Dimension-free path-integral molecular dynamics without preconditioning

Abstract

Convergence with respect to imaginary-time discretization is an essential part of any path-integral-based calculation. However, an unfortunate property of existing non-preconditioned numerical integration schemes for path-integral molecular dynamics (PIMD) - including ring-polymer molecular dynamics (RPMD) and thermostatted RPMD (T-RPMD) - is that for a given MD timestep, the overlap between the exact ring-polymer Boltzmann-Gibbs distribution and that sampled using MD becomes zero in the infinite-bead limit. This has clear implications for hybrid Metropolis Monte-Carlo/MD sampling schemes. We show that these problems can be avoided through the introduction of "dimension-free" numerical integration schemes for which the sampled ring-polymer position distribution has non-zero overlap with the exact distribution in the infinite-bead limit for the case of a harmonic potential. We show that dimension freedom can be achieved via mollification of the forces from the physical potential and with the BCOCB integration scheme. The dimension-free numerical integration schemes yield finite error bounds for a given MD timestep as the number of beads is taken to infinity; these conclusions are proven for harmonic potential and borne out numerically for anharmonic systems, including water. The numerical results for BCOCB are particularly striking, allowing for three-fold increases in the stable timestep for liquid water with respect to the Bussi-Parrinello (OBABO) and Leimkuhler (BAOAB) integrators while introducing negligible errors in the statistical properties and absorption spectrum. Importantly, the dimension-free, non-preconditioned integration schemes introduced here preserve ergodicity and global second-order accuracy, and they remain simple, black-box methods that avoid additional computational costs, tunable parameters, or system-specific implementations.