Volume 46, Number 6, November-December 2012
|Page(s)||1321 - 1336|
|Published online||30 March 2012|
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.