Surface Hopping Without Momentum Jumps: a Quantum- Trajectory-Based Approach to Nonadiabatic Dynamics

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Cite This: J. Phys. Chem. A 2019, 123, 1110−1128 pubs.acs.org/JPCA

Surface Hopping without Momentum Jumps: A Quantum- Trajectory-Based Approach to Nonadiabatic Dynamics Published as part of The Journal of Physical Chemistry virtual special issue “William P. Reinhardt Festschrift”. Craig C. Martens* University of California, Irvine, California 92697-2025, United States

ABSTRACT: We describe a new method for simulating nonadiabatic dynamics using stochastic trajectories. The method, which we call quantum trajectory surface hopping (QTSH), is a variant of the popular fewest- switches surface-hopping (FSSH) approach, but with important differences. We briefly review and significantly extend our recently described consensus surface-hopping (CSH) formalism, which captures quantum effects such as coherence and decoherence via a collective representation of the quantum dynamics at the ensemble level. Using well-controlled further approximations, we derive an independent trajectory limit of CSH that recovers the FSSH stochastic algorithm but rejects the ad hoc momentum rescaling of FSSH in favor of quantum forces that couple classical and quantum degrees of freedom and lead to nonclassical trajectory dynamics. The approach is well-defined in both the diabatic and adiabatic representations. In the adiabatic representation, the classical dynamics are modified by a quantum-state-dependent vector potential, introducing geometric phase effects into the dynamics of multidimensional systems. Unlike FSSH, our method obeys energy conservation without any artificial momentum rescaling, eliminating undesirable features of the former such as forbidden hops and breakdown of the internal consistency of quantum and ensemble-based state probabilities. Corrections emerge naturally in the formalism that allow approximate incorporation of decoherence without the computational expense of the full CSH approach. The method is tested on several model systems. QTSH provides a surface-hopping methodology that has a rigorous foundation and broader applicability than FSSH while retaining the low computational cost of an independent trajectory framework.

1. INTRODUCTION improve FSSH have mainly focused on corrections to this Trajectory surface hopping is a popular and eﬃcient method problem. Another issue is related to the strict classical energy for simulating the coupled electronic and nuclear dynamics of conservation imposed on the individual trajectories in FSSH. 1−11 When a trajectory undergoes a transition between electronic molecular systems in a quantum-classical framework. The ﬀ most commonly used implementation is the fewest-switches states, the corresponding di erence in electronic-state energies surface-hopping (FSSH) method, originally introduced by at the transition point is accommodated in the nuclear 1 dynamics by an ad hoc rescaling of the momentum along the

Downloaded via UNIV OF CALIFORNIA IRVINE on February 7, 2019 at 22:44:42 (UTC). Tully in 1990, and its many subsequent variants (see, e.g., refs 9−11 for reviews). In the FSSH approach, the evolving nonadiabatic coupling vector. This algorithm, although quite See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. physically reasonable prima facie, has no rigorous foundation multicomponent nuclear quantum wavepacket is approximated fi by an ensemble of independent classical trajectories, each of based on rst principles. The FSSH algorithm also results in which carries its own copy of an electronic Schrödinger practical problems, such as the spurious closing of classically equation that evolves under the influence of the time- forbidden channels allowed by the full quantum evolution and dependent classical variables and determines the probability the presence of “frustrated hops”, transitions that are dictated of sudden stochastic transitions of the trajectory between the to occur by the surface-hopping stochastic process but rejected quantum states. FSSH has proven to be a simple and robust by the ad hoc imposition of classical energy conservation. method for simulating classical molecular dynamics with These events break the consistency of surface hoppingthe agreement between the evolving quantum density matrix quantum electronic transitions. fl The FSSH method has a number of well-known short- probabilities and the state populations re ected by the hopping comings that limit its applicability. In particular, the original trajectory ensemble. Furthermore, the use of the nonadiabatic coupling vector in the momentum rescaling is only well- implementation does not treat quantum coherence, and fi especially decoherence, properly, leading to a representation de ned in the adiabatic representation of electronic states, of the quantum evolution that is overcoherent, in the sense that the off-diagonal quantum density matrix elements of Received: October 28, 2018 individual trajectories can be spuriously large in magnitude Revised: January 11, 2019 compared with the exact quantum coherence. Attempts to Published: January 11, 2019

© 2019 American Chemical Society 1110 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article limiting the applicability of FSSH to dynamics in the adiabatic Ĥ = THq̂ + ̂o(,rq ) (1) representation. Recently, we proposed an alternative surface-hopping Here r and q are the electronic and nuclear coordinates, 12 framework, consensus surface hopping (CSH), which avoids respectively. T̂q is the nuclear kinetic energy, whereas Ĥ o(r,q) the independent trajectory approximation and more rigorously is the electronic Hamiltonian, which depends parametrically on incorporates the nonclassical effects of nonlocality, uncertainty, the nuclear coordinates q. An electronic basis is chosen in 13 and quantum coherence. The advantages of CSH come at a terms of states ϕn(r;q), which are functions of the electronic cost, however, and the method is numerically more expensive coordinates r and may also depend parametrically on the than FSSH due to the interdependence of the trajectories in nuclear coordinates q. Matrix elements of the electronic the ensemble. Its use as a computational approach is thus Hamiltonian are given by limited to low-dimensional model systems. The greatest value VH()qrqrqrq* (; )̂ (, ) (; )dr of the CSH formalism, in our opinion, is not as a numerical mn = ϕϕmno (2) method for simulations but as a framework for developing additional approximations and more economical methodology In the adiabatic representation, the electronic wave functions in a well-controlled and rigorous manner. depend on q, and the derivative coupling matrix element ff In this Article, we describe such an approximate approach, dmn(q) results from o -diagonal matrix elements of the nuclear quantum trajectory surface hopping (QTSH). The theory kinetic energy develops from a rigorous quantum-classical limit of the Ÿ multistate quantum Liouville equation14−20 in the context of d ()qrqrq* (; ) (; )dr mn = ϕϕmn∇q (3) the computationally efficient independent trajectory-based FSSH method. We take an approximate independent trajectory The FSSH formalism approaches the problem of non- limit of the full CSH method, yielding an algorithm that is adiabatic dynamics by using classical trajectories and ensemble equivalent to the standard FSSH stochastic trajectory hopping averaging to approximate the nuclear quantum dynamics of a approach. The main difference with FSSH is the abandonment multicomponent wavepacket. These classical trajectories of ad hoc momentum rescaling to conserve the classical capture theŸ quantum electronic transitions by stochastic kinetic-plus-potential energy at the individual trajectory level “hops” between the electronic surfaces. The electronic degrees and its replacement by quantum forces derived rigorously from of freedom are, in turn, driven by the time-dependent nuclear the semiclassical-limit quantum-classical Liouville equation. trajectories q(t)thatappearinthenuclearcoordinate This feature of the method restores the consistency of surface dependence of the electronic Hamiltonian. For a given hopping that is broken by the frustrated hops of the standard classical path q(t), the electronic wave function can be FSSH approach. In addition, the energetics of the system are expanded in the chosen electronic basis as treated correctly: The full quantum-classical energy is conserved rigorously at the ensemble level. The ensemble ψ(r,tct )= ∑ n ( )ϕj (rqt ; ( )) average energy conservation is the correct behavior required by n (4) quantum mechanics; individual trajectory conservation of the The substitution of this expression into the time-dependent classical energy is a constraint that is too restrictive and too Schrödinger equation yields a set of coupled equations for the classical and so precludes important quantum effects. Further expansion coefficients corrections are developed and implemented to incorporate average ensemble-level decoherence as an approximation to iℏctṁ ()= ∑ ( Vmn −ℏ iqḋ· mn) ct n () the full CSH treatment of coherence. The approach is tested n (5) on standard 1D models. For cases where FSSH works well, the It is convenient to use the quantum density matrix, a = c c*, QTSH approach gives similar results for an equivalent mn m n rather than the wave function amplitudes, cm. The quantum computational cost. In situations where FSSH fails due to equations of motion then become spuriously frustrated hops, the QTSH method continues to give results in close agreement with quantum mechanics. iℏatmṅ ()=[∑ ( Vml −ℏ iqḋ· ml)() a ln−−ℏ aV ml ln iqḋ· ln ] The organization of the rest of this paper is as follows. In l Section 2, we review the standard FSSH methodology. We (6) fl then brie y summarize the full CSH approach as reported Here the following relations hold previously13 and significantly extend its treatment of energy conservation through electronic-state-dependent nonclassical d*ln= −d nl (7) forces. The QTSH approach, the quantum trajectory fi modi cation of FSSH, is developed in both the diabatic and d*nn = 0 (8) adiabatic representations. Numerical results comparing the methods for a number of model systems are presented in The equation of motion for the population of the nth state, Section 3. Finally, a summary and discussion is given in Section represented by the diagonal density matrix element, ann, is then 4. given by

anṅ = ∑ bnl 2. THEORY ln≠ (9) 2.1. Fewest-Switches Surface Hopping. We begin by where brieﬂy reviewing the FSSH method proposed by Tully in 1990.1 The total Hamiltonian describing the electronic and 2 baVanl = Im(nl* nl )− 2Re(nl*qḋ· nl ) nuclear degrees of a molecular system is given by ℏ (10)

