Simulation of quantum processes using entangled trajectory molecular dynamics

Transcription

1 JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 10 8 SEPTEMBER 2003 Simulation of quantum processes using entangled trajectory molecular dynamics Arnaldo Donoso Laboratorio de Física Estadística de Sistemas Desordenados, Centro de Física, Instituto Venezolano de Investigaciones Científicas IVIC, Apartado 21827, Caracas 1020A, Venezuela Yujun Zheng and Craig C. Martens a) Department of Chemistry, University of California Irvine, Irvine, California Received 7 April 2003; accepted 12 June 2003 In this paper, we describe a new method for simulating quantum processes using classical-like molecular dynamics. The approach is based on solving the quantum Liouville equation in the Wigner representation using ensembles of classical trajectories in phase space. The nonlocality of quantum mechanics is incorporated in the trajectory representation as nonclassical interactions between the members of the ensemble, leading to an entanglement of their evolution. The statistical independence of the individual trajectories making up an ensemble in the classical limit is lost when quantum effects are included, and the entire state of the system must be propagated as a unified whole. We develop the formalism and its numerical implementation, and illustrate its application on two model problems of quantum mechanical tunneling: escape from a metastable well and wave packet penetration of the Eckhart barrier American Institute of Physics. DOI: / I. INTRODUCTION a Electronic mail: cmartens@uci.edu Quantum mechanical effects play an essential role in many chemical and physical systems. Ideally, such systems should be treated theoretically using rigorous quantum mechanical methods. Direct solution of the time-dependent Schrödinger equation has become a feasable approach for problems of significant complexity in cases where approximations and their systematic corrections can be implemented efficiently. Particularly good examples of this are the impressive recent applications of the multiconfiguration timedependent Hartree MCTDH method 1,2 to systems with many degrees of freedom. 3 7 As complexity is increased to encompass fully coupled many-body condensed phase systems, direct quantum methods eventually become inapplicable, and alternative approaches based on more severe approximations must be employed. In some cases, classical molecular dynamics MD Ref. 8 itself can be used effectively to treat many-atom systems. Here, an ensemble of classical trajectories is employed to represent a quantum mechanical wave packet, thermal distribution, or other initial equilibrium or nonequilibrium state. The evolution of the state is then simulated by propagating the ensemble in phase space under Hamiltonian dynamics. 9 Classical MD can often do a remarkably good job of modeling the statistics and dynamics of molecular systems and reproducing the results of detailed experiments see, for instance, Ref. 10. In some important cases, however, this classical-limit approximation cannot be made. Examples include systems involving the motion of hydrogen atoms, other light particles, or systems at very low temperatures; here, the quantum nature of the nuclear motion becomes important, and effects such as zero point energy and quantum mechanical tunneling require explicit treatment. 11 Another situation where nonclassical effects become dominant is when a breakdown of the clean separation between nuclear and electronic degrees of freedom occurs. In such systems, two or more Born Oppenheimer surfaces are coupled to each other, and quantum electronic transitions accompany the almost classical nuclear motion In this paper, we describe a general approach to modeling the dynamics of systems for which nonclassical effects are important. Despite the breakdown of the validity of classical mechanics, we work within the context of classical-like molecular dynamics and ensemble averaging. Our formal development is based on a phase space representation of the quantum Liouville equation the Wigner representation and its approximate solution using trajectory ensembles. We describe our methodology and apply it to the problem of quantum mechanical tunneling. In this context, nonclassical effects emerge as the breakdown of the statistical independence of the members of the trajectory ensemble. The organization of the rest of this paper is as follows: In Sec. II the general formalism underlying entangled trajectory molecular dynamics is presented. The implementation of the formalism as a numerical method is described in Sec. III. In Sec. IV the method is applied to two one-dimensional models of quantum mechanical tunneling: escape from a metastable potential and transmission through the Eckhart barrier. The entangled trajectory results are compared with both classical mechanics and exact quantum time-dependent wave packet calculations. Finally, a discussion is given in Sec. V /2003/119(10)/5010/11/$ American Institute of Physics

