Numerically exact O($N^{7/3}$) method for large-scale electronic structure calculations

TL;DR

This paper introduces a numerically exact low-order scaling method for large-scale electronic structure calculations based on density functional theory, capable of handling both insulating and metallic systems efficiently.

Contribution

The paper presents a novel O(N^{7/3}) scaling method that is an exact alternative to traditional diagonalization, suitable for large-scale systems and parallel computing.

Findings

Achieves O(N^{7/3}) scaling in 3D systems

Applicable to both insulating and metallic materials

Demonstrates suitability for massively parallel computation

Abstract

An efficient low-order scaling method is presented for large-scale electronic structure calculations based on the density functional theory using localized basis functions, which directly computes selected elements of the density matrix by a contour integration of the Green function evaluated with a nested dissection approach for resultant sparse matrices. The computational effort of the method scales as O($N(\log_2N)^2$), O($N^{2}$), and O($N^{7/3}$) for one, two, and three dimensional systems, respectively, where $N$ is the number of basis functions. Unlike O($N$) methods developed so far the approach is a numerically exact alternative to conventional O($N^{3}$) diagonalization schemes in spite of the low-order scaling, and can be applicable to not only insulating but also metallic systems in a single framework. It is also demonstrated that the nested algorithm and the well separated data structure are suitable for the massively parallel computation, which enables us to extend the applicability of density functional calculations for large-scale systems together with the low-order scaling.