Convergence of gradient-based algorithms for the Hartree-Fock equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 6, pp. 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 : 10.1051/m2an/2012008
Classification : 35Q40, 65K10
Mots clés : Hartree-Fock equations, Łojasiewicz inequality, optimization on manifolds
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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://archive.numdam.org/articles/10.1051/m2an/2012008/

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

[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

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

[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 | Zbl

[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 | Zbl

[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

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