1111 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

It should be remembered that Vnl(q(t)) and dnl(q(t)) as well d()ρ̂ t iℏ =[Ht̂ ,()ρ̂ ] as q̇(t) all depend on the time-dependent classical path, q(t). dt (12) In the FSSH method, an ensemble of independent where Ĥ is the Hamiltonian of the system. For dynamics on a trajectories (qj(t),pj(t)) (j = 1, 2, ..., N) is sampled from a distribution representing the initial nuclear quantum state, single potential surface, the classical limit of eq 12 is the well- where p is the canonical momentum conjugate to the jth known classical Liouville equation of nonequilibrium statistical j mechanics22 trajectory’s nuclear coordinate qj. Each trajectory so generated is initiated on one of the electronic states and then evolves ∂ρ ={H, ρ } under Hamilton’s equations that correspond to the instanta- ∂t (13) neous occupied state. Stochastic transitions occur between where ρ(q,p,t) and H(q,p,t) are now functions of the 2f- these states with a probability that is proportional to the dimensional (for f nuclear degrees of freedom) phase-space relative rate of change of the quantum populations associated variables Γ =(q,p) and time t, and {H,ρ} is the Poisson with the trajectory. bracket of H and ρ:{H,ρ}=∂H/∂q·∂ρ/∂p − ∂ρ/∂q·∂H/∂p. To illustrate, we consider a system with two electronic states This correspondence can be derived systematically from eq 12 and a trajectory currently evolving on state 1 (here we suppress by performing a Wigner−Moyal expansion23,24 of the the trajectory index j). In the FSSH method, this trajectory has quantum-mechanical Liouville equation. To lowest order in a probability of hopping from surface 1 to surface 2 if a11̇ (t) is ℏ FSSH , this involves replacing commutators by Poisson brackets: negative. In that case, Phop (t), the probability of hopping at [Â,B̂] → iℏ{A,B}+O(ℏ2). time t during a time step of duration Δt, is then given by Diabatic Representation. The semiclassical limit of eq 12 can be generalized to two coupled quantum states coupled to FSSH 1 classical degrees of freedom. The approach is general for mixed P ()t bt() t hop = 12 Δ quantum-classical problems. Here we consider two quantum at11() (11) electronic states in the diabatic electronic representation The hop is realized or not by generating a random number coupled to classical limit nuclear dynamics. The Hamiltonian FSSH and density matrix are given by 2 × 2 matrices between 0 and 1 and comparing it with Phop (t). An analogous procedure is used for trajectories currently on state 2. HV11̂ ̂ Strict energy conservation at the individual trajectory level is Ĥ = imposed by rescaling the momenta at the instant of the hop so VĤ ̂22 (14) that the total kinetic plus potential energy of the trajectory i y remains unchanged during the transition. In multidimensional and j z j z systems, the momentum rescaling is performed along the j z j ()ttz () direction of the nonadiabatic coupling vector, d .If k ρρ11̂ { 12̂ 12 ρ()̂ t = insufficient energy is available for an upward hop in energy, ρρ21̂ ()tt 22̂ () (15) then the event is termed “frustrated” and does not occur respectively.ji The elementszy of these matrices are nuclear despite the stochastic algorithm dictating the transition. Such j z operators. Withj the replacementz of the quantum-mechanical aborted events lead to a breakdown of the consistency between j z the density matrix populations and the trajectory ensemble operators byk the corresponding{ classical phase-space functions, statistics. this becomes a set of coupled classical-like Liouville 14−18,25−30 2.2. Consensus Surface Hopping. The FSSH method is equations a sensible but ad hoc solution to the problem of modeling ∂ρ 2V nonadiabatic dynamics with trajectories. The algorithm was 11 ={HV,,ραβ }+{ }− 11 11 (16) proposed based on physical reasoning rather than derived ∂t ℏ systematically from the underlying exact quantum dynamics. ∂ρ22 2V The CSH approach seeks to go beyond this and build a ={HV22,,ραβ }+{ }+ t 22 (17) trajectory-based method for nonadiabatic dynamics simula- ∂ ℏ 12 tions with a rigorous foundation. The CSH formalism ∂α 1 ={HV0, αωβ }+ + {, ρρ + } focuses on solving the multistate quantum Liouville equation ∂t 2 11 22 (18) for coupled electronic and nuclear dynamics in the semiclassical limit using trajectory ensembles to represent phase- ∂β V ={H0,()βωαρρ }− + − space densities.14−18 These states evolve quantum mechan- ∂t ℏ 11 22 (19) ically, and so the trajectory dynamics must correspondingly Here we have written the coherence ρ12(Γ,t)=α(Γ,t)+ become nonclassical. iβ(Γ,t) in terms of its real and imaginary parts and have An initial description of the approach was given in ref 12. fi de ned the average Hamiltonian H0 =(H11 + H22)/2 and the Here we provide a review of that work and, in addition, frequency ω =(H − H )/ℏ. All higher order terms in ℏ have fi 11 22 signi cantly extend the formalism to give a more rigorous been neglected, leading to a classical-limit formalism that treatment of the energy conservation by the inclusion of retains only the most important nonclassical corrections. nonclassical terms in the phase-space dynamics. This addi- The CSH method employs a trajectory ensemble tional aspect will be a key component of the QTSH approach representation of the phase-space functions describing the that is the focus of this paper and is developed below. density matrix in the coupled semiclassical Liouville equations. The quantum-mechanical Liouville equation for the density Quantum population transfer is represented by stochastic operator ρ̂(t) is given by21 trajectory hops between the diagonal surfaces, whereas

1112 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article quantum coherence is represented collectively at the ensemble nonclassical terms appear. The first are sink and source terms level by interrelationships between nonclassical amplitudes and ±2Vβ/ℏ, which are responsible for the population transfer phases associated with each trajectory. between states. The second type of nonclassical term is the The phase-space densities corresponding to the populations Poisson brackets {V,α}, which appear symmetrically in the of states 1 and 2 are together represented by a single ensemble equations for both ρ11 and ρ22. These interactions modify the of N trajectories, each of which is characterized by a point in shape of the evolving distributions but do not change the total phase space Γj(t)=(qj(t),pj(t)) and a binary integer σj(t), state populations. Conservation of population under these which can take on the values 1 or 0, indicating whether the terms results from the fact that the classical trace (integral over trajectory is associated with quantum state 1 or 2, respectively. phase-space volume) of a Poisson bracket vanishes for The states 1 and 2 phase-space densities are then given by functions satisfying appropriate boundary conditions. N The evolution of the dynamical variables is determined by 1 numerically integrating the ordinary differential equations for ρ (,)Γ t = σδ()(ttΓ−Γ ()) 11 ∑ jj the trajectories and the coefficients. Each time step of duration N j=1 (20) Δt is divided into two parts. First, the coefficients are updated; and then, the phase-space trajectories are propagated forward in N time. 1 We first consider the population sink and source terms ρ (,)Γ t = (1−Γ−Γσδ (tt )) ( ( )) 22 ∑ jj responsible for the evolution of the stochastic variables σ (t). N j 1 (21) j = To derive a probabilistic algorithm for updating the former we respectively. The δ functions represent the discrete points of consider the subsets of trajectories on surfaces 1 and 2 ffi the trajectories in phase space, whereas the coe cients σj separately. For surface 1, substitution into the semiclassical denote which state the trajectory currently occupies. In the Liouville equations yields fi numerical implementation involving nite trajectory ensem- N N bles, the δ functions are smoothed using phase-space 1 1 2()V Γ σσΔΓ−Γg()= − βgt()Γ−Γ Δ Gaussians, as described in ref 12.Thisresultsinthe ∑∑kk k k k N k==11N k ℏ replacement of the delta functions by the Gaussian basis (26) g(Γ): δ(Γ − Γj) → g(Γ − Γj). with a similar expression for surface 2. We can evaluate the left The coherence ρ12(Γ,t) is also represented in terms of the trajectory ensemble. Unlike the populations, however, the and right sides of these expressions at each of the trajectory coherence is a complex quantity, and thus the coefficients of points of interest, Γj, yielding coupled linear equations for the ffi ffi the trajectories are complex numbers. The populations and the change in coe cients. This gives, for the surface 1 coe cients real and imaginary parts of the coherence are given in terms of N N 1 1 2()V Γj the smoothed trajectory ensemble as σσΔΓ−Γg()= − βgt()Γ−ΓΔ N ∑∑kk j k N k jk N k==11k ℏ 1 ρ (,)Γ t = σ()(tgΓ−Γ ()) t (27) 11 N ∑ jj j=1 (22) for j = 1, 2, ..., N. In general, this presents a linear algebra problem for the determination of the Δσj. We can simplify its N 1 solution by making the following approximation that becomes ρ (,)Γ t = (1−Γ−Γσ (tg )) ( ( t )) fi 22 N ∑ jj exact as N becomes in nite and the Gaussian functions, g(Γ), j=1 (23) become localized N N 1 1 α(,)Γ t = α ()(tgΓ−Γ ()) t σσΔΓ−Γ≃⟨⟩Δg() ρ σ N ∑ jj ∑ kk j k11 jj j=1 (24) N k=1 (28)

