Decoherence-induced surface hoppingHeather M. Jaeger, Sean Fischer, and Oleg V. Prezhdo

Citation: The Journal of Chemical Physics 137, 22A545 (2012); doi: 10.1063/1.4757100 View online: http://dx.doi.org/10.1063/1.4757100 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/137/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Performance analysis of boron nitride embedded armchair graphene nanoribbon metal–oxide–semiconductorfield effect transistor with Stone Wales defects J. Appl. Phys. 115, 034501 (2014); 10.1063/1.4862311 Morphology of a graphene nanoribbon encapsulated in a carbon nanotube AIP Advances 3, 092103 (2013); 10.1063/1.4821102 Comparison of isotope effects on thermal conductivity of graphene nanoribbons and carbon nanotubes Appl. Phys. Lett. 103, 013111 (2013); 10.1063/1.4813111 Interfacial thermal conductance limit and thermal rectification across vertical carbon nanotube/graphenenanoribbon-silicon interfaces J. Appl. Phys. 113, 064311 (2013); 10.1063/1.4790367 Regarding the validity of the time-dependent Kohn–Sham approach for electron-nuclear dynamics via trajectorysurface hopping J. Chem. Phys. 134, 024102 (2011); 10.1063/1.3526297

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THE JOURNAL OF CHEMICAL PHYSICS 137, 22A545 (2012)

Decoherence-induced surface hoppingHeather M. Jaeger,1 Sean Fischer,2 and Oleg V. Prezhdo1

1Department of Chemistry, University of Rochester, Rochester, New York 14627-0216, USA2Department of Chemistry, University of Washington, Seattle, Washington 98195-1700, USA

(Received 8 June 2012; accepted 19 September 2012; published online 15 October 2012)

A simple surface hopping method for nonadiabatic molecular dynamics is developed. The methodderives from a stochastic modeling of the time-dependent Schrödinger and master equations for opensystems and accounts simultaneously for quantum mechanical branching in the otherwise classical(nuclear) degrees of freedom and loss of coherence within the quantum (electronic) subsystem dueto coupling to nuclei. Electronic dynamics in the Hilbert space takes the form of a unitary evolution,intermittent with stochastic decoherence events that are manifested as a localization toward (adia-batic) basis states. Classical particles evolve along a single potential energy surface and can switchsurfaces only at the decoherence events. Thus, decoherence provides physical justification of sur-face hopping, obviating the need for ad hoc surface hopping rules. The method is tested with modelproblems, showing good agreement with the exact quantum mechanical results and providing animprovement over the most popular surface hopping technique. The method is implemented withinreal-time time-dependent density functional theory formulated in the Kohn-Sham representation andis applied to carbon nanotubes and graphene nanoribbons. The calculated time scales of non-radiativequenching of luminescence in these systems agree with the experimental data and earlier calculations.© 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4757100]

I. INTRODUCTION

Mixed quantum-classical methodologies provide anintuitive approach to real-time dynamic simulations ofphysical and chemical processes in condensed phases.1

Quantum-classical simulations of solution chemistry,2–4 sur-face chemistry,5–9 and excitons in polymers10 and nano-sizedstructures11–13 prove valuable toward understanding quan-tum systems in a macroscopic environment. Classical equa-tions of motion are applied to a large number of degreesof freedom, avoiding intractable quantum mechanical ba-sis set calculations. Dynamics of a small quantum subsys-tem can be treated utilizing a reduced description of thetotal system. Quantum-classical models are widely used de-spite fundamental obstacles in the interaction between quan-tum and classical regimes.14, 15 The Ehrenfest method16–18 ap-proximates quantum-classical coupling through expectationvalues of quantum variables. An improved treatment of thequantum-classical interaction introduces the quantum backre-action through a distribution of classical variables individu-ally correlated to quantum basis states.19 Renowned surfacehopping methods20, 21 stochastically sample the dynamic evo-lution of this quasi-classical distribution. As the system sizegrows, however, the quantum subsystem responds with in-creasing measure to subtle quantum effects in the classicallytreated degrees of freedom.

A quantum mechanical wave packet of the total sys-tem, accounting for all quantum effects of the bath, splitsinto uncorrelated branches and loses coherence over time.The resulting decoherence can be viewed as an environment-induced destruction of superpositioned quantum states of themicroscopic system.22, 23 In more detail, the microscopic sys-tem evolves quantum mechanically as a superposition of

fluctuating system states in contact with a bath. Concomi-tantly, through the quantum backreaction, the macroscopicbath follows an array of (non-classical) trajectories associ-ated with individual components of the quantum state. Di-vergence among the bath trajectories leads to a loss of phaserelationship among the states of the total system. When thedivergence is slow relative to transitions within the quantumsubsystem, standard quantum-classical approaches model dy-namic evolution quite well. However, when the opposite istrue and the divergence is fast relative to the rates of tran-sitions, quantum-classical approaches must be corrected fordecoherence. Coherence loss is facilitated by a lack of phase-space structure,24 and furthermore, becomes an irreversiblephenomenon in systems with many degrees of freedom. Mod-eling the dynamic evolution of condensed phase systemsrequires a theoretical account of decoherence as either an in-herent quantum effect of the bath or a phenomenological con-sequence of the system-bath interaction.

The importance of decoherence in condensed-phasechemical reactions has long been appreciated.25, 26 Reactionrates carry an explicit dependence on decoherence.27, 28 Theenvironment acts as a source of friction that damps nonadi-abatic effects and, accordingly, reaction probabilities. Mini-mizing environment-induced decoherence has become an ide-alized goal of quantum control reaction mechanisms,29 wherequantum control relies on the existence and manipulationof electronic coherences. In addition to coherence-controlledchemistry,30–32 the rates and mechanisms of energy transferwithin large molecular aggregates,33–37 polymers,38 and or-ganic chromophores39 depend on decoherence. The quantumZeno effect is an extreme example of such dependence, inwhich quantum transitions cease in the limit of infinitely fast

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22A545-2 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

decoherence.40–42 In quantum dots, nano-wires, and nano-tubes, exciton dephasing43–45 is pervasive as the wave packetevolves nonadiabatically throughout a large number of quan-tum states.

Modifications of quantum-classical surface hopping ap-proaches treat the effect of decoherence on the evolutionof the quantum subsystem. Decoherence corrections havebeen designed and applied to electronic46 and vibrational47, 48

excited state dynamics. Interestingly, inclusion of decoher-ence strengthens the quantum-classical correspondence, and,in the context of trajectory surface hopping simulations, ismost readily seen as an increase in the internal stability.49, 50

Many decoherence-corrected approaches utilize informationabout the excited state potential surfaces to estimate the ef-fect of trajectory divergence on the transition probability.51

A computationally simple approach assumes the classicalenvironment establishes an open system in which diffu-sive, coherence-destroying, dynamics can be described byMarkovian evolution.52 Combining wave packet collapse withthe time-dependent Schrödinger equation (TDSE) provides acomplete description of electron dynamics subject to quantumdecoherence. Previous work on the stochastic Schrödingerequation along a mean-field trajectory53 sets the stage for thecurrent work, in which decoherence induces stochastic sur-face hopping.

Decoherence-induced surface hopping (DISH) relies on asimple picture of quantum state diffusion, that is applicable tolarge, condensed-phase systems. The system-bath interactionis treated within a semiclassical approximation, and appearsin both the dynamics of the quantum system coupled to anexplicit classical bath and the model for decoherence. The-oretical background on the TDSE, dissipative dynamics withexplicit connections to DISH, and electronic dephasing is pre-sented in Sec. II. Section III contains the DISH algorithm,and Sec. IV offers an analysis of the method with regardto established effectiveness criteria.21 Next, DISH is testedwith model problems and compared to the exact quantummechanical solutions and fewest switches surface hopping(FSSH). Finally, DISH is implemented within time-domaindensity functional theory in the Kohn-Sham representationand is applied to electron-phonon relaxation in carbon nan-otubes (CNT) and graphene nanoribbons (GNR). The lattercalculations are compared to available experimental data andresults obtained with decoherence-corrected fewest FSSH.

