J. Phys. Chem. 1983, 8 7 , 3387-3400
that the isotropic hyperfine values estimated from the DP scheme correlate particularly well with our proton experimental coupling constants (Table 111). Comparison between observed and calculated fluorine hyperfine couplings is rather poor-a lo4 typical of several fluoro radHowever, the essentially axial anisotropy in the fluorine hyperfine interaction observed in our experiment, AAeXpt= 1/3(Azr- 1/2(Azz+ A,)) = 2.32 X cm-l, correlates nicely with that calculated from the p-orbital spin densities of the anti configuration @Amti = 1.93 X cm-') thereby supporting our earlier conculsion. In the case of .CH2FCH2,INDO/2 calculation reveals that the unpaired electron is mainly localized in the 2p, orbital of the a-carbon atom. In view of this, the observed splitting for the a-proton may be rationalized in terms of spin polarization of the C-H a-bond induced by the unpaired r-electron that is basically in the 2p, orbital of the trigonal a-carbon. Also, our calculation indicates a negative isotropic coupling of 20.13 X lo4 cm-' for the a-protons in -CH2FCH2,the calculated magnitude comparing extremely well with the experimentally derived value (22.03 x lo4 cm-'). We have stated earlier that free rotation about the C-C bond axis is not likely in CH2FCH isolated in the cancrinite matrix. This is supported by our observed 0-proton and fluorine splittings for this radical. For free rotation, the isotropic &proton hyperfine coupling should have a value that is nearly 28 Gel6Examination of the &proton couplings for this radical (Table 11)in terms of the wellknown relationship Aop = B1
+ B2 cos2 4
where 4 is the dihedral angle between the axis of the odd-electron 2p, orbital and the C,-C,-H, plane and B1
-
-
3387
and B2 are constants (B, 0 G; Bz 55 G), reveals that the observed fl-proton splitting (36.27 G or 33.87 X cm-') is much larger than the 28 G (26.15 X cm-') expected for the free rotation of CH2Fabout the C-C bond. Furthermore, the &fluorine hyperfine interaction has been noted to be highly anisotropic and is comparatively much smaller than that of the fluorovinyl radical (A, = 15.54 X cm-' for .CH2FCH2 and A , = 22.91 X cm-' for CHFCH, see Tables I and 11). This reflects the fact that in CH2FCH2the anisotropic contribution to the hyperfine tensor from fluorine 2p orbital is much more effective than that in CHFCH radical. In other words, there is a direct overlap of the fluorine 2p orbital with the carbon 2p, orbitaLg This is most effective when the fluorine is almost in the nodal plane of the carbon 2p, orbital, i.e., when the dihedral angle of C-F bond is about 90°. Conclusion We have effectively stabilized the fluorovinyl and fluoroethyl radicals in the cancrinite matrix at 77 K. The EPR powder spectra have enabled a fairly complete analysis of their spin Hamiltonian parameters. Spin density distributions estimated from semiempirical MO calculations compare favorably with those derived from the EPR data for both species. The finer details of the experimental spectrum of CHzFCHzare suggestive of a model in which torsional oscillations about the C-C bond may be involved, and further experiments on the temperature dependence of the EPR parameters for this radical are under way. Acknowledgment. We thank a referee for several helpful suggestions which have served to clarify our interpretations. Registry No. CHFCH, 86129-81-9; CH2FCH2,28761-00-4; acetylene, 74-86-2; ethylene, 74-85-1; fluorine atom, 14762-94-8.
Classical Solvent Dynamics and Electron Transfer. 1. Continuum Theory Daniel F. Calef*+ and Peter 0. Wolynes' Depertment of Chemishy, Harvard Universw, Cambrwe, Massachusetts 02138 (Received: May 24, 1982; In Final Form: January 14, 1983)
The importance of solvent dynamics in solution-phase chemistry is discussed. The role of solvent fluctuations in determining the rate of adiabatic electron-transfer reactions is investigated. In this first report, a classical continuum dielectric model of the solvent is used. Under certain circumstances,the transfer of charge between two centers reduces to one-dimensionaldiffusion over a barrier. The reaction coordinate is identified and the parameters in the resulting simple closed-form expression for the rate coefficient are evaluated in terms of properties of the reactants and the solvent. The limits of the adiabatic approach are discussed. The results of this investigation are compared with experiment.
