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Chemical Physics 324 (2006) 172–194

Activation entropy of electron transfer reactions

Anatoli A. Milischuk a, Dmitry V. Matyushov a,*, Marshall D. Newton b

a Department of Chemistry and Biochemistry, Center for the Early Events in Photosynthesis, Arizona State University, P.O. Box 871604,

Tempe, AZ 85287-1604, United Statesb Brookhaven National Laboratory, Chemistry Department, Box 5000, Upton, NY 11973-5000, United States

Received 12 June 2005; accepted 25 November 2005Available online 6 January 2006

Abstract

We report microscopic calculations of free energies and entropies for intramolecular electron transfer reactions. The calculation algo-rithm combines the atomistic geometry and charge distribution of a molecular solute obtained from quantum calculations with themicroscopic polarization response of a polar solvent expressed in terms of its polarization structure factors. The procedure is testedon a donor–acceptor complex in which ruthenium donor and cobalt acceptor sites are linked by a four-proline polypeptide. The reor-ganization energies and reaction energy gaps are calculated as a function of temperature by using structure factors obtained from ouranalytical procedure and from computer simulations. Good agreement between two procedures and with direct computer simulations ofthe reorganization energy is achieved. The microscopic algorithm is compared to the dielectric continuum calculations. We found thatthe strong dependence of the reorganization energy on the solvent refractive index predicted by continuum models is not supported bythe microscopic theory. Also, the reorganization and overall solvation entropies are substantially larger in the microscopic theory com-pared to continuum models.� 2005 Elsevier B.V. All rights reserved.

Keywords: Electron transfer; Theory of solvation; Entropy of activation; Structure factors; Polar liquids

1. Introduction

Beginning with work of Marcus on electron transfer(ET) between ions dissolved in polar solvents [1], theunderstanding of the dynamics and thermodynamics ofthe nuclear polarization coupled to the transferred electronhas been viewed as a key component of ET theories. Theconcept of polarization fluctuations as a major mechanismdriving ET [2,3] has been extended over the several decadesof research from simple molecular solvents to a diversity ofcondensed-phase media of varying complexity. A signifi-cant part of the present experimental and theoretical effortis directed toward the understanding of ET in biology,where this process is a key component of energy transportchains [4–7]. Biological systems pose a major challenge to

0301-0104/$ - see front matter � 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2005.11.037

* Corresponding author. Tel.: +1 480 965 0057; fax: +1 480 965 2747.E-mail addresses: dmitrym@asu.edu (D.V. Matyushov), newton@

bnl.gov (M.D. Newton).

theoretical and computational chemistry from at leasttwo viewpoints. First, the solvent, including bulk andbound water [8], membranes, and parts of the polar andpolarizable matrix of the biopolymer, is highly anisotropicand heterogeneous. Second, the geometry of what can beseparated as a solute is often very complex, including con-cave regions of molecular scale occupied by the solvent andregions of the biopolymer with a significant mobility ofpolar and ionizable residues.

Dielectric continuum models accommodate the complexsolute shape by numerical algorithms solving the Poissonequation with the boundary conditions defined by a dielec-tric cavity [9]. The heterogeneous nature of the solvent inthe vicinity of a redox site can in principle be included byassigning different dielectric constants to its heterogeneousparts [10]. Two fundamental problems inevitably arise inthis algorithm. The first has been well recognized overthe years of its application and is related to the ambiguityof defining the dielectric cavity for molecular solutes. This

Fig. 1. Diagram of the polypeptide donor–spacer–acceptor (DSA) com-plex referred to as complex 1 in the text.

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 173

problem is often resolved by proper parameterization ofthe radii of atomic and molecular groups of the solute.The second problem is much less studied. It is related tothe fact that collective polarization fluctuations of molecu-lar dielectrics possess a finite correlation length which maybe comparable to the length of concave regions of the sol-ute or to some other characteristic dimensions significantfor solvation thermodynamics. The definition of the dielec-tric constant for polar regions of molecular length isambiguous and, in addition, once the dielectric constantis defined, it is not clear if the dielectric response can fullydevelop on the molecular length scale.

In addition to the problems in implementing the contin-uum formalism for molecular solutes there are some funda-mental limitations of the continuum approximation itselfthat may limit its applicability to solvation and electrontransfer thermodynamics. On the basic level, the definitionof the molecular cavity should be re-done for each particu-lar thermodynamic state of the solvent [11,12] and/or elec-tronic state of the solute [13]. This precludes the use ofcontinuum theories with a given cavity parametrizationto describe derivatives of the solvation free energy, e.g.,entropy and volume of solvation [14]. In addition, the cal-culation of the free energy of ET activation requires aproper separation of nuclear solvation from the overall sol-vent response. This problem, actively studied by formaltheories in the past [15–18], has been recently addressedby computer simulations [19–21]. Computer simulationshave indicated that continuum recipes for the separationof nuclear and electronic polarization are unreliable, result-ing in too strong a dependence of the solvent reorganiza-tion (free) energy on solvent refractive index. All theselimitations call for an extension of traditional approachesto solvation and ET thermodynamics that would includemicroscopic length-scales of solvent polarization.

Microscopic theories of solvation are not yet sufficientlydeveloped to compete efficiently with continuum models inapplication to solvation of biopolymers. Computer simula-tions provide a very detailed picture of the local solvationstructure, but their application to solvation of large solutesrequires very lengthy computations and often includesapproximations that are hard to control. In particular,the dielectric response is very slowly converging in simula-tions and is potentially affected by approximations made todescribe the long-range electrostatic forces. Several simula-tion protocols in which polarization response is (partially)integrated out by analytical techniques have been proposed[22,23]. In addition, approaches combining electron struc-ture calculations with explicit account of the solvent in afew solvation shells around the solute have been proposedand used to calculate optical charge-transfer spectra [24].Integral equation theories have been successfully appliedto small solutes [25], but examples of their application tosolvation and reactivity of large solutes are just a few [26].

The formulation of the solvation problem in terms ofmolecular response functions holds significant promise, asit combines the molecular length scale of the polarization

response with a possibility to accommodate an arbitraryshape of the solute [27–37]. A recent re-formulation ofthe Gaussian model [31] for solvation in polar solvents[38,39] shows a good agreement with simulations of modelsystems and an ability to conform with experiment whenapplied to ET in biomolecules [40] and charge-transfercomplexes [41] and to solvation dynamics [42]. Testingthe algorithm, referred to as the non-local response func-tion theory (NRFT), on model systems for which bothcomputer simulations and experiment exist is critical forfuture applications to more complex systems. This is theaim of the present contribution.

Testing microscopic solvation theories require compari-son to computer simulations on model, yet realistic,systems. The current experimental database does not pro-vide sufficient accuracy to test various approximationsentering theoretical algorithms. On the other hand, com-puter experiment offers essentially exact (within the accu-racy of simulation protocols) integration of the sameHamiltonian as the one used in the analytical theory.Therefore, the present calculations of the ET thermody-namics are compared to recent very extensive moleculardynamics (MD) simulations [43] of a donor–spacer–accep-tor (DSA) complex consisting of transition-metal donor(D) and acceptor (A) sites linked by a polyproline peptidespacer (S) (Fig. 1)

ðbpyÞ2Ru2þðbpy0Þ–ðproÞ4–O�Co3þðNH3Þ5where in the donor bpy = 2,2 0-bipyridine and bpy 0 = 4 0-methyl-2,2 0-bipyridyl. The spacer is a polyproline chainwhose first member (the N-terminus) is connected to thebpy 0 carbonyl, and whose fourth member is terminatedby a carboxylate moiety bound to the –Co3+(NH3)5 accep-tor. This system, modeling ET in redox proteins, is a repre-sentative member of a homologous series of DSAcomplexes for which ET rates as a function of temperaturehave been reported [44]. This complex will be referred to ascomplex 1 in the text.

The analytical NRFT model is shown to agree excep-tionally well with MD simulations for complex 1 (Fig. 1)in TIP3P water. In order to provide a rigorous comparisonbetween simulations [43] and analytical theory, the set of

174 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

solute charges employed in the simulations was also used inthe analytical calculations. The main result of the analyti-cal formalism is the connection between the solvationresponse function and the polarization autocorrelationfunctions of the pure solvent. These autocorrelation func-tions are proportional to structure factors of dipolar orien-tations in the polar liquid. The polarization structurefactors of TIP3P water were obtained from separate MDsimulations. In this setup, both the solute and the solventare the same in simulations and analytical theory, allowingus a stringent test of the analytical formalism used to calcu-late the solvation response function.

Once the accuracy and robustness of the analytical pro-cedures have been tested on MD simulations, the next stepis to see if the model is capable of reflecting the behavior ofreal systems. The main challenge in this step is to extendthe model from non-polarizable force fields used in the sim-ulations to a realistic description of polarizable molecularsolvents. To this end, we have developed a parameteriza-tion scheme for polarization structure factors which allowsus to approach solvation in polarizable solvents. Once thisis done, the theory can be extended to calculations at vary-ing thermodynamic conditions of the solvent (e.g., temper-ature) and should generate a set of predictions which canbe tested experimentally.

We use the polypeptide DSA to focus on two problem-atic areas of dielectric continuum models: dependence ofthe reorganization energy on the solvent polarizability[19,21] and the entropy of nuclear solvation [14,32]. Forboth areas there is a fundamental, both quantitative andqualitative, disagreement between microscopic modelsand continuum calculations. Unfortunately, no experimen-tal evidence on the dependence of the solvent reorganiza-tion energy on solvent refractive index is available in theliterature. There is, on the other hand, a limited numberof experimental [14,45–55] and simulation [56–58] studieson the entropy of reorganization. Most of the availableexperimental (laboratory and simulation) evidence pointsto a positive reorganization entropy (i.e., a negative slopeof the reorganization energy vs. temperature) in polar sol-vents, in agreement with the prediction of microscopic the-ory [32] and in disagreement with negative entropies fromcontinuum calculations [14,59]. We are aware, however,of a few measurements performed on charged donor–acceptor complexes indicating either zero or negative reor-ganization entropies [45,51,52]. Our current calculations oncomplex 1 (Fig. 1) give absolute values of the reorganiza-tion entropy much higher than continuum calculations.This great discrepancy calls for additional tests of the the-ory against experimental data.

A reliable formalism for calculating solvation entropiesis necessary for obtaining reaction rate preexponents fromkinetic temperature-dependence measurements. In particu-lar, the analysis of experimental data [46,59–61] shows agreat sensitivity of matrix elements of ET to the model usedto describe the activation entropy. Even though this is anold problem of chemical kinetics, entropic effects on chem-

ical reactivity in condensed phases are still often mysteriousand poorly understood, requiring further experimental andtheoretical effort.

2. Golden Rule rate constant

The Golden Rule rate constant of ET is [62]

kET ¼2pV 2

12

�h2FCWDð0Þ; ð1Þ

where FCWD stands for the density-of-states weightedFranck–Condon (FC) factor

FCWDðxÞ ¼Z

dt2pheiH2t=�he�iH1t=�hine�ixt. ð2Þ

Here, h. . .in is an ensemble average over the nuclear degreesof freedom of the system (denoted by subscript ‘‘n’’) inequilibrium with the initial electronic state of the donor–acceptor complex. The manifold of nuclear degrees of free-dom includes N normal vibrational modes of the donor–acceptor complex Q = {q1, . . . ,qN} and the nuclear compo-nent of the dipolar polarization of the solvent Pn. Theensemble average is carried out over the configurations inequilibrium with the initial state. Further, Hi (i = 1, 2)are the diagonal matrix elements of the unperturbed systemHamiltonian H taken on the two-state electronic basis{W1,W2}: Hi = hWi|H|Wii (i = 1 and i = 2 stand for the ini-tial and final electronic states, respectively). The sum of Hand the perturbation V makes the whole system Hamilto-nian, H 0 = H + V, and V12 = hW1|V|W2i is the off-diagonalmatrix element in the Golden Rule expression.

