Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation.

It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.