A test of the possibility of calculating absorption spectra by mixed quantum-classical methods

Abstract

Some of the most efficient methods for studying systems having a large number of degrees of freedom treat a few degrees of freedom quantum mechanically and the remainder classically. Here we examine how these methods fare when used to calculate the cross section for photon absorption by a quantum system imbedded in a medium. To test the method, we study a model which has two degrees of freedom and mimicks the properties of a one-dimensional alkali atom–He dimer. We treat the electron motion quantum mechanically and the distance between the He atom and the alkali ion classically. Light absorption occurs because the electron is coupled to radiation. The calculation of the absorption cross section by quantum-classical methods fails rather dramatically−at certain frequencies, the absorption coefficient is negative. By comparing with exact quantum calculations, we show that this failure takes place because the time evolution of the classical variables influences the dynamics of the quantum degree of freedom through the Hamiltonian only; important information, which a fully quantum treatment would put in the wave function, is missing. To repair this flaw, we experiment with a method which uses a swarm of classical trajectories to generate a ‘‘classical wave function.’’ The results are encouraging, but require substantial computer time when the number of classical variables is large. We argue that in the limit of many classical degrees of freedom, accurate calculations can be performed by using the time-dependent Hartree method and treating some degrees of freedom by exact numerical methods (e.g., a fast Fourier transform procedure) and the others by Gaussian wave packets or any other propagation method that is accurate for a very short time. This procedure leads to a simple time domain picture of dephasing and line broadening in the case of a localized quantum system imbedded in a medium with heavy atoms.