Variational formula for the free energy based on incomplete sampling in a molecular simulation

Abstract

Finite sampling in free-energy perturbation (FEP) calculations by molecular simulation leads to reproducible systematic errors, with sign shown to depend (in a known way) only on which system governs sampling in the simulation. Thus the result of a FEP calculation can be used as a bound on the true free energy. This inequality is of a wholly different nature from established forms such as the Gibbs-Bogoliubov inequality or the second law, in that its origins relate to the performance of a molecular simulation. If one can identify a suitable reference system having a free energy known as a function of some defining parameter, variational schemes based on the finite-sampling inequalities can be implemented. This idea is demonstrated through calculation of the free energy of a hard-sphere solid by perturbing from harmonic references and of a hard-sphere fluid by perturbing from infinitely polydisperse references. The tightness of the bounds increases with the amount of sampling in the simulation and correlates with the entropy difference between the target and reference systems. The bounds are tightest near the point where the entropy difference is least.