Integrating the Car-Parrinello equations. II. Multiple time scale techniques

In this paper, new techniques for integrating the Car-Parrinello equations based on multiple time scale methodology are presented. The formalism of multiple time scale methodology based on operator factorizations of the classical Liouville propagator is reviewed. It is shown how the techniques are applied to Car-Parrinello for use with the velocity Verlet and Gaussian dynamics schemes presented in the preceding paper [M. E. Tuckerman and M. Parrinello, J. Chem. Phys. 101, 1302 (1994)], and a detailed discussion is presented of how a reference system for Car-Parrinello simulations may be chosen. It is shown that the use of such techniques can save up to a factor of 5-10 in cpu time over the standard methods.