The Finite Coulomb Lattice Sum: A Resolution of Conditional Convergence through Exact Shape and Size

This work examines conditionally convergent Coulomb lattice sums under periodic boundary conditions. The recently developed finite lattice sum cleanly decomposes the series into three distinct components: a periodic bulk term ${\nu }_{\mathrm{p}\mathrm{b}\mathrm{c}}$, a shape-dependent non-periodic boundary term ${\nu }_{\mathrm{b}}$, and a finite-size correction term ${\nu }_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$. This rigorous formulation explicitly parameterizes the geometry of a finite lattice by its exact shape and size and takes an effective pairwise form. We analyze it in detail and compare it with various derivations of lattice sums in the literature. Perspectives on future applications are discussed, including analytical developments for arbitrarily shaped crystals and numerical mesh-type algorithms for condensed matter simulations.