Classical trajectory-based approaches to solving the quantum Liouville equation

Classical Trajectory-Based Approachesto Solving the Quantum LiouvilleEquation

ARNALDO DONOSO, CRAIG C. MARTENSDepartment of Chemistry, University of California–Irvine, Irvine, CA 92697-2025

Received 26 February 2002; accepted 26 July 2002Published online 8 October 2002 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.10377

ABSTRACT: The time-dependent quantum mechanics of heavy particles moving ona single potential energy surface can often be represented surprisingly well by theevolution of classical trajectory ensembles. However, manifestly quantum mechanicalphenomena—such as transitions between coupled electronic states, electronic coherenceand its decay, or quantum mechanical tunneling—require fundamental modification ofthe purely classical motion. We introduced and developed an approach to this problemthat is based on solving the quantum Liouville equation using ensembles of classicaltrajectories. In this article, we describe the general approach and its application to theproblems of nonadiabatic dynamics, coherent multistate electronic–nuclear dynamics,and tunneling through potential barriers. When viewed from the trajectory ensembleperspective, quantum effects arise as a breakdown of the statistical independence of thetrajectories in the ensemble and a nonlocal entanglement of their collective evolution.© 2002 Wiley Periodicals, Inc. Int J Quantum Chem 90: 1348–1360, 2002

Key words: Liouville equation; classical trajectory ensembles; tunneling; classicalmolecular dynamics; phase space; nonadiabatic dynamics

Introduction

M olecular-scale physical processes fall withinthe realm of quantum mechanics [1]. Direct

numerical methods for solving the time-dependentSchrodinger equation have become powerful due toadvances in both methodology and computer per-formance. Unfortunately, a rigorous quantum me-chanical treatment is practical only for relatively

simple few-body problems, even when the Born–Oppenheimer approximation allows separation ofthe electronic and nuclear motion [2]. Approximatemethods must be developed to confront complexmany-body systems for which the direct approachremains intractible. A number of such methodshave been investigated, including mean-field theo-ries, semiclassical and mixed classical quantummethods, phenomenological reduced descriptions,and others.

An extreme approximation is to ignore quantumeffects altogether and use classical mechanics toCorrespondence to: C. C. Martens; e-mail: [email protected]

International Journal of Quantum Chemistry, Vol 90, 1348–1360 (2002)© 2002 Wiley Periodicals, Inc.

describe the motion of atoms in molecular systems.This is the basis of the classical molecular dynamics(MD) method [3] and is an appealing alternativeapproach to the simulation of many-body systems,where quantum effects are unimportant. The ad-vantages of classical MD are both computationaland conceptual.

An MD simulation finds a numerical solution ofthe appropriate Hamilton’s or Newton’s equationsfor the constituent particles of the system, giventheir mutual forces of interaction [3]. An individualclassical trajectory for a multidimensional problemis much easier to integrate numerically than thetime-dependent wave packet of the correspondingquantum system. Unless the anecdotal informationrevealed by a single trajectory is sufficient, how-ever, collections of trajectories—ensembles—must ingeneral be considered. Distributions of trajectoriesevolving in phase space are the most direct classicalanalog of evolving quantum wave packets, andstatistical averages of dynamic variables over theclassical ensemble parallel the corresponding quan-tum expectation values of operators. (Note that iflarge ensembles are required to obtain good statis-tics classically some of the legendary computationaladvantage of classical trajectory methods overquantum simulations is lost!)

Classical MD is expected to work well for sys-tems with large atomic masses and at high energiesor temperatures, where quantum effects are sup-pressed or averaged out. For systems involving themotion of light particles such as hydrogen atoms, orat very low temperatures, the quantum nature ofthe nuclear motion becomes important. Here, ef-fects such as zero-point energy or quantum me-chanical tunneling require treatment. Another situ-ation where nonclassical effects cannot be ignoredis when a breakdown of the clean separation be-tween nuclear and electronic degrees of freedomoccurs. Two or more Born–Oppenheimer surfacesbecome coupled to each other, and electronic tran-sitions unavoidably accompany the nuclear motion.

In this article, we describe a general approach tomodeling the dynamics of systems for which thesenonclassical effects are important. Despite thebreakdown of the validity of classical mechanics,we work within the context of classical-like MD andensemble averaging. We treat two general physicalproblems: coherent nonadiabatic MD on multipleelectronic surfaces and quantum mechanical tun-neling on a single potential surface. Our generalapproach to both problems is based on a phase-space representation of the quantum Liouville

equation and its approximate solution using trajec-tory ensembles. Nonclassical effects emerge in ourformalism as the breakdown of the statistical inde-pendence of the members of the trajectory ensem-bles.

