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

Eric Cancès; Rachida Chakir; Yvon Maday

- Volume: 46, Issue: 2, page 341-388
- ISSN: 0764-583X

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topCancès, Eric, Chakir, Rachida, and Maday, Yvon. "Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.2 (2012): 341-388. <http://eudml.org/doc/273250>.

@article{Cancès2012,

abstract = {In this article, we provide a priorierror 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 optimala priori error estimates for the spectral discretization method.},

author = {Cancès, Eric, Chakir, Rachida, Maday, Yvon},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {electronic structure calculation; density functional theory; Thomas-Fermi-von Weizsäcker model; Kohn-Sham model; nonlinear eigenvalue problem; spectral methods; -body Schrödinger equation; approximate solutions; a priori error estimates; Fourier discretization; molecular simulations},

language = {eng},

number = {2},

pages = {341-388},

publisher = {EDP-Sciences},

title = {Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models},

url = {http://eudml.org/doc/273250},

volume = {46},

year = {2012},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2012

PB - EDP-Sciences

VL - 46

IS - 2

SP - 341

EP - 388

AB - In this article, we provide a priorierror 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 optimala priori error estimates for the spectral discretization method.

LA - eng

KW - electronic structure calculation; density functional theory; Thomas-Fermi-von Weizsäcker model; Kohn-Sham model; nonlinear eigenvalue problem; spectral methods; -body Schrödinger equation; approximate solutions; a priori error estimates; Fourier discretization; molecular simulations

UR - http://eudml.org/doc/273250

ER -

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