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] 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
and ,[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] Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quant. Chem. 79 (2000) 82-90.
and ,[5] On the convergence of SCF algorithms for the Hartree-Fock equations. Math. Mod. Numer. Anal. 34 (2000) 749-774. | Numdam | MR | Zbl
and ,[6] Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations. J. Chem. Phys. 128 (2008) 134-108.
and ,[7] Computational quantum chemistry : a primer. Handbook Numer. Anal. 10 (2003) 3-270. | MR | Zbl
, , , and ,[8] The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 (1998) 303. | MR | Zbl
, and ,[9] Globally convergent trust-region methods for self-consistent field electronic structure calculations. J. Chem. Phys. 121 (2004) 10863. | Zbl
, and ,[10] Unique solutions to Hartree-Fock equations for closed shell atoms. Arch. Ration. Mech. Anal. 203 (2012) 883-900. | MR | Zbl
and ,[11] Rate of decay to equilibrium in some semilinear parabolic equations. J. Evol. Equ. 3 (2003) 463-484. | MR | Zbl
, and ,[12] The augmented Roothaan-Hall method for optimizing Hartree-Fock and Kohn-Sham density matrices. J. Chem. Phys. 129 (2008) 124-106.
, , , , and ,[13] A black-box self-consistent field convergence algorithm : one step closer. J. Chem. Phys. 116 (2002) 8255.
, and ,[14] The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185-194. | MR
and ,[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] A “Level-Shifting” method for converging closed shell Hartree-Fock wave functions. Int. J. Quant. Chem. 7 (1973) 699-705.
and ,[21] Expokit : a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24 (1998) 130-156. | Zbl
,Cité par Sources :