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

References in corpus (8)

• Relativistic separable dual-space Gaussian Pseudopotentials from H to Rn
• Higher-order adaptive finite-element methods for Kohn-Sham density functional theory
• A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case
• Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods
• A posteriori error estimation for the non-self-consistent Kohn-Sham equations
• An overview of a posteriori error estimation and post-processingmethods for nonlinear eigenvalue problems
• Practical error bounds for properties in plane-wave electronic structure calculations
• A priori error analysis of linear and nonlinear periodic Schr{ö}dinger equations with analytic potentials