Research Article

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

Antoine Levitt

Université Paris-Dauphine, CEREMADE, Place du Maréchal Lattre de Tassigny, 75775 Paris Cedex 16, France. e-mail: levitt@ceremade.dauphine.fr

Abstract

The numerical solution of the Hartree-Fock equations is a central problem in quantum chemistry for which numerous algorithms exist. Attempts to justify these algorithms mathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod. Numer. Anal. 34 (2000) 749–774], but, to our knowledge, no complete convergence proof has been published, except for the large-Z result of [M. Griesemer and F. Hantsch, Arch. Rational Mech. Anal. (2011) 170]. In this paper, we prove the convergence of a natural gradient algorithm, using a gradient inequality for analytic functionals due to Łojasiewicz [Ensembles semi-analytiques. Institut des Hautes Études Scientifiques (1965)]. Then, expanding upon the analysis of [E. Cancès and C. Le Bris, Math. Mod. Numer. Anal. 34 (2000) 749–774], 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.

(Received September 23 2011)

(Revised January 14 2012)

(Online publication March 30 2012)

Key Words:

• Hartree-Fock equations;
• Łojasiewicz inequality;
• optimization on manifolds

Mathematics Subject Classification:

• 35Q40;
• 65K10

Footnotes

^∗  Support from the grant ANR-10-BLAN-0101 of the French Ministry of Research is gratefully acknowledged