which are the excess polarizability of ions of species i in solution, and the polarizability of the molecule, respectively. i , M and w are the dielectric functions of the ions, molecule and water, respectively. r i and r M are the radius of ion i and molecule. R M is the position of the center of mass of the molecule and r is the position of the grid point. d is the distance between the ion and the solute, that is, d = |R M -r| -r M -r i . Note that d ≥ σ i /2, as dictated by excluded volume interactions.