2 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Entangled trajectory molecular dynamics 5011 II. GENERAL FORMALISM The classical mechanical evolution of a probability distribution (q,p,t) in phase space is described by the classical Liouville equation, 9,23 t H,, 1 where q and p are the canonical coordinate and momentum, respectively, H is the system Hamiltonian, and H, is the Poisson bracket of H and, defined as H, H q p q H p. In the classical MD method, 8 a numerical solution of the Liouville equation is accomplished by generating an ensemble of N distinct initial conditions q k (0) and p k (0) (k 1,2,...,N) sampled from the given initial probability distribution (q,p,0) and then integrating Hamilton s equations, q H p, 3 2 ṗ H q, 4 using the q k (0) and p k (0) as initial data. The evolving distribution (q,p,t) is then approximated by the local phase space density of the evolving trajectories (q k (t),p k (t)) around the point q,p. This relation between the evolution of the classical function (q,p,t) and the trajectory ensemble (q k (t),p k (t)) (k1,2,...,n) in phase space is illustrated schematically in the top panel of Fig. 1. In order to describe the evolution of quantum systems from an analogous ensemble perspective, we adopt a phase space representation of quantum mechanics the Wigner representation Consider a one-dimensional system with mass m evolving under the influence of a potential V(q). The quantum dynamics of such a system can be described by the wave function (q,t), which is a solution of the timedependent Schrödinger equation. 11 An equivalent phase space description is given in terms of the Wigner function W (q,p,t) The Wigner function is related to the wave function (q,t) by W q,p,t 1 * 2 q y 2,t q y 2,t e ipy/ dy and obeys the equation of motion W p W t m q Jq,pW q,,td, where Jq,p i V q y 2 V q y 2e ipy/ dy. 7 FIG. 1. A schematic representation of the evolution of phase space distribution functions and their approximation by classical trajectory ensembles. In the top frame, the evolution of the classical phase space density (q,p,t) is represented by an ensemble of independent classical trajectories. In quantum mechanics, the nonlocality of the underlying evolution equation for the Wigner function leads to a breakdown of this independence, and quantum interactions between members of the ensemble, as indicated schematically in the lower frame. We emphasize that this is an exact and faithful representation of quantum mechanics; for pure states, 11 the Wigner function W (q,p,t) contains the same information about observable quantities as (q,t). It should be noted that the Wigner formalism also provides a phase space representation of more general quantum systems described by a density operator, and in particular, for mixed states that cannot be represented by a wave function. Equation 6 highlights the fundamental nonlocality of quantum mechanics: the time rate of change of W at point q,p depends on W over a range of momentum values p. For a potential with a power series expansion in q, Eq. 6 can itself be expanded to give 19,20 W p W t m q Vq W p 2 24 Vq 3 W p 3, where prime denotes the derivative with respect to q. The higher order terms not shown involve successively higher even powers of, odd derivatives of V with respect to q, and corresponding derivatives of W with respect to p, the general nth term being

3 5012 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Donoso, Zheng, and Martens 1 n 2n d 2n1 Vq 2n1 W q,p 2 2n 2n1! dq 2n1 p 2n1. 9 For a polynomial potential, the series in Eq. 8 contains a finite number of terms while for a general potential, the result is an infinite series. In the classical ( 0) limit, the nonclassical terms vanish and the Wigner function becomes a solution of the classical Liouville equation. A probabilistic interpretation of the Wigner function is complicated by the fact that W, although always real, can assume negative values. Faithful representations of quantum mechanics built on positive-definite phase space probability distributions do exist, however. An example, the Husimi representation, 20 is based on smoothing the Wigner function locally with a minimum uncertainty Gaussian test function in phase space. The nonlocality of quantum mechanics forbids the arbitrarily fine subdivision of the quantum distribution into individual independent elements. Rather, a quantum state must be propagated as a unified whole. If a trajectory ensemble representation of nonlocal quantum motion is to be achieved, the statistical independence of the individual elements the trajectories that represent the state must be given up. Algorithmically, the members of the ensemble must interact with each other as they evolve in phase space. This interdependence, or entanglement, of the trajectory ensemble is depicted schematically in the lower panel of Fig. 1. We now describe a theoretical formalism for treating the time evolution of quantum systems which is based on the concept of an interacting and entangled classical trajectory ensemble. 