II. THEORETICAL BACKGROUND

The quantum-classical approximation alleviates compu-tational requirements on the evaluation of the TDSE by elim-inating terms in the Hamiltonian associated with nuclear de-grees of freedom and introducing a convenient dependence onclassical nuclear trajectories. Nuclei evolve according to theclassical equations of motion,

MR(t) = −∇RVR(R(t)) + FQ(t), (1)

where M is a diagonal matrix of individual particle masses,VR is the interaction potential between classical particles, andFQ is the quantum force. Classical paths of the nuclei, R(t),are determined through the quantum force, which can be com-

puted as an adiabatic Hellmann-Feynman force or a mean-field force.54, 55 Alternatively, the Pechukas method56 offersa self-consistent solution for the nuclear path under the in-fluence of quantum transitions. As coherences are quenched,quantum transition probabilities approach zero, and the “best”classical path is determined by the Hellmann-Feynman forceassociated with a single adiabatic state. Environment-induceddecoherence leads to a conceptually and computationally sim-ple procedure for switching potential energy surfaces.

Introducing decoherence at the microscopic level forcondensed phases can be achieved by an approximate so-lution to the Nakajima-Zwanzig generalized quantum mas-ter equation,57–62 quantum correction factors to classicalreaction rates,63–65 numerical formulations of path inte-gral methods,66–68 or phenomenological equations of motionfor the reduced density matrix.69–72 Phenomenological ap-proaches aim to generate quantum Langevin-like dynamics,in which the dynamics become dissipative due to a “friction”generated by quantum fluctuations in the environment. Wefollow a distinct, yet parallel, procedure toward modeling de-coherence by applying a dissipative extension to the TDSE.Drawing from the methodology of stochastic Schrödingerand master equations,73–78 solutions to the modified equa-tions of motion are found through a computationally efficient,“quantum-jump” procedure. Hence, decoherence results insurface hopping, and, in DISH, quantum transitions only oc-cur at the time of decoherence.

The primary task is to employ a time-evolution operatorthat evolves the initial electronic state according to interac-tions between the electronic subsystem and nuclear bath,

|�(t)〉 = ˆU (t, t0)|�(t0)〉, (2)

ˆU (t, t0) =∏α

[Lα(t)] U (t, t0). (3)

The unitary part, U (t, t0), of the evolution operator resultsfrom solving the TDSE, generating a coherent superpositionof quantum states. The L operator introduces a stochastic lo-calization of the quantum state, destroying coherences and ul-timately damping the dynamics. Adiabatic states serve as botha representation for the TDSE as well as a localization basis.Our treatment of decoherence extends the TDSE through astochastic adaptation of the time-evolution operator for quan-

tum systems coupled to an environment, ˆU , and provides amechanism for surface hopping that models diverging evolu-tions of the environment.

A. Time-dependent Schrödinger equation

Applying the unitary part of the time-evolution operator,U (t, t0), is equivalent to solving the TDSE,

i¯∂

∂tψ(r, R, t) = Hel(r, R(t))ψ(r, R, t), (4)

where Hel is the electronic Hamiltonian,

Hel = −¯2

2

∑i

1

mi

∇2i + V (r, R(t)). (5)

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22A545-3 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

V (r, R) includes the electron-electron repulsion and electron-nuclear attraction. V (r, R) is evaluated for a fixed nuclear po-sition, necessarily neglecting nuclear quantum-effects fromthe evolution of the electronic wave function. Surface hop-ping methods developed by Tully build an on-average descrip-tion of the nuclear wave packet through repeated evaluation of(3) for an ensemble of independent surface hopping trajecto-ries. Decoherence effects are not included in the traditionalsurface hopping methods and have to be added a posteriorias a correction.51 DISH follows a different strategy, in whichquantum nuclear effects are treated concurrently with the evo-lution of the electronic state (Eq. (3)).

The time-dependent wave function is expanded in an adi-abatic basis

ψ(r, t) =∑

j

cj (t)φj (r, R), (6)

where quantum phase factors have been absorbed into thetime-dependent coefficients. Electronic basis states suitablefor condensed-phase nonadiabatic simulations can be de-termined from ab initio electronic structure methods, time-dependent density functional theory, and semi-empirical ap-proaches. Substituting Eq. (6) into the TDSE results in a setof coupled equations of motion for the coefficients,

i¯∂

∂tcj (t) =

∑k

ck(t)(εk + djk · R). (7)

Energies of the adiabatic basis states are εk, and the nonadia-batic coupling vector is given by djk . R is classical particle ve-locity. Nonadiabatic coupling can be computed through finitedifferences along the classical trajectory using the expression,

djk · R = −i¯

⟨φj

∣∣∣∣ ∂

∂t

∣∣∣∣φk

⟩(8)

or analytic evaluation of the nonadiabatic coupling vector,djk = 〈φj |∇R|φk〉.

For many applications, the density of states is quite largeand, as a consequence, dynamics of the quantum system takesplace throughout a large subspace of the total system. In thesingle-particle picture, nonadiabatic couplings can be deter-mined in an orbital basis.79 Solving for adiabatic states φj

in a basis independent of classical coordinates does not re-quire the computation of Pulay, or non-Hellmann-Feynmanterms.80 The computational bottleneck in nonadiabatic molec-ular dynamics simulations is the determination of the adia-batic electronic wave functions and the task of solving theTDSE reduces to a propagation of the expansion coefficients.

Representations other than the adiabatic states can beused to solve the TDSE. In which case, the electronic Hamil-tonian is no longer diagonal, and εk of Eq. (7) is replacedby Hjk = 〈j |Hel|k〉. Solving the TDSE in the diabatic basiseliminates the final term of Eq. (7) given that the basis func-tions are independent of nuclear coordinates, i.e., djk = 0.The electronic representation can be chosen to reflect theasymptotic limits of the system states, at which electroniccoupling vanishes. For systems without clearly defined lim-its, the adiabatic representation provides a suitable basis.

The solution of Eq. (7) defines the time-evolved elec-tronic state as a set of basis coefficients. The coefficients com-

prise the time-dependent quantum state vector. Equivalently,the Schrödinger time-evolution operator, U (t, t0), when ap-plied to the quantum state vector at t = t0, generates a coherentsuperposition of the adiabatic basis states. In Subsection II B,we define an operator, L, that extends U (t, t0) in order to in-corporate quantum decoherence into the dynamics.

B. Quantum dissipative dynamics

Quantum dissipation can be described by a numberof different schemes.57–76 We follow most closely the ap-proach associated with the stochastic Schrödinger equationand related quantum-jump procedures.77, 78, 81, 82 The quantumjumps in the electronic subsystem due to interaction with thenuclei is the source of surface hopping in DISH. The idea ofusing quantum dissipation in order to achieve surface hop-ping was originally proposed in Ref. 53 in the context of thestochastic Schrödinger equation.

