# 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

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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 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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