Impact of Temperature and Non-GaussianStatistics on Electron Transfer inDonor-Bridge-Acceptor Molecules

Morteza M. Waskasi,† Marshall D. Newton,‡ and Dmitry V. Matyushov† ∗,¶

†School of Molecular Sciences, Arizona State University, PO Box 871604, Tempe, AZ85287-1604

‡Brookhaven National Laboratory, Chemistry Department, Box 5000, Upton, NY11973-5000, United States

¶Department of Physics, Arizona State University, PO Box 871504, Tempe, AZ85287-1504

E-mail: dmitrym@asu.edu

1

Abstract

A combination of experimental data and theoretical analysis provides evidence of a bell-shaped kinetics of electron transfer in the Arrhenius coordinates ln k vs 1/T . This kineticlaw is a temperature analog of the familiar Marcus bell-shaped dependence based on ln k vsthe reaction free energy. These results were obtained for reactions of intramolecular chargeshift between the donor and acceptor separated by a rigid spacer studied experimentally byMiller and co-workers. The non-Arrhenius kinetic law is a direct consequence of the solventreorganization energy and reaction driving force changing approximately as hyperbolic func-tions with temperature. The reorganization energy decreases and the driving force increaseswhen temperature is increased. The point of equality between them marks the maximum ofthe activationless reaction rate. Reaching the consistency between the kinetic and thermo-dynamic experimental data requires the non-Gaussian statistics of the donor-acceptor energygap described by the Q-model of electron transfer. The theoretical formalism combines thevibrational envelope of quantum vibronic transitions with the Q-model describing the classi-cal component of the Franck-Condon factor and a microscopic solvation model of the solventreorganization energy and the reaction free energy.

Keywords: Electron transfer, bell-shaped kinetic law, Arrhenius coordinates, polar solvation

Introduction

The Marcus theory of electron transfer1 has pre-dicted the existence of the inverted region, i.e., adrop of the rate for exoergic reactions, which oc-curs after reaching the maximum rate with zeroactivation barrier. This prediction was experimen-tally verified by Miller, Calcaterra, and Closs for aseries of donor-spacer-acceptor molecules in whichan increasingly negative reaction free energy drovethe reaction from the normal region, through themaximum activationless rate, and into the invertedregion.2 Eight different acceptors were used in thatstudy to map the entire bell-shaped Marcus energygap law.This pioneering work was followed by a large

body of experimental evidence supporting the Mar-cus prediction.3–5 Most of these studies followedthe protocol set up by Miller and co-workers,2,6,7 inwhich the reaction free energy, or the driving force(the negative of the reaction free energy, −∆G0),was altered in a set of chemically modified donor-acceptor complexes. A potential drawback of thisstrategy is that it assumes that the rest of the pa-rameters affecting the reaction rate (the solventreorganization energy, electronic coupling, inter-nal reorganization energy, and the frequency of in-tramolecular vibrations) remain unchanged whenchemical modification is introduced. It is not easyto establish the limits of this approximation sincemost of these parameters are not directly accessibleto experimental measurements. Despite potential

limitations, this approach has been widely adoptedas a way of “measuring” the reorganization energyby fitting the kinetic parameters to the Marcus en-ergy gap law and locating the top of the “invertedparabola” at which the reorganization energy λ isequal to the driving force −∆G0.Additional experimental data, temperature-

dependent kinetics8,9 among them, have been ac-cumulated to test the overall consistency of thetheory. Changing the thermodynamic state ofthe system has offered a potentially cleaner ap-proach to sampling the bell-shaped energy gap lawsince neither chemical modification of the donor-acceptor complex nor the change of the solvent10

are involved. Pressure11 or temperature12,13 canbe used to reach the top activationless rate andturn into the inverted region. While the assump-tion of molecule-independent rate parameters canbe avoided in this approach, one still needs to knowhow the solvent reorganization energy and the driv-ing force change when pressure and temperatureare varied. Effective modeling of the dependenceof the activation free energy on thermodynamicvariables becomes a significant component of thisresearch agenda.Fitting kinetics to the energy gap law is not

the only approach to access the reorganization en-ergy experimentally. An alternative route is offeredby spectroscopy of charge-transfer optical transi-tions,14,15 either through direct measurement ofthe Stokes shift between the absorption and emis-sion lines16–18 or by performing the band-shape

2

analysis.19–21 Measurements of the Stokes shift ofcharge-transfer transitions at different tempera-tures17,22 have shown that the reorganization en-ergy depends on temperature much stronger thananticipated from dielectric continuum models typi-cally adopted in evaluating λ by the Marcus equa-tion.23 This general result is a consequence of theliquid state of a polar molecular solvent, produc-ing more structural fluctuations and distinct modesof thermal agitation12,24,25 than allowed by con-tinuum models better suited for modeling solids.This observation, now well supported by both ex-periment17,22,26,27 and numerical simulations,25,28

calls for a critical re-examination of the earlytemperature-dependent kinetic data.8,9 Those weremostly viewed as fully consistent with the Mar-cus picture,8 although some inconsistencies havebeen identified. In particular, the reorganizationenergy found from fitting the kinetics to the en-ergy gap law for Miller’s set of donor-acceptor com-plexes fell significantly below the continuum Mar-cus equation.29 Such observations, even thoughscarce, add to the obvious concern that if thetemperature slope of the reorganization energy issignificantly underestimated by continuum mod-els,12,24,25 a measurable deviation from the contin-uum prediction should be achieved in experimentsperformed in a sufficiently broad range of temper-atures. 2-methyltetrahydrofuran (MTHF) as a sol-vent provides exactly this opportunity. We havetaken an advantage of it here by looking at boththe general problem of how temperature affects theelectron transfer rate and how the individual pa-rameters entering the rate are affected by temper-ature.The standard framework to understand the ef-

fect of temperature on the reaction rate is providedby the Arrhenius law, which predicts a straightline for the rate plotted in the Arrhenius coor-dinates ln k(T ) vs 1/T . Although many of suchdata have been analyzed in the literature, thefunctional simplicity of the law does not allow toclearly discriminate between different theories ad-dressing the effect of the thermodynamic state ofthe thermal bath on the activation barrier. Thesituation changes near the top of the Marcus “in-verted parabola”, where entropic effects becomesufficiently strong to cause curved forms of ln k(T )vs 1/T .12 Although the experimental resolutionis often insufficient, the donor-acceptor complexesfrom Miller’s set near the top of the bell-shapedenergy gap law are good candidates for observ-