I. Introduction Much significant chemistry takes place in condensed phases. It is therefore important to understand how the solvent affects chemical kinetics. Recently the role of the solvent as a heat bath for a dynamical event has been elucidated by many investigators.' In some chemical Current address: Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 01239. *Current address: School of Chemical Sciences, University of Illinois, Urbana, IL 61801.
reactions, however, the solvent plays a less passive part. Notable among these are electron-transfer reactions. In this paper we begin the examination of electron-transfer reactions in the light of modern liquid-state chemical physics. The transfer of an electron is an activated event. By this we mean there is a free energy barrier separating two
f
0022-3654/03/2007-3307~0 1.50/0
(1) (a) J. L. Skinner and P. G. Wolynes, J. Chem. Phys., 69,2143 (1978);(b) E.Helfand, ibid., 54, 4651 (1971);(c) D.Chandler, ibid., 68, 2959 (1978).
0 1903 American Chemical Society
3388
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
minima of an energy surface which represent reactant and product species. The saddle point at the col of this barrier is the so-called “transition state”. For certain simple activated events, such as the rotation about the central bond of butane in model solvents, the dynamical description of the event is straightforward and well understood.2 The motion can be regarded as nearly one-dimensionalmotion, collisionally interrupted by the solvent. When a reaction involves charged species, or any species that is strongly solvated, the description becomes much more complicated. The complete reaction may involve the transition from a solvated reactant to a solvated product. This may be regarded as a collective activated event, where coordinated motion of many particles is required. In this paper we initiate our study of electron-transfer reactions in polar solvents by investigating the effect of classical solvent dynamics using a continuum dielectric model. In another communication we discuss a molecular treatment of the s01vent.~ We consider adiabatic outersphere reactions, where charge is transferred without major structural changes in the reactants (no bonds broken or formed). The static, or thermodynamic, aspects of an adiabatic charge-transfer reaction in a continuum dielectric were discussed by Marcus in his landmark paper^.^ However, there does not seem to exist in the literature a comparable, detailed discussion of dynamics. The majority of previous theoretical studies of adiabatic charge-transfer reactions have focused on the static (thermodynamic) aspects of solvation. The reader is referred to one of the many reviews for more detail^.^ Efrima and Bixon6 and Zusman7 have discussed dynamic solvent effects on nonadiabatic electron-transfer reactions. Zusman7 and Alexandrovs have investigated the role of both dynamics in adiabatic reactions by studying diffusion on one-dimensional adiabatic potential surfaces. One important aspect of this work is to justify such models. In certain limits, our expressions for the rate reduce to those given by Zusman. In a recent publication, Helmang has extended the Alexandrov-Zusman work to include intramolecular modes. The motion of a classical charged particle in a fluctuating polarization field has been discussed by Kurz and Kurz’O and van der Zwan and Hynes” as a model for proton transfer. Tempe, Friedman, and Newton12 also include a brief discussion of solvent dynamics in their recent work. We feel, however, that a systematic discussion of the dynamics would not be an unwelcome contribution to the literature. Specifically, we address the following question: What is the true “reaction coordinate” (so often shown in free energy diagrams), and what equation of motion does this coordinate obey? The simplicity of the final expressions derived from the continuum model allows us to discuss (2)R. M. Levy, M. Karplus, and J. A. McCammon, Chem. Phys. Lett., 65,4 (1979). (3)D. F. Calef and P. G. Wolynes, J. Chem. Phys., 78,470 (1983). (4)(a) R.A. Marcus, J. Chem. Phys., 24,966(1956);(b) ibid.,26,867 (1937);(c) Can. J. Chem., 37, 155 (1959);(d) Discuss. Faraday SOC.,29, 21 (1960);(e) Annu. Reu. Phys. Chem., 15, 155 (1964). (5)(a) P.P.Schmidt, Spec. Period. Rep.: Electrochemistry, 5, 21 (1975);(b) J. Ulstrup, ‘Charge Transfer Processes in Condensed Media”, Lecture Notes in Chemistry 10,Springer-Verlag, New York, 1979; (c) B. Chance, D. C. DeVault, H. Fraunfelder, R. A. Marcus, J. R. Schrieffer, and N. Sutin, Eds., “Tunneling in Biological Systems”, Academic Press, New York, 1979. (6)S. Efrima and M. Bixon, J. Chem. Phys., 70,3531 (1979). (7)L. D.Zusman, Chem. Phys., 49,295 (1980). (8) I. V. Alexandrov, Chem. Phys., 51,449 (1980). (9)A. B. Helman, Chem. Phys., 65,271 (1982). (10)J. L. Kurz and L. C. Kurz, J. Am. Chem. SOC.,94,4451 (1972). (11) G.van der Zwan and J. T. Hynes, J.Chem. Phys., 76,2993 (1982). (12)B. L.Tempe, H. L. Friedman, and M. D. Newton, J. Chem. Phys.. 76,1490 (1982).
