Efficient free energy calculations by variationally optimized metric scaling: Concepts and applications to the volume dependence of cluster free energies and to solid–solid phase transitions

Abstract

Finite-time variational switching is an efficient method for obtaining converging upper and lower bounds to free energy changes by computer simulation. Over the course of the simulation, the Hamiltonian is changed continuously between the system of interest and a reference system for which the partition function has an analytic form. The bounds converge most rapidly when the system is kept close to equilibrium throughout the switching. In this paper we introduce the technique of metric scaling to improve adherence to equilibrium and thereby obtain more rapid convergence of the free energy bounds. The method involves scaling the coordinates of the particles, perhaps in a nonuniform way, so as to assist their natural characteristic evolution over the course of the switching. The scaling schedule can be variationally optimized to produce the best convergence of the bounds for a given Hamiltonian switching path. A correction due to the intrinsic work of scaling is made at the end of the calculation. The method is illustrated in a pedagogical one-dimensional example, and is then applied to the volume dependence of cluster free energies, a property of direct relevance to vapor–liquid nucleation theory. Order-of-magnitude improvements in efficiency are obtained in these simple examples. As a contrasting application, we use metric scaling to calculate directly the free energy difference between face-centered-cubic and body-centered-cubic Yukawa crystals. A continuous distortion is applied to the lattice, avoiding the need for separate comparison of the two phases with an independent reference system.