ing non-trivial temperature effects. We thereforehave chosen three such ASB complexes studied inthe past.9 In these molecules, a 4-biphenylyl (B)donor is connected through 5α-androstane spacer(S) to the acceptor (A). Three acceptors fromMiller’s set were studied here: 2-naphthyl (N), 2-benzoquinonyl (Q), and 5-chloro-2-benzoquinonyl(ClQ) (Figure 1). We have applied the microscopicsolvation model24,30 to analyze the temperature-dependent kinetics of the rate and have reacheda good agreement between the theory and experi-ment. Based on our calculations, we have extendedthe range of temperatures reported in the litera-tures (from 179 to 373 K in MTHF9) and foundthat the rate constant, plotted in the Arrheniuscoordinates, shows a bell-shaped form. This novelphenomenology is a direct consequence of a strong,nearly hyperbolic, variation of the solvation freeenergies entering the activation barrier with tem-perature.12 To ascertain whether the experimentalkinetics displays this form would require data forsomewhat lower temperatures.

NSB

QSB

ClQSB

Figure 1: Structures of ASB complexes, where theanion 4-biphenylyl (B, colored green) is the donorof the electron in the electron shift through thespacer 5α-androstane (S, colored blue) to the ac-ceptor (A, colored grey). Complexes with threeacceptors from Miller’s set2 were studied here: 2-naphthyl (N), 2-benzoquinonyl (Q), and 5-chloro-2-benzoquinonyl (ClQ).

The main agenda of this study is illustratedin Figure 2. It shows fits of the experimental

3

rates (points) for two out of three ASB com-plexes displayed in Figure 1 to theoretical calcu-lations (solid lines). Curved form of the tempera-ture law requires taking a full account of the tem-perature dependence of the activation free energyand cannot be produced by assuming temperature-independent enthalpy and entropy of activation,as is commonly done in applying the Arrheniuslaw. Further, the dielectric continuum theories nei-ther describe the experimental reaction free ener-gies (see below) nor produce curved temperaturelaws.The third complex, NSB does not show a curved

temperature law. The experimental activation en-thalpy is positive,9 in contrast to negative (ClQSB)or rather small (QSB, but with uncertain signand magnitude due to the error bars) activationenthalpies observed experimentally for two othercomplexes (Figure 2). This complex, however,presents us with an opportunity to test the consis-tency of the widely used combination of the Gaus-sian Marcus model with the Bixon-Jortner equa-tion modeling the effect of intramolecular quan-tum vibrations on the reaction rate.31 The ad-ditional source of information is provided by thetemperature dependence of the reaction free en-ergy ∆G0(T ) measured in an independent exper-iment.7,8 We have found that two levels of thetheory improvement are required. First, the useof the dielectric models for the solvation compo-nents of that activation barrier is inconsistent withthe experimental results. One has to replace thedielectric continuum with microscopic models ofsolvation. Such improved theory still preservesthe Gaussian statistics of the donor-acceptor en-ergy gap and will be labeled as “Marcus theory”here. However, this improvement is not sufficientsince its application to the experimental data pro-duced parameters incompatible with experimental∆G0(T ). At the second level of the theory improve-ment, we have found that replacing the Gaussianstatistics of the donor-acceptor energy gap with thenon-Gaussian statistics described by the Q-modelof electron transfer32,33 has eliminated the diffi-culty and produced consistent sets of model pa-rameters.The Q-model was designed to describe electron

transfer reactions characterized by non-Gaussianstatistics of the donor-acceptor energy gap.32 Sev-eral physical scenarios can be identified which re-quire this extension of the Gaussian Marcus theory.Among them are polarizable donor-acceptor com-

19.8

19.4

19.0

18.6

Ln

[k×

s]

876543

ClQSB

20.0

19.5

19.0

18.5

Ln

[k×

s]876543

1000/T (K-1

)

Exp.Q-modelMarcus

QSB

Figure 2: Rate constants in the Arrhenius coor-dinates (k(T ) vs 1/T ) for ClQSB (top) and QSB(bottom) in MTHF. The points represent experi-mental data9 and the solid lines refer to the theo-retical calculations combining the Franck-Condonanalysis with the SolvMol30,34 calculations of thesolvent reorganization energy λ(T ) and the solventcomponent of the reaction Gibbs energy ∆Gs(T ).The Marcus model (solid black lines) and theQ-model (solid red lines), both combined withthe microscopic solvation calculations, were usedto model the classical component of the Franck-Condon factor entering the rate of nonadiabaticelectron transfer (eq 5). The gas-phase componentof the energy gap ∆Eg (eq 13) and the electron-transfer coupling V are the fitting parameters listedin Table 1. The combination of the Gaussian statis-tics of the donor-acceptor energy gap with mi-croscopic solvation models is labeled as “Marcusmodel” here.

4

plexes, in which electron transfer causes not onlythe change in the charge distribution, but also achange in the polarizability of the complex.20,33,35

Other scenarios include nonlinear solvation and acoupling between classical intramolecular modes,such as torsional rotations, with the solvent po-larization.32 Both effects, polarizability and tor-sional rotations, potentially affect electron trans-fer in the ASB complexes studied here. The ap-plication of the Q-model to all three complexes(red lines in Figure 2) produce consistently superiorresults compared to the Gaussian Marcus model(black lines in Figure 2).

Conceptual framework

Reaction rate. The conceptual basis of the Mar-cus theory of electron transfer reactions goes backto Onsager’s principle of microscopic reversibil-ity.36,37 It states that the average regression ofspontaneous fluctuations obeys the same laws asthe corresponding irreversible process. What thispractically means is that one can calculate the re-versible work38 (free energy) required to drive thesystem from equilibrium to a given non-equilibriumstate and that free energy will quantify the prob-ability of the system to reach the same state by aspontaneous fluctuation driven by thermal agita-tion. The two properties, the probability P andthe free energy F , are connected by the Gibbs dis-tribution, P ∝ exp[−βF ], β = 1/(kBT ). Note thatthe Gibbs distribution is defined in terms of theHelmholtz free energy F , which gives the maxi-mum reversible work gained at constant volume.In contrast, the Gibbs energy G = F + pV quan-tifies the work done at constant pressure p. ThepV term can be neglected for most problems deal-ing with condensed materials. We therefore use Gthroughout the paper, with the provision that Fin fact follows from theoretical derivations usingstatistical mechanics.The Marcus theory makes full use of microscopic

reversibility by calculating the probability of spon-taneously reaching the top of the activation barrier∆G†, required for kinetics, as the reversible free en-ergy invested to drive the system to the correspond-ing non-equilibrium state.1 The reasoning of micro-scopic reversibility also largely determines the lan-guage used by practitioners in the field, who drawtheir ideas from either the statistics of stochas-tic fluctuations or from reversible thermodynamics.