Calef and Wolynes
many of the physical-chemicalaspecb of this question, and so we have attempted in this paper to present the results in such a way that the mathematics need not overwhelm the reader. In doing so, it is necessary to discuss some well-known ideas in order to emphasize the new ones. In addition, the calculation requires a certain amount of manipulation which, while not being particularly enlightening, is important for completeness. These manipulations have been put into an Appendix. Our treatment of the solvent dows us to shed some light on certain controversies surrounding electron-transfer reactions. The principle result of this paper is a formula for the prefador (the quantity A in the formula k = Ae-hF/kBr) for an adiabatic reaction. This prefactor differs considerably from both the standard adiabatic and nonadiabatic prefactor expressions, With this expression, and the development behind it, the importance of considering the solvent dynamics in questions of adiabaticity becomes apparent. The prefactor shows the connection between electron-transfer reactions and other transport properties of dielectric fluids. The paper is organized as follows. In section I1 we discuss the particulars of the continuum dielectric approximation. Section I11 details the form of the reaction surface. The nature of the dynamics is covered in section IV. The one-dimensional reduction of the problem is discussed in section V. Section VI contains the expressions for the rate, while modification to these expressions that increase their range of validity are in section VII. We close with a discussion of the various rate expressions and a (limited) comparison to experimental results. 11. Adiabatic Electron-Transfer Theory The adiabatic theory is essentially based on the BornOppenheimer approximation and the Franck-Condon principle. The motion of the electron occurs on a much shorter time scale than the motion of the nuclei. For a radiationless transition to occur, the electron must transfer between states of equal energy. The energy of an electronic state can be found by solving the Schroedinger equation at fixed nuclear position. Certain configurations of the nuclei can equate the energies of two states. We can group the particles in the system into two classes. A “reaction complex” can be defined as the reactants and any strongly bound solvent molecules. The remaining solvent molecules can be regarded as “the solvent”. By strongly bound solvent, we mean to include, in this discussion, the possibility of a region of dielectric saturation around a charged species as well as molecules that may function as ligands. We shall treat the solvent classically. This should be acceptable for a polar fluid where the interactions between the reactant complex and the solvent will be chargedipole interactions. These interactions are affected most by movements of entire solvent molecules, translations and rotations that move the permanent dipole. In most situations of chemical interest, the reaction complex is some “complicated” s t r ~ c t u r e . ’ ~ - ’In~ elucidating the role of the solvent some simplification of the quantum chemical aspects will be necessary. For this initial work we will assume that the electron transfers from a state localized on a donor to a state localized on an acceptor. If the charge cannot be regarded as localized, (13)B.S.Brunschwig, J. Logan, M. D. Newton, and N. Sutin, J.Am. Chem. SOC.,80,5798 (1980). (14)T. J. Meyer, Acc. Chem. Res., 11, 94 (1978). (15)J. L. Walsh, J. A. Baumann, and T. J. Meyer, Inorg. Chem., 19, 2145 (1980).
