Solvent reorganization of electron transitions in viscous solvents

Solvent reorganization of electron transitions in viscous solventsPradip K. Ghorai and Dmitry V. Matyushov Citation: J. Chem. Phys. 124, 144510 (2006); doi: 10.1063/1.2185102 View online: http://dx.doi.org/10.1063/1.2185102 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v124/i14 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 124, 144510 �2006�

Solvent reorganization of electron transitions in viscous solventsPradip K. Ghorai and Dmitry V. Matyushova�

Department of Chemistry, Arizona State University, Tempe, Arizona 85287-1604and The Center for the Early Events in Photosynthesis, Arizona State University,Tempe, Arizona 85287-1604

�Received 19 January 2006; accepted 15 February 2006; published online 14 April 2006�

We develop a model of electron transfer reactions at conditions of nonergodicity when the time ofsolvent relaxation crosses the observation time window set up by the reaction rate. Solventreorganization energy of intramolecular electron transfer in a charge-transfer molecule dissolved inwater and acetonitrile is studied by molecular dynamics simulations at varying temperatures. Weobserve a sharp decrease of the reorganization energy at a temperature identified as the temperatureof structural arrest due to cage effect, as discussed by the mode-coupling theory. This temperaturealso marks the onset of the enhancement of translational diffusion relative to rotational relaxationsignaling the breakdown of the Stokes-Einstein relation. The change in the reorganization energy atthe transition temperature reflects the dynamical arrest of the slow, collective relaxation of thesolvent related to the relaxation of the solvent dipolar polarization. An analytical theory proposed todescribe this effect agrees well with both the simulations and experimental Stokes shift data. Thetheory is applied to the analysis of charge-transfer kinetics in a low-temperature glass former. Weshow that the reorganization energy is substantially lower than its equilibrium value for thelow-temperature portion of the data. The theory predicts the possibility of discontinuous changes inthe dependence of the electron transfer rate on the free energy gap when the reaction switchesbetween ergodic and nonergodic regimes. © 2006 American Institute of Physics.�DOI: 10.1063/1.2185102�

I. INTRODUCTION

Theories of activated chemical dynamics and transportphenomena in condensed phase are often based on transition-state ideas invoking equilibrium thermodynamics to describethe reaction flux across the transition-state surface separatingthe reactants from the products. The Marcus-Hush theory ofelectron transfer �ET� reactions fully relies on the transition-state formalism defining the ET activation barrier in terms oftwo thermodynamic parameters, the free energy gap and thenuclear reorganization energy. The former, �F0, is the differ-ence in free energies between the final and initial ET states,and the latter, �s, determines the curvature of two free energyparabolas.1

The energy gap between the acceptor and donor elec-tronic states makes the collective reaction coordinate for ET�Refs. 2–4�

X = �E�q1, . . . ,qn� , �1�

which depends on a manifold of nuclear coordinatesq1 , . . . ,qn. Two second cumulants ��X�2�i, calculated on sta-tistical equilibrium distributions corresponding to the initial�i=1� and final �i=2� states of the system, define two solventreorganization energies �classical nuclear motions�

�si = ��X�2�i/2, �2�

where �=1/kBT and �X=X− �X�i.The Marcus-Hush theory assumes1,5

a�
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�s = �s1 = �s2. �3�

This assumption, which is equivalent to the linear responseapproximation in statistical mechanics calculations, reducesthe number of independent thermodynamic parameters totwo. The free energy surfaces along the reaction coordinateare then two intersecting parabolas. All three thermodynamicparameters, �F0, �s1, and �s2, can be maintained in a three-parameter model of ET. The condition �s1��s2 then requiresnonparabolic free energy surfaces.6,7

The activation free energy of ET, Fact, follows from theassumption that the stationary statistics of X is Gaussian,

�Fiact = �X�i

2/2��X�2� , �4�

where the angular brackets refer to an equilibrium averageand the subscript i is dropped from the second cumulantaccording to Eqs. �2� and �3�; F1

act and F2act refer to the acti-

vation energies of the forward and backward reactions, re-spectively.

The transition-state description becomes inapplicablewhen the time of passage of the activation barrier is compa-rable to the relaxation time of the condensed medium �sol-vent�. The population of the activated state becomes depletedand one arrives at the friction-affected chemical kinetics de-scribed by Kramers theory8 and its modifications.9 For ET,this regime corresponds to solvent-controlled reactions whenthe preexponential factor of the rate is inversely proportionalto a solvent relaxation time.10 One can anticipate a next stepin this hierarchy of relaxation times when the time of the
reaction itself �not just the time of barrier passage� becomes
© 2006 American Institute of Physics10-1

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144510-2 P. K. Ghorai and D. V. Matyushov J. Chem. Phys. 124, 144510 �2006�

comparable to the solvent relaxation time. The system thenloses ergodicity, which is expected to affect the activationbarrier in contrast to the alteration of the rate preexponent inthe Kramers description.11 A general theory of chemical ratesat such conditions is still missing even though nonergodicbehavior may apply to a broad class of reactions in super-cooled liquids and in biopolymers.12 The latter case is par-ticularly relevant to the problem of nonergodic activationsince the dynamics of biopolymers is characterized by abroad range of relaxation times from a few picoseconds tonanoseconds,13–16 and, for a given reaction rate, at least asubset of nuclear modes may become nonergodic.

For electronic transitions in molecules dissolved incondensed-phase solvents, emission energies are strongly af-fected by freezing of the solvent orientational polarizationthrough liquid crystallization or glass transition.17–22 Al-though some of these effects can be related to the drop of thedielectric constant when a polar fluid freezes,17,23–25 the ef-fect of freezing on ET reactions is actually dynamic innature.20,22,25–27 Slow solvent modes may not achieve equi-librium on the time scale of ET �Refs. 11, 12, and 25–29� oron the lifetime of the emitting state.17–21,30 This dynamicalarrest of nuclear solvation may lead to a significant departureof the measured parameters from their thermodynamicvalues.11,12,25–29 Most experimental studies have focused onsystems undergoing solvent solidification. However, the tem-perature at which the activation barrier departs from its ther-modynamic limit depends on the observation window. Forfast reactions, the onset of the dynamical arrest of a givenslow mode from a distribution of relaxation times may occurat a temperature well above the glass or crystallizationtransition.11,31

Once the system loses ergodicity, the equilibrium aver-age in Eq. �4� should be replaced with the time average overthe observation time �obs, �¯�obs. The actual property whichaffects the activation barrier in Eq. �4� is the second cumu-lant of the energy gap fluctuations. We will, therefore, usethe analogy with Eq. �2� to define the nonergodic reorgani-zation energy

��obs� = ��X�2�obs/2, �5�

which is affected by the observation frequency �obs=1/�obs.The averages entering the activation barrier in the noner-godic regime depend on the observation window dictated bythe rate constant kET��obs. The rate constant itself thenneeds to be calculated from a self-consistent equation11

kET�i� � exp�− �Fi

act�kET�i� �� . �6�

The key portion of this formalism is the description ofthe change in the activation barrier with slowing down thesolvent relaxation at a given observation window of the ex-periment. We approach the problem from two different per-spectives. First, we present an analytical model employingthe equilibrium theory of the Stokes shift dynamics �Sec.II B�. Nonergodicity is introduced by a stepwise frequencyfilter restricting the modes contributing to solvent reorgani-zation by those which are faster than the observation fre-
quency,

��obs� = ��obs

�

��d�/�� , �7�

where the dynamic Stokes shift susceptibility34 �� is de-fined below �Eq. �27��.

The second approach employs computer simulations�Sec. III�. We set up the observation window by the length ofsimulations �1 ns� and change the temperature of the solventto observe how the reorganization energy, defined by Eq. �5�,falls out of ergodicity with lowering temperature. Thesesimulations are done for p-nitroaniline chromophore �Fig. 1�dissolved in force-field water and acetonitrile solvents. Weshow that limiting the range of frequencies according to Eq.�7� gives results in good agreement with simulations. Fur-ther, we apply the theory to the laboratory experiment bycalculating steady-state Stokes shift of fluorescence of com-plex 1 �Ref. 22� and phosphorescence of quinoxaline dye21

�Fig. 1� as a function of temperature.Once the transition to nonergodic behavior at fixed �obs

is specified, this information can be funneled into Eq. �6� inorder to calculate the reaction rates.11 This calculation for-malism, as well as simulation results presented here, apply toa single donor-acceptor complex. This situation is most rel-evant to natural photosynthesis in bacterial reaction centersand to single-molecule measurements.35,36 Modeling ETrates measured on ensembles of donor-acceptor complexesrequires an additional input regarding the heterogeneous dis-tribution of local environments.23,37 This situation is illus-trated in Fig. 2.