Semiclassical Liouville Approach toNonadiabatic Electronic Dynamics

We first consider the problem of MD on coupledelectronic surfaces. For simplicity, we restrict ourdiscussion to 1-D motion on two coupled potentialcurves. The two-component wave function describ-ing the quantum state of such a system is given by

��q, t� � ��1�q, t��2�q, t�� (1)

and the Hamiltonian is the 2 � 2 matrix of opera-tors

H � �H1 VV H2

� . (2)

The diagonal elements of H consist of the kineticplus single-surface potential energy functions

Hj �p2

2m � Uj�q� (3)

for j � 1, 2. We take the off-diagonal element V tobe a real function of the coordinate q; this corre-sponds to a diabatic representation of the electronicproblem [4].

A more direct analogy can be made betweenclassical and quantum mechanics using the Li-ouville representation of quantum mechanics.Here, the state of the system is described by thedensity operator �(t), which obeys the quantumLiouville equation

i�d��t�

dt � �H, ��t��. (4)

For the two-state problem, the density operator isitself a matrix,

��t� � � �11�t� �12�t��21�t� �22�t�� . (5)

SOLVING THE QUANTUM LIOUVILLE EQUATION

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1349

Written out explicitly in terms of components of �,Eq. (4) becomes

i�d�ij

dt � �k�1

2

Hik�kj � �ikHkj. (6)

We now find the classical limit of the multistatequantum Liouville equations of motion. This can beaccomplished by applying the Wigner–Moyal for-malism [5–8], which gives a classical phase-spacerepresentation of the algebra of quantum operatorsin terms of a power series expansion in �. To lowestorder, the product of two operators A and B be-comes

AB � AB � i��A, B � O��2�, (7)

where A(q, p) and B(q, p) are the correspondingfunctions defined on phase space and {A, B} �(�A/�q)(�B/�p) (�B/�q)(�A/�p) is the Poissonbracket [9]. The most extreme classical limit of Eq.(4) for motion on a single electronic surface is justthe classical Liouville equation [9, 10]

��

�t � �H, �, (8)

where �(q, p, t) and H(q, p) are now functions of thecanonical coordinate q and momentum p; these canbe defined rigorously for general operators depend-ing on q and p using the Wigner–Moyal formalism.What is the “most classical” limit of the multistateproblem? We can answer this question by applyingthe expansion in Eq. (7) to Eq. (6). The result is a setof coupled partial differential equations for thesemiclassical phase-space functions correspondingto the matrix elements of �. These are [11]

��11

�t � �11�11 � �V, Re �12 �2V�

Im �12 (9)

��22

�t � �22�22 � �V, Re �12 �2V�

Im �12 (10)

��12

�t � ��0 � i��12 �12 �V, �11 � �22

�iV�

��11 � �22�, (11)

where ��f � {H�, f } (� � 11, 22, 12) defines theclassical Liouville operator �� in terms of the Pois-son bracket with the corresponding Hamiltonian.The average Hamiltonian appears in the equationof motion for the electronic coherence �12(q, p, t):Ho � H12 � (H11 � H22)/2. In addition, an imagi-nary phase factor i� contributes a nonclassicalcomponent to the evolution of the coherence, where� � (H11 H22)/� is the difference potential di-vided by �. The equation of motion for �21 can beobtained from Eq. (11) by complex conjugation. Fornonzero electronic coupling V, sink and sourceterms appear in the equations that couple the evolv-ing generalized phase-space distributions to eachother.

The formal properties of the semiclassical Li-ouville approach to nonadiabatic dynamics and itsimplementation in the context of classical trajectoryintegration and ensemble averaging have been ex-plored in a number of recent publications [11–14].Other authors have also followed a similar ap-proach (see, e.g., [15–18]).

Here, we briefly review our trajectory implemen-tation of the multistate semiclassical Liouvillemethod. The phase-space distribution functions arerepresented by ensembles of classical trajectories,

��q, p, t� �1N �

k�1

N

ak��t��q � qk

��t�� p � pk��t��

(12)

for � � 11, 22, 12. This is a weighted sum of Dirac-functions located at the phase-space positions ofthe trajectories comprising the ensembles. The qk

�(t)and pk

�(t) evolve under the appropriate Hamilton’sequations

qk� �

�H��qk�, pk

��

�p (13)

pk� �

�H��qk�, pk

��

�q . (14)

This part of the evolution is purely classical, and inthe absence of coupling between the electronicstates [V � 0 in Eqs. (9)–(11)] the probability den-sities �11 and �22 solve their respective classical Li-ouville equations. For nonzero coupling, all quan-tum effects are incorporated into the timedependence of the coefficients ak

�(t). These are de-termined at each time step by solving a linear alge-

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1350 VOL. 90, NO. 4/5

bra problem. The details of the algorithm can befound in Ref. [12].

