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

Published online by Cambridge University Press:  24 October 2011

Eric Cancès
Affiliation:
Université Paris-Est, CERMICS, Project-team Micmac, INRIA-École des Ponts, 6 & 8 avenue Blaise Pascal, 77455 Marne-la-Vallée Cedex 2, France. cances@cermics.enpc.fr
Rachida Chakir
Affiliation:
UPMC Univ. Paris 06, UMR 7598 LJLL, 75005 Paris, France CNRS, UMR 7598 LJLL, 75005 Paris, France
Yvon Maday
Affiliation:
UPMC Univ. Paris 06, UMR 7598 LJLL, 75005 Paris, France CNRS, UMR 7598 LJLL, 75005 Paris, France Division of Applied Mathematics, 182 George Street, Brown University, Providence, RI 02912, USA

Abstract

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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 $\Phi^0$ 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 $\Phi^0$ 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2011

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