J. Chem. Phys. 140, 194107 (2014); https://doi.org/10.1063/1.4875702 140, 194107

© 2014 AIP Publishing LLC.

Coherence penalty functional: A simplemethod for adding decoherence in EhrenfestdynamicsCite as: J. Chem. Phys. 140, 194107 (2014); https://doi.org/10.1063/1.4875702Submitted: 27 March 2014 . Accepted: 29 April 2014 . Published Online: 21 May 2014

Alexey V. Akimov, Run Long, and Oleg V. Prezhdo

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THE JOURNAL OF CHEMICAL PHYSICS 140, 194107 (2014)

Coherence penalty functional: A simple method for adding decoherencein Ehrenfest dynamics

Alexey V. Akimov,1,2,a) Run Long,3 and Oleg V. Prezhdo1,a)

1Department of Chemistry, University of Rochester, Rochester, New York 14627, USA2Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973, USA3School of Physics, Complex and Adaptive Systems Laboratory, University College Dublin, Dublin 4, Ireland

(Received 27 March 2014; accepted 29 April 2014; published online 21 May 2014)

We present a new semiclassical approach for description of decoherence in electronically non-adiabatic molecular dynamics. The method is formulated on the grounds of the Ehrenfest dynam-ics and the Meyer-Miller-Thoss-Stock mapping of the time-dependent Schrödinger equation onto afully classical Hamiltonian representation. We introduce a coherence penalty functional (CPF) thataccounts for decoherence effects by randomizing the wavefunction phase and penalizing develop-ment of coherences in regions of strong non-adiabatic coupling. The performance of the methodis demonstrated with several model and realistic systems. Compared to other semiclassical methodstested, the CPF method eliminates artificial interference and improves agreement with the fully quan-tum calculations on the models. When applied to study electron transfer dynamics in the nanoscalesystems, the method shows an improved accuracy of the predicted time scales. The simplicity andhigh computational efficiency of the CPF approach make it a perfect practical candidate for applica-tions in realistic systems. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4875702]

I. INTRODUCTION

Quantum non-adiabatic processes in molecular sys-tems are widespread in nature and technology.1–6 They in-clude electron and proton migration,7–18 charge and energytransfer,16, 19–24 and similar phenomena, which take place innatural and artificial systems. The interest in these processesstimulates development of new theoretical methodologies andsimulation techniques.12–14, 25–46 Quantum dynamics requiressolution of the time-dependent Schrödinger equation (TD-SE). Accurate, fully quantum and advanced semiclassicaland quantum-classical solutions based on numerically exactintegration and wavepacket propagation,18, 35, 47–54 quantum-classical Liouville equation and entangled trajectories,55–59

Bohmian trajectories,60–63 and initial value representation64, 65

have been developed. For a more detailed discussion of theseand related methods for quantum non-adiabatic dynamics werefer the reader to the recent review on the subject.4 Despiteexceptional accuracy and rigorous mathematical foundationof the above methods, they are often restricted to relativelysmall systems. Approximate semiclassical treatments are un-avoidable for realistic, large-scale systems.

The fewest switches surface hopping (FSSH)algorithm25, 26 for molecular dynamics (MD) with elec-tronic transitions is among the most popular semiclassicalapproaches. In this method, the probability of transitionfrom a current electronic state to all other states is computedalong a nuclear evolution trajectory, using solution of theapproximate, semiclassical TD-SE. On the basis of thecomputed wavefunction amplitudes, {ci(t)}, a stochastichop from the current quantum state to one of the accessible

a)Authors to whom correspondence should be addressed. Electronic ad-dresses: alexvakimov@gmail.com and oleg.prezhdo@rochester.edu

states is performed. Making FSSH very popular, its mainadvantages include conceptual and implementation simplicityand, most importantly, ability to correctly describe branchingof quantum trajectories. The latter feature ensures that aftera scattering event the system’s wavefunction collapses ontoone of its pure states (eigenstate). In addition, the methodis capable of approximately reproducing the Boltzmanndistribution of the excited states,66 which is important foraccurate description of relaxation to thermal equilibrium andthermodynamic properties.

Although the FSSH method has been successful in manyapplications, there are cases in which it may be inade-quate or performs less accurately than the mean-field (MF)approaches.27, 67 The FSSH is not straightforward to apply tosystems with a continuum of states. A preliminary discretiza-tion of the continuum is required.30 On the contrary, the MFcalculation can be easily performed.26 In addition, with in-creasing density of states, characteristic of large-scale sys-tems, the chances to encounter conical intersections, as wellas numerically problematic “unavoided” or trivial crossings,increase. They start to dominate the physically interestingavoided crossings, forcing one to keep track of state indexingand to apply corrections to the standard FSSH procedure.68, 69

This problem is naturally overcome by the mean-field Ehren-fest technique.

Neglect of decoherence effects induced in the electronicsubsystem by the nuclear bath constitutes a well-known prob-lem of the most commonly used, original version of FSSH.25

The method is often overcoherent, because solution of thesemiclassical TD-SE, {ci(t)}, preserves coherences, c∗

i cj ,when they should decay to zero and the wavefunction shouldcollapse to an eigenstate according to the fully quantum de-scription of the electron-nuclear system. In his pioneeringwork Tully did point this out and proposed a simple ad hoc

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194107-2 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

scheme that achieved decoherence via damping of the off-diagonal elements of density matrix. Many other decoherencecorrection schemes for FSSH have been proposed later, basedon different principles.37, 44, 45, 70–75 These schemes either in-troduce a stochastic element to the wavefunction evolution ordeal with the density matrix. Stochastic evolution of the wave-function creates additional computational challenges, whileintroduction of the density matrix increases the size of theproblem from N to N2, where N is the basis set size.

