structure than in the gas phase. The other interesting fact is that the transitionally hot CN radicals generated in the ICN photolysis do not react with the adjacent chloroform molecules but are simply cooled by the collisions. The simulations101 show each CN radical to be solvated by about seven chloroform molecules and provide a time scale for the translational cooling in the range of a few hundred femtoseconds. They also confirm that the slow rate of reaction of this barrierless reaction is a result of the improbability of the reactants achieving the transition state. This structure must therefore have a nearly collinear H-C-N configuration, with bond lengths closer to those in HCN. These results show that a solution phase reaction can be diffusion controlled even though the reactants are at van der Waals distance and there is no barrier. Diffusion-controlled reactions and caging effects have been studied by physical chemists for many years.102 The effects of solvent caging, the distance and orientation dependence of reactivity,103 and hydrodynamic interactions101 were examined by a variety of statistical methods. A dramatic difference between gas phase and solution phase reactivity, even for bimolecular reactions for which the diffusive encounter process itself is not the ratelimiting step, can be expected solely on the basis of the differences in rotational motion. In the gas phase, chloroform molecules at 300 K have a mean rotational period of 300 fs, which is comparable with the period of a collision having an impact parameter of a few angstroms. In liquid chloroform a molecule requires 12 ps to turn end-over-end by rotational diffusion, which is hundreds of times slower than the collision period. Clearly the liquid structure and dynamics, as characterized by the translational and rotational diffusion parameters and the specific forces between reactants, can greatly influence the ability of the reactants to achieve the configurations necessary for reaction to occur. Only recently have the new techniques of ultrafast spectroscopy made it possible for experimentalists to probe the detailed predictions of statistical theories of diffusion-controlled reactions involving chemically anisotropic species. A next step must be the introduction of first principles quantum techniques into the theory in order to describe the

J. Phys. Chem., Vol. 100, No. 31, 1996 13045 potential surfaces on which the solute and nearby solvent molecules are moving. A classic way of examining bimolecular transition state dynamics in the condensed phase is by means of the geminate recombination of two fragments created by a unimolecular dissociation reaction. The prototype of this approach is iodine, which can be photodissociated with visible light into two I atoms whose recombination rates and yields can be studied by timeresolved spectroscopy and other photochemical methods. It was the experimental study and theoretical simulations of this system that gave rise originally to the concepts of caging and encounter complexes105 in solution kinetics. More recently, a detailed examination of this reaction using femtosecond laser methods106 has shown that most of the iodine atoms are released in a very fast predissociation process, and they recombine efficiently within a few picoseconds to form hot ground state molecules. Molecular dynamics simulations have indicated that most of the dissociations do not result in the I atoms leaving the solvation shell.107 The studies of I2 have also yielded insights into vibrational relaxation for a highly excited, anharmonic, nonpolar diatomic molecule.106,108 These results relate to barrier crossing dynamics because the hot I2 molecules formed in the geminate process have large internuclear separations, and their relaxation is similar to that expected for a system close to a potential barrier region. The relaxation process occurs on quite different time scales for large and small internuclear separations because they are controlled by different parts of the solvent force-force spectrum when the potential is anharmonic.109 XII. Quantum Activated Rate Processes and Solvent Effects Much of the discussion up to now has been classical in nature, but for a full discussion of reacting systems classical mechanics is not sufficient. This is particularly true for proton and electron transfer reactions, as well as for reactions involving highfrequency vibrations. The exact quantum activated rate constant is given by the Yamamoto reactive flux correlation function expression9

kf )