N where ⟨ρ ⟩ , the local density at point Γ on surface 1, is given 1 11 j j β(,)Γ t = β()(tgΓ−Γ ()) t by N ∑ j j j=1 (25) N 1 ﬃ ⟨ρσ⟩j = kjg()Γ−Γ k The coe cients σj(t) are stochastic binary integers, whereas 11 ∑ N k=1 (29) αj(t) and βj(t)(j = 1, 2, ..., N) are continuous real numbers. (We note that it is not necessary to make such a distinction; in Similarly, we evaluate the value of the coherence at point j as a recent paper, we describe an alternative approach to the N general problem of quantum-state hopping that represents 1 both populations and coherence in terms of separate stochastic ⟨ββ⟩j = ∑ kg()Γ−Γjk N (30) processes.31) k=1 The equations of motion for the trajectories, Γj(t), and state The equation for updating the coeﬃcients of trajectories parameters, (σj(t), αj(t), βj(t)) (j = 1, 2, ..., N), are determined currently on surface 1 becomes by substituting the trajectory representations, eqs 22−25, into the semiclassical Liouville eqs 16−19. In the uncoupled (V = 1 2()V Γj t 0) case, the evolution of the populations reduces to purely Δσj = − ⟨⟩Δβ j ⟨⟩ρ j ℏ (31) classical Liouvillian dynamics corresponding to trajectory 11 ensemble evolution under the appropriate electronic-state For trajectories currently evolving on surface 2, the Hamiltonian. In the presence of coupling, two types of corresponding result is

1113 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article ff ω 1 2()V Γj di erence potential, (Γk), over the ensemble, then this Δσj = − ⟨⟩Δβ j t summation will be “decayed” by decoherence. The individual ⟨⟩ρ j ℏ (32) 22 βk values may be quite large; it is only the weighted sum of β These are then identified as the hopping probabilities for the their values, ⟨ ⟩j, that becomes small with decoherence. In trajectories in the ensemble. For instance, for the jth trajectory contrast, FSSH determines hopping probabilities by using the σ independent individual values of each trajectory’s quantum currently evolving on state 1, if Δ j(t) is negative, then the ff trajectory has a nonzero probability of undergoing a hop to density matrix. This di erence is the origin of the over- state 2 during the time step Δt. The CSH probability for this coherence problem of FSSH. event is The rest of this section describes further developments of the CSH formalism that were not included in our previous publication.12 CSH 2 Phop = Vt()Γ⟨⟩Δjjβ We now consider the terms in the evolution equations that ℏ⟨ρ11 ⟩j (33) involve trace-preserving Poisson brackets. These include both the homogeneous classical phase-space evolution terms of the These equations form the basis of a stochastic hopping form {H,ρ} and the inhomogeneous nonclassical terms {V,α} ξ algorithm. A random number between 0 and 1 is generated coupling the density matrix elements. for each trajectory at each time step and compared with the CSH It is convenient to consider the total nuclear density ρ = ρ11 appropriate value of Phop = |Δσj| corresponding to the + ρ22 occupied state. The value of σj(t) is changed by ±1 or kept at its current value depending on the outcome. ρ(,qp ,)ttt=+ρρ (, qp ,) (, qp ,) The result in eq 33 is strongly reminiscent of the FSSH 11 22 N hopping probability given in eq 11. But we emphasize the 1 ff = ∑ gt(Γ−Γj ( )) essential di erence between this approach and the FSSH N method. Here the ensemble collectively determines the j=1 (36) stochastic hopping probabilities of each of its members. The This quantity is independent of the stochastic parameters σj(t) local densities ⟨ρ11⟩j and ⟨ρ22⟩j and the coherence ⟨β⟩j at point (j = 1, 2, ..., N) (although we will see below that its evolution Γj depend on the ensemble of evolving trajectories Γk(t)(k = depends on the quantum-state parameters). The total nuclear 1, 2, ..., N). They are not independent dynamical variables density ρ(q,p,t) obeys the partial differential equation obtained associated with independent trajectories, as in the FSSH by adding eqs 16 and 17 formalism. Quantum transitions are thus determined by a “consensus” among the members of the ensemble representing ∂ρ ={HH11,,2,ρρα }+{22 }+ { V } the full entangled electronic-nuclear quantum state rather than ∂t 11 22 (37) by the independent trajectories of FSSH. Note that the terms involving Vβ responsible for population The electronic coherence evolves in parallel with, and transfer between states 1 and 2 cancel from the evolution coupled to, the evolving population densities. A similar analysis equation. Equation 37 conserves the total population, given by that includes the approximate neglect of terms in eqs 18 and 19 the phase-space trace of ρ, as it should. that leave the trace of ρ unchanged yields expressions that 12 The equations of motion for q (t) and p (t)(j = 1, 2, ..., N) describe the evolution of the coefficients over the time step j j 12 are derived by substituting eq 36 into eq 37. We have for the Δt. Because these equations are solved deterministically left-hand side of the resulting expression rather than by a stochastic hopping algorithm, the limit Δt → 0 can be taken, yielding the coupled differential equations N 1 ∂Γ−Γgg()k ∂Γ−Γ ()k LHS = − q̇ · + ṗ · () ∑ k k αjj̇ = ωβΓ j (34) N k=1 ∂q ∂p Ä É (38) Å Ñ 1 Å Ñ β̇ = −Γωα() + V()(2Γ− σ 1) The right side of theÅ equation becomes Ñ j jj j j (35) Å Ñ ℏ N ÇÅ ÖÑ 1 ∂H ∂Γ−Γg() These diÄfferential equations are integratedÉ numerically using k k Å Ñ RHS = ∑ − · standardÅ methods. Ñ N 1 ∂pq∂ Å Ñ k= The CSHÇÅ equations for the coherencesÖÑ are identical to the Ä HVÅ () g() FSSH density matrix equations for coherences in the diabatic ∂ kkÅ ∂Γ ∂Γ−Γk + Å + 2 αk · representation if we identify the CSH parameters α and β with ∂qqÅ ∂ ∂ p j j ÇÅ (39) the real and imaginary parts of the jth independent trajectory ÑÉ where i y Ñ coherence in the FSSH method. j z Ñ fi j z Ñ We emphasize that no arti cial decoherence is added to the Ñ evolving system in the CSH formalism. The role played by Hkk= σσHH11k()Γ k+ (1−Γ k )22 (){ k ÖÑ coherence and its decay is treated accurately through the p2 k UU()qq (1 )() collective nature of the method, as highlighted by eq 30. In =+σσk 12k +− k k (40) particular, decoherence is represented naturally via the 2m fi cancellation of the signed terms βk in the summation over Γk which de nes the diagonal diabatic potentials Un(q)(n = 1, 2). ffi ∂ ∂ in the local vicinity of the hopping trajectory, j, to yield ⟨β⟩j. If Equating the coe cients of the terms g(Γ − Γk)/ q and ∂ ∂ these terms exhibit destructive interference due to either the g(Γ − Γk)/ p of the LHS and RHS expressions yields the nature of the pure-state evolution of the multicomponent modified classical equations of motion for the trajectory nuclear wavepacket or by environmental fluctuations in ensemble