The form of the Lα(t) operators in Eq. (3) followsfrom the quantum mechanics with spontaneous localization(QMSL)81, 82 model of environment-induced decoherence.Similar to Lindblad theory,60 the QMSL model applies to theBorn-Markov limit of condensed phase dynamics, which ischaracterized by weak system-bath coupling and memory-independent processes. In a Markovian environment, we as-sume that the time-dependent wave function spontaneouslylocalizes under the influence of the decoherence operators,Lα(t),

|ψ red(r, t)〉 = Lα(t)|ψ(r, t)〉. (9)

Upon localization, a reduced state vector, |ψ red(r, t)〉 is gen-erated. Lα(t) is a nonlinear, norm-conserving operator, whichdepends on the specific times, τα , at which the adiabatic basisstate, φα(r, R), decoheres from the electronic wave packet.Stochastic application of the operator, Lα(t), bypasses theneed for mixed-state densities in the equations of motion fordynamic systems undergoing decoherence83 and introducesdecoherence effects directly at the level of the wave function.

Similar to QMSL, stochastic Schrödinger equations de-scribe the evolution of pure quantum states subject to dissipa-tion induced by a quantum environment. A probability is asso-ciated with each resulting, reduced state vector, |ψ red(r, t)|2,such that the ensemble average over many realizations re-covers the reduced density matrix. A stochastic Schrödingerequation can be expressed using the stochastic Ito calculus as

d|ψ(z(t))〉 = [−iH dt + L · dz]|ψ(z(t))〉. (10)

It contains a deterministic element, −iH dt that evolves thequantum subsystem subjected to the classical fields gener-ated by the environment and a randomized element, L · dz,that depends on specific realizations of the Wiener process,z(t), reflecting the influence of quantum fluctuations in theenvironment. Ensuring that the probability distribution gener-ated by z(t) reproduces quantum mechanical norms requiresadditional modification to Eq. (10).82 DISH utilizes a nu-merical surface hopping procedure, in which each realiza-tion, ψ(z(t)), is renormalized following state reduction. Withproper normalization, the “raw” processes (Eq. (10)) are rep-resentative of a mixed-state density matrix, and reduction, or

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22A545-4 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

localization, of the state vector becomes synonymous withdecoherence and provides a physical mechanism for surfacehopping.

The stochastic differential equation (10) is one route to-ward unraveling the Lindblad master equation for the reduceddensity matrix of the electronic subsystem. It is unique, ifthe evolution of the wave function is required to be contin-uous. If the continuity requirement is not imposed, a familyof stochastic processes can be generated that produce iden-tical ensemble averaged results. An equation describing dis-continuous processes is the ordinary TDSE, whose solutionis stochastically interrupted by discontinuous jumps.82 Thestochastic mean-field approximation developed earlier53 usesthe stochastic Schrödinger equation (10) directly. The DISHapproach formulated here is as a discontinuous jump process,leading to a more typical surface hopping, in which determin-istic nuclear trajectories associated with electronic basis statesare interrupted by occasional stochastic events.

The Markovian approximation involved in QMSL81, 82

and Lindblad theory60 can be lifted by using generalizedLindblad master equations and their stochastic wave functionunraveling.78 Highly non-Markovian quantum dynamics canbe achieved by considering several parallel random processesfor the wave function.

The process of localization depends on the amount oftime each adiabatic basis state remains part of the coherentsuperposition, or the coherence interval for each state. Fol-lowing the QMSL methodology,81, 82 environment-inducedlocalization is coarse-grained in time along the coherence in-tervals. The DISH algorithm defines coherence intervals byeither the decoherence time, τα , or a Poisson process withthe characteristic time τα . The decoherence times for eachcomponent state depend on the Schrödinger evolution of thequantum state and are given by the following equation:

1

τα

(t) =N∑

i �=α

|ci(t)|2 rα i . (11)

Here, the sum is computed over all adiabatic states except thestate under consideration. The coefficients, ci(t), are found bysolving the quantum-classical TDSE (Eq. (7)). Decoherencerates for each pair of states, rα i, can be determined by a num-ber of methods discussed in Subsection II C. This expressionapplies to a multi-state system and is directly related to Eq.(2.29) of Ref. 82. Thus, our multi-state definition of deco-herence time corresponds closely to one of the most standardmodels of decoherence. In DISH, each state can decoherefrom the rest of the wave function on the time scale deter-mined by Eq. (11). Any pair of states can lose coherence dueto interaction with the environment, and therefore, the deco-herence rate for each pair of states should be determined. Asshown in the Appendix, the decoherence time of a two-statesystem becomes the inverse of the decoherence rate, rαi. Asa result of Eq. (11), wave functions that are localized havelonger coherence intervals than wave functions delocalizedover a large number of basis states, coinciding with the phys-ical intuition that stronger delocalization results in faster de-coherence. Equation (11) also demonstrates that the stochas-tic or diffusive nature of the quantum coefficients vanishes

as the wave function approaches the adiabatic eigenstates, τα

→ ∞ as cα → 1, and, accordingly, dissipative dynamics canbe described as irreversible localization in the Hilbert space.

A distribution of decoherence events sets the frameworkfor dynamic reduction of the wave function in DISH. At eachdecoherence event, the wave function reduces according toone of two projections. With the probability of |cα|2, the wavefunction is projected onto the decohering state, φα . Alterna-tively, the decohering state is projected out. At the time of adecoherence event, Lα(t) takes on the form of the correspond-ing projection operators,

Lα(t) ={

1N

(Pα) ζ ≤ |cα|2,1

1−N(I − Pα) ζ > |cα|2.

(12)

Pα = |φα〉〈φα| is the projection operator, and (I − Pα) is itscomplement. ζ is a random number between 0 and 1. Thenormalization constant, N, is |cα|2. The stochastic treatmentof coherence loss defined by Eqs. (11) and (12) is a form ofsurface hopping, in which surface hops coincide with deco-herence events.

Decoherence, and hence surface hopping, can occur forany basis state and are not limited by the particular surfaceon which the nuclei are propagating. This unique feature ofDISH arises from the fact that environment-induced decoher-ence affects the entire quantum wave packet. However, sur-face hops are still regulated by quantum probabilities, and, asa result, Eq. (12) generates low probabilities for surface hopsbetween weakly coupled surfaces.

Designation of the possible outcomes follows from mea-surement theory, where the measuring apparatus is the en-vironment. The system-environment interaction projects thequantum subsystem onto the eigenstates of the interaction op-erator, L. At the decoherence time, the environment “detects”either a quantum state that has localized to the decoheringstate or a quantum state in which the decohering state is ab-sent. Equivalently, progression of the wave function throughthe coherence intervals of each adiabatic basis state gener-ates coarse-grained quantum probabilities that reflect quan-tum evolution in the presence of decoherence.

C. Electronic dephasing and quantum decoherence

Quantum decoherence can be succinctly described by thedecay of the overlap function,

Jij (t) = 〈�i(t)|�j (t)〉. (13)

�(t) is a wave function for the total system, and Jij(t) rep-resents an off-diagonal element of the total density matrix.The decay rate of the norm of J(t) gives the rate of coherenceloss. Determination of the overlap function typically involvessemiclassical approximations, in which coherence informa-tion can be extracted from classical trajectory simulations. Inthese scenarios, Jij is associated with the bath contribution toan off-diagonal matrix element of the reduced density ma-trix of the electronic subsystem.40, 51 In the following, a num-ber of approaches to the computation of dephasing rates arepresented. Any one of these formulas may be applied to theDISH algorithm. The DISH simulation can be performed with

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22A545-5 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

the dephasing rates between all pairs of states determined apriori or evaluated on-the-fly in conjunction with the DISHsimulation.

J(t) plays the role of the response function in nonlinearoptical spectroscopy, and the decay of J(t) provides a mea-sure of the pure dephasing contribution to the single particlespectral linewidths resulting from interactions between quan-tum system and the environment. The response function canbe determined from quantum-classical simulations and can beexpressed by the following:84

Jij (t)≈〈Jij (R(t))〉T = exp(iωij t)

⟨exp

[− i

¯

∫ t

0�Eij (τ )dτ

]⟩T

.

