The Role of Collective Solvent Coordinates and Nonequilibrium

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FEATURE ARTICLE The Role of Collective Solvent Coordinates and Nonequilibrium Solvation in Charge-Transfer Reactions Gregory K. Schenter,*,† Bruce C. Garrett,*,† and Donald G. Truhlar*,‡ EnVironmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352, and Department of Chemistry and Supercomputer Institute, UniVersity of Minnesota, 207 Pleasant Street S.E., Minneapolis, Minnesota 55455-0431 ReceiVed: May 22, 2001

In this article we discuss how the solvent energy-gap reaction coordinate arising in weak-overlap chargetransfer theory may be generalized to include nonequilibrium solvation effects in variational transition state theory (VTST) for arbitrary reactions of neutral or charged species. The discussion also indicates how the parameters of weak-overlap charge-transfer theory may be related to the potential energy surface of a reaction and the potential of mean force. This synthesis of aspects of weak-overlap charge-transfer theory and variational transition state theory provides a clearer picture of the usefulness and limitations of the energy-gap coordinate in Marcus theory or in more general contexts. Furthermore, it furnishes a new view of the role of solvent coordinates in electronically adiabatic reactions. As an example, we show how VTST provides a way to treat solute vibrational coordinates and collective solvation coordinates in a comparable way in the computational modeling of aqeuous proton-transfer reactions including quantum mechanical vibrations and multidimensional tunneling effects.

Introduction The energy-gap coordinate has proven to be a powerful and versatile concept for elucidating the influence of solvent reorganization on charge-transfer reactions. The power of this coordinate is that the effect of multidimensional solvent dynamics on reactions is described by a one-dimensional variable equal to the energy or free energy to move the solute along the reaction coordinate in the presence of a fixed solvent ensemble. Marcus first introduced the energy-gap coordinate as a central construct of his theory of outer-sphere electron transfer (also known as weak-overlap charge-transfer) in condensed-phase systems.1-4 Weak-overlap charge-transfer theory1-16 and its strong-overlap generalizations17-25 are now widely used for qualitative and quantitative descriptions of a wide variety of processes, especially for condensed-phase reactions for which relaxation of the environment of the reaction center plays a significant role in the dynamics. Another central construct of charge-transfer theory is the use of reactant and product diabatic states, and charge-transfer theory uses an electronically nonadiabatic basis to provide one theoretical framework for utilizing the energy-gap coordinate to describe nonequilibrium solvent effects on chemical reactions. Transition state theory (TST),26-29 which originated in the description of electronically adiabatic gas-phase reactions, also provides a general theoretical framework for describing solvent effects on reaction dynamics. The thermodynamic or quasiequilibrium formulation of TST27 provides a convenient construct for describing solvent effects on reaction rate constants in terms of solvation effects on the free energy of activation.30,31 From this † ‡

Pacific Northwest National Laboratory. University of Minnesota.

perspective, charge-transfer theory can be viewed as a form of TST.2,4,6,30,32-34 The dynamical formulation of TST26,35 provides a general framework for examining the approximations of the theory and for developing improvements to it in a systematic manner, such as the variational version of the theory, which is widely used for quantitative calculations. In this paper we focus on the dynamical formulation of variational transition state theory36-40 (VTST) and how it can be used to provide a broad perspective for understanding nonequilibrium solvent effects on reaction dynamics. Originally the energy-gap coordinate represented the solvent, although more generally it can represent a “bath” consisting of solvent or microsolvent plus a subset of the solute coordinates. It can quite generally be called a solvent coordinate, a bath coordinate, or a solvation coordinate, and when specific models are introduced for the solute-solvent coupling, it may also be given a more specific name such as polarization coordinate. The goal of the present paper is to show how an energy-gap solvent or bath coordinate like the one used in weak-overlap charge transfer theory may be incorporated into variational transition state theory (VTST) and how the resulting formalism may be used to interpret charge-transfer reactions. The focus of our presentation is on the essential character of the solvation coordinate in weak-overlap charge-transfer theory (i.e., the most widely used version of Marcus theory), without complicating the discussion with a full treatment of other aspects of the theory. There are many excellent discussions of charge-transfer theory in the literature that provide more detailed descriptions of charge-transfer theory in its original and extended forms.1-25,32-34 Our objectives here are to present a different perspective on charge-transfer reactions, based on VTST, to provide a better

10.1021/jp011981k CCC: $20.00 © 2001 American Chemical Society Published on Web 09/12/2001

Feature Article appreciation of the physical model that underlies the use of a collective solvation coordinate, and also to provide a general computational framework that has some advantages for practical calculations and modeling. We hope that the perspective presented herein will be useful for advancing the qualitative and quantitative theory of reactions in solution, which is a very active field. First a comment on notation: transition-state theory and weak-overlap charge-transfer theory (WOCTT) both exist in many variants and generalizations; even though they differ in important details, we use these terms to refer to all of them. When necessary to make a distinction, one can say conventional transition state theory or Marcus weak-overlap electron transfer theory to refer to the original1-4,26,27 forms. Background In 1956, Marcus introduced a weak-overlap model1-4 for outer-sphere electron transfer (ET) reactions, and this model, with a variety of extensions, has evolved into a variety of more general interacting-states theories that have proved useful not only for weak-overlap ET reactions1-16 but also for systems with a larger resonance interaction, such as strong-overlap electron transfer,17 hydrogen atom transfer,18 proton transfer,13,19-22 hydride transfer,23,24 SN2 reactions,25 and more generally for all kinds of charge-transfer and reaction dynamics in solution and the gas phase. A central element of the original theory and all its extensions is a pair of interacting states or intersecting energy curves corresponding to diabatic initial and final states of the reaction. The energy curves are approximated as quadratic and represent the free energy of the system (for each diabatic state) as a function of a solvent dielectric polarization variable or the potential energy of the system as a function of some collective set of all the coordinates of the system.1,2 The difference in free energy between the intersecting energy curves corresponding to these states is used to define a collective solvent coordinate,1,2 a concept that is more general than the theory in which it arose.22g,41-43 In generalizations of the theory, these curves may represent the potential energy of the system as a function of some collective set of all the coordinates of the system; the flexibility or ambiguity of the independent variable allows the generalization of the theory to a myriad of processes. Another critical element is that reducing the physics of solvation to a one-dimensional picture yields tractable equations even for complex processes. One very general perspective on charge-transfer theory, summarized elsewhere,30,44 is that it corresponds to a quadratic free-energy relationship with physical interpretations of the parameters. Linear free-energy relationships, in which the free energy of activation is assumed to be a linear function of the free energy of reaction (or, equivalently, in which log k is assumed to be a linear function of log K, where k is a rate constant and K is an equilibrium constant), have proven enormously successful,45 but have obvious limitations. The central equation of charge-transfer theory is one in which the free energy of activation is a quadratic function of the free energy of reaction. The general picture of interacting states leads to relationships between the parameters of the theory, e.g., a cross relation involving the rate constant for a degenerate rearrangement (i.e., a reaction in which the initial and final states are the same, e.g., Fe2+ + Fe3+ f Fe3+ + Fe2+ or Cl- + CH3 f ClCH3 + Cl-), which makes the free energy relationship a useful one. From this perspective, an advantage of chargetransfer theory is its ability to use empirical data to construct a

