Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

TL;DR

This paper provides theoretical error estimates for planewave discretizations of the Thomas-Fermi-von Weizs"acker and Kohn-Sham models, crucial for accurate electronic structure calculations in condensed matter physics.

Contribution

It offers the first rigorous a priori error analysis for spectral and pseudospectral planewave methods applied to these models, including the Kohn-Sham model under certain conditions.

Findings

Error estimates are established for large energy cut-offs.

Existence and uniqueness of minimizers are proven under coercivity assumptions.

The analysis applies to models used for ground state energy computations in molecular systems.

Abstract

We provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizs\"{a}cker (TFW) model and for the spectral discretization of the Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the ground state energy and density of molecular systems in the condensed phase. The TFW model is stricly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. Under a coercivity assumption on the second order optimality condition, we prove that for large enough energy cut-offs, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of any Kohn-Sham ground state, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.