Convergence of gradient-based algorithms for the Hartree-Fock equations

Levitt, Antoine

Levitt, Antoine (2012), Convergence of gradient-based algorithms for the Hartree-Fock equations, Modélisation mathématique et analyse numérique, 46, 6, p. 1321-1336. http://dx.doi.org/10.1051/m2an/2012008

Type

Article accepté pour publication ou publié

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http://hal.archives-ouvertes.fr/hal-00626060/fr/

Date

2012

Nom de la revue

Modélisation mathématique et analyse numérique

Volume

Numéro

Éditeur

EDP Sciences

Pages

1321-1336

Résumé (EN)

The numerical solution of the Hartree-Fock equations is a central problem in quantum chemistry for which many algorithms exist. Attempts to justify these algorithms mathematically have been made, notably by Cancès and Le Bris in 2000, but no algorithm has yet been proved to convergence satisfactorily. In this paper, we prove the convergence of a natural gradient algorithm, using a gradient inequality for analytic functionals due to Lojasiewicz. Then, expanding upon the analysis of Cancès and Le Bris, we prove convergence results for the Roothaan and Level-Shifting algorithms. In each case, our method of proof provides estimates on the convergence rate. We compare these with numerical results for the algorithms studied.

Mots-clés

Łojasiewicz inequality; Hartree-Fock equations; optimization on manifolds