A comparison of some variational formulas for the free energy as applied to hard-sphere crystals

Abstract

We examine several variational methods for determining bounds on the free energy of model crystalline phases, as applied to hard spheres in one and three dimensions. Cell- and harmonic-based reference systems are considered. Methods that provide the tightest bounds on the free energy are similar in form to free-energy perturbation, and are prone to inaccuracy from inadequate sampling. Gibbs–Bogoliubov formulas are reliable but weaker. For hard potentials they can give only a lower bound, indicating that their ability to provide upper bounds for other potentials is limited. Nevertheless, bounds given by Gibbs–Bogoliubov when applied with the optimal harmonic system prescribed by Morris and Ho [Phys. Rev. Lett. 74, 940 (1995)] yields impressive results; for hard spheres at higher density it is, within confidence limits, equal to the exact hard-sphere free energy.