Calculation of the Chemical Potential beyond the First-Order Free-Energy Perturbation: From Deletion to Reinsertion

The estimation of the chemical potential is crucial in a variety of applications involving phase equilibrium. The heart of such calculation lies in the evaluation of a free-energy difference usually in the form of a partition function ratio. In this work, a general formulation is proposed for the calculation of the chemical potential from molecular simulation based on an integrated form of the first-order free-energy perturbation theory, which is able to overcome the main obstacle of the traditional first-order free-energy perturbation theory. The formulation is based on a novel scheme, where the perturbation is performed in an integral over a set of degrees of freedom. Beyond the general formalism, a specific example is presented leading to a reinsertion scheme for the evaluation of the chemical potential. Calculations based on this scheme are in excellent agreement with predictions from an accurate equation of state and equivalent to the test particle insertion scheme (Widom insertion scheme) for the pure Lennard-Jones fluid at high densities from NVT Monte Carlo simulations. The proposed method has a straightforward implementation and can be combined with the traditional test particle insertion method (Widom, B. J. Chem. Phys. 1982, 86, 869.). Furthermore, since test particle insertion estimates the excess chemical potential as a forward difference and the proposed reinsertion scheme as a backward difference, their combination can be used as a consistency check to ensure efficient sampling.