Equivalence of M- and P-Summation in Calculations of Ionic Solvation Free Energies

Condensed-phase simulations commonly use periodic boundary conditions (PBCs) to represent the thermodynamic limit. For the vapor to liquid transfer of an ion, the gas/liquid boundary and its associated potential change are then missing. Furthermore, the electric potential and field at a given point are given by conditionally convergent infinite series, for which different summation schemes give different results. Nevertheless, standard simulation protocols can be used to compute experimental quantities unambiguously. In particular, using an auxiliary test particle and a multistep solvation path, we show that particle-based, Ewald, and common molecule-based summation schemes for the potential and field are all essentially equivalent. However, all methods require prior knowledge of the gas/liquid boundary potential to compute ionic solvation free energies using PBC protocols for both force-field and quantum-mechanical models.