Both views are indeed helpful, particularly whenmicroscopic models of electron transfer are devel-oped.12,39 Fluctuations tell the mechanistic story ofwhat are the molecular motions involved in the ac-tivation events, while equilibrium thermodynamicsaddresses the question of how the rates are affectedby changing the thermodynamic state of the sys-tem. We will follow this traditional dual view ofthe problem when addressing the questions of howtemperature and the statistics of fluctuations affectthe rate of electron transfer.The departure point is the definition of the reac-

tion coordinate of electron transfer. Modern the-ories, following Warshel,40 use the energy gap be-tween the acceptor and donor electronic states ofthe transferred electron as the reaction coordinateX.32,41,42 The probability to find a given value ofthe gap, due to a spontaneous fluctuation in themedium, is a Gaussian function,

PG(X) =[

2πσ2X]−1/2

e−(X−X0)2/2σ2

X (1)

The average energy gap X0 = λ+∆G0 is given inthe Marcus theory in terms of the reorganizationenergy λ and the reaction free energy ∆G0. Thereorganization energy also enters the variance ofthe energy gap

σ2X = ⟨(X −X0)2⟩ = 2kBTλ (2)

The probability distribution follows one of thedual views on the problem of electron transfer men-tioned above, the fluctuation approach. The proba-bility PG(0) of reaching the transition state X = 0when electron tunneling becomes possible definesthe free energy of activation

∆G† =X20

2βσ2X=

(λ+∆G0)2

4λ(3)

This probability is a factor in the Golden Rule, ornon-adiabatic, rate of electron transfer43,44

kET ∝ V 2PG(0) (4)

where V is the electron transfer coupling. Quan-tum vibrations modify the picture, particularly inthe inverted region of electron transfer (X0 < 0).Vibrations of the donor-acceptor complex add aseparate vibronic channel to each vibrational exci-tation with the energy m!ωv. This energy is addedto X0 as X0 → X0 + m!ωv. In the inverted re-gion with X0 < 0, adding positive m!ωv leads to

5

a lower barrier in eq 3. The amplitude of each vi-bronic channel is determined by the correspondingFranck-Condon factor and the final result for therate constant is given by the Bixon-Jortner expres-sion31

kET =2π

!V 2e−S

∑

m≥0

Sm

m!PG(−m!ωv) (5)

Here, S = λv/!ωv is the Huang-Rhys factor and λvis the reorganization energy of quantum vibrationswith the characteristic frequency ωv. Equation 3for the activation barrier sets the position of theactivationless transition, ∆G† = 0, at −∆Gmax0 =λ. This condition is modified by intramolecularvibrations and the maximum of the rate shifts to

−∆Gmax0 ≃ λ+ λv (6)

The reorganization energy λ in eq 2 incorporatesall classical modes affecting the donor-acceptor en-ergy gap, which typically include the solvent modesand classical intramolecular vibrations. For or-ganic complexes studied here, quantum intramolec-ular vibrations with frequencies !ωjv > 2kBT domi-nate in the reorganization energy of intramolecularvibrations λv =

∑

j λjv (see below). The latter is

a sum of contributions λjv from all normal modesof vibration with the frequencies ωjv. In the ap-proach adopted here, we replace the manifold ofintramolecular quantum vibrations with an effec-tive vibrational frequency24 ωv =

∑

j ωjv(λ

jv/λv).

The definition of individual normal-mode compo-nents λjv require resonance Raman spectroscopy.45

In the absence of such data for the donor-acceptormolecules studied here, a generic frequency of ωv =1500 cm−1 is used, as is typically assigned to or-ganic charge-transfer complexes.31

In the present calculations, λ in eq 2 is assignedto the solvent reorganization energy λs and the for-mer symbol is used throughout below. An alter-native approach would involve identifying classicalnormal modes within the donor-acceptor complexand combining them with λs in the total classicalreorganization energy λ. This approach causes sig-nificant difficulties, both computational and fun-damental. The calculation of the reorganizationenergy components of separate normal modes iscomputationally challenging compared to the cal-culation of the overall vibrational reorganizationenergy achieved in terms of average gas-phase ver-tical energies of the complex in two charge-transfer

states (see the supporting Information (SI)). A no-ticeable reorganization energy of λφ ≃ 0.13 eV waspreviously assigned to the torsional rotation of thebenzene rings in 4-biphenyl upon electron trans-fer.46 Our calculations produce a close number ofλφ ≃ 0.12 eV (see SI). This rotation does not, how-ever, correspond to a normal mode of the donor-acceptor complex and cannot be separated fromthe overall reorganization energy without account-ing for cross coupling to all normal modes of themolecule and the solvent.Q-Model. The cross coupling to the solvent

might potentially be the most significant qualita-tive effect of torsional flexibility, coupled to chargetransfer, in the molecules considered here and po-tentially other charge-transfer complexes.47,48 In-tramolecular torsional motions alter the electricfield of the solute charges, thus providing a cou-pling mechanism between the torsional mobilityand the solvent polarization.49 When such cou-pling is introduced into the model of electron trans-fer, it results in a linear-quadratic dependence ofthe donor-acceptor energy gap on the solvent po-larization.32 Only a linear dependence of the en-ergy gap on the solvent polarization is consideredin the Marcus model; the extension to the linear-quadratic dependence is covered by the Q-model ofelectron transfer.32

The linear Marcus model results in a Gaussiandistribution for the energy gap X (eq 1). Thisstatistics is well satisfied when the solute-solventcoupling is established through the Coulomb in-teraction of the solute charges with the solventdipoles.50 The wave function of the solute is fixedin each state in this approximation and is not de-formed by the fluctuations of the thermal bath.This is generally not correct since the wave func-tion is obviously deformable and any molecular sys-tem is electronically polarizable. The first-orderperturbation correction to the zeroth-order non-deformable wave function, accounting for the so-lute polarizability, leads to a self-polarization termin the energy gap changing quadratically with theelectrostatic field of the solvent. Combined withthe standard linear coupling between fixed chargesand the solvent polarization, it leads, like in thecase of torsional mobility, to a linear-quadratic de-pendence of the energy gap on the solvent polar-ization.Both solvent-coupled torsional mobility and

varying polarizability result in the linear-quadraticdependence of the energy gap on the solvent coor-

