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Convergence of gradient-based algorithms for the Hartree-Fockequations∗

Published online by Cambridge University Press:  30 March 2012

Antoine Levitt
Antoine Levitt*
Affiliation:
Université Paris-Dauphine, CEREMADE, Place du Maréchal Lattre de Tassigny, 75775 Paris Cedex 16, France. e-mail: levitt@ceremade.dauphine.fr

Abstract

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The numerical solution of the Hartree-Fock equations is a central problem in quantumchemistry for which numerous algorithms exist. Attempts to justify these algorithmsmathematically have been made, notably in [E. Cancès and C. Le Bris, Math. Mod.Numer. Anal. 34 (2000) 749–774], but, to our knowledge, nocomplete convergence proof has been published, except for the large-Zresult 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 agradient inequality for analytic functionals due to Łojasiewicz [Ensemblessemi-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 Roothaanand Level-Shifting algorithms. In each case, our method of proof provides estimates on theconvergence rate. We compare these with numerical results for the algorithms studied.

Keywords

Hartree-Fock equationsŁojasiewicz inequalityoptimization on manifolds

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 6 , November 2012, pp. 1321 - 1336
Copyright
© EDP Sciences, SMAI, 2012

References

Alouges, F. and Audouze, C., Preconditioned gradient flows for nonlinear eigenvalue problems and application to the Hartree-Fock functional. Numer. Methods Partial Differ. Equ. 25 (2009) 380–400. Google Scholar
Bacskay, G.B., A quadratically convergent Hartree-Fock (QC-SCF) method. Application to closed shell systems. Chem. Phys. 61 (1981) 385–404. Google Scholar
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).
Cancès, E. and Le Bris, C., Can we outperform the DIIS approach for electronic structure calculations? Int. J. Quant. Chem. 79 (2000) 82–90. Google Scholar
Cancès, E. and Le Bris, C., On the convergence of SCF algorithms for the Hartree-Fock equations. Math. Mod. Numer. Anal. 34 (2000) 749–774. Google Scholar
Cancès, E. and Pernal, K., Projected gradient algorithms for Hartree-Fock and density matrix functional theory calculations. J. Chem. Phys. 128 (2008) 134–108. Google ScholarPubMed
Cancès, E., Defranceschi, M., Kutzelnigg, W., Le Bris, C. and Maday, Y., Computational quantum chemistry : a primer. Handbook Numer. Anal. 10 (2003) 3–270. Google Scholar
Edelman, A., Arias, T.A. and Smith, S.T., The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 (1998) 303. Google Scholar
Francisco, J.B., Martínez, J.M. and Martínez, L., Globally convergent trust-region methods for self-consistent field electronic structure calculations. J. Chem. Phys. 121 (2004) 10863. Google ScholarPubMed
Griesemer, M. and Hantsch, F., Unique solutions to Hartree-Fock equations for closed shell atoms. Arch. Ration. Mech. Anal. 203 (2012) 883–900. Google Scholar
Haraux, A., Jendoubi, M.A. and Kavian, O., Rate of decay to equilibrium in some semilinear parabolic equations. J. Evol. Equ. 3 (2003) 463–484. Google Scholar
Høst, S., Olsen, J., Jansík, B., Thøgersen, L., Jørgensen, P. and Helgaker, T., The augmented Roothaan-Hall method for optimizing Hartree-Fock and Kohn-Sham density matrices. J. Chem. Phys. 129 (2008) 124–106. Google ScholarPubMed
Kudin, K.N., Scuseria, G.E. and Cancès, E., A black-box self-consistent field convergence algorithm : one step closer. J. Chem. Phys. 116 (2002) 8255. Google Scholar
Lieb, E.H. and Simon, B., The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys. 53 (1977) 185–194. Google Scholar
Lions, P.L., Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33–97. Google Scholar
S. Łojasiewicz, Ensembles semi-analytiques. Institut des Hautes Études Scientifiques (1965).
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.
Pulay, P., Improved SCF convergence acceleration. J. Comput. Chem. 3 (1982) 556–560. Google Scholar
Salomon, J., Convergence of the time-discretized monotonic schemes. ESAIM : M2AN 41 (2007) 77–93. Google Scholar
Saunders, V.R. and Hillier, I.H., A “Level-Shifting” method for converging closed shell Hartree-Fock wave functions. Int. J. Quant. Chem. 7 (1973) 699–705. Google Scholar
Sidje, R.B., Expokit : a software package for computing matrix exponentials. ACM Trans. Math. Softw. 24 (1998) 130–156. Google Scholar