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Decoherence and Quantum-Classical Master Equation Dynamics

Robbie Grunwald and Raymond KapralChemical Physics Theory Group, Department of Chemistry,

University of Toronto, Toronto, ON M5S 3H6 Canada

(Dated: October 7, 2018)

The conditions under which quantum-classical Liouville dynamics may be reduced to a masterequation are investigated. Systems that can be partitioned into a quantum-classical subsysteminteracting with a classical bath are considered. Starting with an exact non-Markovian equation forthe diagonal elements of the density matrix, an evolution equation for the subsystem density matrixis derived. One contribution to this equation contains the bath average of a memory kernel thataccounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating aMarkovian approximation on this term in the evolution equation. The resulting subsystem densitymatrix equation is still non-Markovian due to the fact that bath degrees of freedom have beenprojected out of the dynamics. Provided the computation of non-equilibrium average values orcorrelation functions is considered, the non-Markovian character of this equation can be removed bylifting the equation into the full phase space of the system. This leads to a trajectory description ofthe dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees offreedom. The results are illustrated by computations of the rate constant of a model nonadiabaticchemical reaction.

I. INTRODUCTION

When investigating quantum relaxation processes inthe condensed phase, one often partitions the full quan-tum system into a subsystem whose dynamics is of in-terest and an environment or bath with which the sub-system interacts. There is a large literature dealing withsuch open quantum systems.1,2 A number of differentequations of motion for the density matrix of the sub-system have been derived, including the Lindblad3 andRedfield4 equations and a variety of generalized quantummaster equations.5,6,7,8,9,10,11 Effects due to the environ-ment typically enter these equations through couplingterms involving parameters that characterize the bathrelaxation processes. Such equations have been used toinvestigate aspects of decoherence in the quantum sub-system arising from interactions with bath degrees offreedom.

Sometimes it is convenient to suppose that the dynam-ics of certain degrees of freedom is described by quantummechanics while other degrees of freedom may be treatedby classical mechanics to a good approximation. This isthe case if one considers systems involving light particlesinteracting with a bath of heavy particles. Proton andelectron transfer processes in the condensed phase andin biomolecules fall into this category, as do many vibra-tional relaxation processes. One is then led to study thedynamics of quantum-classical systems where the entiresystem is partitioned into quantum and classical subsys-tems.12 Equations of motion for the quantum subsystemdensity matrix, where the classical bath is modeled as adissipative environment have been derived.13,14,15 Suchdescriptions are useful for many applications; however,there are situations where the quantum subsystem evo-lution depends explicitly on the details of the bath dy-namics. This is important since specific features of bathmotions can influence quantum rate processes.16 To de-

scribe such specific bath dynamical effects one must usethe full quantum-classical equation of motion.

Another partition of the system is required when thequantum subsystem is directly coupled to a small subsetof the environmental degrees of freedom. For example, inproton transfer within a bio-molecule, the quantum sub-system may be taken to be the proton, which interactsdirectly with a specific set of functional groups of a largermolecule immersed in a solvent. In this case, we maydefine a quantum-classical subsystem comprising bothquantum degrees of freedom (the proton) and a subset ofthe classical variables that directly couple to these quan-tum degrees of freedom (the specified functional groups).The remaining classical variables constitute the bath. Insuch a partition, the bath may be treated either explicitlyor as a dissipative environment. Equations of motion forthe density matrix of a quantum-classical subsystem in-teracting with a dissipative bath have been derived.22 Inthis article we consider quantum-classical systems of thistype but instead retain the details of the bath dynamicsand study the conditions under which the dynamics canbe reduced to a master equation.

The reduction of quantum-classical dynamics to a sub-system master equation hinges on the decoherence in thesubsystem induced by interactions with the bath degreesof freedom.23 Consequently, we focus on how decoher-ence is described in quantum-classical systems and theconditions under which it is strong enough to eliminatethe off-diagonal elements of the density matrix on shorttime scales. The result of the analysis is a non-Markoviangeneralized master equation for the density matrix of thequantum-classical subsystem. When this equation is usedto compute non-equilibrium averages or correlation func-tion expressions involving subsystem properties, we showthat the subsystem dynamics may be lifted to the fullphase space, including the bath degrees of freedom, torecover a Markovian master equation for the quantum

2

and all classical degrees of freedom. Solutions of thisequation may be obtained from an ensemble of surface-hopping trajectories, each member of which incorporatesthe effects of quantum decoherence.The outline of the paper is as follows: The starting

point of our analysis is the quantum-classical Liouvilleequation for the entire system.25,26,27,28,29,30,31,32,33,34,35

In Sec. II we show how this equation can be cast into theform of a generalized master equation for the diagonalelements of the density matrix. This equation involvesa memory kernel operator that contains all informationon quantum coherence in the system. We discuss the ex-plicit form of the memory kernel operator that governsthe evolution in off-diagonal space and present its func-tional form. The analysis in Sec. III and Appendix Bshows that an average over an ensemble of trajectoriesevolving on coherently coupled surfaces with differentbath initial conditions can be used to reduce the full-

system generalized master equation to a non-Markoviansubsystem generalized master equation. We show thatthis equation can then be lifted back to the full phasespace to obtain a Markovian master equation. We applythis formalism in Sec. IV and calculate the nonadiabaticrate constant for a model system. Numerical results us-ing full quantum-classical Liouville dynamics and mas-ter equation dynamics are compared. In the concludingsection we comment on the relationship of our masterequation dynamics to other surface-hopping methods.

II. GENERALIZED MASTER EQUATION

Starting from the quantum-classical Liouville equa-tion, it is not difficult to derive a generalized master equa-tion for the diagonal elements of the density matrix. Asdescribed in the Introduction, the position and momen-tum operators, (q, p), of the quantum degrees of freedomare assumed to be coupled directly to a set of classicalphase space variables, X0 ≡ (R0, P0). Together thesemake up the quantum-classical subsystem. The classicalX0 variables are, in turn, directly coupled to the remain-der of the classical phase space variables, Xb ≡ (Rb, Pb),that constitute the bath. (Our formulation must be mod-ified If the quantum degrees of freedom couple directly toall other variables in the system.) The total Hamiltonianof the system is,

H(X) =P 2b

2M+

P 20

2M+

p2

2m+ V (q, R0, Rb)

≡P 2b

2M+

P 20

2M+ h(R) . (1)

The potential energy operator, V (q, R0, Rb), includes thecontributions from the quantum-classical subsystem, thebath, and the interaction between the two. It is oftenconvenient to represent the dynamics in the adiabatic

basis given by, h(R0, Rb)|α;R0〉 = Eα(R0, Rb)|α;R0〉,

where h(R0, Rb) is the quantum Hamiltonian for a fixed

configuration of the classical particles. The adiabaticeigenfunctions depend only on the coordinates R0 sincethe dependence on the bath coordinates enters throughthe potential as an additive constant. In this basis, theequation of motion for the full density matrix is33,

