A coherent derivation of the Ewald summation for arbitrary orders of multipoles: The self-terms

TL;DR

This paper presents a comprehensive mathematical framework for deriving the Ewald summation for arbitrary multipole orders, emphasizing the self-term expressions crucial for polarizable force fields.

Contribution

It offers a unified derivation of Ewald summation for multipoles of any order, clarifying the self-term expressions used in electrostatics under periodic boundary conditions.

Findings

Provides a general framework for Ewald splitting of potentials.

Ensures mathematical well-posedness for all multipole orders.

Clarifies the self-term expressions essential for polarizable force fields.

Abstract

In this work, we provide the mathematical elements we think essential for a proper understanding of the calculus of the electrostatic energy of point-multipoles of arbitrary order under periodic boundary conditions. The emphasis is put on the expressions of the so-called self parts of the \Ewald\, summation where different expressions can be found in literature. Indeed, such expressions are of prime importance in the context of new generation polarizable force field where the self field appears in the polarization equations. We provide a general framework, where the idea of the \Ewald\ splitting is applied to the electric potential and subsequently, all other quantities such as the electric field, the energy and the forces are derived consistently thereof. Mathematical well-posedness is shown for all these contributions for any order of multipolar distribution.