The semiclassical Liouville method, as imple-mented with ensembles of classical trajectories andthe ansatz Eq. (12) interacting via Eqs. (9)–(11), hasproven to give accurate, even nearly quantitative,results for model 1-D problems. To illustrate this,we consider a simple model of photodissociation ofa diatomic molecule on two coupled electronic sur-faces. The first consists of a repulsive potential

U1�q� � Ae�qq1� � B (15)

that is coupled to a second bound excited state,represented by a Morse oscillator

U2�q� � D�e2��qq2� � 2e��qq2��. (16)

The electronic coupling V(q) is taken to be a Gauss-ian centered at qc, the crossing point of the twopotentials,

V�q� � Voec�qqc�2. (17)

The potential curves are shown in Figure 1. Theparameters of the model and further details of themethod can be found in Ref. [12].

Multistate electronic problems for which the ki-netic energy is diagonal while the coupling is of apotential-like form are in a diabatic electronic rep-resentation [4]. We restrict our discussion here to adiabatic description, but note that the formalismcan also be applied effectively in the adiabatic rep-

resentation [13], where the electronic coupling isdue to off-diagonal terms in the kinetic energy.

In Figure 2, the populations of the diabatic elec-tronic states 1 and 2 are shown as functions of time.These quantities are given by the sum of the coef-ficients:

P��t� � Tr ��t� �1N �

k�1

N

ak��t�. (18)

In the quantum wave packet calculations, a mini-mum uncertainty Gaussian wave packet with q� �qo and p� � 0, given by

g�q� � �m�

�� 1/4

exp�m��q � qo�

2

2� �, (19)

is initiated at t � 0 on the repulsive state 1. Thecorresponding Wigner function in phase space is a2-D Gaussian centered at q � qo and p � 0. Thisphase-space distribution is represented by the ini-tial trajectory ensemble. Approximately 60% of thestate 1 population is transferred to the bound state2 as the wave packet passes through the regionwhere the potentials cross, as indicated in the fig-ure. Exact quantum wave packet results obtainedby using the method of Kosloff [19] are comparedwith the semiclassical trajectory simulations. As theresults illustrate, nearly quantitative agreement be-tween the two is obtained for this simple modelproblem.

FIGURE 1. Diabatic potential curves used for multi-state semiclassical Liouville simulations. Also shownare schematic wave packets initiated on one or bothelectronic states at t � 0.

FIGURE 2. Populations of the diabatic states 1 and 2vs. time. The results of the semiclassical multistate Li-ouville simulation are compared with exact wavepacket calculations.



Other accurate and efficient trajectory-basedmethods for simulating electronic population trans-fer between coupled states have been developed,such as the trajectory surface hopping approachpioneered by Tully and coworkers [20–22]. Thisalgorithm is based on sudden stochastic transitionsof trajectories between the surfaces, and providesan excellent description of electronic relaxation in arange of physical systems. More challenging arenonadiabatic processes that involve quantum co-herence between the coupled electronic states. Forthese manifestly nonclassical processes, the surfacehopping approach has trouble treating the creation,evolution, and decay of coherence properly. This isdue to the requirement imposed by the method thateach hopping trajectory separately determine itsown fate. Quantum effects such as electronic coher-ence and decoherence are, when viewed from theperspective of classical trajectory ensembles, aproperty of the ensemble as a whole. The semiclas-sical Liouville method can treat these coherent ef-fects accurately.

We illustrate the simulation of coherent elec-tronic dynamics by modeling an idealized ultrafastcontrol experiment, which we term nonadiabaticwave packet interferometry [14]. Here, a superpositionof two excited electronic states of a molecule iscreated at t � 0 with ultrafast laser pulses. Thephase between these two states is assumed to becontrollable in the experiment. The initial two-com-ponent wave packet is then given by

��q, 0� � � �P1�0�g�q��1 � P1�0�g�q�ei � , (20)

where P1(0) is the initial population of state 1, g(q) isa normalized minimum uncertainty Gaussian, and is the controllable relative phase. As discussed indetail in Ref. [14], the initial nonzero coherencebetween the electronic states amplifies the effect ofsmall electronic coupling V, leading to populationtransfer that is first order in V and strongly depen-dent on the phase . This is in contrast with theincoherent case, where �12 � 0 initially. There, pop-ulation transfer is proportional to V2 in the small Vlimit. This makes the nonadiabatic wave packetinterferometry scheme a potentially useful one fordetecting and measuring small electronic couplingmatrix elements [14].