(14)

The average energy difference between the optically coupledstates divided by ¯ is given by ωij. �Eij(τ ) is the energy dif-ference evaluated along a classical trajectory, and 〈. . . 〉T in-dicates canonical averaging. The optical response functiondescribes the dissolution of photo-induced superpositions ofelectronic levels (coherences) due to interactions with a disor-dered environment, providing a quantitative measure of elec-tronic dephasing.

The canonical averaging in the above expression can beapproximated with a second-order cumulant expansion84, 85

resulting in

〈Jij (R(t))〉T ≈ exp

[− 〈(δU )2〉T

¯2

∫ t

0dτ2

∫ τ2

0dτ1C(τ1)

].

(15)

The energy gap correlation function, C(t), characterizes therate of linear response of the environment to electronic tran-sitions and is expressed through the classical distribution ofenergy fluctuations,

C(t) = 〈δU (t) · δU (t0)〉T〈(δU (t0))2〉T . (16)

δU is the deviation of the energy gap from the average value,

δU (t) = �Eij (R(t)) − 〈�Eij (R(t)〉T . (17)

Equations (14) and (15) are directly amenable to quantum-classical simulations and describe pure dephasing as a resultof classical fluctuations in the environment. Dephasing ratesfor a pair of states are determined by the time constant of thedecay of Jij(t). These formulas provide an excellent estimateof the time scale for quantum decoherence and can be testeddirectly against spectral linewidths in single particle experi-ments when pure dephasing is fast compared to nonadiabaticrelaxation processes.

The prototypical view of decoherence as diverging wavepackets can be captured by frozen Gaussians in the short-timelimit.86, 87 Schwartz and co-workers employ the frozen Gaus-sian approximation to define a quantum decoherence rate at-tainable by quantum-classical simulations,88, 89

τ−2i =

∑n

(Fn(0) − Fn

i (0))2

4an¯2. (18)

The sum is over nuclear coordinates. Fn(0) − Fni (0) is the

difference between the Hellmann-Feynman forces associated

with the decohering state, i, and the mean-field force, and an isa parameter that depends on the width of the Gaussian associ-ated with a particular nuclear coordinate n. The decoherencetime can also be related to the difference between instanta-neous quantum forces, using the expression90

τ−1ij =

∑n

|Fi(t) − Fj (t)|2¯

√an

. (19)

Equation (19) provides a rough estimate of the decoherencetime by evaluating the difference between forces on divergingclassical trajectories, Fi − Fj . The Gaussian width parame-ter for the nuclear coordinate n is an. Within the Gaussianapproximation, the decoherence time is proportional to theshort-time solvent response. In the high-temperature limit, thedecoherence time can then be estimated from C(t) (Eq. (16))and experimentally accessible Stokes shifts,40

[τD

τg

]2

= 6kBT

λ. (20)

τD is the decoherence time, and τ g is short-time, Gaussian de-cay component of C(t). The Stokes shift is 2λ and is measuredbetween the absorption and emission maxima in equilibriumsolvent configurations.

The decoherence time employed in the decay of mixingapproaches by Truhlar and co-workers91, 92 is

τij =∣∣∣∣∣ ¯�Eij

E

Tvib

∣∣∣∣∣. (21)

�Eij is the potential energy difference associated with statesi and j in the diabatic representation. E is the total energy ofthe system, and Tvib is the kinetic energy of the nuclei. Deco-herence relies on diverging classical trajectories. If either thediabatic states are degenerate or the nuclei stop moving, thedecoherence time becomes infinite.

Treating electronic wave packets as Gaussians, Yang andco-workers developed an expression for the decoherence rateby directly associating electronic momenta with energy dif-ferences between adiabatic states;46

1

τij

≈ Re[αij�xij�xij ]. (22)

Here, decoherence depends on Gaussian width parameters, αi,the spatial separation between Gaussians, �xij, and the rate ofseparation between Gaussian centers, �xij . Gaussian widthsenter the expression through αij = αiαj/(αi + αj).

In DISH, stochastic reductions in the Hilbert space area consequence of quantum decoherence. The time scale fordecoherence of a given electronic state from the electronicwave packet is determined by the pure dephasing rates be-tween the given state and all other states (Eq. (11)). Eachpair-wise dephasing rate is weighted by the electronic pop-ulation of the complementary state. DISH can employ, and isnot limited to any one of the dephasing rates presented here.Specifically, application of DISH to carbon nanotubes andgraphene nanoribbons uses the pure dephasing formulas fromoptical response theory (Eq. (15)) in order to define the deco-herence time. One-dimensional model systems made use of aGaussian approach (Eq. (19)) to estimate pairwise dephasing

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22A545-6 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

Set ψ(t0) and R(t0)

Compute U(Δt)ψ(tm-1)

Compute τα(tm)

Generate Pois(τα)

Is Pois(τα) ≤ tm

Yes No

Is ζ ≤ |φ α|2 ?

Evo

lve

Cla

ssic

al P

arti

cles

Set ψ(tm) = φ α

Generate ζ

NoProject out φ α

No

Yes

Hop to φ α

Is ΔE elec ≤ Tclassical?

Yes

FIG. 1. A block scheme of the DISH algorithm.

rates. The decoherence time determined within DISH definesthe time at which electronic coherences have dissolved dueto interaction with the environment. The physical picture ofcondensed-phase dynamics generated by DISH is that elec-tronic coherences fade resulting from diverging nuclear tra-jectories, and quantum jumps reflect the probabilities of ob-serving distinct states associated with diverging pathways.

III. DISH ALGORITHM

The state vector, [c1(t), c2(t), . . . , cN(t)], comprised of thetime-dependent coefficients of the (adiabatic) basis states con-tains all the dynamic information of the quantum subsystem.L(tm)U (�t) operates directly on the state vector during thesimulation. U (�t) continuously evolves the state along thenuclear trajectory according to the TDSE, and L(tm) stochas-tically projects the coherent quantum state onto the statescomprising the mixed-state density matrix. Evolution of thestate vector deviates from traditional quantum-classical meth-ods by application of L(tm), which induces surface hopping atdecoherence events. A block scheme of the DISH method isshown in Figure 1.

1. Set the state vector for t = t0. The initial state can bedefined as the adiabatic state that results from photo-excitation. For example,

ψ(t0) =

⎡⎢⎣

0...1

⎤⎥⎦ (23)

sets the initial state to the N th adiabatic basis state. Al-ternatively, one can initially evolve the wave functionwith the TDSE from the ground state while includingthe interaction with a laser pulse that creates the photo-excitation.

2. Propagate the classical particles on the initial (adiabatic)potential energy surface, Eq. (1). In the case that ψ(t0) isa superposition of adiabatic states, the adiabatic surfaceis chosen according to the quantum mechanical proba-bilities, |ci|2.

3. Integrate the Schrödinger equation (7) to compute thecoefficients of the time-evolved state vector, [c1(tm),c2(tm), . . . , cN(tm)]. tm is the time at the mth time stepwithin the current coherence interval. The wave functionevolves according to

ψ(tm) = U (tm−1 + �t ; tm−1)ψ(tm−1). (24)

Integration over the time step describes coherent evolu-tion. In order to obtain physically meaningful results, thetime step cannot be larger than the smallest coherenceinterval.

