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://www.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://www.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://www.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://www.numdam.org/articles/10.1051/m2an/2012008/
[1] and , Preconditioned gradient flows for nonlinear eigenvalue problems and application to the Hartree-Fock functional. Numer. Methods Partial Differ. Equ. 25 (2009) 380-400. | MR | Zbl
[2] , A quadratically convergent Hartree-Fock (QC-SCF) method. Application to closed shell systems. Chem. Phys. 61 (1981) 385-404.
[3] , SCF algorithms for Hartree-Fock electronic calculations, in Mathematical models and methods for ab initio quantum chemistry, edited by M. Defranceschi and C. Le Bris. Lect. Notes Chem. 74 (2000). | MR | Zbl
[4] and , Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quant. Chem. 79 (2000) 82-90.
[5] and , On the convergence of SCF algorithms for the Hartree-Fock equations. Math. Mod. Numer. Anal. 34 (2000) 749-774. | Numdam | MR | Zbl
[6] and , Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations. J. Chem. Phys. 128 (2008) 134-108.
[7] , , , and , Computational quantum chemistry : a primer. Handbook Numer. Anal. 10 (2003) 3-270. | MR | Zbl
[8] , and , The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 (1998) 303. | MR | Zbl
[9] , and , Globally convergent trust-region methods for self-consistent field electronic structure calculations. J. Chem. Phys. 121 (2004) 10863. | Zbl
[10] and , Unique solutions to Hartree-Fock equations for closed shell atoms. Arch. Ration. Mech. Anal. 203 (2012) 883-900. | MR | Zbl
[11] , and , Rate of decay to equilibrium in some semilinear parabolic equations. J. Evol. Equ. 3 (2003) 463-484. | MR | Zbl
[12] , , , , and , The augmented Roothaan-Hall method for optimizing Hartree-Fock and Kohn-Sham density matrices. J. Chem. Phys. 129 (2008) 124-106.
[13] , and , A black-box self-consistent field convergence algorithm : one step closer. J. Chem. Phys. 116 (2002) 8255.
[14] and , The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185-194. | MR
[15] , Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33-97. | MR | Zbl
[16] , Ensembles semi-analytiques. Institut des Hautes Études Scientifiques (1965).
[17] R. McWeeny,. The density matrix in self-consistent field theory. I. Iterative construction of the density matrix, in Proc. of R. Soc. Lond. A. Math. Phys. Sci. 235 (1956) 496. | MR | Zbl
[18] , Improved SCF convergence acceleration. J. Comput. Chem. 3 (1982) 556-560.
[19] , Convergence of the time-discretized monotonic schemes. ESAIM : M2AN 41 (2007) 77-93. | Numdam | MR | Zbl
[20] and , A “Level-Shifting” method for converging closed shell Hartree-Fock wave functions. Int. J. Quant. Chem. 7 (1973) 699-705.
[21] , Expokit : a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24 (1998) 130-156. | Zbl
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