In Figure 3, we show a family of state 2 popula-tions as a function of time for the coherent initialcondition in Eq. (20) with P1(0) � P2(0) � 0.5. Theresults corresponding to several values of aregiven. The coupling V has been chosen to be weakso that only ca. 5% of the population is transferredfor the incoherent initial condition. The strong de-pendence of the population transfer on the phase is clearly visible. The magnitude of the effect is alsolarger than the incoherent case. In Figure 4, the dependence of the final population transfer [de-fined as the value of P2(t) at the end of the simula-tion, t � 1100 a.u.] is shown. The semiclassicaltrajectory approach again gives good agreement

FIGURE 3. Population of state 2 vs. time. The figurecompares the exact quantum and semiclassical resultsfor the case with P1(0) � P2(0) � 0.5 and several val-ues of the relative phase .

FIGURE 4. Final populations vs. the relative phaseangle . The figure compares the quantum and semi-classical state populations at the final time of T �1100 a.u.

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1352 VOL. 90, NO. 4/5

with the exact quantum calculations, even for thismanifestly quantum coherent process.

For the case of weak nonadiabatic coupling andnonzero initial electronic coherence, the resultingpopulation transfer can be calculated using slightlyaugmented classical dynamics and perturbationtheory. In addition to providing reasonably accu-rate numerical results with minimal computationalcost, this approach highlights the relation betweenclassical ensemble dynamics and quantum coher-ence and decoherence.

In the absence of electronic coupling (V � 0), thezeroth-order coherence obeys the classical-like Li-ouville equation

��12�0�

�t � ��0 � i��12�0�, (21)

where the classical averaged Hamiltonian Liouvilleoperator �0 is augmented by the imaginary phaseterm i�. The formal solution is

�12�0��q, p, t� � et��0i��q��12

�0��q, p, 0�. (22)

The exponential propagator can be disentangled togive [11]

�12�0��q, p, t� � ei��q,p,t�et �0�12

�0��q, p, 0�, (23)

where the phase �(q, p, t) is the time integral of thedifference potential divided by �:

��q, p, t� � �0

t

e� �0��q� d�. (24)

The first-order population transfer can be calcu-lated perturbatively from the evolving zeroth-ordercoherence �12

(0) and the equations of motion in Eqs.(9) and (10). Focusing on the state 1 population andtaking V to be constant for simplicity (this can berelaxed easily in practice), perturbation theoryleads to the inhomogeneous equation for �11

(1), thefirst-order contribution to the evolving state 1 prob-ability density:

��11�1�

�t � �11�11�1� �

2V�

Im �12�0�, (25)

where �12(0)(t) is given by Eq. (23). Noting that the

phase-space trace of a Poisson bracket vanishes, the

time-dependent first-order state 1 population canbe derived, with the result

P1�1��t� �

2V�

Im �o

t

Tr �12�0��s� ds, (26)

where

Tr �12�0��s� � �� 12

�0��q, p, s� dq dp. (27)

We now define the � 0 coherence as �12(0)(q, p, t).

This allows us to show the explicit dependence ofP1

(1)(t):

P1�1��t� �

2V�

Im�ei �o

t

Tr �12�0��s� ds�. (28)

Denoting the final time of the simulation as t � Tand writing the time integral of the trace as a com-plex number in terms of its amplitude and phase,�o

T Tr �12(0)(s) ds � Aei�, the final first-order state 1

population at t � T can be written as

P1�1��T� �

2VA�

sin� � � �. (29)

The final population is thus a trigonometric func-tion of . This qualitative behavior is indeed exhib-ited in Figure 4.

The perturbation theory formalism can be imple-mented easily in the context of an almost conven-tional classical MD simulation. In terms of the tra-jectory ensemble ansatz of Eq. (12), the zeroth-ordercoherence becomes

�12�0��q, p, t� �

1N �

k�1

N

ei�k�t��q � qk12�t�� p � pk

12�t��.

(30)

The members of the ensemble evolve under theHamilton’s equations of the average HamiltonianH0. The nonclassical phase �k(t) is calculated byintegrating the difference potential divided by �along the kth trajectory. The trace of �12

(0) is thengiven simply by summing up the individual phasefactors:



Tr �12�0��t� �

1N �

k�1

N

ei�k�t�. (31)

The only extra effort required beyond that of astandard MD simulation is the computation of thephases �k(t).

