Appropriate methods to combine forward and reverse free-energy perturbation averages

We consider the accuracy of several methods for combining forward and reverse free-energy perturbation averages for two systems ~labeled 0 and 1!. The practice of direct averaging of these measurements is argued as not reliable. Instead, methods are considered of the form b(A12A0) 52ln@^w(u)exp(2bu/2)&0 /^w(u)exp(1bu/2)&1# , where A is the free energy, b51/kT is the reciprocal temperature, u5U12U0 is the difference in configurational energy, w(u) is a weighting function, and the angle brackets indicate an ensemble average performed on the system indicated by the subscript. Choices are considered in which w(u)51 and 1/cosh@(u2C)/2#; the latter being Bennett’s method where C is a parameter that can be selected arbitrarily, and may be used to optimize the precision of the calculation. We examine the methods in several applications: calculation of the pressure of a square-well fluid by perturbing the volume, the chemical potential of a high-density Lennard-Jones system, and the chemical potential of a model for water. We find that the approaches based on Bennett’s method weighting are very effective at ensuring an accurate result ~one in which the systematic error arising from inadequate sampling is less than the estimated confidence limits!, and that even the selection w(u)51 offers marked improvement over comparable methods. We suggest that Bennett’s method is underappreciated, and the benefits it offers for improved precision and ~especially! accuracy are substantial, and therefore it should be more widely used. © 2003 American Institute of Physics. @DOI: 10.1063/1.1537241#