The MF or Ehrenfest approach provides an alternativeformulation of non-adiabatic MD (NA-MD). Here, the evo-lution of a mixed quantum-classical system is treated self-consistently on the average potential energy surface.28, 31, 76–78

The trajectories generated in the Ehrenfest dynamics are con-tinuous, and there are no stochastic changes of quantumstates. Although simple, the approach has a number of seri-ous limitations. One of the commonly discussed examples isits inability to collapse system’s wavefunction on a pure state.For instance, after an interaction event has occurred, the re-sulting wavefunction remains a mixture of nearby adiabaticstates, even in regions with negligible coupling between thestates.14 One can resolve this deficiency by a proper interpre-tation. Namely, the results given by the Ehrenfest solutionsshould not be explained from the point of view of instanta-neous properties of a system. Rather, it is a solution for theaverage properties and their distributions. Similarly, quantummechanics can be formulated in the Heisenberg representa-tion, in which one solves the Heisenberg equation for the ex-pectation values (averages), 〈A〉, of a given operator A. Thesequantities cannot be interpreted in terms of instantaneous tra-jectories and wavefunction collapse. They have only a proba-bilistic and statistical interpretation.

Keeping in mind the above discussion, one finds that thesemiclassical Ehrenfest solution of the TD-SE still missesimportant physical effects. In particular, the dynamics doesnot account for decoherence. This is because the nuclearcomponent of the wavefunction reduces to a classical delta-distribution (phase space point), when the transition from thefully-quantum TD-SE to the semiclassical TD-SE is made.In a 2-level system, the complete loss of coherence betweenthe pair of electronic states is only possible in the Ehrenfestmethod if and only if the population of one of the states iszero.

Neglect of electronic decoherence in FSSH and relatedschemes is known to overestimate electronic transition rates.This problem has been mitigated by ad hoc decoherence cor-rection schemes.38, 75, 79 Application of decoherence correc-tions to the Ehrenfest dynamics has led to approaches, suchas the stochastic MF approximation,80 that contain a stochas-tic element and resemble surface hopping rather than the fullydeterministic MF evolution.37, 45, 80, 81

There has been a number of mixed MF/SH schemes tar-geted at accounting for decoherence effects.42, 72–74 For ex-ample, in the self-consistent decay of mixing73 and in thesubsequent coherent switching with decay of mixing,74 thedensity matrix is propagated deterministically to conservethe angular momentum of the mapped electronic variables –the condition not necessarily satisfied in the standard Ehren-fest formulation. The equations lead to deterministic decay

of the off-diagonal elements of the density matrix. Stochasticsurface hops are then undertaken, similarly to FSSH. There-fore, they retain the stochastic flavor of SH and require av-eraging over a large number of stochastic realizations. Simi-lar in spirit to the MF approaches are the methods based onthe quantum-classical Liouville equation of motion for den-sity matrix.55, 56, 82–84 These techniques are fully deterministicand capable of describing decoherence, but are sophisticatedand computationally demanding. The Liouvillian formulationof equations of motion does not necessarily imply existenceof a Hamiltonian structure for the considered dynamical sys-tem.

In this work, we propose a simple empirical decoherencecorrection scheme for the Ehrenfest dynamics. In addition,we formulate this method within the fully Hamiltonian frame-work. The effects of decoherence are introduced in on aver-age sense directly via the wavefunction, to affect the dynam-ics of the expectation values. Unlike many other approachescorrecting for decoherence, we achieve this goal via contin-uous unitary dynamics of the wavefunction, rather than viastochastic, non-unitary, or density matrix techniques.

II. THEORY

A. Formulation of the coherence penaltyfunctional method

We start by referring to the works of Meyer andMiller,85–87 and Stock and Thoss,88, 89 who established a map-ping between the TD-SE and the fully classical Hamiltonianformulation of the equations of motion for quantum variables.The original work of Meyer and Miller used a specific choiceof the variables that represent quantum degrees of freedom.The choice is not unique, and the variables may be selectedquite arbitrarily.90 Depending on the representation of poten-tial, it may be more convenient to choose one set over another.In particular, it can be shown that in the variables qi = Re (ci)and pi = Im (ci), the semiclassical TD-SE,

i¯∂ci (t)

∂t=

∑j

[Ei (R (t)) δi,j − i¯

P

Mdij

]cj (t), (1)

is equivalent to the Hamiltonian equations of motion:

pi = −∂H

∂qi

, (2a)

qi = −∂H

∂pi

(2b)

with the effective mapped Hamiltonian given by

H =∑

i

Ei

2¯

(q2

i + p2i

) − p

m

∑i,j

dijpiqj . (3)

To account for decoherence effects in the dynamics gen-erated by Hamiltonian Eq. (3), we introduce the followingaugmented Hamiltonian:

H = H +∑i,ji �=j

λij

(q2

i + p2i

) (q2

j + p2j

), (4)

194107-3 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

The term (q2i + p2

i )(q2j + p2

j ) = |c∗i cj |2 is simply a square of

the magnitude of coherence between the pair of states i and j.This term penalizes coherence development, each pair havinga different penalty determined by the constant λij. Therefore,we call the sum in the right-hand side of Eq. (4), and the over-all methodology, the coherence penalty functional (CPF). Therole of the functional is easy to understand by considering theeffective energy surface in the phase space spanned by thequantum variables {(qi, pi)}. Each term (q2

i + p2i )(q2

j + p2j )

creates an additional potential that biases system’s dynam-ics to choose its evolution pathways such that they minimizethe developed coherences (provided λij > 0). When the sys-tem accumulates coherence between the pair of states i andj, the point in the phase space of the dynamical system is lo-cated uphill the bias potential. The gradient of the effectiveenergy surface acts in the opposite direction and moves thephase space point away from the regions of large coherences.Therefore, the system avoids developing large coherences. Inthe on average sense, the effective coherences are minimized.Such avoidance of the regions with large coherences can af-fect the average populations of the electronic states as func-tions of time, by changing the rates of transitions betweenthem.

