Linear Response in Theory of Electron Transfer Reactions As an Alternative to the Molecular Harmonic Oscillator Model Yuri Georgievskii, Chao-Ping Hsu, and R

JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 11 15 MARCH 1999

Linear response in theory of electron transfer reactions as an alternative to the molecular harmonic oscillator model Yuri Georgievskii, Chao-Ping Hsu, and R. A. Marcus Noyes Laboratory of Chemical Physics, Mail Code 127-72, California Institute of Technology, Pasadena, California 91125 ͑Received 21 October 1998; accepted 16 December 1998͒ The effect of solvent fluctuations on the rate of electron transfer reactions is considered using linear response theory and a second-order cumulant expansion. An expression is obtained for the rate constant in terms of the dielectric response function of the solvent. It is shown thereby that this expression, which is usually derived using a molecular harmonic oscillator ͑‘‘spin-boson’’͒ model, is valid not only for approximately harmonic systems such as solids but also for strongly molecularly anharmonic systems such as polar solvents. The derivation is a relatively simple alternative to one based on quantum field theoretic techniques. The effect of system inhomogeneity due to the presence of the solute molecule is also now included. An expression is given generalizing to frequency space and quantum mechanically the analogue of an electrostatic result relating the reorganization free energy to the free energy difference of two hypothetical systems ͓J. Chem. Phys. 39, 1734 ͑1963͔͒. The latter expression has been useful in adapting specific electrostatic models in the literature to electron transfer problems, and the present extension can be expected to have a similar utility. © 1999 American Institute of Physics. ͓S0021-9606͑99͒01511-1͔

I. INTRODUCTION approach of Levich and Dogonadze, subsequently used as a common model to treat quantum aspects of ET reactions,14,15 Interaction with an environment plays a crucial role in required justification. One way to resolve this difficulty was many nonadiabatic processes in condensed phases. Electron given by Ovchinnikov and Ovchinnikova,16 who used a 1–3 transfer reactions provide a major example in which quantum field theoretic method17 to show that the result of strong electrostatic interaction of a reacting species with a Levich and Dogonadze can be recovered in the long-wave polar solvent can control both the energetics and the dynam- approximation without explicitly invoking a molecular har- ics of the process. The theory of nonadiabatic transitions, in monic oscillator model. In their derivation Ovchinnikov and the presence of strong interaction with the environment Ovchinnikova considered uniform systems, so dielectric im- treated quantum mechanically, originates from the papers of age effects and any effect of the solute on the properties of Lax4 and Kubo and Toyozawa.5 Their approach was applied the nearby solvent were not included. Image effects have by Levich and Dogonadze6 to calculate the rate of an elec- been included earlier7–9 on the assumption that the slow and tron transfer reaction in liquids. In their theory Levich and fast solvent modes allow for a clear separation. Dogonadze, like Lax and Kubo and Toyozawa, used a col- Kornyshev investigated the models with nonlocal dielec- lection of harmonic oscillators as a model of the environment tric response,18–20 which was applied earlier to electron whose fluctuations are responsible for the electronic transi- transfer reactions in the framework of a molecular harmonic tion. 14,21,22 23 In a seemingly different approach Marcus7–9 used a di- oscillator model. Song, Chandler, and Marcus con- electric continuum approach and showed that the polarity of sidered the effect of the solvent inhomogeneity due to the the solvent is important for understanding the energetics of solute presence using the Gaussian field model, which could be viewed as a continuum equivalent of a molecular har- an ET reaction. Marcus later assumed only a linear dielectric 24–26 response of the solvent in the vicinity of the reacting species, monic oscillator model. Fleming and co-workers used and did not use any specific molecular model of the solvent.9 nonlinear spectroscopy to study the solvation dynamics of Vorotyntsev et al.10 related the rate constant to a complex, chromophoric molecules in the variety of solvents and space- and time-dependent electric susceptibility ⑀(␻,k)in glasses. They related the spectral density of the harmonic the framework of a harmonic oscillator model, thus general- modes to the correlation function of the energy difference in izing Marcus’ theory in this respect, which was purely clas- the excited and the ground electronic states of the chro- 27 sical. Jortner and co-workers11–13 extended the treatment7 of mophore and used a treatment developed by Mukamel to Marcus and Levich and Dogonadze6 by using the former for express the measured spectra in tems of the spectral density low frequency modes and the latter for high frequency of the effective harmonic oscillators. Berne and co-workers28 modes. have investigated the solvation dynamics and vibronic spec- One factor influencing the dielectric properties of polar tra of chromophoric molecules for models in which the har- solvents is the reorientational motion of the solvent mol- monic bath frequencies and normal modes are different, ecules, a motion which is strongly anharmonic. Thus, the rather than the same, for the initial and final electronic states.

Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 5308 J. Chem. Phys., Vol. 110, No. 11, 15 March 1999 Georgievskii, Hsu, and Marcus

In this paper we show that an expression for the rate effect in the present formulation is related to the one of constant in terms of dielectric dispersion properties, which Ref. 23. was originally derived using a molecular harmonic oscillator model,10,14 is also applicable to molecular systems which are strongly anharmonic, in particular to polar solvents. The II. THEORY 29 present calculation is based on linear response theory and a A. Hamiltonian and molecular polarization operator second-order cumulant29,30 expansion. The calculation then relates the rate constant to the dielectric and some structural The Hamiltonian operator of the system reactants properties of the system. An expression is also obtained ϩsolvent in either electronic state can be written as Eqs. 17 , 23 , and 38 generalizing to frequency space ͓ ͑ ͒ ͑ ͒ ͑ ͔͒ H jϭh jϩH0ϩHЈj , jϭ1,2, ͑1͒ and in quantum mechanical terms a result obtained earlier.31,32 The latter related the free energy barrier for reac- where h j is the Hamiltonian of the ‘‘free reactants,’’ H0 is the Hamiltonian of the ‘‘free solvent,’’ and H is the inter- tion to a difference in ‘‘free energy’’ of two hypothetical Јj action between them. The HЈ depends on the electronic state systems. These two systems have a charge distribution which j of the reactants. The Hamiltonian h includes the intramo- is the difference between those of the products’ and the re- j lecular vibrational modes of the reactants. Purely for simplic- actants’, but now one system interacts via the solvent dielec- ity of notation, since the present article focuses on the sol- tric dispersion at frequency ␻ and the other at the solvent’s vent effect which is associated with HЈ and H , we consider optical frequency. j 0 structureless reactants and so treat h as just numbers. The The cumulant expansion approach is the same as one j role of the intramolecular vibrational modes has been exten- developed by Kubo29 and Mukamel33 to describe the tran- sively discussed in the literature12,13,37–39 and can be easily sient fluorescence of large aromatic molecules. A similar ap- incorporated into the present scheme. The operators H and proach was used earlier by Hizhnyakov and co-workers34–36 0 HЈj act on the nuclear wave function of the solvent. We do to describe the spectra of impurity centers in solids. There is not use carets in their notation in contrast with other opera- perhaps also a pedagogical interest in such a treatment, since tors ͓e.g., in Eq. ͑2͒ below͔ since they are used only in this it provides a simplification and extension over the original sense and so no ambiguity is caused by such notation. pioneering and stimulating quantum field theoretic17 deriva- 16 The interaction between the reacting species and the sol- tion of Ovchinnikov and Ovchinnikova. Another alterna- vent is approximately separated into electrostatic and non- tive approach to the present one is that given by Chandler, 15 electrostatic interactions. Nonelectrostatic interaction in- who used a path integral formulation. The common har- cludes van der Waals attraction, short range repulsion, and monic oscillator bulk model can be viewed as one way of hydrogen bonding. This interaction is assumed here to be visualizing these more rigorous results. independent of the electronic state of the solute and is in- The paper is organized as follows: cluded in H0. However, it should be noted that the last as- The molecular Hamiltonian and the molecular solvent sumption probably breaks down for the reactions in which a ˆ polarization operator P0(r) are described in Sec. II A. In Sec. bond of the solute to a solvent molecule ͑including a hydro- II B the standard expression for the nonadiabatic reaction gen bond͒ is formed or broken when the electronic state of rate constant is given and a generalized coordinate X is in- the reacting species changes. Such an effect could probably troduced. ͑Use of the latter provides a way of avoiding the be included later, instead, as a state-specific contribution field theoretical treatment.͒ The correlation function in the to h j . rate constant expression is simplified using a second-order Because of the complex interactions of dipoles, perma- cumulant expansion and a correlation function of X. Linear nent and induced, of the solvent there are many subtleties in response theory is introduced in Sec. II C to evaluate this their analysis. In an excellent article written in 1958, Mandel correlation function. The X is then related in Sec. II D to the and Mazur40 have described these terms as well as previous ˆ errors in the literature. The electrostatic interaction between solvent polarization operator P0(r) and to a certain hypo- (0) the reactants and the solvent can be written, as shown in thetical electric field D12 (r) and is used to evaluate the sol- Appendix A ͓cf. Eq. ͑A6͔͒,as16 vent response function ␣␻ . The ␣␻ is expressed in Sec. II E in terms of a frequency-dependent ‘‘free energy’’ difference HЈϭϪ D͑0͒ r Pˆ r d3r, jϭ1,2, 2 of two hypothetical systems, Eq. ͑38͒, and the equation for j ͵ j ͑ ͒• 0͑ ͒ ͑ ͒ the rate constant is then given ͓Eq. ͑39͔͒. Some specific sol- (0) ute models are discussed in Sec. II F. The relation of the where Dj (r) is the electric field at point r, created by the present results to previous work and some other features of reacting species in vacuum for the jth electronic state and ˆ the results are considered in the Discussion, Sec. III. Several P0(r) is the operator representing the molecular solvent po- features are treated in more detail in the appendices: The larization at the same point, when the solute charge distribu- various electrostatic operators are treated in Appendix A, the tion ␳(r) is set equal to zero. We use a caret notation for ˆ relation of a term in the rate expression to the standard free P0(r) to stress that it is an operator acting on the nuclear energy of reaction is derived in Appendix B, and the relation wave function of the solvent. Although we do not explicitly ˆ between the electrostatic operators and the time-dependent use the following expression for P0(r) in Eq. ͑2͒ it could be macroscopic properties is given in Appendix C. In Appendix defined in a long-wave approximation as a weighted sum of D it is shown how the treatment of the excluded volume the individual dipole moments of the solvent molecules:

ϩϱ ˆ ˆ ˆ ˆ ˆ 0 ˆ 2 itH1 ϪitH2 P0͑r͒ϭ͚ ␦͑rϪrn͒dnϵ͚ ␦͑rϪrn͒͑dnϩdnЈ͒. ͑3͒ kϭ⌬ dt͗e e ͘, ͑4͒ n n ͵Ϫϱ ͑throughout units in which បϭ1 are used͒, where the aver- In this equation the coordinates of the center of mass of the aging is taken over the initial state: ˆ nth solvent molecule are denoted by rn and its net dipole Tr •••eϪ␤H1 moment by dˆ . This dˆ is not only a function of the internal ͓ ͔ n n ͗•••͘ϭ , ␤ϭ1/kBT. ͑5͒ nuclear configuration of the nth solvent molecule but also of Tr͓eϪ␤H1͔ the other solvent molecules, since they induce an electronic It is convenient to rewrite the expression for the corre- polarization in it. As a result, the solvent polarization opera- lation function in Eq. ͑4͒ in terms of the formalism of time- ˆ tor P0(r) itself consists of two parts, one part arising from all ordered exponents:4 ˆ 0 the unperturbed dipole operators dn of the solvent molecules, t Ϫi͐ X͑␶͒d␶ ˆ eitH1teϪitH2ϭe 0 , 6 and the second part related to dnЈ being a collective electronic ϩ ͑ ͒ response to the first part. where ˆ ˆ rn and dn are quantum mechanical operators which act on the wave function in the configurational space of all XϭH2ϪH1 ͑7͒ nuclear degrees of freedom of the solvent, after an averaging is the energy difference of the system in the final and initial over the total electronic wave function of the solvent has electronic states of the solute ͑the nuclear kinetic energy op- been performed. This averaging assumes a fast electronic erator cancels in X) and where the time-evolution occurs in response. It may be stressed that no averaging over any the initial state, nuclear motion of the solvent, such as over the intermolecu- lar or intramolecular vibrations of the solvent molecules, is X͑␶͒ϭexp͑i␶H1͒X exp͑Ϫi␶H1͒. ͑8͒ performed in Eqs. ͑2͒ and ͑3͒. Instead, the molecular solvent One can then use a cumulant expansion29 to represent ˆ itH ϪitH polarization operator P0(r) is a complicated function of all the correlation function ͗e 1e 2͘: ˆ nuclear coordinates of the solvent. P0(r) is a part of the total ϱ ͑Ϫi͒n ϩϱ ϩϱ polarization operator Pˆ (r)ϭPˆ (r)ϩP (r) ͓cf. Eq. ͑C4͔͒, itH1 ϪitH2 0 op ͗e e ͘ϭexp ͚ ••• d␶1•••d␶n ͭ nϭ1 n! ͵Ϫϱ ͵Ϫϱ which also includes the electronic response Pop(r) to the solute charge distribution. The total polarization operator Pˆ (r) is distinguished from the standard41 macroscopic polar- ϫ͗T͓X͑␶1͒,•••,X͑␶n͔͒͘cͮ, ͑9͒ ization P(r), which is an average of Pˆ (r) over local solvent nuclear fluctuations ͓cf. Eq. ͑C1͔͒. where T͓•••͔ denotes the time-ordering operator and the av- ˆ Since the molecular solvent polarization operator P0(r) erage ͗•••͘c in the nth term of the power of exponent de- in Eq. ͑2͒ was defined in Eq. ͑3͒ using a hypothetical neutral notes the cumulant of the nth order. As an approximation, state of the solute with a charge distribution ␳(r)ϭ0, it is, using a cumulant expansion and retaining only the terms up 30 therefore, a function of the solvent state only. In particular, it to second order, we have does not depend on the electronic state of the solute. In Ap- t ␶1 itH1 ϪitH2 pendix A it is shown that the expression for the electrostatic ͗e e ͘ϭexp Ϫit͗X͘Ϫ͵ d␶1 ͵ d␶2 C͑␶1Ϫ␶2͒ , interaction between the solute and the solvent, given by Eq. ͭ 0 0 ͮ ͑2͒, is valid under rather general conditions. ͑10͒ where C͑␶͒ϭ͗␦X͑␶͒␦X͑0͒͘, ͑11͒ B. Rate constant in terms of correlation function of a generalized coordinate X and ␦XϭXϪ͗X͘ is the variation of X from its average value ͗X͘ in the initial state. The expressions for the first two cu- The Hamiltonians H j , jϭ1,2, describe the collective mulants were used: motion of the system in the initial ͑reactant͒ and final ͑prod- uct͒ electronic states of the solute. To describe the transition ͗X͘cϭ͗X͘, between the two one must introduce the nondiagonal matrix 2 ͗X͑␶͒X͑0͒͘ ϭ͗X͑␶͒X͑0͒͘Ϫ͗X͘ ϭ͗␦X͑␶͒␦X͑0͒͘. ͑12͒ element ⌬ which couples the two electronic states. This ma- c trix element is sometimes small and is assumed to be inde- Equation ͑10͒ shows that in this approximation the cor- Ϫ pendent of the nuclear configuration for statistically impor- relation function ͗eitH1e itH2͘ is expressed in terms of the tant configurations ͑Condon approximation͒. Non-Condon correlation function C(t). In a molecular harmonic oscillator 4,5 effects could be of importance for solvated electrons and model the higher cumulants vanish exactly if X is a linear solutes with weekly localized electronic clouds,42,43 but not combination of the normal modes and Eq. ͑10͒ then becomes for tight redox couples. Then, in first-order perturbation exact for that model. theory the rate constant k for a nonadiabatic transition be- To express Eq. ͑11͒ in terms of a spectral density, tween the two electronic states of a reactant or reactants fixed namely Eq. ͑15͒ below, a Fourier transform C˜ (␻) is intro- in position can be written as5 duced,

Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 5310 J. Chem. Phys., Vol. 110, No. 11, 15 March 1999 Georgievskii, Hsu, and Marcus

ϩϱ where the generalized susceptibility ␣(␶) is given in terms ˜ i␻t 29 C͑␻͒ϭ C͑t͒e dt, ͑13͒ of a commutator of X(␶) and X(0) ͵Ϫϱ ␣ ␶͒ϭ ␦X ␶͒,␦X 0͒ ϭϪ2ImC ␶͒. ͑22͒ which is a real and positive function of ␻. It can be shown ͑ ͓͗ ͑ ͑ ͔͘ ͑ that C˜ (␻) satisfies the relation The second part of Eq. ͑22͒ follows directly from the definition of the correlation function C(␶) in Eq. ͑11͒. In passing, ␻␤ C˜ ͑␻͒ϭe C˜ ͑Ϫ␻͒. ͑14͒ we note that we have been setting បϭ1. With ordinary units The last equation is a direct consequence of the fluctuation- of ប, the left side of Eq. ͑22͒ would read ␣(␶)ប. dissipation theorem.29 Comparing Eqs. ͑15͒, ͑16͒, and ͑22͒ the spectral density The spectral density J(␻) can be defined by J(␻) is seen to coincide with the negative imaginary part of the Fourier component of the linear response function ␣␻ 1 ˜ ˜ ϱ J͑␻͒ϭ 2͓C͑␻͒ϪC͑Ϫ␻͔͒. ͑15͒ ϭ͐0 ␣(t)exp(i␻t)dt:

The following standard expression for the correlation func- J͑␻͒ϭϪIm ␣␻ . ͑23͒ tion C(t) is then readily obtained: D. Relation of C t and J to dielectric properties 1 ϱ „ … „␻… C͑t͒ϭ d␻ J͑␻͓͒coth͑␤␻/2͒cos ␻tϪi sin ␻t͔. ͑16͒ ␲ ͵0 C(t) and J(␻) are molecular statistical mechanical properties. We turn next to their calculation in terms of the From Eqs. ͑4͒, ͑10͒, and ͑16͒ a standard expression for the dielectric properties of the solvent, in the long wavelength 5,6,14,16 nonadiabatic reaction rate follows: ͑local response͒ approximation for the solvent. Upon substi- ϱ tuting Eqs. ͑1͒ and ͑2͒ into Eq. ͑7͒ the following expression kϭ⌬2 dt exp Ϫit⌬G0 for the reaction coordinate X is obtained: ͵Ϫϱ ͫ

ϱ ͑0͒ ˆ 3 1 J͑␻͒ cosh͑␤␻/2Ϫi␻t͒Ϫcosh͑␤␻/2͒ XϭϪ͵ D21 ͑r͒•P0͑r͒d r, ͑24͒ ϩ d␻ 2 , ͑17͒ ␲͵0 ␻ sinh͑␤␻/2͒ ͬ (0) where D21 is the difference of the vacuum electric fields in where ⌬G0 is shown in Appendix B to be the standard free the initial and final states of the solute, energy of reaction. ͑0͒ ͑0͒ ͑0͒ D21 ͑r͒ϭD2 ͑r͒ϪD1 ͑r͒. ͑25͒ Equations ͑18͒ and ͑24͒ yield C. Relation of the correlation function of X, C„t…,to H ϭH Ϫ D͑0͒͑r,t͒ Pˆ ͑r͒d3r, ͑26͒ the generalized susceptibility ␣␻ eff 1 ͵ • 0 The correlation function C(t) for the anharmonic mo- where lecular system is next expressed in terms of the dielectric ͑0͒ ͑0͒ properties of the solvent. To this end linear response theory D ͑r,t͒ϭD21 ͑r͒f ͑t͒, ͑27͒ is used, after introducing an effective Hamiltonian Heff(t): is the vacuum electric field formed by the external, time-

Heff͑t͒ϭH1ϪXf͑t͒. ͑18͒ dependent charge distribution ␳(r,t). The latter is a linear combination of the charge distributions of the solute in its Here, the energy difference X plays the role of a generalized initial and final electronic states: coordinate ͑reaction coordinate͒ and f (t) is a generalized dimensionless force. Thereby, Heff can evolve from H1 to ␳͑r,t͒ϭ␳21͑r͒f ͑t͒, ␳21͑r͒ϭ␳2͑r͒Ϫ␳1͑r͒. ͑28͒ H2,iff(t) is chosen to tend to zero as t Ϫϱ and to be- The effective Hamiltonian ͑26͒ describes the interaction come unity as t ϩϱ. → → of the solvent with the external charge distribution ␳(r,t). A nonequilibrium density matrix R(t) which evolves ac- Thus, the calculation of the correlation function C(t) is re- cording to a Liouville equation with the Hamiltonian Heff(t), duced to finding the linear dielectric response of the solvent Eq. ͑18͒, is next introduced, to the time-dependent external electric field D(0)(r,t) given 44 d by Eq. ͑27͒. The standard ‘‘macroscopic’’ treatment can R͑t͒ϭϪi͓H ͑t͒,R͑t͔͒, ͑19͒ dt eff then be used to calculate the average electric field E(r,t) and the average electric displacement D(r,t) which arise in the and used to define a time-dependent average ¯X(t) at time t, solvent in response to the external time-dependent electric field D(0)(r,t). In Appendix C it is shown how E(r,t) and ¯ X͑t͒ϭTr͓XR͑t͔͒. ͑20͒ ˆ D(r,t) are related to the solvent polarization operator P0(r) In linear response theory, the average value of the gen- in terms of the molecular model of the solvent. These fields eralized coordinate ¯X(t) is linearly related to the generalized E(r,t) and D(r,t) coincide with the corresponding macro- force:29 scopical fields in the standard electrostatic and electrody- namic treatments.41,45 ϱ ¯ ¯ As a result the standard electrostatic equations for E(r,t) X͑t͒Ϫ͗X͘ϭ␦X͑t͒ϭ ␣͑t͒f ͑tϪ␶͒d␶, ͑21͒ 45 ͵0 and D(r,t) can be used:

D͑r,t͒ϭ4␲␳͑r,t͒, ٌ؋E͑r,t͒ϭ0. ͑29͒ On comparing Eqs. ͑34͒ and ͑21͒ an expression follows–ٌ for the Fourier component of the linear response function E(r,t) and D(r,t) are coupled via the solvent electric sus- ␣ ϭ͐ϱ␣(t)exp(i␻t)dt: ceptibility ⑀(r,rЈ␶), which is generally nonlocal in both the ␻ 0 space and time domains, ͑0͒ 3 ͑0͒ 3 ␣␻ϭ ͵ D21 ͑r͒–Pop͑r͒d r Ϫ ͵ D21 ͑r͒•P␻͑r͒d r. ͑35͒ ϱ D͑r,t͒ϭ d␶ drЈ⑀͑r,rЈ,␶͒E„r,tϪ␶). ͑30͒ Equations ͑32͒ and ͑35͒ permit the calculation of the linear ͵0 ͵ response function ␣␻ and, using Eq. ͑23͒, the spectral den- We will use a standard, long-wave approximation for sity J(␻). ⑀(r,rЈ,␶) in which ⑀͑r,rЈ,␶͒Ӎ⑀͑r,␶͒␦3͑rϪrЈ͒, ͑31͒ E. Representation of ␣␻ in terms of a ‘‘free energy’’ where ␦3(r) is the three-dimensional delta-function. difference

