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The Journal of Chemical Physics

Time correlation functions via forward-backward quantum dynamics using Hamilton’s law of varying action

Jonathan Chen, Nancy Makri

Published Sep 23, 2009• 3 citations
Publisher

Abstract

We introduce a stable numerical procedure for solving Bohm's equations of motion to compute quantum trajectories in the forward-backward quantum dynamics (FBQD) formulation of zero-temperature time correlation functions. Rather than integrating the differential equations forward in time, our method is based on a series expansion of the quantum trajectory, exploiting Hamilton's law of varying action to determine the expansion coefficients. Because in FBQD the quantum trajectories generally are smooth and the quantum potential is well behaved, our method allows accurate determination of time correlation functions in strongly anharmonic bound systems over several oscillation periods.

DOI
10.1063/1.3224494

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We introduce a stable numerical procedure for solving Bohm's equations of motion to compute quantum trajectories in the forward-backward quantum dynamics (FBQD) formulation of zero-temperature time correlation functions. Rather than integrating the differential equations forward in time, our method is based on a series expansion of the quantum trajectory, exploiting Hamilton's law of varying action to determine the expansion coefficients. Because in FBQD the quantum trajectories generally are smooth and the quantum potential is well behaved, our method allows accurate determination of time correlation functions in strongly anharmonic bound systems over several oscillation periods.
Several theoretical methods for the computation of quantum dynamical quantities are formulated, implemented, and applied, in the overall context of overcoming the exponential scaling property of quantum mechanics. The convergence of the approximate forward-backward semiclassical dynamics (FBSD) method has been accelerated by working with only the imaginary part of the real-time position and momentum time correlation functions. Simple manipulations allow the computationally favorable limit of a single factorization (“bead”) of the Boltzmann operator to yield greatly improved results. Numerical examples for a one-dimensional anharmonic oscillator and Lennard-Jones fluids are presented. Bohm's trajectory formulation of quantum mechanics is an exact mapping of the time-dependent Schrödinger equation onto classical fluid dynamics, plus one extra term responsible for all non-classical effects, the dreaded quantum potential. A computational procedure has been introduced to stably solve Bohm's hydrodynamic equations of motion in the forward-backward quantum dynamics (FBQD) formulation of time correlation functions, and applied to one-dimensional bound systems at zero and low temperature. A key step in the formulation of this procedure was the use of Hamilton's law of varying action, a generalization of Hamilton's principle of stationary action applicable to initial-value problems. This has resulted in the first reported synthetic computation of non-stationary quantum trajectories for a fully anharmonic oscillator. Success in one dimension has laid the groundwork for future extension to a rigorous quantum-classical scheme, which is briefly discussed. Other unpublished methods for obtaining time correlation functions and propagators in quantum dynamics are also discussed.
The de Broglie-Bohm formulation of the Schrodinger equation implies conservation of the wave function probability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase and amplitude near each trajectory similar to expansions used in semiclassical methods. Computation of the quantities of interest in this procedure is linear with respect to the number of trajectories. The introduced approximations are consistent with approximate quantum potential dynamics method. The projection technique is applied to model chemical systems and to the H+H(2) exchange reaction in three dimensions.
Bohmian mechanics, developed by Bohm in 1952, provides an alternative interpretation to nonrelativistic quantum mechanics [1-3]. In the hydrodynamic formulation of quantum mechanics, the continuity equation and the quantum Hamilton-Jacobi equation (QHJE) are obtained by substituting the wave function expressed in terms of the real amplitude and the real action function into the time-dependent Schrödinger equation. Bohm’s analytical approach has been used to compute and interpret realvalued quantum trajectories from a precomputed wave function for a diverse range of physical processes [3-8]. In the synthetic approach, the quantum trajectory method has been developed as a computational tool to generate the wave function by evolving ensembles of real-valued quantum trajectories through the integration of the hydrodynamic equations on the fly [9]. Remarkable progress has been made in the use of T&F Cat # K11224, Chapter 18, Page 284, 15-9-2010 real-valued quantum trajectories for solving a wide range of quantum mechanical problems [10].
Quantum trajectories, originating from the de Broglie–Bohm hydrodynamic description of quantum mechanics, are used to construct time-correlation functions in an initial value representation. The formulation is fully quantum mechanical and the resulting equations for the correlation functions are similar in form to their semiclassical analogs but do not require the computation of the stability or monodromy matrix or conjugate points. We then move to a local trajectory description by evolving the cumulants of the wave function along each individual path. The resulting equations of motion are an infinite hierarchy, which we truncate at a given order. We show that time-correlation functions computed using these approximate quantum trajectories can be used to accurately compute the eigenvalue spectrum for various potential systems.
In this paper we present a new quantum-trajectory based treatment of quantum dynamics suitable for dissipative systems. Starting from a de Broglie/Bohm-like representation of the quantum density matrix, we derive and define quantum equations-of-motion for Liouville-space trajectories for a generalized system coupled to a dissipative environment. Our theory includes a vector potential which mixes forward and backwards propagating components and non-local quantum potential which continuously produces coherences in the system. These trajectories are then used to propagate an adaptive Lagrangian grid which carries the density matrix, ρ(x,y)ρ(x,y)ρ(x,y), and the action, A(x,y)A(x,y)A(x,y), thereby providing a complete hydrodynamic-like description of the dynamics.