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International Journal of Quantum Chemistry

Two‐dimensional wavepacket dynamics with quantum hydrodynamics

Daisuke Matsumoto, K. Hayashi, Tomonori Ida, Motohiro Mizuno, Kazuhiko Endo, Kiyoshi Nishikawa

Published Jan 1, 2007• 6 citations
Publisher

Abstract

Abstract Wyatt and coworkers introduced a new approach (Wyatt, Quantum Dynamics with Trajectories Introduction to Quantum Hydrodynamics, Springer, New York, 2005; Lopreore and Wyatt, Phys Rev Lett, 1999, 82, 5190; Wyatt et al., J Chem Phys, 2001, 114, 5113; Lopreore and Wyatt, J Chem Phys, 2002, 116, 1228; Trahan and Wyatt, J Chem Phys, 2003, 118, 4784; Trahan et al., J Chem Phys, 2005, 122, 164104; Babyuk and Wyatt, J Chem Phys, 2006, 125, 64112) which is based on quantum hydrodynamics to the quantum wavepacket dynamics and applied to several systems including tunneling and nonadiabatic transition. This approach is expected to overcome some disadvantages of the conventional wavepacket dynamics, such as order‐N problem and use of absorbing potential. We applied this method to two‐dimensional model of quantum tunneling system and compared the new approach with the conventional wavepacket simulation since the validity of this method to multidimensional systems is not well known. In the simulation, we adopted two different model potentials (Model I and Model II) and calculated time dependence of the probability density and the tunneling probability in both models, where we showed explicitly that the results of the quantum hydrodynamics are identical with those of the conventional method. Using the moving grid and the regridding algorithm in the quantum hydrodynamics, we can effectively avoid order‐N problem and use of the absorbing potential. In addition to these computational advantages, the flux vector is easily obtained by the quantum hydrodynamics and these vectors field helps us to intuitively understand detailed tunneling dynamics. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

DOI
10.1002/qua.21470

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Abstract Wyatt and coworkers introduced a new approach (Wyatt, Quantum Dynamics with Trajectories Introduction to Quantum Hydrodynamics, Springer, New York, 2005; Lopreore and Wyatt, Phys Rev Lett, 1999, 82, 5190; Wyatt et al., J Chem Phys, 2001, 114, 5113; Lopreore and Wyatt, J Chem Phys, 2002, 116, 1228; Trahan and Wyatt, J Chem Phys, 2003, 118, 4784; Trahan et al., J Chem Phys, 2005, 122, 164104; Babyuk and Wyatt, J Chem Phys, 2006, 125, 64112) which is based on quantum hydrodynamics to the quantum wavepacket dynamics and applied to several systems including tunneling and nonadiabatic transition. This approach is expected to overcome some disadvantages of the conventional wavepacket dynamics, such as order‐N problem and use of absorbing potential. We applied this method to two‐dimensional model of quantum tunneling system and compared the new approach with the conventional wavepacket simulation since the validity of this method to multidimensional systems is not well known. In the simulation, we adopted two different model potentials (Model I and Model II) and calculated time dependence of the probability density and the tunneling probability in both models, where we showed explicitly that the results of the quantum hydrodynamics are identical with those of the conventional method. Using the moving grid and the regridding algorithm in the quantum hydrodynamics, we can effectively avoid order‐N problem and use of the absorbing potential. In addition to these computational advantages, the flux vector is easily obtained by the quantum hydrodynamics and these vectors field helps us to intuitively understand detailed tunneling dynamics. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
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.
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.
In this paper we compute quantum trajectories arising from Bohm’s causal description of quantum mechanics. Our computational methodology is based upon a finite-element moving least-squares method (MWLS) presented recently by Wyatt and co-workers [Lopreore and Wyatt, Phys. Rev. Lett. 82, 5190 (1999)]. This method treats the “particles” in the quantum Hamilton–Jacobi equation as Lagrangian fluid elements that carry the phase, S, and density, ρ, required to reconstruct the quantum wave function. Here, we compare results obtained via the MWLS procedure to exact results obtained either analytically or by numerical solution of the time-dependent Schrödinger equation. Two systems are considered: first, dynamics in a harmonic well and second, tunneling dynamics in a double well potential. In the case of tunneling in the double well potential, the quantum potential acts to lower the barrier, separating the right- and left-hand sides of the well, permitting trajectories to pass from one side to another. However, as probability density passes from one side to the other, the effective barrier begins to rise and eventually will segregate trajectories in one side from the other. We note that the MWLS trajectories exhibited long time stability in the purely harmonic cases. However, this stability was not evident in the barrier crossing dynamics. Comparisons to exact trajectories obtained via wave packet calculations indicate that the MWLS trajectories tend to underestimate the effects of constructive and destructive interference effects.
The de Broglie-Bohm hydrodynamic equations of motion are solved using a meshless method based on a moving least squares approach and an arbitrary Lagrangian-Eulerian frame of reference. A regridding algorithm adds and deletes computational points as needed in order to maintain a uniform interparticle spacing, and unitary time evolution is obtained by propagating the wave packet using averaged fields. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. The methodology is applied to a two-dimensional model collinear reaction with an activation barrier. Reaction probabilities are computed as a function of both time and energy, and are in excellent agreement with those based on the quantum trajectory method.
The hydrodynamic equations of motion associated with the de Broglie-Bohm formulation of quantum mechanics are solved using a meshless method based upon a moving least-squares approach. An arbitrary Lagrangian-Eulerian frame of reference and a regridding algorithm which adds and deletes computational points are used to maintain a uniform and nearly constant interparticle spacing. The methodology also uses averaged fields to maintain unitary time evolution. The numerical instabilities associated with the formation of nodes in the reflected portion of the wave packet are avoided by adding artificial viscosity to the equations of motion. A new and more robust artificial viscosity algorithm is presented which gives accurate scattering results and is capable of capturing quantum resonances. The methodology is applied to a one-dimensional model chemical reaction that is known to exhibit a quantum resonance. The correlation function approach is used to compute the reactive scattering matrix, reaction probability, and time delay as a function of energy. Excellent agreement is obtained between the scattering results based upon the quantum hydrodynamic approach and those based upon standard quantum mechanics. This is the first clear demonstration of the ability of moving grid approaches to accurately and robustly reproduce resonance structures in a scattering system.
The quantum hydrodynamic equations associated with the de Broglie-Bohm formulation of quantum mechanics are solved using a new methodology which gives an accurate, unitary, and stable propagation of a time dependent quantum wave packet [B. K. Kendrick, J. Chem. Phys. 119, 5805 (2003)]. The methodology is applied to an N-dimensional model chemical reaction with an activation barrier. A parallel version of the methodology is presented which is designed to run on massively parallel supercomputers. The computational scaling properties of the parallel code are investigated both as a function of the number of processors and the dimension N. A decoupling scheme is introduced which decouples the multidimensional quantum hydrodynamic equations into a set of uncoupled one-dimensional problems. The decoupling scheme dramatically reduces the computation time and is highly parallelizable. Furthermore, the computation time is shown to scale linearly with respect to the dimension N=2,...,100.