6

dinates and the corresponding non-Gaussian statis-tics of the energy gap.20,32,33 It is likely that bothof these factors are influencing the statistics ofthe energy gap in the complexes considered here.The Q-model provides a closed-form solution forall such problems in terms of the standard pa-rameters of the Marcus theory and an additionalparameter α quantifying the deviation from theGaussian statistics

PQ(X) = βA

√

λα3

|X −X∗|e−β(α|X−X

∗|+α2λ)

I1(2β√

α3λ|X −X∗|)(7)

The parameter α can be either positive or negative.In order to simplify the equations, we assume α >0 for most of the discussion below. Further, thenormalization constant A = (1− exp[−βα2λ])−1 ineq 7 can be omitted in most cases and I1(x) is themodified Bessel function of the first order.51

The distribution of the energy gap in the Q-model is based on three parameters, instead of twoparameters in the Gaussian distribution in eq 1.In addition to λ and ∆G0, the Q-model introducesthe non-Gaussian parameter α. It can be deter-mined by combining λ, as defined by the varianceof the energy gap in eq 2, with the reorganizationenergy λSt = |X01 −X02|/2 obtained from the av-erage energy gaps in the initial, X01, and final,X02, states. This reorganization energy is equal tohalf of the Stokes shift16,20 associated with chargetransfer (see also eq 22 below). In the GaussianMarcus model either definition gives the same re-sult, λSt = λ, but this equality breaks down for anon-Gaussian statistics of the energy gap. The de-viation between λ and λSt allows one to determineα (eq 22).The non-Gaussian statistics of the energy gap

also leads to the asymmetry of the solvent reor-ganization energies between the forward and back-ward reactions. If λ = λ1 as defined by eq 2 speci-fies the reorganization energy of the forward reac-tion, then the reorganization energy of the back-ward reaction is given by an analogous relation,λ2 = (β/2)⟨(δX)2⟩2, δX = X −X02, in which theaverage is now taken over the statistics of the ther-mal bath in the final state of the donor-acceptorcomplex. The reorganization energies λ2 and λ canbe related in terms of the parameter α as follows:λ2 = λα3/(1 + α)3.To summarize, any pair of reorganization ener-

gies out of three, λSt, λ1, λ2, provides the value ofα. The three parameters of the model specify theactivation barrier along the reaction coordinate X,which replaces eq 3 of the Marcus model with therelation

∆G† = α(√

|λα2/(1 + α)−∆G0|−√αλ)2 (8)

The Marcus theory and the Gaussian distributionin eq 1 are restored at α → ∞, when λSt = λ1 = λ2.One arrives at eq 3 from eq 8 in that limit, as canbe shown trough a Taylor expansion performed interms of 1/α in eq 8.The parameter X∗ in eq 7 specifies the low-

est magnitude of the energy gap, the “fluctuationboundary”, allowed in the Q-model. The probabil-ity of reaching X < X∗ (when α > 0 as assumedin eq 7) is identically zero, PQ(X) = 0. The fluc-tuation boundary is given in terms of λ and ∆G0as X∗ = ∆G0 − α2λ/(1 + α). Note that the onlyrestriction on the magnitude of α is to fall outsidethe range −1 < α < 0 of mechanical instability.32For negative values of α, one has to take its ab-solute magnitude, α → |α| in eqs 7 and 8. Thecondition X > X∗ at α > 0 changes to X < X∗ atα < 0.The extension of the Q-model to transitions in-

volving intramolecular vibrational excitations isstraightforward. One needs to replace the Gaus-sian distribution PG(−m!ωv) with PQ(−m!ωv) ineq 5. One of significant consequences of changingthe statistics of the energy gap fluctuations is theshift of the position of the maximum of the ratefrom the one given by eq 6 to

−∆Gmax0 ≃ (λ+ λv)α

1 + α−→α→0

0 (9)

The significance of this result is that the maximumrate at the top of the Marcus bell-shaped energygap law can be achieved at a substantially lowerdriving force when α becomes small (see below).Temperature effect. As is clear from eq 2,

the reorganization energy entering both the Mar-cus theory and the Q-model is determined by thevariance of the donor-acceptor energy gap, i.e.,by the breadth of thermal fluctuations modulat-ing the gap. The fact that the thermal noise isenhanced with increasing temperature is known asthe Nyquist theorem52 and is reflected in the tem-perature factor in front of λ in eq 2. The questionwe consider next is whether λ should be treated asa temperature-independent coefficient or as carry-

7

ing its own dependence on temperature.This question can be addressed by first noting

that the solvent reorganization energy is identifiedin the Marcus theory with the free energy of po-larizing the continuum dielectric representing thesolvent. The notion of free energy already re-quires one to pay attention to possible effects ofchanging temperature. However, those are usuallysmall in the continuum model and can often beneglected (see below). They arise from the Pekarfactor c0(T ), which is a combination of the refrac-tive index n(T ) and the dielectric constant ϵs(T ),c0(T ) = n(T )−2 − ϵs(T )−1. As λ is proportionalto c0 in the Marcus theory, λ(T ) ∝ c0(T ), all ef-fects of temperature on λ come from two dielectricconstants, ϵ∞ = n2 and ϵs.This physical picture changes substantially when

the continuum dielectric is replaced with a liquid ofmolecules carrying dipoles.12,39 Thermal agitationnow translates into fluctuations of dipolar orien-tations and dipolar positions (density). The dif-ference between these two modes can be appreci-ated by turning again to Onsager’s microscopic re-versibility. A spontaneous rise of a fluctuation oforientations of molecular dipoles can be viewed asa reversible work invested in rotating the dipolesagainst the field of the donor-acceptor complex.This work will mostly require an input of energy.If one instead considers a fluctuation of moleculardensity, it will require rearranging the moleculesagainst their repulsive cores, i.e., a local re-packingof the liquid. In contrast to the work invested inchanging orientations, the corresponding free en-ergy of re-packing is mostly entropic. One there-fore concludes that the orientational and densityfluctuations carry different thermodynamic signa-tures, energetic (enthalpic) for the former and en-tropic for the latter.This physics displays itself directly in the liquid-

state theory of electron-transfer reorganization.12

The overall reorganization energy becomes a sumof the orientational λp and density components

λ = λp + σ2d/(2kBT ), (10)

where σ2d is the variance of the energy gap pro-duced by the density fluctuations. The reason λ isgiven in this form is that λp and σ2d depend on tem-perature through density and are essentially con-stant parameters at the constant volume. Whenthe continuum limit is taken for the solvent, whichis achieved in the theory by shrinking the molecular

size to zero, λp turns into the continuum expressionand σ2d vanishes. There is no density reorganiza-tion in a continuum medium and only orientationsof dipoles determine λ.