The system Hamiltonian of a donor–acceptor complexin a condensed-phase solvent can be separated into thegas-phase component, Hg, the solute–solvent interaction,H0s (‘‘0’’ stands for the solute, ‘‘s’’ stands for the solvent),and the bath Hamiltonian, HB, describing thermal fluctua-tions of the solvent

H ¼ H g þ H 0s þ H B. ð3ÞThe gas-phase Hamiltonian is the sum of the kinetic energyof the electrons, kinetic energy of the nuclei, and the fullelectron-nuclear Coulomb energy. The solute–solventHamiltonian for ET in dipolar solvents is commonly givenby the coupling of the operator of the solute electric fieldbE0 to the dipolar polarization of the solvent P

H 0s ¼ �bE0 � P. ð4ÞThe bath Hamiltonian represents Gaussian statistics of thecollective mode P with the linear response function v(r,r 0)

HB ¼1

2P � v�1 � P. ð5Þ

The asterisk between the bold capital letters denotes tensorcontraction (scalar product for vectors) and space integra-tion over the volume X occupied by the solvent

E � P ¼Z

XE � Pdr. ð6Þ

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 175

Assuming that the intramolecular vibrations are decoupledfrom solvent nuclear modes allows one to cast the FCWDas a convolution of the vibrational, Gv(x), and solvent,Gs(x), FC densities [63]

FCWDðxÞ ¼Z 1

�1dx0Gvðx0ÞGsðx� x0 � DG=�hÞ; ð7Þ

where the diabatic equilibrium free energy gap is the sum ofthe gas-phase component DGg and difference in solvationenergies DGs

DG ¼ DGg þ DGs. ð8ÞIn the absence of vibrational frequency change, the formercomponent is equal to the 0–0 transition energy in the gasphase.

The FC density for each nuclear mode is given in termsof a broadening function gn(t) [64,65]

Gnðx0Þ ¼Z 1

�1

dt2p

exp½iðkn=�h� x0Þt � gnðtÞ�; ð9Þ

where

gnðtÞ ¼1

p

Z 1

0

dzz2ð1� cos ztÞv00nðzÞ coth

�hz2kBT

þ i

p

Z 1

0

dzz2ðzt � sin ztÞv00nðzÞ. ð10Þ

In Eq. (9), kn is the nuclear reorganization energy

kn ¼�hp

Z 1

0

dzz

v00nðzÞ ð11Þ

and v00nðzÞ is the imaginary part of the frequency-dependentlinear response function (spectral density) corresponding tothe nuclear mode n (in general, many such modes contrib-ute to the solvent (s) and vibrational (v) FC densities).

For a set of vibrational normal modes with frequenciesxq and reorganization energies kq, the spectral density is[65]

v00vðzÞ ¼ pX

q

Sqx2q½dðz� xqÞ � dðzþ xqÞ�; ð12Þ

where Sq = kq/�hxq is the Huang–Rhys factor. When all nu-clear modes are classical, gn(t) = kBTknt2/�h2 and onereaches the classical, high temperature limit of the Marcustheory

FCWDðxÞ ¼ ½4pðks þ kvÞkBT ��1=2

� exp �ðDGþ ks þ kv � �hxÞ2

4kBT ðks þ kvÞ

" #; ð13Þ

where kv ¼P

qkq is the total vibrational reorganization en-ergy and the solvent reorganization energy ks is discussed inSection 3.4 below.

When the solvent mode is classical and the vibrationsare quantized, one can use the small t expansion in Eq.(9), valid in the limit when xq is much smaller than the ver-tical energy gap [|kv � x 0| in Eq. (9)]. With �hxq/kBT� 1,one gets

gvðtÞ ’ ðt2=2�hÞxvkv; ð14Þ

where

xv ¼ ðkvÞ�1X

q

xqkq ð15Þ

is the effective vibrational frequency. With the vibrationalbroadening function in the form of Eq. (14) the FCWD be-comes [66–69]

FCWDðxÞ ¼ ½pð4kskBT þ 2�hxvkvÞ��1=2

� exp �ðDGþ ks þ kv � �hxÞ2

4kBTks þ 2�hxvkv

" #. ð16Þ

The above equation, present in some early papers on ET[67,68], is not very accurate as was pointed out by Jortner[70]. The set of equations given below, which can be foundin work by Lax [71], Davydov [72], and Kubo and Toyoz-awa [62], provides a better description of the vibronicenvelope.

When the normal mode vibrations are represented by asingle effective vibration with frequency defined by Eq.(15), the vibrational FCWD is a weighted sum of resonantvibrational transitions

GvðxÞ ¼X1

m¼�1Amdðx� mxvÞ; ð17Þ

where

Am ¼ e�S coth vvþmvv ImS

sinh vv

� �; ð18Þ

S = kv/�hxv, vv = �hxv/2kBT, and Im(x) is the modified Bes-sel function of order m.

The FCWD for the classical nuclear modes of the sol-vent is given by the expression

Gsðx� DG=�hÞ ¼ �hhdðDEðPnÞ � �hxÞi; ð19Þwhere

DEðPnÞ ¼ DGþ ks � DE0 � dPn ð20Þand h. . .i denotes an ensemble average over the fluctuationsdPn of the nuclear solvent polarization Pn. The ensembleaverage is taken over the states in equilibrium with the ini-tial electronic state of the donor–acceptor complex andDE0 = E02 � E01 is difference in the electric field of the do-nor–acceptor complex in the final and initial electronicstates.

With the Gaussian Hamiltonian for the polarizationfluctuations (Eq. (5)), Gs(x � DG/�h) is a Gaussian functionleading to a total FCWD in the form of a weighted sum ofGaussians

FCWDðxÞ ¼ ½4pkskBT ��1=2X1

m¼�1Am

� exp � DGþ ks þ m�hxv � �hxð Þ2

4kskBT

!. ð21Þ

176 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

When the energy of vibrational excitations is much greaterthan kBT (vv� 1 in Eq. (18)) the FC envelope turns into asum of Gaussians with weights given by the Poisson distri-bution [63]

Am ¼ e�S Sm

m!; m > 0. ð22Þ

a b

Fig. 2. Two approaches to the calculation of the response function: aspolarization response to an external electric field perturbation (a) and ascorrelation of polarization fluctuations near the solute hard core fromwhich the polarization field is excluded (b).

3. Solvation thermodynamics

Inserting a solute into a molecular solvent results in per-turbation of the solvent that can roughly be split into twocomponents with drastically different length scales. The firstcomponent is due to the repulsion of the solvent from thesolute core caused by short-range, but strong repulsiveforces. This perturbation creates a local density profile inthe solvent around the solute which may or may not inducea polarization field acting on the solute charges. The electricfield of the solute charges creates yet another perturbation.The solute electric field is sufficiently long-ranged to inducethe dipolar polarization P(r) in a quasi-macroscopic regionof the solvent around the solute. Gradients of the solutefield couple to the higher-order (quadrupolar, etc.) polariza-tion, but this interaction is more short-ranged [41,73–75].

The dipolar polarization is caused by alignment of thepermanent and induced solvent dipoles along the solutefield. This alignment occurs on two quite different timescales: ’10�15 s for induced dipoles and ’10�11 � 10�12 sfor permanent dipoles. Accordingly, the polarization fieldsplits into a fast relaxing electronic polarization (induceddipoles, Pe) and a much slower nuclear polarization(permanent dipoles, Pn) [76]. The electronic solvent polar-ization is always in equilibrium with the changing distribu-tion of the electronic density in the donor–acceptorcomplex. The energy conservation condition of the GoldenRule formula is thus imposed on the energies with equili-brated electronic polarization. Therefore, before being usedin the Golden Rule expression, the density matrix exp[�Hi/kBT] should be averaged over the fast electronic compo-nent of the dipolar polarization [17,77]. For the energyEi[Pn] depending on the instantaneous configuration ofthe nuclear subsystem one gets

e�Ei½Pn�=kBT ¼ Trel½e�Hi=kBT �; ð23Þ

where Trel denotes the statistical average over the electronicdegrees of freedom of the solvent.

In reality, the trace over the electronic degrees of free-dom cannot be taken exactly and some approximationshave to be used. The electronic polarization Pe is normallyassumed to be a Gaussian quantum field [17,77] over whichthe integration has to be done in order to determine Ei[Pn]

e�Ei½Pn�=kBT ¼Z

e�Hi ½Pe;Pn�=kBTDPe; ð24Þ

where DPe denotes the functional integral over the fieldPe(r). The dependence on the electronic polarization inHi[Pe,Pn] is generally a bilinear functional of Pe(r) [17,19]

H e½Pe� ¼1

2Pe � v�1

e � Pe þ Pe � v�1en � Pn. ð25Þ

with corresponding response functions ve and ven. Beforegoing into the details of separate calculations for electronicand nuclear components of the polarization, we outline thegeneral formalism of polarization response to an externalelectric field.

3.1. Linear response approximation

In the linear response approximation, the solvent polar-ization P(r) is a linear functional of the perturbing electricfield E0 (vacuum electric field of the solute for solvation)

PðrÞ ¼ v � E0 ¼Z

vðr; r0Þ � E0ðr0Þdr0. ð26Þ

Here, v(r,r 0) is a two-rank tensor describing the inhomoge-neous, non-local linear response of the solvent to the soluteelectric field, dot denotes tensor contraction over the com-mon Cartesian projections. This function is different fromdielectric susceptibility appearing in Maxwell’s equationsin two respects. First, v(r,r 0) describes the polarization re-sponse to the field of external charges and not to the totalelectric field E = E0 + EP combining the external field withthe electric field EP created by the solvent polarization (vcorresponds to v0 of Madden and Kivelson [78]). Second,v(r,r 0) is affected by the presence of the solute and thusv(r1,r 0) is generally not equal to v(r2,r00) even if r1 � r 0 =r2 � r00.

Eq. (26) determines the response function in terms of anexternal electrostatic perturbation and polarizationinduced by it (Fig. 2(a)). An alternative view of theresponse function is through the fluctuation–dissipationtheorem which relates the response to the correlation func-tion of polarization fluctuations in the solute vicinity

vðr; r0Þ ¼ ðkBT Þ�1hdPðrÞdPðr0Þi. ð27ÞAn important result of the linear response approximationis that this correlation function does not depend on thelong-range electrostatic field of the solute. The ensembleaverage h. . .i in the presence of the real solute with itscharge distribution is equivalent (within the linear

200 300 400 500T/K

0.12

0.14

0.16

0.18

λ i/eV

0 4y

0.2

0.3

0.4

0.5

λ i/(e2 /σ

)

::

+ - SPC/Ewater

dipolar hard spheres

SPC/E water

2 6

Fig. 3. Upper panel: k1 for a neutral diatomic D–A (circles) and k2 for apolar diatomic D+–A� (diamonds) for a donor–acceptor complexrepresented by two contact spheres with radii R0/r = 0.9. Solvent is afluid of dipolar hard spheres of diameter r and dipole moment m. Thesimulations, reported in Ref. [39], were carried out at different m and atconstant density qr3 = 0.8. The change in solvent polarity is reflected bythe dipolar density y = (4p/9)m2q/kBT. The vertical arrow indicates thevalue of the polarity parameter for SPC/E water. Lower panel: solventreorganization energy of p-nitroaniline in the ground (k1, circles) andcharge-transfer (k2, diamonds) states from Ref. [57]. Data points areobtained from NVE MD simulations at varying temperature of SPC/Ewater. The dotted line is the linear regression through the circles.

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 177

response) to the ensemble average h. . .i0 in the presence ofa fictitious solute which has the geometry of the real solute(and therefore the complete repulsion potential) but nopartial charges. This notion provides a convenient routeto the calculations of the response function for complexsolutes. Instead of calculating the polarization in responseto a non-trivial field E0(r), one can calculate the correlationof polarization fluctuations in the presence of a fictitioussolute with only the hard repulsive core of the real soluteretained. The correlation function is then calculated withthe requirement of zero polarization within the solute(Fig. 2(b))

vðr; r0Þ ¼ ðkBT Þ�1hdPðrÞdPðr0Þi0; ð28Þwhere subscript ‘‘0’’ is distinct from the notation of vari-ables attributed to the solute. This is the essence of the ap-proach adopted in the present formalism, making theresponse function solely determined by the molecular struc-ture inherent to the pure solvent and the short-range per-turbation produced by the repulsive core of the solute.The mean-field solution for v(r,r 0) introduced below in Sec-tion 3.2 incorporates the dependence of the response func-tions on the direction of the electric field of the solute in thequasi-homogeneous response functions v(k) depending ona single wavevector k.