B. Analogy of the CPF method withexchange-correlation functionals in DFT

The CPF may be interpreted from a different viewpoint.The idea of the CPF for the Ehrenfest dynamics has a lot incommon with the density functional theory (DFT). Partiallybecause of this analogy, the name of our current approachcontains the word “functional.” As any other MF theory, theEhrenfest dynamics has a serious problem accounting forcorrelation/entanglement effects. The dynamics of electronicstates is not correlated with nuclear evolutions to yield desiredasymptotic properties. Overcoherence can also be considereda consequence of the lack of dynamic correlation. In the con-ventional electronic structure theory, the lack of correlation inthe MF Hartree-Fock method is well recognized, and multi-ple approaches for incorporating correlation have been devel-oped. Dynamic correlation arises in the electronic structuretheory due to interactions of electrons, leading them to un-dergo correlated motions. (No analogies are drawn here withrespect to the non-dynamic correlation of the electronic struc-ture theory.)

Having discussed the relationship between the Ehrenfestdynamics of electronic states in the mixed quantum-classicalevolution and the mean-field dynamics of electrons in theelectronic structure theory, and having identified the missingcorrelation, we can consider how theories in one field canhelp develop theories in the other field. Namely, introduc-tion of universal exchange-correlation functional constitutesthe central idea of the DFT approach to the electronic struc-ture theory. It is true that the exact form of such functional isnot known, in general, and its approximations are difficult toconstruct. At the same time, a wide variety of density func-tionals has been developed over the last few decades, rang-ing in accuracy, efficiency, and applicability. One can regard

CPF as a counterpart of density functionals, which is aimed atcorrecting one of the correlation-related issues (decoherence)with the Ehrenfest dynamics. The CPF plays the same rolein the Ehrenfest dynamics as the correlation functionals inDFT. Similar to the electronic structure density functionals,the CPF depends on the density (matrix), is hard to derive,in general, and provides a correction to correlation-relatedproblems. The functional form of the CPF given by Eq. (4)is one of the possible approximations. It may be much morecomplex.

C. Choice of parameters and functional form

The parameters λij determine the strength with which co-herences of each pair of states i and j are suppressed. From thedimensional analysis, the variable λij has units of inverse time,and therefore, it has the meaning of rate. Because λij controlscoherences, it is natural to assume that the rate we refer tois proportional to the decoherence rate. Determination of de-coherence rates is not a simple task. An attempt to describedecoherence effects by a single parameter constitutes a strongapproximation, which becomes accurate in large systems withfast decoherence.

There exists a variety of ways to define the decoherencerate, and there are certain conditions under which those def-initions work. Expressions for decoherence times are basedon consideration of wavepacket dynamics, often in the frozenGaussian short-time limit.38–40, 44, 79, 81, 91, 92 The common re-sult is that the decoherence time can be estimated as thetimescale of the decay of overlap of frozen Gaussians repre-senting nuclear degrees of freedom correlated with differentelectronic states. An example of a frozen Gaussian expressionis given by Schwartz:81, 92

1

τij

=∑

n

|Fni (t) − Fn

j (t)|2¯

√an

, (5)

where Fi is the force in the adiabatic level i and an is the effec-tive width of the nuclear wavepacket. According to Schwartzand co-workers, the width of the nuclear wavepacket can beapproximated by

an =[

(w/a0)2

2λD (t)

]2

= m (w/a0)4

2h

mv2

2= m (w/a0)4

2hEkin,

(6)where m is the effective mass of the particle, w is the spa-tial extent of non-adiabatic coupling, a0 is the Bohr radius,h is the Planck constant, vn is the velocity of the particle,and λD (t) = h

mvn(t) is the instantaneous de Broglie wave-length. Note that the quantities m and w are defined semi-

quantitatively; therefore, the entire prefactor m(w/a0)4

2hcan be

considered an empirical parameter.A more general approach to the derivation of the de-

coherence times has been developed by Subotnik and co-workers on the basis of the Wigner transform of the electron-nuclear wavefunction and the quantum-classical Liouvilleequation.45, 59 The general result for a 2-level system suggests

194107-4 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

the state-dependent decoherence rates:

1

τ(1)12

≈ 1

2

∑n

(Fn

11 − Fn22

) 1

A(1)12

∂A(1)12

∂Pn

, (7a)

1

τ(2)12

≈ −1

2

∑n

(Fn

11 − Fn22

) 1

A(2)12

∂A(2)12

∂Pn

. (7b)

Here, Fnij ( �R) = −〈�i( �R)| ∂V

∂ �Rn

|�j ( �R)〉 are the generalizedforces: the diagonal elements are the standard forces, whilethe off-diagonal elements are proportional to the non-adiabatic coupling. A

(λ)12 is the element of the Wigner trans-

form of the joint electron-nuclear wavefunction for λth activeelectronic state. Pn is the momentum of the nth nuclear mode.The results for the Gaussian wavepacket propagation, similarto Eq. (5), can be obtained as a specific case of Eq. (7), byproper ansatz of the nuclear wavefunction.

In the decay of mixing scheme, Truhlar and co-workersutilized the phenomenological expression for the decoherencerates:

τij = ¯

|Ei − Ej |(

1 + C

Ekin

), (8)

where Ei are the energies of the electronic states, Ekin is thekinetic energy of nuclei, and C is an empirical parameter, typ-ically set to 0.1 Hartree.

In general, an approximation describing decoherence bya single parameter should become more accurate in large sys-tems and with fast decoherence. Strong motivation for the de-coherence correction arose from NAMD applications of thistype.39, 40, 79, 93 In such cases, decoherence arises due to thecombined effect of fluctuations in many degrees of freedom,and the decoherence rate can be evaluated by the optical re-sponse theory,94, 95 which gives reliable results and can bebenchmarked against optical measurements of homogeneouslinewidths:96, 97

λ2ij t

2 =(

t

τij

)2

= 1

¯2

t∫0

dt ′t ′∫

0

dt ′′Cij

(t ′′

)

=⟨δE2

ij

⟩¯2

t∫0

dt ′t ′∫

0

dt ′′Cij

(t ′′

), (9a)

Cij (t) = Cij⟨δE2

ij

⟩ = 〈δEij (t) δEij (0)〉⟨δE2

ij

⟩ , (9b)

Cij (t) = 〈δEij (t) δEij (0)〉. (9c)

The approach does not require adjustable parameters and isactively utilized by Prezhdo and co-workers to study quantumdynamics in nanoscale systems.75, 95, 98–102

In the present work, we utilize the optical response for-malism, Eqs. (9), for the realistic system and the modifiedwavepacket result, Eq. (5), for the model. The modification isas follows:

λij = f1

τij

= f|Fi(t) − Fj (t)|

2¯√

a0. (10)

The function f is a proportionality factor between the deco-herence rate, 1

τij, and the strength of decoherence control, λij.

