Model system-bath Hamiltonian and nonadiabatic rate constants for proton-coupled electron transfer...

Model system-bath Hamiltonian and nonadiabatic rate constants for proton-coupled electron transfer at electrode-solution interfacesIrina Navrotskaya, Alexander V. Soudackov, and Sharon Hammes-Schiffer

Citation: J. Chem. Phys. 128, 244712 (2008); doi: 10.1063/1.2940203 View online: http://dx.doi.org/10.1063/1.2940203 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v128/i24 Published by the American Institute of Physics.

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Model system-bath Hamiltonian and nonadiabatic rateconstants for proton-coupled electron transferat electrode-solution interfaces

Irina Navrotskaya, Alexander V. Soudackov, and Sharon Hammes-SchifferaDepartment of Chemistry, 104 Chemistry Building, Pennsylvania State University,University Park, Pennsylvania 16802, USA

Received 16 April 2008; accepted 16 May 2008; published online 26 June 2008

An extension of the AndersonNewnsSchmickler model for electrochemical proton-coupledelectron transfer PCET is presented. This model describes reactions in which electron transferbetween a solute complex in solution and an electrode is coupled to proton transfer within the solutecomplex. The model Hamiltonian is derived in a basis of electron-proton vibronic states definedwithin a double adiabatic approximation for the electrons, transferring proton, and bath modes. Theinteraction term responsible for electronic transitions between the solute complex and the electrodedepends on the proton donor-acceptor vibrational mode within the solute complex. This modelHamiltonian is used to derive the anodic and cathodic rate constants for nonadiabaticelectrochemical PCET. The derivation is based on the master equations for the reduced densitymatrix of the electron-proton subsystem, which includes the electrons of the solute complex and theelectrode, as well as the transferring proton. The rate constant expressions differ from analogousexpressions for electrochemical electron transfer because of the summation over electron-protonvibronic states and the dependence of the couplings on the proton donor-acceptor vibrationalmotion. These differences lead to additional contributions to the total reorganization energy, anadditional exponential temperature-dependent prefactor, and a temperature-dependent term in theeffective activation energy that has different signs for the anodic and cathodic processes. This modelcan be generalized to describe both nonadiabatic and adiabatic electrochemical PCET reactions andprovides the framework for the inclusion of additional effects, such as the breaking and forming ofother chemical bonds. 2008 The American Physical Society. DOI: 10.1063/1.2940203

I. INTRODUCTION

Proton-coupled electron transfer PCET reactions formthe basis of a wide range of electrochemical processes. Un-derstanding the fundamental physical principles underlyingthese processes is technologically important for the design offuel cells and chemical sensors. The theoretical treatment ofelectrochemical PCET is challenging because of the quantummechanical behavior of the transferring proton, as well as theelectrons in the electrode and the solute complex, and thestrong coupling among the electrons, proton, and solvent.The modulation of the vibronic coupling by the protondonor-acceptor vibrational mode leads to additional compli-cations. Although substantial efforts have been directed to-ward the development of theoretical models for electro-chemical electron transfer ET,125 less effort has beendirected toward electrochemical PCET.2634

One theoretical approach that has been used to studynonadiabatic electrochemical PCET is based on Fermisgolden rule formalism. This approach has also been appliedto electron and proton transfer, as well as homogeneousPCET.9,10,28,29,3544 Recently, we applied the golden rule for-malism to nonadiabatic electrochemical PCET reactions and

systematically derived a series of rate constant expressionsthat are valid in specified regimes.45 This approach leads to ageneral rate constant expression in terms of time correlationfunctions that can be calculated from molecular dynamicssimulations in conjunction with a realistic potential energysurface. Analytical expressions for the rate constants are ob-tained when the harmonic and high temperature approxima-tions are applied to the solvent and the proton donor-acceptorvibrational mode. The resulting expressions are similar tothose used previously by Costentin et al.28,29

An alternative theoretical approach for electrochemicalPCET is based on the AndersonNewns Hamiltonian origi-nally developed to study localized magnetic states inmetals46 and chemisorption.47 These models have been usedto describe ET between a redox center in solution and anelectrode in terms of an electronic subsystem linearlycoupled to a harmonic bath.57,1114 Schmickler andco-workers15,16,30,31 applied this type of model to ET reac-tions involving the breaking of a bond and to combined elec-tron and proton transfer reactions in electrochemical sys-tems. Their model of electrochemical PCET Refs. 30 and31 is based on the AndersonNewns Hamiltonian with theremoval of the electron repulsion terms and the addition ofterms describing the harmonic bath linearly coupled to theelectron and proton, the proton motion in an analytical one-dimensional proton potential, and the electron-proton inter-

aAuthor to whom correspondence should be addressed. Electronic mail:[email protected].

THE JOURNAL OF CHEMICAL PHYSICS 128, 244712 2008

0021-9606/2008/12824/244712/15/$23.00 2008 The American Physical Society128, 244712-1


action. The two-dimensional adiabatic potential energy sur-faces were generated for a model that includes two collectivesolvent coordinates, where one is coupled to the proton andone is coupled to the electron. Stochastic molecular dynam-ics simulations for the solvent coordinates were performedon these two-dimensional adiabatic potential energy sur-faces. In addition, the proton dynamics on these surfaces wasinvestigated.

In this paper, we propose an extension of the AndersonNewnsSchmickler Hamiltonian for electrochemical PCETreactions. This model describes reactions in which ET be-tween a solute complex in solution and an electrode iscoupled to proton transfer within the solute complex. Wederive this Hamiltonian in second quantization representa-tion in a basis of electron-proton vibronic states definedwithin a double adiabatic approximation for the electrons,the transferring proton, and the bath modes. We also includethe effects of the proton donor-acceptor vibrational modewithin the solute complex. An important feature of thismodel is that the interaction term responsible for electronictransitions between the solute complex and the electrode de-pends on the proton donor-acceptor vibrational mode. In ad-dition to developing this model Hamiltonian, we use it toderive the forward and backward rate constants for nonadia-batic electrochemical PCET. The derivation is based on themaster equations for the reduced density matrix of theelectron-proton subsystem, which includes the electrons ofthe solute complex and the electrode, as well as the transfer-ring proton. In contrast to the golden rule formalism, theextended AndersonNewnsSchmickler model developed inthe present paper can be used to describe adiabatic as well asnonadiabatic electrochemical PCET reactions and providesthe framework for the inclusion of additional effects, such asthe breaking and forming of other chemical bonds.