The applicability of the linear response approximation tosolvation of large donor–acceptor complexes common forET research in molecular solvents is well supported by exist-ing evidence from computer simulations [39,57,79–83]. Thedirect consequence of the linear response are the followingrelations for the moments of the solute–solvent interactionpotential v0s:

�kBT hv0si ¼ hðdv0sÞ2i ¼ hðdv0sÞ2i0. ð29ÞWhen the solute–solvent interaction is limited to the cou-pling of the solute charges to the solvent dipolar polariza-tion, v0s = hW|H0s|Wi in Eq. (4).

The independence of the response function with respectto the solute charge is propagated into equality of the vari-ance of v0s in equilibrium with fully charged solute, h. . .i,and in equilibrium with uncharged solute, h. . .i0. The upperpanel in Fig. 3 shows the results of simulations from Ref.[39] for a model diatomic donor–acceptor complex D–Ain a dense solvent of hard sphere point dipoles. The systemis designed to mimic the charge separation, D–A! D+–A�, and charge recombination, D+–A� ! D–A, reactions.The reorganization energies of the complex D–A, k1 =h(dv0s)

2i0/2kBT, and the complex D+–A�, k2 = h(dv0s)2i/

2kBT, turn out to be very similar over a broad range of sol-vent polarities monitored by the dipolar density parameter

y ¼ ð4p=9Þm2q=kBT ; ð30Þwhere q is the solvent number density and m is the perma-nent dipole moment of the solvent molecules.

The lower panel in Fig. 3 shows the results on NVE MD[84] simulations from Ref. [57]. These simulations studysolvation of p-nitroaniline in the ground and charge-trans-

fer states in SPC/E water at different temperatures. In thiscase too, k1 ’ k2 even though the hydrogen bond couplingbetween the oxygens of p-nitroaniline and water changessignificantly with charge transfer [57]. Note that the tworeorganization energies in hard-sphere dipolar solventsstart to deviate from each other in the polarity range char-acteristic of water (vertical arrow in the upper panel ofFig. 3) in contrast to the results in SPC/E water. Thisbehavior is expected since solvation in hard-sphere dipolarsolvents is subject to stronger non-linear solvation effectsdue to the depletion of water density (dewetting) in the firstsolvation sphere of the solute [58]. This effect, resulting in adownward turn of k1, is partially compensated by Lennard-Jones (LJ) interactions in force-field models.

The thermodynamic state of the fluid of dipolar hardspheres is fully determined by two parameters: q* = qr3

(r is the hard-sphere diameter) and y. At constant density,the increase of y in the upper panel of Fig. 3 is equivalent tothe decrease of temperature. These results thus suggest anegative slope of the reorganization energy with increasingtemperature at constant volume. The results of computerexperiment shown in the lower panel of Fig. 3 (NVE MD[84]) are of course consistent with this conclusion. Labora-tory experiments are normally done at constant pressure

178 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

when alteration of density with changing temperatureneeds be taken into account. These constant-pressure cal-culations, presented in Section 5.1 below, also result in anegative slope of the reorganization energy.

Even though the linear response approximation for sol-vation in dense polar solvents is very robust, deviationsfrom it have been observed in computer experiments of sol-vation in hydrogen-bonding solvents [85]. In addition, con-ventional solvatochromism sometimes demonstratessignificant deviations from the expected behavior in sol-vents forming hydrogen bonds with the solute [86–88],although direct testing of Eq. (29) is often prohibited bythe lack of Stokes shift and spectral width data. One shouldkeep in mind that the Marcus two-parabolas constructionrelies on the linear response approximation and, once itfails, one has to resort to more general schemes of con-structing the free energy surfaces. In particular the Q-model of ET [89] provides a general framework of treatingcases with k1 5 k2 and was shown to handle well non-lin-ear solvation effects [58]. The results of both simulationsand solvatochromic experiments suggest that non-linearsolvation due to hydrogen bonds would be expected in sit-uations when charge population of solute atoms, partici-pating in strong coupling to the solvent, changes in thecourse of electronic transition. Therefore, such atoms areexposed to the solvent. The DSA complex considered herehas the advantage of having the atomic charges changing inthe course of ET buried well inside the solute. We thereforedo not expect complications from specific interactions,although this assumption has not been directly tested insimulations and remains an approximation of our analysis.

3.2. Response functions

The inhomogeneous character of the response functions(Eqs. (27) and (28)) is retained after transformation to k-space. The function v(k,k 0) then depends on two wavevec-tors in contrast to the dependence on a single wave-vectorfor the homogeneous dielectric response. The calculation ofv(k,k 0) is still a major challenge for microscopic theories ofpolar solvation. Despite some very active research in thisarea for the last 80 years since the formulation of the Bornmodel for solvation of spherical ions [90], no microscopicsolution applicable to solutes of arbitrary shape has beenpresented so far. A promising strategy, adopted alreadyin the Born [90] and Onsager [91] models, is to calculatethe response functions in terms of properties of the puresolvent. This connection can be achieved by consideringthe polarization correlation function in the presence ofthe repulsive core of the solute (Eq. (28)).

The exclusion of the polarization field from the solutevolume (Fig. 2(b)) is provided by the Li–Kardar–Chandlerapproach [31,92], in which the trajectories defining theresponse function in its path integral representation arerestricted from entering the solute. The result of this proce-dure is an integral equation relating v(k,k 0) to the non-localsusceptibility of the pure solvent vs(k) (with a single k-vec-

tor for the homogeneous response) and the shape of thesolute. The equation for the response function is thenequivalent to the Ornstein–Zernike equation for the sol-ute–solvent correlation function with the Percus–Yevickclosure for the solute–solvent direct correlation function[31].

No general solution for v(k,k 0) in the Li–Kardar–Chan-dler integral equation has been obtained so far. One can,however, employ analytical properties of the responsefunctions to obtain the solvation chemical potential [38].The solvation chemical potential l0s is given in terms ofthe polarization field induced by the solute by the followingrelation:

�l0s ¼1

2P � E0; ð31Þ

which, after substitution of Eq. (26), results in

�l0s ¼1

2E0 � v � E0. ð32Þ

The Fourier transform of the vacuum field E0 then leads tothe relation which is the starting point of our formalism

�l0s ¼1

2

Zdk dk0

ð2pÞ6eE0ðkÞ � vðk; k0Þ � eE0ð�k0Þ. ð33Þ

Here, eE0ðkÞ denotes the Fourier transform of the electricfield of the solute calculated on the volume X obtainedby excluding the solute from the solvent

eE0ðkÞ ¼Z

XE0ðrÞeik�r dr. ð34Þ

We will discuss the definition of the solute volume furtherin Section 4.2 (also see Fig. 10).

Closed-form results for l0s can be obtained only whenthe Fourier transform of the electric field eE0ðkÞ is knownin analytical functional form [38,39,93]. This is not the casefor many real problems, when the distribution of molecularcharge is given by force fields or quantum calculations andthe Fourier transform of the field is calculated numerically.Unfortunately, the analytical solution is given by the differ-ence of two large numbers almost canceling each other. Ittherefore becomes not very practical in strongly polar sol-vents because of accumulation of numerical errors. Tofacilitate numerical applications, a mean-field solution forv(k,k 0) was offered in Ref. [39]. This solution results inthe response function depending on a single wave-vector

vðk; k0Þ ¼ ð2pÞ3dðk� k0Þ½vLðkÞJL þ vTðkÞJT�. ð35ÞIn Eq. (35), JL ¼ kk and JT ¼ 1� kk are, respectively, thelongitudinal and transverse projections of a 2-rank tensorwith the axial symmetry established by the direction ofthe wavevector, k ¼ k=k. These two projections always ap-pear in the k-space representation of polarization correla-tion functions of pure polar solvents [78] since thewavevector introduces axial rotational symmetry in theotherwise isotropic solvent. Therefore, the representationof Eq. (35) makes the solvent response quasi-isotropic with

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 179

the response functions constructed to account for the pres-ence of the solute. The 6D integral of Eq. (33) is then re-duced to the computationally tractable 3D integral.

The transverse, vT(k), and longitudinal, vL(k), projec-tions in Eq. (35) are related to the homogeneous solventsusceptibility as follows:

vTðkÞ ¼ vTs ðkÞ

vLs ð0Þvtr

� fsvLs ðkÞ

f0 � JL � eE0ðkÞf0 � JT � eE0ðkÞ

ð36Þ

and

vLðkÞ ¼ vLs ðkÞ. ð37Þ

In Eq. (36), vtr = (1/3)Tr[vs(0)] and

fs ¼2½vT

s ð0Þ � vLs ð0Þ�

3vtr

. ð38Þ

The mean-field elimination of the dependence on two sepa-rate wave vectors in Eq. (35) can be achieved by introducingthe dependence of the response function on the direction ofthe solute electric field, which plays the role of boundaryconditions at dielectric cavities in continuum electrostatics.The quasi-homogeneous (depending on one wave-vector)linear response function of the solvent to the electric fieldof the solute does not depend on the magnitude of the elec-tric field but does depend on its symmetry broadly under-stood here in terms of expansion of the field in rotationalinvariants [94]. The best example of this is the transforma-tion of the response function of a spherical solute fromthe Born / (1 � 1/�s) form to the Onsager / (�s � 1)/(2�s + 1) form when the symmetry of the field is changedfrom that of a point charge to the symmetry of a point di-pole. The symmetry of the solute electric field is reflectedin Eq. (36) in terms of the Fourier transform eE0ðkÞ andthe field F0 given by the following relation:

F0 ¼vTð0Þ � vLð0Þ

4pvtr

ZX

E0ðrÞ �Dr

dr

r3; ð39Þ

where

Dr ¼ 3rr� 1 ð40Þis the dipolar tensor. Further, in Eq. (36),

f0 ¼ F0=F 0 ð41Þis the unit vector in the direction of F0. Each field eE0ðkÞ en-ters both the nominator and denominator in the secondsummand in Eq. (36). The resulting response function doesnot, therefore, depend on the field strength, but the infor-mation on the field symmetry is retained in the contractionof the tensors.

The electric field F0 is a generalization of the Onsagerreaction field for the case of non-spherical solutes withnon-dipolar charge distribution. When the solute is a spher-ical ion with charge q0, one gets F0 = 0 corresponding to thecontinuum result, implying that the reaction potential

u0 ¼ �q0

R1� 1

�s

� �ð42Þ

is constant within the cavity of radius R, and the reactionfield is zero. When the solute is a point dipole m0 at the cen-ter of a spherical cavity, F0 reduces to the Onsager reactionfield

F0 ¼2m0

R3

�s � 1

2�s þ 1. ð43Þ

Eq. (39) introduces the solvent reaction field inside a sol-ute of arbitrary shape and charge distribution. The approx-imation adopted in deriving Eq. (36) is that of neglect ofthe gradients of the reaction field which amounts to takingthe dipolar projection (2-rank tensor Dr) of the solute fieldin Eq. (39). Finally, the molecular shape of the solutecomes into the solution through the Fourier transform ofthe electric field which is taken over the volume outsidethe solute (Eq. (34)) and through the reaction field F0

which is also defined by integration over the solvent vol-ume outside the solute (Eq. (39)). The solute molecularshape thus defines the integration volume.

The mean-field renormalization of the transverse com-ponent of the response function in Eq. (36) resolves thefundamental difficulty of microscopic solvation theoriesarising from the fact that the short-range repulsive pertur-bation caused by the solute produces a major change inthe polarization response functions compared to thoseof the pure solvent. For instance, a direct replacementof v(k) with vs(k) in the homogeneous approximation(see Ref. [95] for discussion) results in divergent behaviorof ks with increasing solvent dipole moment [96]. Thedivergence arises from the transverse component of theresponse (‘‘transverse catastrophe’’) which has to beincluded once the solute surface does not coincide withan equipotential surface of the solute charge distribution[97]. In continuum calculations, the divergent behavioris eliminated by imposing boundary conditions at thedielectric cavity and infinity on the solution of the Poissonequation.

