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

Eric Cancès; Rachida Chakir; Yvon Maday

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

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

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 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.

How to cite

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Cancès, 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 46.2 (2011): 341-388. <http://eudml.org/doc/222166>.

@article{Cancès2011,
abstract = { 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. },
author = {Cancès, Eric, Chakir, Rachida, Maday, Yvon},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
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; electronic structure calculation; a priori error estimates; Fourier discretization; molecular simulations},
language = {eng},
month = {10},
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/222166},
volume = {46},
year = {2011},
}

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
DA - 2011/10//
PB - EDP Sciences
VL - 46
IS - 2
SP - 341
EP - 388
AB - 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.
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; electronic structure calculation; a priori error estimates; Fourier discretization; molecular simulations
UR - http://eudml.org/doc/222166
ER -

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