Phase-space overlap measures. II. Design and implementation of staging methods for free-energy calculations

We consider staged free-energy calculation methods in the context of phase-space overlap relations, and argue that the selection of work-based methods should be guided by consideration of the phase-space overlap of the systems of interest. Stages should always be constructed such that work is performed only into a system that has a phase-space subset relation with the starting system. Thus multiple stages are required if the systems of interest are not such that one forms a phase-space subset with the other. Three two-stage methods are possible, termed umbrella sampling, overlap sampling, and funnel sampling. The last is appropriate for cases in which the subset relation holds, but only in the extreme, meaning that one system's important phase space constitutes a very small portion of the others. Umbrella sampling is most suitable for nonoverlap systems, and overlap sampling is appropriate for systems exhibiting partial phase-space overlap. We review recently introduced metrics that characterize phase-space overlap, showing that the performance of the single- and two-stage methods is consistent with the phase-space picture. We also demonstrate that a recently introduced bias-detection measure is effective in identifying inaccuracy in single- and multistage calculations. The examples used are the chemical-potential calculation for a Lennard-Jones liquid at moderate and at high densities, the same for model water at ambient conditions, and a process of charging a neutral ion in water.