Calculation of the aqueous solvation free energy of the proton

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.

Synthesis, Characterization, and Linkage Isomerism in Mononuclear Ruthenium Complexes Containing the New Pyrazolate-Based Ligand Hpbl

“…On the other hand, for the aqueous solvation free energy of the protons released, we assumed the value of 262.5 kcal mol -1 from the literature. 24 Results and Discussion 1. Synthesis.…”

Calculation of the Solvation Free Energy of the Proton in Methanol

“…For the aqueous proton solvation free energy, the contribution of this term was less than 0.5 kcal/mol. 27 In order to get insight into basis set effects, the studies were repeated with four other basis sets (cc-pVDZ, aug-ccpVDZ, cc-pVTZ(-f), and aug-pVTZ(-f)), using initial geometries obtained with 6-31++G** basis set.…”

“…For the gas phase free energy of the proton, ∆G(H + , g), the value of -6.28 kcal/mol was used from the consideration of the entropy at 298 K and 1 atm pressure using the SackurTetrode equation. 27 The solvation free energy, ∆G sol (A) of neutral and protonated methanol clusters was calculated using the implicit Poisson-Boltzmann solvation model. 42 A solute was depicted as a set of atomic charges located in a cavity immersed in a continuum solvent with high dielectric constant and the solute-solvent boundary was represented by the surface of closest approach as a probe sphere was rolled over the van der Waals envelope of the solute.…”

“…We used the similar procedure as used for evaluating proton solvation free energy in aqueous solution. [27][28][29] Preliminary results at the B3LYP/6-31++G** level 30,31 were reported elsewhere. 32 In the present study, we extend the calculation to the other basis sets and report the structure and energies of the clusters in detail.…”

“…26 Tawa and coworkers calculated the free energy of proton solvation in water. 27 They used the hybrid representation of the solvent and found that the free energy was converged to be -262.23 kcal/mol when the number of explicit solvents is larger than four. Mejias and Lago used the clusters up to 13 water molecules in their evaluation of the hydration free energy.…”