In Figures 5 and 6, we show the results of apply-ing the perturbation theory approach to the non-adiabatic wave packet interferometry problem. Fig-ure 5 shows the time-dependent state 2 populationcalculated using Eqs. (26) and (31) [P2(t) � 0.5 P1

(1)(t)]. The dependence of the final state popula-tions is given in Figure 6. These results agree qual-itatively with the full simulation results shown inFigures 3 and 4.

The trace of the off-diagonal density matrix ele-ment, Tr �12(t), is a measure of the level of electroniccoherence exhibited by the system. The zeroth-or-der result given in Eq. (31) highlights the coopera-tive role played by the entire ensemble in determin-ing this quantum mechanical quantity. From thisperspective, a single trajectory is “completely co-herent” in the sense that it has associated with it anondecaying complex phase factor. The persistenceor decay of coherence results from the interplay ofmany such phases. If all the trajectories follow aroughly similar history when moving throughphase space, as would be expected for a low-dimen-sional problem, then the ensemble phase factorsadvance with roughly the same rate. The result is along-lived constructive interference in the sum ofEq. (31). In a many-body system, however, the in-volvement of many “bath” degrees of freedom isexpected to modify the time history of each ensem-

ble member’s evolving difference potential. Thiscauses the phase factors to lose their synchroniza-tion. The result is decoherence.

Even in this simple approximation, the quantummechanical electronic coherence can be readilyidentified as an ensemble-level property of the clas-sical limit MD. The full nonadiabatic dynamics inthe semiclassical Liouville method involves notonly a cooperative effect in the calculation of ob-servables but an irreducible entanglement of theensemble members in the dynamic evolution itself.In the absence of coupling (V � 0), the ensemblesrepresenting �11, �22, and �12 evolve as an indepen-dent collection of trajectories, just as in classicaldynamics. The addition of coupling, however,causes the coefficients of the trajectories to changewith time. These changes depend, directly or indi-rectly, on all the trajectories in all of the ensembles.

Figure 7 depicts this interdependence schemati-cally. From a computational perspective, the loss oftrajectory independence creates algorithmic chal-lenges, as the full ensembles must be propagatedtogether and their interactions incorporated in thesimulation. From a fundamental perspective, how-ever, a novel insight into the nature of quantummechanics emerges as the breakdown of the statisticalindependence of the trajectories in a quantum ensemble.

Quantum Tunneling Using EntangledTrajectory Ensembles

The interactions between members of the evolv-ing ensembles that emerge as a necessary conse-quence of quantum effects in the semiclassical Li-

FIGURE 6. Same as Fig. 4, except calculated usingthe perturbation theory expression, Eq. (29).

FIGURE 5. Same as Fig. 3, except calculated usingthe perturbation theory expression, Eq. (28).

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1354 VOL. 90, NO. 4/5

ouville treatment of nonadiabatic electronicprocesses is suggestive of a more general interpre-tation of quantum mechanics in the context of tra-jectories and ensembles. In this section, we explorethis issue more deeply. We focus our discussion onthe problem of tunneling in one dimension througha potential barrier—one of the paradigms of non-classical physics at the atomic scale. The quantumdynamics of such a system moving under a poten-tial V(q) can be described by the wave function �(q,t), which is a solution of the time-dependent Schro-dinger equation. As noted above, an equivalentphase-space description more closely analogous toclassical mechanics is given in terms of the Wignerfunction �W(q, p, t) [5–8]. The Wigner function isrelated to the wave function �(q) by

�W�q, p� �1

2��

�

�*�q �y2��q �

y2�eipy/� dy (32)

and obeys the partial integrodifferential equation

��W

�t � pm

��W

�q � ��

�

J�q, p � ��W�q, �, t� d�, (33)

where

J�q, p� �i

2��2 ��

� �V�q �y2� � V�q �

y2��eipy/� dy.

(34)

We emphasize that this is an exact and faithfulrepresentation of quantum mechanics; the Wignerfunction �W(q, p, t) contains the same informationabout observable quantities as does �(q, t).

For a potential with a power series expansion inq, Eq. (33) can itself be expanded to give [5, 8]

��W

�t � pm

��W

�q � V��q��W

�p ��2

24 V��q��3�W

�p3 � · · · ,

(35)

where prime denotes the derivative with respect toq. The higher-order terms not shown involve suc-cessively higher even powers of �, odd derivativesof V with respect to q, and �W with respect to p. Inthe classical (� 3 0) limit, the Wigner functionbecomes a solution of the classical Liouville equa-tion for probability distributions in phase space.The probabilistic interpretation of the Wigner func-tion is complicated for quantum systems, however,by the fact that �W, although always real, can as-sume negative values.