One feature of Eqs. ͑29͒–͑31͒ is the inhomogeneity of To facilitate the calculation of ␣␻ it is first noted that the dielectric environment reflected in ⑀(r,␶), since the each term in Eq. ͑35͒ is related to a ‘‘free energy’’ of the Hamiltonian H1 includes the nonelectrostatic interaction be- dielectric with the dielectric response function ⑀␻(r)or tween the solvent and the reacting species which defines the (0) ⑀op(r) in the external electric field D12 (r). Specifically, the structural properties of the solvent in the vicinity of the re- ‘‘free energy’’ ␻ associated with a charge distribution ␳12 actants. F interacting with a dielectric with polarization P␻(r) and a The solution of Eqs. 29 – 31 is facilitated by taking 45 ͑ ͒ ͑ ͒ dielectric response function ⑀␻(r), is given by the time-Fourier transform and so going over to the fre- 1 quency domain. As a result, Eqs. ͑29͒–͑31͒ are reduced to ϭ D ͑r͒ E ͑r͒d3r electrostatic-like equations in which the usual static electric F␻ 8␲ ͵ ␻ – ␻ susceptibility ⑀0 is replaced by its frequency analog ⑀␻(r) ϱ 1 ϭ͐ ⑀(r,␶)exp(i␻␶)d␶ ϭ ⑀ ͑r͓͒E ͑r͔͒2 d3r. ͑36͒ 0 8␲ ͵ ␻ ␻ ,D ͑r͒ϭ4␲␳ ͑r͒, ٌϫE ͑r͒ϭ0 ٌ – ␻ 21 ␻ It can be shown45 that this equals ͑32͒ F␻ D␻͑r͒ϭ⑀␻͑r͒E␻͑r͒, 1 1 ͑0͒ 2 3 ͑0͒ 3 ␻ϭ ͓D21 ͑r͔͒ d rϪ P␻͑r͒•D21 ͑r͒d r, ͑37͒ where E␻(r) and D␻(r) denote the corresponding Fourier F 8␲ ͵ 2 ͵ components of the electric field and electric displacement per and so Eq. ͑35͒ can be rewritten as unit f ␻ : ␣ ϭ2͑ Ϫ ͒, ͑38͒ 1 ϱ ␻ F␻ Fop E ͑r͒ϭ E͑r,t͒exp͑i␻t͒dt, ␻ ͵ where op is the free energy, Eq. ͑36͒, of the charge distri- f ␻ Ϫϱ bution F␳ (r) in the dielectric environment with the dielectric ͑33͒ 21 1 ϱ response function ⑀op(r). Equation ͑38͒ was derived from D␻͑r͒ϭ ͵ D͑r,t͒exp͑i␻t͒dt. molecular considerations and so still represents a molecular f ␻ Ϫϱ statistical mechanical expression, in the long wavelength ap- ϱ Here, f ␻ϭ͐Ϫϱ f (t)exp(i␻t)dt is the Fourier component of proximation for treating the solvent. the generalized force f (t) ͓cf. Eq. ͑21͔͒. E␻(r) and D␻(r)so Equation ͑38͒ is a useful generalization in two respects, defined are independent of the function f (t).46 in frequency domain and in treating the problem quantum The spatial inhomogeneity of the solvent enters into Eqs. mechanically, of a result obtained earlier.31,32 The latter re- ͑32͒ via the spatial dependence of ⑀␻(r). In the presence of lated the reorganization energy of the reaction to the ͑free͒ any sharp boundaries in the problem, the appropriate bound- energy difference of two hypothetical systems, one of which ary conditions for the electric field E␻(r) and electric dis- responds to a difference in charge distributions of the reac- placement D␻(r) must be used to solve Eqs. ͑32͒. tant and product states via a static dielectric response and Using the expression for the reaction coordinate X, Eq. another via an electronic one. ͑24͒, and Eqs. ͑C4͒–͑C10͒, the following expression for the Only the frequencies which correspond to the solvent Fourier component of the averaged reaction coordinate, nuclear motion must be considered, since the solvent elec- ¯ ϱ ¯ ␦X␻ϭ͐Ϫϱ␦X(t)exp(i␻t)dt, is readily obtained: tronic motion has been taken into account implicitly in the form of the solvent electronic dielectric response ⑀op(r). As ¯ ͑0͒ 3 a result, the rate constant k of the reaction, Eq. ͑4͒, can be ␦X␻ϭϪf␻ D21 ͑r͒•͓P␻͑r͒ϪPop͑r͔͒d r, ͑34͒ ͵ rewritten as where P␻ is the Fourier component of the total average sol- ϱ 2 ␻op Ϫ2 2 0 vent polarization, Eq. ͑C10͒, and Pop(r) is a part of the elec- kϭប ⌬ dt exp Ϫit⌬G /បϪ d␻ tronic polarization which is induced by the external charge ͵Ϫϱ ͫ ␲͵0 distribution, Eq. ͑C5͒. The difference P␻(r)ϪPop(r) reflects Im ␻ cosh͑ប␤␻/2Ϫi␻t͒Ϫcosh͑ប␤␻/2͒ the induced nuclear polarization of the solvent. P␻(r) tends ϫ F , ͑39͒ to P when ␻ approaches the optical frequency region. ប␻2 sinh͑ប␤␻/2͒ ͬ op

Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 5312 J. Chem. Phys., Vol. 110, No. 11, 15 March 1999 Georgievskii, Hsu, and Marcus where the standard units, in which បÞ1, have been used. distant from each other, r0ӷa, so that their dielectric image The frequency ␻op corresponds to the transparency region effect can be neglected. Then using Eqs. ͑38͒ and ͑41͒ it can which separates the frequency region of the solvent nuclear readily be shown that ␣␻ is motion and the one of the solvent electronic motion. ␣ ϭ2q2 ⑀Ϫ1Ϫ⑀Ϫ1͒ aϪ1ϪrϪ1͒. ͑43͒ The above expression for the rate, Eq. ͑39͒, can be con- ␻ ͑ op ␻ ͑ 0 siderably simpliﬁed by estimating the time-integral in Eq. Using this equation, the analytical properties of ⑀␻ , and the 16,47 ͑39͒ using a saddle point method and by dividing the expression for the reorganization energy ␭ in terms of the 10,48 whole frequency range of integration into two regions: spectral density J(␻),24,48,49 the low frequency region corresponding to ␻Ͻ4kBT/ប and 48 1 ϱJ͑␻͒ the high frequency region with ␻Ͼ4kBT/ប. We leave this ␭ϭ d␻, ͑44͒ calculation for a future work. ␲ ͵0 ␻ the standard expression for ␭ is obtained7 2 Ϫ1 Ϫ1 Ϫ1 Ϫ1 F. Speciﬁc solute models ␭ϭq ͑⑀op Ϫ⑀0 ͒͑a Ϫr0 ͒, ͑45͒

where ⑀0 is the static dielectric constant of the solvent. Calculations based on ␻ can be used for realistic solute charge distributions, i.e., thoseF not containing point charges. The dielectric properties of the solvent enter into Eq. In some idealized models based on point charges, Eq. ͑36͒ ͑43͒ in the form of a factor 1/⑀␻ . Such a form corresponds to a homogeneous solvent and has been commonly used in the would yield an infinity which cancels when ␻Ϫ op is cal- 1,2,14 culated. To avoid this infinity, it is useful toF introduceF the theory of electron transfer reaction rates and nonlinear spectroscopy.27 This form, however, is modified when ac- energy ␻Ј of the ‘‘dielectric’’ in the external electric field D(0)(r):F45 count is taken of the inhomogeneity of the system near the 21 solute molecule.50 1 Another example is the Onsager model,51 which has Ј ϭ Ϫ ͓D͑0͒͑r͔͒2 d3r F␻ F␻ 8␲ ͵ 21 been used in the spectroscopy of static and dynamical solvatochromism.52 In this model the solute has a spherical 1 ͑0͒ 3 shape but the charge distribution is approximated by a dipole ϭϪ P␻͑r͒–D21 ͑r͒d r. ͑40͒ 2͵ d0 in the center of a polarizable sphere of radius a and the electronic dielectric constant ⑀c inside the sphere. The effec- That is, Ј differs from only by the frequency- F␻ F␻ tive dipole which would be seen from outside of such a independent self-interaction term in Eq. ͑40͒. When the sol- 53 sphere in vacuum is dϭ3d0 /(⑀cϩ2). The solution of the ute can be modeled with point charges q j , the representation electrostatic equations for this problem is well known.51,53 Ј in terms of an electrostatic potential ␾(r) is convenient, F␻ Using Eq. ͑42͒ one then obtains that the ␣␻ in Eq. ͑35͒ in Eq. ͑40͒ corresponds to this case is54

1 2 2 ͑⑀cϩ2͒ d 1 1 ␻Ј ϭ ͚ q j␾␻Ј ͑rj͒, ͑41͒ ␣ ϭ Ϫ . ͑46͒ F 2 j ␻ 3 3 ͫ2⑀ ϩ⑀ 2⑀ ϩ⑀ ͬ a op c ␻ c where ␾␻Ј (r) is the part of electrostatic potential created by Comparing Eqs. ͑46͒ and ͑43͒ one can see that the factor the solvent polarization with the dielectric function ⑀ (r). In ␻ 1/⑀␻ for the homogeneous medium is changed to the factor the case when the solute is modeled by a point dipole d ,it 0 1/(2⑀␻ϩ⑀c) in the Onsager model. This change is a result of is more convenient to express Ј in terms of an electric F␻ an inhomogeneity in the system, i.e., it is due to the presence field, of the solute cavity in the solvent. It can modify the longitu- 33,55 1 dinal relaxation time in the dynamics of the Stokes shift. ␻Ј ϭϪ 2d0R␻ , ͑42͒ 41 F Often ⑀cӍ2 is used. where R is the reaction field, equal to Ϫٌ␾␻Ј (r), created by the solvent at the site of the dipole. Any spatial dependence of ⑀␻ reflects the structural in- III. DISCUSSION homogeneity of the solvent in the vicinity of the reacting species. In the simplest approximation, the polarizability of Linear response theory has frequently been used to relate the solute is neglected and it is so assumed that ⑀␻(r)ϭ1in the fluctuations of the electromagnetic field in the entire en- the region occupied by the reactants. An improved approxi- vironment, with the dielectric properties of the environment, mation for the electronic polarizability of the solute would be e.g., as in Refs. 16, 17, and 56. The approach in the present ⑀cӍ2 inside the cavity, corresponding approximately to an paper is somewhat different and simpler, because attention is electronic polarizability of the solvent.41 As an example, the focused on the calculation of the fluctuations of only a single donor and acceptor have been modeled as spheres of radii a variable, the reaction coordinate X. In this sense our ap- 33 separated by distance r0. The charge distribution ␳21 consists proach is similar to the one used by Mukamel and by Flem- of two point charges q (q is an elementary charge͒ which are ing and co-workers26 to treat the state-specific solvation dy- situated at the centers of the spheres. It will also be assumed namics of chromophoric molecules excited by a sequence of for simplicity that the donor and acceptor are sufficiently ultrashort laser pulses. The reaction coordinate X, defined in