We found empirically that with the parameter a0 set to theconstant value a0 = 1 Bohr, the optimal choice for the pro-portionality factor, f, is

f = 100/p0, (11)

where p0 is the initial nuclear momentum. The constant 100in Eq. (11) was found empirically and produces reasonable re-sults for different model systems. By choosing the fixed valuefor all systems, we pursue the goal of transferability of thefactor f. A functional dependence of the factor f on the initialmomentum and other system parameters can be considered.

Keeping in mind the discussed elaborations in the form ofEq. (11), the physical meaning of the choice given by Eq. (11)is straightforward to rationalize. It represents a linear depen-dence of the nuclear wavepacket width on system tempera-ture, which can be related to the (initial) nuclear kinetic en-ergy. Using Eq. (11), the factor f√

a0can be rewritten as

f√a0

= 1p0

100

√a0

= 1√a0

(p0

100

)2∼ 1√

a0Ekin,0

∼ 1√a0T

∼ 1√a (T )

, (12)

which is consistent with Eq. (6). Note that for the nuclearmass m = 2000 a.u. used in the model calculations, the firstproportionality in Eq. (12) is close to the equality, because

( p0

100 )2 = p20

2m·5/2 = 25Ekin.

D. Analysis of the CPF method

It is illustrative to analyze the limiting behavior of thedecoherence control parameters, λij. In the high-temperaturelimit, T → ∞, the parameter a → ∞, which is equivalent tof → 0, and λij → 0.92 Therefore, the coherence penalty termvanishes, and the original Ehrenfest dynamics is recovered. Itis known that in the high-temperature limit the results of theFSSH and Ehrenfest dynamics coincide. Hence, our schemeis well behaved in the high-temperature limit.

Let us show how decoherence arises from the penaltyterm in the modified Hamiltonian, Eq. (4). The dynamics ofthe phase space variables z = {(qi, pi), i = 1, ...N} is gov-erned by the propagator exp(iLH · dt), where iLH = z ∂

∂z.

The Hamiltonian H can be split into two parts:

H = H + H ′, (13)

where H is the original (closed) system Hamiltonian,Eq. (3), and H ′ = ∑

i,ji �=j

λij (q2i + p2

i )(q2j + p2

j ) is the penalty

functional. The Liouvillian can be split in a similar way:iLH = iLH + iLH ′ , followed by the Trotter factorization:103

exp(iLH · dt) = exp

(iLH ′ · dt

2

)exp (iLH · dt)

× exp

(iLH ′ · dt

2

). (14)

194107-5 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

Expression (14) is a composition of two Hamiltonian flowsmaps. Because Hamiltonian flow maps are symplectic, andbecause the compositions of Hamiltonian flow maps are alsoHamiltonian, the overall flow map is symplectic. This is animportant property for numerical integration of Eq. (2).

From the physical point of view, the picture is clear – theoperator exp (iLH · dt) corresponds to dynamics of the closedsystem, which is described by the Hamiltonian H. The actionof this operator generates the standard Ehrenfest dynamics.The action of the operator exp

(iLH ′ · dt

2

)deserves a closer

examination. Writing iLH′ explicitly, we obtain

iLH ′ =∑

i

(qi

∂

∂qi

+ pi

∂

∂pi

)=

∑i

ϕi

(pi

∂

∂qi

− qi

∂

∂pi

),

(15)where

ϕi = 2∑j �=i

(λij + λji)(q2

j + p2j

). (16)

Equation (16) has a great similarity with the expression for theinverse of the decoherence time intervals used in the Lindbladformulation.75, 104, 105 According to this formulation, the aver-age of the inverse decoherence time, 〈 1

τi〉, for state i is given by

the microscopic decoherence rates between state i and otherstates, ri→j, weighed by the populations of the other states,|cj|2, j �= i:⟨

1

τi

⟩=

∑j �=i

ri,j |cj |2 =∑j �=i

ri,j

(q2

j + p2j

). (17)

The close similarity of Eqs. (16) and (17) reinforces the earlierhypothesis on the physical meaning of parameters λij.

Further application of the Trotter factorization gives

exp (iLH ′ · dt) =N∏

i=1

exp

(ϕi

(pi

∂

∂qi

− qi

∂

∂pi

)dt

2

)

×1∏

i=N

exp

(ϕi

(pi

∂

∂qi

− qi

∂

∂pi

)dt

2

).

(18)

The quantum state of a system is altered by a series of opera-tors of the type

exp

(ϕi

(pi

∂

∂qi

− qi

∂

∂pi

)dt

). (19)

The action of such operator is well known – it propagates thephase of state i:

qi (t + dt) = cos (ϕidt) qi (t) − sin (ϕidt) pi (t) , (20a)

pi (t + dt) = sin (ϕidt) qi (t) + cos (ϕidt) pi (t) , (20b)

or

ci (t + dt) = qi (t + dt) + ipi (t + dt)

= [cos (ϕidt) + i sin (ϕidt)] qi (t)

+ [− sin (ϕidt) + i cos (ϕidt)] pi (t)

= exp (ϕidt) (qi (t) + ipi (t))

= exp (ϕidt) ci (t) . (21)

An operator of a similar type is encountered in the Trotter de-composition of iLH,98 in which the standard dynamical phaseis used: ϕi = Ei

¯. In the above derivation, we assume that ϕi

is constant during any given infinitesimal period of time. Thisapproximation can further be supported by noting that the so-lution given by Eqs. (20) and (21) preserves the norm of thewavefunction. The operator equation (18) causes no popula-tion transfer between the pair of states i and j. The phase shiftϕi of a given electronic state i does not depend on the elec-tronic degrees of freedom of the same electronic state i. It de-pends only on the variables corresponding to all other statesj �= i.