This paper is organized as follows. In Sec. II, we derivethe model Hamiltonian for electrochemical PCET. In Sec.III, we use this model vibronic Hamiltonian to derive expres-sions for the oxidation and reduction PCET rate constants.We also calculate the rate constants for a model system andanalyze the dependence of the rate constant on the frequencyof the proton donor-acceptor mode. Concluding remarks arepresented in Sec. IV.

II. MODEL HAMILTONIAN FOR ELECTROCHEMICALPCET

In this section, we develop a model Hamiltonian forelectrochemical PCET. In this model Hamiltonian, the react-ing molecule or hydrogen-bonded complex in solution is as-sumed to be located at a fixed distance from the surface ofthe metal electrode. The electron exchange between the re-dox center of the reacting complex and the electrode is as-sumed to be strongly coupled to a proton transfer reactionwithin the reacting complex. For typical PCET reactions, theproton transfer interface is characterized by an asymmetricpotential energy curve along the proton coordinate with aminimum near the proton donor or acceptor when the elec-tron is localized on the electron donor or acceptor, respec-tively. Thus, typically the ground state proton vibrationalwave function is localized near the proton donor when the

redox center is in the reduced state and near the proton ac-ceptor when the redox center is in the oxidized state. Figure1 illustrates this general model system. In this model, theoxidation or reduction of the redox center occurs simulta-neously with the proton transfer reaction.

The total Hamiltonian of the entire system, which con-sists of the electrode, the reacting complex, and the solvent,is

Htot = Te + Tp + TQ + Tq + Ure,rp,Q,q , 1where re and rp are the electron and proton coordinates, re-spectively, Q corresponds to the proton donor-acceptor modecoordinate, and q denotes the set of solvent nuclear degreesof freedom. In this Hamiltonian, Te, Tp, TQ, and Tq are ki-netic energy operators for the electrons, proton, Q mode, andsolvent modes, respectively. The total Hamiltonian definedabove includes only the essential degrees of freedom for thedescription of a PCET reaction. For simplicity, we includeonly the proton coordinate and the proton donor-acceptordistance within the reacting complex, neglecting the otherintramolecular nuclear coordinates within this complex. We

FIG. 1. Color online Schematic picture of the electrochemical PCET sys-tem. In the PCET reaction from the initial state I to the final state II, theelectron transfers from the one-electron state r localized on the electrondonor De of the reacting complex to the metal electrode labeled with an M.As a result, the number of electrons on the metal electrode increases fromNM to NM +1, and the number of electrons on the reacting complex de-creases from NR+1 to NR. Concurrently, the proton transfers from the protondonor Dp to the proton acceptor Ap within the reacting complex. The protonpotential energy profile along the proton donor-acceptor axis is asymmetricwith the minimum near the proton donor for the initial, reduced state andwith the minimum near the proton acceptor for the final, oxidized state.Thus, the ground state proton vibrational wave function is localized near theproton donor for the initial, reduced state and near the proton acceptor forthe final, oxidized state. The Q mode corresponds to the change in the DpAp distance relative to its equilibrium value.

244712-2 Navrotskaya, Soudackov, and Hammes-Schiffer J. Chem. Phys. 128, 244712 2008


also assume that the solvent electrons are infinitely fast onthe time scale of the PCET reaction, so their solvatingeffects are incorporated into the solute-solvent interactionpart of the total potential Ure ,rp ,Q ,q.48

A. Electronic Hamiltonian

This subsection focuses on the electronic BornOppenheimer Hamiltonian Hel. This Hamiltonian can be splitinto three parts corresponding to the electrode M, the re-acting complex R, and the electrode-complex interactionMR,

Hel = Te + Ure,rp,Q,q = HMel + HRel + HMRel . 2For the metal electrode, we adopt a quasi-free electron

model and represent the electrode Hamiltonian HMel as an

effective one-electron Hamiltonian built in the basis of thedelocalized spin orbitals kre characterized by the quasi-momentum k. For simplicity, we restrict the model to asingle conduction band of the metal. In this case, the elec-trode Hamiltonian can be expressed as

HMel

= kkck

ck. 3

Here k are the energies of the metal one-electron states kwithin the conduction band and are independent of thenuclear coordinates of the reacting complex and the solventmolecules, and ck

and ck are second quantization creationand annihilation operators, respectively, corresponding to themetal one-electron states k.

The electronic Hamiltonian of the reacting complex isconstructed in the basis of one-electron molecular spin orbit-als irerp ,Q ,q for the solvated reacting complex in itsreduced state. Note that these orbitals depend parametricallyon the nuclear coordinates rp, Q, and q. The reduced reactingcomplex has NR+1 electrons. We assume that the highestoccupied molecular orbital HOMO rrerp ,Q ,q is local-ized on the redox center of the reacting complex and is theonly MO with a nonvanishing resonance integral with thedelocalized metal spin orbitals kre. The electronic Hamil-tonian for the reacting complex can be written in the follow-ing form:

HRel

= rrp,Q,qcrcr + iirp,Q,qcici, 4

where irp ,Q ,q are the MO energies for an effective one-particle Hamiltonian, and ci

and ci are the second quantiza-tion creation and annihilation operators, respectively, corre-sponding to the one-electron states i. In the aboveexpression, we isolated the term for the HOMO for clarity.Throughout the rest of the paper, the subscript r will be usedto denote the HOMO. Note that the energy of the HOMO canbe calculated in the framework of the KohnSham formalismof density functional theory DFT.49 In this framework, theenergy of the highest occupied KohnSham orbital, which isassociated with the one-electron state r, corresponds to thefirst ionization potential of the reacting complex according toa DFT version of Koopmans theorem.50 In general, the MOenergies of the reacting complex can be calculated with any

appropriately defined effective one-particle Hamiltonian.The interaction term responsible for electron exchange

between the electrode and the HOMO localized on the redoxcenter is written in the following standard form:

HMRel

= k

Vrkrpcrck + Vkrrpck

cr , 5

where Vkr=Vrk is the electronic coupling term i.e., reso-

nance integral, which is assumed to depend on only theproton coordinate rp.

The electronic Hamiltonian defined by Eqs. 25 isidentical to the modified AndersonNewns Hamiltonianwidely used in the theory of electrochemical ET Ref. 51with the exception of the additional term iirp ,Q ,qcicidescribing the electronic subsystem of the reacting complex.In addition, the energies rrp ,Q ,q and irp ,Q ,q aretreated as functions of the nuclear degrees of freedom ratherthan as constants.