1 ∫dτ 〈h˙ P(-iτ) hP(tpl)〉 xRpβ

(21)

where hP(t) is the Heisenberg product state population operator. As opposed to the classical case, the t f 0+ limit of this expression is always equal to zero.110 This result ensures that an entirely different approach than the classical one outlined above must be adopted in order to formulate a quantum TST, as well as a theory for its dynamical corrections. The article by Truhlar et al. in the present issue describes many of the efforts over the past 60 years to develop quantum versions of TST. Clearly, a quantum TST that is useful for condensed phase reactions is a desirable theoretical goal since a direct numerical attack on the time-dependent Schro¨dinger equation for manybody systems is computationally prohibitive. (The latter fact is true perhaps even in the fundamental sense; i.e., there is an exponential scaling of numerical effort with system size.) As a result of several complementary theoretical efforts, primarily the path integral centroid perspective,111-113 the periodic orbit,114 or instanton115 approach, and the “above crossover” quantum activated rate theory,116 an interesting candidate for a unifying perspective on TST, both quantum and classical, has recently emerged as a synthesis of ideas from refs 117-120. In this theory, the quantum TST expression for the forward rate constant is expressed as117

kf ≈ ν

Im Qb QR

(22)

where ν is a simple frequency factor, QR is the reactant partition

13046 J. Phys. Chem., Vol. 100, No. 31, 1996

Voth and Hochstrasser

function, and Qb is the barrier “partition function” which is to be interpreted in the asymptotic limit.117-120 The frequency factor has the piecewise continuous form117

ν ) λ‡0/2π; pβλ‡0 < 2π ) (pβ)-1; pβλ‡0 g 2π

(23)

while the barrier partition function is defined under most conditions as117-119

Qb )

qcfiqc

∫dqc Fc(qc)

(24)

The quantity Fc(qc) is the Feynman path integral centroid density121 which is understood to be expressed asymptotically as

Fc(qc) ≈ Fc(q*) exp[-βV′′c(q*)(qc - q*)2/2]

(25)

where the quantum centroid potential of mean force is given by Vc(qc) ) -kBT ln[F(qc)] and q* is defined to be the value of the reaction coordinate that gives the maximum value of Vc(q) in the barrier region (i.e., it may differ111,113 from the maximum of the classical PMF along q). The path integral centroid density along the reaction coordinate is given by the Feynman path integral expression

Fc(qc) )

∫...∫Dq(τ) Dx(τ) δ(qc - q˜ 0) exp{-S[q(τ),x(τ)]/p}

(26)

which is a functional integral over all possible cyclic paths of the system coordinates weighted by the imaginary time action functional121

{

}

N m m i q˘ (τ)2 + ∑ x˘ i(τ)2 + V[q(τ),x(τ)] 2 i)1 2 (27) The key feature of eq 26 is that the centroids of the reaction coordinate Feynman paths are constrained to be at the position qc. The centroid q˜ 0 of a particular reaction coordinate path q(τ) is given by the zero-frequency Fourier mode, i.e.,

S[q(τ),x(τ)] ) ∫0 dτ pβ

q˜ 0 )

1 pβ ∫ dτ q(τ) pβ 0

(28)

Under most conditions, the sign of V′′c(q*) in eq 25 is negative. In such cases, the centroid variable appears naturally in the theory,117 and the equation for the quantum thermal rate constant from eqs 22-25 is then given by117

kf ≈ V

x2π/β|V′′c(q*)| e-βV (q*)

∫-∞dqc e-βVc(qc)

c

q*

(29)

We note that in the alternative cases where V* (q*) > 0, the c centroid variable becomes irrelevant to the quantum activated dynamics as defined by eq 22. The instanton approach115 can then be used to evaluate Qb based on the steepest descent approximation to the path integral. In the limit of reasonably high temperatures, i.e., pβλ‡0 < 2π, the above formula can be simplified further and approximately written as

kf ≈ κGH

(2πmβ)-1/2

∫-∞dqc e q*

-βVc(qc)

e-βVc(q*)

(30)

This expression, aside from the prefactor κGH, is often referred to as the path integral quantum transition state (PI-QTST) formula.111 The strength of this formula is its clear analogy with the classical TST formula in eq 6. In turn, this allows for