Eq. ͑7͒ as an energy difference of the system in two elec- ity of the solute. At present, however, only nonlocal spatial tronic states has been used by earlier workers37,57,58 in more dependence ͑or, equivalently, a k-dependence in the Fourier classical treatments of the rate constant. space͒ of the static electric susceptibility has been estimated, Equation ͑16͒ formally coincides with the correlation namely from neutron diffraction measurements.64–66 Use of function of the collection of molecular harmonic oscillators the spatially nonlocal dielectric response function for the in- if one introduces for J(␻) the spectral density of their nor- homogeneous solvent would, of course, complicate the solu- mal modes,49,59 which justifies the usage of this title ͑spectral tion of Eqs. ͑32͒.67 The static counterpart of Eq. ͑38͒ has density͒ for J(␻). The spectral density of the harmonic sys- been applied to various problems, e.g., Refs. 68 and 69. tem is temperature-independent. For real nonlinear systems the spectral density was found to depend strongly on the ACKNOWLEDGMENTS temperature for low frequency component of J(␻) ascribed to diffusive motion of the solvent molecules.24 It is worth- It is a pleasure to acknowledge the support of the Na- while noting also that while for a harmonic oscillator solvent tional Science Foundation and the Office of Naval Research. the spectral density is, of course, the same in classical and One of us ͑Y.G.͒ would like to acknowledge the support of quantum mechanics, classical and quantum mechanics will the James W. Glanville Postdoctoral Scholarship in Chemis- give different results for J(␻) of a real nonlinear solvent. try at Caltech. We would also like to thank Shaul Mukamel, The spectral density which enters into Eq. ͑16͒ can be Bruce Berne, and David Chandler for helpful comments. readily rewritten as a Fourier transform of the imaginary part of the correlation function C(t): APPENDIX A: DERIVATION OF EXPRESSION FOR ϱ SOLUTE-SOLVENT INTERACTION ENERGY J͑␻͒ϭ2 dt Im͓C͑t͔͒sin ␻t. ͑47͒ ͵0 Equation ͑2͒ describes the electrostatic interaction between the solute characterized by the charge distribution Using this equation as a definition of a spectral density of ␳ j(r) and the solvent in the particular nuclear configuration, effective harmonic oscillators a real anharmonic system which is characterized by the solvent polarization operator could be mapped, if one wished, onto the harmonic oscillator Pˆ (r). To justify it one can calculate the work W which must model.26,30 The present considerations show that the results 0 be done to charge the solute from the hypothetical neutral obtained in the framework of such a model would be valid in state with ␳(r)ϭ0 to the actual jth state with the charge the second-order cumulant expansion approximation. distribution ␳(r)ϭ␳ (r) while the solvent nuclear configura- In the present paper we considered electron transfer re- j tion is kept fixed. At any specified solvent nuclear configu- actions. Our considerations, however, are applicable to other ˆ nonadiabatic processes in a polar environment in which the ration the solvent polarization operator P0(r), which refers change of a charge distribution occurs in the process of a to the hypothetical neutral state of the solute, can be viewed 30 as a real function of r and not only as an operator. nonadiabatic transition. Mukamel showed in a similar fash- el ion that the correlation function C(t), Eq. ͑11͒, enters the The electronic part of the Hamiltonian operator H , final expressions for different order nonlinear optical pro- which acts on the electronic wave function ⌿ of the solvent cesses. The present Eqs. ͑16͒, ͑35͒, and ͑23͒ can be used to and which describes the solvent interacting with itself and calculate J(␻) and C(t) for an appropriate model of a chro- with the charge distribution ␳(r), can be written as mophoric solute molecule. ͑el͒ ͑el͒ ˜ 3 In the dielectric continuum models, such as those in Sec. H ϭH0 ϩ ͵ ␳͑r͒␾͑r͒d r, ͑A1͒ II F, the inhomogeneity of the system is taken into account in the simplest possible way, namely the solvent is assumed to where ␾˜ (r) is the electrostatic potential. The ␾˜ (r) in Eq. be homogeneous up to the boundary of the solute cavity with ͑A1͒ must be considered as a quantum mechanical operator ͑actually, as an operator function parametrically dependent ⑀␻(r)ϭ⑀␻ , which does not depend on r. The last assumption does not take into account molecular nature of the sol- on r) acting on the electronic wave function ⌿. The tilde vent. As a result, one could expect that more realistic ⑀ (r) notation is used in this section to denote operators acting on ␻ (el) changes smoothly on a molecular length scale from its bulk the electronic wave function ⌿.InEq.͑A1͒ H0 is an operator which refers to the solvent and includes the interaction value inside the solvent to ⑀c at the solute boundary ͑we neglect polarizability of the solute͒. One approximate way of of the electrons with the nuclei and with themselves and the including molecular effects is given in Ref. 23. As a possible kinetic energy operator of the electrons of the solvent. The way of dealing with this problem, the mean spherical part of the solute-solvent interaction which is not solute 60–63 (el) (el) approximation may be applied. The present treatment state-specific is also treated as being included in H0 . H0 also includes an excluded volume effect as discussed in Ap- does not include the kinetic energy operator for the nuclei, in pendix D and as was discussed earlier by Song et al.23 accord with the Born–Oppenheimer approximation. By using Eq. ͑32͒ it is assumed that the dielectric re- The solvent electronic wave function ⌿ will change in sponse which appears in this equation in terms of the electric the process of charging. As a result, the energy of the system (el) susceptibility ⑀␻ is local in space. This assumption may not ͗⌿͉H ͉⌿͘ will change too. To calculate this change one be accurate for a field which varies considerably on a mo- notes that the electrostatic potential ␾˜ (r) is actually a ͑func- lecular scale and one can use a spatially nonlocal dielectric tional͒ derivative of Hel over ␳(r) ͓cf. Eq. ͑A1͔͒. Then, using response function18 to calculate the electric field in the vicin- the Hellman–Feynman theorem on the derivative of the en-

Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 5314 J. Chem. Phys., Vol. 110, No. 11, 15 March 1999 Georgievskii, Hsu, and Marcus ergy with respect to a parameter for the system with a and HЈj in Eq. ͑1͒. They obtained Eq. ͑A8͒ for the total parameter-dependent Hamiltonian,70 one readily obtains the electric energy of molecules having permanent and induced following expression for the work W to charge the solute in dipoles ͑we omit our carets for notational simplicity͒: this case, 1 0 0 1 0 0 Vϭ 2␮ –T–␮Ϫ␮–D Ϫ 2D –A–D , ͑A8͒ ␳ϭ␳ j 3 ˆ 0 Wϭ ͵ d r͵ ␦␳͑r͒␾␳͑r͒, ͑A2͒ where ␮ and ␮ are 3N-dimensional vectors ͑whose compo- ␳ϭ0 nents in the three-dimensional subspace of solvent molecule ˆ ˜ n are d and d0). D0 isa3N-dimensional vector with three where ␾␳(r)ϭ͗⌿͉␾(r)͉⌿͘ is the average of the electro- n n (0) static potential ␾˜ (r) over the ground state electronic wave components of D (rn) at each molecule n and arises from function ⌿ corresponding to the solute charge distribution the solute charges, T is a symmetric tensor of order 3N -␳(r). It is an operator which acts on the nuclear wave func- ϫ3N with elements Tmn͓ϭٌmٌn(1/rmn)ifmÞn] in three tion. dimensional subspace, and with Tmmϭ0. A is a symmetric tensor related to T by Aϭ␣ (Iϩ␣ T ؊1, where ␣ is the The electrostatic potential ␾ˆ (r), i.e., the average of – – … ␳ polarizability of a solvent molecule and I is a unit tensor of ˜ ␾(r) over ⌿, can be written as a sum of two terms, order 3Nϫ3N. ͓The expression is readily generalized when ␾ˆ r͒ϭ␾ˆ r͒ϩ␾Ј r͒, ͑A3͒ ␣ is a tensor of order 3Nϫ3N, and one obtains the tensorial ␳͑ 0͑ ␳͑ Ϫ1 -product ␣–(Iϩ␣–T) for A.] ٌn is the gradient with re ˆ where the electrostatic potential ␾0(r) corresponds to ␳(r) spect to the coordinates of molecule n. ˆ ϭ0 and arises from P0„r…: In Eq. ͑A8͒ the second term on the right is equivalent to a term Ϫ͐Pˆ r D(0)(r)d3r, when one notes the definition of 1 0„ …– ˆ ˆ 3 Pˆ (r) in Eq. ͑3͒. The first term in the right is independent of ␾0͑r͒ϭ P0͑rЈ͒–ٌ d rЈ. ͑A4͒ 0 ͵ rϪr „0… ͉ Ј͉ D „r… and so is the electrostatic contribution to the H0 in Eq. ͑1͒. The last term in Eq. ͑A8͒ does not contain the per- The second term in Eq. ͑A3͒, ␾␳Ј(r), describes the additional 0 electronic solvent response to the external electric field cor- manent dipoles dn and so is insensitive to nuclear position. It ˆ contributes to the h j in Eq. ͑1͒. responding to the charge distribution ␳(r). ͓P0(r) contains the electronic response in the absence of that field.͔ As a consequence of Eq. ͑A3͒, the work W can be writ- APPENDIX B: EXPRESSION FOR THE STANDARD ten as FREE ENERGY OF REACTION WϭW0ϩWЈ, ͑A5͒ The constant ⌬G0 in Eq. ͑17͒ is seen from Eqs. ͑10͒, ˆ ͑16͒ and ͑44͒ to be where W0 is the contribution from ␾0(r) and WЈ is the con- 0 tribution from ␾␳Ј(r). Reexpressing W0 in terms of the ⌬G ϭ͗X͘Ϫ␭. ͑B1͒ vacuum electric field D(0)(r) and the solvent polarization j It can be shown that this ⌬G0 coincides with the standard operator Pˆ (r), using arguments similar to those used in 0 free energy of reaction ⌬G0, as follows. The standard free macroscopic electrostatics,45 it follows that energy of reaction ⌬G0 is defined by