The phase space of a reacting system, which is schemati-cally represented in Fig. 2 by a two-dimensional plane of the

FIG. 1. Molecules used in simulations �p-nitroaniline�, in the analysis ofexperimental Stokes shift data �quinoxaline �Ref. 21� and complex 1 �Ref.22��, and in the calculations of the charge-separation rates �complex 2�. Thecharge-transfer donor-acceptor molecule is composed of tetracene �TET�donor and pyromellitimide �PI� acceptor connected by a bridge. FollowingRefs. 32 and 33, this donor-acceptor molecule will be referred to as complex2.

conjugate momentum p and coordinate q, is fully explored


144510-3 Electron transitions in viscous solvents J. Chem. Phys. 124, 144510 �2006�

by the system in equilibrium, when the observation time isset to infinity. For a finite observation time �obs, a part of thephase space, shown by the shaded area, is dynamically ar-rested. The unshaded part of the phase space is responsiblefor the fluctuations of the reaction coordinate �E�q1 , . . . ,qn�and the reorganization energy given by Eqs. �5� and �7�. Theshaded part of the phase space represents heterogeneouslydistributed local environments of the donor-acceptor com-plexes frozen in different parts of the energy landscape. Ob-servables measured for ensembles of donor-acceptor com-plexes then require heterogeneous average �¯�het overdifferent local environments. The overall rate observed for anensemble of donor-acceptor complexes is then

kEThet � �kET�het, �8�

where kET is obtained from solving Eq. �6� and index i isomitted for simplicity.

The part of the energy gap fluctuations which are dy-namically frozen on the time scale of ET is determined bythe corresponding reorganization energy

�het = �s − ��obs� . �9�

The average in Eq. �8� can then be taken over a Gaussiandistribution of vertical gaps of ET transitions with the widthrepresenting the shaded area in Fig. 2

�het2 = 2kBT�het. �10�

We realize this program in Sec. III where we apply thistheory to the kinetics of ET in a large donor-acceptor mol-ecule �complex 2, Fig. 1� reported for a broad range of sol-

32,33

FIG. 2. Schematic representation of the phase space of the system in termsof the conjugate momentum p and coordinate q �upper part�. The shadedpart indicates the dynamically arrested slow modes. The unshaded part isresponsible for the energy gap fluctuations and the reorganization energygiven by Eqs. �5� and �7�. The arrows indicate the expansion of the dynami-cally frozen part with lowering temperature. The plot in the lower part of thediagram shows the decay of the reorganization energy ��obs�, correspond-ing to the unshaded part of the phase space, and a simultaneous rise of �het,responsible for the heterogeneous distribution of the donor-acceptor energygaps, Eqs. �9� and �10�.

vent temperatures. The main outcome of these calcula-


tions is the demonstration that the charge-transfer reactionfalls in the nonergodic regime for a substantial portion of therecorded data.

II. ERGODIC AND NONERGODIC SOLVENTREORGANIZATIONS

A. Equilibrium reorganization energy

Thermodynamics of solvent reorganization can be under-stood by considering a fictitious solute with the geometry ofthe real solute, but the charge distribution obtained as a dif-ference of atomic charges in the final and initial charge-transfer states, �qj =q2j −q1j. The solute-solvent interactionpotential is then the difference of interaction potentials in thefinal and initial states, �U0s= j�v0s�j�, �v0s�j�=v0s,2�j�−v0s,1�j�, where summation is over the solvent moleculesand “0” stands for the solute and s refers to the solvent. Thechemical potential of solvation of such a solute can be ob-tained from the thermodynamic �-integration38

�0s = �0

1

d�� v0s�1�g0s��;1�d�1. �11�

In addition, the average solute-solvent interaction energy is

e0s = � �v0s�1�g0s�� = 1;1�d�1. �12�

Here, g0s�� ;1� denotes the solute-solvent distribution func-tion defined on the scaled solute-solvent interaction poten-tial, ��v0s�1�, and d�1 denotes the integration over the phasespace of the solvent including the solvent positions and ori-entations. The combination of position and orientation for asolvent molecule is denoted by “1” and is the solvent num-ber density.

The linear response approximation assumes that the pairdistribution function g0s�� ;1� can be linearly expanded inthe potential ��v0s�1� relative to a reference state which isindependent of the solute-solvent interaction and is thus in-dependent of solvent orientations,

g0s��;1� = g0s�0��r1� − ��g0s

�0��r1��v0s�1� − ��U0s��

− �� v0s�2�g0s�0��12�d�2, �13�

where g0s�0��r1� and g0s

�0��12� are the two-particle, solute-solvent and three-particle, solute-solvent-solvent distributionfunctions of the reference system. Since the average

��U0s� = � g0s�0��r1��v0s�1�d�1 = 0 �14�

is equal to zero, the chemical potential becomes

2�0s = e0s = − �C�0� , �15�

where C�0� is the t=0 value of the equilibrium Stokes shiftcorrelation function

C�t� = ��U0s�t��U0s�0�� . �16�

Once the donor-acceptor energy gap X=�E is related to
�U0s by a constant shift, one gets from Eqs. �2� and �15�


�s = �C�0�/2. �17�

Equations �15� and �17� reflect two fundamental predic-tions of the picture of equal-curvature parabolas.1,5 First isthe equality of two reorganization energies corresponding tothe initial and final states of the donor-acceptor complex� Eq. �3��. Second is the relation between the second cumu-lant of the energy gap fluctuations and the energy � Eq. �15��.This latter relation leads to the connection between the reor-ganization energy and the Stokes shift, ��st, of charge-transfer optical lines

2�s = ��st. �18�

Equation �3� implies that the average �¯� in the defini-tion of C�t� in Eq. �16� can be calculated on configurations inequilibrium with the solute in either ground or excited state�Eq. �3��.39 Computer simulations of solutes with permanentpartial charges immersed in molecular solvents reportedpreviously40–43and presented here �see below� supportthis assumption. Note, however, that some simulations withhighly charged ions44–46 and small optical dyes47 report�s1��s2.

Equation �15�, and its experimental consequence givenby Eq. �18�, relies on the disappearance of the first cumulantcalculated on the reference distribution with no charges onthe solute � Eq. �14��. This relation may break down in asso-ciated solvents where insertion of the solute repulsive corecreates a specific orientation of solvent molecules in the firstsolvation shell. In such cases, the direct connection betweenthe Stokes shift and the reorganization energy � Eq. �18��might break down even though the equality of reorganizationenergies � Eq. �3�� is still maintained.48

From Eq. �17�, the solvent reorganization �free� energycan be split into the energy and constant-volume entropycomponents

�s = E − TSV, �19�

where

E = 2�s + � �20�

and

SV = �s/T + �/T . �21�

In Eqs. �20� and �21�, � denotes the component of entropyand internal energy that arises from the alteration of thesolvent-solvent interaction energy induced by the solute.49–51

This component does not affect the solvation chemical po-tential as it identically cancels out in Eq. �19�. However, itcan significantly affect the observed entropies obtained fromthe temperature derivative of the reorganization energy

SV = − ��s/�T�V. �22�

In solvation literature, the analog of the energy �, whichappears in the energy of solvation, is often called the solventreorganization energy.52 In order to avoid confusion with theET reorganization energy considered here, we will call � theenergy of solvent restructuring. One can calculate � by not-ing that the variance of the potential difference �v0s can be
connected to �s by the following relation:

2�s = �

� ��U0s�2e−�H0d�

� e−�H0d�

, �23�

where H0 is the Hamiltonian of the system when the solute-solvent potential is switched off; d�=d�1¯d�N. By differ-entiating this relation over temperature one immediately gets

� = − ��2/2��U0s�2�H0� . �24�

The energy of solvent restructuring is represented bycorrelated fluctuations of the solute-solvent electrostatic po-tential and the Hamiltonian H0, the main part of which is thesolvent-solvent interaction. Note that the solvation reorgani-zation energy, appearing in the energy of solvation, is deter-mined by a second-order correlator49–51 in contrast to thethird-order correlator in the expression for the energy of sol-vent restructuring �. Equation �24� clarifies the fundamentalorigin of the failure of models neglecting the solventstructure1,53–55 to describe the entropy of solvation:56 theproblem of solvent structure alteration is simply not ad-dressed by such models.

The constant-pressure entropy normally observed in ex-periment can be obtained from Eq. �22� by adding the de-rivative of �s over density,

SP = SV + �p ��s

� �

T, �25�

where �p is the constant-pressure expansivity.

