Efficient and Accurate Linear Algebraic Methods for Large-scale Electronic Structure Calculations with Non-orthogonal Atomic Orbitals

TL;DR

This paper presents efficient algebraic methods, including generalized Lanczos and Arnoldi algorithms, for large-scale electronic structure calculations, demonstrating their accuracy and suitability for molecular dynamics simulations.

Contribution

It introduces and compares generalized shifted conjugate orthogonal conjugate gradient, Lanczos, and Arnoldi methods for solving large linear equations in electronic structure calculations, highlighting their efficiency.

Findings

Methods are equally accurate to exact calculations.

Lanczos and Arnoldi are most suitable for large-scale simulations.

CPU time and memory usage are optimized with these methods.

Abstract

The need for large-scale electronic structure calculations arises recently in the field of material physics and efficient and accurate algebraic methods for large simultaneous linear equations become greatly important. We investigate the generalized shifted conjugate orthogonal conjugate gradient method, the generalized Lanczos method and the generalized Arnoldi method. They are the solver methods of large simultaneous linear equations of one-electron Schr\"odinger equation and maps the whole Hilbert space to a small subspace called the Krylov subspace. These methods are applied to systems of fcc Au with the NRL tight-binding Hamiltonian (Phys. Rev. B {\bf 63}, 195101 (2001)). We compare results by these methods and the exact calculation and show them equally accurate. The system size dependence of the CPU time is also discussed. The generalized Lanczos method and the generalized Arnoldi method are the most suitable for the large-scale molecular dynamics simulations from the view point of CPU time and memory size.