Decoherence-Induced Surface Hopping

Decoherence-induced surface hopping

Cite as: J. Chem. Phys. 137, 22A545 (2012); https://doi.org/10.1063/1.4757100 Submitted: 08 June 2012 . Accepted: 19 September 2012 . Published Online: 15 October 2012

Heather M. Jaeger, Sean Fischer, and Oleg V. Prezhdo

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J. Chem. Phys. 137, 22A545 (2012); https://doi.org/10.1063/1.4757100 137, 22A545

Decoherence-induced surface hopping Heather M. Jaeger,1 Sean Fischer,2 and Oleg V. Prezhdo1 1Department of Chemistry, University of Rochester, Rochester, New York 14627-0216, USA 2Department 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 method derives from a stochastic modeling of the time-dependent Schrödinger and master equations for open systems and accounts simultaneously for quantum mechanical branching in the otherwise classical (nuclear) degrees of freedom and loss of coherence within the quantum (electronic) subsystem due to 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 (adiabatic) basis states. Classical particles evolve along a single potential energy surface and can switch surfaces only at the decoherence events. Thus, decoherence provides physical justiﬁcation of surface hopping, obviating the need for ad hoc surface hopping rules. The method is tested with model problems, showing good agreement with the exact quantum mechanical results and providing an improvement over the most popular surface hopping technique. The method is implemented within real-time time-dependent density functional theory formulated in the Kohn-Sham representation and is applied to carbon nanotubes and graphene nanoribbons. The calculated time scales of non-radiative quenching 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 ﬂuctuating system states in contact with a bath. Concomi- Mixed quantum-classical methodologies provide an tantly, through the quantum backreaction, the macroscopic intuitive approach to real-time dynamic simulations of bath follows an array of (non-classical) trajectories associ- physical and chemical processes in condensed phases.1 ated with individual components of the quantum state. Di- Quantum-classical simulations of solution chemistry,2–4 sur- vergence among the bath trajectories leads to a loss of phase face chemistry,5–9 and excitons in polymers10 and nano-sized relationship among the states of the total system. When the structures11–13 prove valuable toward understanding quan- divergence is slow relative to transitions within the quantum tum systems in a macroscopic environment. Classical equa- subsystem, standard quantum-classical approaches model dy- tions of motion are applied to a large number of degrees namic evolution quite well. However, when the opposite is of freedom, avoiding intractable quantum mechanical ba- true and the divergence is fast relative to the rates of tran- sis set calculations. Dynamics of a small quantum subsys- sitions, quantum-classical approaches must be corrected for tem can be treated utilizing a reduced description of the decoherence. Coherence loss is facilitated by a lack of phase- total system. Quantum-classical models are widely used de- space structure,24 and furthermore, becomes an irreversible spite fundamental obstacles in the interaction between quan- phenomenon in systems with many degrees of freedom. Mod- tum and classical regimes.14, 15 The Ehrenfest method16–18 ap- eling the dynamic evolution of condensed phase systems proximates quantum-classical coupling through expectation requires a theoretical account of decoherence as either an in- values of quantum variables. An improved treatment of the herent quantum effect of the bath or a phenomenological con- quantum-classical interaction introduces the quantum backre- sequence of the system-bath interaction. action through a distribution of classical variables individu- The importance of decoherence in condensed-phase ally correlated to quantum basis states.19 Renowned surface chemical reactions has long been appreciated.25, 26 Reaction hopping methods20, 21 stochastically sample the dynamic evo- rates carry an explicit dependence on decoherence.27, 28 The lution of this quasi-classical distribution. As the system size environment acts as a source of friction that damps nonadi- grows, however, the quantum subsystem responds with in- abatic effects and, accordingly, reaction probabilities. Mini- creasing measure to subtle quantum effects in the classically mizing environment-induced decoherence has become an ide- treated degrees of freedom. alized goal of quantum control reaction mechanisms,29 where A quantum mechanical wave packet of the total sys- quantum control relies on the existence and manipulation tem, accounting for all quantum effects of the bath, splits of electronic coherences. In addition to coherence-controlled into uncorrelated branches and loses coherence over time. chemistry,30–32 the rates and mechanisms of energy transfer The resulting decoherence can be viewed as an environment- within large molecular aggregates,33–37 polymers,38 and or- induced destruction of superpositioned quantum states of the ganic chromophores39 depend on decoherence. The quantum microscopic system.22, 23 In more detail, the microscopic sys- Zeno effect is an extreme example of such dependence, in tem evolves quantum mechanically as a superposition of which quantum transitions cease in the limit of inﬁnitely fast

