Convergence of gradient-based algorithms for the Hartree-Fock equations∗

Antoine Levitt

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

  • Volume: 46, Issue: 6, page 1321-1336
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Levitt, Antoine. "Convergence of gradient-based algorithms for the Hartree-Fock equations∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.6 (2012): 1321-1336. <http://eudml.org/doc/222111>.

@article{Levitt2012,
abstract = {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.},
author = {Levitt, Antoine},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Hartree-Fock equations; Łojasiewicz inequality; optimization on manifolds},
language = {eng},
month = {3},
number = {6},
pages = {1321-1336},
publisher = {EDP Sciences},
title = {Convergence of gradient-based algorithms for the Hartree-Fock equations∗},
url = {http://eudml.org/doc/222111},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Levitt, Antoine
TI - Convergence of gradient-based algorithms for the Hartree-Fock equations∗
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2012/3//
PB - EDP Sciences
VL - 46
IS - 6
SP - 1321
EP - 1336
AB - 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.
LA - eng
KW - Hartree-Fock equations; Łojasiewicz inequality; optimization on manifolds
UR - http://eudml.org/doc/222111
ER -

References

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