The main effect of the coherence penalty term definedin Eq. (4) is to include additional phase shifts to phases ofall quantum states. Because the phase shifts are different andtime-dependent, the total phase of each adiabatic state di-verges from that of all other states – the states dephase, de-cohere. It should be emphasized that the effects included viathe coherence penalty term constitute only a partial correctionof the Ehrenfest dynamics – no spatial redistribution or statecollapse of wavefunctions is included, all effects arise solelydue to the dynamic phase shifts. Therefore, the method canfail in some cases. Still, it is expected that the CPF approachcan outperform the standard Ehrenfest and even FSSH tech-niques.

It is illustrative to analyze the proposed method from thepoint of view of the mapped QHD theory.106–108 Specifically,one can utilize the Ehrenfest Hamiltonian, Eq. (3), to derivea hierarchy of Heisenberg equations of motion, similar to theone leading to the QHD-2 Hamiltonian.106, 109, 110 Under cer-tain approximations, the equations of motion can be derivedfrom the classically mapped QHD-2 Hamiltonian. In such anextension, quantum effects of nuclei, including decoherence,arise via the wavepacket width variable, s, and its conjugatemomentum, ps. The QHD-2 Hamiltonian can also be obtainedeffectively via the following substitution:

Ei( �R) → Ei( �R) + E(2)i ( �R)s2

i + ..., (22a)

dij ( �R) → dij ( �R) + d(2)ij ( �R)sisj + .... (22b)

In the simplest case, we truncate the Taylor expansions atthe second order of energy and at the zeroth order of non-adiabatic coupling. We also add the corresponding conjugatemomenta and width kinetic energy to express the final modelHamiltonian:

H =∑

i

(Ei+E

(2)i s2

i

)2¯

(q2

i +p2i

)− p

m

∑i,j

dijpiqj

+∑

i

(p2

s,i

2m+ ¯2

8ms2i

)

= H +∑

i

E(2)i s2

i

2¯

(q2

i + p2i

)

+∑

i

(p2

s,i

2m+ ¯2

8ms2i

)= H + H ′, (23a)

194107-6 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

H ′ =∑

i

E(2)i s2

i

2¯

(q2

i + p2i

) +∑

i

(p2

s,i

2m+ ¯2

8ms2i

). (23b)

The equations of motion generated by the Hamiltonian H′ are

qi = ∂H ′

∂pi

= E(2)i s2

i

¯pi, (24a)

pi = −∂H ′

∂qi

= −E(2)i s2

i

¯qi, (24b)

si = ∂H ′

∂ps,i

= ps,i

m, (24c)

ps,i = −∂H ′

∂si

= ¯2

8ms3i

− E(2)i

¯

(q2

i + p2i

)si . (24d)

The term∑

i

E(2)i s2

i

2¯ (q2i + p2

i ) contributes to dephasing ofwavepackets, as it is clear from Eqs. (24a) and (24b). Thevariable s changes in time and is different for each electronicstate, but its derivative depends on the state population. In thecase when the population of the state is 0, the wavepacketmonotonically spreads out, since ps,i > 0 and at some pointsi > 0.

The QHD-2 dynamics is similar to the Ehrenfest dynam-ics in the sense that the expectation values are propagated di-rectly. Performing calculations according to Eqs. (24) con-stitutes one option, but it may be rather elaborate, especiallyfor realistic multi-dimensional systems. At the same time, thephysics of Eqs. (24) is clear – the phase of each electronicstate is corrected by an additional time-dependent term that,in turn, depends on populations of electronic states. This samephysics is modeled by the CPF dynamics, but the latter iscomputationally more tractable for realistic systems. In thepresent work, we only point out to the similarity of the twoapproaches, as the QHD-2 dynamics may be somewhat morerigorous and can be thought of as a basis for and an extensionof the CPF method.

III. RESULTS AND DISCUSSION

In this section, we demonstrate performance of the CPFmethod with two types of systems. First, we apply it to twomodel 1D systems – the famous Tully’s double avoided cross-ing (DAC) and extended coupling with reflection (ECWR)problems.25 We utilize slightly modified parameters for thefirst model, in order to enhance the effects of quantum inter-ference and decoherence. In the second type of application,we study electronic energy relaxation dynamics in two realis-tic nanoscale systems.

A. Model systems

We apply the CPF method to study the scattering proba-bilities for Tully’s DAC and ECWR potentials. The potentialsare defined by

V00 = A exp(−Bx2), (25a)

V11 = E − V00, (25b)

V01 = V10 = C exp(−Dx2) (25c)

with parameters A = 0.1, B = 0.028, C = 0.015, D = 0.06, E= 0.05 for DAC, and

V00 = A, (26a)

V11 = −A, (26b)

V01 = V10 ={

B exp (Cx) , x ≤ 0B exp (−Cx) , x > 0

(26c)

with parameters A = 0.0006, B = 0.1, C = 0.9, D = 1.0 forECWR. Atomic units of energy and length are adopted in allcases. The potentials are depicted in Fig. 1. In our studies wemodify the original DAC potential, to increase the effectivesize of the potential well and enhance decoherence effects.The parameters for the ECWR potential are the same as in theoriginal Tully’s prescription.

Because the CPF approach has a lot in common withthe Ehrenfest method and does not include stochastic trajec-tory branching, it cannot be expected to resolve accuratelyreflection vs. transmission. Further, in multi-dimensional po-tentials, which can correspond to realistic systems, the terms“reflection” and “transition” may be not well defined. Oneis often interested in total populations of different quantumstates rather than directionally resolved scattering probabili-ties. Therefore, in the present work we focus on descriptionof total populations.

