  Building on Paper I of this series, which introduced path integral Monte
Carlo (PIMC) estimators for the derivative of the potential of mean force
(PMF), we propose two path integral molecular dynamics (PIMD) integrators that
can make use of these estimators. These integrators, c-OBABO and c-BAOAB, are
based on the path integral Langevin equation (PILE) integrator, which has seen
widespread success in PIMD applications, but they include support for holonomic
constraints. When the reaction coordinate is the distance between two centers
of mass, we find that several exact expressions are accessible: the Fixman
correction, the position constraint Lagrange multiplier, and various
derivatives with respect to the reaction coordinate. It is observed that
c-BAOAB tends to have a smaller time step error than c-OBABO, which is
consistent with previous studies on integrator step ordering in molecular
dynamics with holonomic constraints and in PIMD. Further, we show that both the
PMF of a water dimer and its derivative may be obtained from PIMD simulations
using c-BAOAB, yielding results in agreement with the path integral umbrella
sampling method previously used for this system.