We now describe a method for solving the quan-tum Liouville equation in the Wigner representa-tion in the context of a classical trajectory simula-tion [23]. We represent the time-dependent state ofthe system �(q, p, t) as an ensemble of trajectories. Inclassical mechanics, the ensemble members evolveindependently of each other. A quantum state,however, is a unified whole, and the uncertaintyprinciple prohibits an arbitrarily fine subdivisionand independent treatment of its constituent parts.Inspired by what was learned in the work describedin the previous section, we incorporate the nonclas-sical aspects of quantum mechanics explicitly as abreakdown of the statistical independence of themembers of the trajectory ensemble. We derive non-classical forces acting between the ensemble membersthat model the quantum effects governing the evolu-tion of the corresponding nonstationary wave packet.

The continuous distribution function � is repre-sented by a finite ensemble of trajectories:

��q, p, t� �1N �

j�1

N

�q � qj�t�� p � pj�t��. (36)

FIGURE 7. Schematic depiction of the interdepen-dence of trajectories in the ensemble representation ofthe dynamics of the generalized distribution functions�11, �22, and �12. The evolution of one of the functionsat point (q, p) in phase space depends on the values ofthe other functions at that point. In a trajectory imple-mentation, the evolution of the individual trajectoriesthus becomes intertwined with each other.



This ansatz is an approximate one, as the exactWigner function �W can become negative, as notedabove. The assumed positive-definite form of thesolution in Eq. (36) thus cannot capture the fullquantum dynamics in the Wigner representation.Faithful representations of quantum mechanics doexist, however, that are compatible with this ansatz.An example is that based on the Husimi distribu-tion [5]—a Wigner function that is smoothed with aminimum uncertainty Gaussian in phase space. Os-cillations in �W average out, resulting in a distribu-tion function that has the desired nonnegative prop-erty and can thus be interpreted probabilistically. Inour method, we identify the continuous phase-spacefunction resulting from smoothing Eq. (36) with anequivalent positive-definite smoothing of the under-lying Wigner function �W. There are a number ofways to implement the smoothing in practice [24].

The semiclassical Liouville-based trajectorymethod incorporated quantum effects by append-ing amplitudes to trajectories that evolved underpurely classical mechanics [see Eq. (12)]. In thepresent approach, we seek a trajectory representa-tion of quantum mechanics that alters the motion ofthe trajectories themselves. The instantaneous forceacting on a particular member of the ensemble willdepend not only on the gradient of the potential buton the phase-space locations of all the other mem-bers of the ensemble. Their evolution will thus be-come mutually entangled. The distinction betweenclassical and quantum dynamics as represented bytrajectory ensembles is illustrated schematically inFigure 8.

To implement this idea as a numerical method,equations of motion for the trajectories must bederived. The phase-space trace of the Wigner func-tion is conserved: Tr �W � � � dq dp � 1, a propertyshared by its approximation in Eq. (36) (and thesmoothed versions used in practice). In terms of thephase-space flux j� � �v�, the ensemble must evolvecollectively so that the continuity equation

��

�t � �� j� � 0 (37)

is obeyed, where �� is the gradient in phase space.We impose this continuity condition on the inter-acting ensemble. In particular, the evolution of thetrajectories comprising Eq. (36) is accomplished byintegration along the vector field (q, p) � v� � j�/� inphase space.

We first consider the strict classical limit. Here, the�-dependent terms in Eq. (35) vanish and the phase-

space density obeys the classical Liouville equation[9, 10]:

��

�t � �� j� � �H, �, (38)

where {H, �} is the Poisson bracket of H and �. Bynoting that �q/�q � �p/�p � 0, we can identify thephase-space current vector as

j� � � �H/�p�H/�q ��. (39)

The usual situation of independent evolution underconventional Hamiltonian’s equations q � �H/�p, p ��H/�q results, as expected. This occurs because ofthe cancellation of � from the expression for thephase-space vector field v� when j� in Eq. (39) is dividedby �. It does not have to do so in general, however.

FIGURE 8. Schematic representation of trajectory-based evolution of classical and quantum states inphase space. In the classical case (top), the individualtrajectories evolve independently. In the quantum case(bottom), a trajectory-based representation unavoidablyleads to a breakdown of the statistical independence ofthe ensemble members. A caricature of the resultinginteractions is shown.

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1356 VOL. 90, NO. 4/5

We now turn to the quantum Liouville equationin the Wigner representation. The continuity con-dition then involves the full equation of motion, Eq.(35). We can identify the divergence of the current as

�� j� ��

�q ��H�p ��

��

�p �V��q�� 2

24 V��q��2�

�p2 � · · ·�. (40)

Then, dividing the corresponding current by �, wearrive at the equations of motion for the trajectoryat point (q, p):

q � vq �pm

p � vp � V��q� ��2

24 V��q�1�

�2�

�p2 � · · · . (41)

Note that the distribution � does not cancel out ofthe equations! In marked contrast with the classicalHamilton’s equations, the vector field now dependsnot only on the point (q, p) but on the state of thesystem � as well.

