Convergence of gradient-based algorithms for the Hartree-Fock equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, p. 1321-1336

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.

DOI : https://doi.org/10.1051/m2an/2012008
Classification:  35Q40,  65K10
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {6},
     year = {2012},
     pages = {1321-1336},
     doi = {10.1051/m2an/2012008},
     zbl = {1269.82008},
     mrnumber = {2996329},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_6_1321_0}
}
Levitt, Antoine. Convergence of gradient-based algorithms for the Hartree-Fock equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 6, pp. 1321-1336. doi : 10.1051/m2an/2012008. http://www.numdam.org/item/M2AN_2012__46_6_1321_0/

[1] F. Alouges and C. Audouze, Preconditioned gradient flows for nonlinear eigenvalue problems and application to the Hartree-Fock functional. Numer. Methods Partial Differ. Equ. 25 (2009) 380-400. | MR 2483772 | Zbl 1166.65039

[2] G.B. Bacskay, A quadratically convergent Hartree-Fock (QC-SCF) method. Application to closed shell systems. Chem. Phys. 61 (1981) 385-404.

[3] E. Cancés, 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 1857459 | Zbl 0992.81103

[4] E. Cancès and C. Le Bris, Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quant. Chem. 79 (2000) 82-90.

[5] E. Cancès and C. Le Bris, On the convergence of SCF algorithms for the Hartree-Fock equations. Math. Mod. Numer. Anal. 34 (2000) 749-774. | Numdam | MR 1784484 | Zbl 1090.65548

[6] E. Cancès and K. Pernal, Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations. J. Chem. Phys. 128 (2008) 134-108.

[7] E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational quantum chemistry : a primer. Handbook Numer. Anal. 10 (2003) 3-270. | MR 2008386 | Zbl 1070.81534

[8] A. Edelman, T.A. Arias and S.T. Smith, The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 (1998) 303. | MR 1646856 | Zbl 0928.65050

[9] J.B. Francisco, J.M. Martínez and L. Martínez, Globally convergent trust-region methods for self-consistent field electronic structure calculations. J. Chem. Phys. 121 (2004) 10863. | Zbl 1110.92069

[10] M. Griesemer and F. Hantsch, Unique solutions to Hartree-Fock equations for closed shell atoms. Arch. Ration. Mech. Anal. 203 (2012) 883-900. | MR 2928136 | Zbl 1256.35101

[11] A. Haraux, M.A. Jendoubi and O. Kavian, Rate of decay to equilibrium in some semilinear parabolic equations. J. Evol. Equ. 3 (2003) 463-484. | MR 2019030 | Zbl 1036.35035

[12] S. Høst, J. Olsen, B. Jansík, L. Thøgersen, P. Jørgensen and T. Helgaker, The augmented Roothaan-Hall method for optimizing Hartree-Fock and Kohn-Sham density matrices. J. Chem. Phys. 129 (2008) 124-106.

[13] K.N. Kudin, G.E. Scuseria and E. Cancès, A black-box self-consistent field convergence algorithm : one step closer. J. Chem. Phys. 116 (2002) 8255.

[14] E.H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185-194. | MR 452286

[15] P.L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33-97. | MR 879032 | Zbl 0618.35111

[16] S. Łojasiewicz, 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 81755 | Zbl 0071.42302

[18] P. Pulay, Improved SCF convergence acceleration. J. Comput. Chem. 3 (1982) 556-560.

[19] J. Salomon, Convergence of the time-discretized monotonic schemes. ESAIM : M2AN 41 (2007) 77-93. | Numdam | MR 2323691 | Zbl 1124.65059

[20] V.R. Saunders and I.H. Hillier, A “Level-Shifting” method for converging closed shell Hartree-Fock wave functions. Int. J. Quant. Chem. 7 (1973) 699-705.

[21] R.B. Sidje, Expokit : a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24 (1998) 130-156. | Zbl 0917.65063