1114 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

pj space. Momentum rescaling to impose energy conservation q̇ = j m (41) induces a spurious shift of the probability in the phase space upon a transition that is unjustified by, and in conflict with, the ṗ = −∇UV()qq − 2α ∇ () mathematical form of the underlying semiclassical Liouville j j j j j (42) equation. for j = 1, 2, ..., N, where ∇≡∂/∂q. In addition to the classical Adiabatic Representation. The CSH method can be force acting on the jth trajectory resulting from the implemented equally well in the adiabatic representation, where electronic-state coupling appears through off-diagonal instantaneous Hamiltonian, Hj, an additional quantum force 16,41,42 appears, which depends on both the gradient of the off- terms in the kinetic energy. We start with the quantum- mechanical Hamiltonian and density matrix in the adiabatic diagonal diabatic coupling, V(qj), and the real part of the representation. These are given by coherence parameter, αj(t), corresponding to that trajectory. The nonclassical force contributes whenever coupling and ̂ ̂ coherence are present. In CSH, these continuous quantum HW++ Ĥ = forces replace the sudden impulsive momentum rescaling of ̂ † ̂ FSSH. Each trajectory does not conserve the classical energy, WH−− (43) H (t). Rather, the total energy expectation value, E(t)= ji zy j and j z Tr(Ĥρ(t)), is conserved on average over the ensemble. We j z ̂ j z have discussed the energy budget of nonadiabatic dynamics in k ρρ++̂ ()tt{ +̂ − () detail in a recent paper.13 We will examine this point in more ρ()̂ t = ()tt () detail below in the description of the approximate QTSH ρρ−̂ + −−̂ (44) method. ji zy | | fi respectively.j The adiabatic eigenstatesz { +⟩, −⟩} are de ned in The expression LHS = RHS resulting from equating eqs 38 terms of thej diabatic basisz {|1⟩,|2⟩} as and 39 is a single equation for the 2f unknowns q and p (j = 1, j z ̇j ̇j k { 2, ..., N). One possible solution is given by our quantum |+⟩ =|1cos(/2)⟩ ϕϕ+| 2sin(/2)⟩ (45) trajectory equations of motion, eqs 41 and 42. Other solutions are also possible. This is related to the general ambiguity |−⟩ = −|1sin(/2)⟩ ϕϕ+| 2cos(/2)⟩ (46) associated with trajectory representations of quantum-state where the mixing angle ϕ(q) is given by evolution in any quantum trajectory approach. We have discussed this issue in other contexts in previous publica- 2()V q tions.32−35 tanϕ (q ) = UU12()qq− () (47) We note that the quantum trajectory equations of motion, eqs 41 and 42, (as well as the corresponding adiabatic Here U1(q)andU2(q)arethediagonaldiabatic-state ff expressions described below) also appear in other non-surface- potentials, and V(q) is the o -diagonal diabatic coupling. ff hopping trajectory-based approaches to nonadiabatic dynam- In terms of these states, the o -diagonal nonadiabatic ics, such as in the Meyer−Miller classical analogue couplings are 36−38 39,40 approach and in the recent work of Tao. 2 iℏ ℏ 2 We emphasize that no momentum rescaling is performed in WT̂ = ⟨+| ̂|−⟩ = ∇ϕϕ()pqq· ̂ + ∇ () the CSH method when electronic transitions occur. In general, 2mm4 (48) individual trajectories representing a quantum system have no and requirement to separately conserve energy,13,32,34 and they do † not do so in this method. We believe that energy conservation WT̂ = ⟨−| ̂|+⟩ = − Ŵ (49) of individual trajectories imposed by momentum rescaling is The nonadiabatic coupling vector matrix element, d(q), is too classical from a physical perspective. The trajectories in a defined as surface-hopping ensemble comprise a statistical representation of an underlying quantum density matrix and should not d()q ≡⟨+|∇|−⟩ (50) separately be overinterpreted as being “real”. In particular, there is no reason why they should individually conserve This can be evaluated for the nonadiabatic states in terms of energy. In quantum mechanics, it is the expectation value of the diabatic states and position-dependent angle, ϕ(q), the Hamiltonian (and its moments) that should be conserved yielding the result by the time evolution. Whereas the adoption of momentum 1 d()qq= −∇ϕ() rescaling can lead to accurate results in some situations by 2 (51) imposing correct asymptotic properties by hand, as it were, it ff ̂ also leads to serious problems such as spuriously frustrated In terms of this quantity, the o -diagonal element W = ⟨+| ̂ hops and corresponding forbidden processes that are allowed H|−⟩ can be written as by exact quantum dynamics. iℏ pp̂ ̂ From a mathematical perspective, momentum rescaling is Ŵ = − dq()· + ·dq () undesirable because it violates the phase-space locality of the 2 mm (52) underlying coupled semiclassical Liouville equations. This with Ŵ † = ⟨−|Ĥ|+⟩ = −Ŵ . In the semiclassical limit employed ji zy locality is readily apparent in eqs 16 and 17, which highlights below, this becomesj z the symmetrical appearance of the transition-inducing terms in k p { the equations: Every element of population that is induced to Wi()dq () Γ = −ℏ · (53) leave surface 1 by the term −2V(Γ)β(Γ)/ℏ appears on surface m 2 as +2V(Γ)β(Γ)/ℏ at the same point Γ =(q,p) in phase with W*(Γ)=−W(Γ).

1115 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

By evaluating the Wigner transform of the quantum The trajectory equations of motion for qj(t) and pj(t) can be Liouville equation in the adiabatic representation, eq 12, to derived from the equation of motion for the total nuclear lowest order in ℏ, we obtain the corresponding semiclassical density ρ = ρ++ + ρ−− using the same procedure employed Liouville equations in the adiabatic representation12,16,41 above in the diabatic case. The result is

∂ρ++ p p pj dq()j ={H++,,2ρβα++ }−ℏ d· − d· q̇ = −ℏ2 β ∂t {}mm(54) j mmj (65)

∂ρ−− p p 2ℏ ={H−−,,2ρβα }−ℏ d·+·d ṗ = −∇Uj()qpdq+ βj()()·∇ ∂t −− {}mm(55) j j m j j (66) for j = 1, 2, ..., N. The second term can also be written in terms ∂α p ={Ho,()αωβρ }+ +d · ++ − ρ−− of the time derivative of the nonadiabatic coupling vector along ∂t m (56) the resulting trajectories ḋ =(v·∇)d, where v = p/m is the trajectory velocity ∂β ℏ p ={H , βωα }−− d·+, ρ ρ o ++ −− (57) p = −∇Uq()q + 2ℏβ d()̇ ∂t 2 { m } j̇ j j j j (67) 2 2 where H++(Γ)=p /2m + E+(q), H−−(Γ)=p /2m + E−(q), In numerical implementation, it is much easier to determine 1 (), and ω(Γ)=(E (q) − E (q))/ℏ; here Ho =+2 HH++ −− + − the time derivative of d along a trajectory than to evaluate the E+(q) and E−(q) are the adiabatic potentials, the position- spatial derivatives directly. dependent eigenvalues of the diabatic potential matrix. The These equations of motion are closely related to those density matrix elements ρ ( ,t)arenowphase-space appearing in the Miller−Meyer treatment of coupled mn Γ 36−38 functions, and we have written the coherence ρ+− = α + iβ electronic-nuclear dynamics. In ref 38,Millerand in terms of its real and imaginary parts. coworkers introduce a non-Hamiltonian “kinematic momen- The phase-space-generalized densities in the adiabatic tum”, given in our notation by pkin,j = pj − 2ℏβjd(qj), and show fi representation are written in terms of an ensemble of N that its use simpli es numerical calculations by avoiding the trajectories as appearance of ∇d. This approach may also be useful in the numerical application of the present method. N 1 The quantum forces acting on the classical trajectories in the ρ (,)Γ t = σ()(tgΓ−Γ ()) t ++ N ∑ jj adiabatic representation are in the form of Hamilton’s j=1 (58) equations in the presence of a vector potential A(q,β(t)) N 2 1 ((,()))pAq− β t ρ (,)Γ t = (1−Γ−Γσ (tg )) ( ( t )) H(,Γ σβ , )= + U ()q −− ∑ jj 2m σ (68) N j=1 (59) where A(q,β(t)) = 2ℏβ(t)d(q) (neglecting terms of order ℏ2). N 1 This vector potential depends on the quantum subsystem α(,)Γ t = ∑ αjj()(tgΓ−Γ ()) t dynamics through the appearance of the imaginary part of the N j 1 ff = (60) coherence, βj(t). Interesting geometric phase e ects resulting from these nonclassical forces may result in systems with two N 1 or more dimensions in the presence of, for example, conical β(,)Γ t = β()(tgΓ−Γ ()) t 43−47 N ∑ j j intersections. This will be explored in future work. j=1 (61) The CSH method is based on a systematic derivation of the A similar analysis to the one performed above for the diabatic equations of motion for a trajectory ensemble representation of case then yields the CSH equations of motion for the the nonadiabatic dynamics from the underlying quantum quantum-state parameters and phase-space trajectories. Liouville equation in the semiclassical limit. CSH eliminates The stochastic parameters {σj}(j = 1, 2, ..., N) are updated the ad hoc instantaneous momentum rescaling and strict as follows. For the jth trajectory at phase-space point Γj = energy conservation of FSSH by incorporating continuous (qj(t),pj(t)) currently occupying state |+⟩, the probability of state-dependent quantum forces into the trajectory equations hopping to state |−⟩ at time t during a time interval Δt is given of motion. In addition, a correct treatment of quantum by coherence emerges naturally in the CSH formalism through the collective and interdependent nature of the trajectories dq()· p CSH 2 j j across the ensemble in determining hopping probabilities. The Phop = Δσj = ⟨⟩Δα j t numerical implementation of the method can be quite accurate m ⟨⟩ρ++ (62) for model systems. However, the interdependent nature of the with an analogous expression for hops from |−⟩ → |+⟩. The trajectories greatly increases the numerical cost of the method equations of motion for the coherence parameters yield the in direct implementations. For multidimensional systems, the differential equations CSH method quickly becomes prohibitively expensive with increasing size. Furthermore, the complexity of the method can

dq()j · pj lead to errors if conditions and parameters such as the () (2 1) Gaussian smoothing width are not chosen carefully. The main αjj̇ = ωβΓ j + σj − (63) m value of CSH is not as a practical method per se but as a framework for introducing further approximations in a well- ̇ ()()t βj = −Γωαjj (64) controlled manner

