The numerical solution of the Hartree-Fock equations is a central problem in quantumchemistry for which numerous algorithms exist. Attempts to justify these algorithmsmathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod.Numer. Anal. 34 (2000) 749–774], but, to our knowledge, nocomplete convergence proof has been published, except for the large-Zresult 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 agradient inequality for analytic functionals due to Łojasiewicz [Ensemblessemi-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 Roothaanand Level-Shifting algorithms. In each case, our method of proof provides estimates on theconvergence rate. We compare these with numerical results for the algorithms studied.