0021-9606/2012/137(22)/22A545/14/$30.00137, 22A545-1 © 2012 American Institute of Physics 22A545-2 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012) decoherence.40–42 In quantum dots, nano-wires, and nano- puted as an adiabatic Hellmann-Feynman force or a mean- tubes, exciton dephasing43–45 is pervasive as the wave packet field force.54, 55 Alternatively, the Pechukas method56 offers evolves nonadiabatically throughout a large number of quan- a self-consistent solution for the nuclear path under the in- tum states. fluence of quantum transitions. As coherences are quenched, Modifications of quantum-classical surface hopping ap- quantum transition probabilities approach zero, and the “best” proaches treat the effect of decoherence on the evolution classical path is determined by the Hellmann-Feynman force of the quantum subsystem. Decoherence corrections have associated with a single adiabatic state. Environment-induced been designed and applied to electronic46 and vibrational47, 48 decoherence leads to a conceptually and computationally sim- excited state dynamics. Interestingly, inclusion of decoher- ple procedure for switching potential energy surfaces. ence strengthens the quantum-classical correspondence, and, Introducing decoherence at the microscopic level for in the context of trajectory surface hopping simulations, is condensed phases can be achieved by an approximate so- most readily seen as an increase in the internal stability.49, 50 lution to the Nakajima-Zwanzig generalized quantum mas- Many decoherence-corrected approaches utilize information ter equation,57–62 quantum correction factors to classical about the excited state potential surfaces to estimate the ef- reaction rates,63–65 numerical formulations of path inte- fect of trajectory divergence on the transition probability.51 gral methods,66–68 or phenomenological equations of motion A computationally simple approach assumes the classical for the reduced density matrix.69–72 Phenomenological ap- environment establishes an open system in which diffu- proaches aim to generate quantum Langevin-like dynamics, sive, coherence-destroying, dynamics can be described by in which the dynamics become dissipative due to a “friction” Markovian evolution.52 Combining wave packet collapse with generated by quantum fluctuations in the environment. We the time-dependent Schrödinger equation (TDSE) provides a follow a distinct, yet parallel, procedure toward modeling de- complete description of electron dynamics subject to quantum coherence by applying a dissipative extension to the TDSE. decoherence. Previous work on the stochastic Schrödinger Drawing from the methodology of stochastic Schrödinger equation along a mean-field trajectory53 sets the stage for the and master equations,73–78 solutions to the modified equa- current work, in which decoherence induces stochastic sur- tions of motion are found through a computationally efficient, face hopping. “quantum-jump” procedure. Hence, decoherence results in Decoherence-induced surface hopping (DISH) relies on a surface hopping, and, in DISH, quantum transitions only oc- simple picture of quantum state diffusion, that is applicable to cur at the time of decoherence. large, condensed-phase systems. The system-bath interaction The primary task is to employ a time-evolution operator is treated within a semiclassical approximation, and appears that evolves the initial electronic state according to interac- in both the dynamics of the quantum system coupled to an tions between the electronic subsystem and nuclear bath, explicit classical bath and the model for decoherence. The- | = ˆ˜ | oretical background on the TDSE, dissipative dynamics with (t) U(t,t0) (t0) , (2) explicit connections to DISH, and electronic dephasing is presented in Sec. II. Section III contains the DISH algorithm, ˆ˜ = ˆ ˆ and Sec. IV offers an analysis of the method with regard U(t,t0) [Lα(t)] U(t,t0). (3) to established effectiveness criteria.21 Next, DISH is tested α with model problems and compared to the exact quantum The unitary part, Uˆ (t,t0), of the evolution operator results mechanical solutions and fewest switches surface hopping from solving the TDSE, generating a coherent superposition (FSSH). Finally, DISH is implemented within time-domain of quantum states. The L operator introduces a stochastic lo- density functional theory in the Kohn-Sham representation calization of the quantum state, destroying coherences and ul- and is applied to electron-phonon relaxation in carbon nan- timately damping the dynamics. Adiabatic states serve as both otubes (CNT) and graphene nanoribbons (GNR). The latter a representation for the TDSE as well as a localization basis. calculations are compared to available experimental data and Our treatment of decoherence extends the TDSE through a results obtained with decoherence-corrected fewest FSSH. stochastic adaptation of the time-evolution operator for quantum systems coupled to an environment, Uˆ˜ , and provides a II. THEORETICAL BACKGROUND mechanism for surface hopping that models diverging evolu- tions of the environment. The quantum-classical approximation alleviates computational requirements on the evaluation of the TDSE by elim- inating terms in the Hamiltonian associated with nuclear de- A. Time-dependent Schrödinger equation grees of freedom and introducing a convenient dependence on Applying the unitary part of the time-evolution operator, classical nuclear trajectories. Nuclei evolve according to the Uˆ (t,t0), is equivalent to solving the TDSE, classical equations of motion, ∂ ¨ =−∇ + i¯ ψ(r, R,t) = Hˆ (r, R(t))ψ(r, R,t), (4) MR(t) RVR(R(t)) FQ(t), (1) ∂t el M where is a diagonal matrix of individual particle masses, where Hêl is the electronic Hamiltonian, VR is the interaction potential between classical particles, and ¯2 1 FQ is the quantum force. Classical paths of the nuclei, R(t), Hˆ =− ∇2 + V (r, R(t)). (5) el 2 m i are determined through the quantum force, which can be com- i i 22A545-3 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