J. Phys. Chem. B, Vol. 105, No. 40, 2001 9673 rate constant without building a microscopic description of the system. Two parameters, the reorganization energy, λ, and the standard reaction free energy, ∆G°, play a very prominent role. A quantum mechanical transmission coefficient κ can be included in charge-transfer theory to account for dynamical effects, in particular, tunneling or nonadiabatic reflection at the transition state or both. In weak-overlap charge-transfer theory, the activation step is treated classically, whereas the particle transfer step is assumed to occur along a one-dimensional cut and can be treated quantum mechanically. For the truly weak overlap case one would expect a transmission coefficient significantly less than unity. The transmission coefficient is sometimes approximated by Landau-Zener theory4,6,8,11,21,46-48 based on the probability PLZ that a system reaching the transition region along the reactant diabatic potential will switch to the product diabatic potential; however, this requires further assumptions about the nature of the process, and even if these are valid, one still has significant reservations about a onedimensional theory of nonadiabatic transitions in multidimensional systems.49 A general framework for estimating κ requires not only the diabatic potential curves that cross but also the coupling between the diabatic states, which allows specification of adiabatic potential curves that do not cross. Reaction requires an adiabatic traversal of the transition state region in which the system switches from the reactant diabat to the product one. In general, the transmission coefficient should account for tunneling through the adiabatic potential, adiabatic recrossing of the transition state, nonclassical adiabatic reflection by the barrier top, and diabatic reflection by coupling to the upper adiabat, and κ is related to the overlap of localized diabatic wave functions. For electron transfer, these would be the tails of the diabatic electronic wave functions, and one would be considering electronic tunneling or the “resonance” interaction of reactant and product valence bond wave functions.4 For proton transfer, these would be the tails of protonic wave functions,21,22 and only the lowest BornOppenheimer potential energy surface would be involved. Variational transition state theory36-40 provides another exceedingly general approach to chemical reactivity. Transition state approaches are based on adiabatic electronic states (in particular on the Born-Oppenheimer ground state), and they have a close relation to the adiabatic theory of reactions.50,51 Many workers have commented on the relationships between weak-overlap charge-transfer theory and transition state theory,2,4,6,30,32-34 and a goal of the present paper is to present new perspectives on these relationships. In addition, we wish to show how the energy-gap solvation coordinate can be used in VTST and to illustrate how the resulting approach, based upon adiabatic electronic states, is related to charge-transfer theory, which is based upon diabatic interacting states. As popularized by Eyring, transition state theory (TST) postulates a quasiequilibrium between reactants and transition states and a universal rate of conversion (k˜T/h, where k˜ is Boltzmann’s constant, T is temperature, and h is Planck’s constant) of transition states to products.52 The transition states, although sometimes called activated complexes, are recognized to be unstable species as opposed to true complexes or intermediates. In fact, the transition states have only 3N - 1 degrees of freedom (where N is the number of atoms), as opposed to 3N for the reactants or a real complex; a system that is missing one degree of freedom is a hypersurface or, for short, a surface (such as a 2-D plane in a 3-D volume). The calculation of a reaction rate is reduced to the calculation of the free energy difference between the transition state and the

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reactants; since these species have a different number of degrees of freedom, this is not a real free energy (although it is precisely defined mathematically) and hence it is called a free energy of activation. One of the great original triumphs of TST was the explanation of kinetic isotope effects by the isotopic dependence of the free energy of activation.53 Another attractive feature of TST, still largely a dream for most reactions but almost sure to guide a considerable amount of 21st century research, is its ability to predict absolute reaction rates from the properties of limited regions of the potential energy surface.29 One can get a truer appreciation of the meaning of transition state theory from Wigner’s approach.26,54 Wigner formulated transition state theory more dynamically as the equilibrium flux of phase points (also called trajectories) through a hypersurface (the “dividing surface”) that separates reactants from products. The equilibrium flux through the hypersurface has a form identical to k˜T/h times the Boltzmann factor of the free energy difference ∆G‡ between a hypothetical (3N - 1)-degree-offreedom system in the hypersurface and the 3N-degree-offreedom reactants. That is

kTST )

κ exp(-β∆G‡) βh

(1)

where β ≡ 1/k˜ T and κ is a transmission coefficient that is unity according to this derivation. (Pedagogical derivations of this equality are available elsewhere.35,40) This justifies Eyring’s formula and enables us to understand when the theory does or does not break down, at least in a classical world: if reactants are at equilibrium and if all trajectories that pass through the transition state dividing surface in the direction of products have come directly from reactants and will proceed directly to products (the first “directly” means without having already crossed the transition state dividing surface, and the second “directly” means without crossing it again), then TST is exact. Thus, the transition state should be located at a dynamical bottleneck that is a point of no return on the way from reactants to products. Conventional transition states are located at saddle points; in an attempt to improve the theory one can consider other transition states, which are called generalized transition states to be precise, but sometimes are just called transition states. Variational transition state theory28-30,36-40,51 (VTST) has its roots in the early days of TST.36 In VTST, one locates the transition state in a place that minimizes the equilibrium flux through it. A consequence of the background summarized above is that this is equivalent to maximizing the generalized free energy of activation. Equation 1 is also used in the standard version2,25c of WOCTT, although sometimes one finds alternative prefactors for electron transfer reactions for two reasons: First, for weak-overlap electron transfer, the charge localized states provide unambiguous diabatic states which may be used to define coupling elements and estimate the extent of nonadiabatic behavior. Second, nonadiabatic reflection is a more important concern for weak-overlap electron transfer than for heavy-body transfers. In contrast, when WOCTT is applied to heavy-particle charge transfer (e.g., proton transfer), the diabatic states are not uniquely defined and any attempt to improve on eq 1 involves additional approximations that require justification on a case-by-case basis. Thus we use eq 1. Note that eq 1 strictly applies only for dividing surfaces that are hyperplanes in rectilinear coordinates.40 For more general dividing surfaces such as interatomic distances there are additional small terms arising from the spatial derivatives of the dividing surface. These are small and almost always neglected.

It has been found that to obtain quantitatively accurate estimates of rate constants, particularly for reaction coordinates involving motion of a light particle (such as H, H+, or H-), zero point vibrational requirements and quantum mechanical effects on reaction-coordinate motion must be included.51,55,56 Zero point effects and discrete vibrational energy spacings can be added to ∆G‡ by using the quantized forms of the traces,51 and multidimensional tunneling contributions can be added by using a transmission coefficient that accounts for the competition between tunneling and overbarrier processes.55,56 In conventional WOCTT, only the reaction coordinate is treated explicitly and it is taken to be a pure energy gap coordinate. In recent years, a number of multidimensional theories have been proposed that combine an energy gap coordinate with explicit treatment of one or more other solute coordinates, and this allows a gradation of dynamical approximations. For example, if the energy gap collective solvent coordinate is assumed to adjust adiabatically (i.e., instantaneously) to a reaction coordinate consisting entirely of solute coordinates, we obtain the equilibrium solvation limit. In general, though, one can allow the reaction coordinate to be a linear or nonlinear combination of solute coordinates and the energy gap coordinate, which allows for nonequilibrium solvation of the reacting solute when the solvent time scale is not infinitely fast compared to the solute one. At the other extreme, the solute may be assumed to adjust infinitely rapidly to the energy gap coordinate, in which case the reaction coordinate reduces to the energy gap coordinate, and, with further assumptions, one can rederive WOTCC. In fact, such a rederivation is a central element of the present paper. We employ the framework of VTST, which in principle can describe the reacting system with a full set of atomic coordinates of the solute plus an energy gap coordinate, to describe the dominant collective effect of the solvent. We show how an energy gap coordinate suitable for this purpose may be defined in a completely general way without reference to valence bond structures or other system-specific models such as various charge-localized states. We then show how a systematic series of approximations leads to WOCTT. The motivation of this derivation is two-fold. First, we believe it provides new insight into the assumptions implicit in WOCTT. Second, the theory presented here paves the way for more general applications of the extremely powerful energy gap coordinate to systems where solute motion is not infinitely rapid compared to solvent reorganization. A Multidimensional Test Case The present article shows how an energy-gap coordinate can be treated by VTST. Because the goal of this paper is to clarify the underlying dynamical assumptions rather than to duplicate the many excellent papers presenting the full mathematical details of TST and weak-overlap charge-transfer theory, we will present the ideas in terms of a single example reaction, namely the proton-transfer reaction:

AH+ + B f A + BH+

(2)

in solution, where A and B are arbitrary molecules, with NA and NB atoms, respectively. This model will allow us to define the energy-gap coordinate in terms of the microscopic potential and reduce the model to a two-dimensional system from which we can derive the parameters required in weak-overlap chargetransfer theory and those entering variational transition-state theory. We assume that the reaction is not so fast as to be

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diffusion limited, and we recognize an encounter complex as a reaction precursor. Thus we consider

