Generalized quantum Liouville equation: Its solution by wave packet dynamics

A generalized quantum Liouville equation of the form iℏ(∂ρ̂/∂t)=Ĥuρ̂−ρ̂Ĥl for the density operator ρ̂(t) is introduced; the quantum mechanical Hamiltonians Ĥu and Ĥl are, in general, different operators. Such an equation is of interest for a variety of problems involving systems at finite temperatures, including the calculation of vibrational and electronic spectra of molecules that are initially distributed over a range of eigenstates. It is shown that the generalized Liouville equation can be solved by exploiting its equivalence to a time-dependent Schrödinger equation in the coordinate space representation. In particular, this equivalence makes it possible to utilize techniques of Schrödinger wave packet propagation to compute the time evolution of the desired operator. Application of Gaussian wave packet dynamics and its extensions is considered and shown to be justified when the coordinate representation of ρ̂(t) remains essentially Gaussian throughout the course of the relevant dynamics. As an illustration of the method, the finite temperature electronic absorption of a model molecule with two degrees of freedom is calculated. Other applications of the method are discussed.