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The Journal of Chemical Physics

An arbitrary Lagrangian–Eulerian approach to solving the quantum hydrodynamic equations of motion: Equidistribution with “smart” springs

Corey Jason Trahan, Róbert E. Wyatt

Published Feb 28, 2003• 58 citations
Publisher

Abstract

Recently, the quantum trajectory method (QTM) has been utilized in solving several quantum mechanical wave packet scattering problems including barrier transmission and electronic nonadiabatic dynamics. By propagating the real-valued action and amplitude functions in the Lagrangian frame, only a fraction of the grid points needed for Eulerian fixed-grid methods are used while still obtaining accurate solutions. Difficulties arise, however, near wave function nodes and in regions of sharp oscillatory features, and because of this many quantum mechanical problems have not yet been amenable to solution with the QTM. This study proposes a hybrid of both the Lagrangian and Eulerian techniques in what is termed the arbitrary Lagrangian–Eulerian method (ALE). In the ALE method, an additional equation of motion governing the momentum of the grid points is coupled into the quantum hydrodynamic equations. These new “quasi-” Bohmian trajectories can be dynamically adapted to the emergent features of the time evolving hydrodynamic fields and are non-Lagrangian. In this study it is shown that the ALE method applied to an uphill ramp potential that was previously unsolvable by the current Lagrangian QTM not only yields stable transmission probabilities with accuracies comparable to that of a high resolution Eulerian method, but does so with a small number of grid points and for extremely long propagation times. To determine the grid point positions at each new time, an equidistribution method is used that is constructed similar to the stiffness matrix of a classical spring system in equilibrium. Each “smart” spring is dependent on a local function M(x) called the monitor function which can sense gradients or curvatures of the fields surrounding its position. To constrain grid points from having zero separation and possible overlap, a new system of equations is derived that includes a minimum separation parameter which prevents this from occurring.

