Interfacial electrostatic potential gradients arise from nonuniform charge distributions encountered crossing the interface. The charges involved can include the molecular charges predominantly bound to each neutral solvent molecule and distributions of ions (or electrons) free to move in the interfacial region. This paper focuses on the solvent contribution to the interfacial potential. Quasichemical theory (QCT) provides a physical framework for the analysis of near-local (chemical) and far-field contributions to ion solvation free energies. Here, we utilize QCT to analyze cavity net potentials that contribute to the single-ion real solvation free energy. In particular, we discuss the results of molecular dynamics simulations of water droplets large enough to exhibit bulklike behavior in the droplet interior. A multipolar analysis of the cavity potential illustrates the importance of the solvent molecular quadrupole due to the near-cancellation of the dipolar contributions from the cavity–liquid and liquid–vapor interfaces. The results reveal the physical origin of the previously observed strong classical model dependence of the cavity potential.

Ionic solvation phenomena in liquids involve intense interactions in the inner solvation shell. For interactions beyond the first shell, the ion–solvent interaction energies result from the sum of many smaller-magnitude contributions that can still include polarization effects. Deep neural network (DNN) methods have recently found wide application in developing efficient molecular models that maintain near-quantum accuracy. Here we extend the DeePMD-kit code to produce accurate molecular multipole moments in the bulk and near interfaces. The new method is validated by comparing the DNN moments with those generated by ab initio simulations. The moments are used to compute the electrostatic potential at the center of a molecular-sized hydrophobic cavity in water. The results show that the fields produced by the DNN models are in quantitative agreement with the AIMD-derived values. These efficient methods will open the door to more accurate solvation models for large solutes such as proteins.

<div>We report a calculation scheme on water molecular dipole and quadrupole moments in the liquid phase through a Deep Neural Network (DNN) model. Employing the the Maximally Localized Wannier Functions (MLWF) for the valence electrons, we obtain the water moments through a post-process on trajectories from \textit{ab-initio} molecular dynamics (AIMD) simulations at the density functional theory (DFT) level. In the framework of the deep potential molecular dynamics (DPMD), we develop a scheme to train a DNN with the AIMD moments data. Applying the model, we calculate the contributions from water dipole and quadrupole moments to the electrostatic potential at the center of a cavity of radius 4.1 \AA\ as -3.87 V, referenced to the average potential in the bulk-like liquid region.</div><div>To unravel the ion-independent water effective local potential contribution to the ion hydration free energy, we estimate the 3rd cumulant term as -0.22 V from simulations totally over 6 ns, a time-scale inaccessible for AIMD calculations. </div>

<div>We report a calculation scheme on water molecular dipole and quadrupole moments in the liquid phase through a Deep Neural Network (DNN) model. Employing the the Maximally Localized Wannier Functions (MLWF) for the valence electrons, we obtain the water moments through a post-process on trajectories from \textit{ab-initio} molecular dynamics (AIMD) simulations at the density functional theory (DFT) level. In the framework of the deep potential molecular dynamics (DPMD), we develop a scheme to train a DNN with the AIMD moments data. Applying the model, we calculate the contributions from water dipole and quadrupole moments to the electrostatic potential at the center of a cavity of radius 4.1 \AA\ as -3.87 V, referenced to the average potential in the bulk-like liquid region.</div><div>To unravel the ion-independent water effective local potential contribution to the ion hydration free energy, we estimate the 3rd cumulant term as -0.22 V from simulations totally over 6 ns, a time-scale inaccessible for AIMD calculations. </div>