In this paper we develop a new approach to semiclassical dynamics which exploits the fact that extended wavefunctions for heavy particles (or particles in harmonic potentials) may be decomposed into time−dependent wave packets, which spread minimally and which execute classical or nearly classical trajectories. A Gaussian form for the wave packets is assumed and equations of motion are derived for the parameters characterizing the Gaussians. If the potential (which may be nonseparable in many coordinates) is expanded in a Taylor series about the instantaneous center of the (many−particle) wave packet, and up to quadratic terms are kept, we find the classical parameters of the wave packet (positions, momenta) obey Hamilton’s equation of motion. Quantum parameters (wave packet spread, phase factor, correlation terms, etc.) obey similar first order quantum equations. The center of the wave packet is shown to acquire a phase equal to the action integral along the classical path. State−specific quantum information is obtained from the wave packet trajectories by use of the superposition principle and projection techniques. Successful numerical application is made to the collinear He + H2 system widely used as a test case. Classically forbidden transitions are accounted for and obtained in the same manner as the classically allowed transitions; turning points present no difficulties and flux is very nearly conserved.

Time‐dependent approach to semiclassical dynamics

A fourier method solution for the time dependent Schrödinger equation as a tool in molecular dynamics

The semiclassical (SC) initial value representation (IVR) provides a potentially practical way for adding quantum mechanical effects to classical molecular dynamics (MD) simulations of the dynamics of complex molecular systems (i.e., those with many degrees of freedom). It does this by replacing the nonlinear boundary value problem of semiclassical theory by an average over the initial conditions of classical trajectories. This paper reviews the background and rebirth of interest in such approaches and surveys a variety of their recent applications. Special focus is on the ability to treat the dynamics of complex systems, and in this regard, the forward−backward (FB) version of the approach is especially promising. Several examples of the FB-IVR applied to model problems of many degrees of freedom show it to be capable of describing quantum effects quite well and also how these effects are quenched when some of the degrees of freedom are averaged over (“decoherence”).

The Semiclassical Initial Value Representation: A Potentially Practical Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations

Theory and application of the quantum phase-space distribution functions

A new method for solving the close coupled equations of inelastic scattering is presented. The method is based on Johnson’s log derivative algorithm, and uses the same quadrature for the solution of the corresponding integral equations. However it differs from the original method in the use of a piecewise constant diagonal reference potential. This results in a reduction in matrix operations at subsequent energies, and an improved convergence of the solution with respect to the number of grid points. These advantages are clearly demonstrated when our method is applied to an atom–diatom rotational excitation problem.

An improved log derivative method for inelastic scattering

Exact quantum‐mechanical calculations of the transition probabilities for the collinear collision of an atom with a diatomic molecule are performed. The diatomic molecule is treated as a harmonic oscillator. A range of interaction potentials from very hard to very soft are considered. It is found that for ``realistic'' interaction potentials the approximate calculations of Jackson and Mott are consistently high, even when the transition probabilities are low and good approximate results are expected. In some cases double and even triple quantum jumps are more important than single quantum jumps. Comparisons are made with exact classical calculations. A semiempirical formula is given for computing quantum‐mechanical transition probabilities from classical calculations.

Exact Quantum-Mechanical Calculation of a Collinear Collision of a Particle with a Harmonic Oscillator

Using the method of amplitude densities, we derive the equations for computing the scattering T matrix in the total‐angular‐momentum representation. We also show that in the appropriate limit this method reduces to a first‐order differential equation for the T matrix. In an Appendix a similar discussion is presented for the reaction matrix. Using the T‐matrix approach, we have obtained numerical solutions to the helium‐atom–hydrogen‐molecule scattering problem in both the close‐coupling and distorted‐wave approximations. Cross sections were computed for the J = 0 to J = 2 transitions in parahydrogen and the J = 1 to J = 3 transition in orthohydrogen. These results were calculated using both an interaction potential computed by Roberts and a potential computed by Krauss and Mies. The close‐coupling and distorted‐wave results are compared, and it is found that, in general, the distorted‐wave cross sections are about 20% too high when Roberts potential is used and are about 10% too high when the less anisotropic Krauss and Mies potential is used.

Quantum‐Mechanical Calculations of the Inelastic Cross Sections for Rotational Excitation of Para and Ortho H2 upon Collision with He

The method of ``amplitude density functions'' is a new formalism which allows a scattering problem to be broken up into problems with weaker interaction potentials. These simpler problems may be solved separately and be ``added'' together to give the total solution. A numerical method is discussed which takes advantage of this property. The formulas are given for the use of this method in the solution of the one‐dimensional atom‐molecule collision and the e+−H collision. A numerical example is discussed.

B. Robert Johnson

Papers

Exact Quantum-Mechanical Calculation of a Collinear Collision of a Particle with a Harmonic Oscillator

Quantum‐Mechanical Calculations of the Inelastic Cross Sections for Rotational Excitation of Para and Ortho H2 upon Collision with He

The Solution of the Nonrelativistic Quantum Scattering Problem without Exchange