Multipole-Preserving Quadratures for Discretization of Functions in Real-Space Electronic Structure Calculations

TL;DR

This paper introduces a novel quadrature scheme that preserves moments of analytic functions in real-space discretizations, enhancing stability and accuracy in electronic structure calculations, especially at coarser grid resolutions.

Contribution

A new quadrature method that exactly preserves moments of functions, improving stability in real-space electronic structure calculations at larger grid spacings.

Findings

Improves stability of DFT calculations at coarse grids.

Reduces the number of degrees of freedom needed.

Enhances reliability of low-accuracy discretizations.

Abstract

Discretizing an analytic function on a uniform real-space grid is often done via a straightforward collocation method. This is ubiquitous in all areas of computational physics and quantum chemistry. An example in Density Functional Theory (DFT) is given by the external potential or the pseudo-potential describing the interaction between ions and electrons. The accuracy of the collocation method used is therefore very important for the reliability of subsequent treatments like self-consistent field solutions of the electronic structure problems. By construction, the collocation method introduces numerical artifacts typical of real-space treatments, like the so-called egg-box error, that may spoil the numerical stability of the description when the real-space grid is too coarse. As the external potential is an input of the problem, even a highly precise computational treatment cannot cope this inconvenience. We present in this paper a new quadrature scheme that is able to exactly preserve the moments of a given analytic function even for large grid spacings, while reconciling with the traditional collocation method when the grid spacing is small enough. In the context of real-space electronic structure calculations, we show that this method improves considerably the stability of the results for large grid spacings, opening the path towards reliable low-accuracy DFT calculations with reduced number of degrees of freedom.