The Journal of Physical Chemistry, Vol. 87, No. 18, 1983 3388
Classical Solvent Dynamics and Electron Transfer
it may not be possible, or useful, to speak of a chargetransfer reaction. We will discuss this “two-state approximation” in slightly more detail later. In an outer-sphere electron-transfer reaction it is assumed that major structural changes (bonds broken or formed) are not necessary for reaction. Decreasing or increasing the charge on a molecule or complex always has some structural effects. For example, a recent EXAFS study of the Fe-0 bond distance in Fe2+(H O ) , and This Fe3+(H20),shows that the distance changes 0.08 can result in large contributions to the activation energy. If these changes occur on a much faster time scale than the necessary solvent motions, which should be the case for such a bond length adjustment, the effect on the rate will be a modification of the energy surface. The higher frequencies of the inner-sphere modes normally require them to be treated quantum-mechanically. I t may be possible to “fold in” these modes in a manner similar to that discussed by Kestner.” A recent article by Marcus and Siders also discusses the static contribution of the inner modes.I8 In conjunction with the ”frozen-reactant”approximation is the assumption that the orientation of the reactants can be regarded as fixed. Again we are assuming that the formation of some relative geometry of the reactants which allows reaction is fast compared to electron transfer. For divalent metal ions with rigid molecular bridging groups, this is clearly not an approximation. In general, once we have derived an expression for the rate of charge transfer a t an arbitrary fixed geometry, this expression can be averaged over a distribution of geometries. The distribution will be proportional to e - W I k B T , where W is the work necessary to bring the reactants to the given configuration. If the formation of the reaction complex is much slower than the charge transfer, the reaction will become diffusion controlled. The reaction complex will be described by a complicated many-body (both nuclei and electron) Hamiltonian. Within our model, we can write
The electronic Hamiltonian will be changing with time because of thermal motion of the solvent. The solvent can interact with the reaction complex in a variety of ways. The first way, which we will focus on in this report, is through charge-dipole interactions. There are other possible electrostatic effects. There will be quadrupole and higher multipole effects. These effects will be of shorter range than dipolar effects. There are also charge-induced dipole effects. The time scale for the distortion of the solvent molecules’ electron cloud is much faster than the time scale for solvent molecule reorientation. In a polar fluid, the polarizable nature of the solvent screens the charge-dipole interaction by the high-frequency dielectric constant. Optically anisotropic molecules can change the dielectric properties of the fluid by reorienting. This effect will be small compared to the charge-dipole interaction but may be important in a nonpolar fluid. Frequently the reactants are highly charged species and there are counterions in solutions that, in this example, may be considered part of the solvent. Motion of these counterions relative to the reaction complex will cause changes in the energy. In a fluid the diffusion of these ions can usually be considered much slower than the reorientation of a solvent molecule. The highly charged nature of many reactants can also cause significant changes in the solvent structure in the vicinity of the ion. Specifically, within the context of a continuum model, the solvent density in the vicinity of the ion will be higher than the bulk. This change in solvation energies due to these effects is usually small but can be important in nonpolar fluids. We will treat only the charge-dipolar interactions which will be the most important for the time scales considered in this problem. The possible effects of repulsive interactions should also be considered. An electron can be “pushed” between centers. For example, in the EXAFS study mentioned previously the change in bond length between the two valence states of the iron indicates that collisions could change the relative energies of the two states. These effects are strongly dependent on the nature of the reaction complex and cannot be treated in this simplified theory. The interactions that we will be focusing on are the electrostatic interactions between the charge to be transferred and the dipoles on the surrounding solvent molecules. Thermal motion will cause movements of the solvent molecules, causing both translational motion and reorientation relative to the reaction complex. The events of interest for charge transfer cause changes in the relative energies of the two diabatic states. In most common polar solvents, translational effects are relatively unimportant compared to rotational effects. The reorientation of a dipole normally occurs on a faster time scale than the translational motion, and the reorientation is more effective