In all our simulations, the system starts in a pure state 0(lower energy state), at coordinate x = −20 a.u., in the regionof negligible non-adiabatic coupling between the states. Aninitial positive momentum is assigned to the classical degreeof freedom. The magnitude of the initial momentum is variedas the initial condition parameter. Integration of the equationsof motion is performed over a sufficiently long time interval,to allow the classical trajectory to escape the region of strongcoupling by the end of the evolution. Typically, 10 000 inte-gration time steps of 10 a.u. (∼0.5 fs) satisfy this requirement.The mass of the classical particle is set to m = 2000 a.u.,which is close to the mass of the hydrogen atom.

In order to understand the improvements and possible pit-falls of the new method, we compare its performance withFSSH, Ehrenfest, and exact numerical solutions. The latteris obtained using the split-operator technique combined withthe fast Fourier transform, similar to the procedure of Kosloffet al.47

For the CPF calculations we utilize the decoherence rates,λij, given by Eq. (10). The parameters entering Eq. (10)are defined as a = a0p0 and f = K

p0, where p0 is the ini-

tial wavepacket momentum, a0 is the reference width of thewavepacket, set to 1 a.u., and K is the empirical proportional-ity factor, set to 100 a.u. of momentum. Only a single trajec-tory is needed for the CPF method, similar to the Ehrenfest

194107-7 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

FIG. 1. Model 1D potentials for the 2-level systems: (a) double avoided crossing, DAC; (b) extended coupling with reflection, ECWR.

method. For the FSSH calculations, a swarm of 2000 trajec-tories was used to calculate statistical properties (observableprobabilities).

The total population of the initial quantum state is shownin Fig. 2 as a function of the initial nuclear momentum, p0.The CPF method yields notable improvements over the stan-dard Ehrenfest technique for the DAC potential (Fig. 2(a)).It is known that in the limit of large kinetic energies, so-lutions given by the Ehrenfest and FSSH methods becomeindistinguishable. One can, therefore, expect that the CPFhigh-temperature limit also coincides with the Ehrenfest re-sult. This is partially satisfied, but one can still observe a slightshift of the exact numerical solution with respect to the Ehren-fest/FSSH. The CPF method perfectly matches the exact solu-tion, asymptotically approaching the Ehrenfest/FSSH resultsin the high-energy/high-temperature limit.

Although the high-temperature/high-energy limit israther straightforward to describe, the intermediate range ofkinetic energies presents a bigger challenge. One can observethat positions of the maxima of the Stueckelberg oscillationsobtained with the FSSH and Ehrenfest methods are shiftedtoward higher kinetic energies, in contrast to the exact numer-ical solution. For the range of initial momenta p0 > 15 a.u.,the CPF method yields solutions that are much closer to the

FIG. 2. Demonstration of performance of the CPF method. Total populationof the lower energy level is shown as a function of the initial nuclear momen-tum for the two Tully model problems: (a) modified DAC and (b) ECWR.25

exact numerical results. The solutions generated by the othertwo methods show notable deviation with respect to the exactresult. Starting at p0 > 25 a.u., the CPF and numerical solu-tions practically coincide. Therefore, it can be concluded thatthe CPF method successfully reproduces the expected “lower-energy” shift of the maxima.

In the region of low kinetic energies, p0 < 12 a.u., inwhich the Ehrenfest method breaks down as manifested bylarge-amplitude oscillations, the CPF method works espe-cially well, approaching both the FSSH and exact solutions.Because of the inverse dependence on the initial momentumvia factor f, Eq. (11), the coherence penalty is larger in thelow-energy region. As a result, the dephasing is fast. On theother hand, fast dephasing often leads to the quantum Zeno ef-fect – frequent observations of the system by external meanslead to projection of the quantum state onto the dominanteigenstate, slowing down interstate transitions.100, 111–115 Thismechanism explains the decreased transition rates betweenthe quantum states and the smaller magnitude of fluctuationsobserved in the low kinetic energy region. From the entirerange of the initial momenta, perhaps only the region with12 < p0 < 15 is the most problematic – the CPF results de-viate notably from the exact numerical results. At the sametime, the solutions are close to those obtained with the FSSHmethod.

Performance of the CPF method for the ECWR poten-tial is illustrated in Fig. 2(b). As expected, in the high-energylimit (p0 > 30 a.u.) all methods produce identical results. Forlower kinetic energies, the CPF results show no artificial os-cillation of FSSH and are intermediate between the Ehren-fest and exact solutions. Because the CPF method is a correc-tion to the Ehrenfest dynamics, the shape of the CPF curvein Fig. 2(b) is similar to the one obtained with the originalEhrenfest method. The CPF results show an improvement inon average sense. For small initial momenta, the total popula-tion of the lower energy state is systematically larger than theone obtained with the plain MF description, and correspondsto the FSSH results if they are averaged over the period of theartificial oscillation.

Comparison of the panels in Fig. 2 provides additionalinformation on the prerequisites for successful applicationof the CPF method. Most notably, CPF provides an impor-tant correction in cases when complex interference patternsof wavefunctions can develop. The two avoided crossing re-

194107-8 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

FIG. 3. Evolution of coherence measure, |c∗i cj |2, along the reaction coordinate X. Comparison of the Ehrenfest and CPF methods. Development of coherences

is suppresses by CPF at the points of strong non-adiabatic coupling. Conditions are as follows: (a) DAC, p0 = 15 a.u.; (b) DAC, p0 = 20 a.u; (c) ECWR,p0 = 5 a.u.; (d) ECWR, p0 = 15.

gions provide the mean for developing such interference pat-terns in the DAC problem. In the ECWR problem there isonly one point of strong non-adiabatic coupling, thus, theinterference effects are less pronounced. Hence, correctionsprovided by CPF are less essential. Second, being an exten-sion of the Ehrenfest approach, the CPF method breaks downwhen the wavefunction scatters and loses its localization, asin the ECWR problem. The above analysis has important im-plication to realistic systems, in which the number of avoidedcrossing regions can be large, while the nuclear wavepacketoften remains localized. Under these circumstances, the CPFmethod is likely to be applicable, rendering itself a usefulpractical approximation for studies of realistic systems.