B. Stokes shift susceptibility

The equilibrium reorganization energy �s is definedthrough the t=0 value of the equilibrium Stokes shift corre-lation function �Eqs. �15� and �17��. This implies that theequilibrium value is given by the sum of all fluctuatingmodes contributing to the frequency Fourier transform C��,

�s = ��0

�

C��d� . �26�

C�� can be related to the imaginary part of the Stokes shiftdynamic susceptibility by the fluctuation-dissipationtheorem34

�� = ��C�� . �27�

Therefore, from Eqs. �26� and �27�,

�s = �0

�

��d�/�� . �28�

Equation �28� applies to equilibrium reorganization char-acterized by an infinite observation window. A finite obser-vation window �obs cuts the frequencies below �obs=�obs

−1 ,thus resulting in the nonergodic reorganization energy givenby Eq. �7�. Equation �7� is merely a definition imposing astepwise filter on the frequency integral in Eq. �28�. It is thegoal of the present paper to show that Eq. �7� describes the
results of both the computer and laboratory experiment. In


this section, we outline the connection between the dynamicStokes shift susceptibility �� and microscopic propertiesof polar liquids.

An optical transition, instantaneous on the time scale ofnuclear fluctuations creates the variation of the electric fieldof the solute in the solvent �E0�r��t�, where ��t� is a stepfunction. This field perturbation results in the dipolar polar-ization in the solvent related to the solute field through thelinear response function ��r ,r� , t�

P�r,t� = �−�

t

��t��dt�� r,r�,t − t��E0�r��dr�. �29�

The time-dependent solute-solvent interaction energy is then

e0s�t� = −� P�r,t� · �E0�r�dr . �30�

Its Laplace transform then becomes

e0s�s� = − s−1F�s� , �31�

where

F�s� =� dk1dk2

�2�6 �E0�k1� · ��k1,k2,s� · �E0�− k2� . �32�

Here, �E0�k� is the Fourier transform of the solute fieldtaken over the volume outside the repulsive core of the sol-ute and the solvation response function ��k1 ,k2 ,s� is givenin wave-vector–Laplace space �k ,s . Then, from Eqs. �15�and �31� the dynamic susceptibility in Eqs. �7� and �28� is

�� = F��− i�� , �33�

where F��z� denotes the imaginary part of F�z� obtained byanalytic continuation of the real function F�s�.

A microscopic theory for calculating the Laplace re-sponse function for solutes of atomistic structure dissolved inpolar solvents has been recently proposed by one of us.57 Inshort, this procedure provides a connection between��k1 ,k2 ,s� and the dipolar susceptibility of the polar solvent�s�k ,s�. Inverted space for isotropic liquids has axial sym-metry set up by the wave vector. The second-rank tensor ofsolvent susceptibility is then fully characterized by two pro-jections, longitudinal �L� and transverse �T� as follows:

sL�k,s� = k · �s�k,s� · k ,

�34�s

T�k,s� = 12 �Tr��s�k,s�� − s

L�k,s�� ,

where k=k /k and ”Tr” refers to the trace over the Cartesiancomponents of a second-rank tensor. These projections canbe obtained in terms of the frequency-dependent dielectricconstant ��s� and the high-frequency dielectric constant��:57,58

sL,T�k,s� = s

L,T�0,0��SL,T�0�SL,T�k�

+1

1 + p��k��2

sL,T�0,0� − s

L,T�0,s�s

L,T�0,s� �−1

. �35�

L,T
In Eq. �35�, S �k� are the structure factors of the solvent

dipolar polarization.43,59 They are calculated according to theformalism based on the mean-spherical approximationfor dipolar liquids reparametrized to fit the results ofsimulations.43 Finally, the susceptibilities s

L,T�0,0� at �k ,s =0 in Eq. �35� are expressed in terms of static, �s=�s�0�, andhigh-frequency, ��, dielectric constants.57The parameter

p� = DT/�2�2DR� �36�

is given by the ratio of the translational, DT, and rotational,DR, diffusion coefficients of the solvent and the diameter ofthe solvent molecules �. This parameter quantifies the effectof translational diffusion on polarization dynamics.60,61

III. SIMULATION RESULTS

A. Simulations in SPC/E water

Figure 3�a� shows the solvent reorganization energy ofp-nitroaniline �Fig. 1� in extended simple point charge�SPC/E� water calculated according to Eq. �17� from con-figurations produced by molecular dynamics �MD� simula-tions �Appendix A�. The difference solute-solvent potential�U0s in Eq. �16� is taken between the charge-transfer �S4�and ground �S0� electronic states.62 The average is performedover the simulation trajectories generated for the solute in theground state. The circles indicate constant-volume simula-tions and the squares refer to simulations at constant pressureof P=1 atm. The reorganization energy increases with low-ering temperature in the high-temperature liquid43 and thenturns down and drops sharply at T*�219 K. Shown in Figs.

FIG. 3. Reorganization energy �a�, heat capacity �b�, and dielectric suscep-tibility �c� vs T. In �a�, the circles refer to NVE simulations at =0.997 cm3/g, the squares indicate the NPT simulations at P=1 atm, andthe triangles refer to the fast Gaussian component of the solvent reorgani-zation energy, �G. The dotted lines are drawn to guide the eye.

3�b� and 3�c� are also the heat capacity,



cV/kB = 3/2 + �2��E�2�obs/N �37�

and dielectric susceptibility

P = ��/V��M�2�obs �38�

of SPC/E water �E is the total energy and M is the totaldipole moment of a sample of liquid with volume V contain-ing N molecules�. All these properties, given as second cu-mulants of certain physical properties �cf. Eqs. �2�, �37�, and�38��, show clear similarities in their temperature depen-dence. The decay of cV and P at T�T* should be under-stood as a kinetic transition occurring at the crossing of therelaxation times given by the peaks of cV�� and P�� withthe observation time of the simulation experiment �1 ns�.Similarly, the dip in the reorganization free energy is relatedto the crossing of the slow relaxation time of the Stokes shiftdynamics with the observation time of the computer experi-ment.

The equilibrium Stokes shift correlation function C�t�from MD simulations is biphasic, typically for solvation dy-namics in soft media including polar solvents63 andproteins14,15 �Fig. 4�. The correlation function is composed of

FIG. 4. Equilibrium Stokes shift correlation function � Eq. �16�� of ground-state p-nitroaniline in SPC/E water at temperatures specified in the plot. Thedotted line indicates the fit to Eq. �39�.

FIG. 5. Imaginary part of the Stokes shift susceptibility of p-nitroaniline �a�and dielectric loss of SPC/E water �b�. The data are collected at the tem-
peratures indicated in the plot.

a fast Gaussian component and a slow exponential decaywith corresponding two peaks seen in the imaginary part ofthe Stokes shift susceptibility �� Fig. 5�a��. The fre-quency Fourier transform C�� is obtained from C�t� fittedto a biphasic form containing the Gaussian �G� and stretchedexponential �E� parts

C�t� = AGe−�t/�G�2+ �1 − AG�e−�t/�E��

. �39�

The stretching exponent � obtained from the fit is equal toone above 272 K and starts to drop below this temperaturereaching the value of 0.34 at 219 K.

The high-frequency Gaussian peak is caused by the fastinertial dynamics of the solvent molecules around the solute.This component is absent in the polarization dynamics of thepure solvent, as is seen from the comparison of �� Fig.5�a�� with the solvent dielectric loss function �� Fig.5�b��. The Gaussian relaxation time63 �G�70 fs is weaklytemperature dependent and is approximately given by a lin-ear function of T �K�: ln��G�T� /ps�=−1.34−0.0042�T,220�T�509 K �Fig. 6�a��. In contrast, the exponential re-laxation time �E�T� increases sharply with lowering T andcan be approximated by the Vogel-Fulcher �VF� law

ln��T�/ps� = �0 + D/�T − T0� , �40�

with �0=−3.73, D=447 K, and T0=161 K �220�T�509 K�. The Gaussian reorganization component,�G=AG�s, turns out to be weakly dependent on temperature�Fig. 3�a�, triangles�. The combination of the weak tempera-ture dependence of �G with ultrafast relaxation of the Gauss-ian component of C�t� clearly indicates that it is the slow

FIG. 6. �a� Relaxation times �G, �E �solvation dynamics, Eq. �39��, �1 �one-particle orientational relaxation, Eq. �41��, and �D �Debye dielectric relax-ation� vs T. The solid lines indicate the VF fits. The closed diamonds refer to�constant pressure� experimental dielectric relaxation of supercooled water�Ref. 66� and the half-open triangles refer to the experimental data from Ref.68. �b� Diffusion coefficient of SPC/E water from current simulations�points� compared with experimental self-diffusivity of supercooled water�solid line� given by the VF fit: ln�D�s /m2�=−15−892/ �T−118�. �Ref.69�. The dashed and dash-dotted lines refer to the power-law and VF fits ofthe simulation data, respectively.

exponential component of C�t� that becomes dynamically ar-



rested at the transition to nonergodicity. The drop of ��obs�at T�T* is thus equal to �E= �1−AG��s.