in changing the relative energy of the two localized states. Reorientation takes place on a time scale related to the rotational diffusion coefficient. This time scale is normally measured in picoseconds. Near the end of this report we will discuss what happens when reorientation can no longer be regarded as purely diffusional. The simplest possible description of the solvent that will allow us to treat the dynamics in a reasonable fashion is to treat it as a macroscopic continuum dielectric, with the reaction complex imbedded in cavities in the continuum. Since this is very similar to the Born approximation for the solvation energy, which is known to require modification in many circumstances, the results derived from this approximation must be used carefully. In future papers
i>6
H = Ho + Vsolv(r)
(1)
where VmlV(r)is the interaction with the classical solvent and Hois the Hamiltonian for the reaction complex in the absence of the solvent. The electron is assumed to transfer between a state localized on the donor to a state localized on the acceptor. The simplest possible treatment of the quantum-mechanical aspects of this problem is to expand the wave function in terms of two such localized states. These states, which in general are not eigenstates of the Hamiltonian, are referred to as “diabatic states”. They may be, for example, eigenstates of two different Hamiltonians, such as eigenstates of the infinitely separated donor and acceptor Hamiltonians. The choice of diabatic or localized states is far from unique. Newton has performed calculations on Fe2+(H20)6/Fe3+(H20)6showing that two such states can be identified,lgbut we emphasize that some care must be taken in such an identification. Regarding the adiabatic electron state, I$),a t the time t as a linear combination of the two diabatic states, we can write
I+) = C1lrC.i) + ~ 2 l + f )
(2)
(16) T. K. Shaw, J. B. Hastings, and M. L. Perlman, J. Am. Chem.
102,5904 (1980). (17) N. R. Kestner, J. Phys. Chem., 84,1270 (1980). (18)P.Siders and R. A. Marcus, J. Am. Chem. SOC.,103, 741,748 (1981). (19)M. D.Newton in ‘Proceedings of the International Symposium on Atomic Molecular and Solid State Theory”, Flagler Bench, FL, 1980. SOC.,
3390 The Journal of Physical Chemistv, Vol. 87,No. 18, 1983
we will move away from the continuum approximation and introduce the molecular character of the solvent using the modern integral equation theories of fluid structure. The continuum calculation will serve as a limiting case for the future calculations. We will treat the case, at least initially, where the dielectric properties of the solvent can be described by a Debye law for the frequency-dependent dielectric constant. This is normally written as €0
€(W)
Calef and Wolynes
In this formula the $o is the potential due to the charge distribution
-1
- 1= 1- iWTD
(3)
for (4)
In this expression rDis the dielectric relaxation time, a measure of how long it takes the polarization to relax to zero after the electric field has been turned off. tois the static, or low-frequency, dielectric constant. From nonequilibrium statistical mechanics, we know that the presence of dielectric relaxation, which involves dissipation, will imply the presence of fluctuations in the polarization. It is the dynamics of these fluctuations that we will need to study to understand the dynamics of the charge transfer. In the continuum approximation, the interaction term in the Hamiltonian Vsolv(r)will be given by
where p(79 is the polarization at point ?’in space. Because the polarization can fluctuate, it need not be in equilibrium with the field from the charges in the reaction complex. In summary we have the following picture of the electron-transfer process in a polar fluid. An electron is initially localized in one part of the reaction complex. The polarization of the surrounding solvent is on the average the equilibrium polarization appropriate to that charge distribution. The solvent polarization is fluctuating around that average. Occasionally an arrangement of solvent molecules will occur that will equate the energies of the initial and final localized electronic states. This fluctuation, which persists for a long time on the time scale of electron motion, allows the charge to transfer without emitting or absorbing radiation. The fluctuations in the polarization continue, and, if the polarization relaxes to a state solvating the product, the electron-transfer reaction has been completed. To understand the rate of electron transfer, we will need to know the rate at which these fluctuations occur. 111. Reaction Energy Surface For a macroscopic dielectric, the free energy of a given polarization in the presence of an electric field has been discusjed by Felderhof.20 We will write the free energy as S[P(?)], using square brackets to denote that 3 is a functional of P ( 8 ; $ is a real number that is calculated from the function P ( 3 . The probability of a given polarization will be proportional to the exponential of this free energy, or W [ P ( ~ ) ]e-m[j+ql
The form of 3 given by Felderhof is (20) B. U. Felderhof, J. Chem. Phys., 67,493 (1977).