AH+ ... B f A ... H+B 1 2

(3)

where the product is a successor complex (an encounter complex of the reverse reaction). The bimolecular rate constant for AH+ + B involves an additional free energy contribution to bring the reactants from AH+ + B to AH+...B (this is usually called the “work term” in charge-transfer theory, and it is similar to the free energy of formation of the Michaelis complex in enzyme kinetics modeling), but we do not consider this. Instead, we focus on the unimolecular rate constant of the process in reaction 3. Let r be the protonic coordinate, which has three dimensions; let R be the collection of all the coordinates of A and B; and let S denote the collection of the coordinates of the solvent molecules. The dimension of R is (3NA + 3NB), the combined dimensions of r and R is F ) 3(NA + NB + 1), and the dimension of S is very large (call it NS; then log 3NS ≈ log No, where No is Avogadro’s number for our example in solution, although NS would be smaller for reactions in clusters or even 0 in the absence of solvent). Our starting point is the full potential of the system, V(r,R,S). In a brute force treatment, VTST puts all F + NS degrees of freedom on an equal footing. Starting from the full potential, a minimum energy path connecting reactants to products through a saddle point is constructed. The signed distance along this path defines a one-dimensional reaction coordinate s that is free to contain all degrees of freedom, both solute and bath. In contrast, charge-transfer theory involves the separation of the solute charge-transfer coordinate from other degrees of freedom and introduces a collective bath coordinate that effectively represents solute noncharge-transfer coordinates and the bath coordinates. The solute charge-transfer coordinate and the collective bath coordinate play privileged roles in the theory. An advantage of weak-overlap charge-transfer theory is the ability to use empirical data to construct a rate constant without building a microscopic description of the system. A parameter called the reorganization energy, λ, which is introduced along with the collective bath coordinate, is used to replace a considerable amount of microscopic detail. However, VTST can also use collective coordinates. One can define these rather generally16,22,24,42,43,57-65 and retain only the few most important ones or even only the single most important one. (Thus the bath may have F + NS - 1 degrees of freedom, as in the original weak-overlap charge-transfer theory, but it may also have less.) Furthermore, in VTST one can map the Hamiltonian such that the equilibrium free energy of solvation is added to the solute potential for zero values of the collective solvation coordinate or coordinates; this allows one to incorporate equilibrium solvation effects fully and to model nonequilibrium solvation effects realistically.63-66 For purposes of discussion, after the general treatment is presented, we will consider a two-dimensional model that contains the most important qualitative features necessary to show the connection between a microscopic potential, the parameters required in weak-overlap charge-transfer theory, and the nonequilibrium formulation of VTST. First, though, we proceed further with the general treatment. For the general treatment, V(r,R,S) is the potential energy function of a system having the two equilibrium states of reaction 3, including their associated equilibrium solvation structure. In 1, the proton is localized in the vicinity of A, corresponding to the reactant state,

and in 2 it is localized in the vicinity of B, corresponding to the product state. Equilibrium Solvation Path We will first assume that a solute reaction coordinate, q, can be defined as a function of r and R alone, q ) q(r,R). This section discusses a procedure for doing this. The potential of mean force (PMF) surface, W(r,R), is defined to within a constant by

e-βW(r,R) ) TrS[e-β[TS+V(r,R,S)]]

(4)

where TrS[...] denotes a trace over the S coordinates with the solute coordinates fixed, β ) k˜T, and TS is the kinetic energy in the solvent coordinate. In classical mechanics, the trace operation over a set of coordinates corresponds to integrating over the entire range of these coordinates and their conjugate momenta. In quantum mechanics, the trace of an expression containing a Hamiltonian may be evaluated by summing its value over all its eigenstates. Alternatively, a quantum mechanical trace may be evaluated by path integrals. A trace over solvent coordinates is evaluated by an integral over the entire solvent phase space. Readers unfamiliar with these operations should keep in mind that Trx is an addition operation that converts e-βH(x) to e-βFE, where x is an arbitrary coordinate, H(x) is a Hamiltonian, and FE is a free energy G (if x denotes all coordinates) or a potential mean force W (if x is a subset of all the coordinates). Thus, it is like an averaging step without the normalization constant; relative free energies and potentials of mean force can be evaluated in classical mechanics by the use of appropriate sampling methods, using Monte Carlo integration67 or thermalized molecular dynamics,68 for example. Note that equally good names for W would be equilibriumsolvation potential or reduced-dimensionality free energy function, but we will adhere to the PMF language since it is well established. Note that when x denotes the solvent coordinates, W equals the sum of the solute potential energy function and the fixed-solute free energy of equilibrium solvation. The reactant state 1, the product state 2, and the saddle ‡, correspond to stationary points of W, i.e., points where ∂W/∂r ) ∂W/∂R ) 0. A minimum energy path, called the equilibrium solvation path63,66 (ESP), connecting these points may then be constructed by considering the steepest decent path in massscaled coordinates51,69 in both directions from the equilibrium solvation saddle point. The reaction path so defined extends from ‡ to 2 and from ‡ to 1, but for graphical purposes we will extend it beyond 1 and 2 in a reasonable way. At each point along the equilibrium solvation path, it is possible to define a local coordinate system where q measures signed distance along the path, and coordinates transverse to the q coordinate are collected in coordinates Q of dimension F - 1. The coordinates q and Q are mass-scaled; that is, they are defined so that they have a common effective mass mq; this can be accomplished, for example, by multiplying the Cartesian coordinates of each atom X by (mq/mx)1/2 where mx is the mass of X. By convention, we take the location of the saddle point to correspond to q ) 0. (One possible choice for Q, at least in the vicinity of the ESP, is the set of local generalized normal modes that are obtained from the second derivative matrix of W with the reactive motion removed by construction38 or by projection.70) We assume a well-behaved one-to-one transformation from (r,R) to (q,Q) and define the potential of mean force surface in the (q,Q) space as W(q,Q) ) W[r(q,Q), R(q,Q)].

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Schenter et al. a vertical excitation energy, or difference in diabatic potentials, with fixed values for all coordinates in the system except for the charge-transfer coordinate, which in this case describes the proton motion. In our case, the diabatic potentials are defined by the potential energy surfaces V(q1,Q,S) and V(q2,Q,S), which are evaluated at the same fixed values of Q and S, but for values of the charge-transfer coordinate q at two characteristic solute configurations corresponding to the reactant and product minima, q1 and q2. The energy gap is then defined by

∆E(Q,S) ) V(q1,Q,S) - V(q2,Q,S)

(6)

(Note that in general ∆E should also include contributions from kinetic energy terms, but to obtain weak-overlap charge-transfer theory we assume there is no momentum coupling between q and Q and S so that this contribution vanishes.) The central theoretical construct of Marcus theory, the energy gap coordinate, x, is a collective bath coordinate that describes all configurations for which x ) ∆E(Q,S). Diabatic PMFs in this collective bath coordinate may be defined by

e-βWi(x) ) TrQ,S[e-β[TQ+TS+V(qi,Q,S)]F∆E(x,Q,S)] Figure 1. Potential energies for a 2D model of a solute coordinate q coupled to an energy gap bath coordinate x. See eqs 43, 44, and 48 for a description of the model. (a) Solute potential of mean force WESP(q) along the equilibrium solvation path (dashed curve), and potential in the solute coordinate for solvent coordinate fixed at x ) 0, W(q,x ) 0) (solid curve). (b) Diabatic potential in the energy gap coordinate, x, for solute coordinate q fixed at q1 and q2 (i.e., the Marcus parabolas). Note that the figure is rotated by 90° so the x axis coincides with that in part (c). The difference in the equilibrium solvation potential between reactants and products, ∆W, solvent reorganization energy, λ, and difference in PMF between the crossing point and reactants ∆W‡, are indicated in the figure. (c) Contours of equal potential energies for the 2D model. The straight dashed line connecting the reactants, saddle point, and products is the equilibrium solvation path. The potential along this dashed line is depicted as a dashed curve in part (a). The thick solid lines show the 3-step pathway of weak-overlap charge-transfer theory. The vertical dashed lines are at the minima, q1 and q2, of the equilibrium solvation potential, and the horizontal dashed lines are at the minima, x1 and x2, of the Marcus parabolas.

It will be convenient in the following development to put the transverse coordinates Q on the same footing as the solvent coordinates S. The PMF for q motion can be rewritten

e-βWESP(q) ) TrQ,S[e-β[TQ+TS+V(q,Q,S)]]

(5)

where TrQ,S[...] denotes a trace over the (Q,S) coordinates, TQ is the kinetic energy for Q coordinates, and V(q,Q,S) ) V[r(q,Q),R(q,Q),S]. The general structure of WESP(q) is displayed as a dashed curve in part (a) of Figure 1 (parts (b) and (c) of Figure 1 and the solid curve in part (a) are discussed below). Notice that we have labeled the locations of the minima q1 and q2 corresponding to 1 and 2 in reaction 3. Energy-Gap Coordinate in Charge-Transfer Theory In the original versions of weak-overlap charge-transfer theory, coordinates (Q,S) are replaced by a single effective solvent coordinate, which we call x, defined in terms of the energy gap. For the model discussed above, the energy gap in charge-transfer theory is the instantaneous change in solvation energy upon moving the proton from A to B or in going from 1 to 2 in reaction 3. Generally the energy is written in terms of

(7)

where F∆E (x,Q,S) is a probability density, which is proportional to δ[x - ∆E(Q,S)]. By construction, the two diabatic potential curves cross at x ) 0 [i.e., W1(x ) 0) ) W2(x ) 0)], since V(q1,Q,S) ) V(q2,Q,S) for x ) 0. The curves Wi(x) are usually approximated by parabolas in terms of their quadratic expansions

