In this article, 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 periodic Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the electronic ground state energy and density of molecular systems in the condensed phase. The TFW model is strictly 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. We prove that, for any local minimizer of the Kohn-Sham LDA model, and under a coercivity assumption ensuring the local uniqueness of this minimizer up to unitary transform, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of for large enough energy cut-offs, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.
Mots-clés : electronic structure calculation, density functional theory, Thomas-Fermi-von Weizsäcker model, Kohn-Sham model, nonlinear eigenvalue problem, spectral methods
@article{M2AN_2012__46_2_341_0, author = {Canc\`es, Eric and Chakir, Rachida and Maday, Yvon}, title = {Numerical analysis of the planewave discretization of some orbital-free and {Kohn-Sham} models}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {341--388}, publisher = {EDP-Sciences}, volume = {46}, number = {2}, year = {2012}, doi = {10.1051/m2an/2011038}, mrnumber = {2855646}, zbl = {1278.82003}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an/2011038/} }
TY - JOUR AU - Cancès, Eric AU - Chakir, Rachida AU - Maday, Yvon TI - Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 341 EP - 388 VL - 46 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2011038/ DO - 10.1051/m2an/2011038 LA - en ID - M2AN_2012__46_2_341_0 ER -
%0 Journal Article %A Cancès, Eric %A Chakir, Rachida %A Maday, Yvon %T Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 341-388 %V 46 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2011038/ %R 10.1051/m2an/2011038 %G en %F M2AN_2012__46_2_341_0
Cancès, Eric; Chakir, Rachida; Maday, Yvon. Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 341-388. doi : 10.1051/m2an/2011038. http://archive.numdam.org/articles/10.1051/m2an/2011038/
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