(6)
For an incmite-system (no bouqdary-effects) the last term is just -SP(r).Eo(r)d3r, where Eo= V(Po(?) is the bare field due to the charges. This term is the interaction between the polarization and the electric field. The important thing about this free energy is that $(?) can be a nonequilibrium polarization. It can be chosen independsnt of the bare gield. In the absence of any charges, Eo(?) = 0, and 3[P(?)]allows us to calculate the probability of observing a nonzero polarization. The form given by Felderhof differs slightly from the form presented by Marcus in his 1955 paper2I
The Marcus form is completely local, whereas the Felderhof form has a term that depends on the polarization at two points in space. The difference occurs because Marcus considers the nonequilibrium polarization to be created by turning on fictitious charges, such that the equilibrium response to these charges is the desired polarization. This polarization is frozen, and the charges are removed. The free energy is the sum of the free energy of the two steps. This limits the validity of the Marcus expression to polarizations that can be written as equilibrium responses to charge distributions. If there does not exist a function f(r) such that
fi(?)
=
a/(?)
then the Marcus expression is not valid. We shall show, however, that this difference does not affect our calculation. The continuum approximation uses a macroscopic free energy expression at a microscopic level. The quality of this approximation depends on the system and what quantities are being investigated. Generally, a continuum approximation will be valid when the solvent molecules are small and the interactions between the solvent and solute (here reaction complex) are long-ranged. If the potential is long ranged then “cells” can be defined such that the potential does not vary over the size of the cell, but the cells are large enough to contain many solvent molecules. Some sort of averaging can be done within the cells, resulting in a “coarse-grained” or continuum approximation. Our choice of free energy functional is an extreme case of this, where all molecular detail has been washed out. For example, this free energy functional does not account for dielectric saturation. Near a small ion the strong electric fields will completely align the nearby dipoles; increases in the local polarization are not possible. As mentioned previously, the compression of the fluid near an ion will usually not be important, but in any event it is not treated with this free energy functional. In future publications a more sophisticated functional, from the integral equation theory of fluids, will be discussed. (21)R. A. Marcus, J. Chem. Phys., 24,979 (1956).
Classical Solvent Dynamics and Electron Transfer
The free energy functional should have a minimum corresponding to the macroscopic equilibrium relationship between the bare field and the polarization. Near such a minimum, a small change in the polarization will not lower the free energy. This requires ~ 3 / ~ f i (=ro) (10)
The Journal of Physical Chemktry, Vol. 87, No. 18, 1983 3391
3
where we have introduced the functional derivative notation. Performing the functional derivative we easily obtain
which is the value of the polarization given by classical electrostatics. The importance of the functional 3 [P(r)] is that it also allows us to find the probability of fluctuations away from the minimum. For the electron-transfer problem, we start, as done originally by Marcus,22with an approximate free ene_rgy for the entire system when the polarization is given by P(r) and the adiabatic electronic wave function by I$). Combining Felderhof’s continuum free energy functional with the quantum energy, we have
fi(r) d3r d3r’ (12) This is now a functional of two functions. We can now uae the separation of time scales between solvent motion (polarizationchanges) and electronic events to reduce this to a functional of the polarization alone. Electronic motion is rapid enough for the electron to adjust to the solvent configuration. The resulting state _will minimize the free energy, with respect to I$),at fiied P(8. Mathematically this means ~ ~ [ ~ , I $ >=I0/ W (13) subject to S$(r) $*(r) dr = 1 for a fixed fi(i). Minimizing the free energy with respect to $(r) yields the Schroedinger equation and can now be-regarded as the equation that for a given P(?).Substituting the resultant specifies $(i) wave functio: into the free energy functional resqlts in a [unction of P(i) alone. We will write this as 3[P(?),$(e P(r311. This functional can be thought of as defining a surface. A point on the surface is specified by functions P(8,which, with a certain lack of rigor, can be thought of as a point in a many-dimensional space. In general, the structure of such a surface may be very complicated, but we would expect that the surface will have (at least) two local minima, one corresponding to the electron localized on the donor, and the polarization in equilibrium with that charge configuration, and another corresponding to the electron localized on the acceptor with the polarization in equilibrium with that charge configuration. The regions on the surface near these minima will be the configurations that characterize the reactants and products. Since these regions are local minima, going from one mimimum to the other will involve passing through regions of higher free energy. The dynamics of crossing the ridge between the two minima will determine the kinetics of the electrontransfer reaction. The ridge between the two minima will have some lowest point, a saddle point. The local minima and the saddle (22) R. A. Marcus, Faraday Symp. Chem. SOC.,10,60 (1975).