V (r, R) includes the electron-electron repulsion and electron- prise the time-dependent quantum state vector. Equivalently, nuclear attraction. V (r, R) is evaluated for a fixed nuclear po- the Schrödinger time-evolution operator, Uˆ (t,t0), when ap- sition, necessarily neglecting nuclear quantum-effects from plied to the quantum state vector at t = t0, generates a coherent the evolution of the electronic wave function. Surface hop- superposition of the adiabatic basis states. In Subsection II B, ping methods developed by Tully build an on-average descrip- we define an operator, Lˆ , that extends Uˆ (t,t0) in order to in- tion of the nuclear wave packet through repeated evaluation of corporate quantum decoherence into the dynamics. (3) for an ensemble of independent surface hopping trajectories. Decoherence effects are not included in the traditional B. Quantum dissipative dynamics surface hopping methods and have to be added a posteriori as a correction.51 DISH follows a different strategy, in which Quantum dissipation can be described by a number quantum nuclear effects are treated concurrently with the evo- of different schemes.57–76 We follow most closely the ap- lution of the electronic state (Eq. (3)). proach associated with the stochastic Schrödinger equation The time-dependent wave function is expanded in an adi- and related quantum-jump procedures.77, 78, 81, 82 The quantum abatic basis jumps in the electronic subsystem due to interaction with the nuclei is the source of surface hopping in DISH. The idea of ψ(r,t) = c (t)φ (r, R), (6) j j using quantum dissipation in order to achieve surface hop- j ping was originally proposed in Ref. 53 in the context of the where quantum phase factors have been absorbed into the stochastic Schrödinger equation. time-dependent coefficients. Electronic basis states suitable The form of the Lˆ α(t) operators in Eq. (3) follows for condensed-phase nonadiabatic simulations can be de- from the quantum mechanics with spontaneous localization termined from ab initio electronic structure methods, time- (QMSL)81, 82 model of environment-induced decoherence. dependent density functional theory, and semi-empirical ap- Similar to Lindblad theory,60 the QMSL model applies to the proaches. Substituting Eq. (6) into the TDSE results in a set Born-Markov limit of condensed phase dynamics, which is of coupled equations of motion for the coefficients, characterized by weak system-bath coupling and memory- ∂ independent processes. In a Markovian environment, we as- i¯ c (t) = c (t)( + d · R˙ ). (7) ∂t j k k jk sume that the time-dependent wave function spontaneously k localizes under the influence of the decoherence operators, Energies of the adiabatic basis states are k, and the nonadia- Lˆ α(t), batic coupling vector is given by d . R˙ is classical particle ve- jk |ψred(r,t)=Lˆ (t)|ψ(r,t). (9) locity. Nonadiabatic coupling can be computed through finite α differences along the classical trajectory using the expression, Upon localization, a reduced state vector, |ψred(r,t) is generated. Lˆ α(t) is a nonlinear, norm-conserving operator, which ∂ djk · R˙ =−i¯ φj φk (8) depends on the specific times, τ α, at which the adiabatic basis ∂t state, φα(r, R), decoheres from the electronic wave packet. or analytic evaluation of the nonadiabatic coupling vector, Stochastic application of the operator, Lˆ α(t), bypasses the djk =φj |∇R|φk. need for mixed-state densities in the equations of motion for For many applications, the density of states is quite large dynamic systems undergoing decoherence83 and introduces and, as a consequence, dynamics of the quantum system takes decoherence effects directly at the level of the wave function. place throughout a large subspace of the total system. In the Similar to QMSL, stochastic Schrödinger equations de- single-particle picture, nonadiabatic couplings can be deter- scribe the evolution of pure quantum states subject to dissipa- 79 mined in an orbital basis. Solving for adiabatic states φj tion induced by a quantum environment. A probability is asso- in a basis independent of classical coordinates does not re- ciated with each resulting, reduced state vector, |ψred(r,t)|2, quire the computation of Pulay, or non-Hellmann-Feynman such that the ensemble average over many realizations re- terms.80 The computational bottleneck in nonadiabatic molec- covers the reduced density matrix. A stochastic Schrödinger ular dynamics simulations is the determination of the adia- equation can be expressed using the stochastic Ito calculus as batic electronic wave functions and the task of solving the d|ψ(z(t))=[−iH dt + Lˆ · dz]|ψ(z(t)). (10) TDSE reduces to a propagation of the expansion coefficients. Representations other than the adiabatic states can be It contains a deterministic element, −iH dt that evolves the used to solve the TDSE. In which case, the electronic Hamil- quantum subsystem subjected to the classical fields gener- tonian is no longer diagonal, and k of Eq. (7) is replaced ated by the environment and a randomized element, Lˆ · dz, by Hjk =j|Hêl|k. Solving the TDSE in the diabatic basis that depends on specific realizations of the Wiener process, eliminates the final term of Eq. (7) given that the basis func- z(t), reflecting the influence of quantum fluctuations in the tions are independent of nuclear coordinates, i.e., djk = 0. environment. Ensuring that the probability distribution gener- The electronic representation can be chosen to reflect the ated by z(t) reproduces quantum mechanical norms requires asymptotic limits of the system states, at which electronic additional modification to Eq. (10).82 DISH utilizes a nu- coupling vanishes. For systems without clearly defined lim- merical surface hopping procedure, in which each realiza- its, the adiabatic representation provides a suitable basis. tion, ψ(z(t)), is renormalized following state reduction. With The solution of Eq. (7) defines the time-evolved elec- proper normalization, the “raw” processes (Eq. (10)) are rep- tronic state as a set of basis coefficients. The coefficients com- resentative of a mixed-state density matrix, and reduction, or 22A545-4 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012) localization, of the state vector becomes synonymous with as the wave function approaches the adiabatic eigenstates, τ α decoherence and provides a physical mechanism for surface →∞as cα → 1, and, accordingly, dissipative dynamics can hopping. be described as irreversible localization in the Hilbert space. The stochastic differential equation (10) is one route to- A distribution of decoherence events sets the framework ward unraveling the Lindblad master equation for the reduced for dynamic reduction of the wave function in DISH. At each density matrix of the electronic subsystem. It is unique, if decoherence event, the wave function reduces according to 2 the evolution of the wave function is required to be contin- one of two projections. With the probability of |cα| ,thewave uous. If the continuity requirement is not imposed, a family function is projected onto the decohering state, φα. Alterna- of stochastic processes can be generated that produce iden- tively, the decohering state is projected out. At the time of a tical ensemble averaged results. An equation describing dis- decoherence event, Lα(t) takes on the form of the correspond- continuous processes is the ordinary TDSE, whose solution ing projection operators, 82 is stochastically interrupted by discontinuous jumps. The 1 2 53 (Pˆα) ζ ≤|cα| , stochastic mean-field approximation developed earlier uses Lˆ (t) = N (12) α 1 ˆ − ˆ | |2 the stochastic Schrödinger equation (10) directly. The DISH 1−N (I Pα) ζ> cα . approach formulated here is as a discontinuous jump process, =| | ˆ − ˆ leading to a more typical surface hopping, in which determin- Pα φα φα is the projection operator, and (I Pα)isits istic nuclear trajectories associated with electronic basis states complement. ζ is a random number between 0 and 1. The | |2 are interrupted by occasional stochastic events. normalization constant, N,is cα . The stochastic treatment The Markovian approximation involved in QMSL81, 82 of coherence loss defined by Eqs. (11) and (12) is a form of and Lindblad theory60 can be lifted by using generalized surface hopping, in which surface hops coincide with deco- Lindblad master equations and their stochastic wave function herence events. unraveling.78 Highly non-Markovian quantum dynamics can Decoherence, and hence surface hopping, can occur for be achieved by considering several parallel random processes any basis state and are not limited by the particular surface for the wave function. on which the nuclei are propagating. This unique feature of The process of localization depends on the amount of DISH arises from the fact that environment-induced decoher- time each adiabatic basis state remains part of the coherent ence affects the entire quantum wave