The frequency-dependent dielectric constant �� wasobtained from simulations, as described in Appendix B. Thevariation of �� with lowering temperature �Fig. 5�b�� ischaracteristic of many polar glass formers64,65 and super-cooled water.66,67 In particular, the peak �max of the dielectricloss �� shifts to lower frequencies with cooling the sol-vent. The corresponding Debye relaxation time, �D=1/�max

�diamonds in Fig. 6�a��, compares well to experimentallyreported values66,68 �closed diamonds and half-open trianglesin Fig. 6�a��. A similarly good agreement was reported pre-viously by Rønne et al.68 for MD simulations in a narrowertemperature range �271.5�T�368.15� on a smaller system�216 SPC/E molecules, 4 ns runs with the time step of 2 fs�.The temperature dependence of �D�T� is given by the VF law�Eq. �40�� with �0=−2.35, D=961 K, and T0=88.2 K.

The nanosecond observation window of our simulationsputs the turnover temperature T* in the range of temperaturesat which properties of glass formers undergo some dramaticchanges. This is seen already from the similarity of the tem-perature dependencies of solvation energies and susceptibili-ties of the pure solvent in Fig. 3. An additional insight intothe underlying dynamics can be gained from the comparisonof the single-particle rotational dynamics with self-diffusivity.

FIG. 7. One-particle rotational correlation function of SPC/E water �Eq.�41��. The dots refer to the fit to Eq. �41�. The curves �from down up� markthe following temperatures: 507, 442, 420, 390, 359, 337, 301, 272, 240,219, 207, 198, and 190 K.


Rotational dynamics is reflected by the single-particlecorrelation function of the molecular dipole vector m�t��Fig. 7�,

C1�t� = �m�t� · m�0��/�m�0�2� � e−t/�1�T�. �41�

The relaxation time �1�T� for C1�t� shows a non-Arrheniustemperature dependence �Fig. 6�a�� with the VF parameters�Eq. �40��: �0=−2.15, D=578 K, and T0=137 K. Our resultsfor �1�T� are consistent with the earlier reports by Sciortinoet al.70 obtained for 216 SPC/E water molecules.

Figure 8 shows a plot of the evolution of the mean-squared displacement �r2�t�� for different temperatures.�r2�t�� is a straight line during the first 400 ps suggesting thediffusive behavior. The translational self-diffusivity is thenobtained from the Einstein relation71

DT = �r2�t��/6t . �42�

The self-diffusion coefficient �Fig. 6�b�� can be fitted by boththe power-law function �ln�DT�T��s /m2�=−19+2.5 ln�T /181−1�� and VF function �ln�DT�T��s /m2�=−16.5−540/ �T−136��, with the latter providing a betterglobal fit. The power law has been previously reported72–75

to give poor fits for temperatures below the critical tempera-ture Tc of the mode-coupling theory �MCT�,76 while the VFformula sometimes fails at T�Tc where the power lawapplies.77 The diffusivity data from simulations compare fa-vorably with the experimental self-diffusivity of supercooledwater reported in Ref. 69 �Fig. 6�b��.

The temperature T* at which �s�T� and solvent suscepti-bilities start to drop �Fig. 3� also marks the onset of theseparation of translational and rotational diffusions signalingthe breakdown of the Stokes-Einstein relation �Fig. 9�77–79

DT�T� � �1�T� � const. �43�

Indeed, the product of the diffusion coefficient and the rota-tional relaxation time is approximately constant down to thetemperature Tc�207 K.80 If 165 K is adopted as the glasstransition temperature for water,81 the onset of translationalenhancement falls in the range of 1.2–1.3Tg found in labo-ratory experiment.78 Apart from the MCT critical tempera-ture, the separation of diffusivity and rotational relaxationcan be related to the crossover temperature of the scalingof Rössler et al.,82 the bifurcation of the � and �relaxations,79and the Stickel TB temperature.75

FIG. 8. Mean squared displacement for SPC/E watermolecules over the first 400 ps of MD runs at the tem-peratures indicated in the plot.



The rise of DT�1 is commonly explained by either staticspatial83–86 or dynamic87 heterogeneity. In the frustration-limited domain picture of Tarjus and Kivelson83 andKivelson and Tarjus84 the turning temperatures in Figs. 3 and9 can be associated with the onset of domain formation. Thisinterpretation is questionable, however, given the small sizeof our simulation box which cannot incorporate mesoscopicdomains. Note also that no discontinuous change in pair dis-tribution functions is seen at T*, except the commonlyobserved70,88 sharpening of solvation shell peaks.

The results of our simulations better fit the picture ofspatially heterogeneous dynamics which assumes the pres-ence in a supercooled liquid of groups of mobilemolecules.87,89,90 These groups may represent clusters of hy-drogen bond defects in SPC/E water,90 chains of mobile par-ticles in monoatomic Lennard-Jones �LJ� fluids,89 or someother structures. The collective motion of such mobile het-erogeneities provides the enhancement of translational diffu-sion correlated with the increase of the non-Gaussianparameter88–90 �Fig. 10�

�2�t� = 3�r4�t��/5�r2�t��2 − 1. �44�

The maximum value �2�t*� �Figs. 9 and 10� is reached attime t*. Both �2�t*� and t* grow with cooling, indicatingincreasing dynamic heterogeneity of molecular translations.On the other hand, dielectric loss data show no heterogeneitysuggesting a more homogeneous distribution of molecularrotations.

FIG. 9. The product of the translational diffusion coefficient and the rota-tional one-particle relaxation time �in Å2, circles� and the maximum of thenon-Gaussian translational diffusion parameter �squares, Eq. �44�� forSPC/E water.

FIG. 10. Non-Gaussian parameter �Eq. �44�� of SPC/E water at different
temperatures indicated on the plot.

The width of the dielectric spectrum does not changewith changing temperature and the dielectric loss data at dif-ferent temperatures can be superimposed on one mastercurve by proper rescaling �Fig. 11�. The increase in the widthof dielectric loss is commonly associated with liquidheterogeneity.90–92 The Debye dielectric loss shown in Fig.11 thus indicates that dielectric response is homogeneous.This conclusion agrees with the recent dielectric data byRichert et al.93 and Richert94 which do not show any increasein spatial heterogeneity in the region of temperaturesTg�T�1.2Tg where previous reports86 had indicated thebreakdown of the Stokes-Einstein relation. Although nobroadly accepted explanation of this phenomenon currentlyexists, the onset of translation/rotation decoupling is nor-mally associated with the critical MCT temperature.77,79 Wewill therefore resort to the interpretation of temperature T*, atwhich �s dips to its Gaussian component, as the point ofkinetic transition to nonergodicity with many features of thecritical temperature of an ideal glass transition predicted byMCT.76

The reorganization energy is essentially ergodic �equilib-rium� above T* with a negative temperature slope consistentwith the predictions of microscopic solvation theories.95,96 InFig. 12 we report the direct calculation of the constant-volume reorganization entropy from Eqs. �21� and �24�. Theenergy of solvent restructuring, given by the three-particlecorrelator in Eq. �24�, was calculated directly from MD tra-jectories in equilibrium with the solute in the ground elec-tronic state. We compare these results with NVT Monte Carlo�MC� simulations of a spherical point dipole in fluids ofdipolar hard spheres42 �line marked “D” in Fig. 12� and toNVT MC simulations of charge-transfer diatomic, D+−A−, indipolar hard-sphere liquids43 �marked “DA” in Fig. 12�. Inorder to minimize the effect of the solute size and shape, weconsider the reduced parameter

TSV

�s= 1 − �

��v0s�2�H0��v0s�2�

, �45�

which directly quantifies the relative contribution of the sol-vent restructuring to the reorganization entropy.

In the case of solvation of dipolar and diatomic solutes

FIG. 11. Scaled dielectric loss �� /��max� of SPC/E water vslog10�� /�max� calculated at 198, 219, 242, and 300 K; �max is the frequencymaximum of the dielectric loss function. The data points at different tem-peratures cannot be distinguished on the plot scale.

in hard-sphere dipolar solvents �D and DA in Fig. 12�, the



entropy SV was calculated by direct temperature differentia-tion of the simulation data obtained at different values ofdipolar strength

y = �4/9�� m2, �46�

where m is the solvent dipole moment and is the numberdensity. The thermodynamic state of dipolar hard-sphere flu-ids is fully characterized by y and the reduced density �3

�� is the hard-sphere diameter�. Therefore, the parameter ycan be varied by changing either the dipole moment �as itwas done in simulations� or temperature. SPC/E water doesnot share this universality, but the result of varying tempera-ture of SPC/E water is generally consistent, in terms of thereduced entropy in Eq. �45�, with the variation of the dipolemoment in hard-sphere solvents.

Some experimental data obtained from temperature-dependent Stokes shifts of optical lines97,98 are also shown inFig. 12 �closed points�. The most significant result of ourcalculations is the dramatic effect of the solvent restructuring�� in Eq. �21�� on the reorganization entropy. The observedentropy is the result of mutual cancellation of two large num-bers �the first and the second summands in Eq. �21�� with theresultant entropy as small as 15% of each component.