an interpretation of solvent effects on quantum activated rate constants in terms of the quantum centroid potential of mean force in a fashion analogous to the classical case. The quantum activation free energy for highly nontrivial systems can be directly calculated with imaginary time path integral Monte Carlo techniques.122 Many such studies have now been carried out, but two examples are described in the following sections. Furthermore, it has been shown that the accuracy of the PIQTST formula can be enhanced through variations of the centroid dividing surface.111,123 It should also be noted that the path integral centroid perspective has also recently been shown to provide a powerful way to study a variety of equilibrium and dynamical processes in condensed matter,124 not just for quantum activated rate processes. XIII. Solvent Effects in Quantum Charge Transfer Processes In this section, the results of a recent computational study125 will be used to illustrate the effects of the solventsand the significant complexity of these effectssin quantum charge transfer processes. The particular example described here is a proton transfer reaction in a polar solvent. This study, while useful in its own right, also illustrates the level of detail and theoretical formalism that is likely to be necessary in the future to accurately study solvent effects in condensed phase charge transfer reactions. Obvious targets for quantum activated rate studies are proton, hydride, and hydrogen transfer reactions because they are of central importance in solution phase and acid-base chemistry, as well as in biochemistry. These reactions are particularly interesting to the theorist because they can involve large quantum mechanical effects, and since there is usually a redistribution of solute electronic charge density during the reaction, a substantial contribution to the activation free energy may come from the solvent. It is thought that intramolecular vibrations may also play a crucial role in modulating the reactive process by lowering the intrinsic barrier for the reaction. Many of these effects have been studied computationally using the PI-QTST approach. One such study125 has examined the model three-body proton transfer reaction

A-0.5-H+0.5...A f [A-0.25...H+0.5...A-0.25]‡ f A...H+0.5-A-0.5 in a polar (Stockmayer) fluid with dipoles chosen to model methanol. After some straightforward manipulations of eq 30, the PI-QTST estimate of the proton transfer rate constant is given by125

kfPI-QTST )

ωc,0 exp(-β∆F*c) 2π

(31)

where ωc,0 ) [V*c(q0)/m]1/2 and the quantum activation free energy is given by125,126

∆F* c ) -kBT ln[Fc(q*)/Fc(q0)] ) -kBT ln[Pc(q0fq*)]

(32)

The probability Pc(q0fq*) to move the reaction coordinate centroid variable from the reactant configuration to the transition state is readily calculated125 by path integral Monte Carlo techniques122 combined with umbrella sampling.125-127 From the calculations on the model PT system above, the quantum activation free energy curves are shown in Figure 6 for both a rigid and nonrigid (vibrating) intracomplex A-A distance. Shown are the activation curves for the complex both in isolation and in the solvent. The effect of the solvent in the total activation free energy is immediately obvious, contributing

Dynamics and Relaxation Processes in Solutions

J. Phys. Chem., Vol. 100, No. 31, 1996 13047 were also studied using path integral centroid methods124 and found to be significant as well.129 Another interesting application of the PI-QTST approach has been to calculate the solvent activation free energy in a realistic heterogeneous electron transfer (ET) process across the electrode/electrolyte interface126 [i.e., the Fe3+(aq) + e- f Fe2+(aq) reaction with a Pt(111) electrode]. In this study, a full quantization of all water molecules was carried out. The difference between the classical and quantum solvent activation free energies for water, even at 300 K, was found to have a significant effect on the predicted ET rate. XIV. Conclusions

Figure 6. Quantum versus classical dynamics of proton transfer. Quantum activation free energy curves calculated the model A-H-A proton transfer reaction described in ref 125. The solid line is for the classical limit of the proton transfer solute in isolation, while the other curves are for different fully quantized cases. The rigid curves were calculated by keeping the A-A distance fixed. An important feature here is the direct effect of the solvent activation process on both the solvated rigid and flexible solute curves. Another feature is the effect of a fluctuating A-A distance which both lowers the activation free energy and reduces the influence of the solvent. The latter feature enhances the rate by a factor of 20 over the rigid case.

