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The Journal of Chemical Physics

Simulation of quantum processes using entangled trajectory molecular dynamics

Arnaldo Donoso, Yujun Zheng, Craig C. Martens

Published Aug 21, 2003• 85 citations
Publisher

Abstract

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.

DOI
10.1063/1.1597496

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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.
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 demonstrate a different method of simulation of nonstationary quantum processes, considering the tunneling of two interacting identical particles, represented by wave packets. The used method of quantum molecular dynamics (WMD) is based on the Wigner representation of quantum mechanics. In the context of this method ensembles of classical trajectories are used to solve quantum Wigner-Liouville equation. These classical trajectories obey Hamiltonian-like equations, where the effective potential consists of the usual classical term and the quantum term, which depends on the Wigner function and its derivatives. The quantum term is calculated using local distribution of trajectories in phase space, therefore, classical trajectories are not independent, contrary to classical molecular dynamics. The developed WMD method takes into account the influence of exchange and interaction between particles. The role of direct and exchange interactions in tunneling is analyzed. The tunneling times for interacting particles are calculated.
Quantum tunneling effects of Gaussian wave packet in one- and two-dimensional double well potentials are investigated using quantum Liouville equation for time evolution of Wigner distribution in phase space. It is shown that a trajectory-based solution of this problem can be constructed by the entangled trajectory ensemble simulating the evolving quantum state. Quantum effects arise in this approach as a breakdown of the statistical independence of the trajectory ensemble. © 2014 Wiley Periodicals, Inc.
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 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.
Quantum phase space theory is widely used to investigate physical, and chemical questions. It gives us a powerful tool to study the relations between the classical and quantum world. In quantum phase space theory, entangled molecular dynamics method is developed based on quantum Liouville equation. It is different from classical Hamilton dynamics, the trajectories are entangled i.e. not independent with each other. The quantum effect in the system can be described using the semiclassical theory, and give a vivid physical picture. The quantum tunneling, photodissociation, and self-correlation are investigated using the entangled trajectory. In this paper, we review the recent development in this field.