∂

∂tραα

′

W (X, t) = −∑

ββ′

iLαα′,ββ′ρββ′

W (X, t), (2)

where ραα′

W (X, t) is a density matrix element which de-pends on the full set of phase space coordinates X =(R,P ) ≡ (X0, Xb). The quantum-classical Liouvillesuper-operator is given by,33

− iLαα′,ββ′ = −i(ωαα′ +Lαα′)δαβδα′β′ + Jαα′,ββ′ , (3)

where ωαα′ = ∆Eαα′/~, with ∆Eαα′ = Eα − Eα′ , andthe classical Liouville operator, iLαα′ is given by

iLαα′ =P

M·

∂

∂R+

1

2

(

Fα + Fα′

)

·∂

∂P. (4)

The Hellmann-Feynman force36 for state α isFα = 〈α;R0|∂V (q, R0, Rb)/∂R|α;R0〉, and the op-erator Jαα′,ββ′ , defined in the next section, accountsfor quantum transitions and corresponding momentumchanges in the environment.We shall often simplify the notation in what fol-

lows. Above, ρW (X, t) refers to the partially Wignertransformed density matrix whose matrix elements areραα

′

W (X, t). Since we use partially Wigner transformedvariables throughout this article, we shall drop the sub-script W.The quantum-classical Liouville evolution operator can

be partitioned into diagonal, off-diagonal and couplingcomponents by defining the super-operators: Ld, Ld,o,Lo,d, and Lo, where the d, and o superscripts denotediagonal and off-diagonal, respectively. We define the di-agonal part of the density matrix ρd(X, t), with matrixelements, ραα(X, t)δαα′ ≡ ραd (X, t). Similarly, the off-diagonal part of the density matrix, ρo(X, t), has matrix

elements ραα′

(X, t)(1 − δαα′) ≡ ραα′

o (X, t). Using thesedefinitions, the quantum-classical Liouville equation maybe expressed formally as the following set of coupled dif-ferential equations,

∂

∂tρd(X, t) = −iLdρd(X, t)− iLd,oρo(X, t) (5)

∂

∂tρo(X, t) = −iLoρo(X, t)− iLo,dρd(X, t). (6)

By substituting the formal solution of Eq. (6) intoEq. (5), we obtain the evolution equation for ρd(X, t),

∂

∂tρd(X, t) = −iLd,oe−iLotρo(X, 0)− iLdρd(X, t) (7)

+

∫ t

0

dt′iLd,oe−iLo(t−t′)iLo,dρd(X, t′)

For the remainder of this analysis we will assume thatρo(X, 0) = 0, and thus the first term vanishes. This

3

amounts to initially preparing the system in a pure stateor incoherent mixture of states. Although equation (7)is general and may be used to study systems that areinitially prepared in coherent states, we shall not con-sider such situations here. Using Eq. (3), the explicitdefinitions of the matrix elements of the Liouville super-operators are,

iLdαα′,ββ′ ≡ iLαδαβδαα′δββ′

iLd,oαα′,ββ′ ≡ −J d,o

α,ββ′δαα′(1 − δββ′) (8)

iLo,dαα′,ββ′ ≡ −J o,d

αα′,β(1− δαα′)δββ′

iLoαα′,ββ′ ≡ iLαα′,ββ′(1− δαα′)(1 − δββ′),

where iLα = iLαα and J d,oα,ββ′ = Jαα,ββ′ , for β 6= β′, with

a similar definition for J o,d. Using these definitions andthe initial condition discussed above, we obtain the gen-eralized master equation for the evolution of the diagonalelements of the density matrix.

∂

∂tραd (X, t) = −iLαρ

αd (X, t)

+

∫ t

0

dt′∑

β

Mαβ(t′)ρβd (X, t− t′) , (9)

where the memory kernel operator Mαβ(t) is given by,

Mαβ(t) =∑

νν′,µµ′

J d,oα,µµ′

(

e−iLo(X)(t))

µµ′,νν′

J o,dνν′,β ,

(10)and acts on all of the classical degrees of freedom that ap-pear in functions to its right. Next, we analyze the formof the memory kernel operator (10) in order to cast itinto a form that is suitable for the derivation of a masterequation.

Memory kernel

The explicit form of the J operator was derived pre-viously33 and is given by

Jαα′,ββ′ = Cαβδα′β′ + C∗α′β′δαβ , (11)

where

Cαβ = −Dαβ(X0)

(

1 +1

2Sαβ ·

∂

∂P0

)

, (12)

the nonadiabatic coupling matrix element is given bydαβ = 〈α;R0|∇R0

|β;R0〉, Sαβ = ∆Eαβdαβ/Dαβ(X0),and Dαβ(X0) = (P0/M0) · dαβ . We have shown in earlierwork that the action of the C operator on phase spacefunctions may be computed using the momentum-jumpapproximation20,37

Cαβ(X0) ≈ −Dαβ(X0)jαβ(X0) , (13)

where the momentum shift operator, jαβ(X0), is a trans-lation operator in momentum space,

jαβf(P0) ≡ e∆EαβM0∂/∂(P0·dαβ)2

f(P0)

= f(P0 +∆P0αβ), (14)

∆P0αβ = dαβ

(

sgn(P0 · dαβ) (15)

×

√

(P0 · dαβ)2 +∆EαβM0 − (P0 · dαβ))

.

Since momentum shifts occur in conjunction with quan-tum transitions, they depend on the quantum states in-volved in the transition. Consequently, we use the follow-ing notation: X0αβ = (R0, P0αβ) = (R0, P0 +∆P0αβ), so

that, jαβ(X0)f(X0) = f(X0αβ). It is worth noting herethat the momentum shift operators do not act on thefull classical environment. They only act on the classicalvariables X0 that directly couple to the quantum degreesof freedom. We also observe that the argument of thesquare root in Eq. (15) must be positive. This conditionprevents quantum transitions when there is insufficientenergy in the classical degrees of freedom to effect thetransition.The time evolution in Eq. (10), is given by the propaga-

tor, e−iLo(X)t. The quantum-classical Liouville operatorin this expression acts on the entire phase space X andaccounts for the following processes: classical evolutionof the bath coordinates Xb and evolution of the classicalsubsystem coordinates X0 on the mean potential surface(Eµ+E′

µ)/2 with an associated phase factor. This evolu-tion is interspersed with quantum transitions taking thesubsystem to other coherently coupled states where evo-lution is again on mean surfaces with associated phasefactors. In the course of this evolution the system neverreturns to a diagonal state involving evolution on a sin-gle adiabatic surface. For this reason we refer to suchevolution as being in “off-diagonal space”.If one considers the explicit action of the propagators

appearing in the memory kernel, one can show how thisoperator acts on an arbitrary function of the phase spacecoordinates X . The details are given in Appendix A.Using the results obtained there, one can show that thegeneralized master equation can be written as,

∂

∂tραd (X, t) = −iLαρ

αd (X, t) (16)

+

∫ t

0

dt′(

∑

β

Mαβαβ (X, t′)ρβd (X

αβ0αβ,t′ , Xb,t′ , t− t′)

+∑

ν

Mνααν (X, t′)ραd (X

να0αν,t′ , Xb,t′ , t− t′)

)

.

