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The Journal of Chemical Physics

Bohmian dynamics on subspaces using linearized quantum force

Vitaly A. Rassolov, Sophya V. Garashchuk

Published Mar 29, 2004• 49 citations
Publisher

Abstract

In the de Broglie-Bohm formulation of quantum mechanics the time-dependent Schrodinger equation is solved in terms of quantum trajectories evolving under the influence of quantum and classical potentials. For a practical implementation that scales favorably with system size and is accurate for semiclassical systems, we use approximate quantum potentials. Recently, we have shown that optimization of the nonclassical component of the momentum operator in terms of fitting functions leads to the energy-conserving approximate quantum potential. In particular, linear fitting functions give the exact time evolution of a Gaussian wave packet in a locally quadratic potential and can describe the dominant quantum-mechanical effects in the semiclassical scattering problems of nuclear dynamics. In this paper we formulate the Bohmian dynamics on subspaces and define the energy-conserving approximate quantum potential in terms of optimized nonclassical momentum, extended to include the domain boundary functions. This generalization allows a better description of the non-Gaussian wave packets and general potentials in terms of simple fitting functions. The optimization is performed independently for each domain and each dimension. For linear fitting functions optimal parameters are expressed in terms of the first and second moments of the trajectory distribution. Examples are given for one-dimensional anharmonic systems and for the collinear hydrogen exchange reaction.

DOI
10.1063/1.1669385

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In the de Broglie-Bohm formulation of quantum mechanics the time-dependent Schrodinger equation is solved in terms of quantum trajectories evolving under the influence of quantum and classical potentials. For a practical implementation that scales favorably with system size and is accurate for semiclassical systems, we use approximate quantum potentials. Recently, we have shown that optimization of the nonclassical component of the momentum operator in terms of fitting functions leads to the energy-conserving approximate quantum potential. In particular, linear fitting functions give the exact time evolution of a Gaussian wave packet in a locally quadratic potential and can describe the dominant quantum-mechanical effects in the semiclassical scattering problems of nuclear dynamics. In this paper we formulate the Bohmian dynamics on subspaces and define the energy-conserving approximate quantum potential in terms of optimized nonclassical momentum, extended to include the domain boundary functions. This generalization allows a better description of the non-Gaussian wave packets and general potentials in terms of simple fitting functions. The optimization is performed independently for each domain and each dimension. For linear fitting functions optimal parameters are expressed in terms of the first and second moments of the trajectory distribution. Examples are given for one-dimensional anharmonic systems and for the collinear hydrogen exchange reaction.
Solution of the Schrodinger equation within the de Broglie-Bohm formulation is based on propagation of trajectories in the presence of a nonlocal quantum potential. We present a new strategy for defining approximate quantum potentials within a restricted trial function by performing the optimal fit to the log-derivatives of the wave function density. This procedure results in the energy-conserving dynamics for a closed system. For one particular form of the trial function leading to the linear quantum force, the optimization problem is solved analytically in terms of the first and second moments of the weighted trajectory distribution. This approach gives exact time-evolution of a correlated Gaussian wave function in a locally quadratic potential. The method is computationally cheap in many dimensions, conserves total energy and satisfies the criterion on the average quantum force. Expectation values are readily found by summing over trajectory weights. Efficient extraction of the phase-dependent quantities is discussed. We illustrate the efficiency and accuracy of the linear quantum force approximation by examining a one-dimensional scattering problem and by computing the wavepacket reaction probability for the hydrogen exchange reaction and the photodissociation spectrum of ICN in two dimensions.
We present a time-dependent semiclassical method based on quantum trajectories. Quantum-mechanical effects are described via the quantum potential computed from the wave function density approximated as a linear combination of Gaussian fitting functions. The number of the fitting functions determines the accuracy of the approximate quantum potential (AQP). One Gaussian fit reproduces time-evolution of a Gaussian wave packet in a parabolic potential. The limit of the large number of fitting Gaussians and trajectories gives the full quantum-mechanical result. The method is systematically improvable from classical to fully quantum. The fitting procedure is implemented as a gradient minimization. We also compare AQP method to the widely used semiclassical propagator of Herman and Kluk by computing energy-resolved transmission probabilities for the Eckart barrier from the wave packet time-correlation functions. We find the results obtained with the Herman–Kluk propagator to be essentially equivalent to those of AQP method with a one-Gaussian density fit for several barrier widths.
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.
The quantum nature of nuclei often affects molecular structure and properties, which are associated with the motion of protons and other light nuclei at low temperature. However, incorporation of the nuclear quantum effects into theoretical studies of large molecular systems is an outstanding challenge in theoretical chemistry. In this paper, the de Broglie-Bohm formulation of the time-dependent Schrödinger equation is used as a framework for the development of approximate quantum corrections to the dynamics of a trajectory ensemble, representing a time-dependent wave function. Specifically, the trajectory-centered local Least Square Fit (L-LSF) to the trajectory attributes is introduced, formally analyzed and illustrated on model potentials, typically used to benchmark approximate and semiclassical dynamics methods. The L-LSF method improves the dynamics accuracy, compared to the globally defined approximations, developed previously, and is shown to converge to the exact (generally impractical for large systems) limit of quantum dynamics.
The hydrodynamic, or the de Broglie-Bohm, formulation provides an alternative to the conventional time-dependent Schrödinger equation based on quantum trajectories. The trajectory dynamics scales favorably with the system size, but it is, generally, unstable due to singularities in the exact quantum potential. The approximate quantum potential based on the fitting of the nonclassical component of the momentum operator in terms of a small basis is numerically stable but can lead to inaccurate large net forces in bound systems. We propose to compensate errors in the approximate quantum potential by applying a semiempirical friction-like force. This significantly improves the description of zero-point energy in bound systems. Examples are given for one-dimensional models relevant to nuclear dynamics.
A novel time-dependent hybrid quantum/classical propagation scheme based on Bohmian quantum trajectories is presented. The quantum subsystem is described by a wave packet depending on the quantum variables x and, via the total potential energy of the system, parametrically on the classical trajectories X(t). The wave packet is used to calculate de Broglie–Bohm quantum trajectories x(t) which are used to calculate the force acting on the classical variables. Quantum corrections of the classical equation of motion are also included. The method is applied to a simple case of two coupled oscillators. Comparison between exact quantum and approximate results demonstrates that these MQCB (Mixed Quantum/Classical Bohmian) trajectories provide a good description of the energy exchange between the two oscillators.