Wi(x) ) Wi° + (1/2)Ki(x - xi)2, i ) 1, 2

(8)

about their local minima at xi, where Wi° are the Values of WESP at the two minima, Wi° ) WESP(qi). Note that WESP(qi) can be obtained from eq 5. The quadratic expansions in eq 8 are the “Marcus parabolas,” and they have the form depicted by the curves in Figure 1b. The difference in PMFs from reactants to products is

∆W ) W2(x2) - W1(x1) ) W2° - W1°

(9)

We have now identified a single “solute” progress variable, q, and a single “bath” coordinate, x, in terms of r, R, and S. Constrained averages over a full thermal distribution of r, R, and S with constraints placed by q or x, have been used to define effective potentials in both the q and x coordinates, e.g., see eqs 5 and 7. Whereas the PMFs in q are intuitively clear because q is a familiar solute coordinate, the PMFs in x are less intuitive because they are collective coordinates (one value of x describes many possible values of Q and S), which have units of energy. The PMFs in x are the central element of weak-overlap chargetransfer theory. In the original form of weak-overlap charge-transfer theory,1 the rate constant was estimated as a collision frequency times exp(-β∆G‡). In subsequent work,2 the rate constant was expressed as in eq 1. We will use the latter formulation of weakoverlap charge-transfer theory. Thus, since both VTST and weak overlap charge-transfer theory use eq 1, we can compare their predictions for rate constants by comparing their predictions for ∆G‡, and this is the way we will proceed. The free energy of activation from the reactants at x ) x1 to the crossing point at x ) 0 is given by

∆G‡ ) W1(x ) 0) - G1

(10)

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where the reactant free energy is written in terms of the reactant partition function by

e-βG1 ) Q1 ) Trr,R,S[e-βHθ[-q(r,R)]] ) Trq,Q,S[e-βHθ(-q)] ) Trq[e-β[Tq+WESP(q)]θ(-q)]

(11)

The reactant partition function is a sum over all reactant configurations, which we define as those with q < 0; and θ(z) is a Heaviside step function [θ(z) ) 0, z < 0, θ(z) ) 1, z > 0]. The free energy of activation can be rewritten as

˜1 ∆G‡ ) W1(x ) 0) - W1° + k˜T lnQ

(12)

where Q ˜ 1 is the reactant partition function with the zero of energy taken as W1°

Q ˜ 1 ) Q1/exp(-βW1°) ) Trq[exp{-β[Tq + WESP(q) - W1°]}θ(-q)]

(13)

In weak-overlap charge-transfer theory, the contribution from the partition function is often neglected, in which case ∆G‡ is approximated by

∆W‡ ) W1(x ) 0) - W1°

(14)

This expression for the activation free energy is usually written in terms of two other quantities: the free energy of reaction, ∆G, and the reorganization energy, λ. The absolute free energy difference between reactants and products is given by

Q ˜2 ∆G ) G2 - G1 ) ∆W - k˜T ln Q ˜1

(15)

where G2 is the product free energy and is given by the same expression as eq 11 but with θ(-q) replaced by θ(q), and the product partition function Q ˜ 2 is defined by an expression similar to eq 13, but with W1° replaced by W2° and θ(-q) replaced by θ(q). In weak-overlap charge-transfer theory, the ratio of partition functions in eq 15 is often neglected, and ∆G is approximated by ∆W. The reorganization energy is a measure of the cost in energy to change the solvent around the reactant solute geometry from an equilibrium solvent configuration to a solvent configuration that would be in equilibrium with the solute at the product geometry:

λ ) W1(x2) - W1(x1)

(16)

In the case where W1(x) and W2(x) are harmonic with the same force constant (as assumed by Marcus), eqs 8, 9, 14, and 16 yield

∆W‡ )

(λ + ∆W)2 4λ

(17)

Since ∆W‡ and ∆W are assumed to approximate ∆G‡ and ∆G, respectively, one often sees this written as

(λ + ∆G) ∆G ) 4λ ‡

2

(18)

In a TST calculation with a unit transmission coefficient, it is sufficient to specify the transition state dividing surface, and

the global definitions of the reaction path and reaction coordinate are irrelevant. However, in practice, the choice of reaction path or reaction coordinate can be used to motivate or justify the location and orientation of the transition state dividing surface, and it plays a critical role in developing models for transmission coefficients. (TST with unit transmission coefficient is a local theory, but the transmission coefficient is intrinsically global). In charge-transfer theory, solvent reorganization or motion along the collective bath coordinate, x, plays the role of a reaction coordinate.1-5,13,16,42 By treating the pure bath coordinate as the reaction coordinate for the initial and final segments of the path, the reaction can be viewed as proceeding in three steps. The first step involves activation at fixed q equal to q1, along the bath coordinate, taking the system from a reactant bath configuration, x ) x1, to the configuration where the two Marcus parabolas cross, x ) 0. This step represents overcoming the free energy of activation through solvent rearrangement; it is often described as a fluctuation of the solvent environment. Next, at a fixed set of bath configurations corresponding to x ) 0, the system progresses along the q coordinate from a state that is localized on the reactant side at q ) q1 to a state that is localized on the product side at q ) q2. This corresponds to tunneling from one diabatic state to another. Finally, the system deactivates along the bath coordinate with q fixed at q2 to rest in a product configuration with x ) x2, representing deactivation through solvent motion. In general, the rate constant for such a process is a product of a factor associated with the probability of bath activation and a factor associated with the central, tunneling step. The Energy-Gap Coordinate in Variational Transition-State Theory The application of VTST to reactions in solution has been previously discussed from Wigner’s perspective, in which the rate constant is proportional to the reactive flux through a dividing surface separating reactants from products.29,63 In the current work we summarize this perspective because it provides a convenient procedure for introducing the energy-gap coordinate in VTST and it will allow us to make a more direct comparison of the VTST approach with weak-overlap chargetransfer theory. Taking Wigner’s perspective, the activation free energy in eq 1 can be expressed as a trace over configurations of the system that are constrained to lie on a dividing surface defined by f(r,R,S) ) f(q,Q,S) ) 0:

e-β∆G ) Trr,R,S[e-βHFDS(r,R,S)]/Q1 ‡

) Trq,Q,S[e-βHFDS(q,Q,S)]/Q1

(19)

where H is the total Hamiltonian, FDS(r,R,S) and FDS(q,Q,S) are distribution functions that are proportional to δ[f(r,R,S)] and δ[f(q,Q,S)], and Q1 is given in eq 11. In the equilibrium solvation approximation, the dividing surface is chosen to be a function of the solute coordinates only, and for conventional TST the dividing surface is simply given by f(q,Q,S) ) q. For this choice of dividing surface, the activation free energy is given by

∆G‡ ) WESP(q ) 0) - G1

(20)

In VTST the dividing surface is chosen to minimize the rate constant or maximize ∆G‡. In the equilibrium solvation approximation, dividing surfaces that are functions of only solute coordinates are considered, whereas nonequilibrium solvation

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Schenter et al.

treatments consider dividing surfaces that are also functions of the solvent coordinates.29,60-65,71 Treating nonequilibrium solvation within VTST on the full anharmonic potential V(r,R,S) is often not feasible, and a convenient way to proceed is to first evaluate an equilibrium solvation potential, W(r,R) or WESP(q), then introduce dynamical solvent effects through a generalized Langevin equation57,59,62-64 (GLE) with effective solvent coordinates y. The advantage of the GLE approach is that the effective solvent coordinates are harmonic coordinates that are bilinearly coupled to the solute coordinates. (This is sometimes called “linearized solvent dynamics” since the Hamiltonian contains terms that are at most quadratic in effective solvent coordinates, and hence the forces are linear in these coordinates.) For the case of n solvent coordinates, the nonequilibrium solvation potential may be written

VNES(q,Q,y) ) W(q,Q) + ∆V(q,Q,y)

(21)

where W is still the PMF, and n

∆V(q,Q,y) )

kj

F

˜ i)2 (yj - ∑CjiQ ∑ j)1 2 i)1

(22)

˜ i ≡ Qi where kj is a force constant, Cji is a coupling constant, Q for i ) 1, ..., F - 1, and Q ˜ F ≡ q. Note that the coupling terms (those terms in eq 22 that contain both solvent coordinates and solute coordinates) are bilinear, i.e., linear in both solvent and solute coordinates. For this reason, the generalized Langevin approach is a special case of the linear response72 approximation. The parameters of the GLE model, kj and Cji, are related to the friction tensor of the GLE by the relationship

∑

j)1

(x ) kj

n

ηii′(t) )

CjikjCji′cos

my

t

(23)

where my is the mass used for the bath oscillator coordinates. The friction tensor is given in terms of the force autocorrelation function

ηii′(t) ) β〈δFi(t)δFi′(0)〉q,Q

(24)

where the constrained average is given by

〈‚‚‚〉q,Q )

