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The Journal of Physical Chemistry A

Forward−Backward Quantum Dynamics for Time Correlation Functions

Nancy Makri

Published Jan 7, 2004• 36 citations
Publisher

Abstract

A novel quantum mechanical formulation of time correlation functions is derived, based on Bohmian trajectories integrated along a forward−backward time contour. The derived expression involves a smooth integrand amenable to Monte Carlo integration with a readily available position space density. The Bohmian potential governing the motion of the quantum trajectories is weak and well-behaved, facilitating the computation of Bohmian trajectories.

DOI
10.1021/jp0308615

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A novel quantum mechanical formulation of time correlation functions is derived, based on Bohmian trajectories integrated along a forward−backward time contour. The derived expression involves a smooth integrand amenable to Monte Carlo integration with a readily available position space density. The Bohmian potential governing the motion of the quantum trajectories is weak and well-behaved, facilitating the computation of Bohmian trajectories.
It is shown that the quantum force in the Bohmian formulation of quantum mechanics can be related to the stability properties of the given trajectory. In turn, the evolution of the stability properties is governed by higher order derivatives of the quantum potential, leading to an infinite hierarchy of coupled differential equations whose solution specifies completely all aspects of the dynamics. Neglecting derivatives of the quantum potential beyond a certain order allows truncation of the hierarchy, leading to approximate Bohmian trajectories. Use of the method in conjunction with Bohmian initial value formulations [ J. Chem. Phys. 2003, 119, 60] gives rise to simple position-space representations of observables or time correlation functions. These are analogous to approximate quasiclassical expressions based on the Wigner or Husimi phase space density but involve lower dimensional integrals with smoother integrands and avoid the costly evaluation of phase space transforms. The lowest-order version of the truncated hierarchy can capture large corrections to classical mechanical treatments and yields (with fewer trajectories) results that are somewhat more accurate than those based on quasiclassical phase space treatments.
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.
We report on a Monte Carlo method that generates one-dimensional trajectories for Bohm's formulation of quantum mechanics that doesn't involve differentiation or integration of any equations of motion. At each time, t=nδt (n=1,2,3,...), N particle positions are randomly sampled from the quantum probability density. Trajectories are built from the sorted N sampled positions at each time. These trajectories become the exact Bohm solutions in the limits N->\infty and δt -> 0. Higher dimensional problems can be solved by this method for separable wave functions. Several examples are given, including the two-slit experiment.
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.
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.
A rigorous, yet practical semiclassical formulation of time correlation functions or expectation values is presented. The main idea is to combine the forward and backward propagation steps into a single semiclassical propagator for those degrees of freedom that are not probed in a calculation, while retaining an explicit two-propagator description of the observable low-dimensional system. The combined forward-backward treatment of the environment naturally leads to extensive cancellation and results in an action that is small on the scale of Planck's constant. As a consequence, the integrand is smooth and thus amenable to Monte Carlo sampling. At the same time, important nonclassical effects terms, arising from interference among multiple bounce solutions of the system component, are fully accounted for.