In order to understand better the mechanics of the CPFmethod, we study in detail representative trajectories for eachmodel – at high and low initial nuclear momenta. Figure 3presents the coherence measure (CM), defined by |c∗

i cj |2,as a function of the nuclear coordinate. The CM trajecto-ries are superimposed with the profiles of the adiabatic en-ergy levels. The CM changes very rapidly in regions of strongnon-adiabatic coupling – at each of the two avoided crossingpoints for the DAC potential and at the maximum of the non-adiabatic coupling for the ECWR potential. The CMs remainpractically constant away from these points. In the vicinity ofthe avoided crossing points the CM exhibits decaying oscilla-tions. CPF contributes the most of its effect in these regions bydecreasing coherences, as clearly illustrated in Fig. 3. Coher-ences in the Ehrenfest method are notably larger, or, at least,not smaller than in the CPF method.

The FSSH and Ehrenfest methods are often called over-coherent. After a stochastic hop is made in FSSH, the coher-ences developed in the solution of the semiclassical TD-SEremain unchanged, even though the population of the quan-tum levels is changed. The CPF method provides a mean ofcorrecting for overcoherence. The penalty terms suppress de-velopment of coherence in the solution of the semiclassicalTD-SE.

B. Realistic systems

To demonstrate applicability of the CPF method to real-istic systems, we compute relaxation time scales for two in-terfacial systems involving a quantum dot (QD) and a sub-strate. We consider the Au(QD)/TiO2 and PbSe(QD)/TiO2

heterojunctions (Fig. 4). The interfacial electron transfer dy-namics in these systems is studied experimentally.116–118

The electron-hole recombination is considered for theAu(QD)/TiO2 system. The measured time-scales span a rel-atively wide range due to charge diffusion. The fastest com-ponent corresponding to the recombination of charges thatalready reached the interface is of the order of 10 ps forlarge nanoparticles and is faster for smaller particles.116 Thenanoparticle used in the present calculations is small, and theextrapolated timescale of 1 ps is expected.119 The quantumtransition leads to recombination of the electron from the low-est unoccupied molecular orbital (LUMO), localized on theTiO2 surface, with the hole in the highest unoccupied molec-ular orbital (HOMO), localized on the gold nanoparticle. For

194107-9 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

FIG. 4. Molecular structure of the studied systems: (a) Au(QD)/TiO2; (b) PbSe(QD)/TiO2. Diagrams to the right of each structure illustrate the nature of theelectron transfer process. The dashed line represents the Fermi level of the entire system.

the PbSe(QD)/TiO2 system, the experimentally studied pro-cess is electron transfer from LUMO+1, which is localized onthe TiO2 substrate, to LUMO, which is localized on the PbSeQD. The reported timescale for this process is 1.6 ps.117, 118

The simulation cells and ab initio calculations are set upas follows. The stoichiometric rutile (110) surface is createdto represent the substrate material. A bigger (6 × 2) surfaceand a smaller (5 × 2) surface are used for the PbSe(QD)/TiO2

and Au(QD)/TiO2 systems, respectively, as motivated by thesize of the adsorbed QDs–Pb16Se16 bigger than Au20. EachTiO2 (110) surface comprises six atomic layers of TiO2, withthe bottom three layers frozen in the bulk configuration. Tomodel the interfacial systems, slab geometry is utilized with20 Å of vacuum added in the direction normal to the surface.The PbSe QD binds strongly to the TiO2 surface via two O-Pb bonds between the QD and the surface, while the goldnanoparticle interacts weakly with the TiO2 surface withoutcovalent bonding.

Geometry optimization, electronic structure, and adia-batic molecular dynamics calculations are carried out us-ing the Vienna ab initio simulation package (VASP).120, 121

Valence electrons are treated explicitly, while the coreelectrons are accounted for via pseudopotentials generatedwith the projector augment wave method.120 The Perdew–Burke–Ernzerhof (PBE) functional122, 123 is used to describethe exchange and correlation terms in electronic energy.The DFT+U124 approach is utilized to improve accuracy ofthe electronic structure calculations. The on-site Coulombterms U = 6.0 eV and J = 0.5 eV are applied to the 3d orbitalsof Ti atoms. These parameters are known to reproduce theelectronic structure and properties of bulk TiO2.125 The ini-tially optimized structures are thermalized by MD with veloc-ity rescaling to temperatures of 300 K for the Au(QD)/TiO2

system and 100 K for the PbSe(QD)/TiO2 system, the tem-peratures used in the corresponding experiments.116–118 3 psand 2 ps Born-Oppenheimer MD trajectories are computedwithin the microcanonical ensemble for Au(QD)/TiO2 andPbSe(QD)/TiO2, respectively. The velocity Verlet126 schemewith 1 fs integration time-step is utilized to integrate theequations of motion. Additional details of simulations can befound elsewhere.119, 127

The NA-MD calculations have been performed withthe PYXAID package.98, 128 The dynamics is computed withthe Ehrenfest129, 130 and FSSH25 schemes. We also considertwo semiclassical schemes that account for decoherence ef-

fects – the recently developed decoherence-induced surfacehopping (DISH)75 technique and FSSH with decoherence(D-FSSH).38, 131

The computational schemes that account for decoher-ence (CPF, DISH, and D-FSSH) utilize the microscopic de-coherence rates computed using the optical response for-mula, Eq. (9). The calculated energy gaps are adjusted tomatch the experimental values, by utilizing a constant shift– the scissor-operator132–134 approach. Specifically, the aver-age HOMO-LUMO gap in the Au(QD)/TiO2 system is ad-justed to 1.1 eV reported in the experiment.116 The averageenergy gap between the LUMO and LUMO+1 levels of thePbSe(QD)/TiO2 system is adjusted to the experimental valueof 0.3 eV.117, 118

The time-scales for the interfacial electron transfer com-puted with the different methods are summarized in Ta-ble I. It can be observed that in all cases the plain Ehren-fest method yields longer relaxation times in comparisonto those obtained with FSSH. The DISH scheme increasesthese timescales even further. One can also observe that theDISH scheme tends to overestimate the relaxation timescales.This discrepancy is especially pronounced in the case of thePbSe(QD)/TiO2 system.