Although the problem with the transverse response haslong been recognized in the literature [96–98], many micro-scopic formulations of solvation thermodynamics anddynamics have avoided the problem by neglecting thetransverse response [29,99,100]. This component is alsoneglected in some continuum calculations, e.g., the Gener-alized Born approximation [101]. Eqs. (33)–(39) provide ageneral solution of the problem which agrees well withavailable simulations of polar solvation [38,39] and exper-iment on solvation dynamics [42]. The formalism is basedon the homogeneous solvent susceptibility as input and,once the susceptibility is defined from computer experimentor liquid-state theories, can be applied to solvation in anarbitrary isotropic dielectric.

3.3. Polarization structure factors

The dipole moment at a given molecule j in a polar-polarizable solvent is a sum of the permanent dipole mj

and the induced dipole pj

4

180 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

lj ¼ mj þ pj. ð44Þ

The total induced dipole then splits into p0j created by the

external electric field E0(rj) and pRj induced by the reaction

field (superscript ‘‘R’’) caused by the dipole mj itself (Fig. 4)

pj ¼ p0j þ pR

j . ð45Þ

The reaction field caused by the dipole mj relaxes on thetime-scale of translational–rotational motion of moleculej. Therefore, the induced dipole pR

j , which follows adiabat-ically the reaction field, should be attributed [102] to theslow nuclear polarization of the solvent Pn. In contrast,the component p0

j , following adiabatically the externalfield, is attributed to the fast solvent polarization Pe. Thesum of the permanent dipole mj and the induced dipolepR

j makes the effective condensed-phase dipole [103]

m0j ¼ mj þ pRj ¼ m0ej; ð46Þ

where ej is the unit vector along the direction of mj. The di-pole moment m 0 in principle depends on the instantaneousconfiguration of the liquid [104]. However, we will not con-sider fluctuations of m 0 here and, following self-consistentmodels of polarizable liquids [103], will replace m 0 withits statistical average value.

The combination of the permanent dipole mj and theinduced dipole pR

j defines the slow nuclear polarization ofthe solvent

PnðrÞ ¼X

j

m0jdðr� rjÞ. ð47Þ

This definition of the nuclear polarization, which includesthe component pR of the induced dipole, is an essential partof the Pekar partitioning of the solvent polarization intofast and slow components [105,106]. Other partitioningschemes have been proposed [107], but they all lead tothe same value of the solvation energy when correctlyimplemented [108]. Computer simulation protocols inwhich the induced polarization is self-consistently adjustedto the instantaneous nuclear configuration [109–111] pro-vide direct access to the slow polarization in Pekar’s defini-tion. Self-consistent simulations of polarizable solvents are

pj

R

pj

0p

j

mj

E0(r

j)

+

rj

Fig. 4. Components of the induced dipole moment at solvent molecule j.p0

j is produced by the external electric field E0(rj); pRj is produced by the

reaction field induced in the solvent by the permanent dipole mj.

used here to test the analytical procedure employed for theresponse functions of the nuclear polarization (Section 4.1).

The total dipolar response function of the homogeneoussolvent is a 2-rank tensor describing correlations of dipolemoments lj (Eq. (44))

vsðkÞ ¼ ðb=XÞX

j;k

ljlkeik�rjk

* +; ð48Þ

where rjk = rj � rk and brackets refer to an ensemble aver-age. Because of the isotropic symmetry of the solvent, vs(k)splits into longitudinal and transverse components [78]

vsðkÞ ¼ vLs ðkÞJ

L þ vTs ðkÞJ

T. ð49ÞIt is convenient to factor the response function into theeffective density of dipoles yeff, which is mostly affectedby the magnitude of the solvent dipole, and the structurefactor, which reflects dipolar correlations and can be ex-pressed through angular projections of the pair distributionfunction [39]

vsðkÞ ¼3yeff

4p½SLðkÞJL þ STðkÞJT�. ð50Þ

The structure factors SL,T(k) (Fig. 5) are defined based onthe unit vectors uj ¼ lj=lj in the direction of the respectivetotal dipole moments:

SLðkÞ ¼ 3

N

Xi;j

ðui � kÞðk � ujÞeik�rij

* +;

STðkÞ ¼ 3

2N

Xi;j

ðui � ujÞ � ðui � kÞðk � ujÞh i

eik�rij

* +.

ð51Þ

The effective dipole density in Eq. (50) is

yeff ¼ yp þ ð4p=3Þqa; yp ¼ ð4p=9Þqðm0Þ2=kBT . ð52Þ

In the above equation, m 0 is the condensed-phase solventdipole moment (Eq. (46)) and a is the gas-phase dipolarpolarizability. This renormalization scheme, in which only

0 2 4 5

k,Ao -1

0

1

2

3

S(k)

,Sn(k

)

T

L

31

Fig. 5. Longitudinal (L) and transverse (T) polarization structure factorscalculated by using the PPSF with the parameters of ambient water. Thesolid lines refer to the total structure factors SL,T(k), the dashed lines referto the nuclear structure factors SL;T

n ðkÞ.

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 181

the dipole moment is renormalized by the mean field of thesolvent, corresponds to Wertheim’s 1-RPT theory [112]. 2-RPT theory renormalizes the polarizability a to a 0, but the1-RPT version of the theory is in better agreement withcomputer simulations [21].

The nuclear response function reflects correlated orien-tations and positions of dipoles m0j

vnðkÞ ¼ ðb=XÞX

j;k

m0jm0keik�rjk

* +. ð53Þ

Similarly to Eq. (50), vn(k) can be separated into the longi-tudinal and transverse components

vnðkÞ ¼3yp

4p½SL

n ðkÞJL þ ST

n ðkÞJT�. ð54Þ

The nuclear structure factors are defined by Eq. (51), inwhich the unit vectors uj are replaced by the unit vectorsej (Eq. (46)).

The k = 0 values of the structure factors are related tothe macroscopic dielectric properties of the solvent. Thetotal polarization response is defined through the staticdielectric constant �s:

SLð0Þ ¼ �s � 1

3�syeff

;

STð0Þ ¼ �s � 1

3yeff

.

ð55Þ

The nuclear structure factors depend, in addition, on thehigh-frequency dielectric constant �1:

SLn ð0Þ ¼

c0

3yp

;

STn ð0Þ ¼

�s � �13yp

;ð56Þ

where

c0 ¼ 1=�1 � 1=�s ð57Þis the Pekar factor.

Both SL;Tn ðkÞ and SL,T(k) tend to unity at k!1. This

limit is the result of the point multipole approximationfor the intramolecular charge distribution within the sol-vent molecules. In contrast, charge–charge structure fac-tors, defined for interaction-site models of liquids, decayto zero at k!1 (Refs. [25,73,113]). The region of k-val-ues where this distinction becomes important is, however,insignificant for the calculation of solvation thermody-namics (see below). The nuclear and the total structurefactors differ in the range of small k-values and aroundthe longitudinal peak as a result of the influence ofthe high-frequency dielectric constant of the solvent(Fig. 5). The effect of �1 on the longitudinal peak isinsignificant for the calculation of the reorganizationenergy. Therefore, it is the range of small k-values and,in addition, the dependence of the liquid-state dipolemoment m 0 on the solvent polarizability, that ultimatelydetermine the variation of the solvent reorganization

energy with the solvent high-frequency dielectric constant�1 (see below).

3.4. ET thermodynamics

Electronic transitions result in a change in the electricfield of the solute. Therefore, in contrast to the precedingdiscussion, where we have used electric field E0, fromnow on we will be using subscript i to distinguish betweenthe initial (i = 1) and final (i = 2) electronic states in theelectric field E0i. In order to approach the problem of thesolvent effect on ET activation barrier we need to considerthe solvent reorganization energy and the solvent compo-nent of the free energy gap (introduced in Section 2). Theyare defined in terms of the nuclear and total response func-tions by the following relations:

ks ¼1

2

Zdk dk0

ð2pÞ6DeE0ðkÞ � vnðk; k0Þ � DeE0ð�k0Þ ð58Þ

and

DGs ¼ �Z

dkdk0

ð2pÞ6DeE0ðkÞ � vðk; k0Þ � �E0ð�k0Þ. ð59Þ

In Eqs. (58) and (59), D~E0ðkÞ ¼ eE02ðkÞ � eE01ðkÞ andE0ðkÞ ¼ ð~E02ðkÞ þ eE01ðkÞÞ=2; eE0iðkÞ are the Fourier trans-forms of the solute electric field in the initial and final ETstates taken over the volume X occupied by the solvent(Eq. (34)). The mean-field solution for the response func-tions (Eq. (35)) splits both the solvent reorganization en-ergy and the free energy gap into their correspondinglongitudinal and transverse components:

ks ¼ kLs þ kT

s ð60Þ

and

DGs ¼ DGLs þ DGT

s . ð61Þ

Each projection is obtained as a k-integral with the corre-sponding polarization structure factor. For the ‘‘T’’ projec-tions one gets

kTs ¼

3yp

8pSL

n ð0ÞgKn

Zdk

ð2pÞ3jDeET

0 ðkÞj2ST

n ðkÞ ð62Þ

and

DGTs ¼ �

3yeff

8pSLð0Þ

gK

�Z

dk

ð2pÞ3jeET

02ðkÞj2 � jeET

01ðkÞj2

h iSTðkÞ. ð63Þ

In Eqs. (62) and (63),

gKn ¼ ð1=3Þ½SLn ð0Þ þ 2ST

n ð0Þ� ð64Þand

gK ¼ ð1=3Þ½SLð0Þ þ 2STð0Þ� ð65Þare the nuclear and total Kirkwood factors, respectively.

Solvent parameters:m, α, ρ, ε

s, ε∞, σ

Solute parameters:x

0k, y

0k, z

0k, q

0k, a

0k

SL(k), S

T(k) E(k)

~

k-integration

λs, ΔG

s

Fig. 6. Diagram of the calculation algorithm. Solvent parameters include:m (gas-phase dipole moment), a (gas-phase dipolar polarizability), �1(high-frequency dielectric constant), �s (static dielectric constant), and r(effective hard sphere diameter of the solvent molecules). Parameters x0k,y0k, z0k stand for Cartesian coordinated of the solute atoms, q0k are partialcharges, and a0k are atomic vdW radii.

182 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

The longitudinal components of free energies, kLs and

DGLs , include both the longitudinal and transverse projec-

tions of the solute field:

kLs ¼

3yp

8p

Zdk

ð2pÞ3Eeff

D ðkÞSLn ðkÞ ð66Þ

and

DGLs ¼ �

3yeff

8p

Zdk

ð2pÞ3ðEeff

2 ðkÞ � Eeff1 ðkÞÞSLðkÞ. ð67Þ

In Eqs. (66) and (67),

EeffD ðkÞ ¼ jDeEL

0 ðkÞj2 � fnjDeET

0 ðkÞj2 Df0 � JL � D~E0ðkÞDf0 � JT � DeE0ðkÞ

ð68Þ

and

Eeffi ðkÞ ¼ jeEL

0iðkÞj2 � fsjeET

0iðkÞj2 f0i � JL � eE0iðkÞ

f0i � JT � eE0iðkÞ. ð69Þ

The longitudinal and transverse components of the electro-static energy density in Eqs. (62)–(67) are defined as:

jDEL;T0 ðkÞj

2 ¼ DeE0ðkÞ � JL;T � DeE0ð�kÞ;jEL;T

0i ðkÞj2 ¼ eE0iðkÞ � JL;T � eE0ið�kÞ.

