Convergence of gradient-based algorithms for the Hartree-Fock equations∗
Université Paris-Dauphine, CEREMADE, Place du Maréchal Lattre de Tassigny,
Paris Cedex 16,
Revised: 14 January 2012
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.
Mathematics Subject Classification: 35Q40 / 65K10
Key words: Hartree-Fock equations / Łojasiewicz inequality / optimization on manifolds
© EDP Sciences, SMAI, 2012