For convenience, the electronic Hamiltonian can be ex-pressed in a basis of two sets of many-electron states corre-sponding to the initial and final states in the ET reactionbetween the HOMO of the redox center and the metal elec-trode. The many-electron wave function of the model systemcan be represented as an antisymmetric product of the wavefunctions i.e., Slater determinants constructed from themetal electrode spin orbitals and the reacting complex MOs.The initial electronic states corresponding to the reduced re-acting complex are defined to have NM electrons occupyingthe metal orbitals and NR+1 electrons occupying the MOs ofthe reacting complex, where one of these latter electrons oc-cupies the HOMO r. The final electronic states corre-sponding to the oxidized reacting complex are defined tohave NM +1 electrons occupying the metal orbitals and NRelectrons occupying the MOs of the reacting complex. Thetotal number of electrons is N=NM +NR+1. We assume thatall states corresponding to electronic excitations within thesubset of the reacting complex MOs are very high in energyand can be neglected. Thus, our basis set consists of twosubsets K

I and LII for the initial and final states, respec-

tively. In the occupation number representation, these statescan be written as the following antisymmetric products,where the symbol denotes an antisymmetric product:

6

Here K and L are composite indices denoting the occupationnumbers of the metal spin orbitals. In particular, nKi =0,1and nLi =0,1 are the occupation numbers for the metal spinorbitals, and KNM and LNM +1 are the many-electron states i.e., Slater determinants of the metal elec-trode with NM and NM +1 electrons, respectively. The ener-gies of the states KNM and LNM +1 are EK and EL,

244712-3 Proton-coupled electron transfer J. Chem. Phys. 128, 244712 2008


respectively, and will appear in later equations. Finally, Ris the many-electron state i.e., Slater determinant of thereacting complex with NR electrons. Note that the state Ris common to all of the basis states defined in Eq. 6.

By construction, all of the basis states defined in Eq. 6are eigenstates of the operator iirp ,Q ,qcici, which is thesecond term in Eq. 4:

i irp,Q,qciciKI = URrp,Q,qKI , 7where the eigenvalue is

URrp,Q,q = i=1

NR

irp,Q,q . 8

In Eq. 8, the summation is over the lowest NR occupiedMOs of the reacting complex. Equation 7 also holds forL

II with the same eigenvalue. Introducing the identity op-erator in the electronic space,

Ie = K

KI K

I + L

LIIL

II , 9

noting that

iirp,Q,qcici = URrp,Q,qIe, 10

and performing a simple rearrangement, we rewrite the elec-tronic Hamiltonian of the reacting complex in solution as

HRel

= UIrp,Q,qcrcr + UIIrp,Q,qIe crcr . 11Here

UIrp,Q,q = rrp,Q,q + URrp,Q,q ,12

UIIrp,Q,q = URrp,Q,qare the diabatic electronic potentials for the reduced and oxi-dized states of the reacting complex, respectively. Typicallythe slices of these potentials along the proton coordinate rpare highly asymmetric with the minima near the proton do-nor or acceptor for UI or UII, respectively, as depicted inFig. 1.

After these simplifications, the electronic Hamiltonianfor the entire system defined in Eq. 2 has the followingform:

Hel = kkck

ck + UIrp,Q,qcrcr

+ UIIrp,Q,qIe crcr+

kVrkrpcr

ck + Vkrrpckcr . 13

This electronic Hamiltonian is similar to the Hamiltonianused by Schmickler and co-worker15,16 to describe electro-chemical ET reactions involving the breaking of a bond.

B. Electron-proton vibronic Hamiltonian

In this subsection, we develop the quantized form of theelectron-proton vibronic Hamiltonian defined as Hep=Tp+Hel. For this purpose, we introduce two sets of proton vi-

brational basis states obtained in the double adiabatic ap-proximation by solving the following Schrdinger equationsfor the proton motion in the diabatic electronic potentialsdefined in Eq. 12:

Tp + UIrp,Q,qI rpQ,q = I Q,qI rpQ,q ,14

Tp + UIIrp,Q,qIIrpQ,q = IIQ,qIIrpQ,q .The proton vibrational states

I and II form two com-

plete sets corresponding to the reduced and oxidized elec-tronic states of the reacting complex, respectively. Note thatthese states depend parametrically on Q and q. The reso-lution of the identity operator Ip in the space of the protonstates can be expressed as follows:

Ip =

I

I =

II

II . 15

We can also define the second quantization creation annihi-lation operators a1

a1 and a2 a2 corresponding to the

proton vibrational states I and

II, respectively. Theseoperators satisfy the following anticommutation relations:

a1

,a1 + = a2

,a2

+ = a1,a1+ = a2,a2+ = 0,

a1

,a1+ = ,16

a2

,a2+ = ,

a1,a2 + = SQ ,

where SQ= I II are the interset overlap integrals thatare assumed to be independent of the solvent coordinates q.

In this representation, the proton vibrational Hamilto-nians in Eq. 14 can be expanded as

Tp + UIrp,Q,q =

I Q,qa1 a1,

17Tp + UIIrp,Q,q =

IIQ,qa2 a2.

Using the above relations and neglecting the electron-protonnonadiabatic interactions within each of the electronic sub-sets K

I and LII given in Eq. 6, we can express the

electron-proton vibronic Hamiltonian in the following form:

Hep = Ipkkck

ckIp +

I Q,qa1 a1crcr

+

IIQ,qa2 a2Ie crcr

+ k,

VrkQa1 a2crck

+ Vrk Qa2 a1ckcr , 18

where VrkQ= I VrkrpII are vibronic couplings. Weassume that these vibronic couplings are independent of thesolvent coordinates q but depend on Q exponentially:35,38,42



VrkQ = Vrk0 eQ, 19where is a real constant and Vrk

0 is the vibronic cou-pling evaluated at Q=0.

C. Total vibronic Hamiltonian

In this subsection, we develop the quantized form of thetotal vibronic Hamiltonian Htot defined in Eq. 1. This stepinvolves the quantization of the proton donor-acceptor modeQ and solvent modes q. For this purpose, we employ theharmonic approximation for the two sets of electron-protonvibronic surfaces

I Q ,q and IIQ ,q and expand them inTaylor series up to second order around their minima:

I Q,q = JI +

12j mj j

2qj qj2 +12

M2Q Q2,

20

IIQ,q = JII +

12j mj j

2qj2 +

12

M2Q2,

where JI and J

II are the values of I Q ,q and IIQ ,q,

respectively, at their minima. Here M and mj are the effectivemasses of the Q mode and solvent modes, respectively, and and j are the corresponding frequencies. The equilibriumpositions of the Q mode and the solvent modes for the sur-faces

IIQ ,q are chosen to be zero, while those of the sur-faces

I Q ,q are shifted from zero by Q and qj, respec-tively. For simplicity, we assume that the equilibriumpositions are the same for all vibronic states within each set,and the effective masses and frequencies are the same for allelectron-proton vibronic states.