*I
x3
x2
Figure 1. A typical reaction free energy diagram. The free energy 3 Is plotted against x , the reaction coordinate. The reactant states are near x , , the product states are near x 2 , and the transition state is xl.
point will all be points where the functional derivative of the free energy with respect to the polarization vanishes. Inspection of the second functional derivative will tell us the geometric nature of these points. As mentioned in the previous section, we will assume that the wave function can be written as a mixture of two states. This reduces the free energy dependence on the wave function to dependence on a single variable cl. The identification of the critical points on the surface is technical and presented in the Appendix. In brief there are two minima and a saddle point. We will identify quantities evaluated at the saddle point by a superscript and those of the minima by i and f. The saddle point will be the transition state, a transition state that depends on the configuration of the solvent.
*
IV. Dynamics in Many Dimensions The time development of the polarization is governed microscopically by the motions of a very large number of solvent molecules. We will introduce a functional g[P(r),t], defined 8%the probability of observing the polarization given by P ( 8 (or techzically a pol_arizationwithin some small neighborhopd AP(7)around P(8)at time t. In general, we expect g[P(A,t]will depend on time through some rather complicated functional of the polariz_ationat previous times. In the limit of very long times g[P(i),t]should approach its equilibrium value which will be given by a Boltzmann distribution governed by the free energy. It is helpful to consider the problem in relation to a simple one-dimensional model of a particle moving in a double well. The potential, V ( x ) ,will be something like Figure 1. Suppose, in addition to the potential, the particle is in a fluid of molecules much smaller than itself so that it can be regarded as a Brownian particle.23 The motion of such a Brownian particle can be described by a Langevin equation, which is essentially the equation of motion of the particle in the well with a friction term and a random force term added mf
+ {x + dV(x)/dx = frandom(t)
(14)
The random force is defined by its average properties. The damping term and the random force term combined simulate the collisions with the fluid particles. This is a stochastic differential equation. Another way to treat, mathematically, the same system is to introduce a function f ( x , t ) which is the probability (23) N. Wax, Ed., “Selected Papers on Noise and Stochastic Processes”, Dover Publications, New York, 1954.
3392 The Journal of Physical Chemistry, Vol. 87, No. 18, 1983
of observing the Brownian particle at point x (or technid y in some small neighborhood of x ) at the time t. When the friction is large, so that motion is completely overdamped and the internal effects are unimportant, this function will satisfy a Smoluchowski equation
This is a diffusion equation (the first term) with an additional term that accounts for the presence of the potential. When the system i~ not overdamped, this equation must be modified. The diffusion constant D is related to the friction in the Langevin equation by an Einstein relation D = kBT/(
As mentioned previously, the polarization can be thought of as a point in a many-dimessional space. The equation for the time evolution of g[P(?),t]that we will discuss is a Smoluchowski equation in that many-dimensional space. We will call it a functional Smoluchowski equation. The functional Smoluchowski equation can be derived by using projection operators. This technique is standard in nonequilibrium statistical mechanics, and what follows is taken from a paper by Zwanzig written in 1963.% The essence of the method is to separate the variables that describe the system into two group depending on the time scale on which they change. For example, the Brownian particle is a system where the position of the particle changes much slower than the positions and momenta of the solvent particles. The Smoluchowski equation for this system describes the case where, in addition to that separation of time scales, the momentum of the Brownian particle changes rapidly in comparison to ita position. The derivation of a functional Smoluchowski equation will require a similar separation of time scales between the polarization and ita time derivative. We will assume that that holds for now, and return to the case where it does not later. For the Brownian particle the diffusion constant can be found from the time integral of the velocity time correlation function