ð70Þ

The electric field F0i in Eq. (69) is a generalization of theOnsager reaction cavity field [91] to the case of solutes ofnon-spherical shape and non-point-dipole charge distribu-tion. This field is obtained from Eq. (39) by replacing E0

with E0i. Also, DF0 in Eq. (68) is DF0 = F02 � F01 andDf0 ¼ DF0=DF 0. Finally, in Eqs. (68) and (69),

fs ¼2ð�s � 1Þ2�s þ 1

ð71Þ

is the usual Onsager polarity parameter [91]. The polarityparameter for the nuclear polarization fn is obtained anala-gously to fs in Eq. (38) with the vs quantities replaced bythe vn ones

fn ¼2ð�1�s � 1Þ2�1�s þ 1

. ð72Þ

Eqs. (71) and (72) are obtained from Eq. (38) by substi-tuting into it the k = 0 expressions for the solvent responsefunctions (Eqs. (50) and (54)–(56)). In particular, thenuclear polarity parameter in Eq. (72) turns out to be dif-ferent from the Lippert–Mataga polarity parameter f LM

n

often derived from the Onsager reaction field by assumingadditivity of the nuclear and electronic response in theoverall solvent response function [114]

f LMn ¼ 2ð�s � 1Þ

2�s þ 1� 2ð�1 � 1Þ

2�1 þ 1. ð73Þ

At large values of the dielectric constant �s the parameterf LM

n scales as the Pekar factor

f LMn ’ 3

2c0; ð74Þ

whereas fn in Eq. (72) becomes

fn ’ 1� 3

2�1�s

. ð75Þ

The latter has a weaker dependence on �1 than the former,in accord with computer simulations of dipolar solvation inpolarizable solvents [21].

4. Calculation procedure

The formalism outlined above is realized in a computa-tional algorithm sketched in Fig. 6. It includes twobranches, one is for the solvent part of the calculationand another is for the solute part. The two parts are com-bined together in the integration over the inverted space,which yields the reorganization energy (ks) and the totalfree energy of nuclear plus electronic solvation (DGs). Westart with describing the solvent branch followed by theoutline of the solute part.

4.1. Solvent

The calculation of the structure factors in the solventbranch in Fig. 6 requires a set of experimental input param-eters: m (gas-phase dipole moment), a (gas-phase dipolarpolarizability), �1 (high-frequency dielectric constant), �s

(static dielectric constant), and r (effective hard spherediameter of the solvent molecules). The hard sphere diam-eter is obtained from the experimental compressibility ofthe solvent by fitting it to the compressibility found fromthe generalized van der Waals (vdW) equation of state[115]. Based on these parameters, an analytical procedurehas been recently proposed to calculate SL,T(k) [39]. Thisparameterization, called parameterized polarization struc-ture factors (PPSF), makes use of the analytical solutionof the mean-spherical approximation (MSA) for dipolarfluids [116]. The MSA solution gives SL,T(k) in terms ofthe Baxter function Q(kr,g) appearing as solution of Per-cus–Yevick integral equations for hard sphere fluids [94]

0 8kσ

0

1

2

3

4

S n(k)

T

L

PPSF

MC

4

Fig. 7. Nuclear longitudinal (L) and transverse (T) structure factors fromthe PPSF (dashed lines) and MC simulations (solid lines). MC simulationsare carried out for a fluid of 1372 hard spheres with permanent dipole m,diameter r, polarizability a, and density q: (m*)2 = bm2/r3 = 1.0, a* = a/r3 = 0.06, qr3 = 0.8. The dielectric properties from the simulations are:�s = 21.4, yeff = 1.57, and yp = 1.54; �1 = 1.75 is obtained from theClausius–Mossotti equation.

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 183

Sðkr; gÞ ¼ jQðkr; gÞj�2; ð76Þ

where

Qðkr; gÞ ¼ 1� 12gZ 1

0

eikrt½aðgÞðt2 � 1Þ=2þ bðgÞðt � 1Þ�dt

ð77Þand a(g) = (1 + 2g)/(1 � g)2, b(g) = �3g/2(1 � g)2. For afluid of hard sphere molecules, g = (p/6)qr3 is the packingdensity, equal to the ratio of the volume of the solvent mol-ecules to the volume of the liquid. In the MSA, the SL,T(k)are obtained by setting g = 2n for SL(k) and g = �n forST(k) in Eq. (76). Here, n is the MSA polarity parameterwhich can be related either to the dipolar density yeff orto the static dielectric constant �s [116].

Two problems arise when dealing with the reorganiza-tion energy calculations using the polarization structurefactors from the MSA. First, one needs a general procedurewhich would provide the nuclear structure factors SL;T

n ðkÞin polarizable solvents in contrast to total structure factorsSL,T(k) given by the MSA solution. Such a formalismshould thus exclude (quantum) fluctuations of the inducedsolvent dipoles p0

j which are not included in the nuclearpolarization field (Fig. 4). Second, the MSA does not givea consistent description of the dielectric properties of polarsolvents, i.e., the polarity parameters n calculated from yeff

and �s are quite different. The PPSF procedure goes aroundthe second problem by considering yeff and �s as two inde-pendent input parameters used to calculate SL,T(k). A con-venient way to introduce the two-parameter scheme is tospecify two separate polarity parameters which areobtained from the longitudinal and transverse structurefactors at k = 0:

ð1� 2nLÞ4

ð1þ 4nLÞ2¼ SLð0Þ;

ð1þ nTÞ4

ð1� 2nTÞ2¼ STð0Þ.

ð78Þ

Separate definitions of nL and nT in terms of SL(0) andST(0) (Eq. (55)) allows us to incorporate macroscopic prop-erties which affect dielectric constants of real solvents, butare not present in the model of dipolar HS fluids. Specifi-cally, the magnitude of parameter yeff, calculated accordingto Wertheim’s 1-RPT algorithm [112], defines the solventdipolar strength which strongly affects the dielectric con-stant. However, �s also depends on such factors as solventquadrupolar moment [103], solvent non-sphericity, etc. Theinfluence of these factors is incorporated into SL,T(0)through the dielectric constant. Similarly, the polarityparameters nL

n and nTn are calculated from Eq. (78) with

SL,T(0) replaced by SL;Tn ð0Þ taken from Eq. (56).

Dipolar projections of the structure factors of molecularliquids modeled by site–site interaction potentials havebeen studied previously [113,117–119]. The PPSF proce-dure has also been tested against MC simulations of dipo-lar hard sphere fluids [39]. However, the structure factors

arising from the nuclear polarization as well as the applica-bility of the PPSF to non-spherical molecules with site–sitepotentials have not been previously tested. This is the aimof the Monte-Carlo (MC) and MD simulations carried outin this study. The details of the simulation protocol aregiven in Appendix A and here we focus only on the results.

Fig. 7 shows the comparison of the transverse and lon-gitudinal components of the nuclear structure factors cal-culated from the PPSF and from MC simulations. TheMC simulations (dashed lines in Fig. 7) have been per-formed on a fluid of 1372 polarizable dipolar hard spherescharacterized by dipole moment m, diameter r, and isotro-pic polarizability a ((m*)2 = bm2/r3 = 1.0, a* = a/r3 =0.06, Appendix A). Since the simulation protocol generatesthe induced polarization in equilibrium with the nuclearconfiguration of the solvent [21], the generated ensembleyields the nuclear polarization in the Pekar partitioning[105].

The PPSF nuclear structure factors are calculated by therelations:

STn ðkÞ ¼ jQðkr;�nT

n Þj�2 ð79Þ

and

SLn ðkÞ ¼ jQðjkr; 2nL

n Þj�2. ð80Þ

In Eq. (80), j = 0.95 is an empirical parameter introducedfor a better agreement between the PPSF and MC simula-tions of non-polarizable dipolar fluids [39]. The simulationsand the PPSF agree well in the entire range of solvent polar-izabilities a* = a/r3 = 0.01–0.08 studied by simulations.

The MSA solution in Eq. (76) was derived for a modelliquid of dipolar hard spheres. The parameterization intro-duced by the PPSF suggests to use the experimental �s toaccommodate empirically the features which are notincluded in the MSA solution. Two factors, often presentin real polar solvents, molecular quadrupoles and non-sphericity, are expected to affect significantly the form ofthe structure factors. Therefore, we have performed MD

0 1 2 3 4 5

k,Ao -1

0

2.5

5

SL

,T(k

)

0

0.5

1

1.5

2

2.5

3

103 e

V ×

A

T

L

1

2

o

Fig. 8. Upper panel: longitudinal (1), k2hEeffD ðkÞik, and transverse (2),

k2hðDET0 ðkÞÞ

2ik components of the electrostatic energy density of complex1 entering the k-integral in Eqs. (66) and (62), respectively. h. . . ik denotesthe average over the orientations of the wavevector k. Lower panel:longitudinal (L) and transverse (T) structure factors for TIP3P water at298 K. The solid lines refer to the results of MD simulations. Dashed linesindicate the results of PPSF calculations with the input parameterscorresponding to the TIP3P force field (Table 6, m = 2.35 D, �s = 95.4,�1 = 1.0) and r = 2.87 A. The dash-dotted lines refer to the nuclearstructure factors of ambient water. The graphs in the upper and lowerpanels are plotted against the same scale of k-values to indicate that thedetails of the structure factors beyond approximately kr ’ p are insignif-icant for the calculation of the reorganization energy.

0 2 4 6

k,Ao -1

0

1

2

SL

,T(k

) T L

MD nPPSF

Fig. 9. Longitudinal (L) and transverse (T) polarization structure factorsof acetonitrile at 298 K. The solid lines refer to the results of MDsimulations. Dashed lines indicate the results of PPSF calculations withthe input parameters corresponding to ACN3 (Table 6, m = 4.12 D,�s = 29.6, �1 = 1.0) and r = 4.14 A. The dash-doted lines indicate thenuclear structure factors SL;T

n ðkÞ from the PPSF with the parameters ofambient acetonitrile: m = 3.9 D, �s = 35.9, �1 = 1.8, a = 4.48 A3, g =0.424, r = 4.14 A.

184 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

simulations for two solvents with well-developed forcefields, water [120] and acetonitrile [121]. Water is a mole-cule with a relatively symmetric shape and with a very largequadrupole moment Q [122] ((Q*)2 = bQ2/r5 = 1.1 atT ’ 300 K) among commonly used molecular solvents.On the other hand, acetonitrile has a small quadrupolemoment ((Q*)2 = 0.13), but the molecule is very non-spher-ical with the aspect ratio ’3. Therefore, these two extremecases may provide a good test of the ability of the PPSF toincorporate the complications related to molecular specificsof the solvents in terms of their macroscopic dielectricconstants.

Fig. 8 (lower panel) shows the comparison of the simu-lation results for TIP3P water to the PPSF. A slightlywrong positioning of the longitudinal peak may be relatedto a different hard sphere diameter of TIP3P water (seeTable 6 in Appendix A) compared to the hard sphere diam-eter of water at ambient conditions used in scaling wave-vectors in Fig. 8. A downward scaling of r by just 5%results in a very good match between calculated and simu-lated structure factors. As expected, the steric effects ofpacking the solvent molecules in dense liquids is the mainfactor determining the position of the longitudinal peak.This indeed is seen in Fig. 9 for simulations of acetonitrile.The effective hard sphere diameter obtained from solventcompressibility does not accommodate the fact that lineardipoles tend to pack side-to-side pointing in opposite direc-tions. The longitudinal peak thus effectively reflects a lowermolecular diameter. The preferential opposite orientationof the dipoles leads to a low Kirkwood factor, and thedielectric constant is much lower than one would expectfor a dipolar solvent with such large dipole moment(4.12 D for the force field by Edwards et al. [121]). As aresult, the transverse structure factor does not change withk as much as it does for hard sphere dipolar liquids (cf.Figs. 7 and 8 to Fig. 9). As is seen, the PPSF with �s fromMD simulations accommodates this feature of the solventquite well.

Fig. 8 compares on the common scale the k-dependenceof the longitudinal and transverse components of the elec-trostatic energy density of complex 1, k2hEeff

D ðkÞik andk2hjDET

0 ðkÞj2ik, with the longitudinal and transverse compo-

nents of the polarization structure factors (h. . . ik refers tothe average over the orientations of the wavevector k). Thiscomparison shows that details of the molecular structure ofthe polar solvent affecting the range of k-values beyond thelimit of k ’ p/r are insignificant for the calculation of thereorganization energy and the free energy gap. Therefore,the discrepancies in the position of the longitudinal peakbetween the simulations and the PPSF do not noticeablyaffect the results of calculations. This statement also appliesto the range of k-values (k > 2p/ls, where ls is the character-istic distance between partial charges within the solventmolecule) at which the multipolar approximation for thecharge distribution within the solvent molecules breaksdown. The charge–charge structure factors calculated onsite–site interaction potentials [25,73,113,123–126] then

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 185

decay to zero instead of approaching the unity limit(SL,T(k)! 1 at k!1) of multipolar approximations[117,119,125]. The range of k-values where the inaccuracyof the multipolar approximation becomes significant laysbeyond the range of small k-values affecting the calculationof thermodynamic properties unless the solute is muchsmaller than the solvent.