24 We represent the time-dependent phase space distribution W (q,p,t) approximately as an ensemble of trajectories sampling a probability distribution (q,p,t) and incorporate the nonclassical aspects of quantum mechanics explicitly as a breakdown of the statistical independence of the members of the trajectory ensemble. We derive nonclassical forces that act between the ensemble members and approximate the nonlocal quantum effects governing the evolution of the corresponding nonstationary wave packet. The goal is a trajectory representation of quantum mechanics that includes quantum effects by altering the motion of the trajectories themselves. The instantaneous force acting on a particular member of the ensemble will therefore depend on both the classical force V(q) and on the phase space locations of all the other members of the ensemble via a quantum force. Their evolution will become mutually entangled. The phase space trace of the Wigner function is conserved: Tr W dqdp1, a property shared by its approximation as a finite set of trajectories in phase space sampled from a probability distribution. In terms of the phase space flux jv, the ensemble must evolve collectively so that the continuity equation, t "j0 10 is obeyed, where is the gradient in phase space. We exploit this continuity condition in our equations of motion by identifying the current j in the quantum Liouville equation in the Wigner representation, finding the corresponding vector field vj/, and then integrating the trajectories in phase space using (q,ṗ)v. It is instructive to first consider the strict classical limit. Here, the -dependent terms in Eq. 8 vanish, and the phase space density obeys the classical Liouville equation, 9,23 t "jh,. 11 By noting that q /qṗ/p0 by Hamilton s equations, we can identify the phase space current vector as j H/q H/p. 12 Division by then gives the phase spece vector field v and recovers the familiar independent evolution of phase space trajectories under conventional classical Hamiltonian s equations q v q H/p, ṗv p H/q. This comes about because of the cancellation of from the expression for v. We now analyze the quantum Liouville equation in the Wigner representation from the same perspective. The continuity condition involves the full equation of motion, Eq. 8. Writing the divergence of the current as "j q H p p Vq 2 24 Vq 2 p 2 13 and dividing the corresponding current by, we obtain the equations of motion for the trajectory at point (q,p), q v q p m, ṗv p Vq 2 24 Vq 1 2 p In this case does not cancel out of the equations. In marked contrast with the classical Hamilton s equations, the vector field now depends on the global state of the system as well as on the phase point q,p. A consequence of the additional -dependent contribution to the force is that individual trajectory energies are not conserved. This is acceptable and in fact essential if quantum effects are going to be represented by the method. Energy conservation is only required on average, dh/dt 0. We can show that this is indeed the case by calculating the time derivative of the classical Hamiltonian assuming no explicit time dependence and inserting the equations of motion for q and ṗ, dh dt H H q ṗ q p p m 2 24 Vq 1 2 p This quantity vanishes in the classical limit; the nonzero terms on the right-hand side of Eq. 15 are of purely quantum origin. Calculating the ensemble average yields

4 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Entangled trajectory molecular dynamics 5013 dh dt dh dt dqdp p m 2 24 Vq 2 p 2 dqdp. 16 For densities that obey 0 and /p 0 asp, integration by parts then gives the desired result dh/dt 0. In a similar manner, it is straightforward to show from Eq. 14 that the ensemble average ṗtr(ṗ)v, and so the method obeys Ehrenfest s theorem 11 as well. On average, the ensemble behaves classically. The individual trajectories, however, evolve nonclassically as they must if they are to faithfully capture the dynamics of, for example, quantum tunneling. III. METHOD The realization of our formalism in the context of a classical molecular dynamics simulation is accomplished by generating an ensemble of initial conditions representing W (q,p,0) and then propagating the trajectory ensemble using Eq. 14. The continuous phase space distribution function approximating W is represented by a finite and discrete ensemble of trajectories, N q,p,t 1 qq N j tpp j t. j1 17 This ansatz is an approximate one, as the exact Wigner function W can become negative. The assumed positive-definite form of the solution in Eq. 17 therefore cannot capture the full quantum dynamics in the Wigner representation. As noted above, the Husimi distribution, 20 a Gaussian smoothed Wigner function, is compatible with this ansatz. Oscillations in W average out, resulting in a distribution function that has the desired non-negative property and can be interpreted probabilistically. In our method, we identify the continuous phase space function resulting from smoothing Eq. 17 with an equivalent positive-definite smoothing of the underlying Wigner function W. A number of approaches to the problem of constructing a smooth positive function from a finite set of sampled points have been developed, such as the statistical method of density estimation. 