͑0͒ ˆ 3 ˆ 3 Ϫ␤H2 ␤ W0ϭ ␳ j͑r͒␾0͑r͒d rϭϪ Dj ͑r͒•P0„r…d r, ͑A6͒ 0 Tr͓e ͔ Ϫ͐ ˜X͑␶͒d␶ ͵ ͵ eϪ␤⌬G ϭ ϭ͗e 0 ͘, Tr͓eϪ␤H1͔ ϩ which is the term denoted by HЈj in Eq. ͑2͒ in the text. ␶H Ϫ␶H It is usually assumed that the response in ␾␳Ј(r) depends ˜X͑␶͒ϭe 1Xe 1. ͑B2͒ negligibly on the nuclear configuration of the solvent. As a 29 result, the corresponding contribution WЈ to the work W in Using the second-order cumulant expansion one finds from Eq. ͑A2͒ depends negligibly on the solvent nuclear configu- Eq. ͑B2͒ that ration either and therefore can be added to the solute energy ␤ ␶ G0 X Ϫ Ϫ1 d d C˜ , h j , renormalizing it. If one wished, one could readily esti- ⌬ Ӎ͗ ͘ ␤ ␶ ␶Ј ͑␶Ј͒ ͵0 ͵0 mate WЈ in terms of the electronic dielectric response func- 45 tion ⑀op(r), obtaining as a result, C˜ ͑␶͒ϭ͗␦˜X͑␶͒␦˜X͑0͒͘ϭC͑Ϫi␶͒. ͑B3͒ 1 1 Substituting Eq. ͑16͒ into Eq. ͑B3͒ and integrating over ␶Ј WЈϭ ⑀ ͑r͓͒E ͑r͔͒2d3rϪ ͓D͑0͒͑r͔͒2d3r, 8␲ ͵ op j,op 8␲͵ j and ␶ one finds that ⌬G0 in Eq. ͑B2͒ is the same as that ͑A7͒ given by Eq. ͑B1͒. where Ej,op(r) is the electric field in the environment with the dielectric response function ⑀ (r). E (r) satisfies Eq. op j,op APPENDIX C: MACROSCOPIC POLARIZATION P AND ͑C3͒ below, with ␳ (r) replacing ␳(r,t) in the right-hand j MACROSCOPICAL FIELDS E AND D IN TERMS side of the first equation in Eq. ͑C3͒. OF THE MOLECULAR MODEL OF THE SOLVENT It is useful to relate Eqs. ͑1͒ and ͑2͓͒also Eq. ͑A6͔͒ to the insightful molecular treatment of Mandel and Mazur,40 In this Appendix we relate the average electric field and obtain, thereby, the electrostatic contribution to h j , H0, E(r,t) and electric displacement D(r,t) to the relevant op-

Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp J. Chem. Phys., Vol. 110, No. 11, 15 March 1999 Georgievskii, Hsu, and Marcus 5315 erators ͓Eqs. ͑C2͒–͑C8͔͒ in terms of our molecular model. In this equation the first term is an electric displacement We first note that the average polarization P(r)atris given which is associated with the solvent electronic response only. by The last two terms in Eq. ͑C7͒ represent the electric field due to the solvent polarization ͑the second term͒ and the solvent ˆ P͑r,t͒ϭTr͓P0͑r͒R͑t͔͒, ͑C1͒ polarization operator term itself ͑the third term͒. ˆ The average electric field E(r,t) and electric displace- where the operator P0(r) of the solvent polarization is defined by Eq. ͑3͒ and the time-dependent density matrix R(t) ment D(r,t) can then be defined as is given by Eq. ͑19͒. E͑r,t͒ϭTr͓Eˆ ͑r,t͒R͑t͔͒, D͑r,t͒ϭTr͓Dˆ ͑r,t͒R͑t͔͒, ͑C8͒ We next introduce an electric field operator Eˆ (r,t)as where, as noted earlier, the time-dependent density matrix 1 R(t) is given by Eq. ͑19͒. One could use a bar notation for ˆ ˆ 3 E͑r,t͒ϭϪٌ P0͑rЈ͒–ٌ d rЈϩEop͑r,t͒. ͑C2͒ E(r,t) and D(r,t), as in Eq. ͑20͒, but since they coincide ͵ ͉rϪrЈ͉ with the conventionally used macroscopic fields,41 we omit The first term in Eq. ͑C2͒ is an electric field generated by the it. Equation ͑C8͒ completes the definition of E(r,t) and ˆ D(r,t) in terms of the operators defined in Eqs. ͑C2͒ and solvent polarization P0„r…. The second term Eop(r,t)isan additional electric field which is due to the external charge ͑C7͒ and Pˆ (r). They are used in the text in Eqs. ͑29͒–͑30͒. and to the instantaneous electronic response of the solvent to The Fourier component of the average value of the total that external charge distribution ␳(r,t). ͑See also the discus- polarization per unit f ␻ ͓cf. Eq. ͑33͔͒, sion in Appendix A.͒ The electric field E (r,t), which is op 1 ϱ ˆ assumed to be independent on the solvent nuclear configura- P␻͑r͒ϭ Tr͓R͑t͒P͑r,t͔͒exp͑i␻t͒dt, ͑C9͒ tion, satisfies the equations: f ␻ ͵Ϫϱ is related in a standard way to the corresponding Fourier ,Dop͑r,t͒ϭ4␲␳͑r,t͒, ٌϫEop͑r,t͒ϭ0–ٌ ͑C3͒ component of the average electric field ͓cf. Eq. ͑32͔͒ Dop͑r,t͒ϭ⑀op͑r͒Eop͑r,t͒, ⑀␻͑r͒Ϫ1 P ͑r͒ϭ E ͑r͒. ͑C10͒ where ␳(r,t) is defined in Eq. ͑28͒, ⑀op(r) is the electronic ␻ 4␲ ␻ dielectric susceptibility of the solvent at r, and Dop(r,t)is the electronic electric displacement associated with ␳(r,t). Using Eqs. ͑C4͒, ͑C5͒, and ͑C10͒ one readily obtains Eq. 34 . We note that the operator Pˆ (r,t) of the total solvent ͑ ͒ polarization can be written as a sum of two terms APPENDIX D: EQUATION FOR THE GENERALIZED ˆ ˆ P͑r,t͒ϭP0͑r͒ϩPop͑r,t͒. ͑C4͒ SOLVENT SUSCEPTIBILITY ␹˜ „m… IN THE ˆ PRESENCE OF A SOLUTE CAVITY The first term P0„r… is the operator of the molecular solvent ˆ polarization P0„r…, Eq. ͑3͒, which corresponds to a hypo- Two dipolar solvent molecules correlate with each other ˆ not only directly via the dipole–dipole interaction between thetical neutral state of the solute with ␳(r)ϭ0. P0„r… itself consists of two parts, one part arising from all the unper- them, but also indirectly via the interaction with other mol- turbed dipole moments of the solvent molecules, and the ecules of the solvent. As a result, the correlation between second part being a collective electronic response to the first orientations of the two solvent molecules in the vicinity of part ͓cf. the discussion after Eq. ͑3͔͒. the solute is modified in comparison with the bulk because of The second term in Eq. ͑C4͒, Pop(r,t), is an additional the absence of the fluctuating solvent polarization at the electronic polarization at fixed positions of the solvent nuclei place of the solute. This excluded volume effect leads to which arises as a collective response of the solvent electrons modification of the dielectric response of the solvent to the 23 to the solute electric field D(0)(r,t), Eq. ͑27͒. We note that external electric field. In this Appendix we show how this Pop(r,t) in Eq. ͑C4͒ can be written in the conventional way excluded volume effect naturally appears in the present for- as mulation and demonstrate that the resulting equation, Eq. ͑D14͒, coincides with Eq. ͑3.11͒ in Ref. 23. 1 ⑀opϪ1 P ͑r,t͒ϭ ͓D ͑r,t͒ϪE ͑r,t͔͒ϭ E ͑r,t͒. ͑C5͒ For our purposes, it is convenient to rewrite the electro- op 4␲ op op 4␲ op static equations ͑32͒ in a different form. We will assume that each term in Eqs. ͑D1͒–͑D14͒ refers to an arbitrary fre- We next define an electric displacement operator Dˆ (r,t) quency ␻ and will omit the index ␻ for notational simplicity, ͑motivated by the standard macroscopical electrostatic description41,45͒ as P͑r͒ϭ␹E͑r͒,