In this expression the superscripts on the coordinates in-dicate the action of a second momentum shift operator,(jνα(X0αν)f(X0αν , Xb) = f(Xνα

0αν , Xb)). This result pro-vides us with a definition of the memory function,

Mαβαβ (X, t) = 2Re

[

Wαβ(t′, 0)

]

Dαβ(X0)Dαβ(X0αβ,t′),

(17)

4

where the subscripts and superscripts on the memoryfunction label the indices on the first and second D func-tion respectively. The phase factor Wαβ is defined as

Wαβ(t1, t2) = e−iR t2t1

dτ ωαβ(R0αβ,τ ) . (18)

Now the actions of all classical propagators and momen-tum jumps have been accounted for explicitly in Eq. (16)so that the memory kernel is a function of the phase spacevariables.

Given the assumption about the initial condition onthe density matrix (ρo(X, 0) = 0), for a two-level sys-tem the generalized master equation (16) is fully equiv-alent to the quantum-classical Liouville equation fromwhich it was derived. This result is also applicable tomulti-level systems in a weak coupling limit where quan-tum transitions among different coherently coupled statesare neglected in the the off-diagonal propagator. Thisamounts to neglecting terms higher than quadratic orderin the nonadiabatic coupling strength, dαβ , in the evolu-tion operators. The time evolution described by Eq. (16)consists of classical evolution along single adiabatic sur-faces and two memory terms. The memory terms accountfor transitions to the mean surface, evolution along thissurface, and transitions to a new adiabatic surface with

rate Mαβαβ , or transitions back to the original surface with

rate Mνααν . Thus, the dynamics of the generalized mas-

ter equation derived here is separated into diagonal andoff-diagonal components providing a framework withinwhich to investigate decoherence in the quantum-classicalsubsystem induced by the bath.

III. MASTER EQUATION

We next consider the conditions under which the gen-eralized master equation (16) may be reduced to a sim-ple master equation without memory. This reductionhinges on the ability to consider the memory kernel as arapidly decaying function so that a Markovian approxi-mation can be made. From its form in Eq. (17) one cansee thatM(X, t), which contains all information on quan-tum coherence, is an oscillatory function. As a result, aMarkovian approximation to the memory kernel cannotbe made directly on the full phase space equation sincethere is no obvious mechanism for the decay of the mem-ory function. It is the decoherence by the environmentthat provides such a mechanism.

In this analysis we exploit the fact that decoherencehas its origin in interactions with the bath degrees offreedom. We have already observed that we are inter-ested in dynamical properties of the quantum-classicalsubsystem. For instance, nonequilibrium average values

of interest have the form,

A(t) =∑

αβ

∫

dX0

∫

dXbAβα(X0)ρ

αβ(X, t)

=∑

αβ

∫

dX0Aβα(X0)ρ

αβs (X0, t) , (19)

where Aβα(X0) are the matrix elements of a property ofthe subsystem and ραβs (X0, t) ≡

∫

dXbραβ(X, t) is the

subsystem density matrix. If the operator Aβα(X0) isdiagonal, then only the diagonal elements of the subsys-tem density matrix are needed to compute its averagevalue. Alternatively, if decoherence quickly destroys theoff-diagonal subsystem density matrix elements, then, af-ter a short transient, only the diagonal elements will beneeded to compute the expectation value. Later, we shallshow that similar considerations can be used to evaluatecorrelation function expressions for transport propertiesof the subsystem.To compute such average quantities, we see that we

need the subsystem density matrix elements. Startingwith the generalized master equation in full phase space(Eq. (16)), we can introduce a bath projection opera-tor,38

P· = ρcb(Xb;R0)

∫

dXb· , (20)

whose action on the density matrix yields the subsystemdensity matrix. Here ρcb(Xb;R0) is the bath density con-ditional on the subsystem configuration space variablesR0. The bath density function is in general quantummechanical but in some applications it may be replacedby its high temperature classical limit. In the projectionoperator formalism it is not necessary to distinguish be-tween these two cases. The complement of P is Q. InAppendix B, using standard projection operator meth-ods,39 we argue that bath averaged correlations involv-ing the fluctuations of the memory kernel from its bathaverage may be neglected. In this case, the subsystemgeneralized master equation is,

∂

∂tραs (X0, t) = (21)

−〈iLαe−iQLαtQραd (X, 0)〉b − 〈iLα〉bρ

αs (X0, t)

−

∫ t

0

dt′〈iLαe−iQLαt′ iQLα〉bρ

αs (X0, t− t′)

+

∫ t

0

dt′(

∑

β

〈Mαβαβ (X, t′)〉bρ

βd (X

αβ0αβ,t′ , t− t′)

+∑

ν

〈Mνααν (X, t′)〉bρ

αd (X

να0αν,t′ , t− t′)

)

.

In this expression the memory function appears in theform of an average over a bath distribution function con-ditional on the subsystem configuration space variables,

〈Mαβαβ (X, t′)〉b ≡

∫

dXbMαβαβ (X, t′)ρcb(Xb;R0). Since the

5

memory function Mαβαβ (X, t′) involves evolution on the

mean of the α and β adiabatic surfaces, the average overbath initial conditions will result in an ensemble of suchtrajectories, each carrying an associated phase factor. As

a result, the bath ensemble average 〈Mαβαβ (X, t′)〉b will

decay on a time scale characterized by the decoherencetime, τdecoh. If the decoherence time is short comparedto the decay of the populations we can make a Markovianapproximation,

〈Mαβαβ (X, t′)〉b ≈ 2

(∫ ∞

0

dt′〈Mαβαβ (X, t′)〉b

)

δ(t′)

≡ 2mαβ(X0)δ(t′) . (22)

Applying this Markovian approximation to Eq. (21), weobtain,

∂

∂tραs (X0, t) = (23)

−

∫

dXbiLαe−iQLαtQραd (X, 0)− 〈iLα〉bρ

αs (X0, t)

−

∫ t

0

dt′〈iLαe−iQLαt′iQLα〉bρ

αs (X0, t− t′)

+∑

β

mαβ(X0)jα→βρβs (X0, t)−mαα(X0)ρ

αs (X0, t) ,

where

mαα(X0) = −∑

ν

∫ ∞

0

dt′〈Mνααν (X, t′)〉b . (24)

The use of the Markovian approximation results in theinstantaneous action of two momentum shift operatorson the density. Consequently, the penultimate term inEq. (23) was rewritten to incorporate a single momentum

shift operator whose action is jα→βf(X0) = f(Xαβ0αβ) re-

sulting from a transition from one single adiabatic surfaceto another single adiabatic surface:

jα→βf(P0) = jαβ(X0)jαβ(X0)f(P0)

= f(P0 +∆P0αβαβ) , (25)

∆P0αβαβ = dαβ

(

sgn(P0 · dαβ) (26)

×

√

(P0 · dαβ)2 + 2∆EαβM0 − (P0 · dαβ))

.

