Corrections to Wigner type phase space methods
  Over decades, the time evolution of Wigner functions along classical Hamiltonian flows has been used for approximating key signatures of molecular quantum systems. Such approximations are for example the Wigner phase space method, the linearized semiclassical initial value representation, or the statistical quasiclassical method. The mathematical backbone of these approximations is Egorov's theorem. In this paper, we reformulate the well-known second order correction to Egorov's theorem as a system of ordinary differential equations and derive an algorithm with improved asymptotic accuracy for the computation of expectation values. For models with easily evaluated higher order derivatives of the classical Hamiltonian, the new algorithm's corrections are computationally less expensive than the leading order Wigner method. Numerical test calculations for a two-dimensional torsional system confirm the theoretical accuracy and efficiency of the new method. 
Wolfgang Gaim and Caroline Lasser
math.NA