Dˆ ͑r,t͒ϭEˆ ͑r,t͒ϩ4␲Pˆ ͑r,t͒, ͑C6͒ E͑r͒ϭD͑0͒͑r͒ϩ ͵ d3rЈ T͑rϪrЈ͒P͑rЈ͒, ͑D1͒ which can be rewritten using Eqs. ͑C2͒, ͑C4͒, and ͑C5͒ as out where ␹ϭ͑⑀Ϫ1͒/4␲ is the dielectric polarizability of the sol- 1 ͑0͒ ˆ ˆ 3 ˆ vent, D ͑r͒ is the external electric ﬁeld, E͑r͒ is the local .D͑r,t͒ϭDop͑r,t͒Ϫٌ P0͑rЈ͒•ٌ d rЈϩ4␲P0͑r͒ ͵ ͉rϪrЈ͉ electric ﬁeld at the point r, and T͑rϪrЈ͒P͑rЈ͒ is given by Eq. ͑C7͒ ͑D2͒, T being a tensor:

Downloaded 08 Mar 2006 to 131.215.225.174. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp 5316 J. Chem. Phys., Vol. 110, No. 11, 15 March 1999 Georgievskii, Hsu, and Marcus

must be solved. 3P͑rЈ͒•͑rϪrЈ͒͑rϪrЈ͒ P͑rЈ͒ T͑rϪr ͒P͑r ͒ϭ Ϫ . ͑D2͒ Substituting Eq. ͑D11͒ into Eq. ͑D9͒ one arrives at Eq. Ј Ј 5 3 ͉rϪrЈ͉ ͉rϪrЈ͉ ͑3.11͒ in Ref. 23, which in our notation is Equation ͑D2͒ gives the electric field produced at the point r ͑m͒ ͑0͒ Poutϭ˜␹ D , by a dipole P͑rЈ͒ at the point rЈ. The integration in Eq. ͑D1͒ out,out out ͑m͒ Ϫ1 ͑D13͒ is over the volume outside of the solute. ˜␹out,outϭ˜␹out,outϪ˜␹out,in͑˜␹in,in͒ ˜␹in,out Expressing E͑r͒ in terms of P͑r͒ from the first part in or, in the standard notation, is Eq. ͑D1͒ and substituting it into the second part one arrives at the following result relating the solvent polarization P r ͑ ͒ ͑m͒ 3 3 ␹ ͑r,rЈ͒ϭ˜␹͑rϪr͒Ϫ d rЉd rٞ˜␹͑rϪrЉ͒˜ (0) to the external electric field D (r): ͵͵in

͑0͒ 3 Ϫ1 ˜ Ϫ1 ˜ D ͑r͒ϭ d rЈ ˜␹ ͑rϪrЈ͒P͑rЈ͒, ͑D3͒ ϫ␹in ͑rЉ,rٞ͒␹͑rٞϪrЈ͒. ͑14͒ ͵out where ˜␹Ϫ1(rϪrЈ) is a tensor, the inverse generalized sus- 1 M. D. Newton and N. Sutin, Annu. Rev. Phys. Chem. 35, 437 ͑1984͒. 2 ceptibility of the bulk solvent in our model. It is defined as R. A. Marcus and N. Sutin, Biochim. Biophys. Acta 811, 265 ͑1985͒. 3 P. F. Barbara, T. J. Meyer, and M. A. Ratner, J. Phys. Chem. 100, 13148 ˜␹Ϫ1͑rϪrЈ͒ϭ␹Ϫ1I␦3͑rϪrЈ͒ϪT͑rϪrЈ͒, ͑D4͒ ͑1996͒. 4 M. Lax, J. Chem. Phys. 20, 1752 ͑1952͒. where I is a unit tensor. It is convenient to use the following 5 R. Kubo and Y. Toyozawa, Prog. Theor. Phys. 13, 160 ͑1955͒. 6 symbolic shorthand notation for Eq. ͑D4͒ and subsequent V. G. Levich and R. R. Dogonadze, Dokl. Akad. Nauk SSSR 124, 123 ͑1959͒; V. G. Levich, Adv. Electrochem. Electrochem. Eng. 4, 249 equations: ͑1966͒. 7 ͑0͒ ˜Ϫ1 R. A. Marcus, J. Chem. Phys. 24, 966 ͑1956͒. Doutϭ␹out,outPout, ͑D5͒ 8 R. A. Marcus, Can. J. Chem. 37, 155 ͑1959͒. 9 R. A. Marcus, J. Chem. Phys. 43, 679 ͑1965͒. where the two indices ‘‘out’’ indicate that both r and rЈ in 10 M. A. Vorotyntsev, R. R. Dogonadze, and A. M. Kuznetsov, Dokl. Akad. the corresponding integral equation are in a region outside Nauk SSSR 195, 1135 ͑1970͒. the solute cavity. We will also use the index ‘‘in’’ to imply 11 J. Jortner, J. Chem. Phys. 64, 4860 ͑1976͒. 12 J. Ulstrup and J. Jortner, J. Chem. Phys. 63, 4358 ͑1975͒. that the corresponding variable is inside the cavity. The no- 13 ˜Ϫ1 ˜Ϫ1 ˜ Ϫ1 N. R. Kestner, J. Logan, and J. Jortner, J. Phys. Chem. 78, 2148 ͑1974͒. tation ␹out,out means ͑␹ ͒out,out and not ͑␹out,out͒ . 14 J. Ulstrup, Charge Transfer Processes in Condensed Media ͑Springer In Eq. ͑D3͒ both r and rЈ are the points outside the Verlag, Berlin, 1979͒. solute cavity. We can include r and rЈ in the cavity, taking 15 D. Chandler, in Liquids, Freezing and Glass Transition, edited by J. P. P͑rЈ͒ϭ0 for rЈ in the cavity, which in our shorthand notation Hansen, D. Levesque, and J. Zinn-Justin ͑Elsevier Science, Amsterdam, 1991͒, p. 193. is simply Pinϭ0, and treating r as being everywhere, and so 16 A. A. Ovchinnikov and M. Y. Ovchinnikova, Sov. Phys. JETP 29, 688 extend Eq. ͑D3͒ to the whole space, ͑1969͒. 17 A. A. Abrikosov, L. P. Gorkov, and I. Y. Dzyaloshinski, Methods of ͑0͒ 3 ˜ Ϫ1 Quantum Field Theory in Statistical Physics ͑Dover, New York, 1975͒. D ͑r͒ϭ d rЈ ␹ ͑rϪrЈ͒P͑rЈ͒, ͑D6͒ 18 A. A. Kornyshev, in The Chemical Physics of Solvation, Part A, Vol. 38 ͵all space of Studies in Physical and Theoretical Chemistry, edited by R. R. Dogo- which can be rewritten in a shorthand notation as ͑the ab- nadze, E. Ka´lmań, A. A. Kornyshev, and J. Ulstrup ͑Elsevier, Amsterdam, sence of in and out subscripts denotes r and r in all space , 1985͒,p.77. Ј ͒ 19 A. A. Kornyshev and J. Ulstrup, Chem. Phys. Lett. 126, 203 ͑1986͒. 20 D͑0͒ϭ˜␹Ϫ1P. ͑D7͒ A. A. Kornyshev, A. M. Kuznetsov, D. K. Phelps, and M. Weaver, J. Chem. Phys. 91, 7159 ͑1989͒. The last equation can be easily solved by the Fourier trans- 21 R. R. Dogonadze and A. M. Kuznetsov, Elektrokhimia 7, 763 ͑1971͒ ͓English Translation: Sov. Electrochem. 7, 735 ͑1971͔͒. form method. The solution can be written in our shorthand 22 R. R. Dogonadze, A. A. Kornyshev, and A. M. Kuznetsov, Theor. Math. notation as Phys. 15, 4047 ͑1973͒. 23 ͑0͒ X. Song, D. Chandler, and R. A. Marcus, J. Phys. Chem. 100, 11954 Pϭ˜␹D . ͑D8͒ ͑1996͒. 24 S. A. Passino, Y. Nagasawa, and G. R. Fleming, J. Chem. Phys. 107, 6094 The solvent polarization can be then written as ͑1997͒. 25 P ϭ˜ D͑0͒ϩ˜ D͑0͒. D9 Y. Nagasawa, S. A. Passino, T. Joo, and G. R. Fleming, J. Chem. Phys. out ␹out,out out ␹out,in in ͑ ͒ 106, 4840 ͑1997͒. 26 To find D͑0͒ it is noted that G. R. Fleming and M. Ado, Annu. Rev. Phys. Chem. 47, 109 ͑1996͒. in 27 S. Mukamel, Principles of Nonlinear Optical Spectroscopy ͑Oxford Uni- 0ϭP ϭ˜␹ D͑0͒ϩ˜␹ D͑0͒, ͑D10͒ versity Press, New York, 1995͒. in in,out out in,in in 28 J. S. Bader and B. J. Berne, J. Chem. Phys. 104, 1293 ͑1995͒; J. S. Bader, which yields C. M. Cortis, and B. J. Berne, ibid. 106, 2372 ͑1997͒; S. A. Egorov and B. J. Berne, ibid. 107, 6050 ͑1997͒; S. A. Egorov, E. Rabani, and B. J. Berne, ͑0͒ Ϫ1 ͑0͒ ibid. 108, 1407 ͑1998͒; E. Rabani, S. A. Egorov, and B. J. Berne, ibid. Din ϭϪ͑˜␹in,in͒ ˜␹in,outDout. ͑D11͒ 109, 6379 ͑1998͒. Ϫ1 29 It is useful to note that the operator (˜␹in,in) is the inverse R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II ͑Springer, of the operator ˜␹ and cannot be obtained from ˜␹Ϫ1, but Berlin, 1995͒; R. Kubo, J. Phys. Soc. Jpn. 17, 1100 ͑1962͒. in,in 30 S. Mukamel, J. Phys. Chem. 89, 1077 ͑1985͒. that the appropriate integral equation, represented symboli- 31 R. A. Marcus, J. Chem. Phys. 38, 1858 ͑1963͒. cally by 32 R. A. Marcus, J. Chem. Phys. 39, 1734 ͑1963͒. 33 Ϫ1 R. F. Loring, Y. J. Yan, and S. Mukamel, J. Chem. Phys. 87, 5840 ͑1987͒. ͑˜␹in,in͒ ˜␹in,inϭIin,in ͑D12͒ 34 V. Hizhnyakov and I. Tehver, Phys. Status Solidi 21, 755 ͑1967͒.