The approximately linear dependence of �s on tempera-ture at T�T* can be extrapolated into the range of T�T* inorder to determine the nonergodicity function

f��obs,T� = ��obs,T�/�s�T� . �47�

The same function was calculated from the Stokes shift dy-namics of p-nitroaniline according to Sec. II A. The input inthese calculations is the molecular structure and charge dis-tribution of the p-nitroaniline molecule in the ground andexcited states �Table III� and the frequency-dependent dielec-tric constant �� from MD simulations. The results of directMD calculations of f��obs ,T�, shown by points in Fig. 13,agree well with the microscopic theory �solid line�. No ad-

FIG. 12. SV,PT /�s vs y= �4 /9�� m2 for p-nitroaniline in SPC/E water�open circles� from NVE MD simulations in which the three-particle cor-relator in Eq. �24� was directly calculated from trajectories in equilibriumwith the ground-state solute. The dipolar density y was varied by changingthe temperature. Also shown are the results of NVT MC simulations for aspherical dipolar solute �Ref. 42� �solid line, “D”� and a contact diatomicsolute with opposite charges �Ref. 43� D+−A− �dashed line, “DA”� in dipo-lar hard-sphere solvents with varying dipole moment. The two lines areobtained by direct differentiation of the simulated reorganization energies asfunctions of y. The closed points �SP� correspond to the experimental resultsreported in Ref. 97 for diethyl ether �close square� and acetonitrile �closedtriangle� and in Ref. 98 for acetonitrile �closed diamond�.

justable parameters have been used in the calculations.


The dielectric continuum limit for the solvation responsefunction �k1 ,k2 ,s� can be obtained by setting k=0 in thesolvent dipolar susceptibility s�k ,s�. The complete solutionof the boundary Poisson problem representing the continuumresult can be obtained only numerically for a solute of com-plex molecular shape. However, it is generally expected that,at high values of the dielectric constant, the expansion of�� in 1 /�� should be of the general form

�� = a + b/�� + ¯ . �48�

This form, truncated after the first-order term in 1/��, im-mediately leads to the nonergodicity function in the form11

f��obs,T� = I��obs,T�/I�0,T� , �49�

where

I�x,T� = �x

� d�

�

��,T��,T��2

. �50�

For the Debye form of dielectric relaxation, Eqs. �49� and�50� lead to the following form of the nonergodicity func-tion:

f��obs,T� = �2/�cot−1��obs�L�T�� , �51�

where �L�T�= ��T� /�s�T��D�T� is the longitudinal dielec-tric relaxation time. This function, shown by the dashed linein Fig. 13, gives a reasonable estimate of the turning tem-perature for the nonergodicity function.

The dielectric relaxation of glass formers close to theglass transition temperature is often rather inadequately de-scribed by Debye relaxation. The experimental relaxationspectra are commonly better reproduced by the empiricalHavriliak-Negami �HN� equation99

��,T� = ��T� + ��T�/�1 + �i��T��, �52�

where, in addition to the temperature-dependent dielectricconstants and relaxation time, the powers � and � may de-pend on temperature. The use of the HN form in Eqs. �49�and �50� requires numerical integration �dash-dotted line inFig. 13�.

The examination of Fig. 13 shows that the continuum

FIG. 13. Nonergodicity function for p-nitroaniline �Fig. 1� in SPC/E water�Eq. �47�� from MD simulations �points� and from the analytical theory�lines�. The solid line is the result of the microscopic theory �Eqs. �32� and�33��, the dashed line refers to Eq. �51�, and the dash-dotted line refers tonumerical integration in Eqs. �49� and �50� with the Cole-Davidson dielec-tric function.

estimate �Eq. �49�� gives a sharper falloff of the nonergodic-



ity function with temperature compared to the microscopiccalculation. This result is expected since the continuum sol-vation dynamics are faster than the microscopic dynamics.100

Multiexponential dielectric relaxation, represented by theCole-Davidson law ��=0 and ��1 in Eq. �52��, makes thesolvation dynamics even faster, resulting to a shift of thefalloff temperature to lower values. This feature may explainthe failure of the present calculations to reproduce the low-temperature shoulder of the nonergodicity function �Fig. 13�.We have used Debye dielectric relaxation with the relaxationtime extrapolated from the simulation data obtained aboveT*. The experimental evidence suggests that HV and Cole-Davidson laws with � close to 0.5 should be more accurate atlow temperatures approaching the glass transition. Atemperature-dependent � exponent, lowering at T�T*, istherefore expected to bring the calculations in closer agree-ment with the simulations. Note also that the calculations forquinoxaline dye in Sec. IV below, which use �=0.48, per-form better in capturing the low-temperature component ofthe nonergodicity function.

An additional remark is relevant here. The calculationsshown in Fig. 13 were done for the fixed value of the param-eter p�=0.12 obtained from Eq. �36� in which �2DR�−1=�1

�Ref. 101� and DT�1 from the Stokes-Einstein plateau havebeen used. The parameter p� starts to grow below T*, whichgenerally means faster solvation dynamics producing a shal-lower shoulder of the nonergodicity function at low tempera-tures. Unfortunately, our simulation data are insufficient tomodel this effect properly.

B. Simulations in acetonitrile

Calculations of the mean-squared fluctuation of the

FIG. 14. Solvent reorganization energy of p-nitroaniline in acetonitrile �a�.The closed circles refer to the solute ground state ��s1 in Eq. �2�� and theopen circles refer to the solute excited state ��s2 in Eq.�2��. The dotted linesconnect the simulation points. The nonergodic function � Eq. �47�� is shownin �b�. The solid line is the result of the microscopic theory calculations� Eqs. �32� and �33��.

solute-solvent potential difference �Eq. �17�� on configura-


tions of acetonitrile in equilibrium with p-nitroaniline in theground and excited electronic states are shown in Fig. 14�a�.In the equilibrium region, the reorganization energy �s in-creases with lowering temperature resulting in a positive re-organization entropy, in accord with the data for SPC/E wa-ter. It reaches the nonergodic drop at T*�130 K. Note thatthe reorganization energies calculated in equilibrium with theground-state and excited-state p-nitroaniline are almostequal, in support of Eq. �3�. The dielectric data for acetoni-trile were obtained from separate simulations of pure aceto-nitrile resulting in the Debye relaxation time approximatedby the VF law �Eq. �40�� with the parameters �0=−7.27,D=362.8 K, and T0=49 K. These data, combined withthe parameters of the force-field acetonitrile, m=4.146 D,��=1, �=4.141 Å, p�=0.05, and �s�T�=38.212−0.043�Tfrom our simulations, were used in the microscopic calcula-tion of the nonergodicity function shown by the solid line inFig. 14�b�. As in the case of SPC/E water, the theory repro-duces well the position of the turning temperature.

IV. COMPARISON TO EXPERIMENT

A. Stokes shift

Figure 15 shows the temperature dependence of theStokes shift obtained in the laboratory experiment on twocharge-transfer chromophores, oxazine-4 �circles�20 and1-phenyl-4-��5-cyano-1-naphtyl�methylene� �complex 1 inFig. 1, squares�,22 both dissolved in low-temperature2-methyl-tetrahydrofuran �MTHF�. The emission frequencyrecorded experimentally first shifts to the red with loweringtemperature, then sharply shifts to the blue and levels off atthe lowest recorded temperatures. The solvent-inducedStokes shift is formed by subtracting the temperature-dependent emission energy from this low-temperature level;the reorganization energy is then calculated from Eq. �18�.The comparison to our computer experiments shown in Figs.3�a� and 14�a� is instructive. In all cases, the equilibriumreorganization energy increases with lowering temperature,thus resulting in a positive reorganization entropy, in accordwith Fig. 12. The sharp drop occurs when nonergodicity is

FIG. 15. ��obs� vs T for oxazine-4 �circles� �Ref. 20� and 1-phenyl-4-��5-cyano-1-naphtyl�methylene� �squares, complex 1 in Fig. 1� �Ref. 22� inMTHF. The data points represented by circles were multiplied by a factor of5 to bring both sets of data to a comparable scale of values. The dotted linesconnect the experimental points.

reached. Here, we use our theory for the Stokes shift dy-



namic susceptibility to see if the turning point observed inthe experiment can be reproduced by the calculations.

The nonergodicity function is calculated for complex 1and quinoxaline chromophores �Fig. 1�. Both molecules aredissolved in low-temperature MTHF, but the measurementsare done on distinctly different time scales. �obs=8.7 ns isset up by the lifetime of complex 1 �Ref. 22�, whilethe triplet state of quinoxaline has a much longer lifetime;�obs�1–2 ms is fixed by the instrument.21 Correspondingly,the onset of nonergodicity is marked by a higher temperaturefor complex 1 �Fig. 16�a�� than for quinoxaline �Fig. 16�b��.

