In Eq. ( ), • λ denotes an ensemble-average obtained using λ, which is the thermodynamic path variable with λ = 0 corresponding to a state with N w water molecules and λ = 1 corresponding to that with N w +1 water molecules. The integral is numerically approximated using 12-point Gauss-Legendre quadrature. To accomplish the transition in U (λ, q), we use soft-core potentials for both Coulombic and Lennard-Jones potential energy contributions involving the "added" water molecule. Because the simulation box is heterogeneous, we find this preferable from a sampling efficiency perspective because including only Lennard-Jones interactions in the absence of any electrostatics results in the added water molecule predominantly exploring the vast "vapor" phase.