Comptes Rendus
no. 6
Partial differential equations/Numerical analysis
Variational projector augmented-wave method

Presented by: Olivier Pironneau

Xavier Blanc1; Éric Cancès2; Mi-Song Dupuy1
1 Université Paris-Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis-Lions, UMR 7598, UPMC, CNRS, 75205 Paris, France
2 CERMICS, École des ponts and Inria Paris, 6 & 8, avenue Blaise-Pascal, 77455 Marne-la-Vallée, France
Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 665-670.

In Kohn–Sham electronic structure computations, wave functions have singularities at nuclear positions. Because of these singularities, plane wave expansions give a poor approximation of the eigenfunctions. The PAW (projector augmented-wave) method circumvents this issue by replacing the original eigenvalue problem by a new one with the same eigenvalues, but smoother eigenvectors. Here a slightly different method, called VPAW (variational PAW), is proposed and analyzed. This new method allows for a better convergence with respect to the number of plane waves. Some numerical results on an idealized case corroborate this efficiency.

Dans les calculs de structure électronique de type Kohn–Sham, les fonctions d'ondes présentent des singularités aux positions des noyaux. Ces singularités empêchent une bonne approximation de la fonction par des ondes planes. La méthode PAW (projector augmented-wave) vise à contourner cette difficulté en remplaçant le problème aux valeurs propres d'origine par un autre ayant les mêmes valeurs propres, mais des vecteurs propres plus réguliers. Nous proposons et analysons une implémentation différente de cette méthode, baptisée VPAW (variational PAW). Elle permet d'obtenir une meilleure convergence en nombre d'ondes planes. Quelques résultats numériques sur un cas idéalisé confirment son efficacité.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.05.004

Xavier Blanc 1; Éric Cancès 2; Mi-Song Dupuy 1

1 Université Paris-Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis-Lions, UMR 7598, UPMC, CNRS, 75205 Paris, France
2 CERMICS, École des ponts and Inria Paris, 6 & 8, avenue Blaise-Pascal, 77455 Marne-la-Vallée, France 
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Xavier Blanc; Éric Cancès; Mi-Song Dupuy. Variational projector augmented-wave method. Comptes Rendus. Mathématique, Volume 355 (2017) no. 6, pp. 665-670. doi : 10.1016/j.crma.2017.05.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.004/

[1] C. Audouze; F. Jollet; M. Torrent; X. Gonze Projector augmented-wave approach to density-functional perturbation theory, Phys. Rev. B, Volume 73 (2006) no. 23

[2] X. Blanc, E. Cancès, M.-S. Dupuy, in preparation.

[3] P.E. Blochl Projector augmented-wave method, Phys. Rev. B, Volume 50 (1994) no. 24, pp. 17953-17979

[4] S. Fournais; M. Hoffmann-Ostenhof; T. Hoffmann-Ostenhof; T.Ø. Sørensen Sharp regularity results for coulombic many-electron wave functions, Commun. Math. Phys., Volume 255 (2005), pp. 183-227

[5] E. Hunsicker; V. Nistor; J.O. Sofo Analysis of periodic Schrödinger operators: regularity and approximation of eigenfunctions, J. Math. Phys., Volume 49 (2008) no. 8

[6] F. Jollet; M. Torrent; N. Holzwarth Generation of projector augmented-wave atomic data: a 71 element validated table in the XML format, Comput. Phys. Commun., Volume 185 (2014) no. 4, pp. 1246-1254

[7] T. Kato On the eigenfunctions of many-particle systems in quantum mechanics, Commun. Pure Appl. Math., Volume 10 (1957) no. 2, pp. 151-177

[8] G. Kresse; D. Joubert From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B, Volume 59 (1999) no. 3, pp. 1758-1775

[9] A. Levitt; M. Torrent Parallel eigensolvers in plane-wave density functional theory, Comput. Phys. Commun., Volume 187 (2015), pp. 98-105

[10] C. Rostgaard The projector augmented-wave method, 2009 (arXiv preprint) | arXiv

[11] M. Torrent; F. Jollet; F. Bottin; G. Zérah; X. Gonze Implementation of the projector augmented-wave method in the ABINIT code: application to the study of iron under pressure, Comput. Mater. Sci., Volume 42 (2008) no. 2, pp. 337-351

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