This momentum shift differs from that defined earlier bya factor of 2 in front of the energy difference since thisoperator captures the action of two jumps. In the lastterm in Eq. (23) the net effect of two momentum shiftoperators with reversed indices acting simultaneously isjνα(X0αν)jαν(X0)f(X) = f(X). Since these momentumshift operators are inverses of each other, there is no netshift.As discussed in Appendix A, the transition rate

Mαβαβ (X, t) captures the effect of two momentum shift op-

erators. Each of these operators imposes a condition on

the subsystem kinetic energy that ensures transitions toand from the mean surface are allowed. Consequently,the transition rate mαβ(X0) inherits these conditions.For example, if α < β then mαβ(X0) is non-zero only

if (P0 · dαβ)2/2M0 > ∆Eβα. Conversely, if α > β there

is no such restriction. In contrast, the transition rate

mαα(X0) is non-zero only if (P0 · dαν)2/2M0 > ∆Eνα/2.

This condition arises from the fact that this contributionhas its origin from transitions to the mean surface andthen back to the original surface.

Lift to full phase space

Equation (23) is still rather difficult to solve since itcontains a convolution involving bath projected dynam-ics. Often, the non-Markovian character of an equationcan be removed by expanding the space upon which theequation is defined. In the analysis above, the non-Markovian character arose by projecting out the bathvariables to obtain a description in the subsystem phasespace. Consequently, by lifting this equation back intothe full phase space we can recover the Markovian natureof the dynamics. In the full phase space the equation ofmotion is given by

∂

∂tραd (X, t) = −iLαρ

αd (X, t) (27)

+∑

β

mαβ(X0)jα→βρβd (X, t)−mαα(X0)ρ

αd (X, t) .

It is easily verified that applying the projection operatoralgebra to this equation, using the projection operatordefined in Eq. (20), one obtains the subsystem evolu-tion equation (23). Thus, when computing average val-ues like those in Eq. (19), or their correlation functionanalogs discussed below, the master equation lifted tofull phase space yields results identical to those of thenon-Markovian equation (23).Through this analysis we have succeeded in finding a

master equation description of the dynamics in the fullphase space which incorporates the effects of decoher-ence. The first term in Eq. (27) yields dynamics on sin-gle adiabatic surfaces. The other terms correspond tocontributions to the evolution due to nonadiabatic tran-sitions between adiabatic states. The nonadiabatic tran-sition rates in these terms incorporate the effects of de-coherence. Given the nature of the dynamics generatedby this master equation, there is a close connection tomany currently-used surface-hopping schemes which willbe discussed below.It is instructive to compare master equation and

quantum-classical Liouville dynamics. The master equa-tion (27), like the full quantum-classical Liouville equa-tion (2), can be simulated by following an ensemble ofsurface-hopping trajectories. The trajectories that enterin each description are shown in Fig. 1. We see thatin full quantum-classical Liouville dynamics the system

6

makes transitions between single adiabatic surfaces viacoherently coupled off-diagonal states. Coherence is cre-ated when such an off-diagonal state is entered and isdestroyed when it is left. The average over the ensembleaccounts for net destruction of coherence in the systemas it evolves. In contrast, the master equation evolvesthe classical degrees of freedom exclusively on single adi-abatic surfaces with instantaneous hops between them.Transitions from a diagonal state to a coherently coupledstate and then back to the diagonal state, which play animportant role in quantum-classical Liouville dynamics,are accounted for explicitly in master equation dynamicsby mαα(X0). Each single (fictitious) trajectory accountsfor an ensemble of trajectories that correspond to differ-ent bath initial conditions. In this connection the evo-lution in off-diagonal space is crucial: for a given initialsubsystem coordinate, the choice of different bath coor-dinates will result in different trajectories on the meansurface. Thus, it is the average over this collection ofclassical evolution segments that results in decoherence.Consequently, this master equation in full phase spaceprovides a description in terms of fictitious trajectories,each of which accounts for decoherence. When the ap-proximations that lead to the master equation are valid,this provides a useful simulation tool since no oscillatoryphase factors appear in the trajectory evolution.

IV. APPLICATION TO REACTION RATES

In this section we apply the formalism developed aboveto calculate the rate constants of a reaction A B.For this reaction, the quantum-classical forward rate con-stant was derived earlier and is given by40

kAB(t) =1

neqA

∑

α

∑

α′>α

(2 − δα′α) (28)

×

∫

dXRe[

Nαα′

B (X, t)Wα′αA (X,

i~β

2)]

,

where Nαα′

B (X, t) is the time evolved matrix elementof the number operator for the product state B. Att = 0 this operator is diagonal in the adiabatic basisand its value, in this context, depends only on the sub-system coordinates R0. The spectral density function,Wα′α

A (X, i~β/2), accounts for the quantum equilibriumstructure of the entire system.40,41 The spectral densitycan be approximated by the form Wα′α

A (X, i~β/2) ≈

Wα′αA (X0, i~β/2)ρ

cb(Xb;R0), such that it is factorized

into subsystem and bath components.42 Performing theintegration over the bath variables, the rate constant ex-

FIG. 1: Trajectories that enter the solution of the quantum-classical Liouville and master equations. (a) In quantum-classical Liouville dynamics, the system makes transitions be-tween single adiabatic surfaces and coherently coupled statesinvolving evolution on mean surfaces. (b) In master equa-tion evolution, the dynamics is restricted to single adiabaticsurfaces. All off-diagonal evolution is accounted for by thememory kernel.

pression may be written as,

kAB(t) =1

neqA

∫

dX0dXbNαα′

B (X, t)ρcb(Xb;R0)

×Wα′αA (X0,

i~β

2) (29)

=1

neqA

∫

dX0Re

[

〈Nαα′

B (X, t)〉bWα′αA (X0,

i~β

2)

]

.

We see that the calculation of the rate coefficient entailsknowledge of the bath average of the time-evolved speciesvariable and sampling from the subsystem spectral den-sity function. The time evolution of this species variablemay be calculated using mixed quantum-classical dynam-ics.43 In general, the subsystem spectral density containsboth diagonal and off-diagonal components; therefore,both diagonal and off-diagonal components of the speciesoperator contribute to the computation of the rate coef-ficient. Previous work has shown that the off-diagonalcontributions to the rate constant are negligible,44 allow-ing one to consider only diagonal contributions.The computation of the time evolution of the bath av-

eraged species variable is completely analogous to thecalculation of the subsystem density matrix leading toEq. (23); however, now the analysis must be carried outstarting with the quantum-classical Heisenberg equation

7

of motion,33

d

dtAαα′

(X, t) =∑

ββ′

iLαα′,ββ′Aββ′

(X, t). (30)

The rate coefficient can be computed from the expressionin the first line of Eq. (29) using the lifted form of the evo-lution equation for the diagonal elements of a dynamicalvariable,

d

dtAαα(X, t) = iLα(X0)A

αα(X, t) (31)

+∑

β

m†αβ(X0)jα→βA

ββ(X, t)−m†αα(X0)A

αα(X, t) ,

where the memory function, m†, is the adjoint of m de-fined previously in Eq. (22). The effects of decoherencethat lead to this expression restrict the evolution of theobservable to its diagonal components. Therefore, oneonly needs to consider the diagonal terms of the subsys-tem spectral density in the calculation of the rate coeffi-cient.

