Suitability of umbrella- and overlap-sampling methods for calculation of solid-phase free energies by molecular simulation.

We examine the ability of two-stage free-energy perturbation methods to yield solid-phase free energies using a system of harmonically coupled particles as a reference. We consider two ways to construct a reference system, one based on derivatives of the intermolecular potential of the target system of interest (the conventional choice in lattice dynamics), and the other based on analysis of pairwise configurational correlations observed in simulations of the target system. For each case, we consider two perturbation techniques that compute the free energy difference between the target and reference systems while avoiding lengthy thermodynamic integration procedures. The methods are overlap sampling as optimized by Bennett, and umbrella sampling optimized in a similar fashion. Such methods require at most two simulations to yield a result, but they can fail if the target and reference do not share a sufficiently large set of relevant configurations. In particular, failure can be expected for large systems, and we examine the question of how large a system can be before this point is reached. Our test case is a face-centered cubic system of r(-12) soft spheres, and we find that for systems of up to 108 particles the methods are accurate for all temperatures up to melting; for systems of 256 particles the methods begin to break down at about half the melting temperature. Significantly, we observe that the correction to the harmonic reference is only weakly dependent on system size, suggesting an N-hybrid technique in which perturbation is applied to a small system and the result added to a large-system harmonic reference to obtain a good estimate of the correct large-system free energy. We also examine these approaches, along with thermodynamic integration in temperature, with respect to their computational efficiency. We find that Bennett's method using a derivative-based harmonic reference is the most efficient of all those examined, particularly when employed in the N-hybrid method.