several kcal/mol to the overall value. One effect of the A-A distance fluctuations is a lowering the quantum activation free energy (i.e., increased tunneling) both when the solvent is present and when it is not. A second interesting effect becomes evident from a comparison of the curves for the systems with the rigid versus flexible A-A distance. The contribution to the quantum activation free energy from the solvent is reduced when the A-A distance can fluctuate, resulting in a rate which is 20 times higher than in the rigid case. This novel behavior125 arises from a nonlinear coupling between the intracomplex fluctuations and the solvent activation, resulting in a reduced dipole moment of the solute when there is an inward fluctuation of the A-A distance. From these path integral studies, it was found that in order to fully quantify the solvent effects for even a “simple” model PT reaction, one must deal with a number of complex, nonlinear interactions. Examples of other such interactions include the nonlinear dependence of the solute dipole on the position of the proton and the intrinsically nonlinear interactions arising from both solute and solvent polarizability effects. In the latter context, it was found that the solvent electronic polarizability modes must be treated quantum mechanically when studying their influence on PT activation free energy.125 (A similar conclusion has been reached analytically for electron transfer processes.128) These detailed calculations, while only for a model PT system, clearly illustrate the significant challenge that lies ahead for those who hope to quantitatiVely predict the rates of liquid phase chemical reactions through computation. In passing, it is worth noting two more recent examples of PI-QTST calculations on realistic condensed phase systems.126,129 One calculation has focused on the quantum activation free energy along the asymmetric stretch PT coordinate in the aqueous H3O+-H2O dimer at 300 K.129 In this study, it was found that there is a small (∼0.5 kcal/mol) classical barrier to the PT in aqueous solution. However, when the hydrogen nuclei were quantized with path integral methods, the barrier along the centroid reaction coordinate completely disappears. This PI-QTST result therefore suggests that the nuclear quantization will be important in the aqueous proton transport problem. The quantum dynamical effects in the system

While much progress has been achieved in the quest to understand dynamical corrections to the TST rate constant in the condensed phase, there are several quite significant issues that remain to be explored theoretically. For example, even if the GLE is a valid model for calculating the dynamical corrections, can an accurate and predictive microscopic theory be developed for the friction kernel η(t) so that one does not have to resort to molecular dynamics simulation20 to calculate this quantity? After all, if one goes to all the trouble to compute the solvent friction along the reaction coordinate in such a manner, why not instead just calculate the exact rate constant using the reactive flux formalism? A microscopic theory for the friction is also needed to relate the friction along the reaction coordinate to the parameters varied by experimentalists such as pressure or solvent viscosity. No complete test of Kramers theory will ever be possible until such a theoretical effort is complete. Two candidates in the latter vein are the instantaneous normal mode theory of liquids, reviewed by Stratt and Maroncelli in the present issue, and the recently developed damped normal mode theory130 for liquid state time correlation functions. Another key issue is whether a one-dimensional GLE as in eq 11 is the optimal choice of dynamical model in the case of strong damping or whether a two- or multidimensional GLE which explicitly includes coupling to solvation and/or intramolecular modes is better or more insightful. Such an approach might, for example, allow closer contact with nonlinear optical experiments which could measure the dynamics of such additional modes. Finally, in many cases the GLE may not even be a good approximation to the true dynamics because, for example, the friction strongly depends on the position of the system along the reaction coordinate. In fact, a strong solvent modification of the PMF usually ensures that the friction will be spatially dependent.131 Recent analytical studies have dealt with this issue (see, e.g., refs 132-134 and literature cited therein). Spatially dependent friction is found to have an important effect on the dynamical correction in some instances, but in others the Grote-Hynes estimate is predicted to be robust.135 Nevertheless, the question of nonlinearity and the accurate modeling of real activated rate processes by the GLE remains open. As a final point, it should again be noted that many of the quantities which are measured experimentally such as relaxation rates, coherences, and time-dependent spectral features are complementary to the thermal rate constant. Their information content in terms of the underlying microscopic interactions may only be indirectly related to the value of the rate constant. A better theoretical link is clearly needed between experimentally measured properties and the common set of microscopic interactions, if any, which also affect the more traditional solution phase chemical kinetics. From the experimental standpoint the field of transition state dynamics in solution is wide open for new discovery. In the past, each advance in ultrafast laser technology has enabled chemists to sharpen their understanding of molecular dynamics.

