Practical error bounds for properties in plane-wave electronic structure calculations

TL;DR

This paper introduces practical, accurate error bounds for key quantities in plane-wave electronic structure calculations, enhancing reliability and efficiency in computational materials science.

Contribution

It develops new computable error bounds based on residuals and inverse Jacobian analysis, with improved bounds via small linear problems involving Schur complements.

Findings

Bounds are accurate for silicon, gallium arsenide, and titanium dioxide.

The method efficiently estimates errors in energies and forces.

Numerical results demonstrate the bounds' effectiveness.

Abstract

We propose accurate computable error bounds for quantities of interest in plane-wave electronic structure calculations, in particular ground-state density matrices and energies, and interatomic forces. These bounds are based on an estimation of the error in terms of the residual of the solved equations, which is then efficiently approximated with computable terms. After providing coarse bounds based on an analysis of the inverse Jacobian, we improve on these bounds by solving a linear problem in a small dimension that involves a Schur complement. We numerically show how accurate these bounds are on a few representative materials, namely silicon, gallium arsenide and titanium dioxide.