Systematic and statistical error in histogram-based free energy calculations

A common technique for the numerical calculation of free energies involves estimation of the probability density along a given coordinate from a set of configurations generated via simulation. The process requires discretization of one or more reaction coordinates to generate a histogram from which the continuous probability density is inferred. We show that the finite size of the intervals used to construct the histogram leads to quantifiable systematic error. The width of these intervals also determines the statistical error in the free energy, and the choice of the appropriate interval is therefore driven by the need to balance the two sources of error. We present a method for the construction of the optimal histogram for a given system, and show that the use of this technique requires little additional computational expense. We demonstrate the efficacy of the technique for a model system, and discuss how the principles governing the choice of discretization interval could be used to improve extended sampling techniques.