13048 J. Phys. Chem., Vol. 100, No. 31, 1996 We are now moving into an era of all-solid-state laser devices, routine 15 fs pulse widths at any desired center frequency, and turn-key operation. With this situation, important new discoveries in solution dynamics will no longer be driven by advances in technology. The main effort can now involve the creation of new and more incisive experiments on the molecular dynamics of systems than can be addressed by theory. That is not to say there will be no technological challenges: Molecular vibrational periods extend down to 8 fs, and no appropriately versatile devices are yet available in the few femtoseconds range; the generation of femtosecond pulses in the X-ray regime for time resolved diffraction is certainly not yet an individual investigator option. However, there is no shortage of scientific challenges. There are very few examples of the experimental determination of potential surfaces in solutions; one hopes that this situation can now change. There are even fewer examples of the determination of reactive surfaces or transition state structures; this situation can also improve from experiments using the new technology which can measure product state energy distributions and solvent effects on them and even allow direct measurements of the structures in the transition state regime. A key question concerns the solvent effects on the nuclear motions in the region of the transition state; experiments should now be able to characterize these effects and establish microscopic descriptions of the dynamics that can be related to theory. Clearly, one important factor to examine is the strength of the coupling between solvent and solute, so we should anticipate new experiments not only on neutral and dipolar molecules but also on ions in polar solvents such as water. Also much remains to be learned before relaxation processes become predictable; this is an area where the new ultrafast infrared methods, which are capable of examining vibrational state dynamics in essentially any frequency regime, can be important. By enabling the rational fine tuning of reactive processes in solution, we can expect significant discovery to be made at this new frontier. Acknowledgment. This research was supported by grants from NSF (to R.M.H. and G.A.V.), NIH (to R.M.H.) and the Office of Naval Research (to G.A.V.). We wish to thank S. Gnanakaran, N. Pugliano, and K. Wynne for helpful suggestions and assistance in preparing figures. References and Notes (1) Herschbach, D. R. In Les Prix Nobel; Elsevier: Amsterdam, 1986. (2) Lee, Y. T. Science (Washington, D.C.) 1987, 236, 793. (3) Polyani, J. C. Science (Washington, D.C.) 1987, 236, 680. Polyani, J. C. Acc. Chem. Res. 1972, 5, 161. (4) Eyring, H. J. Chem. Phys. 1934, 3, 107. (5) Wigner, E. J. Chem. Phys. 1937, 5, 720. (6) Kramers, H. A. Physica 1940, 7, 284. (7) Grote, R. F.; Hynes, J. T. J. Chem. Phys. 1980, 73, 2715; 1981, 74, 4465. (8) Ha¨nggi, P.; Talkner, P.; Borkovec, M. ReV. Mod. Phys. 1990, 62, 250. (9) Yamamoto, T. J. Chem. Phys. 1960, 33, 281. (10) Chandler, D. J. Chem. Phys. 1978, 68, 2959. Montogomery, Jr., J. A.; Chandler, D.; Berne, B. J. J. Chem. Phys. 1979, 70, 4056. Rosenberg, R. O.; Berne, B. J.; Chandler, D. Chem. Phys. Lett. 1980, 75, 162. (11) Keck, J. J. Chem. Phys. 1960, 32, 1035. Anderson, J. B. J. Chem. Phys. 1973, 58, 4684. Bennett, C. H. In Algorithms for Chemical Computation; ACS Symposium Series No. 46, Christofferson, R. E., Ed.; American Chemical Society: Washington, DC, 1977. Hynes, J. T. In The Theory of Chemical Reactions; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. 4, p 171. Berne, B. J. In Multiple Timescales; Brackbill, J. V., Cohen, B. I., Eds.; Academic Press: New York, 1985. Truhlar, D. G.; Hase, W. L.; Hynes, J. T. J. Phys. Chem. 1983, 87, 2664. (12) For reviews of theoretical work on the corrections to classical TST, see: Hynes, J. T. Annu. ReV. Phys. Chem. 1985, 36, 573. Hynes, J. T. In The Theory of Chemical Reactions; Baer, M., Ed.; CRC Press: Boca Raton, FL, 1985; Vol. 4, p 171. Berne, B. J.; Borkovec, M.; Straub, J. E. J. Phys. Chem. 1988, 92, 3711. Nitzan, A. AdV. Chem. Phys. 1988, 70 (2), 489. Onuchic, J. N.; Wolynes, P. G. J. Phys. Chem. 1988, 92, 6495.

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