Multidimensional Classical Liouville Dynamics with Quantum Initial Conditions

Wigner representations of v=4 vibrational eigenfunctions of a harmonic oscillator (left) and of a Morse oscillator (right). Upper: Numerically exact. Lower: Gaussian decomposition

Illia Horenko, Burkhard Schmidt, and Christof Schütte

A simple and numerically efficient approach to Wigner transforms and classical Liouville dynamics in phase-space is presented.

  1. The Wigner transform can be obtained with a given accuracy by optimal decomposition of an initial quantum-mechanical wavefunction in terms of a minimal set of Gaussian wavepackets.

  2. The solution of the classical Liouville equation within the locally quadratic approximation of the potential energy function requires a representation of the density in terms of an ensemble of narrow Gaussian phase-space packets. The corresponding equations of motion can be efficiently solved by a modified Leap-Frog integrator.

For both problems the use of Monte-Carlo based techniques allows numerical calculation in multidimensional cases where grid-based methods such as fast Fourier transforms are prohibitive. In total, the proposed strategy provides a practical and efficient tool for classical Liouville dynamics with quantum-mechanical initial states.

J. Chem. Phys. 117 (10), 4643-4650 (2002)