The formulation of quantum statistical mechanics based on the Feynman path centroid density. III. Phase space formalism and analysis of centroid molecular dynamics

The formulation of quantum statistical mechanics based on the path centroid variable in Feynman path integration is generalized to a phase space perspective, thereby including the momentum as an independent dynamical variable. By virtue of this approach, operator averages and imaginary time correlation functions can be expressed in terms of an averaging over the multidimensional phase space centroid density. The imaginary time centroid-constrained correlation function matrix for the phase space variables is then found to define the effective thermal width of the phase space centroid variable. These developments also make it possible to rigorously analyze the centroid molecular dynamics method for computing quantum dynamical time correlation functions. As a result, the centroid time correlation function as calculated from centroid molecular dynamics is shown to be a well-defined approximation to the exact Kubo transformed position correlation function. This analysis thereby clarifies the underlying role of the equilibrium path centroid variable in the quantum dynamical position correlation function and provides a sound theoretical basis for the centroid molecular dynamics method.