1116 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

N 2.3. Quantum Trajectory Surface Hopping. We now 1 describe a new surface-hopping approach based on an ⟨ρσ⟩≃ ⟨⟩g() Γ−Γ = ⟨⟩⟨⟩ σρ 11 j N ∑ jj k jj independent trajectory limit of the full CSH formalism. We k=1 (71) provide a derivation of the individual trajectory quantum where ⟨ρ⟩j is the local value of the total nuclear density ρ = ρ11 electronic-state density matrix dynamics and stochastic + ρ at point Γ . Now consider the local value of the phase- hopping algorithm for the independent trajectories of the 22 j fi space function representing the imaginary part of the FSSH method by employing additional well-de ned approx- coherence at Γ imations to CSH. The ad hoc impulsive momentum jumps of j N FSSH are abandoned, however, and replaced by the 1 continuous quantum forces that emerge from the CSH ⟨ββ⟩j = g()Γ−Γjk N ∑ k formalism. We call the resulting method QTSH, a quantum k=1 (72) trajectory-based variant of FSSH with a rigorous foundation. Here we make the simplifying assumption that the system is In the CSH methodology, the underlying focus is on solving fully coherent, in the sense that the parameters βk are slowly the coupled evolution of the phase-space functions ρ ( ,t) mn Γ varying in the vicinity of Γj with values of the index k using a trajectory ensemble representation. This naturally leads corresponding to |Γj − Γk| ≤ ΔΓ. For small enough ΔΓ, we can to the appearance of ensemble-level quantities in the equations then make the replacement βk → βj in the expression, giving of motion for the phase-space trajectories and quantum the approximation parameters. Consider, for example, the stochastic hopping of N the jth trajectory from state 1 to state 2 in the diabatic 1 representation. The local values of the functions representing ⟨ββ⟩≃j ∑ jg() Γ−Γjk= βρj⟨⟩j N k=1 (73) state 1 population density ρ11(Γ,t) = (1/N)∑jσj(t)g(Γ − Γj(t)) and the imaginary part of the coherence β(Γ,t) = (1/ Under these approximations, the CSH hopping probability N)∑jβj(t)g(Γ − Γj(t)) at the phase-space point Γ = Γj becomes determine the CSH hopping probability of trajectory j through the expression CSH 1 2()V Γj Phop ()t ≃ βρj⟨⟩Δj t ⟨⟩⟨⟩σρjj ℏ 1 2()V Γj PCSH = ⟨⟩Δβ t hop j 1 2()V Γj ⟨⟩ρ11 j ℏ (69) = βjΔt ⟨⟩σ j ℏ (74) where ⟨ρ11⟩j = ρ11(Γj) and ⟨β⟩j = β(Γj). This “consensus” involvement of the entire ensemble in the with the total nuclear density at point Γj canceling from hopping decision making emerges systematically from first numerator and denominator. principles. The rigor of the methodology comes with a The connection between the independent limit of CSH and relatively high cost of numerical effort, however, as the local the conventional FSSH formalism can now be made. Within the consistency assumption underlying FSSH, the populations values of these quantities at every trajectory point Γj for (j = 1, 2, ..., N) must be determined from the ensemble as a whole at of the auxiliary density matrix elements of each trajectory each time step. It is therefore desirable to introduce further should agree with the state population statistics of the well-controlled approximations to seek a “disentangling” of the trajectory ensemble. We assume that the proper correspond- ensemble to yield an approximate independent trajectory ence should hold locally in the phase space, so the independent method, perhaps with ensemble-level corrections. One such trajectory population a11,j(t) should be equated with the approximate method, QTSH, will now be described. appropriate local average behavior of the ensemble. In our The FSSH algorithm proposed by Tully relies on an notation assumption of consistency between two complementary a11,jj()tt= ⟨⟩σ () (75) representations of the quantum evolutionthe trajectory populations of the electronic states and the corresponding With these identifications, we finally arrive at the QTSH individual trajectory auxiliary quantum density matrix pop- hopping probability expression ulations.1 We seek to establish a rigorous connection between Tully’s ensemble of auxiliary density matrices and the local QTSH 1 2()V Γj values of the CSH phase-space functions and its relation to Phop ()t = βjΔt a ℏ surface-hopping consistency. 11,j (76) Returning to the hopping of the jth trajectory in our which is identical to the corresponding FSSH result, eq 11. example, recall that the ensemble representation of the local The consistency assumption of FSSH further assumes that the phase-space population density at phase-space point = on Γ Γj akl(t) parameters can be computed by solving the auxiliary state 1 is quantum equations of motion for each trajectory, eq 6.

N A similar line of reasoning gives the QTSH hopping 1 probability in the adiabatic representation ⟨ρσ⟩ = g()Γ−Γ 11 j N ∑ kj k k=1 (70) dq()p· QTSH 2 j j Phop = αjΔt We now make the assumption that the stochastic variables, σk, am ﬁ ++,j (77) have a well-de ned local average ⟨σ⟩j in the phase-space region |Γj − Γk| ≤ ΔΓ, where ΔΓ is the width of the Gaussian In this independent trajectory limit, ⟨ρ++⟩j → a++,j⟨ρ⟩j and ⟨α⟩j function, g(Γ). This suggests the approximation → αj⟨ρ⟩j

1117 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

The numerical implementation of the QTSH method uses N dia 2 the following procedure (given here for the adiabatic E = V()Γ α coh N ∑ j j representation): The continuous equations, eq 6, for the j=1 (85) quantum subsystem of each trajectory are integrated to The diagonal energy is the sum E = E + E . It should be determine the smoothly varying quantities a , a , α , and diag 1 2 ++,j −−,j j noted that the total energy, E, is not equal to E . This β . In addition, the stochastic variable σ is propagated using the diag j j diagonal energy is the quantity that FSSH rigorously conserves probabilistic algorithm in eq 77. The classical variables are at the individual trajectory level by momentum rescaling. propagated under the influence of the instantaneous When coherence αj ≠ 0 and the coupling V(Γj) is present, a Hamiltonian Hj = σjH++ + (1 − σj)H−− augmented by the dia third coherence energy term, Ecoh, is required to balance the CSH nonclassical terms derived above, eqs 65 and 66. The 13 energy budget. classical forces in the equations of motion change discontin- We can write the total energy as the sum over single uously at the points of transition, whereas the nonclassical trajectory contributions forces are continuous there. The resulting phase-space path N (qj(t),pj(t)) is continuous, unlike in the FSSH method, as we 1 do not rescale the momenta to impose energy conservation. Et()= Et() N ∑ j Energy Conservation. The FSSH method for surface j=1 (86) hopping imposes strict conservation of the classical energy of The QTSH method conserves this energy on average at the each independent trajectory Hj(Γj(t),t)=Ej, where in our level of the consistency of the FSSH approach. To prove this, σ σ notation Hj(Γ,t)= j(t)H1(Γ) + (1 − j(t))H2(Γ) and Ej is the we take the time derivative of eq 82. This gives initial classical energy. This is accomplished by the ad hoc N rescaling of the individual trajectory momenta at the time of 1 each hop, which imposes energy conservation on each Et()̇ = Etj̇ () N ∑ trajectory by hand. j=1 (87) Quantum mechanics, of course, requires energy conservation where as well but at the state level. Furthermore, the full Hamiltonian Ĥ and density matrix ρ̂ are involved, not just the diagonal pj Eṫ ()= ṗ ·+σ̇[UU()()qq− ] elements. The total conserved energy of the quantum system is j j m j 12jj the operator trace E = Tr(Ĥρ̂), and its quantum-classical limit is given by the corresponding classical trace + qq̇j·[σσj∇UU12()j + (1−∇j ) () qj 2()VVqq 2() Et() Tr H H ( ) ( ,t )d + ∇ j ααj]+ j j̇ (88) ==ρρ∫ ΓΓ Γ (78) From the equations of motion for the density matrix elements, where the integral is over the 2f-dimensional phase space. Here we have both H(Γ) and ρ(Γ,t) are 2 × 2 matrices of the corresponding classical-limit phase-space functions. Writing this out in terms 2()V qj of the matrix elements in the diabatic representation gives σβjj̇ ≃ a11,̇ = − ℏ j (89) Et()= Tr( Hρ ) 1 ()qqqUU() () αj̇ ==ωβj j [ 12jj− ]βj (90) =+Tr(HH11ρρ11 ) Tr(22 22 ) + 2ReTr( V ρ12 ) (79) ℏ where we have used ω =(H − H )/ℏ and have indicated or 11 22 that the first equation holds on average. For the phase-space variables (q ,p ), we have the quantum Et()=+ Tr( H11ρρα11 ) Tr( H22 22 ) + 2Tr( V ) (80) j j trajectory equation of motion The total energy consists of three terms pj dia q̇j = EE=++12coh E E (81) m (91) In terms of the trajectory representation, this becomes pj̇ = −[σσj∇UUV12()qqqj + (1−∇j ) ()jj] −∇ 2 () αj N 1 (92) E = σσjjHHV11()Γ + (1−Γ j )22 () j+ 2()Γ j αj ∑ By eliminating q̇j, ṗj, σ̇j, and α̇j from the equation for Ėj, we can N j=1 show the time derivative of each term vanishes on average, Ėj ≃ (82) 0, so that which defines the terms Et()̇ = 0 (93) N 1 It should be noted that this energy conservation, which holds E1 = ∑ σjjH11()Γ rigorously if σj(t) evolves continuously, is not strictly obeyed at N j 1 (83) = the individual trajectory level when a stochastic algorithm is N employed to propagate σj. A sudden “hop” of σj(t) = 0 to σj(t) 1 = 1, for instance, leads to an instantaneous change in the E2 = ∑ (1−Γσjj )H22 ( ) N Hamiltonian Hj = σjH11 + (1 − σj)H22 from H11 to H22. j=1 (84) ff However, on average, σj obeys the smooth di erential and equation, and so averaged over the ensemble the energy