packet. However, sur- superposition, or the coherence interval for each state. Fol- face hops are still regulated by quantum probabilities, and, as lowing the QMSL methodology,81, 82 environment-induced a result, Eq. (12) generates low probabilities for surface hops localization is coarse-grained in time along the coherence in- between weakly coupled surfaces. tervals. The DISH algorithm defines coherence intervals by Designation of the possible outcomes follows from mea- surement theory, where the measuring apparatus is the en- either the decoherence time, τ α, or a Poisson process with vironment. The system-environment interaction projects the the characteristic time τ α. The decoherence times for each component state depend on the Schrödinger evolution of the quantum subsystem onto the eigenstates of the interaction op- ˆ quantum state and are given by the following equation: erator, L. At the decoherence time, the environment “detects” either a quantum state that has localized to the decohering 1 N state or a quantum state in which the decohering state is ab- = | |2 (t) ci (t) rαi. (11) sent. Equivalently, progression of the wave function through τα i=α the coherence intervals of each adiabatic basis state gener- Here, the sum is computed over all adiabatic states except the ates coarse-grained quantum probabilities that reflect quantum evolution in the presence of decoherence. state under consideration. The coefficients, ci(t), are found by solving the quantum-classical TDSE (Eq. (7)). Decoherence rates for each pair of states, rα i, can be determined by a num- C. Electronic dephasing and quantum decoherence ber of methods discussed in Subsection II C. This expression applies to a multi-state system and is directly related to Eq. Quantum decoherence can be succinctly described by the (2.29) of Ref. 82. Thus, our multi-state definition of deco- decay of the overlap function, herence time corresponds closely to one of the most standard J (t) = (t)| (t). (13) models of decoherence. In DISH, each state can decohere ij i j from the rest of the wave function on the time scale deter- (t) is a wave function for the total system, and Jij(t) rep- mined by Eq. (11). Any pair of states can lose coherence due resents an off-diagonal element of the total density matrix. to interaction with the environment, and therefore, the deco- The decay rate of the norm of J(t) gives the rate of coherence herence rate for each pair of states should be determined. As loss. Determination of the overlap function typically involves shown in the Appendix, the decoherence time of a two-state semiclassical approximations, in which coherence informa- system becomes the inverse of the decoherence rate, rαi.As tion can be extracted from classical trajectory simulations. In a result of Eq. (11), wave functions that are localized have these scenarios, Jij is associated with the bath contribution to longer coherence intervals than wave functions delocalized an off-diagonal matrix element of the reduced density ma- over a large number of basis states, coinciding with the phys- trix of the electronic subsystem.40, 51 In the following, a num- ical intuition that stronger delocalization results in faster de- ber of approaches to the computation of dephasing rates are coherence. Equation (11) also demonstrates that the stochas- presented. Any one of these formulas may be applied to the tic or diffusive nature of the quantum coefficients vanishes DISH algorithm. The DISH simulation can be performed with 22A545-5 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012) the dephasing rates between all pairs of states determined a with the decohering state, i, and the mean-field force, and an is priori or evaluated on-the-fly in conjunction with the DISH a parameter that depends on the width of the Gaussian associ- simulation. ated with a particular nuclear coordinate n. The decoherence J(t) plays the role of the response function in nonlinear time can also be related to the difference between instanta- optical spectroscopy, and the decay of J(t) provides a mea- neous quantum forces, using the expression90 sure of the pure dephasing contribution to the single particle − |Fi (t) − Fj (t)| spectral linewidths resulting from interactions between quan- τ 1 = √ . (19) ij 2¯ a tum system and the environment. The response function can n n be determined from quantum-classical simulations and can be Equation (19) provides a rough estimate of the decoherence expressed by the following:84 time by evaluating the difference between forces on diverging t classical trajectories, Fi − Fj . The Gaussian width parame- ≈ = − i Jij (t) Jij (R(t)) T exp(iωij t) exp Eij (τ)dτ . ter for the nuclear coordinate n is a . Within the Gaussian ¯ n 0 T approximation, the decoherence time is proportional to the (14) short-time solvent response. In the high-temperature limit, the The average energy difference between the optically coupled decoherence time can then be estimated from C(t)(Eq.(16)) 40 states divided by ¯ is given by ω . E (τ) is the energy dif- and experimentally accessible Stokes shifts, ij ij ference evaluated along a classical trajectory, and ... in- 2 T τ 6k T dicates canonical averaging. The optical response function D = B . (20) describes the dissolution of photo-induced superpositions of τg λ electronic levels (coherences) due to interactions with a disor- τ D is the decoherence time, and τ g is short-time, Gaussian de- dered environment, providing a quantitative measure of elec- cay component of C(t). The Stokes shift is 2λ and is measured tronic dephasing. between the absorption and emission maxima in equilibrium The canonical averaging in the above expression can be solvent configurations. 84, 85 approximated with a second-order cumulant expansion The decoherence time employed in the decay of mixing resulting in approaches by Truhlar and co-workers91, 92 is (δU)2 t τ2 J (R(t)) ≈ exp − T dτ dτ C(τ ) . ¯ E ij T 2 2 1 1 = ¯ 0 0 τij . (21) Eij Tvib (15) Eij is the potential energy difference associated with states The energy gap correlation function, C(t), characterizes the i and j in the diabatic representation. E is the total energy of rate of linear response of the environment to electronic tran- the system, and Tvib is the kinetic energy of the nuclei. Deco- sitions and is expressed through the classical distribution of herence relies on diverging classical trajectories. If either the energy fluctuations, diabatic states are degenerate or the nuclei stop moving, the δU(t) · δU(t ) decoherence time becomes infinite. C(t) = 0 T . (16) 2 (δU(t0)) T Treating electronic wave packets as Gaussians, Yang and co-workers developed an expression for the decoherence rate δU is the deviation of the energy gap from the average value, by directly associating electronic momenta with energy dif- 46 δU(t) = Eij (R(t)) − Eij (R(t)T . (17) ferences between adiabatic states; Equations (14) and (15) are directly amenable to quantum- 1 ≈ Re[αij xij x˙ij ]. (22) classical simulations and describe pure dephasing as a result τij of classical fluctuations in the environment. Dephasing rates Here, decoherence depends on Gaussian width parameters, α , for a pair of states are determined by the time constant of the i the spatial separation between Gaussians, x , and the rate of decay of J (t). These formulas provide an excellent estimate ij ij separation between Gaussian centers, x˙ . Gaussian widths of the time scale for quantum decoherence and can be tested ij enter the expression through α = α α /(α + α ). directly against spectral linewidths in single particle experi- ij i j i j In DISH, stochastic reductions in the Hilbert space are ments when pure dephasing is fast compared to nonadiabatic a consequence of quantum decoherence. The time scale for relaxation processes. decoherence of a given electronic state from the electronic The prototypical view of decoherence as diverging wave wave packet is determined by the pure dephasing rates be- packets can be captured by frozen Gaussians in the short-time tween the given state and all other states (Eq. (11)). Each limit.86, 87 Schwartz and co-workers employ the frozen Gaus- pair-wise dephasing rate is weighted by the electronic pop- sian approximation to define a quantum decoherence rate at- ulation of the complementary state. DISH can employ, and is tainable by quantum-classical simulations,88, 89 not limited to any one of the dephasing rates presented here. Fn(0) − Fn(0) 2 Specifically, application of DISH to carbon nanotubes and τ −2 = i . (18) i 4a ¯2 graphene nanoribbons uses the pure dephasing formulas from n n optical response theory (Eq. (15)) in order to define the deco- n − n The sum is over nuclear coordinates. F (0) Fi (0)isthe herence time. One-dimensional model systems made use of a difference between the Hellmann-Feynman forces associated Gaussian approach (Eq. (19)) to estimate pairwise dephasing 22A545-6 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