Model system

As an application of this formalism, we consider a sim-ple model for a quantum rate process that has been stud-ied earlier using mixed quantum-classical dynamics.44

The investigation of this model allows us to assess thevalidity of the Markovian approximation, Eq. (22), andthe utility of the master equation for calculating the ratecoefficientThe model is a two-level system to which we couple

ν oscillators. The subsystem consists of the two-levelquantum system bilinearly coupled to a non-linear oscil-lator with phase space coordinates (R0, P0) governed bya symmetric quartic potential, Vq(R0) = aR4

0/4− bR20/2.

The bath consists of ν − 1 = 300 harmonic oscillatorswhose frequencies ωj, are distributed with Ohmic spec-tral density that depends on ξK , the Kondo parameter.45

The bath is bilinearly coupled to the subsystem oscillatorsuch that the quantum system does not directly interactwith the bath; it only feels its effects through the cou-pling to the quartic oscillator. As discussed elsewhere,it has been argued that this model captures many of theessential features of condensed phase proton transfer pro-cesses.20,44

Using a diabatic representation, the Hamiltonian forthis system is,

H =

(

Vq(R0) + ~γ0R0 −~Ω−~Ω Vq(R0)− ~γ0R0

)

(32)

+

P 20

2M0+

ν−1∑

j=1

P 2j

2Mj+

Mjω2j

2

(

Rj −cj

Mjω2j

R0

)2

I.

The solution of the eigenvalue problem for this Hamilto-nian yields the adiabatic eigenstates, |α;R0〉, and eigen-

values Eα(R) = Vq(R0)+ Vb(Rb;R0)∓ ~√

Ω2 + (γ0R0)2,

−5 0 5−30

−20

−10

0

10

20

R0

W(R

0)

Strong Coupling

−5 0 5−6

−4

−2

0

2

4

R0

W(R

0)

Weak Coupling

FIG. 2: Plots of free energy vs R0 for strong and weak cou-pling cases. The parameters are: γ0 = 10.56 for strong cou-pling and γ0 = 2.64 for weak coupling between the two-levelquantum system and the quartic oscillator. The other pa-rameters are the same for both cases: Ω = 0.51, β = 0.5,ξK = 2, A = 0.5 and B = 1. These parameters were chosento give a well-defined rate process with a significant num-ber of re-crossing events. The small energy gap ensures thatthe majority of trajectories satisfy the energetic requirementsfor nonadiabatic transitions given by (15) and (26), and theparameter β was chosen to be small enough to satisfy thehigh temperature approximation. All other parameters in theOhmic spectral density are the same as those used in earlierstudies,44 and the results are presented in the same dimen-sionless units as those used in previous studies.56.

where 2Ω is the adiabatic energy gap. The adiabatic freeenergy surfaces,Wα(R0) = Vq(R0)∓~

√

Ω2 + (γ0R0)2 aresketched in Fig. 2. In this figure we also show the meanfree energy surface, W12(R0) = (W1 +W2)/2 = Vq(R0),which plays an essential role in the calculation of thememory function.

The simulations of quantum-classical Liouville dynam-ics were carried out using the sequential short-time prop-agation algorithm46 in conjunction with the momentum-jump approximation20,37 and a bound on the observ-able.20 The initial positions and momenta of the quar-tic oscillator and bath were sampled from the classicalcanonical density function. The details of these methodscan be found elsewhere.20,44,46 The simulations of themaster equation consist of two parts which we describebelow. First we computemαβ(X0) in an independent cal-culation involving evolution on the mean surface. Thenwe use this result in the sequential short-time propaga-tion algorithm restricted to single adiabatic surfaces.

Calculation of mαβ(X0)

In order to investigate the validity of the Markovian

approximation we calculate 〈Mαβαβ (X, t)〉b, as a function

of time. From Eq. (29), this average is weighted by

8

0 0.5 1 1.5 2 2.5 3−2

0

2

4

6

8

10

t

<M

1212(X

, t)>

b

FIG. 3: Plot of the bath averaged memory function〈M12

12 (X, t)〉b versus time for γ0 = 2.64, solid line: R0 =−0.55, P0 = 3.2, dotted line: R0 = 0.4, P0 = 2.4, dashed line:R0 = −0.25, P0 = −3.4, dot-dash line: R0 = 0.6, P0 = −2.2.Here we see that for a range of choices of X0, this functiondecays quickly.

ρcb(Xb;R0), the Wigner representation of the quantumbath distribution conditional on the subsystem coordi-nate. In general the determination of the quantum dis-tribution function is a difficult problem; however, it isknown for a harmonic bath,47 and may be used to ac-count for quantum bath effects. In the Conclusions wecomment briefly on quantum bath effects in our formal-ism. In our calculations we use the high temperaturelimit where the classical canonical equilibrium density,conditional on the subsystem configuration, provides agood approximation.41

The quantity 〈Mαβαβ (X, t)〉b involves the product of the

initial value of Dαβ, the phase factor Wαβ , and Dαβ ata time-evolved phase point. The latter two quantitiesmay be obtained from adiabatic dynamics on the meansurface for a given X0. The bath averaged memory func-

tion, 〈Mαβαβ (X, t)〉b may be computed from an average

over an ensemble of trajectories, each with a fixed ini-tial value of X0 and bath coordinates drawn from thephase space distribution ρcb(Xb;R0). As discussed above,the bath average of this oscillatory function provides amechanism for its decay, characterized by the decoher-ence time, τdecoh. This time will depend on the subsys-

tem coordinate X0. In Fig. 3 we plot 〈Mαβαβ (X, t)〉b as a

function of time for several subsystem coordinate valuesand show that the bath averaged memory function doesindeed decay on a rapid time scale. Figure 4 shows howthe decoherence time, taken as the first zero crossing of

〈Mαβαβ (X, t)〉b, depends on the phase space coordinateX0.

In the allowed phase space regions, the decoherence timeis a relatively weak function of the phase space coordi-nates, with the exception of some localized regions whereit varies strongly. From these results we may computethe mean decoherence time and find τdecoh = 0.41± 0.09

−1

0

1

−4−2

02

40

1

2

R0P

0

τ deco

h

FIG. 4: Plot of τdecoh corresponding to upward transitions1 → 2 vs R0 and P0 for γ0 = 2.64.