4.2. Solute

The solute branch of the calculation algorithm (Fig. 6)consists of the numerical calculation of the Fourier trans-form of the electric field outside the solute placed in thevacuum. The direct-space electric fields in the initial andfinal states of the solute are given by a superposition ofelectric fields produced by partial charges qi

0k

E0iðrÞ ¼XM0

k¼1

qi0k

r� r0k

jr� r0kj3; ð81Þ

where the sum runs over M0 partial charges localized onsolute atoms. The field E0i(r) is Fourier transformed inthe region X accessible to the solvent molecules (Eq.(34)). The region X is generated by assigning vdW radii(CHARMM24 [127]) to all atoms of the solute and thenadding the hard sphere radius r/2 of the solvent(r = 2.87 A for water and 4.14 A for acetonitrile). This cre-ates the solvent-accessible surface (SAS), see Fig. 10. Thedefinition of the solute field thus requires atomic coordi-nates (x0k, y0k, z0k in Fig. 6) and vdW radii (a0k inFig. 6) of N0 atoms of the solute and M0 partial chargesq0k (q0k in Fig. 6) to be used in Eq. (81).

The infinite-space Fourier transform of the Coulombelectric field (Eq. (34)) is numerically divergent [39]. Thisnumerical problem is obviated by splitting the region ofintegration into the inner part between the SAS and a cut-off sphere and the region outside the cutoff sphere. TheFourier transform within the sphere is calculated numeri-cally by the Fast Fourier Transform (FFT) technique[128] on a cube with the center at the geometrical centerof the DSA complex

rc ¼ N�10

XN0

k¼1

r0k. ð82Þ

SAS

VdW

σ

solvent

Fig. 10. Solvent-accessible surface (SAS) and van der Waals (vdW)surface of the donor–acceptor complex.

The length of the cube is chosen by multiplying the maxi-mum extension of the molecule measured from rc by a fac-tor of 9. This choice yields a sufficiently small increment ofthe k-grid necessary for the inverted-space integration and,at the same time, avoids numerical errors arising from arti-ficial periodicity imposed by a finite-size numerical FFTtechnique. The FFT calculation was done on a grid ofdimension 256 · 256 · 256 and the step of 0.5 A. Calcula-tions on complex 1 involved 143 atoms holding partialcharges qi

0k. The charge shifts ðDqk ¼ q20k � q1

0kÞ and coordi-nates used in the solvent reorganization and free energycalculations are the same as those reported in Ref. [43].The individual (i.e., initial and final state) charges used inthe reaction free energy calculations are also taken fromRef. [43]. The field eE0iðkÞ obtained by combining thenumerical and analytical parts is used to calculate the lon-gitudinal and transverse components of the electrostaticenergy density in Eqs. (69) and (70). These componentsare then used in the k-integrals with the polarization struc-ture factors (Eqs. (62)–(67); also see Fig. 6).

5. Results and comparison to experiment

5.1. Solvent reorganization energy

The solvent reorganization energy of complex 1 was pre-viously obtained from MD simulations of this complex inTIP3P water [43]. The permanent dipole moment in thisforce field is enhanced from the vacuum dipole of 1.87 Dto 2.35 D to account for water polarizability. This resultsin a dielectric constant of �s = 97.5 from our simulations,which agrees well with �s = 97.0 found in the literature[129]. Table 1 lists the results of calculations of the reorga-nization energy with structure factors from the PPSF (col-umn 5) and from MD simulations (column 7). The densityof the solvent in the NVT simulations was adjusted at eachtemperature in order to reproduce the expansivityap = 2.96 · 10�4 K�1 of TIP3P water [130]. The tempera-ture derivative of the reorganization energy thus gives theconstant-pressure reorganization entropy correspondingto conditions normally employed in experiment,

Sk ¼ �ðoks=oT ÞP . ð83ÞOverall, there is an exceptionally good agreement betweenthe reorganization energies calculated by using the struc-ture factors from PPSF and MD simulations. This is notsurprising in view of the very good agreement betweenthe two sets of structure factors shown in Fig. 8.

The PPSF result at 298 K, ks = 64.11 kcal/mol, alsocompares well with the direct calculation of the reorgani-zation energy from MD simulations, where the value of60.9 kcal/mol was reported [43]. The electrostatic forcesin those simulations were cut off at distances greater than10.1 A. The cutoff is expected to lower the reorganizationenergy compared to that of an infinite system. In orderto estimate the effect of the interaction cutoff, we havecalculated the reorganization energy for a fictitious solute

Table 1Reorganization energy (kcal/mol) of 1 in water

T (K) ga ypb �s

c kpd kp

e kpf kpq

f kqf kp

g

288 0.4110 6.44 107.7 64.93 39.60 64.35 1.55 4.37 45.88293 0.4104 6.32 102.1 64.52 39.58 64.26 1.45 4.33 45.47298 0.4098 6.21 97.5 64.11 39.56 63.93 1.38 4.25 45.07303 0.4092 6.09 96.0 63.67 39.55 63.52 1.42 4.13 44.68308 0.4085 5.99 93.7 63.25 39.54 62.98 1.22 4.14 44.30

All calculations except those in the last column are done with �1 = 1.0.a Packing fraction calculated with r = 2.87 A and the isobaric expansion coefficient ap = 2.96 · 10�4 K�1.b TIP3P water has the permanent dipole of 2.35 D scaled up from the vacuum dipole moment of water, 1.83 D, to account for the mean-field effect of the

induced dipoles.c Calculated from MD simulations as described in Appendix A.d Calculations with the PPSF structure factors with �1 = 1.0 and �s from MD simulations. kp stands for the reorganization energy arising from the

interaction between the solute electric field and the solvent dipoles, kq comes from the interaction between the solute field gradient and solventquadrupoles, kpq is the mixed term from correlated fluctuations of dipoles and quadrupoles on different solvent molecules, see Eq. (84).

e Continuum limit SL;Tn ðkÞ ¼ SL;Tð0Þ at �1 = 1.0 and �s from MD simulations.

f Calculations with the structure factors from MD simulations.g Calculations with the PPSF structure factors with the solvent parameters of ambient water.

186 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

with the distance 10.1 A added to the radius of eachatom exposed to the solvent. This contribution amountsto 7.1 kcal/mol. Column 6 in Table 1 shows the resultsof calculations when the k-dependent polarization struc-ture factors are replaced by their k = 0 values. The gapin ks values between columns 5 and 6 thus quantifiesthe contribution of the non-local part of solvent responseto the reorganization energy. The last column (10) inTable 1 shows the PPSF calculations using parametersof ambient water. In these calculations, the gas phasedipole moment m = 1.87 D is renormalized by the polar-izability effect to give m 0 = 2.43 D (Wertheim’s 1-RPTformalism [21,112]). Despite this renormalization, ks inthis calculation is substantially (’30%) smaller than inthe calculations using parameters of TIP3P water. TIP3Pwater thus appears to produce stronger solvation thanambient water.

Table 1 also presents two components of the solventreorganization energy produced by solvent quadrupoles:kq is the second cumulant of the coupling of the solute elec-tric field gradient to solvent quadrupole moment [41,75]whereas kpq is a mixed term arising from correlated fluctu-ations of dipoles and quadrupoles positioned at different

solvent molecules [75,93]. The resulting solvent reorganiza-tion energy is the sum of the dipolar component kp and twoquadrupolar components

ks ¼ kp þ kpq þ kq. ð84ÞThe problem of quadrupolar solvent reorganization has

recently attracted much attention [73–75,131] in connectionwith new experimental data showing appreciable solventreorganization in non-dipolar solvents [132–136]. However,the components kq and kpq constitute only a small fractionof the overall reorganization energy despite a relativelyhigh reduced quadrupole of water, bQ2/r5 = 1.1 (cf. to(Q*)2 = 0.13 of acetonitrile). For the rest of the paper wewill therefore assume

ks ’ kp. ð85Þ

We note that the value ks = 69.7 kcal/mol calculated forTIP3P water with the account of water quadrupoles is inremarkable agreement with ks = 68 kcal/mol obtained bycorrecting the simulated values [43] by the finite-size cutoffeffects.

The dependence of ks and the reorganization entropy on�1 are given in Table 2. In these calculations, the vacuumdipole moment of water, 1.83 D, was held constant alongwith the total dielectric constant �s = 78.0. The change in�1 was achieved by varying the polarizability a accordingto the Clausius–Mossotti equation

�1 � 1

�1 þ 2¼ 8ga=r3; ð86Þ

where g = (p/6)qr3 is the solvent packing fraction.Two drastically different predictions for the effect of sol-

vent polarizability on ks can be found in the literature. Theclassical Marcus two-sphere model [1] predicts a drop of ks

by about a factor of 0.6 when going from �1 = 1.0 to�1 = 1.8. On the other hand, simulations using non-polar-izable and polarizable versions of the water force field pre-dict almost no dependence of ks on solvent polarizability[19,137]. The actual situation is in between of the twoextremes. The reorganization energy does drop withincreasing �1, but not as much as is predicted by contin-uum models [21]. On the other hand, the change is suffi-cient to make simulations based on non-polarizablesolvent models unreliable.

The situation for the dependence of ks on �1 is illus-trated in Fig. 11, where continuum results for complex 1

obtained with the DelPhi Poisson–Boltzmann solver [138]are compared to the calculations within the NRFT. Thedielectric calculations with the vdW dielectric cavity(denoted ‘‘cont./vdW’’ in Fig. 11, see also Fig. 10) showa substantial drop of ks with �1. The dependence on �1is much weaker in the NRFT (see also Table 2). The weakdependence of ks on �1 is the result of the cancellation oftwo competing factors: the decrease of the longitudinalstructure factor in the range of small k-values with increas-

Table 2Reorganization energy (kcal/mol) and reorganization entropy (e.u., cal K�1 mol�1) at T = 298 K of complex 1 (experimental parameters for ambientwater) calculated with the PPSF for the structure factors

�1 �s ksa Sk

a ksb Sk

b ksc ks

d

1.0 78.0 52.97 46.50 39.49 �8.14 81.12 43.641.2 78.0 48.84 52.17 32.84 �4.85 69.61 37.441.4 78.0 46.41 61.52 28.11 �2.88 61.39 33.011.6 78.0 45.32 70.66 24.52 �1.60 55.23 29.691.8e 78.0 45.15 80.01 21.74 �0.71 50.44 27.112.0 78.0 45.68 90.12 19.52 �0.15 46.60 26.05

1.0f 97.5g 64.11h 84.13i 39.56b 2.96b 81.44 43.79

a NRFT with the PPSF for ambient water with varying polarizability a. The temperature derivatives are calculated with account for the temperaturedependence of the density of water characterized by expansivity ap = 2.6 · 10�4 K�1. The static and high frequency dielectric constants are linearly variedwith temperature with the temperature coefficients of ambient water: d�s/dT = �0.398 K�1 and d�1/dT = �2.75 · 10�4 K�1.

b Calculated with SL(k) = SL(0) and ST(k) = ST(0).c DelPhi calculation with the vdW cavity.d DelPhi calculation with the solvent-accessible cavity.e Parameters corresponding to water at ambient conditions.f TIP3P water.g From present MD simulations. This value is in good agreement with �s = 97.0 reported in the literature [129].h Calculated for TIP3P water using the PPSF.i d�s/dT = �0.654 K�1 from MD simulations is used; this value turns to be higher than experimental d�s/dT = �0.398 K�1. Also, for TIP3P water, d�1/

dT = 0 is adopted.

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 187

ing �1 (Fig. 5) compensated by an increase in yp due tohigher solvent dipole m 0 in more polarizable solvents. Wenote that this cancellation is strongly affected by the k-dependence of the polarization structure factors in therange of small k-values contributing to the k-integral andcannot be reduced to the cancellation of the yp factor inks (Eqs. (62) and (66)) with yp in the denominator inEq. (56), resulting in the Pekar factor of continuumelectrostatics.

