A derivation of centroid molecular dynamics and other approximate time evolution methods for path integral centroid variables

Several methods to approximately evolve path integral centroid variables in real time are presented in this paper, the first of which, the centroid molecular dynamics (CMD) method, is recast into the new formalism of the preceding paper and thereby derived. The approximations involved in the CMD method are thus fully characterized by mathematical derivations. Additional new approaches are also presented: centroid Hamiltonian dynamics (CHD), linearized quantum dynamics (LQD), and a perturbative correction of the LQD method (PT-LQD). The CHD method is shown to be a variation of the CMD method which conserves the approximate time dependent centroid Hamiltonian. The LQD method amounts to a linear approximation for the quantum Liouville equation, while the PT-LQD method includes a perturbative correction to the LQD method. All of these approaches are then tested for the equilibrium position correlation functions of three different one-dimensional nondissipative model systems, and it is shown that certain quantum effects are accounted for by all of them, while others, such as the long time coherence characteristic of low-dimensional nondissipative systems, are not. The CMD method is found to be consistently better than the LQD method, while the PT-LQD method improves the latter and is better than the CMD method in most cases. The CHD method gives results complementary to those of the CMD method.