2. Propagate the classical particles on the initial (adiabatic) Set ψ(t0) and R(t0) potential energy surface, Eq. (1). In the case that ψ(t0)is a superposition of adiabatic states, the adiabatic surface Compute U(Δt)ψ(t ) is chosen according to the quantum mechanical proba- m-1 2 bilities, |ci| . 3. Integrate the Schrödinger equation (7) to compute the Compute τα(tm) coefﬁcients of the time-evolved state vector, [c1(tm),

Generate Pois(τα) c2(tm),..., cN(tm)]. tm is the time at the mth time step within the current coherence interval. The wave function τ ≤ Is Pois( α) tm evolves according to

Yes No Classical Particles Evolve ψ(tm) = Uˆ (tm−1 + t; tm−1)ψ(tm−1). (24) Generate ζ No Integration over the time step describes coherent evolu- ζ ≤ φ 2 φ Is | α| ? Project out α tion. In order to obtain physically meaningful results, the Yes time step cannot be larger than the smallest coherence No interval. Is ΔE ≤ T ? elec classical 4. Evaluate the coherence of each basis state. The decoher- Yes ence time for state φα is determined by weighting pure dephasing rates, rαi, by quantum coefficients, given by Hop to φ α ψ φ Set (tm) = α Eq. (11). Dephasing rates are determined by one of the formulas presented in Sec. II C. They may or may not de- FIG. 1. A block scheme of the DISH algorithm. pend on time, depending on the model chosen to represent the dephasing process. Decoherence times, τ (t ), rates. The decoherence time determined within DISH defines α m are computed from either t or the time of the previ- the time at which electronic coherences have dissolved due 0 ous decoherence event, in which case m indexes the time to interaction with the environment. The physical picture of steps of the current coherence interval for state φ . condensed-phase dynamics generated by DISH is that elec- α 5. Determine if the current time step is within the coher- tronic coherences fade resulting from diverging nuclear tra- ence interval for each basis state. A random number hav- jectories, and quantum jumps reflect the probabilities of ob- ing a Poisson distribution with parameter τ (t ) is gen- serving distinct states associated with diverging pathways. α m erated. If the random number is larger than the time since the previous decoherence event, the state vector is re- III. DISH ALGORITHM duced. If more than one state undergoes decoherence at The state vector, [c1(t), c2(t),...,cN(t)], comprised of the the same time step, a random number is generated to se- time-dependent coefficients of the (adiabatic) basis states con- lect one of the decohered states. Alternatively, one can tains all the dynamic information of the quantum subsystem. reduce the time step in order to achieve a better time- Lˆ (tm)Uˆ ( t) operates directly on the state vector during the resolution of decoherence events. simulation. Uˆ ( t) continuously evolves the state along the The following steps are conditional. If one or more nuclear trajectory according to the TDSE, and Lˆ (t ) stochas- m of the basis states at the given time step decoheres from the tically projects the coherent quantum state onto the states evolving quantum state, then the wave function undergoes a comprising the mixed-state density matrix. Evolution of the stochastic reduction, and the nuclear trajectory can experience state vector deviates from traditional quantum-classical meth- a surface hop. If evolution does not result in decoherence, the ods by application of Lˆ (t ), which induces surface hopping at m simulation continues with Schrödinger evolution (step 2). decoherence events. A block scheme of the DISH method is shown in Figure 1. 6. Compute the reduced wave function corresponding to the decoherence of φ . Lˆ is applied to the state vector 1. Set the state vector for t = t . The initial state can be α α 0 (Eq. (12)). A random number with a uniform distribu- defined as the adiabatic state that results from photo- tion is compared to the quantum probability of observ- excitation. For example, | |2 ⎡ ⎤ ing adiabatic state φα at time tm given by cα(tm) .If 0 the random number is less than or equal to the quantum ⎢ ⎥ = . probability, then the nuclear trajectory hops to the poten- ψ(t0) ⎣ . ⎦ (23) tial surface of φα, and the wave function is reset to ψ(tm) 1 2 = φα. If the random number is greater than |cα(tm)| , φα sets the initial state to the Nth adiabatic basis state. Al- is projected out from the current state vector, the wave ternatively, one can initially evolve the wave function function is renormalized, and the simulation continues with the TDSE from the ground state while including with step 2. If the projected out state corresponds to the the interaction with a laser pulse that creates the photo- potential energy surface occupied by the nuclear trajec- excitation. tory, the trajectory hops to one of the remaining surfaces 22A545-7 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

that is chosen according to the probabilities given by the tial energy surface must equal the quantum probability of ob- squares of the coefficients for the remaining states. serving the associated basis state. Evolving trajectories self- 7. Reset nuclear velocities. The change in electronic energy consistently with the quantum state through either a mean- resulting from the hop is accommodated by the nuclear field or Pechukas force56 eliminates the internal consistency velocity component that lies in the direction of the nona- issue, since each evolution of the wave function gives a single diabatic coupling vector at the time of the hop.21, 93 Tra- nuclear trajectory. In these cases, there are no surface hops, jectory hops that increase electronic energy are rejected and trajectory ensembles are not generated. FSSH benefits if insufficient nuclear kinetic energy is available. Return from a proof, showing that the surface hopping prescription to step 2. is consistent with the standard quantum probability rules.21 However, rejecting hops to higher energy electronic states due to insufficient nuclear kinetic energy needed to accommodate IV. DISCUSSION the electronic energy gain causes the rift between quantum probabilities and surface hopping statistics.49, 93, 102 Compared Nonadiabatic dynamics of large systems involves a non- to FSSH, DISH uses the standard quantum mechanical rules trivial redistribution of energy between the quantum subsys- to determine hopping probabilities, and therefore, DISH is in- tem and its environment. Energy is exchanged across the ternally consistent. At the same time, similar to FSSH, DISH system-environment interface and dissipated throughout the uses velocity rescaling and encounters hop rejection that per- many, diverging degrees of freedom of the bath. As a re- turbs internal consistency. Additional studies of the internal sult, the quantum subsystem undergoes decoherence, becom- consistency violation due to hop rejection in DISH are re- ing resolved into macroscopically distinct states that cannot quired. One would expect the violation to be smaller in DISH be described by Schrödinger dynamics. In order to capture than FSSH, since in DISH the occupied surface is determined decoherence, quantum properties of the bath must be taken 94, 95 at each hopping event directly by the quantum mechanical oc- into account. Unlike quantum Liouvillian approaches that cupation of the corresponding state, while in FSSH the state can simulate dynamics of mixed-state densities resulting from occupation and surface hopping probability are connected in quantum decoherence, DISH builds decoherence effects di- a less direct way. rectly into the wave function. Decoherence effects are re- Even though hop rejection violates the principle of alized through stochastic reductions of the quantum sub- internal consistency, it leads to the principle of detailed system, yielding a surface hopping approach for quantum balance103–105 by making transitions upward in energy less evolution in condensed-phase systems. Many surface hop- likely than transition downward in energy. Assuming that ping approaches, including FSSH, rely on physically moti- energy rapidly equilibrates within the classical subsystem, vated models for hopping probabilities, whereas DISH uti- the velocity-rescaling/hop-rejection scheme is equivalent to | |2 lizes standard quantum probabilities, ci , to predict hopping a Boltzmann factor biasing the probability to hop to a higher dynamics. A conceptual advantage of DISH, as compared to energy potential surface.106, 107 In the long-time limit, the dif- FSSH and its decoherence-corrected variants, is that branch- ference in the rates of transitions upward and downward in en- ing appears due