D AS

+ _

Figure 3: Schematic representation of the donor-spacer-acceptor complex. The positive hole on thedonor and the negative electron on the acceptorshow the “electron transfer dipole” correspondingto the difference electric field∆E0 obtained by sub-tracting the electric fields of the complex in thefinal and initial states.

The Marcus notion of the reorganization energyas the free energy of solvation might seem to havebeen lost in this discussion focused primarily onfluctuations. This is not correct, one just needs toturn again from the fluctuation view to the equiva-lent (within Onsager’s paradigm) thermodynamicview. In the thermodynamic view, one consid-ers the free energy of solvation of the “electron-transfer dipole”, i.e., the difference electric field∆E0 = E02 −E01 produced by the donor-acceptorcomplex in the acceptor, E02, and the donor, E01,states. Since the electric field of all other chargescancels out in the difference, it is effectively givenby the sum of the fields of the negative electron atthe acceptor and a positive hole at the donor (adipole), as is schematically depicted in Figure 3.In solvation theories, one can perform the pertur-

bation expansion in terms of the interaction poten-tial v∆ of the electron-transfer dipole with the sur-rounding solvent. This is a long-ranged Coulombinteraction, while the reference system in the per-turbation expansion is the repulsive core of the so-lute excluding the solvent from its volume throughthe repulsive solute-solvent interaction potential.53

The result of the perturbation expansion is thechemical potential of solvation54 (eq (32.3) in ref38)

µ∆ = ⟨v∆⟩0 − (β/2)⟨(δv∆)2⟩0 (11)

where δv∆ = v∆ − ⟨v∆⟩0 and the higher order ex-pansion terms disappear for a Gaussian medium.54

The average ⟨. . . ⟩0 here is over the statistics of themedium fluctuations around the donor-acceptor

8

complex with v∆ turned off, i.e., the positive andnegative charges in Figure 3 removed. For mostpolar liquids, a repulsive molecular core of the so-lute carrying no charges creates no polarization inthe liquid and one has ⟨v∆⟩0 = 0. One arrives atthe equation

µ∆ = −λ (12)

implying that λ defined through the variance of theenergy gap does carry the meaning of the solvationfree energy according to the standard prescriptionof the Marcus theory. Where this prescription failsis in assigning dielectric continuum to a molecularsolvent, which, being a model for a solid material,eliminates the entropic character of some of themolecular motions.55,56

The derivation performed above also implies thatthe distinct effects of the orientational and densitymotions of the solvent reflected in the reorganiza-tion energy should be shared by the solvent partof the reaction free energy ∆Gs. This componentadds to the gas-phase energy difference ∆Eg in theoverall reaction free energy ∆G0

∆G0 = ∆Eg +∆Gs (13)

Here, ∆Eg is the energy separation between theminima of the Born-Oppenheimer surfaces in thegas phase. It can be associated with the energy of0-0 transition in spectroscopy.We show below that the temperature dependence

of ∆Gs essentially traces that of λ. The differencesare mostly mechanistic: ∆Gs includes solvation byinduced dipoles excluded from λ, and it is givenby the difference in solvation chemical potentialsof the final and initial states of the donor-acceptorcomplex, ∆Gs = µ2 − µ1, instead of the chem-ical potential µ∆ of the electron-transfer dipole.This observation also implies that deficiencies ofthe continuum dielectric in describing the temper-ature dependence of λ should be equally exposed inthe corresponding difficulties of continuum modelsto describe the entropy of solvation, as is now wellestablished.17,55–57

In order to connect the thermodynamic pictureto the fluctuation arguments presented above, onecan define the entropy of solvation

S∆ = − (∂µ∆/∂T )V = −λ/T − Φ/T (14)

where the second part of this equation followsfrom eq 11. The term Φ in the second sum-mand is a third-order statistical correlator,25 Φ ∝

⟨(δv∆)2δH0⟩, where δH0 is the fluctuation of the to-tal energy of the reference system unperturbed bythe Coulomb interaction of the medium with the“electron transfer dipole” (Figure 3). The term Φis responsible for what is known in solvation the-ories as the “solvation reorganization energy”58,59

(not to be confused with the electron transfer reor-ganization energy λ). It describes restructuring ofthe solvent caused by the field of the solute. Obvi-ously, there is no restructuring in continuum mod-els, which therefore fail to describe the solvationentropy. What is worth stressing here is a concep-tual, even though not a direct mathematical, con-nection between “restructuring” and “density reor-ganization”, both leading to substantial changes ofλ and ∆Gs with temperature.Rate in the Arrhenius coordinates. Equa-

tion 4 gives the reaction rate in a form formallyconsistent with the Arrhenius law

kET ∝ e−β∆G†(T ) (15)

There is, however, an important distinction fromthe classical Arrhenius law,

k = Ae−βEa (16)

which assumes that the activation energy (en-thalpy) Ea is constant and is given by the slopeof the linear plot in the Arrhenius coordinates ln kvs 1/T . The entropy in this simplified descriptionenters the rate preexponent A. Since the activationenergy Ea in the Arrhenius equation is associatedwith the height of the activation barrier in the tran-sition state theory,60,61 Ea has to be non-negative.In contrast to this common view, ∆G†(T ) in eqs

3 and 8 involves a fairly complex dependence ontemperature arising from the solvation componentof the reaction free energy ∆Gs(T ) and the reorga-nization energy λ(T ). Both the enthalpy and en-tropy of activation are functions of temperature,instead of the commonly assumed constant pa-rameters. The entropy of activation follows from∆G†(T ) as

∆S†(T ) = −(

∂∆G†(T )/∂T)

p(17)

where the constant pressure conditions are applied.Correspondingly, the enthalpy of activation is

∆H†(T ) =(

∂β∆G†(T )/∂β)

p(18)

9

∆

∆

∆

− β∆ ∆

Figure 4: (a) −β∆G†(T ) and (b) ∆H† and T∆S†vs 1/T . Calculations are performed for the classicalMarcus activation free energy in eq 3 with λ(T ) and∆Gs(T ) calculated for ClQSB complex in MTHF(Figure 1). T∆S†(T ) and∆H†(T ) are given by eqs17 and 18, respectively. The arrows in (a) and (b)show the points of activationless electron transferat T = T ∗ (eqs 19 and 20) when ∆G† = ∆H† =∆S† = 0. The results of using the Marcus modelfor ∆G†(T ) (eq 3) are shown in black, red marksthe results of applying the Q-model (eq 8).