25,26 Here we adopt a different approach: the nonclassical -dependent force is determined in our implementation from a smooth local Gaussian representation of the instantaneous ensemble. We assume that, around a given trajectory (q k,p k ) at a particular time t, the phase space distribution has the form, q,pae q qq k 2 p pp k 2 qq k pp k q qq k p pp k. 18 We then determine the parameters of this function from the evolving ensemble. We emphasize that this is a local representation, and the parameters q, p, q, p, and depend on the point (q k,p k ) and on time t. These parameters are determined from the ensemble of trajectories; as described in more detail below, trajectories near the reference point are weighted more than distant ones in the determination. The entangled trajectory equations of motion depend on the derivatives of evaluated at the phase point in question. Once the local Gaussian approximation to the phase space density at the point (q k,p k ) has been determined, the required values of 1 j /p j ( j2,4,6,...) at that point can be calculated easily. By differentiating the Gaussian ansatz in Eq. 18 the first few expressions required can be written explicitly as 1 2 p 2 2 p 2 p, 1 4 p 4 4 p 12 2 p p 12 2 p, 1 6 p 6 6 p 30 4 p p p 2 p p, and so forth. By determining the parameters of the Gaussian ansatz, we can find the quantum force for a general potential possessing a convergent power series expansion in q. The local parameters of the Gaussian approximation are determined numerically by calculating local moments of the ensemble around the reference point (q k,p k ). These local moments consist of sums of appropriate powers of the dynamical variables over the ensemble, weighted by a Gaussian test function, qq k,pp k e h q qq k 2 h p pp k 2 22 centered at the point under consideration, where h q and h p are chosen to give a minimum uncertainty. This choice seems to work best in practice, and is consistent with the smoothing requirement for a positive quantum phase space distribution that forms the basis of the Husimi distribution. 20 From this calculation, the parameters p and p can be inferred at the point (q k,p k ). The local nature of the fit allows nontrivial densities with, for instance, multiple maxima to be represented by the discrete ensemble in an accurate, efficient, and numerically stable manner. We now describe the numerical extraction of the local Gaussian parameters from the instantaneous trajectory ensemble. Consider the integral Ĩ( q, p, q, p,), defined as

5 5014 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Donoso, Zheng, and Martens Ĩ e q 2 p 2 q p hq,hp,dd, 23 where qq k, pp k, and hq,h p,exph q 2 h p This integral corresponds to a normalization of the density (q,p), but with a Gaussian weighting with a smoothing function hq,h p centered at the phase space point in question. It can be evaluated analytically to give Ĩ q h q p h p 2 4 q exp1 4 2 q h p 2 p q h q q p q h q q h p The integral Ĩ can then be employed to generate the modified first and second moments of the distribution around the reference point. These modified moments are defined formally in terms of and by,,dd,,dd, 2 2 2,,dd,,dd,,,dd,,dd, ,,dd,,dd, 29,,dd,,dd, 30 where qq k and pp k and f f (,)(,)dd. Analytic expressions for the modified moments in terms of the Gaussian parameters are then obtained by differentiation of the function Ĩ; 1 Ĩ 2 1 Ĩ 1 Ĩ 2 1 Ĩ Ĩ q, 2 Ĩ q 2, Ĩ p, 2 Ĩ p 2, Ĩ. 35 Ĩ q p Other equivalent definitions are also possible. We define the modified variances in terms of the modified moments, 2 2 2, 2 2 2, After some algebra, the required parameters of the Gaussian ansatz are found in terms of the modified moments and variances, p 2 2 4, p h p. 40 The preceding calculations show how the underlying parameters of the local Gaussian ansatz describing the distribution can indeed be extracted from moments calculated, not from the distribution itself, but from a distribution that is modified by multiplication by the function. This improves the quality of the local Gaussian fit, as local information about the full distribution is employed preferentially in the description. These expressions give the ingredients for the quantum force in terms of the properties of the distribution of trajectories. These can be found from the corresponding definitions of the modified moments in terms of the ensemble of trajectories as follows: N j1 2 N j1 N j1 2 N j1 q j q k q j q k,p j p k N, 41 q j q k,p j p k j1 q j q k 2 q j q k,p j p k N, 42 q j q k,p j p k j1 p j p k q j q k,p j p k N, 43 q j q k,p j p k j1 p j p k 2 q j q k,p j p k N, 44 q j q k,p j p k j1 N j1q j q k p j p k q j q k,p j p k N, 45 j1 q j q k,p j p k where these expressions refer to the phase point (q k,p k ). Unlike alternative approaches we have considered, such as algorithms based on numerical differentiation of the function, this method yields a stable and accurate representation of the quantum dynamics in terms of interacting trajectory ensembles. IV. NUMERICAL RESULTS In this section, we illustrate the general approach by considering two one-dimensional model problems exhibiting quantum mechanical tunneling. The first system models es-

6 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Entangled trajectory molecular dynamics 5015 FIG. 2. Snapshots of a trajectory ensemble evolving under unmodified classical mechanics in the metastable potential given in Eq. 46 with initial energy E0.75V. Four times are shown: t0, 250, 750, and 1500 a.u. Each frame shows the ensemble of N900 trajectories as well as the separatrix defined as the curve in phase space EV. The trajectories at t0 are chosen on a nonuniform grid in phase space such that the local density is proportional to the initial minimum uncertainty phase space Gaussian, as described in the text. The phase space evolution of the trajectories is stopped as they pass qq c to allow visualization of their asymptotic momenta and energies. cape from a metastable potential well