4. Evaluate the coherence of each basis state. The decoher-ence time for state φα is determined by weighting puredephasing rates, rαi, by quantum coefficients, given byEq. (11). Dephasing rates are determined by one of theformulas presented in Sec. II C. They may or may not de-pend on time, depending on the model chosen to repre-sent the dephasing process. Decoherence times, τα(tm),are computed from either t0 or the time of the previ-ous decoherence event, in which case m indexes the timesteps of the current coherence interval for state φα .

5. Determine if the current time step is within the coher-ence interval for each basis state. A random number hav-ing a Poisson distribution with parameter τα(tm) is gen-erated. If the random number is larger than the time sincethe previous decoherence event, the state vector is re-duced. If more than one state undergoes decoherence atthe same time step, a random number is generated to se-lect one of the decohered states. Alternatively, one canreduce the time step in order to achieve a better time-resolution of decoherence events.

The following steps are conditional. If one or moreof the basis states at the given time step decoheres from theevolving quantum state, then the wave function undergoes astochastic reduction, and the nuclear trajectory can experiencea surface hop. If evolution does not result in decoherence, thesimulation continues with Schrödinger evolution (step 2).

6. Compute the reduced wave function corresponding tothe decoherence of φα . Lα is applied to the state vector(Eq. (12)). A random number with a uniform distribu-tion is compared to the quantum probability of observ-ing adiabatic state φα at time tm given by |cα(tm)|2. Ifthe random number is less than or equal to the quantumprobability, then the nuclear trajectory hops to the poten-tial surface of φα , and the wave function is reset to ψ(tm)= φα . If the random number is greater than |cα(tm)|2, φα

is projected out from the current state vector, the wavefunction is renormalized, and the simulation continueswith step 2. If the projected out state corresponds to thepotential energy surface occupied by the nuclear trajec-tory, the trajectory hops to one of the remaining surfaces

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22A545-7 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

that is chosen according to the probabilities given by thesquares of the coefficients for the remaining states.

7. Reset nuclear velocities. The change in electronic energyresulting from the hop is accommodated by the nuclearvelocity component that lies in the direction of the nona-diabatic coupling vector at the time of the hop.21, 93 Tra-jectory hops that increase electronic energy are rejectedif insufficient nuclear kinetic energy is available. Returnto step 2.

IV. DISCUSSION

Nonadiabatic dynamics of large systems involves a non-trivial redistribution of energy between the quantum subsys-tem and its environment. Energy is exchanged across thesystem-environment interface and dissipated throughout themany, diverging degrees of freedom of the bath. As a re-sult, the quantum subsystem undergoes decoherence, becom-ing resolved into macroscopically distinct states that cannotbe described by Schrödinger dynamics. In order to capturedecoherence, quantum properties of the bath must be takeninto account. Unlike quantum Liouvillian approaches94, 95 thatcan simulate dynamics of mixed-state densities resulting fromquantum decoherence, DISH builds decoherence effects di-rectly into the wave function. Decoherence effects are re-alized through stochastic reductions of the quantum sub-system, yielding a surface hopping approach for quantumevolution in condensed-phase systems. Many surface hop-ping approaches, including FSSH, rely on physically moti-vated models for hopping probabilities, whereas DISH uti-lizes standard quantum probabilities, |ci|2, to predict hoppingdynamics. A conceptual advantage of DISH, as compared toFSSH and its decoherence-corrected variants, is that branch-ing appears due to decoherence rather than ad hoc transitionprobabilities, which have been later corrected for decoherenceeffects.

A number of surface hopping approaches with deco-herence have been developed. Decoherence corrections havebeen suggested for FSSH.69, 96, 97 Introducing the Liouville-von Neuman equations, Rossky and co-workers have modi-fied mean field surface hopping98 by damping off-diagonalelements of the reduced density matrix according to the de-coherence time scale.89, 99 Pioneered by the stochastic mean-field approximation,53 other approaches including coherentswitching with decay of mixing,92 mean-field with stochas-tic decoherence,88, 100 and simultaneous-trajectory surfacehopping101 induce surface hops through wave function col-lapse. Similar to the stochastic mean field approach,53 DISHis built upon the standard model of decoherence and system-bath interaction. By evolving classical trajectories in purestates and using the standard quantum mechanical expressionto determine surface hopping probabilities, DISH is, arguably,the simplest and most physically justified surface hoppingtechnique that includes decoherence.

A. Internal consistency and detailed balance

For an ensemble of independent surface-hopping trajec-tories, the fraction of trajectories evolving on a given poten-

tial energy surface must equal the quantum probability of ob-serving the associated basis state. Evolving trajectories self-consistently with the quantum state through either a mean-field or Pechukas force56 eliminates the internal consistencyissue, since each evolution of the wave function gives a singlenuclear trajectory. In these cases, there are no surface hops,and trajectory ensembles are not generated. FSSH benefitsfrom a proof, showing that the surface hopping prescriptionis consistent with the standard quantum probability rules.21

However, rejecting hops to higher energy electronic states dueto insufficient nuclear kinetic energy needed to accommodatethe electronic energy gain causes the rift between quantumprobabilities and surface hopping statistics.49, 93, 102 Comparedto FSSH, DISH uses the standard quantum mechanical rulesto determine hopping probabilities, and therefore, DISH is in-ternally consistent. At the same time, similar to FSSH, DISHuses velocity rescaling and encounters hop rejection that per-turbs internal consistency. Additional studies of the internalconsistency violation due to hop rejection in DISH are re-quired. One would expect the violation to be smaller in DISHthan FSSH, since in DISH the occupied surface is determinedat each hopping event directly by the quantum mechanical oc-cupation of the corresponding state, while in FSSH the stateoccupation and surface hopping probability are connected ina less direct way.

Even though hop rejection violates the principle ofinternal consistency, it leads to the principle of detailedbalance103–105 by making transitions upward in energy lesslikely than transition downward in energy. Assuming thatenergy rapidly equilibrates within the classical subsystem,the velocity-rescaling/hop-rejection scheme is equivalent toa Boltzmann factor biasing the probability to hop to a higherenergy potential surface.106, 107 In the long-time limit, the dif-ference in the rates of transitions upward and downward in en-ergy leads to the Boltzmann distribution of state populations,maintaining the canonical ensemble under the surface hop-ping dynamics. Note that detailed balance cannot be satisfiedin the quantum Zeno regime of infinitely fast decoherence,when quantum transitions cease to occur.40–42 The principleof detailed balance is particularly important for simulation ofelectron-phonon relaxation processes.13, 79, 106, 108, 109

B. Physical observables

In mixed quantum-classical simulations, physical ob-servables are computed over the course of a (nonadiabatic)trajectory to generate time-dependent properties of the quan-tum system. Both classical phase-space variables and quan-tum wave functions factor into the computation of physicalobservables. Classical equations of motion introduce a de-pendence on initial positions and momenta of the environ-ment, requiring that averages be taken with respect to a setof classical trajectories with distinct initial coordinates. Un-like FSSH, which constructs a distribution of surface hoppingpopulations to represent the quantum density, DISH employsquantum probabilities directly, and expectation values are de-termined by the wave function rather than a statistical con-struction. At every time-step, stationary-state observables, Oi,

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22A545-8 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

are weighted by DISH coefficients, yielding a quasi-classicalexpression for time-dependent expectation values,

〈�(r; R)|O(t)|�(r; R)〉T = 1

Ntraj

Ntraj∑R=1

∑i

∣∣c(R)i (t)

∣∣2Oi.

(25)The sum over Ntraj builds an average over nuclear trajecto-ries. Coefficients, c

(R)i (t), result from DISHs equation of mo-

tion (3) and reflect the distribution of quantum states due todecoherence. Electronic decoherence and trajectory branch-ing are simultaneously ingrained into the computation ofobservables.