The continuum limit of the NRFT is obtained when thedependence on the wavevector k is neglected in the solventstructure factors and one assumes SL,T(k) ’ SL,T(0) andSL;T

n ðkÞ ’ SL;Tn ð0Þ. When this assumption is incorporated

in the microscopic calculations (marked NRFT/S(0) inFig. 11), the resultant reorganization energy gains the

1 1.2 1.4 1.6 1.8 2

∞

20

40

60

80

s , kc

al/m

ole

cont./vdW

cont./SASNRFT/S(0)

NRFT/S(k)

TIP3Pambient water

Fig. 11. kp vs. �1 calculated for complex 1 by using the non-localpolarization response theory (NRFT, solid lines). The dashed lines refer tothe numerical solution of the Poisson equation with the vdW (cont./vdW)and SAS (cont./SAS) cavities. The diamond and square indicate TIP3Pand ambient water, respectively.

strong dependence on �1 characteristic of continuum theo-ries. The continuum limit of the microscopic theory corre-sponds, however, to the dielectric cavity coinciding with theSAS (Fig. 10). The corresponding DelPhi calculation(marked cont./SAS in Fig. 11) indeed goes parallel withthe continuum limit of the NRFT. The distinction betweenthese two results arises from the mean-field approximationused in the NRFT formulation and different handling ofthe polarizability effects in the two formulations (additivein the continuum and non-additive in the microscopic for-mulation, cf., e.g., Eqs. (72) and (73)). Note that the mean-field approximation is more accurate, when compared tothe exact solution of the Li–Kardar–Chandler equation,in the full microscopic formulation than in its continuumlimit [39]. The exact formulation of the theory, which doesnot involve the mean-field approximation, gives the solu-tion of the Poisson equation in its continuum limit.

The numerical values for the reorganization energiesshown in Fig. 11 are given in Table 2. The comparisonbetween the microscopic and continuum calculations isinstructive. At �1 = 1, ks from the vdW continuum is muchhigher than the microscopic calculation, while ks from theSAS continuum is close to the microscopic result. Withincreasing �1, on the other hand, ks from the vdW contin-uum falls down almost to the microscopic value. The con-tinuum calculation with the vdW cavity may thus appear ina reasonable accord with microscopic calculations orexperiment due to the mutual cancellation of errors pro-ducing a similar effect on the calculated solvation free ener-gies in different solvents due to the close proximity of �1values commonly found for dense molecular solvents.

Along with reorganization energies, Table 2 lists reorga-nization entropies Sk (Eq. (83)). Note that Sk obtainedfrom the PPSF calibrated on TIP3P water is in a reason-

188 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

able agreement with the corresponding value obtained withthe structure factors from MD simulations: 84.1 e.u. and69.9 e.u., respectively. The dielectric continuum calculationgives the wrong sign for the entropy in accord with previ-ous reports [14,32]. Also, the magnitude of Sk is substan-tially higher in the microscopic theory than in thecontinuum calculation (cf. columns 4 and 6 in Table 2).

A similar trend is seen for the reaction free energy gap(Table 3) for which the reaction entropy is defined as

DSs ¼ �ðoDGs=oT ÞP . ð87ÞAlthough the sign of DSs is correct in the continuum calcu-lations, the entropy magnitude is much lower than in theNRFT, similar to a previous report for a different ET system[14], where DSs was experimentally obtained from temper-ature-dependent absorption and emission charge-transferbands. Since the analytical theory seems to be consistentwith the computer experiment, one needs a test againstexperimental data. Unfortunately, experimental evidenceon the solvent entropic effects on ET reactions is very lim-ited (see Ref. [60] for a recent review).

5.2. ET rate constant

The calculations of the temperature-dependent reorgani-zation energy and equilibrium energy gap can be comparedto experimental Arrhenius law measurements [44] for com-plex 1. Transition metal-based charge-transfer complexesare commonly characterized by metal–ligand vibrationalfrequencies [43] in the range xv ’ 300–500 cm�1, substan-tially lower than frequencies xv ’ 1100–1500 cm�1 nor-mally assigned to C–C skeletal vibrations of organicdonor–acceptor complexes. Therefore, Eqs. (1), (2) and(21) with the full quantum-mechanical description of vibra-tions and temperature-induced populations of vibrationalstates should be used for the ET rate in complex 1. Unfor-tunately, our calculations provide only the solvent compo-nent of the free energy gap. Its gas-phase component isunknown and the electronic coupling entering the GoldenRule ET rate in Eq. (1) is known with uncertainty [43].

Table 3Solvation Gibbs energy (kcal/mol) and solvation entropy (e.u., cal K�1 mol�1

�1 Gs,1a Ss,1

a Gs,2b

1.0 �236.01 �198.30 �183.1.2 �245.13 �228.90 �189.1.4 �255.10 �262.71 �195.1.6 �266.00 �299.28 �202.1.8d �277.88 �337.52 �210.2.0 �290.77 �378.20 �218.

1.8e �325.1 �29.3 �242.1.8f �197.0 �14.6 �165.

a Gibbs energy and solvation entropy in the initial ET state.b Gibbs energy and solvation entropy in the final ET state.c DGs = Gs,2 � Gs,1, DSs = Ss,2 � Ss,1.d Parameters corresponding to water at ambient conditions.e Continuum calculation (DelPhi [9]) with solute’s vdW cavity.f Continuum calculation (DelPhi [9]) with solute’s SA cavity.

These two parameters (DGg and V12) were varied in fittingthe experimental activation enthalpy DH� = 9.5 kcal/moland the experimental activation entropy DS�/kB =�5.6 e.u. [44]. Note that the experimental quantity is aneffective entropy, including contributions due to the elec-tronic coupling element as well as solvation and inner-sphere vibrational modes [43]. The Arrhenius analysiswas performed by the linear regression of ln(kET/T) vs. 1/T based on the transition-state expression

kET ¼kBT

he�DGyðT Þ=kBT . ð88Þ

The rate constant was calculated based on Eqs. (1) and(21), with ks and DGs varied linearly with temperature usingthe calculated entropies (Tables 2 and 3). The results of cal-culations are listed in Table 4. The fitted electronic cou-pling V12 falls in the range of values given by electronicstructure calculations [43] using the semi-empiricalINDO/s model by Zerner and co-workers [139]. The equi-librium gap obtained from the fit is appreciably more neg-ative than DG ’ �25.4 kcal/mol estimated from the redoxpotentials of separate donor and acceptor sites, based onthe high spin ground state of the Co2+ product (it has beenargued [43] that the less exothermic low spin Co2+ productmay be the relevant one in the experimentally observedprocess). Neglecting the vibrational excitations in the anal-ysis (0–0 transition only) results in a much lower activationenthalpy and a substantially more negative activation en-tropy (second row in Table 4).

The relatively low frequency of metal–ligand vibrationsin transition metal complexes results in a dense manifold ofvibrational levels (Fig. 12) which are partially populated atroom temperature. The change of the vibrational popula-tions with temperature may result in a contribution tothe overall activation entropy [140]. This, however, doesnot happen for complex 1 when ks and DGs are fixed attheir 298 K values. The dashed lines in Fig. 13 show theenthalpy and entropy of activation as a function of thevibrational frequency at constant temperature and kv.Increasing the vibrational frequency makes vibrational

)

Ss,2b DGs

c DSsc

36 �132.68 52.65 65.6228 �151.94 55.85 76.9675 �173.15 59.35 89.5679 �195.60 63.21 103.6843 �219.15 67.45 118.3766 �243.41 72.11 134.79

1 �20.0 83.0 9.22 �12.2 31.6 2.4

Table 4Parameters for complex 1 at T = 298 K

Level V12 (cm�1) DG (kcal/mol) �hxv (cm�1) kv (kcal/mol) DS� (e.u.) DH� (kcal/mol)

Full 0.07a �41.2a 400 16.1b �5.5c 9.4c

kv = 0d 0.07 �25.4e – 0.0 �25.0d 2.0d

a Obtained from fitting the experimental Arrhenius dependence. Sk and DSs are calculated within the PPSF with parameters corresponding to ambientwater.

b From Ref. [43].c Experimental values from Ref. [44].d Obtained by neglecting intramolecular vibrations (kv = 0) in Eq. (21). In this limiting case, the V12 value was taken from the value obtained from the

full analysis.e Estimated from redox potentials of separate donor and acceptor, Ref. [44].

-4 -2 0 2 4X/λ

s

0

5

F1,

2(X)/

λ s

m=0

m=80

n=0

n=80

Fig. 12. Diabatic initial (solid lines) and final (dashed lines) curvesobtained from Eq. (21) for parameters of DSA 1. Curves marked with n

and m indexes refer to vibrational states of the initial and final states,respectively. Relative energies are drawn to scale, based on ks=45.15 kcal/mol, DG = �41.2 kcal/mol, �hxv = 400 cm�1.

0 1000 2000 3000 4000

ωv, cm

-1

-40

-30

-20

-10

0

ΔS✝, e

.u.

0 1000 2000 3000 4000

0

2.5

5

7.5

10

ΔH✝, k

cal/m

ol

h-

Fig. 13. Enthalpy and entropy of activation of DSA complex 1 vs. �hxv.The solid lines indicate the calculations according to Eq. (21) with ks andDGs varying linearly with temperature based on the correspondingentropies from Tables 2 and 3. The dashed line indicate the samecalculation with ks and DGs fixed at their 298 K values. The dash-dotted(Sk = 0, DSs = 0) and dotted (Sk 5 0, DSs 5 0) lines for the activationenthalpy and entropy refer to the xv!1 limit corresponding to Eq. (13)with kv = 0 (no vibrational excitations).

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 189

excitations less accessible, but this is seen to have littleeffect on the activation entropy and enthalpy.

This situation changes when the temperature depen-dence of ks and DGs is included in the calculations of theArrhenius activation parameters. In this case, the tempera-ture dependence of the ET energy gap results in a change ofthe vibrational quantum numbers corresponding to themaximum vibronic contribution. The splitting of the acti-vation barrier into the entropic and enthalpic contributionthen becomes sensitive to the choice of xv (Fig. 13, solidlines). This sensitivity may be important for the interpreta-tion of experimental data since the correct definition of theeffective vibrational frequency (Eq. (15)) increases inimportance once the temperature dependence of the solva-tion parameters is introduced into the analysis of reactionrates. The classical Marcus–Hush equation with kv = 0replaces the sum over all possible vibronic transitions witha single 0–0 transition. The result is a significantly lowerenthalpy and more negative entropy of activation (Table4).

6. Discussion

Table 5 summarizes the main features of the NRFT the-ory presented here and compares it to the available contin-uum approximations. The solute input is similar to that

used in continuum calculations with the Poisson equationincluding charges, coordinates, and vdW radii of the atomsmaking the donor–acceptor complex. The use of the SAS inthe NRFT (Fig. 10), however, requires the solvent radius tobe added to the atoms exposed to the solvent (a0k + r/2 inTable 5). The solvent is specified by six parameters, m, a, q,�s, �1, r, all of which are either directly measured in exper-iment or can be parameterized based on the experimentalinput. The calculation of the reaction entropy requires, inaddition, temperature derivatives of q, �s, and �1 (r is inprinciple also temperature dependent, but this effect issmall [14] and has been neglected in the present calcula-tions). The result of the NRFT procedure is the responsefunction vL,T(k) which is integrated with the solute field

Table 5Hierarchy of theories for the thermodynamics of polar solvation

Theory Solute Solvent v(k) Dependence on E0 symmetry

NRFT r0k, q0k, a0k + r/2 m, a, q, �s, �1, r vL,T(k) Explicit through f0 and eE0 (Eq. (36))Poisson equation r0k, q0k, a0k �s, �1 vL,T(0) Implicit through boundary conditionsOnsager equation m0, a0 �s, �1 vL,T(0) Implicit through boundary conditionsBorn equation q0, a0 �s, �1 vL(0) No dependence

0 1 2 3 4 5 6

k,Ao -1

0

2

4

6

SL,T

(k)

L, microscopic

T, microscopic

L, continuum

T, continuum

Fig. 14. Microscopic structure factors and their continuum limits.