One therefore generally anticipates that the mea-sured rates will be given by a curved function in theArrhenius coordinates and the local slope ∆H†(T )will change in a sufficiently wide range of temper-atures (Figure 4b).This general phenomenology finds its dramatic

confirmation in the bell-shaped form of the rateconstant plotted in the Arrhenius coordinatesas predicted theoretically12 and observed exper-imentally for a number of charge-transfer sys-tems.13,48,62–64 While the standard, Arrhenius-typereaction kinetics is typically found for electrontransfer, anti-Arrhenius portions of this overallbell-shaped temperature law have been observed inthe past and discussed in terms of a negative activa-tion enthalpy.65–67 This feature was also reportedfor two out of three Miller’s complexes (ClQSB andQSB in Figure 1), as is shown in Figure 2 and dis-cussed in more detail below.The basic phenomenology of the bell-shaped rate

law is sketched in Figure 4a. This curved form ofthe rate law is the projection of the Marcus energygap law on the 1/T coordinate.12,68 It arises asthe consequence of crossing the point X0(T ∗) = 0at the temperature T ∗ when activationless electrontransfer is reached by changing the temperature.The reorganization energy is equal to the drivingforce at T ∗

−∆G0(T ∗) = λ(T ∗) (19)

in the Marcus theory. The Q-model predicts analternative result

−∆G0(T ∗) = αλ(T ∗)/(1 + α) (20)

where, as above, α > 0 is assumed. The activationbarrier is positive on both sides of T ∗, but boththe activation enthalpy and the activation entropychange from negative values at T > T ∗ to positivevalues below T ∗ (Figure 4b).

Results

Here we apply the general ideas discussed aboveto the calculation of the rates of three complexesshown in Figure 1. All calculations have been doneby using eqs 1, 5, and 7 in which the solvent reor-ganization energy λ(T ) and the solvent part of thereaction free energy ∆Gs(T ) are calculated theo-retically with the software package SolvMol.34 Thisapproach leaves two parameters in the rate equa-

10

tion unspecified, the electron-transfer coupling Vand the gas-phase energy gap ∆Eg, when the Mar-cus model is used to describe the classical distri-bution of energy gaps entering the Franck-Condonfactor. The use of the Q-model adds one additionalparameter, α, quantifying the deviation from theGaussian statistics. These parameters have beenvaried to produce best fits to experimental kET(T )and the results are listed in Table 1. The qual-ity of the fits is shown in Figures 2 and 5. Thevacuum energy gaps ∆Eg for all complexes havealso been calculated with CDFT.69 The accuracyof such calculations is insufficient for kinetic model-ing (see SI) and fits to experimental rates are morereliable. The vibrational reorganization energies λvwere calculated from DFT and CDFT (see SI) andare also listed in Table 1.

16

14

12

10

8

Ln

[k×

s]

5.04.03.0

1000/T (K-1

)

Exp. Q-model Marcus

Figure 5: Rate constants in the Arrhenius co-ordinates (k(T ) vs 1/T ) for NSB in MTHF. Thepoints represent experimental data8 and the solidlines refer to the theoretical calculations. The blackline indicates the Marcus model (eq 1) used in theBixon-Jortner equation (eq 5), the red line refers tothe Q-model (eq 7) used in eq 5. SolvMol30,34 wasused to calculate the solvent reorganization energyλ(T ) and the solvent component of the reactionGibbs energy ∆Gs(T ). The gas-phase componentof the energy gap ∆Eg and the electron-transfercoupling V are the fitting parameters listed in Ta-ble 1. The parameter α = 1.14, quantifying thenon-Gaussian character of the energy gap statis-tics, was determined from the fit; α → ∞ leads tothe Marcus limit with the Gaussian statistics of X.

The fits of experimental rates by using the Gaus-sian Marcus model in eq 5 are shown by the blacksolid lines in Figures 2 and 5; the fits to the Q-model are shown by the red solid lines. While thequality of the fits is consistently better in the caseof the Q-model, this success can be attributed to a

higher mathematical flexibility of the equations dueto an additional fitting parameter α. A consistencytest is required and it is provided by experimentaldata for the temperature dependent reaction freeenergy ∆G0(T ) available for the NSB complex.7,8

Experimental data for ∆G0(T ) of NSB were ob-tained from the equilibrium constant for the re-action of charge shift from biphenylyl− (B−) to2-naphthyl (N) group in the NSB complex inMTHF.7,8 The measurements were a combinationof BC/NC and BS/NS equilibria, where cyclo-hexane (C) spacer was used at low temperatures.Nevertheless the two sets of data were consistent,with nearly temperature independent free energies∆G0(T ) (Figure 6, points), as one would anticipateif the free energy of solvation of B− and N− wereclose in the magnitude.The main distinction between the results of fits to

the Marcus theory and the Q-model comes as themagnitude of the gas-phase energy gap ∆Eg calcu-lated from the fit of the kinetic data. The fit to thestandard Marcus theory used in eq 5 leads to ∆G0significantly more negative than experiment (blackline in Figure 6). In contrast, the fit of the ki-netic data to the Q-model produces a nearly perfectagreement with the experimental ∆G0(T ) withoutadditional fitting parameters (red line in Figure 6).

-0.2

-0.1

0.0

0.1

0.2

∆G

0(e

V)

360320280240200

T(K)

-0.70

Exp. Q-model Marcus

Figure 6: Temperature dependence of the reactionfree energy for the NSB complex. The points areexperimental data7,8 and the solid line is the the-oretical calculation combining ∆Eg from the fit ofthe experimental rates with ∆Gs from SolMol30,34

(eq 13). The black solid line is obtained with ∆Egfrom the fit of the experimental rates to the Mar-cus model (eqs 1 and 5). The red solid line refers to∆G0 with its gas-phase component ∆Eg obtainedfrom fitting experimental rates with the Q-model(eqs 5 and 7).

The ability to fit the experimental kinetic data

11

Table 1: Parameters of electron transfer reactions (eV) at T = 300 K (solvation parametersrefer to MTHF).

