The problem of computing free energy differences due to isotopic substitution in chemical systems is discussed. The shift in the equilibrium properties of a system upon isotopic substitution is a purely quantum mechanical effect that can be quantified using the Feynman path integral approach. In this paper, we explore two developments that lead to a highly efficient path integral scheme. First, we employ a mass switching function inspired by the work of Ceriotti and Markland [ J. Chem. Phys. 2013, 138, 014112] that is based on the inverse square root of the mass and which leads to a perfectly constant free energy derivative with respect to the switching parameter in the harmonic limit. We show that even for anharmonic systems, this scheme allows a single-point thermodynamic integration approach to be used in the construction of free energy differences. In order to improve the efficiency of the calculations even further, however, we derive a set of free energy derivative estimators based on the fourth-order scheme of Takahashi and Imada [ J. Phys. Soc. Jpn. 1984, 53, 3765]. The Takahashi-Imada procedure generates a primitive fourth-order estimator that allows the number of imaginary time slices in the path-integral approach to be reduced substantially. However, as with all primitive estimators, its convergence is plagued by numerical noise. In order to alleviate this problem, we derive a fourth-order virial estimator based on a transferring of the difference between second- and fourth-order primitive estimators, which remains relatively constant as a function of the number of configuration samples, to the second-order virial estimator. We show that this new estimator converges as smoothly as the second-order virial estimator but requires significantly fewer imaginary time points.

The problem of predicting polymorphism in atomic and molecular crystals constitutes a significant challenge both experimentally and theoretically. From the theoretical viewpoint, polymorphism prediction falls into the general class of problems characterized by an underlying rough energy landscape, and consequently, free energy based enhanced sampling approaches can be brought to bear on the problem. In this paper, we build on a scheme previously introduced by two of the authors in which the lengths and angles of the supercell are targeted for enhanced sampling via temperature accelerated adiabatic free energy dynamics [T. Q. Yu and M. E. Tuckerman, Phys. Rev. Lett. 107, 015701 (2011)]. Here, that framework is expanded to include general order parameters that distinguish different crystalline arrangements as target collective variables for enhanced sampling. The resulting free energy surface, being of quite high dimension, is nontrivial to reconstruct, and we discuss one particular strategy for performing the free energy analysis. The method is applied to the study of polymorphism in xenon crystals at high pressure and temperature using the Steinhardt order parameters without and with the supercell included in the set of collective variables. The expected fcc and bcc structures are obtained, and when the supercell parameters are included as collective variables, we also find several new structures, including fcc states with hcp stacking faults. We also apply the new method to the solid-liquid phase transition in copper at 1300 K using the same Steinhardt order parameters. Our method is able to melt and refreeze the system repeatedly, and the free energy profile can be obtained with high efficiency. © 2014 AIP Publishing LLC.

Enhanced sampling techniques that target a set of collective variables and that use molecular dynamics as the driving engine have seen widespread application in the computational molecular sciences as a means to explore the free-energy landscapes of complex systems. The use of molecular dynamics as the fundamental driver of the sampling requires the introduction of a time step whose magnitude is limited by the fastest motions in a system. While standard multiple time-stepping methods allow larger time steps to be employed for the slower and computationally more expensive forces, the maximum achievable increase in time step is limited by resonance phenomena, which inextricably couple fast and slow motions. Recently, we introduced deterministic and stochastic resonance-free multiple time step algorithms for molecular dynamics that solve this resonance problem and allow ten- to twenty-fold gains in the large time step compared to standard multiple time step algorithms [P. Minary et al., Phys. Rev. Lett. 93, 150201 (2004); B. Leimkuhler et al., Mol. Phys. 111, 3579-3594 (2013)]. These methods are based on the imposition of isokinetic constraints that couple the physical system to Nosé-Hoover chains or Nosé-Hoover Langevin schemes. In this paper, we show how to adapt these methods for collective variable-based enhanced sampling techniques, specifically adiabatic free-energy dynamics/temperature-accelerated molecular dynamics, unified free-energy dynamics, and by extension, metadynamics, thus allowing simulations employing these methods to employ similarly very large time steps. The combination of resonance-free multiple time step integrators with free-energy-based enhanced sampling significantly improves the efficiency of conformational exploration.