Absolute proton hydration free energy, surface potential of water, and redox potential of the hydrogen electrode from first principles: QM/MM MD free-energy simulations of sodium and potassium hydration

The absolute intrinsic hydration free energy $GH+,wat◦$ of the proton, the surface electric potential jump $χwat◦$ upon entering bulk water, and the absolute redox potential $VH+,wat◦$ of the reference hydrogen electrode are cornerstone quantities for formulating single-ion thermodynamics on absolute scales. They can be easily calculated from each other but remain fundamentally elusive, i.e., they cannot be determined experimentally without invoking some extra-thermodynamic assumption (ETA). The Born model provides a natural framework to formulate such an assumption (Born ETA), as it automatically factors out the contribution of crossing the water surface from the hydration free energy. However, this model describes the short-range solvation inaccurately and relies on the choice of arbitrary ion-size parameters. In the present study, both shortcomings are alleviated by performing first-principle calculations of the hydration free energies of the sodium (Na⁺) and potassium (K⁺) ions. The calculations rely on thermodynamic integration based on quantum-mechanical molecular-mechanical (QM/MM) molecular dynamics (MD) simulations involving the ion and 2000 water molecules. The ion and its first hydration shell are described using a correlated ab initio method, namely resolution-of-identity second-order Møller-Plesset perturbation (RIMP2). The next hydration shells are described using the extended simple point charge water model (SPC/E). The hydration free energy is first calculated at the MM level and subsequently increased by a quantization term accounting for the transformation to a QM/MM description. It is also corrected for finite-size, approximate-electrostatics, and potential-summation errors, as well as standard-state definition. These computationally intensive simulations provide accurate first-principle estimates for $GH+,wat◦$⁠, $χwat◦$⁠, and $VH+,wat◦$⁠, reported with statistical errors based on a confidence interval of 99%. The values obtained from the independent Na⁺ and K⁺ simulations are in excellent agreement. In particular, the difference between the two hydration free energies, which is not an elusive quantity, is 73.9 ± 5.4 kJ mol⁻¹ (K⁺ minus Na⁺), to be compared with the experimental value of 71.7 ± 2.8 kJ mol⁻¹. The calculated values of $GH+,wat◦$⁠, $χwat◦$⁠, and $VH+,wat◦$ (−1096.7 ± 6.1 kJ mol⁻¹, 0.10 ± 0.10 V, and 4.32 ± 0.06 V, respectively, averaging over the two ions) are also in remarkable agreement with the values recommended by Reif and Hünenberger based on a thorough analysis of the experimental literature (−1100 ± 5 kJ mol⁻¹, 0.13 ± 0.10 V, and 4.28 ± 0.13 V, respectively). The QM/MM MD simulations are also shown to provide an accurate description of the hydration structure, dynamics, and energetics.