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.
Mots-clés : Hartree-Fock equations, Łojasiewicz inequality, optimization on manifolds
@article{M2AN_2012__46_6_1321_0,
author = {Levitt, Antoine},
title = {Convergence of gradient-based algorithms for the {Hartree-Fock} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1321--1336},
publisher = {EDP-Sciences},
volume = {46},
number = {6},
year = {2012},
doi = {10.1051/m2an/2012008},
mrnumber = {2996329},
zbl = {1269.82008},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2012008/}
}
TY - JOUR AU - Levitt, Antoine TI - Convergence of gradient-based algorithms for the Hartree-Fock equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1321 EP - 1336 VL - 46 IS - 6 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2012008/ DO - 10.1051/m2an/2012008 LA - en ID - M2AN_2012__46_6_1321_0 ER -
%0 Journal Article %A Levitt, Antoine %T Convergence of gradient-based algorithms for the Hartree-Fock equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1321-1336 %V 46 %N 6 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2012008/ %R 10.1051/m2an/2012008 %G en %F M2AN_2012__46_6_1321_0
Levitt, Antoine. Convergence of gradient-based algorithms for the Hartree-Fock equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 1321-1336. doi : 10.1051/m2an/2012008. http://archive.numdam.org/articles/10.1051/m2an/2012008/
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