Substituting Eqs. 19 and 20 into Eq. 18 and addingthe kinetic energy operators

TQ =P2

2M,

21

Tq = j

pj2

2mj,

where P and pj are the conjugate momenta for coordinates Qand qj, respectively, we obtain the following expression forthe total vibronic Hamiltonian:

Htot =

EI a1

a1crcr +

EIIa2

a2Ie crcr

+ kkck

ck +12j pj

2

mj+ mj j

2qj2

+12P

2

M+ M2Q2 j mj j2qjqj

+ M2QQcrcr + k,

Vrk0 a1

a2crck

+ Vrk0 a2

a1ckcreQ. 22

Here we have omitted the identity operators for notationalsimplicity. In the above equation,

EI

= JI + q + Q, E

II= J

II, 23

where the reorganization energies q and Q for the bathoscillators and Q mode, respectively, are defined as

q =12j mj j

2qj2, Q =12

M2Q2. 24

Finally, introducing the harmonic oscillator ladder opera-tors bj

bj and B B for the solvent modes and Q mode,respectively, we obtain the following expression for the totalvibronic Hamiltonian:

Htot =

EI a1

a1crcr +

EIIa2

a2Ie crcr

+ kkck

ck + j

jbjbj + 12 + BB + 12

j gj jbj + bj + GB + Bcrcr+

k,

Vrk0 a1

a2crck

+ Vrk0 a2

a1ckcreB

+B, 25

where we have defined the following unitless coupling con-stants:

gj = qjmj j2 , G = QM2 ,26

= 2M .

D. Comparison to previous model Hamiltonians

This vibronic Hamiltonian is related to the previousHamiltonian used by Schmickler and co-workers30,31 forelectrochemical PCET, but a number of important differ-ences should be noted. The previous Hamiltonian includedan explicit double well potential for the proton, as well asexplicit electron-proton and proton-solvent interaction terms.In contrast, the present Hamiltonian is expressed in a basis ofelectron-proton vibronic states without the introduction of anexplicit proton coordinate or proton potential. Moreover, thepresent Hamiltonian includes the proton donor-acceptormode coordinate Q, which was not included in the previousHamiltonian. Furthermore, the previous Hamiltonian de-scribed the interaction between the redox potential and theelectrode in terms of purely electronic couplings that wereindependent of the proton and bath vibrational modes. Incontrast, the present Hamiltonian describes this interaction interms of vibronic couplings that depend on the proton donor-acceptor mode coordinate and on the proton vibrationalstates within the electron-proton vibronic states. These addi-tions to the model Hamiltonian impact the qualitative physi-cal behavior of electrochemical PCET systems.



III. REACTION RATE CONSTANTS

In this section, we derive the expressions for the oxida-tion and reduction PCET rate constants using the model vi-bronic Hamiltonian defined in Eq. 25. The rate constantsare derived using the master equations for the reduced den-sity matrix.5254 In the present derivation, we employ secondorder perturbation theory, which is valid in the nonadiabaticlimit.

A. Master equations and nonadiabatic rate constants

To facilitate the derivation, we perform the canonicalLangFirsov transformation,11,55 which partially diagonalizesthe model vibronic Hamiltonian in Eq. 25 by shifting theequilibrium positions of the solvent bath oscillators for thefirst set of vibronic states. For this purpose, we apply theunitary shift operator U=expcr

crv with

v = j

gjbj

bj + GB B . 27

The transformed Hamiltonian has the following form:

H tot = UHtotU = HS + HB + Vpert, 28

where

HS = kkck

ck +

JI a1

a1crcr

+

JIIa2

a2Ie crcr 29

is the system Hamiltonian,

HB = j

jbjbj + 12 + BB + 12 30

is the bath Hamiltonian, and

Vpert = k,

Vrk0 rk + Vrk

0 rk

31

is the perturbation term with system operators

rk = a1 a2cr

ck,

32rk

= a2

a1ckcr,

and bath operators

= exp vexp B + B ,33

= exp B + Bexpv .

Our objective is to calculate the anodic rate constant forET from the redox center to the metal electrode and thecathodic rate constant for ET from the metal electrode to theredox center. The total probability P1t that the system is inone of the initial states I corresponding to the reduced stateof the reacting complex is the sum of the diagonal elementsof the reduced density matrix St=TrBt i.e., the traceover the eigenstates of HB over the vibronic states associ-ated with the electronic subset K

I defined in Eq. 6:

P1t = ,K

I K

I StKI

I ,K

K,KS t . 34

Similarly, the total probability P2t=1 P1t that the sys-tem is in one of the final states II corresponding to theoxidized state of the reacting complex is given by

P2t = ,L

IIL

IIStLII

II ,LL,L

S t . 35

For notational simplicity, the index K is always associatedwith an initial state I, and the index L is always associatedwith a final state II. Thus, a summation over K is over allelectronic states in subset I, and a summation over L is overall electronic states in subset II.

The application of second order perturbation theory tothe transformed model vibronic Hamiltonian defined in Eq.28 leads to the master equations governing the time evolu-tion of the diagonal elements of the reduced densitymatrix:53,54

tK,K

S t = K,KS t

,LWL,K +

,LL,L

S tWK,L,

36

tL,L

S t = L,LS t

,KWK,L +

,KK,K

S tWL,K.

37

Here WL,K and WK,L are the microscopic interstate tran-sition probabilities given in second order perturbation theoryby the following golden rule expressions:

WL,K =1

2

dt eit/JII+ELJI EK

k

k

Vrk0 V

rk0

t

I KI rkL

IIII

IIL

IIrk K

I I , 38

WK,L =1

2

dt eit/JII+ELJI EK

k

k

Vrk0 V

rk0

t

IILII

rk K

I I

I K

I rkLII

II . 39

Throughout this paper, the time evolution of the Heisenbergoperators is governed by the bath Hamiltonian HB i.e.,At=expiHBt /A expiHBt /, and the thermodynamicaverages are defined for the bath equilibrium distributioni.e., TrBexpHB /TrBexpHB. As de-fined previously, EK and EL denote the total energiesof the metal electrode many-electron states KNM andLNM +1 with NM and NM +1 electrons, respectively.



