Gaussian wave packet dynamics expressed in the classical interaction picture

Rex T. Skodje

M. L. Glasser

Kazumoto Ohnami, Kazuya Miyazaki, Y. Mikami

The interaction picture wave packet method is applied to hydrogen scattering with a model Gaussian potential barrier, the linear reaction model. Some new features for the behaviour of the interaction picture wave packet are pointed out. It also splits into two parts according to the separation of the original Schrödinger picture wave packet into transmitted and reflected parts. The interrelation between the two wave packets is discussed in detail. In contrast with the Schrödinger picture wave packet, the reflected part is so clearly defined that we can evaluate the reflection probability as a function of time.

Andrey Pereverzev, Eric R. Bittner

Jakob Petersen, Kenneth G. Kay

Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.

Michael F. Herman, D. F. Coker

The relationship between the diverging of classical trajectories in chaotic many-body systems, the spreading of quantum wave packets, and the validity and use of classical molecular dynamics is explored. This analysis, which is based on the semiclassical description of wave function propagation in terms of a weighted integration over a traveling fixed width coherent state basis, suggests that the exponential divergence of nearby classical trajectories in chaotic many-body systems should result in the rapid delocalization of an initially localized quantum wave packet describing the state of the system. Thus the justification for the use of classical molecular dynamics procedures for these supposedly classical systems cannot be based on the picture of the system wave function remaining localized as its center follows a nearly classical trajectory. The quantum evolution of the system density, on the other hand, requires two propagators, and each of these propagators is represented as an integration over trajectories in the semiclassical picture. The interference between the contributions from these two integrations over classical trajectories focuses the analysis on the most important points in this trajectory pair space, which are shown to occur when both trajectories in the pair are the same. Given reasonable assumptions for the initial density for a system that is expected to be well described by classical molecular dynamics, and given an appropriate choice for the width of the coherent state basis which is employed in the semiclassical description, it is shown that the semiclassical expressions for time dependent observables and correlation functions reduce the purely classical expressions, despite the fact that an initially localized wave packet would rapidly delocalize for the same system.