V(q) described by a polynomial in q consisting of quadratic and cubic terms. For this system, the expansion in powers of terminates after the first nonclassical correction. The second system describes wave packet transmission through an Eckhart barrier. Here, the series resulting from Eq. 8 contains an infinite number of terms. For both cases, we simulate the quantum dynamics of a nonstationary wave packet using entangled trajectory molecular dynamics, and compare the results with both exact quantum wave packet propagation and unmodified classical mechanics. A. Cubic polynomial potential Using atomic units throughout, we consider a particle of mass m2000 moving on the potential, Vq 1 2m 2 o q 2 3bq 1 3, 46 where o 0.01 and b This system has a metastable potential minimum with V0 atq0 and a barrier to escape of height V at q In order to avoid numerical problems, the potential is set equal to a constant, V(q)V c, for qq c, where q c is chosen such that V(q c )V and V(q) is continuous. The system is chosen to roughly mimic a proton bound with approximately two metastable states. The dynamics are therefore expected to exhibit significant quantum effects. A series of minimum uncertainty quantum wave packets and corresponding trajectory ensembles are chosen as initial states. The mean momentum p0 in all cases and the mean energy of the state is varied by selecting a range of initial average displacements up the inner repulsive wall of the potential. The trajectories are then propagated using the ensemble-dependent force given by Eq. 14. For the potential in Eq. 46, V2b is constant, and the higher order terms in Eq. 14 rigorously vanish. The force then becomes ṗ j Vq j 2 b 12 2 /p 2 q j,p j q j,p j 47 for j1,2,...,n. The -dependent factor depends on the parameters p and p at the point (q j,p j ), and so involves summations over the entire trajectory ensemble. In Figs. 2 and 3 we show the evolution of trajectory

7 5016 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Donoso, Zheng, and Martens FIG. 3. Same as Fig. 2, but showing ensemble evolution under entangled trajectory molecular dynamics. See text for discussion. ensembles under strict classical mechanics and the modified entangled trajectory molecular dynamics incorporating the quantum force given in Eq. 47, respectively. The initial mean energy is E0.75V, or equivalently, q o 0.2. Four snapshots of the evolving trajectory ensembles on the q,p phase plane are shown in each case. The dotted line indicates the separatrix dividing bound and unbound motion on the metastable potential. Each point shown is an individual trajectory. The first frame in each figure shows the initial t0 distribution of trajectories. Here, ensemble members are chosen on a grid in phase space, with the local density of points adjusted to be proportional to the initial phase space distribution function, a minimum uncertainty Gaussian with q /2m o and p m o /2. Alternatively, initial conditions can be chosen randomly, sampled from the initial phase space Gaussian Wigner function. This approach gives results that are very similar to the regular nonuniform grid employed here. The instantaneous position and momentum of each trajectory in the evolving ensemble is shown for the four times selected: t0, 250, 750, and 1500 a.u. For trajectories that escape and dissociate, the positions and momenta are frozen as they pass through qq c. The classical ensemble dynamics shown in Fig. 2 exhibit the expected behavior. Each trajectory separately conserves its total energy. Thus, ensemble members that begin inside the bounds of the separatrix are trapped there for all time. Trajectories that are initiated with an energy above V inevitably escape from the bound region of configuration space, and encounter qq c in the phase space region above the asymptotic separatrix. In Fig. 3, the evolution of the entangled trajectory ensemble is shown. The initial ensemble depicted in the first frame (t0) is, by construction, identical to the initial conditions in the classical case in Fig. 2. The remaining frames show the evolution of the ensemble under the equations of motion derived in the previous section. The snapshots of the entangled trajectory ensemble evolution shown in Fig. 3 illustrate the manifestly nonclassical behavior captured by the method. In particular, the individual trajectories in the entangled trajectory MD simulation do not individually conserve total energy. Rather, energy exchange between the ensemble members, mediated by the quantum force, allows trajectories to cross and recross the separatrix, capturing the essential qualitative character of quantum tunneling. Both a flux out through the separatrix in the bound region of the potential and a de-excitation below the energy of the barrier top in the region to the right of the barrier are clearly observed. In Fig. 4 we show time-dependent reaction probabilities P(t). The quantum reaction probability is defined at each