Complex Gas Solvent TotalV × 103 a λvb ∆Egc λd ∆Gsd ∆G0e

Marcus

NSB 0.03 0.41 −0.77 0.86 0.04 −0.73fQSB 0.22 0.50 −1.47g 0.74 0.05 −1.42hClQSB 0.16 0.49 −1.54 0.73 0.06 −1.48h

Q-model

NSB 0.07 0.41 −0.11 0.86 0.04 −0.07fQSB 0.21 0.50 −1.06g 0.74 0.05 −1.01hClQSB 0.16 0.49 −1.27 0.73 0.06 −1.21h

aObtained by fitting the kinetic data for all three complexes. The application of the Q-model resulted inα = 1.14, 1.22 and 1.74 for NSB, QSB and ClQSB, respectively. bThe vibrational reorganization

energies were calculated from the gas-phase vertical transition energies as the mean of two values,λv = (λv1 + λv2)/2, which can be produced from the gas-phase energy surfaces (DFT/B3LYP/6-31G, seeTables S1 and S2 in SI). The characteristic vibrational frequency of ωv = 1500 cm−1 was assigned to all

complexes. c∆Eg values listed in the table were obtained as fitting parameters in the analysis of thekinetic data for ASB complexes; ∆Eg ≃ 0 (NSB), −2.35 eV (QSB), and −2.74 eV (ClQSB) were

calculated with DFT, see Table S1 in SI. dCalculated by using the microscopic solvation model SolvMol.eFrom eq 13. f Experimental ∆G0 in MTHF7 is −0.06 eV at T = 300 K. gRedox potentials measuredfor Q and B in dimethylformamide2 were used to estimate the gas-phase energy gap between separate

fragments. SolvMol calculations of the solvation energies of Q− and B− combined with the experimental∆Gredox0 yield ∆Eg(∞) = −1.68 eV at an infinite donor-acceptor separation (see SI for more details).hThe present calculations of ∆G0 cannot be directly compared with those listed by Miller,2 since they

were based on the redox potentials of the isolated fragments in a different solvent (dimethylformamide).

and to obtain the bell-shaped curve in the Arrhe-nius coordinates (Figure 2) are greatly affected bythe strong dependence of the reorganization en-ergy on temperature predicted by the microscopicsolvation model. The dielectric continuum doesnot capture this physics as illustrated in Figure 7,where we compare the SolvMol calculations withthe dielectric model implemented in the numeri-cal software DelPhi numerically solving the Poissonequation.70,71 As expected, the continuum modelgrossly underestimates the change of the reorga-nization energy with temperature24,25 (the slopeof λ with increasing temperature is nearly zero inFigure 7). The microscopic and continuum resultsare numerically close at T ∼ 300 K, which im-plies that the continuum models were empiricallyparametrized to reproduce solvation free energiesaround this temperature. However, the gross un-derestimate of the solvation entropy by the dielec-tric models17,55–57 eventually leads to incorrect freeenergies in an extended range of temperatures.The top of the bell-shaped kinetic law in the

2.0

1.6

1.2

0.8

λ (

eV

)

350300250200150 T (K)

NSB QSB ClQSB

Figure 7: Solvent reorganization energy λ vs tem-perature for NSB (black), QSB (red), and ClQSB(blue) in MTHF. The solid lines refer to the cal-culations with SolvMol30,34 (microscopic solvationmodel) and the dashed lines show the calculationsusing DelPhi71 (continuum solvation model).

12

Arrhenius coordinates represents the activationlessregime (∆G†(T ∗) = 0), when the average energygap is zero, X0(T ∗) = 0. The temperature depen-dence of the barrier arises from the solvent compo-nent of the driving force −∆Gs(T ) and the solventreorganization energy λ(T ). The point of the max-imum in the Arrhenius coordinates is their cross-ing point in the Marcus theory (eq 19 and Figure8). As mentioned above, using continuum dielec-tric to model the solvent effect on electron transferproduces the reorganization energy nearly indepen-dent of temperature (Figure 7) and no maximumin the Arrhenius coordinates.

1.6

1.2

0.8

Energ

y (e

V)

350300250200150 T (K)

T*

λ -∆G0

Figure 8: Temperature variation of the solventreorganization energy λ(T ) (solid lines) and thedriving force −∆G0 (dashed lines) calculated forelectron transfer in ClQSB (blue) and QSB (red)in 2-methyltetrahydrofuran (MTFH). The calcu-lations are performed with the microscopic solva-tion model SolvMol.30,34 The arrows indicate thetemperature T ∗ at which zero activation barrier isreached in the Marcus theory of electron transfer(eq 19). The condition of maximum rate is alteredin the Q-model (eq 20).

Discussion

The extraordinary predictive power of the Mar-cus theory23 is based on the fundamental Gaussianstatistics of the medium fluctuations projected onthe one-dimensional reaction coordinate. The bell-shaped energy gap law provides the means of ex-perimental sampling of the energy gap statistics byvarying the reaction free energy.2 When this funda-mental view of thermal fluctuations in polar mediais supplemented with microscopic liquid state mod-els, the theory becomes capable of very detailedpredictions of the effect of thermodynamic param-eters on the reaction rate.

Electron transfer with low activation barriersbrings forward entropic effects usually hidden be-hind the Arrhenius phenomenology. The appear-ance of a bell-shaped rate law in the Arrheniuscoordinates (ln(k) vs 1/T ) is a dramatic conse-quence of such entropic effects (Figure 2). Thisnew phenomenology provides an alternative exper-imental approach to sample the statistics of thedonor-acceptor energy gap by varying the temper-ature. The ability to sample the bell-shaped ratelaw by varying temperature relies on a relativelystrong variation of the solvation free energies en-tering the activation barrier. The microscopic sol-vation theory12,30 suggests that the most relevantfunctionality for both −∆G0 and λ is hyperbolic,a + b/T . The top of the bell-shaped dependencemarks the point of activationless electron transfer:the reaction is in the Marcus inverted region at hightemperatures and crosses to the normal region atlower temperatures.

0.6

0.4

0.2

0.0

Gi (

X)

(eV

)

-2 -1 0 1 2 X(eV)

Q-ModelMarcus

Figure 9: G1(X) (red) and G2(X) (blue) for elec-tron transfer in the NSB complex calculated withthe parameters listed in Table 1. The non-Gaussianparameter α = 1.14 is used in the Q-model calcu-lations (solid lines) and α → ∞ is set to producethe Marcus limit (dashed lines). The vertical dot-ted line indicates X = 0 at which the free energysurfaces cross and electron tunneling becomes pos-sible. The vertical dashed line indicates the posi-tion of the “fluctuation border” X∗ of the Q-modelbelow which Gi(X) → ∞. The distance betweenthe minima of the free energy surfaces is equal to2λSt, where λSt is the Stokes shift reorganizationenergy.

