Coulomb potentials in two and three dimensions under periodic boundary conditions

TL;DR

This paper introduces a fast-converging method for summing Coulomb and logarithmic potentials in 2D and 3D with periodic boundary conditions, applicable to various unit cell geometries and slab geometries.

Contribution

It generalizes existing methods to arbitrary unit cells and provides explicit formulas for self-energies and Madelung constants in 2D and 3D.

Findings

Derived expressions converge rapidly across the entire simulation cell.

General formulas applicable to rhombic and triclinic cells.

Results include self-energies and Madelung constants for periodic structures.

Abstract

A method to sum over logarithmic potential in 2D and Coulomb potential in 3D with periodic boundary conditions in all directions is given. We consider the most general form of unit cells, the rhombic cell in 2D and the triclinic cell in 3D. For the 3D case, this paper presents a generalization of Sperb's work [R. Sperb, Mol. Simulation, \textbf{22}, 199-212(1999)]. The expressions derived in this work converge extremely fast in all region of the simulation cell. We also obtain results for slab geometry. Furthermore, self-energies for both 2D as well as 3D cases are derived. Our general formulas can be employed to obtain Madelung constants for periodic structures.