8 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Entangled trajectory molecular dynamics 5017 FIG. 4. Time-dependent reaction probabilities in the metastable cubic potential system. Three initial states or ensembles are considered, and the results of the entangled trajectory ensemble simulations E are compared with purely classical C and exact quantum Q calculations. See text for details. FIG. 5. Quantum tunneling rate as a function of initial mean wave packet energy for the metastable cubic potential system. Entangled trajectory and exact quantum results are compared. time as the integral of (q,t) 2 from q to, while the classical and entangled trajectory quantities are defined as the fraction of trajectories with qq at time t. The curves are labeled C, Q, and E indicate classical, quantum, and entangled trajectory ensemble results, respectively. The entangled trajectory simulations are compared with purely classical results generated with the same number of trajectories but in the absence of the quantum force and the results of numerically exact quantum wave packet calculations performed using the method of Kosloff. 27 The trajectory results shown here correspond to ensembles containing N900 trajectories; similar results are obtained with N400 trajectories. Three initial states are considered, numbered 1 3, each corresponding to an initial minimum uncertainty wave packet or ensemble. Case 1 corresponds to a mean energy E o Ĥ0.75 V. For case 2, E o 1.25 V, while for case 3 E o 2.0 V. Increasing the mean energy increases the short time transfer across the barrier, both classically and quantum mechanically. The classical reaction, however, ceases immediately after the first sharp rise, as the trajectories in the ensemble with energy below the barrier initially are trapped there for all time. The quantum wave packet, however, continues to escape from the metastable well by tunneling, and the reaction probability continues to grow slowly with time following the initial rapid classical-like rise. This growth is modulated by the oscillations of the wave packet in the potential well. The entangled trajectory calculation tracks the exact quantum results quite well. Although these results slightly overestimate the exact instantaneous probability, the qualitative dynamics are described quite satisfactorily. In particular, the nonclassical longer time growth of the reaction probability is correctly described. In Fig. 5, we examine the agreement between quantum and entangled trajectory predictions of the nonclassical tunneling dynamics in more detail. Here, the decay of the survival probability 1P(t) at times longer than the initial rapid classical barrier crossing is fit to an exponential exp(kt), and the resulting effective tunneling rate constant k is plotted in the figure as a function of mean wave packet energy. The overall correspondence is very good, especially considering that a nonzero k is a purely quantum mechanical quantity. B. Eckhart barrier The metastable cubic potential treated above leads to a rigorous termination of the series in Eq. 14 after the first nonclassical term. In order to test the approach for general potentials, we treat here nonclassical tunneling of a particle with mass m2000 again using atomic units through the Eckhart barrier. We choose the potential to be given by VqV o sech 2 aq 48 with numerical parameters V o and a1.0. The initial wave packet is initiated well to the left of the barrier with mean position q o 4.1 with a well-defined kinetic energy and positive momentum, and minimum uncertainty widths in q and p are chosen corresponding to a harmonic oscillator with frequency o 0.01, as done in the previous case. The nonlocal quantum contribution to the force acting on the jth trajectory contains, in principle, an infinite number of terms. The total force is then ṗ j Vq j n1 1 W q j,p j 1 n 2n d 2n1 Vq j 2 2n 2n1! dq 2n1 2n W q j,p j p 2n1. 49 This series is truncated in practice, and the dependence of the results on the number of terms is assessed. We find that inclusion of four terms in Eq. 49 yields converged results for this system. Time-dependent reaction probabilities for three initial mean energies are shown in Figs Each figure compares the results of entangled trajectory MD with exact quantum

9 5018 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Donoso, Zheng, and Martens FIG. 6. Time-dependent reaction probabilities for the Eckhart barrier, with E/V o 0.5. The results of the entangled trajectory ensemble method dotted line are compared with exact quantum wave packet calculations solid line and pure classical mechanics dashed line. See text for details. wave packet propagation and classical MD. Figures 6, 7, and 8 correspond to energies relative to the barrier height of E/V o 0.5, 0.8, and 1.0, respectively. For each energy, the entangled trajectory molecular dynamics results agree well with the exact quantum calculations. This agreement extends from the lowest energy case E/V o 0.5, where the barrier crossing is small and essentially forbidden classically, to the highest energy E/V o 1.0, where the behavior of the system is well-represented by purely classical mechanics. V. DISCUSSION In this paper, we have described a new method for simulating the dynamics of quantum systems using classical trajectories. This entangled trajectory molecular dynamics approach is based on solving the quantum Liouville equation in the Wigner representation using ensembles of trajectories. Quantum effects arise through forces of interaction between the evolving trajectories, which lose the statistical independence that characterizes trajectory ensembles in classical me- FIG. 7. Same as Fig. 6, but with E/V o 0.8.