C. Favorable attributes

DISH preserves the sought-after computational facilityafforded by traditional surface hopping approaches. DISHrequires only a single trajectory for each member of theclassical ensemble. In comparison to the stochastic meanfield approach,53 which pioneered the use of decoherence toachieve trajectory branching and employed a mean-field tra-jectory continuously modified for decoherence effects, DISHtrajectories evolve according to Hellmann-Feynman forcesdue to a coarse-grained account of decoherence. The use ofHellmann-Feynman rather than mean-field forces is particu-larly advantageous with ab initio electronic structure calcu-lations, in which the force evaluation often constitutes themost time consuming step. Similar to FSSH,21 computation ofdecoherence-induced transition probabilities does not encum-ber the simulations. In contrast to FSSH, transition probabili-ties in DISH are determined by |ci|2, according to the standardquantum mechanical prescription.

DISH can be applied to systems in which the dynamicsis described by a large number of strongly coupled quantumstates and a large number of classical degrees of freedom. De-coherence is treated as an inherent trait of a single adiabaticstate, and, consequently, decoherence-induced transitions arepertinent for as long as the adiabatic state carries non-zeropopulation. DISH does not rely on pre-defined regions, inwhich decoherence interactions are turned either on or off.Regions of extremely high density of states, where the adia-batic representation is questionable, coincides with frequentdecoherence events and surface hops, producing a mean-fieldtype of evolution. Because decoherence-induced localizationis an irreversible process, DISH results in adiabatic moleculardynamics in the limit of small nonadiabatic coupling.

V. TWO-STATE MODEL SYSTEMS

In systems with many classical degrees of freedom, de-coherence occurs rapidly and has a large impact on scatter-ing probabilities. The effects of decoherence, however, canbe observed in two-state, one-dimensional systems, in whichdecoherence events are facilitated by non-parallel surfaces.The dual avoided crossing (DAC) and extended coupling withreflection (ECWR) models21 are particularly challenging formixed quantum-classical approaches due to regions of strongnonadiabatic coupling followed by diverging surfaces. Indi-

vidual surface hopping in trajectories in FSSH evolve coher-ently, and the ensemble of FSSH trajectories, collectively, ap-proximates divergence of the wave packet. In DISH, individ-ual trajectories experience decoherence through applicationof the decoherence operator, L.

In the DAC model system, low energy quantum par-ticles (wave packets) are rigorously 100% transmitted, yetoverly coherent approaches introduce finite reflection prob-abilities. Coherence loss precludes the generation of upperstate transients,46 which appear as reflected trajectories. Withthe ECWR potential, the wave packet evolves coherently inthe region of strong coupling and, immediately following,branches into non-interacting reflected and transmitted com-ponents. DISH accounts for decoherence on a per-trajectorybasis, and, consequently, the ensemble of quantum-classicaltrajectories does not generate artificial reflection probabilitieswith the DAC potential nor undue interference effects withthe ECWR potential. DISH predicts scattering probabilitiesto greater accuracy than FSSH and does not require ad hochopping prescriptions, while maintaining the computationalbenefit of traditional surface hopping approaches.

In each system, a particle having 2000 a.u. of massevolves on two coupled potential energy surfaces correspond-ing to upper and lower quantum states. DISH is comparedwith FSSH and quantum wave packet propagation. Scatter-ing probabilities from DISH are determined from the squaresof the quantum trajectory amplitudes resulting from Eq. (3)averaged over a spread of momenta (Eq. (25)). In both DACand ECWR, the spread of momenta is defined as 20/k0, corre-sponding to the width of a Gaussian with k0 a.u. of momen-tum. The classical position coordinate is set to −10 bohr andinitially evolves on the lower state potential energy surface.The initial quantum state vector is set to the lower state, andthe adiabatic representation is employed. Classical equationsof motion are integrated using the velocity Verlet algorithmwith a time step of 0.1 as. The quantum subsystem is propa-gated numerically using a Trotter expansion of the quantumpropagator at every classical time step.

DISH requires the computation of pure dephasing rateseither a priori or concurrently with the simulation. Pure de-phasing rates for DAC and ECWR systems are determinedby Eq. (19), which is a concurrent approach. Instantaneousadiabatic forces as well as the width of a diverging Gaussianwave packet contribute to the dephasing rate. The expressionfor the Gaussian width parameter, an, is the same as that usedin the approach developed by Schwartz and co-workers.88 Intotal, one phenomenological parameter, w, is employed, andresulting scattering probabilities are shown to be insensitiveto modest differences in w.

A. Dual avoided crossing

Quantum potentials and state coupling in the diabatic rep-resentation of the DAC model are given by the following:

V11(x) = 0,

V22(x) = −A exp(−Bx2) + E, (26)

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−0.2

−0.1

0.0

0.1

0.2

−10 −5 0 5 10

Ene

rgy

(Har

tree

)

x (Bohr)

−0.06

−0.04

−0.02

0.00

0.02

0.04

0.06

−10 −5 0 5 10

Ene

rgy

(Har

tree

) (a)

(b)

FIG. 2. Adiabatic potential energy surfaces for the (a) dual avoided cross-ing and (b) extended coupling with reflection. Corresponding diabaticpotentials and coupling of these one-dimensional systems are given inEqs. (26) and (27), following the prescription of Tully.21

V12(x) = V21(x) = C exp(−Dx2).

The adiabatic potentials are shown in Figure 2, with A = 0.10,B = 0.28, C = 0.0015, D = 0.06, and E = 0.05. The width pa-rameter, w, is understood to be the width of the nonadiabaticcoupling region, w = 1/

√D. As the quantum wave packet

evolves through the DAC potential, interference between up-per and lower states cause Stueckelberg oscillations in thetransmission probabilities, and the probability for reflectionis zero for quantum particles with loge(E) > −3.

The quantum mechanical interference pattern is repro-duced by mixed quantum-classical simulations, includingDISH and FSSH, most accurately at high energy (Figure 3).In the top panel of Figure 3, reflection probabilities of DISHare compared to FSSH and the quantum mechanical solution.Quantum results are obtained from solution of the TDSE ona grid. FSSH simulations involve the same integration tech-niques as described above for DISH. DISH accurately pro-duces the ratio of reflected to transmitted components, and,similar to FSSH, deviates from the quantum interference pat-tern at low energy. For incident momentum <14.1 a.u., clas-sical trajectories can evolve on the upper surface temporarilybut not asymptotically. Deviations between quantum-classicaland purely quantum transmission probabilities is attributed tostrong quantum mechanical effects stemming from long livedresonance states. DISH shows an improvement over FSSH inthe low-energy reflection probabilities. Here, decoherence ef-

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

1.0

1.0

0.05

0.10

0.15

−4 −3 −2 −1 0 1

loge E (Hartree)

ytilibaborP

ytilibaborP

Tran

smis

sion

Upp

erTr

ansm

issi

on L

ower

Ref

lect

ion

Low

erDISH

FSSH

Quantum

FIG. 3. Comparison of the scattering probabilities of DISH, FSSH, and theexact quantum mechanical solution for the dual avoided crossing potential.Stueckelberg oscillations in the transmission probabilities are reproduced byDISH and FSSH. DISH improves upon FSSH, since the latter generates ar-tificial reflection on the lower state (top). Upper state and lower state trans-mission probabilities of DISH and FSSH deviate from the quantum result atlow incident energy and reproduce the quantum probabilities at high incidentenergy.

fectively shuts down quantum transitions resulting in reflec-tion and can be understood as correcting the classical trajec-tories for quantum effects.