The master equations given in Eqs. 36 and 37 werederived from the exact von Neumann equation for the den-sity matrix in the weak system-bath coupling limit under thefollowing assumptions. First, the bath is assumed to alwaysremain at equilibrium. Second, the system is assumed to fol-low Markovian dynamics i.e., the memory in the systemdynamics is completely destroyed by the fast bath fluctua-tions. Third, all of the purely oscillatory i.e., secular termsin the master equations are neglected. This corresponds tothe secular approximation in the BlochRedfield quantumrelaxation theory.53,56,57

This formalism becomes more complicated if the equi-librium averages and

do not vanish. In this case,Eqs. 36 and 37 include additional terms proportional to and

.54 These terms are easily removed bychoosing the initial density to be the equilibrium density ofthe reactant or product state. Another difficulty arises, how-ever, because the time correlation functions t

and

t do not decay to zero, so the time integralsin Eqs. 38 and 39 do not converge. For our model, theequilibrium averages

ln = ln

=

12j gj

2 coth j/2 12

G2 coth/2

G +12

2 coth/2 40

are not zero in general. This issue is typically addressed byrewriting the Hamiltonian as follows:

H = H S + HB + V pert, 41

where

H S = HS + k,

Vrk0 rk

+ Vrk0 rk

, 42

V pert = k,

Vrk0 rk + Vrk

0 rk

,

43

and AA A. Then the time correlation functions appear-ing in the transition probabilities, namely, t

and

t, decay to zero at long times, ensuringconvergence of the time integrals. In this procedure, how-ever, Eqs. 38 and 39 must be rewritten for the transitionprobabilities of the new system H S with new eigenstates andeigenvalues. For simplicity, in this paper we assume that theaverages and

vanish, as would be the case for abath with Ohmic dissipation.58 In this case, we can use Eqs.38 and 39 without redefining the system Hamiltonian.

Using the master equations 36 and 37, we obtain thefollowing dynamical equations for the total probabilitiesP1t and P2t:

P 1t = ,K

K,KS t

= ,K

,LK,K

S tWL,K

+ ,K

,LL,L

S tWK,L, 44

P 2t = ,LL,L

S t

= ,K

,LL,L

S tWK,L

+ ,K

,LK,K

S tWL,K, 45

where A A /t. Comparing the above equations with thekinetic equations for the total probabilities P1t and P2t:

P 1t = k21P1t + k12P2t ,46

P 2t = k12P2t + k21P1t ,

we obtain the following general expressions for the anodicand cathodic rate constants:

kat =1

P1t,K

,LK,K

S tWL,K, 47

kct =1

P2t,K

,LL,L

S tWK,L. 48

Note that kat and kct are time dependent because transi-tions between multiple pairs of vibronic states are included.However, our objective is to obtain the rate constants whenthe system is at equilibrium i.e., t=, and St=eqSexpHS /TrSexpHS. In this case, using the gen-eral relation for transition probabilities, Wmnnn

eq=Wnmmm

eq,

we can show that the anodic and cathodic rate constants sat-isfy the detailed balance condition ka /kc= P2

eq / P1eq at equilib-

rium. Here ka and kc denote the rate constants at equilibrium.Using the explicit forms of the perturbation operator

terms given in Eqs. 32 and 33, we can evaluate the matrixelements in Eq. 38 and obtain the following expression forthe transition probability in the anodic process:

WL,K =1

2

Vrk0 2

dt eit/JII

JI +k

evteBt+BteB

+Bev . 49

In this equation, all of the occupied one-electron states forthe many-electron states K

I and LII are the same except

for r in KI and k in L

II. For all other pairs of K andL, the transition probability WL,K is zero. Substituting thisresult into Eq. 47, we obtain the following expression forthe anodic rate constant:



ka =1

2P1eq

k,

K

I KI eq

S KI

I Vrk0 2

dt eit/EkevteBt+BteB+Bev ,

50

where the prime following the third summation denotes thatthe sum is restricted to the electronic states K

I with themetal one-electron state k unoccupied, and

Ek = JII

JI + k. 51

The matrix elements of the reduced density matrix atequilibrium appearing in Eq. 50 can be evaluated as shownin Appendix A. The resulting simplified expression for thesematrix elements is:

1P1

eqK

I KI eq

S KI

I = pI 1 fk . 52

Here pI

=eJI/eJ

Iis the Boltzmann occupation prob-

ability for the vibrational state I , and fk is the Fermi oc-

cupation probability defined as

fk = 1 + ekM1, 53where M is the electrochemical potential of the electrode.Substituting the above relation into Eq. 50, we obtain thefollowing expression for the anodic rate constant:

ka =1

2k,

pI 1 fkVrk0 2

dt eit/EkevteBt+BteB+Bev .

54

Analogously, the cathodic rate constant can be expressedas

kc =1

2k,

pII fkVrk0 2

dt eit/EkeBt+BtevteveB+B ,

55

where pII

=eJII/eJ

IIis the Boltzmann occupation prob-

ability for the proton vibrational state II. Here the factor

pIIfk originates from the following expression derived in Ap-

pendix A:

1P2

eqL

IILIIeq

S LII

II = pII fk, 56

where the double prime following the summation denotesthat the sum is restricted to the electronic states L

II with themetal one-electron state k occupied.

The time correlation functions in Eqs. 54 and 55 con-taining the harmonic oscillator operators can be evaluatedanalytically. The resulting analytical expressions for the an-odic and cathodic rate constants have the following form:

ka =1

2k,

pI 1 fkVrk0 2

dt eit/EkCa t ,

57

where

Ca t = exp j gj2coth j/21 cos jt + i sin jt G2coth/21 cos t + i sin t

+ 2 coth/21 + cos t i sin t 2G1 + cos t i sin t coth/2 58

and

kc =1

2k,

pII fkVrk0 2

dt eit/EkCc t , 59

where

Cc t = exp j gj2coth j/21 cos jt + i sin jt G2coth/21 cos t + i sin t

+ 2 coth/21 + cos t i sin t 2G1 cos t + i sin t coth/2 . 60



As shown in Appendix B, the rate constants ka and kc givenin Eqs. 57 and 59 satisfy the detailed balance condition

kakc

=

P2eq

P1eq . 61

B. High temperature limit for solvent modesIn the high temperature limit for the solvent modes i.e.,

c1, where c is the cut-off frequency for the solventmodes, the rate constant expression can be significantly sim-plified. In this limit, we can use a short-time approximationprovided that the solvent reorganization energy q is largeenough for the strong solvation condition to be satisfied:58,59