(weak coupling) and τdecoh = 0.17 ± 0.02 (strong cou-pling). In order for the Markovian approximation to bevalid, the decoherence time must be short compared tothe characteristic decay times of the correlation functionthat determines the rate constant.Simulation of the master equation requires knowledge

of the transition rates mαβ(X0). These quantities wereobtained by numerically integrating the time dependentmemory function discussed above. In this calculation onemust ensure that for a given X0 the transition is allowed.Otherwise mαβ(X0) is assigned a value of zero for thatchoice of subsystem coordinates. These restrictions werediscussed in Sec. III. This process is repeated for a rangeof X0 values generating the surface, mαβ(R0, P0). Weobtain a different surface for each transition (see Fig. 5).The structure of these transition-rate surfaces is due

entirely to classical evolution of X0 along the mean sur-face. It is precisely this evolution that leads to spread inthe ensemble of trajectories giving rise to decoherence.Thus, even though the evolution we are ultimately in-terested in calculating is entirely in diagonal space, theprobability of the nonadiabatic transitions is calculatedfrom the off-diagonal or coherent evolution segments de-pendent on X0. In this way decoherence is accounted forin the formalism.

Simulation of the master equation

Once the surfaces, mαβ(X0) are obtained, Eq. (31) issimulated using the sequential short-time propagation al-gorithm where the probabilities of nonadiabatic transi-tions are given by Π = |mαβ |∆t/(1 + |mαβ|∆t).46 Notethat the value of Π is determined at each time step usingthe value of mαβ(X0) corresponding to the specific valueof X0 at that time. The initial sampling is taken fromthe spectral density function where the bath distributionis given by the conditional density, ρcb(Xb;R0).

9

FIG. 5: Plots of mαβ(R0, P0) versus R0 and P0 for γ0 =2.64. (a) m12(R0, P0) portions of the surface have a value ofzero, corresponding to regions where transitions are forbid-den due to insufficient kinetic energy in the subsystem. (b)m21(R0, P0) does not have this feature as it corresponds todownward transitions where the subsystem gains kinetic en-ergy.

The results of the calculation of the forward time de-pendent rate coefficient, kAB(t), are shown in Fig. 6. Thefigure compares the rate coefficients using adiabatic, mas-ter equation, and quantum-classical Liouville dynamics.As expected the plots show rapid decay on a time scaleτmic to a plateau region characterized by a much slowerdecay on the macroscopic chemical relaxation time scale,τchem ≈ 67 for weak coupling and ≈ 1.3× 106 for strongcoupling.48

In Fig. 6, we see that the short-time decay portion ofthe rate coefficient given by the master equation simu-lation is in agreement with the quantum-classical result.The time scale of this decay, τmic ≈ 4 in the weak cou-pling case and ≈ 2.5 for strong coupling, is about oneorder of magnitude larger than the average decoherencetime τdecoh ≈ 0.41 for weak coupling, and ≈ 0.17 forstrong coupling as discussed above. From the figures weconclude that indeed τdecoh ≪ τmic ≪ τchem. This in-

equality provides the conditions for the applicability ofthe Markovian approximation used to derive the masterequation. The plateau regions for both quantum-classicalLiouvile and master equation dynamics have lower valuesthan those for adiabatic dynamics. The smaller rate con-stant for nonadiabatic dynamics is due to enhanced bar-rier recrossing as a result of motion on either the excitedstate or mean surfaces. The plateau value using masterequation dynamics in the strong coupling case is slightlyhigher than that obtained using quantum-classical Liou-ville dynamics. This likely arises from the fact that inquantum-classical Liouville dynamics the system evolveson the mean surface for long times, allowing trajecto-ries to re-enter the region of high nonadiabatic couplingwhere quantum transitions take place. Thus, the ratecoefficient is reduced due to recrossings in the barrier re-gion. In general, for both weak and strong coupling, themaster equation provides quite a good description of therate coefficient data.

V. CONCLUSION

The master equation calculations presented above bearmany similarities to surface-hopping schemes that havebeen used previously to simulate nonadiabatic dynam-ics of quantum-classical systems49,50,51,52,53,54. It is use-ful to comment on some of the similarities and highlightthe differences. In most surface-hopping schemes, and inour master equation dynamics, the classical degrees offreedom evolve on single adiabatic potential energy sur-face segments according to Newton’s equations of motiongoverned by Hellmann-Feynman forces. This dynamicsshould be contrasted with the trajectory evolution inquantum-classical Liouville dynamics, where the trajec-tory segments of the classical degrees of freedom evolveon single adiabatic surfaces as well as mean surfaces.The differences between our master equation dynamicsand other surface-hopping methods lie in the prescrip-tion for quantum transitions and the manner in whichdecoherence is incorporated into the theory. For exam-ple, in the fewest-switches surface hopping scheme,49,50,51

the probability of a transition depends on the nonadia-batic coupling matrix elements and the off-diagonal el-ements of the density matrix. In our master equationthe probabilities of quantum transitions are determinedby a Monte Carlo sampling based on the magnitudes ofour phase space dependent transition rates, mαβ(X0),and the sampling algorithm reweights averages so thatno bias is introduced. Decoherence is accounted for inthe fewest-switches simulation by collapsing the densitymatrix onto a diagonal state depending on certain con-ditions such as motion outside a window of strong cou-pling.51 In our calculation decoherence effects have beenincorporated into the calculation of the transition rates.Decoherence has also been incorporated into the for-

mulations of surface-hopping methods using other phys-ical principles. The idea in such methods is to include

10

FIG. 6: Plots of the time dependent rate coefficient as afunction of time. (a) γ0 = 2.64 (weak coupling) and (b)γ0 = 10.56 (strong coupling). In these plots the upper dottedcurve is from adiabatic dynamics, the middle curve is frommaster equation dynamics, and the lowest solid curve is fromquantum-classical Liouville dynamics.

the effects of decoherence in single trajectories, much likethe description our master equation provides. For exam-ple, in the methods developed by Rossky et. al.53,54,55

decoherence is introduced through an additional termin the evolution equation that accounts for the quan-tum dispersion about each classical phase space coordi-nate in the bath. In this way each of the trajectoriesin the full phase space experiences decoherence. Thequantum dispersion of the bath is not included in themodel calculations presented above but it is easily ac-counted for in our theory. In our master equation de-coherence arises through averaging over the bath phasespace variables which were taken to be classically dis-tributed for high temperatures. The rate coefficient for-malism in Eq. (29) involves sampling from the full quan-tum spectral density function thus incorporating quan-tum dispersion in the bath coordinates. Such quantumeffects have already been investigated in the context of

quantum-classical Liouville dynamics43 and for the tem-peratures used in our calculations, these effects are verysmall. Regardless of whether the bath is treated clas-sically or quantum mechanically, decoherence enters ourmaster equation through the forms of the transition ratesand not as an additional term in the equation of motion.Finally, we remark that the simulation scheme for mas-

ter equation dynamics has a number of attractive fea-tures when compared to quantum-classical Liouville dy-namics. The solution of the master equation consists oftwo numerically simple parts. The first is the compu-tation of the memory function which involves adiabaticevolution along mean surfaces. Once the transition ratesare known as a function of the subsystem coordinates,the sequential short-time propagation algorithm may beused to evolve the observable or density. Since the dy-namics is restricted to single adiabatic surfaces, no phasefactors enter the calculation increasing the stability ofthe algorithm. For complex reaction coordinates whichare arbitrary functions of the bath coordinates the cal-culation of the transition rates will be more difficult andtime consuming. Future research will determine if themaster equation can be applied easily to realistic generalmany-body systems. Nevertheless, the results reportedin this paper have served to provide a basis for an un-derstanding of the domain of validity of master equationapproaches to quantum-classical nonadiabatic dynamicsbased on decoherence.

