Fully guaranteed and computable error bounds on the energy for periodic Kohn-Sham equations with convex density functionals

TL;DR

This paper develops fully guaranteed, computable error bounds for the energy in convex nonlinear mean-field models like Kohn-Sham equations, enabling adaptive refinement and demonstrating high accuracy on test cases.

Contribution

It introduces a novel method to derive guaranteed error bounds for Kohn-Sham energies, including decomposition into discretization and iteration errors, applicable to practical and nonconvex functionals.

Findings

Bounds are accurate for rHF and Kohn-Sham models on test cases.

Decomposition allows adaptive refinement strategies.

Error bounds remain reliable even for nonconvex functionals.

Abstract

In this article, we derive fully guaranteed error bounds for the energy of convex nonlinear mean-field models. These results apply in particular to Kohn-Sham equations with convex density functionals, which includes the reduced Hartree-Fock (rHF) model, as well as the Kohn-Sham model with exact exchange-density functional (which is unfortunately not explicit and therefore not usable in practice). We then decompose the obtained bounds into two parts, one depending on the chosen discretization and one depending on the number of iterations performed in the self-consistent algorithm used to solve the nonlinear eigenvalue problem, paving the way for adaptive refinement strategies. The accuracy of the bounds is demonstrated on a series of test cases, including a Silicon crystal and an Hydrogen Fluoride molecule simulated with the rHF model and discretized with planewaves. We also show that, although not anymore guaranteed, the error bounds remain very accurate for a Silicon crystal simulated with the Kohn-Sham model using nonconvex exchangecorrelation functionals of practical interest.