TrS[e-βH‚‚‚]

e-βW(q,Q) ) NTry[e-β[Ty+VNES(q,Q,y)]]

(27)

where Ty is the kinetic energy for y coordinates (with diagonal terms pj2/2my) and

N ) 1/Try[e-β[Ty + ∆V(q)0,Q)0,y)]]

(28)

Therefore,

e-β(∆G +G1) ) N Trq,Q,y[e-β[Tq+TQ+Tt+VNES(q,Q,y)]FDS(q,Q,y)] (29) ‡

For a dividing surface dependent only on solute coordinates, e.g., f(q,Q,y) ) f(q,Q), eq 20 is recovered, that is, conventional TST in the equilibrium solvation approximation. More general dividing surfaces that are also functions of y recover nonequilibrium solvation effects. For any value of the solute coordinates q and Q, the equilibrium solvation potential is recovered from VNES(q,Q,y) by minimizing VNES with respect to the yj coordinates to give the equilibrium solvation path defined by yj ) ∑i)1FCjiQ ˜ i. A nonequilibrium solvation path can be defined by following the path of steepest descent in the mass-scaled coordinates from the saddle point (q ) 0, Q ) 0, y ) 0), to reactants and products. As noted above, coordinates q and Q are defined to have the same mass mq. The solvent coordinates y are not physical coordinates of the solvent, but effective oscillators. The choice of solvent mass my is arbitrary (the force constants and coupling terms in eq 23 will change with a change in the choice of my, but my cancels out in physical observables). For convenience we choose my ) mq so that (q,Q,y) represent a set of massscaled coordinates. A convenient family of dividing surfaces can be defined as those orthogonal to the nonequilibrium solvation path and characterized by a single parameter, the distance s along the nonequilibrium solvation path. We now wish to use the concept of an energy-gap coordinate, as defined by x ) ∆E(Q,S) and eq 6, in VTST. As in weakoverlap charge-transfer theory, the solute is described by a single coordinate q, and we wish to reduce the coordinates Q and S to a single collective coordinate x. In this case, y represents the bound solute coordinates Q as well as the solvent coordinates S, and eqs 21 and 22 simplify to

VNES(q,y) ) WESP(q) + ∆V(q,y)

(25)

TrS[e-βH]

(30)

and

and fluctuations in the force on coordinate i about its mean force are given by

〈

We define the appropriate averages over y so that the average of e-βVNES(q,Q,y) yields e-βW(q,Q), i.e.,

〉

∂V(q,Q,S) ∂V(q,Q,S) δFi ) + ∂Q ˜i ∂Q ˜i

n

∆V(q,y) )

kj

(yj - Cjq)2 ∑ j)1 2

(31)

(26)

q,Q

Equations 23-26 show how the parameters of the GLE model can be related to the underlying potential energy surface for the system. The GLE framework is not exact because it assumes linear response of the solvent to the solute. However, the linear response approximation is often expected to be reasonable, and phenomenologically one rarely finds evidence for deviations from linear response. Furthermore WOCTT also assumes linear response, so it is not an additional approximation when invoked in VTST. Extensive previous work on the combination of VTST with the GLE is reviewed elsewhere.29

where WESP(q) is given by eq 5. The energy gap coordinate x ) ∆E(y) defined in eq 6 takes the form

x ) ∆E(y) ) VNES(q1,y) - VNES(q2,y)

(32)

The nonequilibrium solvation potential is defined by an expression analogous to eq 7, but with q used as a variable rather than set to q1 or q2

e-βVNES(q,x) ) Try[e-β[Ty +VNES(q,y)]F∆E(x,y)]

(33)

where Ty is the kinetic energy for y coordinates (with diagonal

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J. Phys. Chem. B, Vol. 105, No. 40, 2001 9679

terms pj2/2my), and the probability density is proportional to δ[x - ∆E(y)]. Appendix A provides an evaluation of eq 33, and the resulting expression for VNES(q,x) is

K VNES(q,x) ) WESP(q) + [x - x0(q)]2, i ) 1, 2 2

(34)

where

K-1 )

n

kjC2j (q2 - q1)2 ∑ j)1 kj

n

x0(q) ) -∆W + ) -∆W +

(35)

C2j (q2 - q1)(2q - q1 - q2) ∑ j)1 2 (2q - q1 - q2) (2K)-1 (q2 - q1)

(36)

Comparing eqs 7 and 33, we see that the Marcus parabolas are just special cases of VNES(q,x):

Wi(x) ) VNES(qi,x)

(37)

and they reduce to eq 8 with K1 ) K2 ) K and

xi ) -∆W + (-1)i(2K)-1

(38)

Equations 34-38 show how the parameters in charge-transfer theory are related to the parameters of the GLE model, and with eqs 23-26, they show how the parameters in charge-transfer theory can be related to the underlying potential energy surface. We can also obtain an explicit expression for x(y) relating the collective solvent coordinate to the solvent coordinates y (see eq A16 in Appendix A). Using this relationship, dividing surfaces in terms of the solute coordinate q and the solvent coordinate x can be used in the expression for the free energy of activation given by eq 29. The free energy of activation is given by

e-β(∆G +G1) ) ‡

∫ dxN Trq,y[e-β[T +T +V q

y

NES(q,y)]

FDS(q,x)δ(x - x(y))] (39)

Consider, for example, the case where the dividing surface f(q,x) ) x. The distribution function FDS(q,x) is proportional to δ[f(q,x)] so that the free energy of activation is given by ‡ ∆G‡ ) -k˜TlnQx)0 - G1

(40)

where the transition-state partition function is given by ‡ ‡ ) Trq[e-β[Τq+VNES(q,x)0)]] ) e-βWi(x)0)Q ˜ x)0 Qx)0

(41)

‡ where Q ˜ x)0 is the transition state partition function with zero taken as Wi(x ) 0). Using eqs 11, 14, and 41, eq 40 can be rearranged to read

‡ Q ˜ x)0 ∆G ) ∆W - k˜Tln Q ˜1 ‡

‡

It is also interesting to compare eq 42 with eq 12 and eq 14. ‡ set to 1, while it Equation 42 reproduces eq 12 with Q ˜ x)0 ‡ ˜ 1. reproduces eq 14 if Q ˜ x)0 ) Q The neglect of individual vibrational displacement force constants in weak-overlap charge-transfer theory has been estimated to lead to uncertainties of about 103 in estimating rate constants even for outer-sphere electron-transfer reactions,73 and coupling to other modes is also quantitatively important for proton transfer.21,22 Nevertheless, the theory leads to useful free energy relationships,28,29 and the use of a nonequilibrium solvation coordinate is a powerful tool for qualitative understanding as well.1-25 The most common approach to improving the theory quantitatively by including more degrees of freedom has been based on perturbation theory and the introduction of Franck-Condon factors;11,13,74 this approach introduces new physical approximations that are hard to improve systematically. The present synthesis, by showing how weak-overlap chargetransfer theory is related to variational transition state theory including all the vibrational coordinates, provides a route for systematically improving the theory by adding more coordinates in a general and widely applicable framework, both for the determination of ∆G‡ and for the inclusion of multidimensional tunneling effects. In eqs 39-42, the coordinate x is used only in defining the dividing surface, not in specifying a reaction coordinate that would be used in improved dynamical treatments. The coordinate system used in weak-overlap charge-transfer theory, (q,x), is not a convenient one for improved dynamical treatments, since it requires defining the momentum conjugate to x and determining an effective mass for the collective coordinate. Rather than proceeding in that direction, a more convenient approach is to use VTST on the effective nonequilibrium solvation potential in the (q,Q,y) coordinate system to systematically improve the dynamical treatment, including quantum mechanical effects. Qualitative understanding of the importance of nonequilibrium solvation effects can be obtained by transformation to the 2D system (q,x). The main focus of the present paper is to contrast the assumptions behind the weak-overlap charge-transfer approach with those behind VTST, and so we will not give details here about numerically implementing this approach for actual applications. A 2D Model For purposes of illustrating these complementary theories, we now consider the case where the solute is represented by a single coordinate, treated as the distance q along the equilibrium solvation path, so that the nonequilibrium solvation of the solute is described by eqs 30 and 31. Furthermore, we model the solvent with a single effective solvent coordinate y (where we have dropped the subscript 1 for the case of n ) 1). Then eqs 30 and 31 reduce to

VNES(q,y) ) WESP(q) + ∆V(q,y)

(43)

∆V(q,y) ) (k1/2)(y - C1q)2

(44)

(42)

Notice that the transition state partition function is expressed as a trace over q, just as in the reactant partition function in eq 13, for a solvent-averaged potential; however, the effective potential VNES(q,x ) 0) is distinct from the function WESP(q) that occurs in the reactant partition function, so that the ratio of partition functions in eq 42 does not, in general, cancel out.

and eqs 27-29 become

e

-βG‡

e-βWESP(q) ) N Try[e-β[Ty+VNES(q,y)]]

(45)

N-1 ) Try[e-β[Ty+∆V(q)0,y)]]

(46)

) N Try[e-β[Tq+Ty+VNES(q,y)-G1]FDS(q,y)]

(47)

9680 J. Phys. Chem. B, Vol. 105, No. 40, 2001

Schenter et al.

