Calculation of the aqueous solvation free energy of the proton

The value of the proton hydration free energy, ΔGhyd(H+), has been quoted in the literature to be from −252.6 to −262.5 kcal/mol. In this article, we present a theoretical model for calculating the hydration free energy of ions in aqueous solvent and use this model to calculate the proton hydration free energy, ΔGhyd(H+), in an effort to resolve the uncertainty concerning its exact value. In the model we define ΔGhyd(H+) as the free energy change associated with the following process: ΔGH+(gas)+H2nOn(aq)→H+(H2nOn)(aq), where the solvent is represented by a neutral n-water cluster embedded in a dielectric continuum and the solvated proton is represented by a protonated n-water cluster also in the continuum. All solvated species are treated as quantum mechanical solutes coupled to a dielectric continuum using a self consistent reaction field cycle. We investigated the behavior of ΔGhyd(H+) as the number of explicit waters of hydration is increased from n=1 to n=6. As n increases from 1 to 3, the hydration free energy decreases dramatically. However, for n=4–6 the hydration free energy maintains a relatively constant value of −262.23 kcal/mol. These results indicate that the first hydration shell of the proton is composed of at least four water molecules. The constant value of the hydration free energy for n⩾4 strongly suggests that the proton hydration free energy is at the far lower end of the range of values that have been proposed in the literature.