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

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

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

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topLevitt, 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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