Linear Algebraic Calculation of Green's function for Large-Scale Electronic Structure Theory

TL;DR

This paper introduces a linear algebraic iterative method for calculating Green's functions and density matrices in large-scale electronic structure problems, avoiding eigenstate computations and enabling high accuracy.

Contribution

The paper presents the shifted conjugate-orthogonal-conjugate-gradient method, a novel iterative solver for Green's functions that improves efficiency and accuracy in large-scale electronic structure calculations.

Findings

Method is robust against round-off errors.

Calculation can reach machine accuracy.

Applicable to both semiconductor and metal systems.

Abstract

A linear algebraic method named the shifted conjugate-orthogonal-conjugate-gradient method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Green's function and the density matrix without calculating eigenstates.The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is robust against the round-off error and the calculation can reach the machine accuracy. With the observation of residual vectors, the accuracy can be controlled, microscopically, independently for each element of the Green's function, and dynamically, at each step in dynamical simulations. The method is applied to both semiconductor and metal.