Dielectric relaxation slows down with lowering tempera-ture. In the temperature range close to the turning tempera-ture T*, �� is well described by the Cole-Davidson lawwith �=0 and ��0.48 in Eq. �52�. The dielectric constantsare ��=2.99 and �s=19.2, and the dielectric relaxation time��T� fits the VF law �Eq. �40�� with �0=−9.0, D=792.5 K,and T0=71.7 K.102

Experimental dielectric relaxation data were used as in-put for the calculation of the Stokes shift correlation functionaccording to the procedure described in Sec. II B. MD simu-lations of the Stokes shift dynamics of complex 1 were doneby Brown et al.103 who also calculated the excited-state ge-ometry of complex 1. Atomic coordinates and charges, aswell as atomic radii used in our calculations were taken fromRef. 103. The electronic structure and equilibrium geometryof quinoxaline in the ground singlet and exited triplet stateswere obtained from the density functional calculations�B3LYP/6–31G�3d ,3p��, as described in Ref. 57 �see Table

FIG. 16. Stokes shift of complex 1 �Ref. 22� �a� and of quinoxaline �Ref.21� �b� in MTHF. The points represent the experimental results and the linesrefer to theoretical calculations. The solid lines refer to the microscopiccalculation according to Eqs. �32� and �33� with the parameter p�=0.03 �Eq.�36�� and the dash-dotted line refers to the same calculation with p�=0.12.The dashed line is obtained by numerical integration of the Cole-Davidsonfrequency-dependent dielectric constant in Eqs. �49� and �50� and the dottedlines connects the experimental points; Tg=90.7 K, �obs=1.15�108 s−1 �a�,�obs=103 s−1 �b�.

II in Ref. 57 for atomic coordinates and charges�. The mo-


lecular structural data were used to calculate the difference

electric field �E0 and its Fourier transform �E�k�. All themicroscopic calculations shown here have been done on thecubic grid of the size 2563, as described in Ref. 43. Testcalculations on the grid of 5123 points were done for com-plex 2 to check the independence of the results of the gridstep.

The Stokes shift correlation function and the nonergod-icity function are obtained from the microscopic solvationalgorithm with no adjustable parameters �solid lines in Fig.16�. The parameter p� �Eq. �36�� is determined through therelation p�= I /2M�2=0.03, where I is the moment of inertia,M is the mass, and �=5.102 Å is the diameter of the MTHFmolecule.57 This number, used in the calculations shown bythe solid lines, agrees well with p� calculated at T�120 Kfrom the relation p�=DT�D /�2, where DT�D has been re-ported in Ref. 104. The product DT�D starts to rise at lowertemperatures T�120 K, resulting in p��0.12 at T=106 K.The dash-dotted line in Fig. 16�b� was obtained with thislatter magnitude of p�. The rotational diffusion coefficient inEq. �36� reflects one-particle dynamics, whereas �D reflectssome collective features of dipolar correlations in the liquid.It is, therefore, not clear if p� can be calculated by using �D.However, one can construct a temperature-dependent func-tion p��T� which would fit the experiment, including the low-temperature shoulder which is now missing from our calcu-lations.

The dashed line in Fig. 16�b� is the result of using Eq.�51� with the dielectric constant from the experiment. Themicroscopic formulation performs noticeably better than thecontinuum model. The difference between the solid anddashed lines in Fig. 16�b� thus mostly reflects the effect ofdipolar correlations in MTHF on the solvation dynamics.

B. Electron transfer rates

Given the success of the theory in reproducing the tem-perature dependence of the Stokes shift in low-temperatureMTHF, we want to study if the rates of electron transfer inviscous solvents need the involvement of the nonergodic de-scription of solvent reorganization. With this goal we havestudied the kinetic data reported by Wasielewski and co-workers32,33 for the charge-separation reaction in the donor-bridge-acceptor molecule �complex 2� shown in Fig. 1. Inthose experiments, photoexcitation of the donor moiety isfollowed by charge separation, D*−A→D+−A−. The datawere collected in a broad range of temperatures 110�T�300 K in MTHF. A peculiar feature of the observed kinet-ics is the non-Arrhenius temperature dependence of thecharge-separation rate passing through a maximum around210 K �Fig. 17, points�. Charge recombination to the initialground state, D+−A−→D−A, has also been reported.33 We,however, do not perform this analysis since the geometry ofcomplex 2 can reliably be optimized only in the neutral state.We will assume that photoexcitation does not significantlyaffect the geometry. Charge separation is most likely fol-lowed by a conformational change of the molecule, which ishard to describe by standard quantum calculations.

The description of the electron transfer rates in viscous



solvents requires solving a self-consistent equation for thereaction rate �Eq. �6��.11 The reason is the dependence of thefree energy of nuclear solvation on the observation window,which makes the activation barrier sensitive to the magnitudeof the rate. The reorganization energy is then obtained bysetting the observation frequency equal to the rate of electrontransfer, �obs=kET in Eq. �7�.

We will use the semiclassical form for the electron trans-fer rate resulting in the following self-consistent equation:

kET = �2VET2 /��e−S

n=0

�

�Sn/n!�Gn�kET� , �53�

where S=�v /��v is the Huang-Rhys factor and �v is thereorganization energy of intramolecular vibrations of thedonor-acceptor complex characterized by a single effectivefrequency �v. In Eq. �53�, Gn�kET� is the Gaussian inhomo-geneous distribution of energies of a single vibronic transi-tion corresponding to n quanta of vibrational excitation ofthe final electron transfer state,

Gn�kET� = �2��kET�2�−1/2 exp�−�En

2

2��kET�2� . �54�

The Gaussian width � is formed by classical nuclear modesof the solute and the solvent coupled to the transferred elec-tron,

�2�kET� = 2kBT��i + ��kET,T�� . �55�

Here, the reorganization energy �i incorporates fast ballisticmodes of the solvent and classical vibrations and rotations ofthe solute. The component ��kET,T�, as above, reflectsnuclear dipolar polarization of the solvent which may be-come dynamically arrested on the time scale of the reaction.

We will assume that the electrostatic coupling between

FIG. 17. Rates of charge separation vs T for complex 2 �a� and correspond-ing nonergodicity functions �b�. The points in �a� refer to the experimentalresults �Ref. 33�; the line is the fit to Eq. �53� with the fitting parameters:VET=758 cm−1, �Eg=1.63 eV, and �i=0.18 eV. For the vibrational mode�v=0.3 eV and ��v=1500cm−1 �Ref. 33� have been adopted.

the solvent and complex 2 in the ground and photoexcited


states is much weaker than in the charge-transfer state. Thisallows us to write the vertical transition energy �En in Eq.�54� in the following form:

�En = �Eg + �Eind + n��v + �i. �56�

In Eq. �56�, �Eg is the gas-phase 0-0 transition energy and�Eind is the energy of nonpolar solvation normally composedof contributions from dispersion and induction forces.105 Thedispersion energy is related to the change in the solute po-larizability and, in the absence of a substantial change, theinduction stabilization is the main contribution to nonpolarsolvation, as, in fact, is assumed in Eq. �56�. The energy ofinduction solvation can be estimated from the followingrelation:11,106

�Eind = − 3�� − 1

�� + 2�E , �57�

where �E is the difference in electrostatic energies of thedonor-acceptor complex in two charge-transfer states

�E = �8�−1��

�E022 − E01

2 �dr . �58�

In Eq. �58�, � is the volume occupied by the solvent. Oncethe electric field of the solute in the initial neutral state isneglected, only E02 contributes to the integral. In the populartwo-sphere representation of the donor-acceptor complex,

�E = �e2/2��RD−1 + RA

−1 − 2RDA−1 � , �59�

where the donor and acceptor are represented by two spheresof radius RD and RA, respectively, separated by the distanceRDA. Our present calculations do not rely on Eq. �59� andinstead use the complete atomic geometry of complex 2.

Geometry for complex 2 was optimized on the AM1level using GAUSSIAN 03.107 The calculation of the chargedistribution of the neutral D−A state was done by the DFT/CHELP �charges from electrostatic potentials� method usingB3LYP/6-31G level of theory. The atomic charges of thepositively charged TET donor and the negatively charged PIacceptor �Fig. 1� were obtained from separate B3LYP/6-31Gcalculations with the geometry optimized for the entire mol-ecule. The unsaturated bonds resulting from cutting off thedonor and acceptor moieties from the donor-bridge-acceptormolecule were terminated with hydrogens. The atomiccharges obtained for D−A and D+−A− configurations aresubtracted from each other to give the set of differencecharges �qj used to calculate the difference electric field

�E0 and its Fourier transform �E0 taken over the solventvolume outside the repulsive core of the solute. As previ-ously for the calculations involving p-nitroaniline and qui-noxaline, the solvent radius � /2 was added to the soluteatoms exposed to the solvent. This calculation results in�E=1.03 eV in Eq. �58�. The difference electric field is thenused to calculate the Laplace transform F�s� for which ex-perimental dielectric relaxation data are required.