For purposes of illustration, the equilibrium solvation potential is taken to correspond to a double well of the form

WESP(q) ) V0 -

kq 2 k(4) q q (1 + δqq) + q4 2 2

(48)

2 where k(4) q ) kq/8V0, so that WESP equals 0 at the minima when δq ) 0. We consider a representative example of WESP(q) that corresponds to asymmetric charge-transfer and choose two values of the mass associated with the q motion, mq ) 1 and 15 amu (corresponding, for example, to proton transfer and X+ CH3Y f XCH3 + Y-). In the model system considered, we assume that WESP(q) is independent of mass. We set the remaining parameters equal to the following arbitrary but realistic values: V0 ) 6 kcal/mol, kq ) 318 kcal/mol/Å2, δq ) 0.5 Å-1, C1 ) 0.5. We choose a solvent frequency, given by (k1/mq)1/2, equal to 1014 s-1 or 531 cm-1. For mq ) 1 amu k1 ) 0.166 mdyn/Å and for mq ) 15 amu k1 ) 2.49 mdyn/Å. All calculations are for T ) 300 K. For this two-dimensional model, the VTST result for ∆G‡ can be calculated from eq 47, where the dividing surface is defined as the straight line orthogonal to the nonequilibrium reaction path that maximizes δG‡. We can apply the weakoverlap charge-transfer model to this 2D problem using the transformation derived in Appendix A and summarized in eqs 33-38. The expression for K reduces to

K-1 ) k1C12(q2 - q1)2

(49)

We note also that for this model eqs 14 and 16 reduce to

∆W‡ ) Kx21/2 ) (K/2)(∆W + (2K)-1)2

(50)

λ ) (K/2)(x2 - x1)2 ) (2K)-1

(51)

TABLE 1: Critical Geometries and Energies of the Model 2D Potential ωq,ESP (cm-1) q (Å)

and

The transformation between x and y is given by eq A16, which for this simple model reduces to

x(y) ) xi + ik1C1(q2 - q1)(y - C1qi)

Figure 2. Potential energy contours and minimum energy paths (MEPs) for a 2D model of a solute coordinate q coupled to a bath coordinate x. MEPs are shown for solute masses of 15 amu (short dashed curve), 1 amu (long dashed curve), and a very small mass that represents the approach to zero mass (solid curve) and are compared with the 3-step pathways of weak-overlap charge-transfer theory (thick line segments).

(52)

The saddle point occurs at q ) 0 and x ) x0 (q ) 0), and the potential energy at the saddle point is V‡ ) WESP(q ) 0). The equilibrium solvation path is then defined by x ) x0(q). The two-dimensional potential of eq 43 is shown in Figure 1c as a function of q and x. This figure also shows, as the dashed line, the equilibrium solvation path, and, as the thick solid lines, the three-step pathway of weak-overlap charge-transfer theory, in particular, activation along q ) q1 (which is often described as a fluctuation), charge-transfer along x ) 0, and deactivation along q ) q2. The equilibrium solvation path, in contrast, is obtained by following the path of steepest descent from the saddle point in the (q,y) coordinates and then transforming it to the (q,x) coordinates using eq 52. We emphasize that it is not necessary to find the steepest descent path in the (q,x) coordinates, which would require determining the effective mass for x. The equilibrium solvation potential along the equilibrium solvation path WESP(q) has already been shown in Figure 1a and is compared there with the potential along the cut x ) 0, W(q,x ) 0). Stationary point data are summarized in Table 1. The Marcus parabolas obtained by evaluating eq 8 for this model have already been shown in Figure 1b, and the constants characterizing these parabolas are ∆W ) -3.35 kcal/mol, λ ) 16.4 kcal/mol, and ∆W‡ ) 2.6 kcal/mol.

reactant -0.25 saddle 0 product 0.30

x WESP(q) (kcal/mol) (kcal/mol) -13.1 1.7 19.8

1.4 6.0 -1.9

1 amu

15 amu

ω‡ (cm-1) 1 amu

15 amu

2608 673 1936 i 500 i 1614 i 462 i 2891 746

Figure 1 clearly displays the different components that go into weak-overlap charge-transfer theory: the Marcus parabolas that describe the activation and deactivation along the solvent coordinate, and the potential along the x ) 0 curve that describes the charge-transfer. VTST takes a different perspective, one in which the solvent and solute coordinates are treated on an equal footing. Figure 2 shows minimum energy paths for this 2D model. The heavier solute mass corresponding to a methyl cation transfer (15 amu) shows a MEP that is close to the straight line equilibrium solvation path connecting reactant and product configurations. As the mass decreases the MEP goes to the thin solid curve. Notice that the MEP for the limit of zero mass is different from the 3-step pathway assumed in weak-overlap charge-transfer theory (and depicted as the thick solid line segments). We also note that this 3-step pathway does not go through the saddle point in this asymmetric example. Furthermore, even in the zero mass limit, the activation steps are not pure solvent reorganization (i.e., motion only in x); they also include contributions from the solute motion, even in the limit of zero mass. From the above discussion it is clear that the zero-solutemass limit of VTST does not recover electron transfer theory. In particular the activation steps are not pure solvent reorganization (i.e., motion only in x); they also include contributions from the solute motion. Additional modifications of the reaction path must be made for the limit to coincide with weak overlap chargetransfer theory. One could imagine modifying either theory so that the zero mass limit of VTST would recover WOCTT. The

Feature Article more natural approach would be to modify WOCTT because in the VTST approach the reaction path is a consequence of the Hamiltonian, and it is not postulated as a separate assumption, whereas in WOCTT it is an a priori assumption. In particular, WOCTT assumes that activation and deactivation involve only solvent coordinates. In VTST, all degrees of freedom are placed on an equal footing; therefore, the zero mass limit can result, more realistically, in solute degrees of freedom being involved. It turns out that the zero-mass limit of the VTST reaction path corresponds to generalizing the WOCTT formalism so that it includes the additional physical effect of solute participation in the activation and deactivation steps. Thus, whereas WOCTT intrinsically postulates a reaction path corresponding to three one-dimensional, separable segments, the VTST approach allows for either one-dimensional or multidimensional character in the reaction path, and the actual reaction path that results is a consequence of the linear response Hamiltonian of eqs 43 and 44 rather than a direct assumption. We can also explain this from an alternative viewpoint: in the zero-mass limit the light mass will adiabatically follow the heavy-atom motion, so the path will be determined by the contours of the potential and not necessarily be composed of straight-line segments. In particular, in the limit of zero mass, as the reaction path is traced from the transition state toward products in Figure 2, initially the reaction path is horizontal (i.e., pure solute coordinate q), but before it reaches q ) q2, the potential contours of eq 34 reach a local minimum along a horizontal-path cut, and then the reaction path turns toward the product well. Thus, the deactivation is not a pure solvent motion, as usually assumed in WOCTT. The reaction path goes into the product along a normal-mode direction; it is easily shown from a normal model analysis of eq 34 that this direction, even in the limit of zero mass of the transferred particle, is not a pure solvent motion, i.e., not a vertical motion in Figure 2. However, the distance along the path at which the reaction path merges with the product normal mode direction does depend on mass, and the two-dimensional reaction path of the full VTST theory does resemble the piecewise separable segments assumed in standard electron transfer theory in the zero mass limit, although there is still a difference. A similar argument explains that the activation step is not a pure solvent motion, but that it becomes more so as the mass of the transferred particle is reduced. The WOCTT approach would need to use a nonseparable reaction path for the activation and deactivation steps to agree with the multidimensional VTST reaction path even in the limit of the transfer of an infinitely light electron. The significance for the actual calculation of reaction rates by WOCTT of allowing solute degrees of freedom to participate in the activation and deactivation steps is quantitatively unknown and is a possible subject of future research. We next turn to a central objective of this article, namely to compare activation free energies computed using weak-overlap charge-transfer theory and VTST at 300 K. For the first part of this comparison, we first treat vibrations classically and harmonically. For weak-overlap charge-transfer theory, the activation free energy is included with the contribution from the reactant partition function, eq 10, and approximated without it, i.e., ∆G‡ ∼ ∆W‡ given by eq 14. The VTST free energy of activation is computed using four choices of dividing surface: (A) the dividing surface implied by weak-overlap charge-transfer theory, i.e., x ) 0, (B) conventional TST in the equilibrium solvation approximation, i.e., q ) 0, (C) conventional TST for the 2D model, i.e., the dividing surface is a function of q and x, but constrained to pass through the saddle point, and be