2q2c

2 1. 62

The short-time approximation is equivalent to the assump-tion that the dynamics of the solvent fluctuations is fast onthe time scale of the coherent nonadiabatic transitions. Tak-ing the appropriate limits in Eqs. 57 and 59, we obtain thefollowing expression for the anodic rate constant:

ka = k,,

pI 1 fkVrk0 2Bka , 63

where

Bka

=

1

2

exp G2 + 2 coth/2 2G

dt exp qt2

2

it

Ek + q

expcos tcoth/2G2 + 2 2G i sin tG2 +

2 2G coth/2 . 64

Analogously, the expression for the cathodic rate constant is

kc = k,,

pII fkVrk0 2Bkc , 65

where

Bkc

=

1

2

exp G2 + 2 coth/2 2G

dt exp qt2

2

it

Ek + q

expcos tcoth/2G2 + 2 + 2G i sin tG2 +

2 + 2G coth/2 . 66

For electrochemical PCET, the energy differencesEk, and thus the anodic and cathodic rate constants, de-pend on the electrode potential . To introduce this depen-dence explicitly, we first express the energies of the metalelectrode states k in terms of their deviations from the elec-trode electrochemical potential M i.e., k= M + k. Wealso express the energies of the vibronic states J

I and JII for

the reduced and oxidized states of the reacting complex interms of their deviations from the ground state energies J0

I

and J0II, respectively:

JI

= J0I + J

I,

67J

II= J0

II + JII

.

In the double adiabatic approximation for the proton mo-tion, the quantities J

I and JII are simply the energies of the

proton vibrational states relative to their respective groundvibrational states. When the metal electrode has an innerpotential , the energy differences are then given by

Ek = J0II

J0I + J

II J

I + M + k e , 68

where e is the electron charge, and the electrochemical po-tential M is related to the chemical potential M by M

=M e. We define a reference potential 0 as the potentialthat shifts the electrochemical potential so that it exactlycompensates for the difference between the energies J0

II andJ0

I i.e., e0=J0II

J0I +M. Introducing the overpotential

=0, we rewrite the energy difference given in Eq. 68as

Ek = e + k + J, 69

where J=JII

JI. Finally, using the continuum limit for

the electrode states by introducing the density of states=d3kk, we obtain the following expressions forthe anodic and cathodic rate constants:

ka = ,

pI

d1 fV2Ba , ,

70

kc = ,

pII

dfV2Bc , , 71

where f= 1+exp1 is the Fermi distribution func-tion, and



V2 =1 d3kVrk0 2k 72

is the energy-dependent squared vibronic coupling integratedover the metal electrode electronic states in the vicinity ofenergy .10 Furthermore, B

a , and Bc , are ob-

tained from Eqs. 64 and 66 by substituting Eq. 69 forthe energy differences Ek and replacing k in Eq. 69with . These expressions can be further simplified in the

high temperature and low temperature limits for the protondonor-acceptor mode Q.1. High temperature low frequency limit for theproton donor-acceptor mode

When the proton donor-acceptor mode frequency islow enough to satisfy the condition 1, we can use thehigh temperature approximation for this mode as well. In thislimit, the integrals in Eqs. 64 and 66 can be readily evalu-ated, and we obtain the following analytical expressions forthe anodic and cathodic B factors:

Ba , =

1

exp 422

2Qexp J e + + 2Q/24 , 73B

c , =1

exp 422

exp J + e + + 2Q/24 , 74

where we have defined the coupling reorganization energy

= 2 75

and the total reorganization energy

= q + Q +

, 76

which includes the solvent, Q mode, and coupling contribu-tions.

Substitution of these expressions for the B factors intothe expressions for the anodic and cathodic rate constantsgiven in Eqs. 70 and 71 leads to analytical expressions forthe first order rate constants for nonadiabatic electrochemicalPCET. These expressions are analogous to the correspondingexpressions derived previously for homogeneous nonadia-batic proton transfer36,37 and PCET reactions.35 These ex-pressions are also identical to the expressions derived re-cently by the direct application of the golden rule formalismto describe nonadiabatic transitions between two sets ofelectron-proton vibronic states representing electrochemicalPCET.45 Note that Ref. 45 denotes the Q mode as the Rmode and defines R relative to the equilibrium R-mode po-sition in the initial state corresponding to the reduced state ofthe reacting complex. In contrast, here Q is defined relativeto the equilibrium Q-mode position in the final state corre-sponding to the oxidized state of the reacting complex. Thisdifference leads to the opposite signs of the terms involvingQ or R in the effective activation energies for both theanodic and cathodic rate constants. Moreover, the factorexp2Q appears in the cathodic rather than the anodicrate constant in Ref. 45. Finally, the vibronic coupling V isdefined to be calculated for the initial state in Ref. 45 and forthe final state in the present paper.

2. Low temperature high frequency limit for theproton donor-acceptor mode

In the low temperature, or high frequency, limit for theproton donor-acceptor mode, 1, and the proton donor-acceptor vibrational motion exhibits quantum mechanical be-havior. In this limit, the Q mode remains predominantly in itsground state, and the excited states become inaccessible dueto the large vibrational energy level splittings . For thisreason, the Q mode cannot effectively participate in the en-ergy dissipation mechanism, so its reorganization will notcontribute to the activation energy. Thus, the sole effect ofthe Q mode on the rate constant is that the vibronic couplingis averaged over the ground state vibrational wave functionof the Q mode. Despite this simple physical picture, the ana-lytical evaluation of the rate constant expression in this limitis not trivial and can be performed only for limited values ofcertain system parameters. In particular, for every nonadia-batic transition between a pair of vibronic states, the strongsolvation condition must be satisfied i.e., Je+q. Although this condition may appear to be restrictive,many electrochemical PCET systems satisfy this conditionprovided that the absolute value of the overpotential is nottoo large.

Under the conditions mentioned above, the time inte-grals in Eqs. 64 and 66 can be evaluated by the stationaryphase method,37,60 as described in Appendix C. The resultinganalytical expressions for the anodic and cathodic B factorsin this limit are

Ba , =

1

qexp Q

Q

exp J e + + q24q

, 77



Bc , =

1

qexp Q

Q

exp J + e + q24q

. 78As expected, these expressions include temperature-independent exponential prefactors originating from the av-eraging of the Q-dependent vibronic coupling over theground state vibrational wave function of the Q mode, andthe Arrhenius exponential with the effective activation en-ergy includes only the solvent reorganization energy q.