A consequence of these state-dependent equa-tions of motion is that individual trajectory energiesare not conserved. This conservation is only re-quired on average. It is straightforward to showfrom Eq. (41) that the ensemble average p� �Tr( p�) � V��, and thus obeys Ehrenfest’s theo-rem, while the average energy E� � Tr(H�) is in-dependent of time. The individual trajectories,however, can behave nonclassically.

An alternative but related formalism in terms of“Wigner trajectories” has been proposed and ex-plored by a number of authors (see, e.g., [7] andreferences therein). Although reminiscent of ourapproach, the equations of motion for Wigner tra-jectories are not the same as those proposed here.They do not have, for instance, the desirable prop-erty of obeying Ehrenfest’s theorem. We also noteanother formulation of quantum dynamics in termsof trajectories, usually called “Bohmian mechan-ics.” This approach is based on the hydrodynamicformulation of quantum mechanics [25, 26]. Origi-nally proposed by Bohm as a way of constructing aquantum theory without giving up realism [27–29],this representation describes quantum dynamics interms of classical-like trajectories evolving underthe influence of both the classical potential and a

nonclassical wave function-dependent quantumpotential. Although initially developed as an inter-pretive tool, Bohmian mechanics has recently un-dergone a revival as the basis of new numericalalgorithms for simulating quantum phenomena[30–34].

The realization of our formalism in the context ofa classical MD simulation is accomplished by gen-erating an ensemble of initial conditions represent-ing �W(q, p, 0) and then propagating the trajectoryensemble using Eq. (41). As discussed earlier, thesingular distribution � must be smoothed to al-low a faithful representation of the (analogouslysmoothed [5]) quantum dynamics. In our imple-mentation, the nonclassical �-dependent force is de-termined from a smooth local Gaussian representa-tion of the instantaneous ensemble. In particular,the value of �1�2�/�p2 at each phase-space point(qj, pj) is calculated by assuming a local Gaussianapproximation of � around �� j � (qj, pj):

��q, p, t� �oe�� j�t��Aj�t�� j�t��b j�t�� j�t��. (42)

The state �(t) at each trajectory location (qj, pj) inphase space is characterized by the time-dependentparameters in the matrix Aj and vector b�j. We de-termine these numerically in practice by calculatinglocal moments of the ensemble around the referencepoint �� j. These consist of sums of appropriate pow-ers of the dynamic variables over the ensemble,weighted by a Gaussian cut-off (�� ) � exp(�� ) centered at the point under consideration, where is chosen to give a minimum uncertainty , con-sistent with the smoothing requirement for a posi-tive quantum phase-space distribution [5]. Fromthis calculation, parameters Aj and b�j can be in-ferred at each point �� j � (qj, pj). The local nature ofthe fit allows nontrivial densities with, for instance,multiple maxima to be represented by the discreteensemble in an accurate, efficient, and numericallystable manner.

We now turn to our 1-D model of quantummechanical tunneling. Using atomic units through-out, we consider a particle of mass m � 2000 mov-ing in the potential

V�q� �12 m�o

2q2 �13 bq3 (43)

where �o � 0.01 and b � 0.2981. This system has ametastable potential minimum with V � 0 at q � 0and a barrier of height V‡ � 0.015 at q‡ � 0.6709.



The parameters are chosen so that the systemroughly mimics a hydrogen atom bound with ap-proximately two (metastable) bound states. The dy-namics are thus expected to be highly quantummechanical.

A series of minimum uncertainty quantum wavepackets and corresponding trajectory ensembleswith zero mean momenta are chosen as initialstates. The mean energy of the state is varied byselecting a range of (negative) initial mean displace-ments. The trajectories are then propagated usingthe ensemble-dependent force given by Eq. (41). Forthe potential in Eq. (43), V� � 2b is constant, andthe higher-order terms in Eq. (41) rigorously van-ish. The entangled trajectory modification of theforce then becomes

pj � V��qj� ��2b12

�2�/�p2�qj, pj�

��qj, pj�(44)

for j � 1, 2, . . . , N. The �-dependent factor dependson the parameters in Eq. (42), and thus involvessummations over functions of (qk, pk) for k � j.