The CPF method shows the results that do not obey sucha monotonic relationship. On the one hand, in the case ofAu(QD)/TiO2 system the CPF method slows down the relax-ation, relative to the Ehrenfest dynamics. On the other hand, itaccelerates the relaxation with respect to the plain Ehrenfestdynamics for the PbSe(QD)/TiO2 system. In all cases, the re-sulting timescales are in much better agreement with the ex-perimental values.

The D-FSSH method provides results in reasonableagreement with the experiments. This is a well-establishedsemiclassical scheme that accounts for decoherence effects.Therefore, it can serve as an internal reference for assessmentof accuracy of the new CPF scheme. D-FSSH and CPF showthe best agreement with each other and with the experimental

TABLE I. Relaxation time scales calculated with different methods.

Ehrenfest FSSH D-FSSH DISH CPF Expt.

Au(QD)/TiO2 324 fs 133 fs 1.2 ps 1.6 ps 890 fs ∼1.0 psPbSe(QD)/TiO2 3.8 ps 622 fs 1.3 ps 8.4 ps 1.8 ps 1.6 ps

194107-10 Akimov, Long, and Prezhdo J. Chem. Phys. 140, 194107 (2014)

results. Thus, the CPF method is sufficiently accurate and canbe easily applied to realistic systems.

The distinctive features of the CPF method are worthmentioning explicitly. Unlike the FSSH, D-FSSH, or DISHschemes, the present approach is based on a simple Ehrenfest-like dynamics, and hence, it is continuous and preservesHamiltonian structure. No stochastic processes that describehopping are required. Moreover, since the TD-SE is solvedfor the averaged quantities (distributions), only one trajectoryis required, as opposed to the hundreds of stochastic trajecto-ries required in the surface hopping schemes, such as FSSH,D-FSSH, or DISH. Thus, the method is computationally veryefficient. Being an extension of the original Ehrenfest dynam-ics, the CPF approach is easy to implement and, hence, it canbe added into any ab initio code with very little effort.

IV. CONCLUSIONS

We have presented the coherence penalty functionalmethod, CPF, for incorporating decoherence effects intoNA-MD simulations. The method is based on a simple ideato dynamically penalize development of coherences duringevolution of quantum degrees of freedom. The methodologyis formulated on the grounds of the Ehrenfest method forsemiclassical dynamics. Decoherence effects are introducedvia an additional term in the classically mapped Hamilto-nian, thus preserving the overall Hamiltonian structure ofthe equations of motion. The CPF is analogous to exchange-correlation functionals in DFT. It provides an effective wayfor introducing dynamical correlation effects, such as deco-herence, on top of the mean-field Ehrenfest approach.

We illustrate the applicability of the method by per-forming calculations on model and realistic systems. For themodel problems, we showed that the development of coher-ences is suppressed in the regions of strong non-adiabaticcoupling, leading to oscillatory decay of coherences to somesaturation value. The coherences do not completely decay tozero – the effect that can be understood from the on aver-age interpretation of the Ehrenfest dynamics. In all cases,the CPF method leads to smaller coherences than the orig-inal Ehrenfest approach, affecting quantum state transitionrates. Our simulations demonstrate clearly the accuracy im-provement for the DAC transition probabilities, leading tobetter agreement of the peaks of the Stuckelberg oscillationswith the exact numerical results. For the ECWR problem,an on average improvement is observed. For the realisticAu(QD)/TiO2 and PbSe(QD)/TiO2 systems the CPF approachshows a notable improvement of the computed electron trans-fer time scales. The results agree with the decoherence cor-rected FSSH method.

The CPF methodology has a certain relation to the quan-tized Hamiltonian dynamics. The method can be enhancedby allowing some of the variables entering the penalty func-tional to depend on time. For example, the quantities λij can bemapped on the QHD variables s and ps, thus providing addi-tional dynamical coupling between the nuclear and electronicdegrees of freedom. Alternatively, the quantities λij may bevaried in time according to a specifically designed protocol orcan simply be approximated by a certain functional form. The

dependence of the penalty functional on coherences can berather arbitrary, in principle, and other functional forms mayprove more accurate, more general, or both. In the presentwork, we considered the simplest possible CPF of the form ofEq. (4), which provides good results.

It is straightforward to combine the CPF scheme withFSSH and related approaches. In FSSH one solves the TD-SE first, and then uses the coefficients of the evolving wave-function to determine the hopping probabilities. CPF includesdecoherence into the TD-SE. Using the modified TD-SE todetermine the hopping probabilities will include decoherenceeffects into FSSH. These opportunities certainly require addi-tional exploration in the future.

To recapitulate, the presented method provides a fullyHamiltonian formulation of semiclassical non-adiabatic dy-namics. It is simple in implementation and computationallyefficient. The functional form of CPF is flexible: it can be gen-eralized and adjusted to study various classes of systems. Thetechnique is easy to use with any ab initio code or a modelpotential, as has been demonstrated with both model and re-alistic systems. It does not require large ensembles of stochas-tic trajectories, which makes it appropriate for calculations onlarge systems.

ACKNOWLEDGMENTS

A.V.A. was funded by the Computational Materialsand Chemical Sciences Network (CMCSN) project at theBrookhaven National Laboratory under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy andsupported by its Division of Chemical Sciences, Geosciences,and Biosciences, Office of Basic Energy Sciences. R.L.thanks the Science Foundation Ireland (SFI) SIRG Program(Grant No. 11/SIRG/E2172). O.V.P. acknowledges financialsupport of the U.S. Department of Energy, Grant No. DE-SC0006527.

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