3. Dependence of rate constant on the protondonor-acceptor mode frequency

In this subsection, we perform model calculations to il-lustrate the dependence of the anodic rate constant on theproton donor-acceptor mode frequency. For simplicity, weassume that the density of states does not change sig-nificantly in the vicinity of the Fermi level and therefore canbe replaced by a constant F evaluated at the Fermi level.Furthermore, we approximate the vibronic coupling Vrk

0 inEq. 72 as

Vrk0

= VelS0

, 79

where Vel is the electronic coupling assumed to be constantin the vicinity of the Fermi level and S

0 is the overlapintegral for the proton vibrational wave functions evaluatedat the equilibrium position of the Q mode in the final oxi-dized state i.e., at Q=0. With these approximations, Eq.70 can be expressed as

ka = FVel2,

pI S

0 2

d1 fBa , .

80

We chose parameters that correspond to a physically rea-sonable electrochemical PCET system for our model calcu-lations. The overlap integrals S

0 were calculated from theharmonic oscillator wave functions corresponding to two

parabolic potentials along the proton coordinate with minimaseparated by 0.8 . The mass of the proton was 1 amu, andthe frequency associated with the harmonic proton potentialswas 3000 cm1. The parameters were determined bycalculating the derivatives of the natural logarithms of theoverlap integrals with respect to Q at Q=0. The equilibriumproton donor-acceptor distance was assumed to be 0.05 greater in the final oxidized state than in the initial reducedstate i.e., Q=0.05 . The constants G and werecalculated from Eq. 26. The reduced mass M of the Qmode was chosen to be 50 amu, and the solvent reorganiza-tion energy was chosen to be q=1.691 eV, as estimated byTanaka and Hsu11 for electrochemical ET in water with ametal electrode. The calculations were performed at 298 Kwith zero overpotential i.e., =0. The integrals over timeand energy were calculated numerically. We found that twoinitial and two final proton vibrational states were sufficientto converge the results.

Figure 2 depicts the anodic rate constant as a function ofthe Q-mode frequency in the high temperature limit forthe solvent using the expression in Eq. 64, as well as theexpressions in Eqs. 73 and 77 for the high temperatureand low temperature limits with respect to the Q mode. Asexpected, the rate constant in the high temperature low fre-quency limit for the Q mode approaches the general rateconstant at low frequencies, and the rate constant in the lowtemperature high frequency limit for the Q mode ap-proaches the general rate constant at high frequencies.

C. Comparison to previous rate constant expressions

The nonadiabatic rate constant expressions derivedabove are equivalent to those derived by direct application ofthe golden rule to describe nonadiabatic transitions betweentwo sets of electron-proton vibronic states45 and are similarto the expressions utilized by Costentin et al.28,29 A numberof differences between the present rate constant expressionsand those utilized by Costentin et al.28,29 should be noted. Inparticular, the present expressions include the effects of dif-ferences between the reactant and product equilibrium protondonor-acceptor distances i.e., Q0, leading to additionaltemperature-dependent terms in the effective Marcus theoryactivation energy. These terms differ in sign for the anodicand cathodic current densities and thus could lead to asym-metries in the Tafel plots of the total current density as afunction of the overpotential.45 In addition, the total reorga-nization energy in the present formulation includes two ad-ditional terms, Q and

, which reflect changes in the equi-

librium proton donor-acceptor distance and the strongdependence of the vibronic coupling on this distance. Refer-ence 45 describes how the present rate constant expressionscan be used to calculate current densities and heterogeneousrate constants for electrochemical PCET processes.

IV. CONCLUSION

In this paper, we have presented an extension of theAndersonNewnsSchmickler model for electrochemicalPCET reactions. This model describes reactions involvingET between a solute complex and an electrode coupled to

FIG. 2. The dependence of the anodic rate constant on the Q-mode fre-quency in the high temperature limit for the solvent modes, as given in Eq.80. The B factors were calculated using Eq. 64 for the general case solidline, Eq. 73 for the high temperature limit for the Q mode dashed line,and Eq. 77 for the low temperature limit for the Q mode dashed-dottedline. Here k= Vel2F /.



proton transfer within the solute complex. Our treatment dif-fers from previous models for PCET in that the Hamiltonianis expressed in a basis of electron-proton vibronic states, andthe vibronic coupling terms responsible for electronic transi-tions between the solute complex and the electrode dependon the proton donor-acceptor vibrational mode within thesolute complex. The quasi-free electrons of the electrode areincluded explicitly in the model Hamiltonian, thereby allow-ing a detailed description of the electronic structure of theelectrode. In principle, this approach can be extended to in-clude the effects of electron-hole excitations and impuritystates in semiconducting electrodes, as well as localized sur-face states and geometrically confined localized states innanosize electrodes. The vibronic structure of the Hamil-tonian also allows the inclusion of additional degrees of free-dom coupled to the ET, such as bond breaking or othernuclear rearrangements in chemical reactions.

We have used this model Hamiltonian to derive the for-ward and backward rate constants for nonadiabatic electro-chemical PCET occurring by a concerted mechanism, inwhich the ET between the solute complex and the electrodeis concurrent with proton transfer within the solute complex.The expressions for the rate constants were derived by ap-plying second order perturbation theory to the master equa-tions for the reduced density matrix of the electron-protonsubsystem, which includes the electrons of the solute com-plex and the electrode, as well as the transferring proton. Dueto the explicit inclusion of electronic excitations in the metalelectrode, the Fermi distribution function weighting the con-tributions from transitions corresponding to electron ex-change between the reacting complex and the metal elec-trode arises rigorously in this formulation. The resultinganodic and cathodic rate constant expressions differ fromanalogous expressions for electrochemical ET in that theyinclude a summation over electron-proton vibronic states andadditional terms arising from the dependence of the vibroniccouplings on the proton donor-acceptor vibrational motion.In particular, the rate constant expressions include additionalcontributions to the total reorganization energy, an additionalexponential temperature-dependent prefactor, and atemperature-dependent term in the effective activation en-ergy that has different signs for the anodic and cathodicprocesses.

These rate constant expressions for nonadiabatic electro-chemical PCET are equivalent to those derived by direct ap-plication of the golden rule to describe nonadiabatic transi-tions between two sets of electron-proton vibronic states.45As shown in Ref. 45, the current densities can be obtained byexplicit integration of the product of the rate constant and thesolute complex concentration over the distance between thesolute complex and the electrode surface. This procedure ac-counts for the effects of the electrical double layer, as well asextended ET in the diffuse layer and beyond. The resultingexpressions allow the generation of theoretical Tafel plotsdepicting the dependence of the current density on the over-potential for comparison to experimental data.

This extended AndersonNewnsSchmickler model, inconjunction with the powerful master equations approach,can be generalized beyond the scope of the present paper.