In the DAC model, decoherence events are very rare, and,as a consequence, surface hops do not occur. Figure 4 containsrepresentative DISH trajectories of different initial momen-tum, k0. Only the trajectory with k0 = 2 a.u. undergoes a deco-herence event, which causes the upper state to be projected outof the state vector. This feature of DISH essentially resets thequantum state such that propagation through avoided cross-ing are independent processes, yielding equally sized peaksin the transition probabilities centered on the avoided cross-ings. For trajectories with larger incident momenta, the DISHsolution is identical to the TDSE evaluated along the classi-cal coordinates. DISH trajectories do not necessarily hop be-tween upper and lower surfaces but, through quantum prob-abilities, reproduce the correct branching ratio of upper andlower state transmission. FSSH, on the other hand, provideshopping probabilities that can underestimate transmission onthe lower state, Figure 3.

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22A545-10 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

k0 = 2 a.u.

k0 = 14 a.u.

k0 = 30 a.u.

0.2

0.4

0.6

0.8

1.0

−5 0 5 10

Pro

babi

lity

x (Bohr)

FIG. 4. Representative trajectories from the dual avoided crossing simulationfor low incident momentum, 2, 14, and 30 a.u. are shown. Decoherence yieldsindependent transition events, as seen by the two equally sized peaks in thek0 = 2 a.u. trajectory. At k0 = 14 a.u. (loge(E) = −3), DISH underestimateslower state transmission by 10% and is an improvement over FSSH, whichunderestimates this value by 20%, Figure 3.

B. Extended coupling with reflection

Diabatic potentials and the coupling matrix elements ofthe ECWR model system are given by

V11(x) = A,

V22(x) = −A,(27)

V12(x) = V21(x) = B exp(Cx), x < 0,

V12(x) = V21(x) = B [2 − exp(−Cx)], x > 0,

where A = 6 × 10−4, B = 0.1, and C = 0.9. Correspondingadiabatic potentials are shown in Figure 2. The width param-eter, w, is taken to be 1/C in units of bohr.88 Strong nona-diabatic coupling extends over a large region (negative x),abruptly followed by a region, in which upper and lower sur-faces are widely separated. Consequently, a wave packet splits(decoheres) into a component that is transmitted on the lowersurface and a component that is reflected back onto the pair ofquasi-degenerated states at negative x. For energies >28 a.u.,part of the wave packet is transmitted on the upper surface.

Unlike the DAC model, which features Stueckelberg os-cillations, quantum interference between upper and lowerstates is absent in the ECWR model, as wave packets readilyand permanently diverge. A comparison of reflection proba-bilities obtained with DISH, FSSH, and the numerical quan-tum results is given in Figure 5. FSSH reflection proba-bilities incorrectly exhibit large oscillations, resulting fromresidual, artificial coherence effects, i.e., surface hops aftercoherence loss. DISH trajectories that encounter a reflectivepotential (upper state) subsequently undergo a large numberof decoherence events after the quantum state reverses direc-tion and re-enters the region of strong coupling. Each of thesedecoherence events results in projecting out one of the quasi-degenerate states, hopping is quenched, and DISH probabili-ties, |c(t)|2, reproduce the non-oscillating quantum results.

0.2

0.4

Ref

lect

ion

Low

er

5 10 15 20 25 30

momentum (a.u.)

DISH

FSSH

Quantum

Pro

babi

lity

FIG. 5. Probability of reflection in the lower state is shown for the extendedcoupling with reflection model. For momenta <28 a.u., FSSH generates largeoscillations in the reflection probabilities due to unphysical interference withthe transmitting part of the wave packet. By explicitly treating decoherencein the dynamics, DISH follows the quantum mechanical reflection in thisregime.

With initial momenta greater than ∼28 a.u., the quantumwave packet has enough energy to overcome the reflectivepotential and transmit on upper and lower states. In DISH,branching between upper and lower state transmission stemsfrom a single decoherence-induced hop as the surfaces di-verge, −5.0 < x < 0.0. Following the hop, surfaces becomeparallel once again, and states do not mix further. On-averageDISH trajectories displayed in Figure 6 show the progres-sion from coherent state to a decohered mixture. For posi-tive x, population is not exchanged between upper and lowerstates. In order to reproduce the quantum mechanical branch-ing ratio, the singular decoherence event must incur a hopaccording to quantum probabilities. Hopping probabilities inDISH are precisely quantum probabilities. Testing the de-pendence of the branching ratio on dephasing rate, Figure 6displays DISH trajectories for w = 0.1, 0.5, 1.0, and 10.0.Trajectories for w = 0.1 significantly deviate from the other

0.2

0.4

0.6

0.8

1.0

−5.0 0.0

Pro

babi

lity

x (Bohr)

w = 1. 0w = 0. 5

−2.5

Transmission Upper

Transmission Lower

w = 0.1w = 10.0

FIG. 6. Ensemble-averaged DISH trajectories with initial group momen-tum of 30 a.u. As the surfaces diverge, the branching ratios generated byDISH are entirely dictated by a single decoherence induced hopping eventand, for w �= 0.1, accurately reproduce the quantum mechanical result. Forw = 0.5 and greater, transmission probabilities are largely independent ofphenomenological parameter, w.

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22A545-11 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

simulations shown. For such narrow w, decoherence occurstoo rapidly, not allowing quantum transitions probabilities tobuild up. Longer coherence intervals, from w = 0.1 on, re-produce correct quantum mechanical results.

VI. EXCITON DYNAMICS IN CARBON NANOTUBESAND GRAPHENE NANORIBBONS

Carbon nanotubes and GNR exhibit extremely large elec-tron mobilities.110, 111 In a photo-excited state, electron trans-port properties of these nano-materials are affected by diffu-sive processes, in which exciton energy converts to heat anddissipates. Excitons in semi-conducting GNRs and CNTs an-nihilate as a result of both radiative and nonradiative decaymechanisms. Radiative processes are competitive in CNTsbut not predominant, as demonstrated by low fluorescenceyields.112–114 The rate of nonradiative relaxation to the groundelectronic state in GNRs predicates its unique photophysicalproperties.115–117

Nonradiative exciton decay is a direct consequence ofthe exchange of energy between electrons and phonons. Theelectron-phonon interaction mediates the rate at which ex-cited electrons lose energy to lattice vibrations. Quantum-classical simulations treat phonons, or lattice vibrations, clas-sically, and downward transitions in the quantum subspacecoincide with increases in the classical kinetic energy. Appli-cation of DISH to exciton annihilation in CNTs and GNRsdemonstrates that decoherence is an important contributionfor nonradiative processes. Relaxation rates calculated fromDISH simulations agree with those determined by experimentand decoherence-corrected FSSH.

The surface hopping simulations employed here, DISHand decoherence-corrected FSSH,51, 108 are based on a two-state model for ground state recovery. Two adiabatic statesand their nonadiabatic coupling are computed within the time-dependent Kohn-Sham formalism79 and represent the groundand first excited states. A standard decoherence correction isapplied to FSSH,108 in which the wave function is modifiedby resetting the excited state coefficient in accordance withthe pure dephasing rate, which is computed from optical re-sponse functions, Eqs. (14) and (15). Employing a two-statemodel in DISH simplifies the algorithm, such that decoher-ence events equate to state switching at the pure dephasingtime. Details on the two-state limit of DISH are reportedin the Appendix. The primary distinction between DISHand decoherence-corrected FSSH is that DISH introducessurface hopping only at decoherence events, while FSSHtreats decoherence and surface hopping as two independentprocesses.