The dielectric data were obtained from Richert’sgroup.102,108,109 Since the experimental electron transfer ki-netics were recorded over a broader range of temperatures
�110�T�300 K� compared to the results for quinoxaline


�83�T�107.5 K�, a wider range of dielectric parameters isrequired. Experimental static dielectric constant in MTHF iswell approximated by the following empirical function �Ref.108�: �s�T�=1189.3/T0.895−2; ��T� was calculated fromdensity according to the Clausius-Mossotti equation. The di-electric relaxation data were fitted to the HN equation109 �Eq.�51�� resulting in all parameters depending on temperature.However, ��0.92 is approximately constant and the datafor � can be empirically fitted to the following function:��T�=0.56+0.2/ �1+exp�0.3� �112−T��. Also, the relax-ation time follows the VF law �Eq. �40�� with �0=−7.45,D=370.8 K, and T0=81.36 K.

The calculations of the Stokes shift correlation functionwith the dielectric parameters for MTHF and the structure ofthe donor-acceptor complex from quantum calculations donot involve any adjustable parameters. It results in a biexpo-nential decay of S�t� and a reorganization energy with anegative temperature slope96 which can be fitted to the fol-lowing relation: �s�T�=1.141−0.002 08�T eV. The noner-godic reorganization energy then becomes

��kET,T� = �s�T� i=1,2

�2ai/�cot−1�bikET��T�� , �60�

where a1+a2=1 and the constants bi are found from the fit ofFs�s� to the Laplace transform of a biexponential function.

The most puzzling result of experiment is the bell-shaped temperature dependence of the charge-separation rateimplying the activation enthalpy changing from positive atlow temperatures to negative at high temperatures. This non-traditional behavior was explained by conformational dy-namics of the bridge unit in Ref. 33. Here we show thatexperimental data can be explained by invoking the tempera-ture dependence of the activation energy. We first providesome semiquantitative arguments and then give the completequantitative analysis.

The activation enthalpy follows from van Hoff equation

Hact = ��Fact

��

P, �61�

in which we accept the classical activation energy

Fact = �E02/4�0, �62�

where �0=�i+�s and �E0 is given by Eq. �56� with n=0.From Eqs. �61� and �62� one gets

Hact = Fact1 + � ln �0

� ln T�

P� − T ��Eind

�T�

P

�E0

2�0. �63�

The logarithmic derivative of �0 over T is negative, but isnormally less than one �equal to −0.62 for complex 2�, mak-ing the first term in Eq. �63� positive. The derivative of theinduction energy over temperature is positive,

T ��Eind

�T�

P= �p

�� − 1

�� + 2�E � 0, �64�

where �p�0 is the constant-pressure expansivity. Since thefirst term is quadratic in the energy gap �E0 and the second
term is linear in �E0, the enthalpy becomes negative for a

sufficiently small �E0. Now we show that the experimentaldata can actually be fitted with the current theory.

We need to take into account the average over the het-erogeneous distribution of reaction rates, Eq. �8�. This can bedone by adding a stochastic variable x to the vertical energygap �Eg→�Eg+x and averaging Eq. �53� over Gaussianfluctuations of x characterized by the heterogeneous widthgiven by Eq. �9�. It is immediately clear that this procedurewill result in Eq. �53� in which the Gaussian terms are char-acterized by the full equilibrium width which follows fromEq. �55� by replacing the nonergodic reorganization energy��kET,T� with the full equilibrium reorganization energy �s.The simple conclusion of this derivation is that the width ofclassical Gaussians making the Franck-Condon envelope isnot affected by nonergodicity for measurements done on en-sembles of donor-acceptor complexes. It is the vertical gapof predominantly charge-recombination reactions that is af-fected by nonergodicity. Note, however, that the actual het-erogeneous distribution of energy gaps depends on the prepa-ration of the initial ET state and only a part of theheterogeneous width �het might be realized in a particularexperiment.

A few remarks are relevant here. Experimental data re-port Stokes shift, whereas our calculations are done for thesecond cumulant of the solute-solvent potential, i.e., for thereorganization energy. Therefore, given a good agreementbetween experiment and calculations in Fig. 16, the linearresponse relation between the reorganization energy and theStokes shift holds for the homogeneous, nonshaded part ofthe phase space in Fig. 2 for the entire range of temperatures,above and below T*. On the other hand, the reorganizationenergy measured from solvent-induced spectral width,�s=��2 /2, for ensembles of donor-acceptor molecules cor-responds to its equilibrium limit. Therefore, the width andStokes shift data will deviate from Eq. �18� below tempera-ture T*.

We used Eq. �53� with the equilibrium �s from our cal-culations instead of ��kET,T� in Eq. �55� to fit the experimen-tal charge-separation rates. VET, �i, and �Eg are consideredas fitting parameters. The results of the fit are shown in Fig.17�a�. We note that the non-Arrhenius downward turn of therate cannot be reproduced without accounting for the tem-perature dependence of the induction solvation energy andthe solvent reorganization energy. The overall agreement be-tween theoretical modeling and experiment is very good. Wealso note that we do not attach any physical meaning to theET matrix element VET=758 cm−1 coming from the fit. Re-actions in the inverted region are often accelerated by theinfluence of intramolecular modes of the solute and ballisticmodes of the solvent110,111,10 and this value of VET should beconsidered as merely a fitting parameter reproducing theoverall reaction rate.

The analysis of the experimental data suggests that thelow-temperature part of the reaction is significantly in thenonergodic regime: nuclear solvent reorganization, at leastpartially, freezes in over the entire range of temperatures upto 210 K �Fig. 17�b��. Note that the measurements of the ratedo not provide any direct indication of this fact because of
the average over the heterogeneous distribution of the donor-


acceptor complexes. However, the energy gap for charge re-combination is affected by nonergodic effects and they canbe recorded from the energy gap law for the recombinationreactions, as is illustrated in Fig. 18 for the back transitionD+−A−→D*−A. When the reaction slows down by shiftingaway from the top of the inverted parabola, it may switchfrom nonergodic to ergodic behavior rather sharply as is seenin the exthothermic part of the energy gap law. This mayresult in discontinuity in the energy gap law of charge-transfer reactions.

V. CONCLUDING REMARKS

Classical theories of electron transfer reactions due toMarcus112 and Hush5 were developed to describe slow reac-tions in the microsecond time scale.113 The experimentaltechniques have made a leap of at least six orders of magni-tude since then with rates in the picosecond time scale rou-tinely reported for solution114,115 and photosynthetic116,117 re-actions. This dramatic change in experimental resolution hasshifted the observable time of ET down to the time scale ofnuclear fluctuations in liquid solutions. On the other hand,the range of media carrying ET reactions has broadened upto include viscous materials such as proteins,118 liquidcrystals,119 and glass forming liquids.33 All these new devel-opments call for rethinking of the assumptions basic to themodels used to interpret the data. The idea of the equilibriumdistribution of states of the reacting system, basic to thetransition-state theory, is one of such avenues which needs acloser reexamination for fast reactions and/or reactions inslowly relaxing solvents.11,25,27,120

The modeling of ET kinetics in the nonergodic regime,when transition-state description fails, requires the knowl-edge of how the solvation free energy changes with slowing

FIG. 18. �a� Rates of charge recombination, D+−A−→D*−A, at T=200 K�solid line� and 250 K �dashed line� vs the equilibrium free energy gap�F0=F0�D+−A−�−F0�D*−A� for the charge-separation reaction. �b� Noner-godic function �Eq. �47�� calculated at the same temperatures. Other param-eters related to the solute and the solvent are the same as in Fig. 17.

down solvent relaxation or, alternatively, narrowing the ob-


servation window. We have shown that this effect can beadequately described by a stepwise filter imposed on therange of frequencies contributing to the solvent reorganiza-tion energy �Eq. �7��. This approach has allowed us to use thealgorithm developed for the equilibrium Stokes shift dynam-ics to calculate nonergodic solvation energies. The theory hasbeen applied to our simulation data and to steady-stateStokes shifts measured experimentally. Further, we have ap-plied the self-consistent description of nonergodic electrontransfer reactions to experimental charge-transfer kinetics.Our calculations show that the low-temperature portion ofthe data requires nonergodic solvation energies. Our calcula-tions also predict a possibility of sharp changes in the energygap law of electron transfer reactions when they switch be-tween ergodic and nonergodic behaviors.

ACKNOWLEDGMENTS

This research was supported by the National ScienceFoundation �CHE-0304694�. The authors are grateful to Pro-fessor R. Richert for help with the experimental data andencouraging discussions and to Professor Ross Brown forstructural data on complex 1.

