Calculation of the aqueous solvation free energy of the proton

TLDR

The proton hydration free energy has been reported in the literature between –252.6 and –262.5 kcal mol⁻¹. The authors develop a theoretical model to calculate ΔGhyd(H⁺) and use it to resolve this uncertainty. Their model defines ΔGhyd(H⁺) as the free‑energy change for H⁺(gas)+H₂Oₙ(aq)→H⁺(H₂Oₙ)(aq) with the solvent represented by a neutral n‑water cluster in a dielectric continuum, treats all solvated species quantum‑mechanically coupled to a self‑consistent reaction field, and examines n=1–6 explicit waters. The calculations show a dramatic drop in ΔGhyd(H⁺) from n=1 to 3, then a plateau at –262.23 kcal mol⁻¹ for n≥4, indicating the first hydration shell contains at least four waters and supporting the lower end of the reported range.

Abstract

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: ΔG[H+(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.