For example, this approach can describe adiabatic as well asnonadiabatic electrochemical PCET reactions. It can also beused to study reactions at semiconductor as well as metalelectrodes. Furthermore, this model provides the frameworkfor the inclusion of additional effects, such as the coupling ofelectrochemical PCET reactions to the breaking and formingof other chemical bonds.

ACKNOWLEDGMENTS

We are grateful to Arindam Chakraborty, CharulathaVenkataraman, and Michael Pak for helpful discussions. Thiswork was supported by NSF Grant No. CHE-05-01260 andwas sponsored in part by the Division of Chemical Sciences,Geosciences, and Biosciences, Office of Basic Energy Sci-ences, U.S. Department of Energy under contract from OakRidge National Laboratory.

APPENDIX A: DERIVATION OF FERMI DISTRIBUTIONRELATIONS

In this appendix we derive Eqs. 52 and 56. Using thesystem Hamiltonian HS given in Eq. 29, the left side of Eq.52 can be expressed as

1P1

eqK

I KI eq

S KI

I

=

K I KI eHSKI I ,K I KI eHSKI I

=

K eEKNM+JI

,K eEKNM+JI

=

eJI

eJI

K eEKNM

K eEKNM= p

IK eEKNM

K eEKNM. A1

As defined previously, the prime following the summationindicates that the sum is restricted to the electronic stateswith the metal electrode one-electron state k unoccupied.For the appendixes, we use a more explicit notation for theenergies of the metal electrode many-electron states than isused in the main text. Here EKNM denotes the total energyof the metal electrode many-electron state KNM with NMelectrons.

The ratio KeEKNM /KeEKNM in the last line of Eq.A1 is the probability that the one-electron state k in themetal electrode is not occupied. Now we show that this prob-ability is equal to 1 fk, where fk was defined in Eq. 53. Inthe canonical ensemble formalism, the electrochemical po-tential M is defined as the partial derivative of the Helm-holtz free energy AT ,V ,N with respect to the number ofparticles N: M = A /NT,V, where A=ln QN / towithin an additive constant and QN is the canonical partitionfunction. For the electronic electrode subsystem,QN=KeEKN. Since the number of electrons on the elec-trode is very large, we can approximate the electrochemicalpotential as



M = 1

ln QN ln QN1 = 1

lnK eEKN

K eEKN1,

A2

leading to

eM =K eEKN

K eEKN1. A3

This relation enables us to rewrite the ratio in the lastline of Eq. A1 as

K eEKNM

K eEKNM=

K eEKNM

K eEKNM + ekK eEKNM1

=

eM

eM + ek

= 1 1

1 + ekM= 1 fk. A4

For the second equality, we divided the numerator and de-nominator by KeEKNM1 and used the relation in Eq.A3. Note that the summations in Eqs. A1 and A4 ex-clude the one-electron state k from the many-electronstates. For the large number of electrons on the electrode,however, the omission of a single one-electron state has anegligible effect. The combination of Eqs. A1 and A4leads to the expression in Eq. 52.

We can derive Eq. 56 in an analogous manner. The leftside of this equation can be expressed as

1P2

eqL

IILIIeq

S LII

II = pIIL eELNM+1

L eELNM+1. A5

As defined previously, the double prime following the sum-mation indicates that the sum is restricted to the electronicstates with the metal one-electron state k occupied. HereELNM +1 is the total energy of the metal electrode many-electron state with NM +1 electrons. The ratio on the rightside of Eq. A5 is the probability that the one-electron statek in the metal electrode is occupied. This probability canbe written as

L eELNM+1

L eELNM+1=

11 + ekM

= fk. A6

The combination of Eqs. A5 and A6 leads to the expres-sion in Eq. 56.

APPENDIX B: PROOF OF THE DETAILED BALANCECONDITION

In this appendix we prove that Eqs. 57 and 59 satisfythe detailed balance condition given in Eq. 61. The timecorrelation functions C

a t and Cc t satisfy the relation

Ca t=C

c t i. Therefore, the time integrals in Eqs.57 and 59 are related as

dt eit/EkCa t

= eEk

dt eit/EkCc t . B1

Furthermore,

pI 1 fkeEk =

eJI

eJI ekM1 + ekMeJIIJI +k

=

eJII

eJI eM1 + ekM

= pII fk

eJII

eJI e

M. B2

However,

P2eq

P1eq =

,L IILIIeHSLIIII,K I KI eHSKI I

=

eJII

eJI

L eELNM+1

K eEKNM=

eJII

eJI e

M , B3

where we used the relation in Eq. A3 for the last equality.Therefore,

pI 1 fkeEk = pII fk

P2eq

P1eq . B4

Using the expressions for ka and kc given in Eqs. 57 and59, respectively, in conjunction with Eqs. B1 and B4,we obtain the detailed balance condition ka /kc= P2

eq / P1eq

.

APPENDIX C: DERIVATION OF RATES IN LOWTEMPERATURE LIMIT FOR THE Q-MODE

In this appendix, we present the analytical evaluation ofthe B factor for the anodic rate constant in the low tempera-ture limit for the proton donor-acceptor mode. In the limit

1, the expression in Eq. 64 has the form

Bka

=

1

2

exp G2 + 2

2G

dt t , C1

where

t = it

Ek + q

q

2t2

+ G2 + 2

2Geit. C2

The asymptotic value of the time integral can be evaluatedusing the stationary phase method:60



dt t = 2 t

et

, C3

where t is the saddle point in the complex plane obtainedfrom the relation

t =

i

Ek + q

2q

2

t

iG2 + 2

2Geit

= 0. C4

First we assume that the term proportional to expit canbe neglected. In this case, the saddle point is

t = iEk + q

2qC5

and

exp it = exp Ek + q2q . C6This exponential vanishes under the conditionsq Ek and 1. Thus, the saddle point given byEq. C5 is a reasonable estimate, leading to the followingexpressions:

t = it

Ek + q

q

2t2,

C7

t =

2q

2

.

Substituting these expressions into Eq. C3 and using therelation in Eq. 69 for Ek leads to the following expres-sion for the anodic B factor:

Ba , =

1

qexp

2 G2 2G

exp J e + + q24q

. C8An analogous derivation leads to the following expressionfor the cathodic B factor:

Bc , =

1

qexp

2 G2 2G

exp J + e + q24q

. C9Substitution of the definitions for the parameters G and into these expressions leads to Eq. 78.

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Model system-bath Hamiltonian and nonadiabatic rate constants for proton-coupled electron transfer...

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