# Convergence of gradient-based algorithms for the Hartree-Fock equations

- Volume: 46, Issue: 6, page 1321-1336
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topLevitt, Antoine. "Convergence of gradient-based algorithms for the Hartree-Fock equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 46.6 (2012): 1321-1336. <http://eudml.org/doc/273317>.

@article{Levitt2012,

abstract = {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.},

author = {Levitt, Antoine},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Hartree-Fock equations; Łojasiewicz inequality; optimization on manifolds},

language = {eng},

number = {6},

pages = {1321-1336},

publisher = {EDP-Sciences},

title = {Convergence of gradient-based algorithms for the Hartree-Fock equations},

url = {http://eudml.org/doc/273317},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Levitt, Antoine

TI - Convergence of gradient-based algorithms for the Hartree-Fock equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2012

PB - EDP-Sciences

VL - 46

IS - 6

SP - 1321

EP - 1336

AB - 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.

LA - eng

KW - Hartree-Fock equations; Łojasiewicz inequality; optimization on manifolds

UR - http://eudml.org/doc/273317

ER -

## References

top- [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. Zbl1166.65039MR2483772
- [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). Zbl0992.81103MR1857459
- [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. Zbl1090.65548MR1784484
- [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. Zbl1070.81534MR2008386
- [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. Zbl0928.65050MR1646856
- [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. Zbl1110.92069
- [10] M. Griesemer and F. Hantsch, Unique solutions to Hartree-Fock equations for closed shell atoms. Arch. Ration. Mech. Anal.203 (2012) 883–900. Zbl1256.35101MR2928136
- [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. Zbl1036.35035MR2019030
- [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. MR452286
- [15] P.L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys.109 (1987) 33–97. Zbl0618.35111MR879032
- [16] S. Łojasiewicz, Ensembles semi-analytiques. Institut des Hautes Études Scientifiques (1965). Zbl0241.32005
- [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. Zbl0071.42302MR81755
- [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. Zbl1124.65059MR2323691
- [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. Zbl0917.65063

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.