Periodic boundary conditions with 1/ r potential

Periodic boundary conditions (PBC) provide a realistic description for systems ranging from ionic crystals to cosmological models. The potential and its corresponding force field under PBC are naturally accommodated using Fourier methods. In doing so, the Green’s function is considerably modified from a simple $1/r$ form, increasing the computational cost. In this work, we study the approach of approximating PBC with a finite real-space summation for the $1/r$ potential. We derive the associated correction term using purely elementary means, utilizing multipole expansions. To our knowledge, this term has previously been obtained only through Ewald splitting. Our work establishes a natural connection between the lattice sum and the modified Poisson equation.