1118 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article conservation of the state re-emerges. The assumptions required the auxiliary density matrices pj(T) → −pj(T) and βj(T) → for this quantum energy conservation are equivalent to the −βj(T)(j =1,2,...,N)producesanotherensemble. consistency assumption underlying the surface-hopping Propagation for an additional time period, T, then is equivalent method itself. to integrating the system dynamics back to t = 0. Formally, this fi The same approach can be followed to show the average will reproduce a nal state ρf(Γ,2T) that is equivalent to the ff energy conservation in the adiabatic representation by using initial ensemble, ρo(Γ,0) = ρf(Γ,2T), although di ering in the the adiabatic ensemble energy details of each trajectory’sdynamicalvariables.Thisis fi N con rmed in numerical simulations, as we will see below. 1 Decoherence Corrections. The QTSH method is an E = σσHH()Γ + (1−Γ−ℏ ) () 2dq ()· p β N ∑ jj++ j−− j j j j independent trajectory limit of the CSH formalism. This j=1 reintroduces the problem of “overcoherence” that characterizes (94) the standard FSSH approach. In our notation, this corresponds and the corresponding adiabatic equations of motion for σj, αj, to approximating the ensemble-level representation of the βj, qj, and pj. The diagonal and coherence contributions to the coherence by the individual trajectory contributions, for total energy, E, are then example: ⟨α⟩j ≃ αj⟨ρ⟩. In some situations, this is an accurate N approximation, but in cases where pure-state dynamics or adia 1 system-bath interactions lead to significant variation of the E = σσHH()Γ + (1−Γ ) () diag ∑ jj++ j−− j phase of trajectories over the local neighborhoods of N j 1 (95) = trajectories, this approximation will lead to an overestimation and of the hopping probabilities. Many attempts have been made to develop decoherence corrections in the context of the FSSH N adia 2ℏ approach (see, e.g., refs 9 and 10). E = − dq()· pβ coh N ∑ j j j Here we propose a simple empirical ensemble-level j=1 (96) decoherence correction. Rather than employing a theoretical respectively. The equations of motion then lead to Ė = 0 for approach based on approximations to the underlying equations the ensemble within the consistency assumption. of motion, we estimate the quantities ⟨α⟩j and ⟨β⟩j empirically Conventional FSSH surface hopping imposes energy from the statistics of the evolving trajectory ensemble itself. fi conservation by an accompanying rescaling of the momentum We de ne a proxy phase for each trajectory, ϕj (j = 1, 2, ..., N), given by pj → pj + Δpj. We stress here that this is not necessary and, in fact, is incorrect. The nonclassical term in the equations of t fi ()ttt (q ( ))d motion for (qj,pj)smoothlymodies the evolution of ϕωj = ∫ j ′ ′ (97) individual trajectories in the phase space and guarantee the 0 average conservation of the state energy. This is the only This quantity is not identical to the phase of the complex rigorous energetic requirement of quantum dynamics. number αj(t)+iβj(t) that represents the coherence of the jth Time Reversibility. Individual surface-hopping trajectories trajectory, which may be undefined or have a particular are not time-reversible due to the stochastic nature of their nonzero value at t = 0, but rather is a phase-like quantity for evolution. The time reversibility of the FSSH method is further that trajectory that can be compared across the ensemble. To sabotaged by two additional features of the method. First, the do so, we calculate the ensemble averages of the phase and its presence of frustrated hops breaks the consistency between the square auxiliary quantum density matrices of individual trajectories N and the population statistics of the ensemble in a manner that 1 ⟨ϕϕ()t ⟩ = ∑ j()t erodes the consistency of these quantities in a time-irreversible N j=1 (98) manner. Second, the energy-conserving momentum jumps introduce discontinuities in the classical phase-space evolution N 2 1 2 that cannot be back-integrated, even on average. The ⟨ϕϕ()t ⟩ = ()t N ∑ j consequence of these features is that an ensemble of FSSH j=1 (99) trajectories cannot be time-reversed. The lack of time which then allows the time-dependent phase variance δϕ2(t) reversibility of the state evolution represented by a time- over the ensemble to be defined dependent FSSH ensemble has led to much attention and ff 22 2 e ort being expended on exploring important but less rigorous δϕϕ()ttt= ⟨⟩−⟨⟩ () ϕ () (100) 48−50 requirements such as detailed balance. fi Unlike standard FSSH, the QTSH approach is manifestly By assuming Gaussian statistics, we can de ne a global time- χ time-reversible on average, within the consistency assumption. dependent decoherence factor (t) Individual trajectories are stochastic and thus lose strict time- 1/22 (t ) ()t e− δϕ (101) reversal symmetry. These objects, however, are not “knowable” χ = parts of a quantum theoretical description. Quantum Unlike most other approaches, which estimate decoherence mechanics describes the evolution of states that in a trajectory corrections to independent trajectory coherences from local context places constraints only on ensemble behavior with properties, we determine χ(t) from the nonlocal characteristics nothing to say about its individual members. The QTSH of the ensemble as a whole, in line with the lessons learned approach formally satisfies time reversibility: An initial density from the full CSH formalism. matrix ρo(Γ,0) propagated for a given system from t = 0 to t = A decoherence-corrected version of QTSH can then be T will produce an intermediate state ρint(Γ,T). The reversal of implemented by incorporating the correction factor, χ, into the the signs of all intermediate momenta and imaginary parts of hopping probabilities of the individual trajectories.

1119 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

Figure 1. (a,b) Time dependence of the lower adiabatic state population for Tully’s single-crossing model. The FSSH (green), QTSH (blue), and CSH (red) results are compared with exact wavepacket calculations (black). (a) ℏk = 10. (b) ℏk = 15. (c) Final upper state population is shown as a function of the initial momentum ℏk. (d) Coherence for ℏk = 15. Exact (black solid) and QTSH (dashed) results for the real (blue) and imaginary (red) parts of Trρ12 are compared. See the text for details. (e) Energy budget for the ℏk = 10 state is shown. Exact quantum results (black solid) are compared with the QTSH ensemble (dashed colored). See the text for discussion. Calculations are done in the adiabatic representation.

p individual trajectories during their propagation but modiﬁes QTSH 2 j Phop ()t →Δdq()j (χαj )t the probability of hopping during each time step using the a++,j m (102) global-ensemble-level decoherence factor, χ(t). Most other approaches add dissipation to the equations of motion for the ﬁ This modi ed hopping algorithm makes the egalitarian terms αj(t)themselves,introducingapuredephasing approximation ⟨α⟩j ≃ χαj⟨ρ⟩ for j = 1, 2, ..., N. It should be component to the quantum evolution that leads to a mixed- noted that this approach retains the full coherence of the state density matrix even for pure-state dynamics. The