to decoherence rather than ad hoc transition ergy leads to the Boltzmann distribution of state populations, probabilities, which have been later corrected for decoherence maintaining the canonical ensemble under the surface hop- effects. ping dynamics. Note that detailed balance cannot be satisfied A number of surface hopping approaches with deco- in the quantum Zeno regime of infinitely fast decoherence, herence have been developed. Decoherence corrections have 40–42 69, 96, 97 when quantum transitions cease to occur. The principle been suggested for FSSH. Introducing the Liouville- of detailed balance is particularly important for simulation of von Neuman equations, Rossky and co-workers have modi- electron-phonon relaxation processes.13, 79, 106, 108, 109 fied mean field surface hopping98 by damping off-diagonal elements of the reduced density matrix according to the decoherence time scale.89, 99 Pioneered by the stochastic mean- field approximation,53 other approaches including coherent B. Physical observables switching with decay of mixing,92 mean-field with stochas- 88, 100 In mixed quantum-classical simulations, physical ob- tic decoherence, and simultaneous-trajectory surface servables are computed over the course of a (nonadiabatic) hopping101 induce surface hops through wave function col- 53 trajectory to generate time-dependent properties of the quan- lapse. Similar to the stochastic mean field approach, DISH tum system. Both classical phase-space variables and quan- is built upon the standard model of decoherence and system- tum wave functions factor into the computation of physical bath interaction. By evolving classical trajectories in pure observables. Classical equations of motion introduce a de- states and using the standard quantum mechanical expression pendence on initial positions and momenta of the environ- to determine surface hopping probabilities, DISH is, arguably, ment, requiring that averages be taken with respect to a set the simplest and most physically justified surface hopping of classical trajectories with distinct initial coordinates. Un- technique that includes decoherence. like FSSH, which constructs a distribution of surface hopping populations to represent the quantum density, DISH employs A. Internal consistency and detailed balance quantum probabilities directly, and expectation values are de- For an ensemble of independent surface-hopping trajec- termined by the wave function rather than a statistical con- tories, the fraction of trajectories evolving on a given poten- struction. At every time-step, stationary-state observables, Oi, 22A545-8 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012) are weighted by DISH coefficients, yielding a quasi-classical vidual surface hopping in trajectories in FSSH evolve coher- expression for time-dependent expectation values, ently, and the ensemble of FSSH trajectories, collectively, ap- proximates divergence of the wave packet. In DISH, individ- N 1 traj ual trajectories experience decoherence through application (r; R)|Oˆ (t)|(r; R) = c(R)(t)2 O . T N i i of the decoherence operator, L. traj R=1 i (25) In the DAC model system, low energy quantum particles (wave packets) are rigorously 100% transmitted, yet The sum over Ntraj builds an average over nuclear trajecto- overly coherent approaches introduce finite reflection prob- ries. Coefficients, c(R)(t), result from DISHs equation of mo- i abilities. Coherence loss precludes the generation of upper tion (3) and reflect the distribution of quantum states due to state transients,46 which appear as reflected trajectories. With decoherence. Electronic decoherence and trajectory branch- the ECWR potential, the wave packet evolves coherently in ing are simultaneously ingrained into the computation of the region of strong coupling and, immediately following, observables. branches into non-interacting reflected and transmitted components. DISH accounts for decoherence on a per-trajectory C. Favorable attributes basis, and, consequently, the ensemble of quantum-classical trajectories does not generate artificial reflection probabilities DISH preserves the sought-after computational facility with the DAC potential nor undue interference effects with afforded by traditional surface hopping approaches. DISH the ECWR potential. DISH predicts scattering probabilities requires only a single trajectory for each member of the to greater accuracy than FSSH and does not require ad hoc classical ensemble. In comparison to the stochastic mean hopping prescriptions, while maintaining the computational 53 field approach, which pioneered the use of decoherence to benefit of traditional surface hopping approaches. achieve trajectory branching and employed a mean-field tra- In each system, a particle having 2000 a.u. of mass jectory continuously modified for decoherence effects, DISH evolves on two coupled potential energy surfaces correspond- trajectories evolve according to Hellmann-Feynman forces ing to upper and lower quantum states. DISH is compared due to a coarse-grained account of decoherence. The use of with FSSH and quantum wave packet propagation. Scatter- Hellmann-Feynman rather than mean-field forces is particu- ing probabilities from DISH are determined from the squares larly advantageous with ab initio electronic structure calcu- of the quantum trajectory amplitudes resulting from Eq. (3) lations, in which the force evaluation often constitutes the averaged over a spread of momenta (Eq. (25)). In both DAC 21 most time consuming step. Similar to FSSH, computation of and ECWR, the spread of momenta is defined as 20/k0, corre- decoherence-induced transition probabilities does not encum- sponding to the width of a Gaussian with k0 a.u. of momen- ber the simulations. In contrast to FSSH, transition probabili- tum. The classical position coordinate is set to −10 bohr and | |2 ties in DISH are determined by ci , according to the standard initially evolves on the lower state potential energy surface. quantum mechanical prescription. The initial quantum state vector is set to the lower state, and DISH can be applied to systems in which the dynamics the adiabatic representation is employed. Classical equations is described by a large number of strongly coupled quantum of motion are integrated using the velocity Verlet algorithm states and a large number of classical degrees of freedom. De- with a time step of 0.1 as. The quantum subsystem is propa- coherence is treated as an inherent trait of a single adiabatic gated numerically using a Trotter expansion of the quantum state, and, consequently, decoherence-induced transitions are propagator at every classical time step. pertinent for as long as the adiabatic state carries non-zero DISH requires the computation of pure dephasing rates population. DISH does not rely on pre-defined regions, in either apriorior concurrently with the simulation. Pure de- which decoherence interactions are turned either on or off. phasing rates for DAC and ECWR systems are determined Regions of extremely high density of states, where the adia- by Eq. (19), which is a concurrent approach. Instantaneous batic representation is questionable, coincides with frequent adiabatic forces as well as the width of a diverging Gaussian decoherence events and surface hops, producing a mean-field wave packet contribute to the dephasing rate. The expression type of evolution. Because decoherence-induced localization for the Gaussian width parameter, an, is the same as that used is an irreversible process, DISH results in adiabatic molecular in the approach developed by Schwartz and co-workers.88 In dynamics in the limit of small nonadiabatic coupling. total, one phenomenological parameter, w, is employed, and resulting scattering probabilities are shown to be insensitive to modest differences in w. V. TWO-STATE MODEL SYSTEMS In systems with many classical degrees of freedom, decoherence occurs rapidly and has a large impact on scatter- A. Dual avoided crossing ing probabilities. The effects of decoherence, however, can be observed in two-state, one-dimensional systems, in which Quantum potentials and state coupling in the diabatic rep- decoherence events are facilitated by non-parallel surfaces. resentation of the DAC model are given by the following: The dual avoided crossing (DAC) and extended coupling with 21 reflection (ECWR) models are particularly challenging for V11(x) = 0, mixed quantum-classical approaches due to regions of strong V (x) =−A exp(−Bx2) + E, (26) nonadiabatic coupling followed by diverging surfaces. Indi- 22 22A545-9 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