DOI
10.1063/1.1553468

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Recently, the quantum trajectory method (QTM) has been utilized in solving several quantum mechanical wave packet scattering problems including barrier transmission and electronic nonadiabatic dynamics. By propagating the real-valued action and amplitude functions in the Lagrangian frame, only a fraction of the grid points needed for Eulerian fixed-grid methods are used while still obtaining accurate solutions. Difficulties arise, however, near wave function nodes and in regions of sharp oscillatory features, and because of this many quantum mechanical problems have not yet been amenable to solution with the QTM. This study proposes a hybrid of both the Lagrangian and Eulerian techniques in what is termed the arbitrary Lagrangian–Eulerian method (ALE). In the ALE method, an additional equation of motion governing the momentum of the grid points is coupled into the quantum hydrodynamic equations. These new “quasi-” Bohmian trajectories can be dynamically adapted to the emergent features of the time evolving hydrodynamic fields and are non-Lagrangian. In this study it is shown that the ALE method applied to an uphill ramp potential that was previously unsolvable by the current Lagrangian QTM not only yields stable transmission probabilities with accuracies comparable to that of a high resolution Eulerian method, but does so with a small number of grid points and for extremely long propagation times. To determine the grid point positions at each new time, an equidistribution method is used that is constructed similar to the stiffness matrix of a classical spring system in equilibrium. Each “smart” spring is dependent on a local function M(x) called the monitor function which can sense gradients or curvatures of the fields surrounding its position. To constrain grid points from having zero separation and possible overlap, a new system of equations is derived that includes a minimum separation parameter which prevents this from occurring.
Recently, in an effort to solve more realistic problems in quantum dynamics, much attention has been directed into numerically integrating the quantum hydrodynamic equations of motions (QHEM), as opposed to directly solving the time-dependent Schrödinger equation (TDSE). Such efforts have been provoked by the many numerical drawbacks encountered when solving the TDSE on a fixed-grid. In this dissertation, one trajectory method for integrating the QHEM is reviewed, and two novel trajectories methods are described. The first of these, the quantum trajectory method (QTM), was introduced in 1999 and has been used to solve many problems in quantum dynamics since then. However, severe numerical problems are encountered when this method is applied to problems that form wave function nodes. To get around this problem, new methods for numerically integrating the QHEM are needed. In the first novel method described, the arbitrary Lagrangian-Eulerian (ALE) method, particle trajectories are governed by a predetermined equation of motion that is user-supplied. The ALE method remedies inflation and compression problems encountered in the pure Lagrangian QTM. In the second new method discussed, the derivative propagating method (DPM), single quantum trajectories can be calculated one at a time, as opposed to the ensemble propagation of the QTM and ALE method. Using these two methods, new solutions to the QHEM are obtained where the QTM fails. In addition to solving the QHEM, the DPM is also used to solve the classical Klein-Kramers equation in this dissertation. This equation governs the Markovian phase space evolution of a system coupled to an environment such as a heat bath. This marks the first time single trajectories have been used to solve both the QHEM and the Klein-Kramers equations.
The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.
The quantum hydrodynamic equations associated with the de Broglie–Bohm formulation of quantum mechanics are solved using a meshless method based on a moving least squares approach. An arbitrary Lagrangian–Eulerian frame of reference is used which significantly improves the accuracy and stability of the method when compared to an approach based on a purely Lagrangian frame of reference. A regridding algorithm is implemented which adds and deletes points when necessary in order to maintain accurate and stable calculations. It is shown that unitarity in the time evolution of the quantum wave packet is significantly improved by propagating using averaged fields. As nodes in the reflected wave packet start to form, the quantum potential and force become very large and numerical instabilities occur. By introducing artificial viscosity into the equations of motion, these instabilities can be avoided and the stable propagation of the wave packet for very long times becomes possible. Results are presented for the scattering of a wave packet from a repulsive Eckart barrier.
The quantum hydrodynamic approach is based on the de Broglie-Bohm formulation of quantum mechanics. The resulting fluid-like equations of motion describe the flow of probability and an accurate solution to these equations is equivalent to solving the time-dependent Schroedinger equation. Furthermore, the hydrodynamic approach provides new insight into the mechanisms as well as an alternative computational approach for treating tunneling phenomena. New concepts include well-defined 'quantum trajectories', 'quantum potential', and 'quantum force' all of which have classical analogues. The quantum potential and its associated force give rise to all quantum mechanical effects such as zero point energy, tunneling, and interference. A new numerical approach called the Iterative Finite Difference Method (IFDM) will be discussed. The IFDM is used to solve the set of non-linear coupled hydrodynamic equations. It is 2nd-order accurate in both space and time and exhibits exponential convergence with respect to the iteration count. The stability and computational efficiency of the IFDM is significantly improved by using a 'smart' Eulerian grid which has the same computational advantages as a Lagrangian or Arbitrary Lagrangian Eulerian (ALE) grid. The IFDM is also capable of treating anharmonic potentials. Example calculations using the IFDM will be presented which include: a one-dimensional Gaussian wave packet tunneling through an Eckart barrier, a one-dimensional bound-state Morse oscillator, and a two-dimensional (2D) model collinear reaction using an anharmonic potential energy surface. Approximate treatments of the quantum hydrodynamic equations will also be discussed which could allow scaling of the calculations to hundreds of degrees of freedom which is important for treating tunneling phenomena in condensed phase systems.
The quantum trajectory method (QTM) was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for N fluid elements (particles) moving under the influence of both the force from the potential surface and from the quantum potential. In this study, distributed approximating functionals (DAFs) are used on a uniform grid to compute the necessary derivatives in the equations of motion. Transformations between the physical grid where the particle coordinates are defined and the uniform grid are handled through a Jacobian, which is also computed using DAFs. A difficult problem associated with computing derivatives on finite grids is the edge problem. This is handled effectively by using DAFs within a least squares approach to extrapolate from the known function region into the neighboring regions. The QTM–DAF is then applied to wave packet transmission through a one-dimensional Eckart potential. Emphasis is placed upon computation of the transmitted density and wave function. A problem that develops when part of the wave packet reflects back into the reactant region is avoided in this study by introducing a potential ramp to sweep the reflected particles away from the barrier region.
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.