1120 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

Figure 2. Comparison of FSSH and QTSH with an “energy aloof” variant, EASH. (a) Time dependence of the adiabatic lower state population for Tully’s single-crossing model with ℏk = 10. The FSSH (green), QTSH (blue), and EASH (red) results are compared with exact wavepacket calculations (black). (b) Energy budget for the state is shown. Exact quantum results (black solid) are compared with the QTSH (blue) and EASH (red). The total, Etot, and diagonal, Ediag, contributions are shown as a function of time. See the text for discussion. approach here is derived from the full CSH formalism as a varying initial mean momenta po = ℏk is generated and well-defined approximation and, we feel, is more faithful to the propagated using the QTSH method. The results are underlying exact quantum evolution. compared with standard FSSH and corresponding quantum- The ensemble-level correction described here is based on wavepacket calculations using the method of Kosloff.51 the simplest possible assumption regarding the relationship In Figure 1, we show results obtained for Tully’s single- 1 between the evolving phase-space function ρ12(Γ,t)(or crossing system. Here calculations are performed in the ρ+−(Γ,t) in the adiabatic representation) and its representation adiabatic representation for the QTSH method and compared as an ensemble of trajectories: The local interplay of the phases with standard FSSH1 as well as the full CSH approach12 of individual trajectories can be represented uniformly across (incorporating the modified classical equations of motion the ensemble by a single average factor, χ. More elaborate and described above) and quantum-wavepacket calculations. In complicated methods can be imagined, where variations of the ff Figure 1a,b, we show the time dependence of the population of e ect are estimated theoretically or statistically. The advantage the initially occupied lower adiabatic state as a function of time of the current proposal is its simplicity and numerical ffi for initial momenta ℏk = 10 and ℏk = 15, respectively. In e ciency. In particular, if the system under study is simple Figure 1c, the dependence of the final population of the upper enough that the entire ensemble of trajectories can be adiabatic state on the initial momentum ℏk is given. propagated in parallel, then the averages required to estimate All of the methods are in reasonable agreement with the χ lead to a negligible additional cost to the calculations. Large exact quantum results. For this system, the FSSH method gives systems where the trajectories comprising the ensemble must be integrated independently will require a different approach. superior agreement for low momenta (k < 8), presumably due In the next section, we will test this simple decoherence to the explicit imposition of energy conservation; here the correction. presence of frustrated hops improves the results. For higher energies, the CSH method gives slightly better agreement with 3. RESULTS the quantum results. For this system, the QTSH method fl slightly overestimates the extent of nonadiabatic transition. In this section, we brie y illustrate the numerical implementa- The asymptotic energetic constraints imposed by FSSH must tion of the QTSH method and compare it with exact quantum- emerge naturally here from the method itself, and the wavepacket results and standard FSSH for several simple independent trajectory approximation apparently leads to systems. We highlight both the strengths and shortcomings of overcoherence and thus too extensive population transfer the QTSH approach. More thorough numerical benchmarking of CSH, QTSH, and decoherence corrections will be given in a that is corrected for by the FSSH energy constraint. We will future publication. return to this point below in the context of the ensemble-level We apply the QTSH method to two model systems decoherence correction. originally proposed by Tully as benchmark problems and In Figure 1d, we investigate the agreement between exact adopted universally by the surface-hopping community as test quantum and QTSH ensemble dynamics in more detail. For cases.1 The first is Tully’s single-crossing system, which is a 1D the state with an initial momentum of ℏk = 15 we show a model corresponding to two electronic surfaces undergoing a comparison of exact and QTSH values for Tr ρ12, a metric for single crossing. The system has been treated in many previous the total coherence of the evolving states. The exact quantum publications. We adopt the potentials and parameters from values were calculated by taking the time-dependent overlap of Tully’s original paper1 and treat ensembles of 2000 trajectories the adiabatic-state wavepackets. The QTSH values for the real sampled randomly from an initial minimum uncertainty phase- and imaginary parts are given by the ensemble averages of αj space Gaussian distribution with spatial width σq = 1.0 and and βj,respectively.Forthiscase, nearly quantitative with mean initial position qo = −6. A range of ensembles with agreement is observed, indicating that the evolving ensemble

1121 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

Figure 3. (a,b) Time dependence of the diabatic state 1 population for Tully’s single-crossing model. The FSSH (green) and QTSH (blue) results are compared with exact wavepacket calculations (black). (a) ℏk = 10. (b) ℏk = 15. (c) Final state 1 population versus initial momentum ℏk is shown. Calculations are done in the diabatic representation. of hopping trajectories is capturing well this feature of the fully Figure 2b displays the energy budget of total and diagonal coherent quantum dynamics. (e.g., classical) contributions. Removing the imposition of Despite the error in population transfer, the QTSH method energy conservation leads to an overestimate of the non- correctly conserves the quantum-classical energy E = Tr(Hρ). adiabatic probability by EASH, as otherwise frustrated hops are In Figure 1d, we show the energy budget of the ℏk = 10 allowed to occur. The results are not as accurate as FSSH but ensemble by separately plotting the diagonal and coherence are still closer to the exact results than QTSH. A bigger contributions to the total energy, given for the adiabatic difference is seen in Figure 2b), where the energetics of the representation by eqs 95 and 96, as well as their sum Etot, and evolution are displayed. QTSH (blue dashed lines) shows compare with a similar partitioning of the exact quantum nearly quantitative agreement with the exact quantum diagonal energy. The QTSH total energy is constant and in agreement and total energies (solid black lines). Neglecting the with the exact value to within numerical error. The classical nonclassical forces of QTSH yields the EASH method (red diagonal energy is not constant, in contrast with the dashed lines). The breakdown of energy conservation of EASH assumption of the FSSH formalism, but is compensated by is clearly visible. Both the diagonal and total energies deviate the contribution of the coherence energy. The QTSH from the exact values, and asymptotically the system has estimates of these are in essentially quantitative agreement violated energy conservation by a nontrivial amount. For the with the quantum-mechanical results. FSSH method, the final total energy would be constrained to It is instructive to consider an even simpler surface-hopping be conserved by the individual trajectory momentum rescaling, simulation using a pared-down methodology defined by which conserved the diagonal (classical) contribution at all ignoring the energy conservation requirement and momentum times. rescaling of FSSH, or equivalently, by removing the non- This example illustrates the important point that the classical forces from QTSH. We compare this “energy aloof rigorous energy conservation of the QTSH method without surface-hopping” variant, which we denote EASH, with FSSH momentum jumps is independent of the accuracy of the and QTSH in Figure 2 for the Tully 1 ℏk = 10 case in the method. For this particular initial state, the EASH results are adiabatic representation, considered in Figure 1. In Figure 2a, actually more accurate than QTSH, despite its failure to we show the time-dependent lower state population, whereas conserve energy.

1122 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

Figure 4. (a,b) Time dependence of the adiabatic lower state population for Tully’s dual-crossing model. The FSSH (green) and QTSH (blue) results are compared with exact wavepacket calculations (black). (a) ℏk = 30. (b) ℏk = 40. (c) Coherence for ℏk = 40. Exact (black solid) and QTSH (dashed) results for the real (blue) and imaginary (red) parts of Trρ+− are compared. See the text for details. (d) Final upper state population versus initial momentum ℏk is shown. Calculations are done in the adiabatic representation.

In Figure 3, we present results for the same initial states as representation are compared with FSSH and exact quantum- presented in Figure 1 but calculated in the diabatic wavepacket calculations. representation. The QTSH results are compared with FSSH In Figure 4a,b, the populations of the initially occupied and exact quantum-wavepacket calculations. lower adiabatic state as a function of time for initial momenta In Figure 3a,b, we show the time dependence of the ℏk = 30 and ℏk = 40 are shown, respectively. Figure 4c population of the initially occupied diabatic state 1 as a compares the real and imaginary parts of the QTSH coherence fi function of time for initial momentum ℏk = 10 and ℏk = 15, Trρ+− for ℏk = 40. In Figure 4d, the dependence of the nal respectively. In Figure 3c, the dependence of the final population of the upper adiabatic state on the initial population of state 1 on the initial momentum ℏk is given. momentum ℏk is given. Again, close agreement is obtained Again, close agreement between the methods is observed, with except at low k, where both surface-hopping methods are slightly shifted from the exact quantum results. the QTSH results being in better agreement at lower k values Figure 5 presents results obtained by applying the QTSH in this case. method to the 1D three-state superexchange model introduced It should be emphasized that unlike FSSH the applicability 9,52 by Prezhdo and coworkers. In this system, population of the QTSH formalism is independent of the electronic-state transfer from the lowest-lying state 1 to final state 3 is representation. For this 1D problem, the application of the calculated. These states are not directly coupled in the diabatic FSSH in the diabatic representation is unambiguous, but in representation but are each coupled to a high-lying and, for higher dimensions, the absence of the nonadiabatic coupling some initial momenta, classically forbidden state 2. The system vector, d,inthediabaticformulationcomplicatesthe consists of three constant diagonal diabatic potentials coupled momentum rescaling component of the FSSH method. at the coordinate origin by off-diagonal potential couplings. QTSH, on the contrary, can be straightforwardly and The potential functions and system parameters are given in the unambiguously applied in the diabatic representation. original references.9,52 We start an initial minimum uncertainty 1 In Figure 4, we consider Tully’s dual-crossing model. phase-space distribution with coordinate width σq = 1 and Results obtained using the QTSH method in the adiabatic mean momentum ℏk to the left of the coupling region and use