APPENDIX A: MODEL AND SIMULATION METHODS

1. Model

Two solvents, water and acetonitrile, have been used inMD simulations employing the DL�POLY package.121 The ex-tended simple point charge �SPC/E� model122 is adopted forwater �Table I� and the potential parameters by Böhmet al.123 were used for acetonitrile �Table II�. The SPC/Ewater molecule is represented by one O site and two H sitescarrying masses similar to those of respective atoms. Themolecular geometry is defined by O–H bond length of 1.0 Åand HOH angle of 109.47°. A positive charge +q is placed oneach H atom and a negative charge of −2q is placed on O.The interaction potential is composed of a short-range �6–12�Lennard-Jones �LJ� potential restricted to oxygen and theCoulomb potential between the partial charges. Acetonitrileis assumed to be rigid with parameters corresponding to theequilibrium bond lengths and angles124 of an isolated mol-ecule.

TABLE I. Water-water potential parameters for SPC/E model.

SPC/E �OO �Å� �OO �kJ/mol� Charge �e�

C �H2O� 3.169 0.6502 −0.8476H �H2O� 0.4238

TABLE II. Acetonitrile-acetonitrile interaction potential parameters �Ref.126�.

ACN � �Å� � �kJ/mol� Charge �e�

C �CH3� 3.00 0.417 71 −0.577C �CN� 3.40 0.417 72 0.488

H 2.20 0.083 48 0.201N 3.30 0.417 72 −0.514



The solute is a p-nitroaniline molecule. Charge-transferexcitation in this push-pull chromophore results in a changeof dipole moment of about 3.7 D. The solute-solvent andsolvent-solvent interactions between sites i and j are mod-eled by the sum of Coulomb and LJ potentials

�ij = 4�ij��ij

rij�12

− �ij

rij�6� +

qiqj

rij, �A1�

where �ij and �ij are the LJ potential parameters and rij is thedistance between the interacting sites with charges qi and qj.The LJ parameters for solute atoms are based on the OPLS�optimized parameters for liquid simulations�parametrization125 �Table III�. The solute-solvent cross inter-action parameters are obtained from the Lorentz-Berthelotcombination rules.71

The ground-state geometry and charge distribution ofp-nitroaniline were obtained using GAUSSIAN 03 �Ref. 107��MP2, 6–31+G*�. The geometry of p-nitroaniline fromx-ray experiments127 was used as a starting model for thegeometry optimization. The ground-state geometry was usedto calculate the ground-state and excited-state �CIS, configu-ration interaction singlets� atomic charges by fitting the elec-trostatic potential �Table III�. The ground-state, 7.18 D, andexcited-state, 10.88 D, dipoles confirm well to the resultsreported in the literature.62,128 The calculated atomic chargesare used in a 16-site model of p-nitroaniline with the poten-tial defined by Eq. �A1�.

2. Simulation protocol

p-nitroaniline in water. NVE and NPT molecular-dynamics �MD� simulations were done for a system com-posed of one solute �p-nitroaniline� and N=466 water mol-ecules, with periodic boundary conditions. All simulationsare carried out for a cubic simulation cell with the side lengthL=24.075 Å and the density of water 0.997 g/cm3 in therange of temperatures from supercooled region �50 K� to su-perheated region �509 K�.129–132 A spherical cutoff with the

TABLE III. Atomic coordinates, ground- �G� and exeter � and LJ energy �� of p-nitroaniline.

j Atom xj �Å� yj �Å� zj �Å

1 O 5.202 3.693 7.442 O 5.421 5.490 6.193 N�NH2� 10.161 1.971 3.914 N�NO2� 5.764 4.347 6.545 C 9.039 2.534 4.536 C 8.411 1.883 5.607 C 7.328 2.462 6.268 C 6.909 3.734 5.879 C 7.514 4.411 4.81

10 C 8.596 3.818 4.1611 H 8.753 0.895 5.9112 H 6.834 1.954 7.0813 H 7.164 5.397 4.5214 H 9.082 4.344 3.3415 H�NH2� 10.220 0.959 3.9716 H�NH2� 10.346 2.299 2.97

radius L /2=12.0 Å is used for the LJ interactions. The Cou-


lomb interactions are treated by the Ewald summationmethod133 which splits the sum over the periodic images ofthe simulation cell into a damped real space sum and a re-ciprocal space sum.134 Summation in real space is truncatedat L /2 �convergence parameter of 0.2346 Å−1� and summa-tion in reciprocal space involves approximately 2300 vectorswith their magnitudes limited by 5.0 Å−1. The length ofsimulations was 1.4 ns �400 ps equilibration� at T�220 Kand 6.8 ns �5.8 ns equilibration� at T�220 K with a timestep of 1 fs. The observation window for the calculation of�s was taken as 1 ns at the end of each simulation run.

In parallel to solvation simulations, simulations of thepure SPC/E water �N=466� at the same thermodynamic con-ditions have been done. For these simulations, equilibrationwas 400 ps long and production runs were 1.0 ns long at220–509 K. At lower temperatures, the equilibration andproduction times were 800 ps and 6.0 ns, respectively. Con-figurations were stored at the intervals of 0.1 ps during theproduction runs. We have also obtained satisfactory energyand momentum conservation �energy fluctuation less than 1in 105� throughout the production runs. Fluctuations of theaverage temperature were less than 5 K after equilibration ofthe system. Our simulations for pure SPC/E water are con-sistent with the results reported by Starr et al.135 for 8000

�E� state atomic charges, and LJ parameters �diam-

qj �G� qj �E� � �Å� � �kJ/mol�

−0.432 −0.478 2.47 1.610−0.430 −0.487 2.47 1.610−0.781 −0.699 3.25 0.711

0.544 0.415 3.03 1.0020.198 0.102 3.55 0.293

−0.053 0.099 3.55 0.293−0.459 −0.621 3.55 0.293

0.432 0.738 3.55 0.293−0.459 −0.619 3.55 0.293−0.052 0.098 3.55 0.293

0.146 0.134 3.55 0.2930.252 0.248 2.42 0.1250.250 0.247 2.42 0.1250.146 0.134 2.42 0.1250.350 0.346 0.00 0.000.349 0.345 0.00 0.00

FIG. 19. Mean squared fluctuation of the total dipole moment of SPC/E

cited-

�

7088071138336745

water vs the trajectory length in NVE MD simulations.



SPC/E water molecules. They report −52.15 �3 ns equilibra-tion� and −56.40 �4 ns equilibration� kJ/mol for the averageinteraction energy at 215 and 100 K, respectively. Our simu-lations give −51.68 and −56.10 kJ/mol at the same tempera-tures. The minor deviations may be attributed to slightlydifferent densities, 1.0046 �T=245 K� and 1.022�T=100 K� g/cm3 in Ref. 135 compared to the constant den-sity of 0.997 g/cm3 in our simulations.

p-nitroaniline in acetonitrile. NVE-MD simulations weredone for a system composed of one p-nitroaniline solute andN=250 molecules of acetonitrile, with periodic boundaryconditions. All simulations employed a cubic simulation cellwith the side length L=27.99 Å and the density of acetoni-trile 0.782 g/cm3. A spherical cutoff with the radiusL /2=13.995 Å is used for the LJ interactions. The Coulombinteractions are treated similarly to the case of water. Sum-mation in real space is truncated at L /2 �convergence param-eter of 0.2713 Å−1� and summation in reciprocal space in-volves approximately 2300 vectors with magnitudes limitedby 9.0 Å−1. Simulations were done in the range of tempera-tures from 500 to 225 K. The length of simulations was1.5 ns �500 ps equilibration� at T�400 K and 4.0 ns �800 psequilibration� at T�400 K with a time step of 1 fs. Theobservation window for the calculation of �s was taken as1 ns at the end of each simulation run.

APPENDIX B: DIELECTRIC CONSTANT

The dielectric properties of SPC/E water and acetonitrilehave been obtained from MD simulations based on the for-malism described previously in Refs. 136–141 for polar mo-lecular fluids. In particular, static dielectric properties anddielectric dispersion of water from simulations have beenpreviously reported in Refs. 136, 137, 139, and 142. Thestatic dielectric constant is calculated from rms fluctuationsof the overall dipole moment M�t� with the corrections forthe cutoff of electrostatic interactions according toNeumann.143 Figure 19 shows the average �M�0�2� versusthe length of MD simulation. A good convergence of thestatic dielectric constant has been achieved at all tempera-tures studied.

The dielectric spectrum is obtained from the autocorre-lation function

CM�t� = �Nm2�−1�M�t� · M�0�� B1�

calculated on MD trajectories �Fig. 20�. The correlation func-tion CM�t� was fitted to a multiexponential decay function

CM�t� = j=1

P

aje−t/�j �B2�

used to generate the Laplace-Fourier transform

CM�z� = �0

�

CM�t�e−iztdt . �B3�

This function is then used to calculate the dielectric spectrumaccording to the equation given by Neumann and

139,143
Omelyan,

��z� − 1

��z�=

9y�z�1 + 9y�z�

, �B4�

where

�z� = CM�0� − izCM�z� �B5�

and parameter y is defined by Eq. �46�.

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�2006�.



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Solvent reorganization of electron transitions in viscous solvents