Reference systems for computational free energy calculations of binary solutions: role of the constrained center of mass motion

In the computational study of thermodynamic properties of model materials, calculation of the free energy is an unavoidable step. Frenkel and Ladd [J. Chem. Phys. 81 (1984) 3188] introduced a method to calculate absolute free energies of arbitrary solids based on the construction of a reversible path from the solid phase under consideration to an Einstein crystal with the same crystalline structure. Their reversible path method, with some modifications, is one of the most common methods used today, either in Monte Carlo or molecular dynamics. With equivalent methods for the liquid phase, the equilibrium phase diagram of a given model Hamiltonian can be constructed. In this work, we present the analytic expression of the free energy of the reference state of a solid or liquid binary mixture (mixture of Einstein oscillators or of ideal gases) for the case of constrained motion of the center of mass, which are needed in those simulations using thermostats that freeze such degree of freedom. We found expressions which are not direct extension of the cases without the constrains.