0.06 0.15 y

t DISH i l 0.04 (a) i b

a 0.10 FSSH b o

0.02 r Quantum P 0.05 0.00

−0.02

Energy (Hartree) 1.0 −0.04 0.8 −0.06 −10 −5 0 5 10 0.6

0.2 (b) 0.4

0.1 y 0.2 t Transmission LowerTransmission Reflection Lower i l i b a

0.0 b 1.0 o r

P 0.8 −0.1 Energy (Hartree) 0.6 −0.2 0.4 −10 −5 0 5 10 x (Bohr) 0.2 Transmission Upper Transmission

FIG. 2. Adiabatic potential energy surfaces for the (a) dual avoided cross- −4 −3 −2 −1 0 1 ing and (b) extended coupling with reflection. Corresponding diabatic potentials and coupling of these one-dimensional systems are given in loge E (Hartree) Eqs. (26) and (27), following the prescription of Tully.21 FIG. 3. Comparison of the scattering probabilities of DISH, FSSH, and the exact quantum mechanical solution for the dual avoided crossing potential. 2 Stueckelberg oscillations in the transmission probabilities are reproduced by V12(x) = V21(x) = C exp(−Dx ). DISH and FSSH. DISH improves upon FSSH, since the latter generates ar- = tificial reflection on the lower state (top). Upper state and lower state trans- The adiabatic potentials are shown in Figure 2, with A 0.10, mission probabilities of DISH and FSSH deviate from the quantum result at B = 0.28, C = 0.0015, D = 0.06, and E = 0.05. The width pa- low incident energy and reproduce the quantum probabilities at high incident rameter, w, is understood√ to be the width of the nonadiabatic energy. coupling region, w = 1/ D. As the quantum wave packet evolves through the DAC potential, interference between upper and lower states cause Stueckelberg oscillations in the fectively shuts down quantum transitions resulting in reflec- transmission probabilities, and the probability for reflection tion and can be understood as correcting the classical trajec- is zero for quantum particles with loge(E) > −3. tories for quantum effects. The quantum mechanical interference pattern is repro- In the DAC model, decoherence events are very rare, and, duced by mixed quantum-classical simulations, including as a consequence, surface hops do not occur. Figure 4 contains DISH and FSSH, most accurately at high energy (Figure 3). representative DISH trajectories of different initial momen- In the top panel of Figure 3, reflection probabilities of DISH tum, k0. Only the trajectory with k0 = 2 a.u. undergoes a deco- are compared to FSSH and the quantum mechanical solution. herence event, which causes the upper state to be projected out Quantum results are obtained from solution of the TDSE on of the state vector. This feature of DISH essentially resets the a grid. FSSH simulations involve the same integration tech- quantum state such that propagation through avoided cross- niques as described above for DISH. DISH accurately pro- ing are independent processes, yielding equally sized peaks duces the ratio of reflected to transmitted components, and, in the transition probabilities centered on the avoided cross- similar to FSSH, deviates from the quantum interference pat- ings. For trajectories with larger incident momenta, the DISH tern at low energy. For incident momentum <14.1 a.u., clas- solution is identical to the TDSE evaluated along the classi- sical trajectories can evolve on the upper surface temporarily cal coordinates. DISH trajectories do not necessarily hop be- but not asymptotically. Deviations between quantum-classical tween upper and lower surfaces but, through quantum prob- and purely quantum transmission probabilities is attributed to abilities, reproduce the correct branching ratio of upper and strong quantum mechanical effects stemming from long lived lower state transmission. FSSH, on the other hand, provides resonance states. DISH shows an improvement over FSSH in hopping probabilities that can underestimate transmission on the low-energy reflection probabilities. Here, decoherence ef- the lower state, Figure 3. 22A545-10 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

1.0 DISH Quantum 0.4 FSSH 0.8

. 0.6 k0 = 2 a.u 0.2 Probability k0 = 14 a.u.

0.4 k = 30 a.u. Reflection Lower Probability 0

5 10 15 20 25 30 0.2 momentum (a.u.)

FIG. 5. Probability of reflection in the lower state is shown for the extended −5 0 5 10 coupling with reflection model. For momenta <28 a.u., FSSH generates large oscillations in the reflection probabilities due to unphysical interference with x (Bohr) the transmitting part of the wave packet. By explicitly treating decoherence in the dynamics, DISH follows the quantum mechanical reflection in this FIG. 4. Representative trajectories from the dual avoided crossing simulation regime. for low incident momentum, 2, 14, and 30 a.u. are shown. Decoherence yields independent transition events, as seen by the two equally sized peaks in the = = =− k0 2 a.u. trajectory. At k0 14 a.u. (loge(E) 3), DISH underestimates ∼ lower state transmission by 10% and is an improvement over FSSH, which With initial momenta greater than 28 a.u., the quantum underestimates this value by 20%, Figure 3. wave packet has enough energy to overcome the reflective potential and transmit on upper and lower states. In DISH, branching between upper and lower state transmission stems B. Extended coupling with reflection from a single decoherence-induced hop as the surfaces diverge, −5.0 < x < 0.0. Following the hop, surfaces become Diabatic potentials and the coupling matrix elements of parallel once again, and states do not mix further. On-average the ECWR model system are given by DISH trajectories displayed in Figure 6 show the progression from coherent state to a decohered mixture. For posi- V (x) = A, 11 tive x, population is not exchanged between upper and lower V22(x) =−A, states. In order to reproduce the quantum mechanical branch- (27) ing ratio, the singular decoherence event must incur a hop V (x) = V (x) = B exp(Cx),x<0, 12 21 according to quantum probabilities. Hopping probabilities in V12(x) = V21(x) = B [2 − exp(−Cx)],x>0, DISH are precisely quantum probabilities. Testing the dependence of the branching ratio on dephasing rate, Figure 6 − where A = 6 × 10 4, B = 0.1, and C = 0.9. Corresponding displays DISH trajectories for w = 0.1, 0.5, 1.0, and 10.0. adiabatic potentials are shown in Figure 2. The width param- Trajectories for w = 0.1 significantly deviate from the other eter, w, is taken to be 1/C in units of bohr.88 Strong nonadiabatic coupling extends over a large region (negative x), abruptly followed by a region, in which upper and lower sur- 1.0 w = 0.1 w = 1. 0 faces are widely separated. Consequently, a wave packet splits w = 0. 5 w = 10.0 (decoheres) into a component that is transmitted on the lower 0.8 surface and a component that is reflected back onto the pair of quasi-degenerated states at negative x. For energies >28 a.u., Transmission Lower part of the wave packet is transmitted on the upper surface. 0.6 Unlike the DAC model, which features Stueckelberg oscillations, quantum interference between upper and lower Probability 0.4 states is absent in the ECWR model, as wave packets readily Transmission Upper and permanently diverge. A comparison of reflection probabilities obtained with DISH, FSSH, and the numerical quan- 0.2 tum results is given in Figure 5. FSSH reflection probabilities incorrectly exhibit large oscillations, resulting from residual, artificial coherence effects, i.e., surface hops after −5.0 −2.5 0.0 coherence loss. DISH trajectories that encounter a reflective x (Bohr) potential (upper state) subsequently undergo a large number of decoherence events after the quantum state reverses direc- FIG. 6. Ensemble-averaged DISH trajectories with initial group momen- tion and re-enters the region of strong coupling. Each of these tum of 30 a.u. As the surfaces diverge, the branching ratios generated by DISH are entirely dictated by a single decoherence induced hopping event decoherence events results in projecting out one of the quasi- and, for w = 0.1, accurately reproduce the quantum mechanical result. For degenerate states, hopping is quenched, and DISH probabili- w = 0.5 and greater, transmission probabilities are largely independent of ties, |c(t)|2, reproduce the non-oscillating quantum results. phenomenological parameter, w. 22A545-11 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012) simulations shown. For such narrow w, decoherence occurs (6,4) CNT too rapidly, not allowing quantum transitions probabilities to build up. Longer coherence intervals, from w = 0.1 on, reproduce correct quantum mechanical results.

