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Building on Paper I of this series, which introduced path integral MonteCarlo (PIMC) estimators for the derivative of the potential of mean force(PMF), we propose two path integral molecular dynamics (PIMD) integrators thatcan make use of these estimators. These integrators, c-OBABO and c-BAOAB, arebased on the path integral Langevin equation (PILE) integrator, which has seenwidespread success in PIMD applications, but they include support for holonomicconstraints. When the reaction coordinate is the distance between two centersof mass, we find that several exact expressions are accessible: the Fixmancorrection, the position constraint Lagrange multiplier, and variousderivatives with respect to the reaction coordinate. It is observed thatc-BAOAB tends to have a smaller time step error than c-OBABO, which isconsistent with previous studies on integrator step ordering in moleculardynamics with holonomic constraints and in PIMD. Further, we show that both thePMF of a water dimer and its derivative may be obtained from PIMD simulationsusing c-BAOAB, yielding results in agreement with the path integral umbrellasampling method previously used for this system.

On the quantum mechanical potential of mean force. II. Constrained path integral molecular dynamics integrators