Acknowledgements

This work was supported in part by a grant from theNatural Sciences and Engineering Research Council ofCanada.

APPENDIX A: REDUCTION TO MEMORY

FUNCTION

Starting from the form of the memory kernel opera-tor given in Eq. (10), we may reduce this operator to afunction. Without loss of generality, we assume that theadiabatic basis is real so that, Cαβ = C∗

αβ . Furthermore,taking the definition of J , and acting with the operatorjαν(X0) coming from the leftmost J d,o operator on alloperators to its right, the memory kernel operator maybe written as

Mαβ(t)

=∑

νν′

Dαν(X0)2Re[

Uoαν,βν′(X, t) + Uo

αν,ν′β(X, t)]

×Dν′β(X0)jν′β(X0)jαν(X0) . (A1)

In the above expression we have introduced the off-diagonal propagator, Uo

αν,βν′(X, t) =(

e−iLo(X)t)

αν,βν′.

In the case of a two level system this propagator is given

11

exactly by

Uoµµ′,νν′(t) = Wµµ′(t, 0)e−iLµµ′ (X)tδµνδµ′ν′ , (A2)

for µ, µ′, ν, ν′ = 1, 2 and µ 6= µ′, ν 6= ν′. Here we usedthe fact that33

e(−iωµµ′−iLµµ′ )(t−t′) = e−iR

t′

tdτ ωµµ′ (R0µµ′,τ )e−iLµµ′(t−t′)

≡ Wµµ′(t, t′)e−iLµµ′ (t−t′) , (A3)

to express the operator as a product of a phase factorand a classical propagator. We note that the only off-diagonal propagator matrix elements that contribute tothe dynamics here are Uo

12,12 = Uo∗21,21.

Recall from the definition of the momentum shift oper-ator, Eq. (15), that transitions can only occur if there issufficient momentum in the subsystem to make a transi-tion to or from a mean surface. Otherwise the transitionsare not allowed. Using the above form of the off-diagonalpropagator in Eq. (A1), the action of the memory kerneloperator on some arbitrary function of the phase spacevariables, f(X0, Xb), takes the following form:

Mαβ(t)f(X0, Xb) =

δαβ∑

ν

2Re[

Wαν(t, 0)]

Dαν(X0αν)e−iLαν(X),t

×Dνα(X0αν)jνα(X0αν)f(X0αν , Xb)

+2Re[

Wαβ(t, 0)]

Dαβ(X0αβ)e−iLαβ(X)t (A4)

×Dαβ(X0αβ)jαβ(X0αβ)f(X0αβ , Xb) .

The arguments of f reflect the fact that we have actedwith the rightmost momentum shift operator on the func-tion. If we now consider the action of the classical prop-agators that enter in the memory kernel operator we ob-tain,

e−iLαν(Xαν)tDνα(X0αν)jνα(X0αν)f(X0αν , Xb)

= Dνα(X0αν,t)jνα(X0αν,t)f(X0αν,t, Xb,t)

= Dνα(X0αν,t)f(Xνααν,t, Xb,t) . (A5)

In the last line we denoted the indices coming from theaction of the second momentum shift operator as super-scripts (jνα(X0αν)f(X0αν , Xb) = f(Xνα

0αν , Xb)). Substi-tuting Eq. (A5) in the expression (A4) for the memorykernel we obtain,

Mαβ(X, t)f(X0, Xb) = δαβ∑

ν

Mνααν (X, t)f(Xνα

αν,t, Xb,t)

+Mαβαβ (X, t)f(Xαβ

αβ,t, Xb,t) , (A6)

where the definition of M is given in Eq. (17).

APPENDIX B: SUBSYSTEM MASTER

EQUATION

In this appendix we focus on the equation of motionfor the subsystem density matrix. In order to simplify

the notation in the following calculation, it is convenientto write the generalized master equation (16) in a moreformal and compact form. Letting,

∑

β

Mαβαβ (X, t′)ραd (X

αβ0αβ,t′ , Xb,t′ , t− t′)

+∑

ν

Mνααν (X, t′)ραd (X

να0αν,t′ , Xb,t′ , t− t′)

≡(

M(X, t′)ρd(Xt′ , t− t′))

α. (B1)

we can write Eq. (16) as

∂

∂tρd(X, t) = −iLdρd(X, t)

+

∫ t

0

dt′M(X, t′)ρd(Xt, t− t′) . (B2)

Starting from Eq. (B2), we use standard projectionoperator methods39 to obtain the evolution equation forthe subsystem density matrix. If we let ρcb(Xb;R0) bethe bath equilibrium density matrix conditional on theconfiguration of the directly coupled R0 subsystem co-ordinates, we may define the projection operator as inEq. (20) and it’s complement, by Q = 1 − P . Note thatPρd(X, t) = ρcb(Xb;R0)ρs(X0, t).

Applying these projectors to the generalized masterequation (B2) we obtain,

∂

∂tPρd(X, t) = −PiLdPρd(X, t)− PiLdQρd(X, t)

+

∫ t

0

dt′PM(X, t′)Pρd(Xt′ , t− t′) (B3)

+

∫ t

0

dt′PM(X, t′)Qρd(Xt′ , t− t′) ,

∂

∂tQρd(X, t) = −QiLdPρd(X, t)−QiLdQρd(X, t)

+

∫ t

0

dt′QM(X, t′)Pρd(Xt′ , t− t′) (B4)

+

∫ t

0

dt′QM(X, t′)Qρd(Xt′ , t− t′) .

Solving the second equation formally we obtain,

Qρd(X, t) = e−iQLdtQρd(X, 0) (B5)

−

∫ t

0

dt′e−iQLdt′

iQLdPρd(X, t− t′) + Φ(X, t) ,

where the function Φ(X, t) involves fluctuations of thememory function from its bath average, δM ≡ M−〈M〉b,at various time displaced coordinates. Substituting this

12

solution into Eq. (B3) gives,

∂

∂tρs(X0, t) =

−〈iLd〉bρs(X0, t)−

∫

dXbiLde−iQLdtQρd(X, 0)

+

∫ t

0

dt′〈iLde−iQLdt

′

iQLd〉bρs(X, t− t′)

+

∫ t

0

dt′〈M(X, t′)〉bρs(Xt′ , t− t′) (B6)

+

∫ t′

0

∫

dXbδM(X, t′)e−iQLd(t−t′)Qρd(Xt′ , 0)

+

∫ t

0

∫ t−t′

0

dt′dt′′〈δM(X, t′)e−iQLdt′′

iQLd〉b

×ρs(Xt′′ , t− t′ − t′′)

+

∫ t

0

dt′∫

dXbδM(X, t′)Φ(Xt′ , t− t′) .

