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.