An order-N electronic structure theory with generalized eigenvalue equations and its application to a ten-million-atom system

TL;DR

This paper introduces the multiple Arnoldi method, a linear-algebraic approach enabling large-scale electronic structure calculations for systems with up to ten million atoms, demonstrating high efficiency and parallel scalability.

Contribution

The paper presents a novel order-N electronic structure method using generalized eigenvalue equations and Krylov subspaces, implemented in the ELSES package for large-scale simulations.

Findings

Successfully simulated 10 million atoms on a workstation.

Achieved parallel efficiency up to 1,024 CPU cores.

Demonstrated applicability to metallic and insulating materials.

Abstract

A linear-algebraic theory called 'multiple Arnoldi method' is presented and realizes large-scale (order-N) electronic structure calculation with generalized eigen-value equations. A set of linear equations, in the form of (zS-H) x = b, are solved simultaneously with multiple Krylov subspaces. The method is implemented in a simulation package ELSES (http://www.elses.jp) with tight-binding-form Hamiltonians. A finite-temperature molecular dynamics simulation is carried out for metallic and insulating materials. A calculation with $10^7$ atoms was realized by a workstation. The parallel efficiency is shown upto 1,024 CPU cores.