D
= J m ( u ( t ) u(t=O)) dt
The quantity W[P(?)]4s essentially the equilibrium (long time limit) form of g[P(?),t],and so we can write
ag[P(r),tl at
a functional Smoluchowskiequation. Comparison of this equation with the one-dimensional Smoluchowski equation shows that the functional equation is a many-dimensional Smoluchowski equation, where the indexj has been replaced by a continuous variable ?, and S[P(?)]is the potential. Fortunately, the functional equation can be simplified. In order to do this, we expand the polarization using a complete set of orthonormal functions
P(3 = Cx,7,(7) i
where
The functional derivatives become
This reduces the functional Smoluchowski equation to an ordinary Smoluchowski equation
(16)
For the polarization we will define a similar quantity
EeAr,r’)is a tensor. From the theory of dielectrics it can be shown that, when the dielectric constant obeys a Debye law (eq 6) this reduces to
= D,f&33(r- r’)
Calef and Wolynes
(18)
where f is the identity matrix and rDthe dielectric relaxation time. Using this form of the diffusion “constant” and the expressions given by Zwanzig, we can write a g [ ~ ( r ) , t-l at
The important aspect of writing eq 21 in this fashion is that it allows us to reduce the relevant dynamics to one dimension. As discussed in detail in the Appendix, a set of functions can be constructed such that the free energy function will be a s u m of one-dimensional functjons. We are able to isolate a single basis function, call it fl(7), such that the “coordinate” xl passes from the reactant minima to the product minima through the saddle point along the path of steepest descent. The potential in all directions perpendicular to this coordinate is harmonic. Clearly, this coordinate is our “reaction coordinate”. Because the free energy functional is now separable, motion perpendicular to the reaction coordinate cannot influence motion along the reaction coordinate and hence is irrelevant to the rate. For the approximate free energy functional that we have chosen, the electron-transfer reaction has a truly one-dimensional reaction coordinate, with dynamics specified by an equation of the form of eq 15. The potential will have the form shown in Figure 1. ks shown in the Appendix, the relevant function is, aside from the normalization factor
j,(r) = ZOl(a - So2(?) (24) R. Zwanzig, Phys. Reu., 124, 983 (1961).
(22)
where Bo1(?) and go2(?) are the “bare” electric fields when
Classical Solvent Dynamics and
Electron Transfer
The Journal of Physical Chemisrry, Vol. 87, No. 18, 1983 3393
the electron is localized on the donor and the acceptor, respectively. They are
charges centered in two spherical cavities in a continuum dielectric
(23)
To summarize, only changes in the polarization that change
will be important to electron transfer. Since we have restricted the interactions between the reaction complex and the solvent to only the charge dipole, it is not surprising that this is the resulting reaction coordinate. The pleasant surprise is that this coordinate is not coupled to any other coordinate. For a different free energy functional, for example, on that treated dielectric saturation, this might still be the reaction coordinate but the dynamics may be more complicated. We can now understand why, for this problem, the distinction between the Marcus and Felderhof forms of the free energy is not important. The mode in the polarization that is driving the reaction can be written as the response to a charge distribution. The complete decoupling of the other modes from the dynamics of this mode allows us to only consider free energy changes due to the reaction coordinate, which will be the same calculated with either functional.
where Ae is the charge transferred in the reaction, r1 and r2 are the cavity radii, and d is the distance between their centers. Although we have only treated polar nonpolarizable fluids, we have included the high-frequency dielectric constant in this expression. Comparison of this expression with the Born approximation for the free energy of solvation for two similarly arranged ions shows that this quantity is onehalf the difference in the Born free energies between the system in a solvent with dielectric constant to and one with dielectric constant e,. The quantities cI1 and c12 are given by the expressions (for small overlap) c*12
= (A - h F , ) / 2 X
c*22
= (A
+ U,,)/2X
(33) (34)
These formulas show that the transition state becomes more “reactant-like” with increasing exothermicity. , be given by the The activation free energy, U swill standard expression (for H12