1123 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

Figure 5. (a) Time dependence of the the state 1, 2, and 3 populations for the superexchange model. QTSH (solid) is compared with exact wavepacket results (dashed). Initial momentum ℏk = 5. (b) The energy budget for the ℏk = 5 ensemble. (c) Final state 3 population vs. initial momentum ℏk. FSSH (green) and QTSH (blue) are compared with exact quantum results (black). the QTSH and FSSH methods to calculate the population imposition of energy conservation by FSSH is what leads to transfer. These results are compared with corresponding spuriously frustrated hops and the failure of that method. quantum-wavepacket results. All simulations are performed in In Figure 5c, the dependence of the final population of state the diabatic representation. 3 on the initial momentum ℏk is given. QTSH results are For ℏk ≲ 6.3, the mean kinetic energy of the state is compared with FSSH and exact quantum simulations. For insufficient to reach the intermediate state 2; the range of values of k ≳ 7, above the superexchange region, both QTSH initial momenta between the minimum value to reach state 3 and FSSH are in nearly exact agreement with the quantum and this value is known as the superexchange region. results. As ℏk decreases into the classically forbidden Figure 5a shows the state populations versus time for an superexchange region, the FSSH method fails to capture the initial state with momentum ℏk = 5 within the superexchange population-transfer process, whereas QTSH remains in good region. The QTSH results are compared with exact quantum agreement with the exact results. calculations. For this initial state, virtually all FSSH hops are In Figure 6, we demonstrate the time reversibility of the frustrated, leading to no population transfer out of state 1 QTSH method. We consider the state treated in Figure 1 for (results not shown). The QTSH populations are in nearly the Tully single-crossing system in the adiabatic representation. quantitative agreement with the quantum results for this Here we integrate the minimum uncertainty state with initial classically forbidden process. Even the “virtual” state 2 momentum ℏk = 10 from t = 0 until an intermediate time t = populations are accurately represented by the ensemble of 2400, for which the system has left the interaction region. The QTSH trajectories. momentum, pj, and imaginary part of the coherence, βj, of each In Figure 5b, the QTSH energy budget for the state shown trajectory are then reversed in sign, and the ensemble is in panel a is presented and compared with the corresponding integrated to the final time of t =2× 2400 = 4800. The figure quantum-mechanical quantities. The total energy is well- shows the population of the initially occupied lower state as a conserved by the quantum trajectories. Again, we see that the function of time. The QTSH results are compared with the diagonal energy is not conserved by the QTSH method or by FSSH method. The QTSH method demonstrates nearly the exact quantum evolution. This nonconservation is what quantitative reversibility of the initial population, with only a allows the nonclassical superexchange mechanism to pass slight asymmetry around the intermediate time t = 2400 and through state 2 and then populate the final state 3, and artificial recovery of the full initial unit population to within statistical

1124 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article

dispersion across the ensemble is apparent, with significant variance of the trajectory phases during the hopping process. This results in the value of χ dropping from the fully coherent value of 1.0 to ∼0.5 during the transition period. Importantly, though, it is observed that the ensemble recoheres to nearly full coherence by the end of the evolution. This is a characteristic of the underlying pure-state evolution of the system. Alternate methods for decoherence correction that irreversibly cause a decay of the trajectory coherence will miss this important feature of the ensemble dynamics. 4. CONCLUSIONS We reviewed the CSH approach, introduced in a recent publication.12 In addition, we significantly extended the formalism to include nonclassical state-dependent forces that take the place of the physically sensible but ad hoc momentum rescaling and resulting strict classical energy conservation of Figure 6. Time reversibility of surface-hopping methods. The time dependence of the adiabatic lower state population for Tully’s single- the FSSH method. The result is a quantum-trajectory -based crossing model in the adiabatic representation is shown for the FSSH formalism for simulating molecular dynamics with electronic (green) and QTSH (blue) methods for initial momentum ℏk = 10. transitions, where the quantum and classical portions of the The simulation is run until t = 2400 (compare with Figure 1); then, mixed quantum-classical system are correctly entangled with the signs of the momenta, pj, and imaginary part of the coherence, βj, each other. are reversed for each member of the ensemble, and the integration The CSH method is based solidly on a solution of the continued until t = 4800. underlying quantum Liouville equation in a mixed quantum- classical approximation using an ensemble of trajectories. A key uncertainty. The FSSH method, on the contrary, demonstrates aspect of the approach is that the trajectories in the ensemble strong irreversibility resulting from the presence of frustrated are no longer independent, but their evolution is mutually hops and momentum rescaling of the trajectories. coupled by their role in propagating the phase-space In Figure 7, we investigate the efficacy of the ensemble-level representation of the full quantum-classical density matrix. decoherence correction described in the last section. Figure 7a Quantum coherence emerges naturally as a characteristic of the compares the decoherence-corrected QTSH results with the ensemble as a whole via the interrelationships between uncorrected QTSH and exact quantum results, reproduced individual trajectory phases. Decoherence is captured by the from Figure 1a. The incorporation of the decoherence factor method without externally imposed corrections. χ(t) into the hopping probability brings the long-time lower Despite these formal advantages, the CSH method is too state probability into close agreement with the quantum intensive numerically to be a practical method for anything results, although the full time-dependent populations are not beyond simple model systems. To address this shortcoming, superimposable. In Figure 7b, we show the time dependence of we have introduced further well-defined approximations to the the proxy phase variance δϕ2(t)=⟨ϕ2(t)⟩ − ⟨ϕ(t)⟩2 and the CSH approach to derive an independent trajectory limit of the global decoherence factor χ(t) = exp(−δϕ2(t)/2) for the theory, which we call QTSH. This method recovers the fewest- ensemble describing the state in Figure 7a. The effect of phase switches stochastic algorithm for independent trajectories

Figure 7. Ensemble-level coherence correction. (a) Time dependence of the adiabatic lower state population for Tully’s single-crossing model. Exact wavepacket calculations (black) are compared with both the unmodiﬁed QTSH (blue) and QTSH + ensemble-level decoherence-corrected (cyan) results for initial momentum of ℏk = 30. Calculations are done in the adiabatic representation. (b) Time dependence of the phase variance δϕ2(t) and decoherence factor χ(t) for the QTSH ensemble in panel a.

1125 DOI: 10.1021/acs.jpca.8b10487 J. Phys. Chem. A 2019, 123, 1110−1128 The Journal of Physical Chemistry A Article employed in FSSH. However, the momentum rescaling and interacting worlds formalism,63,64 and our work on quantum classical trajectory energy conservation of FSSH are discarded tunneling using entangled trajectories.32,34,65,66 in favor of the rigorously derived quantum forces of the CSH In the surface-hopping context, the independent trajectory formalism. The cost of QTSH is comparable to that of FSSH. basis of both FSSH and QTSH leads them to be local hidden We illustrated the QTSH methodology by treating several variable theories. As such, although they may be accurate for simple model systems commonly used as benchmarks for many problems in practice, they cannot, in principle, give exact surface-hopping approaches. agreement with quantum mechanics generally. CSH, on the QTSH rigorously conserves the correct quantum-classical contrary, is an example of a nonlocal hidden variable theory. energy Tr(Hρ) without ad hoc momentum rescaling. This is Future work will focus on developing the CSH framework into due to the presence of the nonclassical forces in the quantum an exact trajectory representation of nonadiabatic processes, trajectory evolution. Energy conservation results without any not as a practical method, but as a context in which to artificial momentum rescaling, eliminating undesirable features understand fundamental aspects of quantum theory. of FSSH such as forbidden hops and the breakdown of the internal consistency of quantum- and ensemble-based state ■ AUTHOR INFORMATION probabilities. In the adiabatic representation, the classical Corresponding Author fi dynamics are modi ed by a quantum-state-dependent vector *E-mail: [email protected]. ff potential, introducing geometric phase e ects into the ORCID dynamics for multidimensional systems. Another advantage of the QTSH method is that it is time-reversible at the Craig C. Martens: 0000-0002-2350-4346 ensemble level, unlike FSSH, where frustrated hops and Notes momentum rescaling break the time-reversal symmetry of even The author declares no competing financial interest. pure-state quantum evolution. We have proposed further approximate corrections inspired by the underlying CSH ■ ACKNOWLEDGMENTS formalism that allow the incorporation of ensemble-level We are grateful to Shaul Mukamel, Vladimir Mandelshtam, and decoherence without the accompanying computational ex- Filipp Furche for helpful conversations. Stimulating discussions pense of CSH. at the UCI Liquid Theory Lunch (LTL) and the Telluride The present manuscript has mainly focused on the formal Science Research Center (TSRC) are also acknowledged. This development of the CSH and QTSH approaches. Detailed material is based on work supported by the National Science numerical investigations and extensions to higher dimensional Foundation under CHE-1764209. systems and realistic applications, including geometric phase effects resulting from the nonclassical forces in the adiabatic ■ REFERENCES representation, will be presented in future publications. (1) Tully, J. C. Molecular Dynamics with Electronic Transitions. J. The surface-hopping approach to simulating molecular Chem. Phys. 1990, 93, 1061. dynamics in the quantum-classical limit is just one example (2) Tully, J. C. Perspective: Nonadiabatic Dynamics Theory. J. of a “quantum trajectory” formalism. Methods that treat Chem. 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