Continuum electrostatics with capped Coulomb potential

For ensembles of spatially extended ions, electrostatic interaction energies can be calculated more accurately using a capped Coulomb potential, which remains constant within a fixed distance and then decays inversely with distance. When this fixed distance is small, the capped Coulomb potential transitions into a composite Coulomb-Yukawa potential, widely used in modeling dense electrolytes and ionic liquids and associated with a fourth-order derivative in Poisson's equation. To go beyond the fourth order, we develop a theoretical framework for continuum electrostatics based on the capped Coulomb potential. We derive and solve a modified Poisson equation and calculate corresponding electrostatic interaction energies. Solutions of the modified Poisson equation are shown to emerge from applying a differential operator to the potential predicted by the unmodified Poisson equation. We demonstrate that the electrostatic potential satisfying the modified Poisson equation can exhibit discontinuities at interfaces.