The last three terms in this equation involve integralsover the bath of expressions containing fluctuations ofthe memory kernel from its bath average. These expres-

sions consist of δM correlated with dynamical quantitiesevolved under projected dynamics. By definition, δM isinitially zero and, due to the presence of the phase factor,it oscillates strongly for long times. Consequently, thebath integral of the product of the oscillatory functionδM with a time evolved dynamical quantity is expectedto be small. Taking these considerations into account,we neglect the last three terms in Eq. (B6). Making thisapproximation, the subsystem evolution equation takesthe form,

∂

∂tρs(X, t) = (B7)

−

∫

dXbiLαe−iQLdtQρd(X, 0)− 〈iLd〉bρs(X, t)

−

∫ t

0

dt′i〈Lde−iQLdt

′

iQLd〉bρs(X, t− t′)

+

∫ t

0

dt′〈M(X, t′)〉bρs(Xt′ , t− t′) .

This equation, written explicitly in terms of its compo-nents is given in Eq. (21) and forms the basis for thereduction to a master equation.

1 U. Weiss, Quantum Dissipative Systems (World Scientific,1999), 2nd ed.

2 H. P. Breuer and F. Petruccione, The Theory of Open

Quantum Systems (Oxford University Press, 2002).3 G. Lindblad, Rep. Math. Phys. 10, 393 (1976).4 A. G. Redfield, IBM J. Res. Dev. 1, 19 (1957).5 A. O. Caldeira and A. J. Leggett, Physica A 121, 587(1983).

6 W. G. Unruh and W. H. Zurek, Phys. Rev. D 40, 1071(1989).

7 B. L. Hu, J. P. Paz, and Y. Zhang, Phys. Rev. D 45, 2843(1992).

8 R. Karrlein and H. Grabert, Phys. Rev. E 55, 153 (1997).9 V. Romero-Rochin and I. Oppenheim, Physica A 155, 52(1989).

10 Q. Shi and E. Geva, J. Chem. Phys. 119, 12063 (2003).11 M. Esposito and P. Gaspard, Phys. Rev. E 68, 066112

(2003).12 R. Kapral, Annu. Rev. Phys. Chem. 57, 129 (2006).13 M. Toutounji and R. Kapral, J. Chem. Phys. 268, 79

(2001).14 M. Toutounji, J. Chem. Phys. 123, 244102 (2005).15 M. Toutounji, J. Chem. Phys. 125, 194520 (2006).16 An example of this is the use of the solvent polarization

to monitor proton or electron transfer in the condensedphase17,18,19,20,21.

17 R. A. Marcus and N. Sutin, Biochim. Biophys. Acta 811,265 (1985).

18 A. Warshel, J. Phys. Chem. 86, 2218 (1982).19 D. Laria, G. Ciccotti, M. Ferrario, and R. Kapral, J. Chem.

Phys. 97, 378 (1992).20 G. Hanna and R. Kapral, J. Chem. Phys. 122, 244505

(2005).21 P. M. Kiefer and J. T. Hynes, Solid State Ion. 168, 219

(2004).22 R. Kapral, J. Phys. Chem. A 105, 2885 (2001).23 A quantum-classical description of the system can also be

derived from considerations based on decoherence by partof the system24.

24 K. Shiokawa and R. Kapral, J. Chem. Phys. 117, 7852(2002).

25 V. I. Gerasimenko, Repts. Acad. Sci. Ukr.SSR pp. 64–67(1981).

26 V. I. Gerasimenko, Theor. Math. Phys. 50, 49 (1982).27 I. V. Aleksandrov, Z. Naturforsch. 36, 902 (1981).28 W. Boucher and J. Traschen, Phys. Rev. D 37, 3522

(1988).29 W. Zhang and R. Balescu, J. Plasma Physics 40, 199

(1988).30 C. C. Martens and J.-Y. Fang, J. Chem. Phys. 106, 4918

(1997).31 I. Horenko, C. Salzmann, B. Schmidt, and C. Schutte, J.

Chem. Phys. 117, 11075 (2002).32 Q. Shi and E. Geva, J. Chem. Phys. 121, 3393 (2004).33 R. Kapral and G. Ciccotti, J. Chem. Phys. 110, 8919

(1999).34 V. V. Kisil, Europhys Lett 72, 873-879 (2005).35 O. V. Prezhdo, J. Chem. Phys. 124, 201104 (2006).36 R. P. Feynman, Phys. Rev. 56, 340 (1939).37 R. Kapral and G. Ciccotti, Bridging Time Scales: Molec-

ular Simulations for the Next decade (Springer, 2001),chap. 16, p. 445.

38 For a discussion of the bath projection operator for quan-tum systems, see for example, Romero-Rochin and Op-penheim9. For our quantum-classical system we define theprojection operator using the canonical bath density con-ditional upon the subsystem coordinate R0.

39 R. W. Zwanzig, Statistical Mechanics of Irreversibility

13

(Wiley-Interscience, 1961), p. 106.40 H. Kim and R. Kapral, J. Chem. Phys. 123, 194108 (2005).41 A. Sergi and R. Kapral, J. Chem. Phys. 121, 7565 (2004).42 H. Kim and R. Kapral, J. Chem. Phys. 122, 214105 (2005).43 H. Kim and R. Kapral, Chem. Phys. Lett. 423, 76 (2006).44 H. Kim, G. Hanna, and R. Kapral, J. Chem. Phys. 125,

084509 (2006).45 N. Makri, J. Phys. Chem. B 103, 2823 (1999).46 D. MacKernan, R. Kapral, and G. Ciccotti, J. Phys.: Con-

densed Matter 14, 9069 (2002).47 K. Imre and E. Ozizmir and M. Rosenbaum and P. F.

Zweifel, J. Math. Phys. 8, 1097 (1967).48 R. Kapral, S. Consta, and L. McWhirter, Classical

and Quantum Dynamics in Condensed Phase Simulations

(World Scientific, 1998), p. 587.49 J. C. Tully, J. Chem. Phys. 93, 1061 (1990).50 J. C. Tully, Int. J. Quantum Chem. 40, 299 (1991).51 S. Hammes-Schiffer and J. C. Tully, J. Chem. Phys. 101,

4657 (1994).52 D. Coker and L. Xiao, J. Chem. Phys. 102, 496 (1995).53 O. V. Prezhdo and P. J. Rossky, J. Chem. Phys. 107, 5863

(1997).54 E. R. Bittner and P. J. Rossky, J. Chem. Phys. 103, 8130

(1995).55 K. F. Wong and P. J. Rossky, J. Chem. Phys. 116, 8418

(2002).56 A. Sergi and R. Kapral, J. Chem. Phys. 118, 8566 (2003).