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On the convergence of SCF algorithmsfor the Hartree-Fock equations

Published online by Cambridge University Press:  15 April 2002

Eric Cancès  and
Claude Le Bris
Eric Cancès
Affiliation:
CERMICS, École Nationale des Ponts et Chaussées, 6 et 8 avenue Pascal, Cité Descartes, 77455 Champs-sur-Marne Cedex 2, France. ([email protected])
Claude Le Bris
Affiliation:
CERMICS, École Nationale des Ponts et Chaussées, 6 et 8 avenue Pascal, Cité Descartes, 77455 Champs-sur-Marne Cedex 2, France. ([email protected])
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Abstract

The present work is a mathematical analysis of two algorithms, namelythe Roothaan and the level-shifting algorithms, commonly used inpractice to solve the Hartree-Fock equations. The level-shiftingalgorithm is proved to be well-posed and to converge provided the shiftparameter is large enough. On the contrary, cases when the Roothaanalgorithm is not well defined or fails in converging areexhibited. These mathematical results are confronted to numericalexperiments performed by chemists.

Keywords

Nonlinear eigenvalue problemHartree-Fock equationsself-consistent fieldconvergence analysis.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 34 , Issue 4 , July 2000 , pp. 749 - 774
Copyright
© EDP Sciences, SMAI, 2000

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References

G. Auchmuty and Wenyao Jia, Convergent iterative methods for the Hartree eigenproblem. RAIRO Modél. Math. Anal. Numér. 28 (1994) 575-610.
Bach, V., Lieb, E.H., Loss, M. and Solovej, J.P., There are no unfilled shells in unrestricted Hartree-Fock theory. Phys. Rev. Lett. 72 (1994) 2981-2983. CrossRef
Bonač, V. ic-Koutecký and J. Koutecký, General properties of the Hartree-Fock problem demonstrated on the frontier orbital model. II. Analysis of the customary iterative procedure. Theoret. Chim. Acta 36 (1975) 163-180.
Facelli, J.C. and Contreras, R.H., A general relation between the intrinsic convergence properties of SCF Hartree-Fock calculations and the stability conditions of their solutions. J. Chem. Phys. 79 (1983) 3421-3423. CrossRef
Fletcher, R., Optimization of SCF LCAO wave functions. Mol. Phys. 19 (1970) 55-63. CrossRef
D.R. Hartree, The calculation of atomic structures. Wiley (1957).
W.J. Hehre, L. Radom, P.V.R. Schleyer and J.A. Pople, Ab initio molecular orbital theory. Wiley (1986).
Igawa, A. and Fukutome, H., A new direct minimization algorithm for Hartree-Fock calculations. Progr. Theoret. Phys. 54 (1975) 1266-1281. CrossRef
Koutecký, J. and Bonač, V.ic, On convergence difficulties in the iterative Hartree-Fock procedure. J. Chem. Phys. 55 (1971) 2408-2413. CrossRef
Lieb, E.H., Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud. Appl. Math. 57 (1977) 93-105. CrossRef
Lieb, E.H., Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A 29 (1984) 3018-3028. CrossRef
Lieb, E.H. and Simon, B., The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys. 53 (1977) 185-194. CrossRef
Lions, P.L., Solutions of Hartree-Fock equations for Coulomb systems. Comm. Math. Phys. 109 (1987) 33-97. CrossRef
McWeeny, R., The density matrix in self-consistent field theory. I. Iterative construction of the density matrix. Proc. Roy. Soc. London Ser. A 235 (1956) 496-509. CrossRef
R. McWeeny, Methods of molecular Quantum Mechanics. Academic Press (1992).
J. Paldus, Hartree-Fock stability and symmetry breaking, in Self Consistent Field Theory and Application. Elsevier (1990) 1-45.
Pulay, P., Improved SCF convergence acceleration. J. Comput. Chem. 3 (1982) 556-560. CrossRef
M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis. Academic Press (1980).
M. Reed and B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press (1978).
Roothaan, C.C.J., New developments in molecular orbital theory. Rev. Modern Phys. 23 (1951) 69-89. CrossRef
Saunders, V.R. and Hillier, I.H., A ``level-shifting'' method for converging closed shell Hartree-Fock wave functions. Int. J. Quantum Chem. 7 (1973) 699-705. CrossRef
H.B. Schlegel and J.J.W. McDouall, Do you have SCF stability and convergence problems?, in Computational Advances in Organic Chemistry, Kluwer Academic (1991) 167-185.
Seeger, R. R. and J.A. Pople, Self-consistent molecular orbital methods. XVI. Numerically stable direct energy minimization procedures for solution of Hartree-Fock equations. J. Chem. Phys. 65 (1976) 265-271. CrossRef
Stanton, R.E., The existence and cure of intrinsic divergence in closed shell SCF calculations. J. Chem. Phys. 75 (1981) 3426-3432. CrossRef
Stanton, R.E., Intrinsic convergence in closed-shell SCF calculations. A general criterion. J. Chem. Phys. 75 (1981) 5416-5422. CrossRef
Zerner, M.C. and Hehenberger, M., A dynamical damping scheme for converging molecular SCF calculations. Chem. Phys. Lett. 62 (1979) 550-554. CrossRef