10 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Entangled trajectory molecular dynamics 5019 FIG. 8. Same as Fig. 6, but with E/V o 1.0. chanics. We have described the entangled trajectory formalism, its numerical implementation, and illustrated its application by considering two simple one-dimensional models of quantum mechanical tunneling dynamics. The method presented here attempts to solve the quantum equations of motion with ensembles of classical trajectories. In this regard, it resembles many other trajectorybased approximations to exact quantum dynamics. Unlike most other methods, however, which associate quantum phase information with trajectories that evolve under purely classical mechanics, the trajectories in our method follow altered paths through phase space. Our approach is not based on a semiclassical approximation, but rather attempts to find an exact quantum solution within the approximate ansatz of a positive-definite phase space distribution sampled by a finite number of trajectories, each with an equal and timeindependent weight see Eq. 17. The trajectories do not follow purely classical paths; this characteristic of the present method is an essential component which allows nonclassical processes, such as quantum tunneling, to be simulated. The entangled trajectory molecular dynamics method bears a similarity to another quantum trajectory approach currently being pursued, based on the hydrodynamic formulation of quantum mechanics. 28,29 Originally proposed by Bohm as a way of constructing a quantum theory without giving up local realism, the hydrodynamics approach formulates quantum dynamics in terms of classical-like trajectories evolving under the influence of both the classical potential and a nonclassical wave function-dependent quantum potential. Although initially proposed as an interpretive tool, Bohmian mechanics has recently undergone a revival as the basis of new numerical algorithms for simulating quantum phenomena Bohmian mechanics describes the evolution of a quantum state in terms of an ensemble of trajectories that obey the equation of motion mq d dq VqQq,t, 50 where Q(q,t) is the quantum potential, given in terms of the configuration space probability density R 2 (q,t) (q,t) 2 by Qq,t 2 Rq,t 2m Rq,t. 51 Here, prime denotes the derivative with respect to q. The resulting quantum force in Bohmian mechanics differs from the quantum force appearing in entangled trajectory molecular dynamics. The present approach is a dynamical theory in phase space, while the hydrodynamics formulation of quantum mechanics and the resulting Bohmian trajectories are a configuration space description. The momentum field p(q) in Bohmian dynamics has a definite value at each point q, 38 while a description in terms of phase space densities allows a distribution of momentum values at each q. This lack of uncertainty spread in the momentum distribution in Bohmian mechanics leads to a nonvanishing quantum force, even for the free particle and the harmonic oscillator, while the entangled trajectory approach describes these by purely classical dynamics as, indeed, does the exact equation of motion for the Wigner function. An added advantage of the present phase space description is the absence of singularities in the quantum force at nodes of the wave function, which leads to zeros of R(q,t) and technical problems in integrating the equations of motion for Bohmian trajectories. Another approach that is related to the present one is the phase space description of quantum mechanics in terms of so-called Wigner trajectories. 21,39 Wigner trajectories are paths in phase space that obey the ordinary differential equations,

