Fortuitous Cancellation of Competing Effects in the Free Energy of Water Autoionization: Long-Range Electrostatics, System Size, and Nuclear Quantum Effects
Few molecular processes are more fundamental to chemistry than the autoionization of water, whose equilibrium constant, $K_w \approx 10^{-14}$, underlies acid-base equilibria and the chemistry of life. Yet the molecular origins of the free energy cost of forming and separating ions remain incompletely understood. Efforts to explain it have been limited by the difficulty of modeling water reactivity in sufficient detail. Here we combine machine learning interatomic potentials (MLIPs) with spectral adaptive biasing force enhanced sampling to connect proton transfer and water wire configurations to the underlying free energy landscape. We identify competing influences of long-range electrostatics, system size, temperature, and nuclear quantum motion on the predicted ionization thermodynamics. The free energy surface extends from neutral water to ions separated beyond the solvent-separated ion pair (SSIP). It reveals a transition state in which a proton is shared between the second and third molecules of a three-membered water wire. The contact ion pair is unstable and the SSIP only weakly metastable, so determining pK$_w$ requires sampling longer wires. Long-range electrostatics raise the free energy at ion separations beyond $3\,\mathrm{\mathring{A}}$. Models with comparable force errors give substantially different free energy profiles in this region. Increasing system size lowers the calculated pK$_w$, while nuclear quantum effects bring it closer to experiment and reproduce the higher pK$_w$ of D$_2$O relative to H$_2$O. These results establish a baseline for interpreting thermodynamic results from MLIP simulations in which charged and polar groups react and diffuse in finite cells. Our analysis also explains how cancellation among competing effects can yield ionization constants close to experiment even when simulations omit several physical contributions.