Surface hopping simulations are carried out for threeCNTs with (6,4) chirality, including one without atomic de-fects and two systems with different bond-rearrangement de-fects (Figure 7). Two (16,16) GNR systems are considered,including one system with the same 7557 bond-insertiondefect present in the CNT. The GNR is also displayed inFigure 7. These systems are representative of those investi-gated by time-resolved spectroscopy under experimental con-ditions. Monitoring exciton dynamics from DISH in compar-

(6,4) CNT

(16,16) GNR

FIG. 7. Structures of the (6,4) CNT and (16,16) GNR are shown. The Stone-Wales bond-rearrangement defect results from a rotation of a C–C bond inthe hexagonal lattice. The 7557 defect results from the addition of 2 carbonatoms.

ison to FSSH for different systems provides a diagnostic onthe reliability of DISH.

Investigation of electron-hole recombination in CNTsby photoluminescence spectroscopy gives evidence of a re-laxation time scale on the order of 10s of picosecondsto nanoseconds.112 Relaxation times of CNTs determinedthrough decoherence-corrected FSSH, are 147 ps, 67 ps,and 42 ps, for the ideal, Stone-Wales, and 7557 systems,respectively.108 Figure 8 compares the population recovery in

0

0.01

0.02

0.03

0.04

0.05

0.06

0 500 1000 1500 2000 2500 3000

Pop

ulat

ion

Time (fs)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Pop

ulat

ion

Ideal (6,4)7557 DefectSW Defect

Ideal (6,4)7557 DefectSW Defect

FIG. 8. Recovery of the ground state in the ideal CNT and two defect sys-tems occurs over 100 s of ps. (Top) Surface hopping populations of theground electronic state generated by decoherence-corrected FSSH simula-tions. (Bottom) Quantum probabilities of occupying the ground electronicstate determined by DISH.

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22A545-12 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

0

0.005

0.01

0.015

0.02

0.025

0.03

Pop

ulat

ion

Ideal 16,167557 Defect

0

0.005

0.01

0.015

0.02

0.025

0 500 1000 1500 2000 2500 3000

Pop

ulat

ion

Time (fs)

Ideal 16,167557 Defect

FIG. 9. Ground state recovery of GNRs from the first excited state takesplace within 100 s of ps. (Top) Surface hopping populations of the groundelectronic state from decoherence-corrected TDKS-FSSH simulations. (Bot-tom) Quantum probabilities of the ground state determined by DISH.

the ground state, which determines the exciton annihilationrates, from both FSSH and DISH simulations. DISH resultsshow good agreement with the FSSH data. The DISH relax-ation times are determined to be 200 ps, 107 ps, and 22 ps.DISH reproduces the enhancement of nonradiative recom-bination by atomic defects. Radiative lifetimes of CNTs are1–2 orders of magnitude larger, and both DISH and FSSH de-scribe relatively fast nonradiative pathways that account forthe low photoluminescence yields.112–114

Direct experimental evidence on the exciton dynamicshas yet to be attained for substrate free GNRs. Figure 9 dis-plays the interband relaxation of ideal and defect-containingGNRs. Similar to CNTs, nonradiative recombination occurson a relatively short time scale of 100 s of picoseconds.Decoherence-corrected FSSH results in relaxation times of309 ps and 124 ps for the ideal and 7557 defect systems,respectively.109 Similar to CNTs, defects increase exciton re-laxation rates. DISH simulations produce relaxation times of231 ps and 143 ps, in agreement with decoherence-correctedFSSH. Both DISH and FSSH simulations show that defectsopen nonradiative decay channels, increasing relaxation rates.

Similar ground state recovery rates between DISH andFSSH indicates that the stochastic DISH trajectory closelyfollows the dynamics of the decoherence-corrected FSSH en-semble. In the context of DISH formalism, the electronicwave function evolves toward the ground state in the pres-ence of decoherence-inducing interactions with a phonon en-vironment. Stochastic hopping in DISH and decoherence-

corrected FSSH are both governed simultaneously by coher-ences and quantum probabilities, but, only in DISH, are quan-tum transitions directly related to coherence intervals. Ac-curacy in the electron-phonon relaxation rates in CNTs andGNRs depends on a number of factors, particularly, decoher-ence rate.108 DISH provides a unified treatment of decoher-ence and quantum transitions without computational expensebeyond standard surface hopping approaches.

VII. CONCLUDING REMARKS

By combining the computational simplicity of quantum-classical nonadiabatic molecular dynamics with a formaltreatment of quantum decoherence, DISH provides a straight-forward and physically justified trajectory surface hoppingscheme, in which transitions between surfaces occur duringdecoherence events. The transition probabilities and observ-ables are computed according to standard quantum mechani-cal rules. On one hand, DISH can be viewed as a surface hop-ping approach to quantum dynamics in dissipative environ-ments. And, in another sense, DISH unifies decoherence andnonadiabatic transitions, providing a non-phenomenologicalaccount of quantum transitions in condensed-phases. Evolu-tion of the quantum subsystem in DISH is affected by a seriesof decoherence events that converts the wave function intoa statistical mixture of quantum states, justifiable through thefundamental nature of decoherence and the quantum-classicalcorrespondence.

The importance of decoherence in chemical and physi-cal processes in the condensed phase cannot be understated.A considerable amount of attention has been given to smallquantum systems in macroscopic environments. DISH ex-tends the feasibility of quantum-classical simulations to largequantum systems in macroscopic environments. Investiga-tion of charge-transfer through crystals or solutions, excitondynamics in biological macromolecules, long-range electrontransfer, and photo-induced dynamics in condensed phases allrequire a theoretical approach that can simultaneously treatmany quantum degrees of freedom undergoing irreversible,dissipative evolution. As demonstrated with the non-radiativerelaxation of CNTs and GNRs, the DISH approach achievesthis goal.

ACKNOWLEDGMENTS

Financial support of the National Science Foundation(NSF) Grant No. CHE-1050405 is gratefully acknowledged.

APPENDIX: TWO-STATE LIMIT OF DISH

Consider an arbitrary wave function in a two-state basisat time, t,

ψ(t) = ciφi + cf φf . (A1)

Initial and final states are designated as, φi and φf, respec-tively. According to the general equation (11), the decoher-ence times for initial and final states are determined by the

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22A545-13 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

following expressions:

1

τi

= |cf |2rif , (A2)

1

τf

= |ci |2rif . (A3)

Because the dephasing rate, rif, between states i and f isweighted by the coefficients, the decoherence events are notdouble counted. The sum of the decoherence rates 1/τ i and1/τ f for states i and f is equal to rij. For example, state i de-coheres from the superposition at the rate of 1/τ i, state f de-coheres from the superposition at the rate of 1/τ f, and over-all, the pair of states loses coherence with the rate of rij. Inthe limit that ci approaches 1, τ i approaches infinity and τ f

approaches 1rif

, and vice versa. Consequentially, decoherenceevents occur on the time scale determined by the pure dephas-ing rate, rif.

The projection operator acts on the wave function duringthe decoherence events to isolate the quantum system to anadiabatic state and according to the following a special caseof Eq. (12):

L ψ(t) ={

φf , ζ ≤ |cf (t)|2φi, ζ > |cf (t)|2 . (A4)

Evolution of the quantum system introduces non-zero quan-tum probabilities for state switching. At every decoherenceevent, the quantum probabilities are reset to 0 and 1. Thesequence of decoherence events between two states identi-fies with surface hopping. For systems in which decoherenceis much more rapid than the quantum transition, nonadia-batic dynamics are dominated by coherence loss, and the co-efficients remain close to 1 or 0. When decoherence is in-finitely fast, transitions stop, and the quantum Zeno effectensues.40–42, 118 For systems that undergo slow loss of phaseinformation, the state coefficients deviate significantly from1 and 0, the state-switching probabilities are large, and thequantum transition is rapid.

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