11 5020 J. Chem. Phys., Vol. 119, No. 10, 8 September 2003 Donoso, Zheng, and Martens q p m, 52 ṗv eff q,p,t, 53 where V eff (q,p) is an effective quantum potential determined by interpreting the exact equations of motion for the Wigner function in the classical-like form, W p W t m q V eff q,p,t W p. 54 Comparing, for instance, with Eq. 8, the effective force V eff (q,p,t) becomes V eff q,p,tvq W 1 p 2 24 Vq 3 W p Like in the present approach, the result of the Wigner trajectory formalism is a quantum force that depends on the state of the system as a whole via the appearance of W in the trajectory equations of motion. The particular form of the force is not the same, however. For Wigner trajectories, the leading term involves the ratio of the third and first partial derivatives of W with respect to momentum. The leading term in entangled trajectory molecular dynamics is the ratio of the second partial momentum derivative of W and W itself. This key difference ensures the constancy of average energy and proof of Ehrenfest s theorem in the present approach. The Wigner trajectory approach has not been found to be a useful method in practice. 21 The entangled trajectory formalism gives a unique and appealing physical picture of the quantum tunneling process. Tunneling is seen to occur through the social aspects of nonlocality in quantum mechanics when represented in terms of a trajectory ensemble. Rather than the elementary textbook view of tunneling as burrowing through the obstacle, trajectories that successfully escape the metastable well do so by borrowing enough energy from their fellow ensemble members to surmount the barrier. In the classical limit, members of the ensemble are statistically uncorrelated and evolve independently under Hamilton s equations. Each trajectory must separately conserve energy, and an ensemble member with initial energy below the barrier is trapped in the reactant region for all time. When the ensemble is entangled by the interactions between trajectories, this individual energy conservation can be violated, as it is only the average ensemble energy that must be conserved. This less restrictive requirement of energy conservation allows the individual quantum trajectories the freedom to exhibit the nonclassical behavior needed to describe manifestly quantum mechanical effects such as tunneling. ACKNOWLEDGMENTS We thank E. R. Bittner, I. Burghardt, K. B. Møller, and R. E. Wyatt for helpful discussions. This work was supported by the National Science Foundation. 1 H. D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165, M. H. Beck, A. Jäckle, G. A. Worth, and H. D. Meyer, Phys. Rep. 324, A. Hammerich, U. Manthe, R. Kosloff, H. D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 101, G. Worth, H. D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 105, G. Worth, H. D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 109, I. Burghardt, H. D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 111, H. Wang, J. Chem. Phys. 113, M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids Clarendon, Oxford, H. Goldstein, Classical Mechanics, 2nd ed. Addison Wesley, Reading, Z. Li, R. Zadoyan, V. A. Apkarian, and C. C. Martens, J. Phys. Chem. 99, C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics Wiley, New York, E. E. Nikitin, Theory of Elementary Atomic and Molecular Processes in Gases Oxford University Press, Oxford, K. S. Lam and T. F. George, in Semiclassical Methods in Molecular Scattering and Spectroscopy, edited by M. S. Child Reidel, Dordrecht, J. C. Tully, J. Chem. Phys. 93, A. Donoso and C. C. Martens, J. Phys. Chem. A 102, A. Donoso and C. C. Martens, J. Chem. Phys. 112, A. Donoso and C. C. Martens, J. Chem. Phys. 112, C. C. Martens and J. Y. Fang, J. Chem. Phys. 106, E. P. Wigner, Phys. Rev. 40, K. Takahashi, Prog. Theor. Phys. Suppl. 98, H. W. Lee, Phys. Rep. 259, S. Mukamel, Principles of Nonlinear Optical Spectroscopy Oxford University Press, Oxford, D. A. McQuarrie, Statistical Mechanics HarperCollins, New York, A. Donoso and C. C. Martens, Phys. Rev. Lett. 87, B. W. Silverman, Density Estimation for Statistics and Data Analysis Chapman and Hall, London, A. Donoso and C. C. Martens, J. Chem. Phys. 116, R. Kosloff, Annu. Rev. Phys. Chem. 45, L. de Broglie, C. R. Acad. Sci. Paris 183, E. Madelung, Z. Phys. 40, D. Bohm, Phys. Rev. 85, D. Bohm, Phys. Rev. 85, Quantum Theory and Measurement, edited by W. H. Zurek and J. A. Wheeler Princeton University Press, Princeton, B. K. Dey, A. Askar, and H. Rabitz, J. Chem. Phys. 109, C. L. Loperore and R. E. Wyatt, Phys. Rev. Lett. 82, J. C. Burant and J. C. Tully, J. Chem. Phys. 112, E. R. Bittner and R. E. Wyatt, J. Chem. Phys. 113, E. Gindensperger, C. Meier, and J. A. Beswick, J. Chem. Phys. 113, P. R. Holland, The Quantum Theory of Motion Cambridge University Press, Cambridge, H. W. Lee and M. O. Scully, J. Chem. Phys. 77,