VI. EXCITON DYNAMICS IN CARBON NANOTUBES (16,16) GNR AND GRAPHENE NANORIBBONS Carbon nanotubes and GNR exhibit extremely large electron mobilities.110, 111 In a photo-excited state, electron trans- port properties of these nano-materials are affected by diffusive processes, in which exciton energy converts to heat and dissipates. Excitons in semi-conducting GNRs and CNTs an- nihilate as a result of both radiative and nonradiative decay FIG. 7. Structures of the (6,4) CNT and (16,16) GNR are shown. The Stone- mechanisms. Radiative processes are competitive in CNTs Wales bond-rearrangement defect results from a rotation of a C–C bond in but not predominant, as demonstrated by low ﬂuorescence the hexagonal lattice. The 7557 defect results from the addition of 2 carbon yields.112–114 The rate of nonradiative relaxation to the ground atoms. electronic state in GNRs predicates its unique photophysical properties.115–117 ison to FSSH for different systems provides a diagnostic on Nonradiative exciton decay is a direct consequence of the reliability of DISH. the exchange of energy between electrons and phonons. The Investigation of electron-hole recombination in CNTs electron-phonon interaction mediates the rate at which ex- by photoluminescence spectroscopy gives evidence of a re- cited electrons lose energy to lattice vibrations. Quantum- laxation time scale on the order of 10s of picoseconds classical simulations treat phonons, or lattice vibrations, clas- to nanoseconds.112 Relaxation times of CNTs determined sically, and downward transitions in the quantum subspace through decoherence-corrected FSSH, are 147 ps, 67 ps, coincide with increases in the classical kinetic energy. Appli- and 42 ps, for the ideal, Stone-Wales, and 7557 systems, cation of DISH to exciton annihilation in CNTs and GNRs respectively.108 Figure 8 compares the population recovery in demonstrates that decoherence is an important contribution for nonradiative processes. Relaxation rates calculated from DISH simulations agree with those determined by experiment 0.07 and decoherence-corrected FSSH. 0.06 Ideal (6,4) The surface hopping simulations employed here, DISH 7557 Defect and decoherence-corrected FSSH,51, 108 are based on a two- 0.05 SW Defect state model for ground state recovery. Two adiabatic states lation 0.04 and their nonadiabatic coupling are computed within the time- u 79 0.03

dependent Kohn-Sham formalism and represent the ground Pop and first excited states. A standard decoherence correction is 0.02 applied to FSSH,108 in which the wave function is modified by resetting the excited state coefficient in accordance with 0.01 the pure dephasing rate, which is computed from optical re- 0 sponse functions, Eqs. (14) and (15). Employing a two-state model in DISH simplifies the algorithm, such that decoher- 0.06 Ideal (6,4) ence events equate to state switching at the pure dephasing 7557 Defect time. Details on the two-state limit of DISH are reported 0.05 SW Defect in the Appendix. The primary distinction between DISH lation 0.04 and decoherence-corrected FSSH is that DISH introduces u surface hopping only at decoherence events, while FSSH 0.03 Pop treats decoherence and surface hopping as two independent 0.02 processes. Surface hopping simulations are carried out for three 0.01 CNTs with (6,4) chirality, including one without atomic de- 0 fects and two systems with different bond-rearrangement de- 0 500 1000 1500 2000 2500 3000 fects (Figure 7). Two (16,16) GNR systems are considered, Time (fs) including one system with the same 7557 bond-insertion defect present in the CNT. The GNR is also displayed in FIG. 8. Recovery of the ground state in the ideal CNT and two defect sys- Figure 7. These systems are representative of those investi- tems occurs over 100 s of ps. (Top) Surface hopping populations of the ground electronic state generated by decoherence-corrected FSSH simula- gated by time-resolved spectroscopy under experimental con- tions. (Bottom) Quantum probabilities of occupying the ground electronic ditions. Monitoring exciton dynamics from DISH in compar- state determined by DISH. 22A545-12 Jaeger, Fischer, and Prezhdo J. Chem. Phys. 137, 22A545 (2012)

0.03 corrected FSSH are both governed simultaneously by coher- Ideal 16,16 ences and quantum probabilities, but, only in DISH, are quan- 0.025 7557 Defect tum transitions directly related to coherence intervals. Ac- curacy in the electron-phonon relaxation rates in CNTs and 0.02 GNRs depends on a number of factors, particularly, decoher- 108 lation ence rate. DISH provides a uniﬁed treatment of decoher- u 0.015 ence and quantum transitions without computational expense

Pop 0.01 beyond standard surface hopping approaches.

0.005 VII. CONCLUDING REMARKS 0 By combining the computational simplicity of quantum- classical nonadiabatic molecular dynamics with a formal 0.025 Ideal 16,16 treatment of quantum decoherence, DISH provides a straight- 7557 Defect forward and physically justiﬁed trajectory surface hopping 0.02 scheme, in which transitions between surfaces occur during

lation decoherence events. The transition probabilities and observ- u 0.015 ables are computed according to standard quantum mechani- Pop 0.01 cal rules. On one hand, DISH can be viewed as a surface hopping approach to quantum dynamics in dissipative environ- 0.005 ments. And, in another sense, DISH unifies decoherence and nonadiabatic transitions, providing a non-phenomenological 0 account of quantum transitions in condensed-phases. Evolu- 0 500 1000 1500 2000 2500 3000 tion of the quantum subsystem in DISH is affected by a series Time (fs) of decoherence events that converts the wave function into FIG. 9. Ground state recovery of GNRs from the first excited state takes a statistical mixture of quantum states, justifiable through the place within 100 s of ps. (Top) Surface hopping populations of the ground fundamental nature of decoherence and the quantum-classical electronic state from decoherence-corrected TDKS-FSSH simulations. (Bot- correspondence. tom) Quantum probabilities of the ground state determined by DISH. The importance of decoherence in chemical and physical processes in the condensed phase cannot be understated. A considerable amount of attention has been given to small the ground state, which determines the exciton annihilation quantum systems in macroscopic environments. DISH ex- rates, from both FSSH and DISH simulations. DISH results tends the feasibility of quantum-classical simulations to large show good agreement with the FSSH data. The DISH relax- quantum systems in macroscopic environments. Investiga- ation times are determined to be 200 ps, 107 ps, and 22 ps. tion of charge-transfer through crystals or solutions, exciton DISH reproduces the enhancement of nonradiative recom- dynamics in biological macromolecules, long-range electron bination by atomic defects. Radiative lifetimes of CNTs are transfer, and photo-induced dynamics in condensed phases all 1–2 orders of magnitude larger, and both DISH and FSSH de- require a theoretical approach that can simultaneously treat scribe relatively fast nonradiative pathways that account for 112–114 many quantum degrees of freedom undergoing irreversible, the low photoluminescence yields. dissipative evolution. As demonstrated with the non-radiative Direct experimental evidence on the exciton dynamics relaxation of CNTs and GNRs, the DISH approach achieves has yet to be attained for substrate free GNRs. Figure 9 dis- this goal. plays the interband relaxation of ideal and defect-containing GNRs. Similar to CNTs, nonradiative recombination occurs on a relatively short time scale of 100 s of picoseconds. ACKNOWLEDGMENTS Decoherence-corrected FSSH results in relaxation times of 309 ps and 124 ps for the ideal and 7557 defect systems, Financial support of the National Science Foundation respectively.109 Similar to CNTs, defects increase exciton re- (NSF) Grant No. CHE-1050405 is gratefully acknowledged. laxation rates. DISH simulations produce relaxation times of 231 ps and 143 ps, in agreement with decoherence-corrected FSSH. Both DISH and FSSH simulations show that defects APPENDIX: TWO-STATE LIMIT OF DISH open nonradiative decay channels, increasing relaxation rates. Consider an arbitrary wave function in a two-state basis Similar ground state recovery rates between DISH and at time, t, FSSH indicates that the stochastic DISH trajectory closely follows the dynamics of the decoherence-corrected FSSH en- ψ(t) = ci φi + cf φf . (A1) semble. 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