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Physical Chemistry Chemical Physics

Quantum tunneling dynamics using entangled trajectories: general potentials

Ashu Wang, Yujun Zheng, Craig C. Martens, Weiyi Ren

Published Jan 1, 2009• 30 citations
Publisher

Abstract

In this paper, we develop the formalism of entangled trajectory molecular dynamics (ETMD), introduced by Donoso and Martens [Phys. Rev. Lett. 2001, 87, 223202] in a form applicable to the treatment of general (i.e., nonpolynomial) potentials. We formulate our approach directly in terms of the integrodifferential equation obeyed by the Wigner function, without assuming a Taylor series expansion of the potential in powers of the coordinate. This alternative formalism has distinct advantages for propagating distribution functions represented by finite trajectory ensembles and for nonpolynomial potentials. We use a numerical implementation of the new approach to calculate the reaction probabilities for three model systems: the cubic polynomial potential, symmetric Eckart barrier and asymmetric Eckart barrier. Our results are in excellent agreement with the results of exact quantum calculation.

DOI
10.1039/b811509e

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In this paper, we develop the formalism of entangled trajectory molecular dynamics (ETMD), introduced by Donoso and Martens [Phys. Rev. Lett. 2001, 87, 223202] in a form applicable to the treatment of general (i.e., nonpolynomial) potentials. We formulate our approach directly in terms of the integrodifferential equation obeyed by the Wigner function, without assuming a Taylor series expansion of the potential in powers of the coordinate. This alternative formalism has distinct advantages for propagating distribution functions represented by finite trajectory ensembles and for nonpolynomial potentials. We use a numerical implementation of the new approach to calculate the reaction probabilities for three model systems: the cubic polynomial potential, symmetric Eckart barrier and asymmetric Eckart barrier. Our results are in excellent agreement with the results of exact quantum calculation.
In this paper, we extend the entangled trajectory molecular dynamics (ETMD) method to multidimensional systems. The integrodifferential form of the evolution equation for the Wigner function is employed, allowing general potentials not represented as a polynomial to be treated. As the example, the method is applied to a two-dimensional model of scattering from an Eckart barrier. The results of ETMD are in good agreement with quantum hydrodynamics and exact quantum simulations. By comparing the quantum and classical trajectory in phase space, the quantum tunneling phenomenon is interpreted vividly.
In this Letter, we present a new method for simulating quantum processes in the context of classical molecular dynamics simulations. The approach is based on solving numerically the quantum Liouville equation in the Wigner representation using ensembles of classical trajectories. Quantum effects arise in this formulation as a breakdown of the statistical independence of the ensemble. New interaction forces between ensemble members are derived, which require the trajectory ensemble representing the state to evolve as an entangled, unified whole. The method is applied to the simulation of quantum tunneling in a one-dimensional model system, yielding excellent agreement with exact quantum calculations.
We develop an effective quantum force to propagate trajectories in Wigner phase space. In this way, the quantum entanglement between trajectories is represented by the effective quantum force, which is related to the variance of the momentum distribution. Reaction probabilities are obtained for three model potentials. For a cubic potential, the reaction probabilities are found to continuously grow in time long after the classical reaction probabilities have become stationary. This growth is due to tunneling and is in good agreement with numerically exact quantum results. The variance of the momentum distribution is updated at each time step of the trajectories, which is the key to the increase of the reaction probabilities at longer times and allows a nice interpretation of tunneling.
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.
The quantum trajectory method (QTM) for time-dependent wave packet dynamics involves integration of the de Broglie–Bohm hydrodynamic equations for the evolving probability fluid [C. Lopreore and R. E. Wyatt, Phys. Rev. Lett. 82, 5190 (1999)]. The equations of motion for discretized elements of the probability fluid (particles) are integrated in the Lagrangian, moving with the fluid, picture. These fluid elements move under the influence of both the usual potential energy function and the quantum potential, which involves the curvature of the quantum amplitude. The quantum potential and the quantum force are evaluated using a moving weighted least squares algorithm. As a demonstration of applicability, the QTM is applied to a model collinear reaction with an activation barrier. The reaction probabilities are in good agreement with exact quantum results, even for a relatively small number of particles in the ensemble. The QTM accurately describes tunneling using only real valued trajectories. In addition to the reaction probability, plots are presented to show the probability density and the flux distributions at several time steps during the reaction.
A new numerical approach for the calculation of quantum time correlation functions in the canonical and micro-canonical ensembles has been developed within the Wigner formulation of quantum statistical mechanics. For electrons in disordered systems of scatterers the numerical results for the tensor of static electrical conductivity have been obtained according to the quantum Kubo formula in the one-electron approximation in the canonical ensemble. The second quantity considered is the flux position time correlation function characterizing the momentum flow for tunnelling particles at the barrier crossing from one potential well to another. The difficulties with studies of exponentially small effects in the temperature averaging for the canonical ensemble has been overcome by a transition to the micro-canonical ensemble with a fixed total energy. The related integral Wigner-Liouville type equations have been obtained and a numerical approach combining both molecular dynamics and Monte Carlo methods for solving this equation has also been developed. Numerical results have been obtained for the momentum flow of a particle tunnelling through the barrier dividing two potential wells in two- and three-dimensional cases.