Simulations of dielectric permittivity of water by machine learned potentials with long-range Coulombic interactions

The dielectric permittivity of liquid water is a fundamental property that underlies its distinctive behaviors in numerous physical, biological, and chemical processes. Within a machine learning framework, we present a unified approach to compute the dielectric permittivity of water, systematically incorporating various electric boundary conditions. Our method employs a long-range-inclusive deep potential trained on data from hybrid density functional theory calculations. Dielectric response is evaluated using an auxiliary deep neural network that predicts the centers of maximally localized Wannier functions. We investigate three types of electric boundary conditions-metallic, insulating, and Kirkwood-Fröhlich-to assess their influence on correlated dipole fluctuations and dielectric relaxation dynamics. In particular, we demonstrate a consistent methodology for computing the Kirkwood correlation factor, correlation length, and dielectric permittivity under each boundary condition, where long-range electrostatics play a critical role. This work establishes a robust and generalizable machine-learning framework for modeling the dielectric properties of polar liquids under diverse electrostatic environments.