Efficient Isoparametric Integration over Arbitrary, Space-Filling Voronoi Polyhedra for Electronic-Structure Calculations

TL;DR

This paper introduces a fast, accurate integration method over convex Voronoi polyhedra using weighted tessellation and isoparametric quadrature, significantly improving efficiency and precision for electronic-structure calculations.

Contribution

It presents a novel integration scheme combining weighted Voronoi tessellation with Gauss-Legendre quadratures, achieving rapid convergence and high accuracy for complex integrands in electronic-structure computations.

Findings

Achieves machine-precision accuracy in milliseconds.

Over 10^5 times faster than shape-function methods.

More accurate by a factor of 10^7 compared to previous approaches.

Abstract

A numerically efficient, accurate, and easily implemented integration scheme over convex Voronoi polyhedra (VP) is presented for use in {\it ab-initio} electronic-structure calculations. We combine a weighted Voronoi tessellation with isoparametric integration via Gauss-Legendre quadratures to provide rapidly convergent VP integrals for a variety of integrands, including those with a Coulomb singularity. We showcase the capability of our approach by first applying to an analytic charge-density model achieving machine-precision accuracy with expected convergence properties in milliseconds. For contrast, we compare our results to those using shape-functions and show our approach is greater than $10^{5}$ faster and $10^{7}$ more accurate. A weighted Voronoi tessellation also allows for a physics-based partitioning of space that guarantees convex, space-filling VP while reflecting accurate atomic size and site charges, as we show within KKR methods applied to Fe-Pd alloys.