35 V. V. Hizhnyakov and I. K. Rebane, Sov. Phys. JETP 47, 463 ͑1978͒. Eq. ͑42͒ denotes the part of the reaction field which is produced by the 36 V. V. Hizhnyakov and I. J. Tehver, J. Lumin. 18/19, 673 ͑1979͒. solvent with a dielectric response function ⑀␣ around the solute sphere. 37 R. A. Marcus, Discuss. Faraday Soc. 29,21͑1960͒. The other part of the field is the dielectric response of the solute sphere 38 R. R. Dogonadze, A. M. Kuznetsov, and M. A. Vorotyntsev, Phys. Status itself, which does not depend on ␻ and cancels in Eq. ͑38͒. The dipole Solidi B 54, 125 ͑1972͒. 39 moment which inters into Eq. ͑42͒ is a bare dipole moment which should S. Efrima and M. Bixon, Chem. Phys. 13, 417 ͑1976͒. be renormalized, dϭ3d /(⑀ ϩ2). It is readily shown using the results of 40 M. Mandel and P. Mazur, Physica ͑Utrecht͒ 24, 116 ͑1958͒. 0 c Ref. 53 that Rϭ2(⑀ Ϫ1)d/(2⑀ ϩ⑀ )a3. Substituting this equation into 41 C. J. F. Bo¨ttcher, Theory of Electric Polarization ͑Elsevier, Amsterdam, ␻ ␻ c 1973͒, Vol. 1. Eq. ͑42͒ obtains Eq. ͑46͒. 55 42 A. M. Kuznetsov, Nouv. J. Chim. 5, 427 ͑1981͒. B. Bagchi, D. W. Oxtoby, and G. R. Fleming, Chem. Phys. 86, 257 43 A. A. Belousov, A. M. Kuznetsov, and J. Ulstrup, Chem. Phys. 129, 311 ͑1984͒. 56 ͑1989͒. L. D. Landau and E. M. Lifshitz, Statistical Physics II ͑Pergamon, New 44 (0) There is also a static external electric field D1 (r created by the charge York, 1986͒. 57 A. Warshel, J. Phys. Chem. 86, 2218 ͑1982͒. distribution ␳1(r) in the initial electronic state of the solute. It, however, 58 only changes the equilibrium average value of X rather than C(t) and is R. A. Kuharski, J. S. Bader, D. Chandler, M. Sprik, M. L. Klein, and R. not a focus of the present calculation. W. Impey, J. Chem. Phys. 89, 3248 ͑1988͒. 59 45 L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media A. J. Leggett et al., Rev. Mod. Phys. 59,1͑1987͒. 60 ͑Pergamon, New York, 1984͒. P. G. Wolynes, J. Chem. Phys. 86, 5133 ͑1987͒. 61 46 In Eqs. ͑32͒ and ͑33͒ we have made use of the fact that the Fourier I. Rips, J. Klafter, and J. Jortner, J. Chem. Phys. 88, 3246 ͑1988͒. 62 I. Rips, J. Klafter, and J. Jortner, J. Chem. Phys. 89, 4288 ͑1988͒. component f ␻ of the generalized force f (t) enters as a simple factor into 63 the expressions for the corresponding Fourier components of the electric M. L. Horng, J. A. Gardecki, A. Papazyan, and M. Maroncelli, J. Phys. field E(r,t) and dielectric displacement D(r,t) because of the linearity of Chem. 99, 17311 ͑1995͒. 64 Eqs. ͑29͒ and ͑30͒, cf. also Eq. ͑28͒. P. A. Bopp, A. A. Kornyshev, and G. Sutmann, Phys. Rev. Lett. 76, 1280 47 P. Siders and R. A. Marcus, J. Am. Chem. Soc. 103, 741 ͑1981͒. ͑1996͒. 65 48 X. Song and R. A. Marcus, J. Chem. Phys. 99, 7768 ͑1993͒. A. K. Soper and J. Turner, Int. J. Mod. Phys. B 7, 3049 ͑1993͒. 49 Y. Georgievskii, C.-P. Hsu, and R. A. Marcus, J. Chem. Phys. 108, 7356 66 A. K. Soper, J. Chem. Phys. 101, 6888 ͑1994͒. ͑1998͒. 67 Another question which arises is that, to our knowledge, apart from Ref. 50 Ovchinnikov and Ochinnilova ͑Ref. 16͒ considered the case that the di- 23, the theoretical results related to a spatially nonlocal dielectric re- electric response function ⑀␻ always enters in the form of 1/⑀␻ . They sponse, refer, thus far, to a homogeneous environment. The applicability used the electric field in vacuum D(0) instead of the dielectric displace- of these results to the inhomogeneous solvent in the vicinity of the solute

ment D␻ in Eq. ͑32͒, a usage which is only applicable for a homogeneous remains to be explored. A possible way to treat a spatially nonlocal di- environment and for some special cases. electric response in the presence of the solute cavity is described in Ref. 51 L. Onsager, J. Am. Chem. Soc. 58, 1486 ͑1936͒. 23. 52 P. F. Barbara and W. Jarzeba, Adv. Photochem. 15,1͑1990͒. 68 R. A. Marcus, J. Chem. Phys. 43, 1261 ͑1965͒. 53 H. Fro¨hlich, Theory of Dielectrics ͑Oxford University Press, London, 69 R. A. Marcus, J. Phys. Chem. 94, 1050 ͑1990͒. 1949͒. 70 L. D. Landau and E. M. Lifshitz, Statistical Physics ͑Pergamon, New 54 To calculate the solvent response function ␣␣ it is understood that R in York, 1980͒.

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