In Figure 9, we show the time-dependent reac-tion probabilities P(t) for three initial conditions(numbered 1–3), each corresponding to an initialminimum uncertainty wave packet (or ensemble)displaced from the minimum of the potential. Theentangled trajectory simulations are compared with

purely classical results generated with the samenumber of trajectories but in the absence of thequantum force and the results of numerically exactquantum wave packet calculations performed us-ing the method of Kosloff [19].

The trajectory results shown here correspond toensembles containing N � 900 trajectories. Thequantum reaction probability is defined at eachtime as the integral of �(q, t)2 from q‡ to �, whilethe classical and entangled trajectory quantities aredefined as the fraction of trajectories with q � q‡ attime t. The curves labeled C, Q, and E indicateclassical, quantum, and entangled trajectory ensem-ble results, respectively. Case 1 corresponds to aninitial displacement qo � 0.2, with a mean energyEo � �H�� 0.75V‡. For case 2, qo � 0.3 andEo 1.25V‡, while for case 3 qo � 0.4 and Eo 2.0V‡. Increasing the mean energy increases theshort time transfer across the barrier, both classi-cally and quantum mechanically. The classical re-action, however, ceases immediately after the firstsharp rise, as the trajectories in the ensemble withenergy below the barrier initially are trapped therefor all time. The quantum wave packet, however,continues to leak out of the metastable well bytunneling, and the reaction probability growsslowly with time following the initial classical-likerise. This growth is modulated by the oscillations ofthe wave packet in the potential well. The entan-gled trajectory calculation tracks the exact quantumresults well. Although these results overestimatethe exact instantaneous probability by a few per-cent in all cases, the qualitative dynamics are de-scribed satisfactorily. In particular, the nonclassicallonger time growth of the reaction probability iscorrectly described.

In Figure 10, we examine the correspondencebetween quantum and entangled trajectory predic-tions of the nonclassical tunneling dynamics inmore detail. The decay of the survival probability1 P(t) at times longer than the initial rapid clas-sical decay is fit to an exponential exp(kt) and thetunneling rate constant k thus defined is plotted inthe figure as a function of mean wave packet en-ergy. The entangled trajectory results systemati-cally overestimate the rate somewhat, but the over-all agreement is good, especially considering that anonzero k is a purely quantum mechanical quantity.

The computational efficiency of the approach isfavorable: The entangled ensemble calculations re-quire less computer time than needed for the exactquantum simulation for a large trajectory ensemblewith N � 900, used to obtain the results shown

FIGURE 9. Time-dependent reaction probabilities.Three initial mean state energies are considered andthe results of the entangled trajectory ensemble simula-tions (E) are compared with purely classical (C) and ex-act quantum (Q) calculations. See text for details.

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here. The results were only slightly less accurate forensembles with as little as 400 trajectories, whichwere roughly an order of magnitude faster than thequantum simulation.

The entangled trajectory formalism gives an ap-pealing physical picture of the quantum tunnelingprocess. In the classical limit, members of the ensem-ble are statistically uncorrelated and evolve indepen-dently under Hamilton’s equations. Each trajectorymust separately conserve energy, and an ensemblemember with initial energy below the barrier istrapped in the reactant region for all time. When theensemble is entangled by the interactions betweentrajectories, this individual energy conservation canbe violated, as it is only the average ensemble energythat must be conserved. Trajectories can “borrow”energy from the rest of the ensemble and use thatexcitation to surmount the barrier. Detailed analysisof the interacting ensemble reveals that borrowedenergy is repaid to the ensemble as a whole, andtrajectories that successfully tunnel often end up as-ymptotically with a final energy that is well below V‡.

Summary

In this article, we described an approach to thesimulation of quantum processes using trajectoryintegration and ensemble averaging. The generalmethod was illustrated in two contexts: classical

limit MD on multiple coupled electronic states andquantum tunneling through a potential barrier.

The basis of the method is the Liouville repre-sentation of quantum mechanics and its realizationin phase space via the Wigner function formalism.The evolution of the classical phase-space functionsis approximated by the motion of the correspond-ing trajectory ensembles. In the classical limit, themembers of the ensemble evolve independently un-der Hamilton’s equations of motion. When quan-tum effects are included, however, the correspond-ing “quantum trajectories” are no longer separablefrom each other. Rather, their statistical indepen-dence is destroyed by nonclassical interactions thatreflect the nonlocality of quantum mechanics. Theirtime histories become interdependent and the evo-lution of the quantum ensemble must be accom-plished by taking this entanglement into account.

ACKNOWLEDGMENTS

This work was supported by the National Sci-ence Foundation.

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FIGURE 10. Tunneling rate as a function of initialmean wave packet energy, entangled trajectory, andexact quantum results.



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Documents

Classical trajectory-based approaches to solving the quantum Liouville equation