Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Chemical Physics Letters

Wavepacket dynamics on dynamically adapting grids: application of the equidistribution principle

Keith H. Hughes, Róbert E. Wyatt

Published Nov 11, 2002• 41 citations
Publisher
DOI
10.1016/s0009-2614(02)01654-8

Related papers

Wave packet dynamics with adaptive grids: The moving boundary truncation method

Lucas R. Pettey, Róbert E. Wyatt

Chemical Physics Letters•May 2, 2006•24 citations

Application of Wavepacket Dynamics: Illuminating Examples

IISc lecture notes series•Nov 1, 2023•0 citations

Wavepacket Dynamics on Dynamically Adapting Grids: Application of the Equidistribution Principle

Keith H. Hughes, Róbert E. Wyatt

A moving grid approach to wavepacket dynamics is described that enables grid points to be used efficiently in regions where high resolution of the wavepacket is required. The grid movement is based on the principle of equidistribution and by using a grid smoothing technique the grid points trace a path that continuously adapt to reflect the dynamics of the wavepacket. The technique is robust and allows accurate computations to be obtained for long wavepacket propagation times. Results are presented for two systems: tunnelling dynamics in a double well potential and scattering of a wavepacket from a repulsive Eckart barrier.

APS•Mar 1, 2003•0 citations

Classical Aspects of Wavepacket Dynamics

Mordechai Bixon

Jerusalem Symposia on Quantum Chemistry and Biochemistry•Jan 1, 1982•3 citations

Dissipative quantum phase space dynamics on dynamically adapting grids

Keith H. Hughes

A moving grid approach to a dynamical study of dissipative systems is described. The dynamics are studied in phase space for the Caldeira-Leggett master equation. The grid movement is based on the principle of equidistribution and, by using a grid smoothing technique, the grid points trace a path that continuously adapts to reflect the dynamics of a phase-space distribution function. The technique is robust and allows accurate computations to be obtained for long propagation times. The effects of dissipation on the dynamics are studied and results are presented for systems subject to both periodic and nonperiodic multiminimum potential functions.

The Journal of Chemical Physics•Feb 8, 2005•10 citations

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

Corey Jason Trahan, Róbert E. Wyatt

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.

The Journal of Chemical Physics•Feb 28, 2003•58 citations
A moving grid approach to wavepacket dynamics is described that enables grid points to be used efficiently in regions where high resolution of the wavepacket is required. The grid movement is based on the principle of equidistribution and by using a grid smoothing technique the grid points trace a path that continuously adapt to reflect the dynamics of the wavepacket. The technique is robust and allows accurate computations to be obtained for long wavepacket propagation times. Results are presented for two systems: tunnelling dynamics in a double well potential and scattering of a wavepacket from a repulsive Eckart barrier.
A moving grid approach to a dynamical study of dissipative systems is described. The dynamics are studied in phase space for the Caldeira-Leggett master equation. The grid movement is based on the principle of equidistribution and, by using a grid smoothing technique, the grid points trace a path that continuously adapts to reflect the dynamics of a phase-space distribution function. The technique is robust and allows accurate computations to be obtained for long propagation times. The effects of dissipation on the dynamics are studied and results are presented for systems subject to both periodic and nonperiodic multiminimum potential functions.
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.