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

TL;DR

This paper introduces two new path integral molecular dynamics integrators, c-OBABO and c-BAOAB, that incorporate holonomic constraints and enable efficient computation of the potential of mean force and its derivatives.

Contribution

The paper develops and compares two constrained PIMD integrators, demonstrating their effectiveness in calculating PMFs and derivatives, with c-BAOAB showing improved accuracy over c-OBABO.

Findings

c-BAOAB has smaller time step errors than c-OBABO.

Both integrators accurately compute the PMF of a water dimer.

Results agree with previous path integral umbrella sampling studies.

Abstract

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.