190 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

to give the solvation thermodynamic potentials (Fig. 6).The response functions v(k) are affected by the symmetryof the solute electric field E0 realized in terms of the expan-sion of the field in rotational invariants. This symmetryinformation is introduced directly into the NRSF equa-tions (Eqs. (36) and (37)), but is supplied separately,through the boundary conditions, in the continuum calcu-lations (Table 5).

The NRFT algorithm contains the Poisson equationsolution as its k = 0 limit when the overall and nuclear sol-vent susceptibility can be defined through two dielectricconstants (Eqs. (50) and (54)–(56)), thus reducing the num-ber of required solvent parameters to two (Table 5).Finally, two simple analytical limits of the Poisson equa-tion correspond to the Onsager [91] and Born [90] solutionsfor a spherical solute. In the former case, the distinctionbetween the spherical symmetry of the dielectric cavityand the dipolar symmetry of the solute electric fieldrequires taking both the longitudinal and transverse projec-tions of the response function. In the latter case, equal sym-metries of the cavity and of the electric field make itsufficient to use only the longitudinal response function(Table 5).

The most relevant question in comparing microscopicsolvation theories with the dielectric continuum approxi-mation is why the latter has allowed to describe so manysystems after proper parameterization of dielectric cavities,despite drastic approximations involved. The microscopicNRFT formulation contains dielectric continuum as itslimit, allowing us to address this question. The continuumlimit is obtained by neglecting the spatial correlationsbetween solvent dipoles, i.e., by neglecting the k-depen-dence in the polarization response functions. This impliesthat polarization structure factors are replaced by theirk = 0 values (Fig. 14). This replacement is not a goodapproximation for the transverse structure factor, whichchanges quite sharply even for small k-values, but may bea reasonable approximation for the longitudinal structurefactor, which is relatively flat in the range of k-values sig-nificant for solvation thermodynamics. However, for mostcharge configurations, even for the point dipole [38], thecontribution of transverse polarization to the solvation freeenergy is relatively small [39] (’10% in our calculations forcomplex 1 in water). Therefore, the inaccurate continuumapproximation for the transverse structure factor doesnot significantly affect the results of calculations.

The continuum estimates for the polarization structurefactors result in the following inequalities between the con-

tinuum and microscopic longitudinal and transverse com-ponents of the reorganization energy:

kL;conts < kL

s ; kT;conts > kT

s . ð89ÞThe sharp change of the transverse structure factor at smallk-values is responsible for a substantial overestimate of thetransverse component of solvation by continuum models[38,39]. This overestimate manifests itself in solvationdynamics. The transverse polarization dynamics is muchslower than the longitudinal polarization dynamics [30].Therefore, continuum models predict biphasic solvationdynamics with an appreciable slow component due totransverse polarization relaxation. This slow componentis not observed in computer simulations of solvationdynamics [141] and it does not show up in the microscopiccalculations reported in Ref. [42].

The relatively flat form of the longitudinal structure fac-tors at low k-values does not mean that replacing SL,T(k)by SL,T(0) gives accurate numbers for the solvation freeenergy and/or the reorganization energy. A moderateincrease of SL(k) in the range of wavevectors contributingto the k-integral substantially affects the calculated valuesof solvation free energies (cf. columns 5 and 6 in Table1). Moreover, the gap between the microscopic and contin-uum values changes with the solvent dielectric parameters(see, e.g., Fig. 11). This observation practically means thatthere is fundamentally no unique scheme for defining thedielectric cavity applicable to all solvent polarities.

The dominance of longitudinal polarization fluctuationsin solvation thermodynamics is also responsible for exper-imentally observed linear trends of the reorganizationenergy with the Pekar factor [53,142,143] (Eq. (57)). Evenat the continuum level, the polarization response functionfor a solute of complex shape is not represented by the

0 20 40 60 80ε

s

-200

-100

0

Sλ, e

.u.

NRFT

cont./DelPhi

Fig. 16. Reorganization entropies from continuum calculations (DelPhi[9] with the vdW surface, closed circles) and by the NRFT (open squares).Points refer to the same solvents as in Fig. 15.

A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194 191

Pekar factor appearing in the longitudinal projection of thesolvent response function [144]. However, large separationof charges is responsible for the predominantly longitudi-nal response of the solvent, and continuum reorganizationenergies calculated for complex 1 in polar solvents correlatewell with the Pekar factor (Fig. 15(a)). If fact, an equallygood correlation is seen in respect to the Lippert–Matagapolarity parameter (Eq. (73)) commonly used for solvationof dipoles [114] (Fig. 15(b), cf. to Eq. (43)). The use of aparticular parameter does not therefore tell much aboutthe nature of the solute charge distribution and, obviously,reflects a linear relation between c0 and f LM

n for commonsolvents (Eq. (74)).

The results of the current microscopic calculations areshown by triangles in Fig. 15. These numbers do not exhi-bit a linear dependence, although the extent of scatter is notuncommon for ET experiment. The comparison of the con-tinuum and microscopic dependence on the solvent polar-ity does not permit a clear distinction between the twoformulations. Where the distinction becomes clear is forthe reorganization entropy in strongly polar solvents.Fig. 16 shows reorganization entropies Sk calculated incontinuum (DelPhi [9] Poisson–Boltzmann solver) andmicroscopic (NRFT) theories. The continuum calculationreflects the temperature variation of the Pekar factor c0

�ðoc0=oT ÞP ¼ ��21 ðo�1=oT ÞP � ��2

s ðo�s=oT ÞP . ð90Þ

0.2 0.3 0.4 0.5c

0

0

20

40

60

s, kca

l/mol

0.2 0.3 0.4 0.5 0.6

fn

LM

0

20

40

60

s, kca

l/mol

1

2 34

56

a

b

7

Fig. 15. Reorganization energies in polar solvents vs. the Pekar factor(Eq. (57)) (a) and the Lippert–Mataga polarity parameter (Eq. (73)) (b).The open points indicate the DelPhi calculations with vdW (squares) andSAS (circles) molecular surfaces. The closed points (triangles) refer tocalculations with the NRFT. Numbers on the plot indicate: chloroform(1), tetrahydrofuran (2), methylacetate (3), N,N-dimethylformamide (4),acetone (5), acetonitrile (6), water (7).

In low-polarity solvents, c0 is mostly influenced by the sta-tic dielectric constant, which has a negative temperaturederivative. The continuum reorganization entropy (closedcircles in Fig. 16) is positive and is close to the microscopicresult (open squares in Fig. 16). The continuum estimate ofthe temperature variation of ks in low-polarity solventsthus gives a semi-quantitative account of the experimentalobservations [54]. In strong polar solvents, the temperaturederivative of c0 is mostly influenced by the high-frequencydielectric constant, and continuum Sk is negative. In thiscase, the predictions of the continuum model significantlydepart from both the microscopic calculations and manyexperimental measurements [14,46–50,53], showing posi-tive reorganization entropies.

The microscopic calculations presented here show a rel-atively weak dependence of the reorganization energy onthe solvent high-frequency dielectric constant, in qualita-tive accord with available computer simulation data [19–21]. Testing this theoretical prediction experimentallymay become problematic because of the narrow range of�1 values available for common polar solvents. We note,however, that the problem of the weak dependence of thereorganization energy on �1 is related to the problem ofcorrect sign of the reorganization entropy. The strongdependence of the continuum reorganization energy on�1 is one of major factors shifting the continuum reorgani-zation entropy to the range of positive values.

The calculations of the quadrupolar component of thesolvent reorganization energy presented here confirm theconclusion previously reached for Stokes shifts in couma-rin-153 optical dye [145]: quadrupolar solvation is insignif-icant in most commonly used polar solvents, and thedipolar approximation for the solvent charge distributionis sufficient for most practical calculations.

Acknowledgments

D.V.M. thanks The Petroleum Research Fund (39539-AC6) and National Science Foundation (CHE-0304694)for support of this research. M.D.N. was supported by

192 A.A. Milischuk et al. / Chemical Physics 324 (2006) 172–194

DE-AC02-98CH10886 at Brookhaven National Labora-tory. The authors are grateful to Professor G.A. Voth forsharing the structural data on the polypeptide-linked do-nor–acceptor complex and to Dr. P.K. Ghorai for the per-mission to use the unpublished data in Fig. 3. This ispublication #596 from the ASU Photosynthesis Center.

Appendix A. Simulation and analysis

The MC simulations of dipolar-polarizable hard spheresolvents shown in Fig. 7 were done as described in Ref.[21]. Simulations of 6 · 105 cycles long were run for 1372polarizable molecules with periodic boundary conditionsand the reaction field cutoff of dipole–dipole interactions.The MD simulations were carried out with the force fieldof 3-site acetonitrile (ACN3) by Edwards et al. [121] andthe 3-site model of water (TIP3P) by Jorgensen et al.[120] (Table 6). The site–site interaction potential is givenby the sum of the LJ and Coulomb interaction potentials

Eab ¼ 4eabrab

rab

� �12

� rab

rab

� �6" #

þqaqb

rab; ðA1Þ

where the LJ parameters are taken according to the Lor-entz–Bertholet rules: eab ¼

ffiffiffiffiffiffiffiffieaebp

and rab = (ra + rb)/2.All simulations were done with the DL_POLY moleculardynamics package [146]. We run two sets of MD simula-tions in the temperature range from 288 K to 308 K witha 5 K step. The timestep in each simulation is 5 fs. AllMD simulation are 20 ns long.

We used the Nose–Hoover thermostat [147] for theACN3 simulations with the relaxation parameter of0.5 ps. This value ensures good stabilization of the total sys-tem energy. The energy drift for ACN3 is only about 0.1%.The simulation box was constructed to include 256 ACN3molecules in a cube with the side length L = 28.2025 A atT = 298 K to reproduce the experimental mass density ofacetonitrile, qM = 0.777 g/cm3. The side length is adjustedat each temperature to account for temperature expansionwith the experimental volume expansion coefficientap = 1.38 · 10�3 K�1.

In simulations of TIP3P water, 256 molecules reside in acube with the side length of L = 19.7744 A at 298 K. Thesystem is coupled to the Berendsen [148] thermostat withthe relaxation time of 0.1 ps. The drift in total energy ofabout 0.1% is observed. The liquid mass density

Table 6Force field parameters for use in the simulation

Atomic interaction site ra (A) ea · 103 (kcal/mol) qa (e)

TIP3P watera

O 3.15 152.10 �0.834

Acetonitrileb

Me 3.6 379.55 0.269C 3.4 99.36 0.129N 3.3 99.36 �0.398

a rOH = 0.9572 A, \HOH = 104.52�.b rMeC = 1.46 A, rCN = 1.17 A.

qM = 0.9896 g/cm3 and the volume expansion coefficientap = 2.96 · 10�3 K�1 are taken from Ref.[130]. The lattervalue is close to the experimental expansion coefficient ofambient water, ap = 2.6 · 10�3 K�1.

The cutoff for short-range LJ interaction is 13 A forACN3 and 9 A for TIP3P. For long-range Coulomb inter-actions, Ewald summation from DL_POLY [84] is used forACN3 and smoothed particle mesh (SPME) [149] Ewald isadopted for TIP3P. Ewald summation parameters are theconvergence parameter a and the maximum wavenumberkmax

x;y;z. The parameter sets a = 0.24 A�1, kmaxx;y;z ¼ 7 A

�1, and

a =0.35 A�1, kmaxx;y;z ¼ 8 A

�1were used for ACN3 and

TIP3P, respectively.The structure factors have been calculated as the vari-

ance of longitudinal and transverse projections of the k-space solvent polarization

MðkÞ ¼ ð1=mÞXN

i¼1

mie�ik�ri ; ðA2Þ

where mi ¼P

aqarai is a dipole moment of the ith molecule

and the sum runs over the N molecules in the simulationbox. The static dielectric constant is given in terms of thek = 0 variance as follows [150]:

es ¼ 1þ 3yhMð0Þ2i=N ; ðA3Þwhere y = (4p/9)qm2/kBT.

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