J. Phys. Chem. B, Vol. 105, No. 40, 2001 9681 TABLE 2: Harmonic Activation Free Energies (kcal/mol) for the 2D Model classical ∆W‡

WOCTT: WOCTT: ∆G‡ TST (x ) 0) TST/ES TST/NEQ VTST/NEQ VTST/NEQ/µOMT

quantized

1 amu

15 amu

1 amu

15 amu

2.6 1.1 1.5 3.1 3.2 3.2

2.6 1.9 1.5 3.9 3.9 3.9

2.6 -1.1 0.9 0.9 1.0 1.0 0.7

2.6 1.7 1.3 3.6 3.7 3.7 3.6

orthogonal to the imaginary frequency normal mode, and (D) the VTST dividing surface determined by optimization from among dividing surfaces orthogonal to the MEP. These results are summarized in Table 2. The fourth row in the table (which is choice B in the list above) is labeled ES because setting the collective solvent coordinate to zero corresponds to equilibrium solvation. The fifth and sixth rows (corresponding to choices C and D, respectively) are labeled NEQ to denote nonequilibrium solvation because q is not constrained to zero in the entire dividing surface. The last row of the table is discussed in the final paragraph of this section. Table 2 clearly shows that the assumptions of weak-overlap charge-transfer theory are significant and lead to quantitative errors. (For example, a 2 kcal/mol error in ∆G‡ corresponds to a factor of 30 error in the rate.) These errors arise because weakoverlap charge-transfer theory presumes the location of the bottleneck and the orientation of the dividing surface rather than optimizing them. The equilibrium solvation result of VTST is much more accurate, and even the conventional TST result (which is the one called TST/EQ in Table 2) is more accurate than the first two rows. The test example is close to harmonic, and the VTST calculation uses the harmonic approximation for coordinates (for example for x) transverse to q. But this is not necessary, and in principle VTST can retain high accuracy even for anharmonic cases. Also we note that for real systems practical techniques are now available to quantize the vibrations and add consistent multidimensional tunneling calculations in VTST.30,37,49,55,56 In the present 2D model, we use the harmonic approximation and we quantize all vibrational partition functions, but we do not include tunneling; these results are shown in Table 2. Furthermore, VTST never assumes quadratic potentials all the way from reactants to the transition state, but weak-overlap charge-transfer theory does. As a consequence, in weak-overlap charge-transfer theory, one has the freedom to choose K or ∆W‡ on physical grounds, and then eq 50 yields the other. In the present 2D model we effectively chose ∆W‡ by setting the parameters of VNES(q,y) in eqs 43, 44, and 48, and this yielded K. However, in principle, K follows from solvent dynamics, and in a pure application of weak-overlap charge-transfer theory one would evaluate K1 from a calculation on the reactant/solvent system (e.g., using eqs 23-26) and assume linear response up to the crossing point of the curves in eq 8 so that ∆W‡ is given by eq 14. This could lead to considerable additional error, except perhaps for the weak coupling limit of outer-sphere electron transfer. The present formulation of the solvent energy-gap coordinate as a variable in VTST allows one to determine K and ∆W‡ independently, which is more realistic. However, one does require a prescription for obtaining numerical values of the solvent parameters, C1 and k1, and values of parameters of the solute PMF, ∆W, q1, and q2, that determine K and ∆W‡. In eqs 49-52 we showed how the energy-gap solvent coordinate, which is the central construct of weak-overlap

9682 J. Phys. Chem. B, Vol. 105, No. 40, 2001

Schenter et al.

charge-transfer theory, may be related to the generalized Langevin solvent coordinate used in VTST. To actually use solvent energy-gap coordinates directly in VTST would, as mentioned earlier, require specification of the momentum conjugate to x and its associated reduced mass. This is possible in the case of the simplified potential of eqs 30 and 31 since we have explicit expressions for transformations between x and y; however, a simpler approach is to perform the VTST calculations in the (q,y) coordinates and then transform to the (q,x) coordinates. Finally, we consider the effect of tunneling. We will include this by the microcanonical optimized multidimensional tunneling approximation described elsewhere.55c This leads to κ ) 1.59 for the 1 amu case and κ ) 1.23 for the 15 amu case. To put these results on the scale of Table 2 we can follow the practice of experimentalists who equate eq 1 with κ ) 1 to the experimental rate constant in order to evaluate a phenomenological transmission coefficient that we will call ∆Gact. Thus

exp(-β∆Gact) ) κ exp(-β∆G‡)

(53)

The final entries in Table 2 correspond to this ∆Gact evaluated with the µOMT values of κ. The effect is to lower the apparent free energy of activation. The effect is small but not entirely negligible. However, for some proton-transfer reactions, κ may be even an order of magnitude larger. One of the advantages of the VTST formulation is that it is straightforward to add tunneling effects with well validated55a,56a methods. Discussion In this paper we have shown how a theory of weak-overlap charge-transfer reactions, similar to that formulated for electron transfer by Marcus1-4 and for proton transfer by Dogonadze et al.,19 may be related to variational transition state theory by including a collective coordinate (x) and using it to define a transition state dividing surface. The advantage of this formulation is that it suggests a way to systematically improve VTST by including nonequilibrium solvation effects. This may also be viewed as a way to improve charge-transfer theory by including a more realistic treatment of solute vibrational coordinates and multidimensional tunneling effects. The reaction coordinate and dominant tunneling path for actual proton transfers depend on the system; Marcus has suggested that the type of solvent reorganization assumed by the weak-overlap change transfer theory does not occur in systems with smallcurvature tunneling or very exothermic systems, although such a theory may be more appropriate for more symmetric systems with large-curvature tunneling.75 Nevertheless, the theory is often used qualitatively as a general framework for a whole range of reaction types, and this is the context in which we have considered it. Another powerful method for including multidimensional tunneling effects in condensed-phase reactions is the combination of TST with path integral methods.29,76,77 A pedogogical illustration of this combination for proton transfer in a collection of bath harmonic oscillators is available elsewhere.77 Although weak-overlap charge-transfer theory was originally limited to a single solvent coordinate, many improvements have been suggested that allow one to treat more coordinates on an equal footing,11-13,21-24,33,42,74,78 and the present approach is not the only way to do this. Our focus, however, has been on the original single bath coordinate of weak overlap charge-transfer theory. In several places in the presentation of the theory we intentionally used the generic term “bath coordinate” for x, and

we noted that x can include not only the solvent but all the degrees of freedom of the solute except the solute progress variable q. In practice, one often identifies x with just the solvent, and this leads to approximate models that can be very useful when the solvent is dominant among all the structural changes accompanying reaction progress. But the full theory requires that x be replaced by an [F + NS - 1]-dimensional set of coordinates. One can also apply charge-transfer theory when a single solute coordinate, rather than a solvent coordinate, dominates the structural rearrangement. Historically, a critical element in demonstrating the power of the theory was the introduction of a nonequilibrium electric polarization field to define the collective solvent coordinate,1,2,32,79 and this constitutes one of the more general approaches to reducing the theory to practice. But the generality of x is important when one uses the theory as a framework for wide applications to problems in organic chemistry and biochemistry. One objective of the present study was to recast the energygap coordinate representation of weak-overlap charge-transfer theory in terms of concepts employed in variational transition state theory, allowing one to utilize the systematic improvements that are possible in VTST. We have seen, as is well recognized in standard treatments,42 that this requires the assumption that the bath is linearly coupled to the solute, i.e., that the coupling potential ∆V has the pure quadratic form of eq 22. The validity of this linear response assumption for liquid-phase solvents is another issue, which has been widely studied in recent years. One generally expects that the linear response approximation may be reasonably accurate for long-range electrostatic interactions of the solute with bulk solvent polarization, but recent studies show examples of both success65 and failure80 for shortrange solute-solvent interactions. Perhaps because the explicit fluctuation language plays a more prominent role in charge-transfer theory than in TST, the question sometimes arises of the applicability of VTST for processes involving charge fluctuations in solution. In this regard we agree with Warshel:81 “All reactions involve dynamical fluctuations. The chance that the fluctuations will take the system to the transition state however is determined mainly by the relevant activation free energy.” A synthesis of the energy-gap solvent-coordinate concept with VTST provides a powerful, general way to calculate that free energy of activation because VTST without a nonequilibrium solvent coordinate cannot realistically treat those cases (e.g., outer-sphere electron transfer) where the bulk reorganization free energy of a liquid solvent is a major contributor to the reaction coordinate, and chargetransfer theory cannot give reliable results for cation or anion transfer without allowing for the realistic shape of the equilibrium solvation potential and the possible participation of many degrees of freedom. Although nonequilibrium solvation effects can be included in VTST by a many-mode (or single-mode) generalized Langevin approach based on memory friction57,63-65 or a single-mode generalized Langevin approach based on nonequilibrium electrostatics,43 a single-mode treatment based on an energy-gap solvent coordinate provides an appealing synthesis of the widely used VTST and charge-transfer pictures, and it therefore has conceptual advantages for building on existing models. We hope, therefore, that the perspective presented above is useful for understanding the general applicability of VTST for all chemical reactions. Concluding Remarks Many aspects of solvation are best described by explicitly treating the specific interactions of one or a few solvent

