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Numerical quadrature in the Brillouin zone for periodic Schrödinger operators

Cancès, Éric; Ehrlacher, Virginie; Gontier, David; Levitt, Antoine; Lombardi, Damiano (2020), Numerical quadrature in the Brillouin zone for periodic Schrödinger operators, Numerische Mathematik, 144, 3, p. 479-526. 10.1007/s00211-019-01096-w

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1805.07144.pdf (1.618Mb)
Type
Article accepté pour publication ou publié
Date
2020
Journal name
Numerische Mathematik
Volume
144
Number
3
Publisher
Springer
Pages
479-526
Publication identifier
10.1007/s00211-019-01096-w
Metadata
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Author(s)
Cancès, Éric
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Ehrlacher, Virginie
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Gontier, David cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Levitt, Antoine
Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique [CERMICS]
Lombardi, Damiano
Laboratoire Jacques-Louis Lions [LJLL (UMR_7598)]
Abstract (EN)
As a consequence of Bloch’s theorem, the numerical computation of the fermionic ground state density matrices and energies of periodic Schrödinger operators involves integrals over the Brillouin zone. These integrals are difficult to compute numerically in metals due to discontinuities in the integrand. We perform an error analysis of several widely-used quadrature rules and smearing methods for Brillouin zone integration. We precisely identify the assumptions implicit in these methods and rigorously prove error bounds. Numerical results for two-dimensional periodic systems are also provided. Our results shed light on the properties of these numerical schemes, and provide guidance as to the appropriate choice of numerical parameters.
Subjects / Keywords
Schrödinger operators

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