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

Eric Cancès (1, 2) , Rachida Chakir (3) , Yvon Maday (3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MICMAC - Methods and engineering of multiscale computing from atom to continuum
3 LJLL - Laboratoire Jacques-Louis Lions

Abstract: We provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizsäcker (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-os, 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.

Domaines: Analyse numérique [cs.NA]

Eric Cancès, Rachida Chakir, Yvon Maday. Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models. ESAIM: Mathematical Modelling and Numerical Analysis, 2012, 46 (02), pp.341-388. ⟨10.1051/m2an/2011038⟩. ⟨hal-00471938⟩