Feature Article molecules in the first solvation shell in terms of their individual atomic coordinates. However, a critical insight of Marcus theory is that the dominant effect of solvation is in many cases better described by a collective solvent coordinate that does not refer to a specific arrangement of solvent molecules, but rather that measures the electric field they generate.1 At an even higher level of abstraction, the electric field generates or contributes to an energy difference (gap) between two different valence bond structures of the solute that differ in their internal charge distribution. The value of the scalar energy gap may be taken as the value of a collective solvent coordinate, and this is more convenient than using a multidimensional vector field such as the infinite-dimensional instantaneous three-dimensional polarization vector as a function of position in three-dimensional space. A single value of this energy gap coordinate describes more than one set of explicit atomic coordinates of the solvent molecules, i.e., it describes an ensemble of solvent configurations. In the Marcus theory of weak-overlap electron transfer, the energy gap coordinate plays the role of the reaction coordinate, and, in fact, it is the only coordinate receiving explicit attention. This theory has been extremely successful, and it has been extended to the treatment of a variety of condensed-phase charge-transfer processes, typically retaining the weak-overlap assumptions that were originally postulated for outer-sphere electron transfer. We have used the term “weakoverlap charge-transfer theory” (WOCTT) to refer to this whole class of theories, and we have used a VTST formulation of the rate process to show how one can improve on WOCTT by a more evenhanded treatment of the various coordinates. This treatment then allows us to address a related question, namely the participation of the solvent in reaction coordinates for chemical reactions, which is a question of long and broad interest. One interesting aspect is the relationship between weakoverlap charge-transfer theories (loosely speaking, Marcus-like theories), which take a solvent energy-gap coordinate as the reaction coordinate, and Eyring-like transition state theory treatments that take a collection of solute coordinates as the reaction coordinate. We have shown how the energy-gap coordinate of charge-transfer theory can be used in variational transition state theory36-40 (VTST), and thereby the two kinds of coordinates can be treated on a nearly equal footing. Using this approach, we have illustrated the relationship of the diabatic interacting states perspective of weak-overlap charge-transfer theory1-4 to the formalism of VTST. This may be useful for providing a new perspective on charge-transfer theory1-25 and other recent work that takes a diabatic energy-gap coordinate as a reaction coordinate,41,42 and it should be especially useful for understanding the crossover12,13 between bath-controlled dynamics and solvent-independent vibrationally controlled dynamics. Perhaps the most important power of these theories is for organizing our qualitative thoughts on the factors that control relative reaction rates, and increased understanding of their conceptual basis can be useful for developing new ways of thinking about a variety of interesting chemical process. Acknowledgment. This work was supported in part by the National Science Foundation and by the Office of Basic Energy Sciences, U.S. Department of Energy. This research at Pacific Northwest National Laboratory was performed in the William R. Wiley Environmental Molecular Sciences Laboratory. Operation of the EMSL is funded by DOE’s Office of Biological and Environmental Research. Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle.

J. Phys. Chem. B, Vol. 105, No. 40, 2001 9683 Appendix A In this appendix we first define the nonequilibrium solvation potential VNES(q,x) in the energy gap coordinates, as defined in eq 33, starting from the nonequilibrium solvation potential VNES(q,y), in eqs 30 and 31 and the energy gap coordinate defined by eq 32. The probability density is given by

F∆E(x,y) ) C∆Eδ[x - ∆E(y)]

(A1)

where C∆E is a constant that will be determined later to set the zero of energy. Substituting eqs 30-32 and eq A1 into eq 33 and performing the momentum integrals yields the expression

e-βVNES(q,x) ) C′e-βWESP(q)

{

n

∫ dy1‚‚‚∫ dyn∏e(-βk /2)(y -C q) j

j

}

j)1

kj [(yj - Cjq1)2 - (yj - Cjq2)2] 2 j)1 n

δ x - ∆W -

j

∑

2

×

(A2)

where

C′ ) C∆E

( ) mq

n/2

(A3)

2πβp2

Defining the dimensionless variables ξ and zj and the dimensionless constants Rj by

[

n

ξ ) β x + ∆W -

]

kj

C2j (q2 - q1)(2q - q1 - q2) ∑ j)1 2

zj )

( ) βkj 2

1/2

(A4)

(yj - Cjq)

(A5)

Rj ) (2βkj)1/2Cj(q2 - q1)

(A6)

eq A2 can be rewritten as

e-βVNES(q,x) ) C′′e-βWESP(q)π-n/2 n

n

∫ dz1‚‚‚∫dzn∏e-z δ[ξ - ∑Rjzj] 2

j

j)1

j)1

) C′′e-βWESP(q)π-n/2

n-1

∫ dz1‚‚‚∫ dzn-1∏e-z Rn-1 2

j

j)1

exp[-Rn-2(ξ where n

C′′ ) βC∆E

∏ j)1

n-1

Rjzj)2] ∑ j)1

( ) mq

(A7)

1/2

(A8)

p2β2kj

The Gaussian integrals in the second line of eq A7 are most easily evaluated by inserting the identity

1

n-1

∫ dζ ∫ dχ exp[ı˜χ(ζ - ∑Rjzj)] ) 2π j)1

n-1

∫ dζδ(ζ - ∑Rjzj) ) 1 j)1

(A9)

9684 J. Phys. Chem. B, Vol. 105, No. 40, 2001

Schenter et al.

where the notation ˜ı is used to distinguish x-1 from the index i used previously. The delta function relationship is used to n-1 Rjzj in eq A7 to ζ, and the zj integrals are convert ∑j)1 performed by completing the squares. The remaining ζ and χ integrals are also Gaussian and can be evaluated to give

e-βVNES(q,x) ) C′′(π

n

∑ j)1

R2j )-1/2 e-βWESP(q) exp[-ξ2(

n

R2j )-1] ∑ j)1

(A10)

The resulting expression for VNES(q,x) is

K VNES(q,x) ) WESP(q) + [x - x0(q)]2, i ) 1, 2 2

(A11)

where

K-1 ) n

x0(q) ) -∆W +

n

kjC2j (q2 - q1)2 ∑ j)1

(A12)

kj

C2j (q2 - q1)(2q - q1 - q2) ) ∑ j)1 2 -∆W +

(2q - q1 - q2) (2K)-1 (A13) (q2 - q1)

and we have chosen C∆E so that VNES(q,x ) 0) ) WNES(q). The Marcus parabolas are just special cases of VNES(q,x):

Wi(x) ) VNES(qi,x)

(A14)

and they reduce to eq 8 with K1 ) K2 ) K and

xi ) -∆W + (-1)i(2K)-1

(A15)

We have shown that the generalized Langevin model with the potential given by eqs 30 and 31 yields analytic expressions for the two-dimensional nonequilibrium solvation potential in the (q,x) coordinates and for the Marcus parabolas. In addition, the multidimensional solvent coordinates y can be mapped onto a collective solvent coordinate x ) ∆E(y) using eq 32 to give

kj [2Cj(q2 - q1)yj - C2j (q22 - q21)] 2 j)1 n

x(y) ) -∆W +

∑

n

) xi +

kjCj(q2 - q1)[yj - Cjqi] ∑ j)1

(A16)

For any collection of values for yj, a unique value of x can be determined using eq A16, so that multidimensional solvent effects can be viewed in a reduced dimensional model; for example, nonequilibrium solvation paths in (q,y) space can be visualized in (q,x) space, as shown in Figure 2. Appendix B The present article is concept oriented, not prescriptive for calculations. Therefore, we did not always state equations in the most computationally convenient form. For example, eq 5 may be evaluated by

e-βWESP(q*) ) Trr,R,S[e-βH(r,R,S)Fq(q*,r,R)]

(B1)

where Fq(q*,r,R) is a probability density that is proportional to

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