# A local-Gaussian approximation for the propagation of a classical distribution function

## Abstract

A simple approximation method is described for the propagation of a classical Gaussian distribution function. This method is the classical analog of Heller's well-known scheme for the propagation of Gaussian wave packets [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)]. The classical version has the merit that thermal spreading of the distribution function can be accounted for. A simple method is proposed for treating the combined effects of quantum-mechanical and thermal spreading. In particular, it is shown that the classical approach can be used for an arbitrary initial temperature; it is only necessary to correctly specify the initial width of the distribution in configuration space and the initial root-mean-square momentum spread. Also, an improved method is presented which conserves the total energy of the system. The local-Gaussian method is useful when the force field is approximately linear over the width of the distribution function. In general, this criterion is more readily satisfied at low temperatures, where thermal spreading is not too severe, but where quantum effects can become important. The local-Gaussian method is applied to the dissociation of diatomic molecules by a sudden electronic transition, where a well-defined initial distribution function is propagated on a repulsive potential-energy curve. A comparison with "exact" classical and quantum-mechanical calculations is made. © 1991 American Institute of Physics.