Gaussian statistics is not required to obtain thebell-shaped energy gap law. Already the involve-ment of quantum intramolecular vibrations of thedonor-acceptor complex (eq 5) brings about the de-

13

viation of the energy gap shape from the invertedparabola of the Marcus theory.31 Further devia-tions from the inverted parabola are allowed by theQ-model producing non-Gaussian statistics of thedonor-acceptor energy gap.32 However, the maxi-mum rate at the top of the bell-shaped curve is stillreached at the point of zero average energy gap,X0 = 0, producing activationless electron transfer.The Q-model brings a number of new features

to the general phenomenology of electron trans-fer in donor-bridge-acceptor systems. First, sincethe fluctuations of the energy gap are not Gaus-sian, the free energy surfaces of electron trans-fer are non-parabolic. The extent of deviationfrom the parabolic shape is very substantial forα = 1.14 − 1.74 following from the fits of the ki-netic data (from NSB to ClQSB in Table 1). Thefree energy surfaces follow directly from the corre-sponding probabilities Pi(X) as

Gi(X) = G0i − β−1 ln [Pi(X)] (21)

where i = 1 and i = 2 refer to the forward andbackward reactions, respectively. Here, G0i is thefree energy at the minimum such that the reactionfree energy is ∆G0 = G02 −G01.

18

16

14

12

10

Ln[k×

s]

2.52.01.51.00.5

-∆G0(eV)

Q-model Marcus

( λ + λv) α/(1+α)

λ + λv

Figure 10: Energy gap law based on the Marcusmodel (black line) and the Q-model (red line). Theparameters are for the NSB complex at T = 300 Kas listed in Table 1.

The free energy surface resulting from the Mar-cus theory and the Q-model are compared in Fig-ure 9 in the case of the NSB complex. As is clearlyseen, the Q-model predicts a significant decrease ofthe activation barrier at a given value of the driv-ing force −∆G0. The rate of electron transfer cantherefore be substantially increased without sacri-ficing the reaction free energy, a feature significantfor applications to solar energy conversion.72 Apart

1.0

0.9

0.8

0.7

λ (

eV

)

-1.5-1.0-0.50.00.5

∆G0 + 〈λ〉 (eV)

Normal Inverted

1

2 34

5

67

8

Figure 11: Solvent reorganization energies ofelectron transfer in Miller’s set of donor-acceptorcomplexes.2 The molecular charge distributionin the donor-acceptor complexes was calculatedfrom CDFT (see SI) and was used in SolvMol30

to calculate λ. The calculated values are plottedagainst ∆G0 + ⟨λ⟩, where ⟨λ⟩ is the averagereorganization energy in the set and ∆G0 is theexperimental reaction free energy from redoxpotentials.2 The complexes included in the set areof the general ASB form (Figure 1) with the fol-lowing acceptors A: 2-naphthyl (1), 9-phenanthryl(2), pyrenyl (3), hexahydronaphthoquinon-2-yl(4), 2-naphthoquinonyl (5), 2-benzoquinonyl(6), 5-chlorobenzoquinon-5-yl (7), 5,6-dichlorobenzoquinon-2-yl (8). The ASB complexesshown in Figure 1 are marked red in the plot.

14

from the nonlinear shape of Gi(X), the reductionof the barrier is achieved by reducing the distancebetween the minima of the free energy curves from2λ in the Marcus theory to the value

2λSt = λα(1 + 2α)

(1 + α)2(22)

The reorganization energy λSt refers to half ofthe Stokes shift between the absorption and emis-sion maxima of charge-transfer bands.16,20 In theGaussian model, one has λSt = λ, as is seen fromeq 22 at α → ∞. However, the distance betweenthe absorption and emission maxima shrinks as αdecreases.20 In terms of thermally activated elec-tron transfer, one has 2λSt = |X01 −X02| and thesame phenomenology is reflected in closer minimaof the free energy surfaces and a lower activationbarrier (Figure 9).This general result is also clearly seen in the reac-

tion energy gap law shown in Figure 10, where thefitting parameters for the NSB complex are used tocompare the Marcus theory to the Q-model. Sinceα ≃ 1 from our fitting, the top of the bell-shapedenergy gap law, that is the maximum rate, shiftsfrom ≃ λ+λv, per eq 6, to about half of this value(≃ (λ + λv)/2, eq 9). Therefore, donor-acceptorcomplexes satisfying the rules of the Q-model (ei-ther through varying polarizability or the couplingof dihedral rotations to the solvent) achieve themaximum rate of electron transfer at the drivingforce significantly reduced compared to the Gaus-sian (Marcus) model. This observation might helpin designing charge-transfer complexes for efficientsolar energy conversion since fast charge separa-tion, followed by a long-living charge separatedstate, is highly sought in those applications.73–75

The broadly accepted line of thought for achiev-ing such conditions is to perform reactions in low-polarity solvents76–78 with a low value of λ. Analternative approach, suggested by the Q-model,would be to seek conditions for a low magnitude ofthe non-Gaussian parameter α.All these mechanistic details, which are poten-

tially important for both the fundamental under-standing of electron transfer and for applicationsto solar energy conversion, are washed out in thestandard construction of the energy gap law whenln(kET) is plotted vs the driving force −∆G0 un-der the assumption that all other parameters ofthe reaction remain unchanged. The reorganiza-tion energy, which defines the top of the Marcus

bell-shaped law in Miller’s set of donor-acceptorcomplexes, turns out to be equal to 0.75 eV.2 Whilethis value is close to the corresponding results forQSB and ClQSB from our calculations (Table 1),this does not mean that the assumption of a con-stant reorganization energy within the set is welljustified. The results of our calculations of λ forall eight complexes from Miller’s set is shown inFigure 11 (see Table S6 in SI). It is clear that thevalues of λ are spread over a significant range andthe assumption of a constant reorganization energythroughout the set is an approximation.

Acknowledgement This research was sup-ported by the Office of Basic Energy Sciences,

Division of Chemical Sciences, Geosciences, andEnergy Biosciences, Department of Energy (DE-SC0015641). CPU time was provided by the

National Science Foundation through XSEDE re-sources (TG-MCB080116N).

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