Clifford boundary conditions: a simple direct-sum evaluation of Madelung constants

TL;DR

This paper introduces a straightforward method using Clifford torus topology for efficient, accurate calculation of Madelung constants in ionic crystals, avoiding complex transformations and enabling quick convergence.

Contribution

The authors present a novel direct-sum approach utilizing Clifford torus topology for lattice sum evaluation, simplifying calculations without integral transformations or charge renormalization.

Findings

Monotonic convergence towards the infinite system limit.

High accuracy in Madelung constant calculation.

Minimal computational resources required.

Abstract

We propose a simple direct-sum method for the efficient evaluation of lattice sums in periodic solids. It consists of two main principles: i) the creation of a supercell that has the topology of a Clifford torus, which is a flat, finite and border-less manifold; ii) the renormalization of the distance between two points on the Clifford torus by defining it as the Euclidean distance in the embedding space of the Clifford torus. Our approach does not require any integral transformations nor any renormalization of the charges. We illustrate our approach by applying it to the calculation of the Madelung constants of ionic crystals. We show that the convergence towards the system of infinite size is monotonic, which allows for a straightforward extrapolation of the Madelung constant. We are able to recover the Madelung constants with a remarkable accuracy, and at an almost negligible computational cost, i.e., a few seconds on a laptop computer.