# On the convergence of SCF algorithms for the Hartree-Fock equations

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 4, page 749-774
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topCancès, Eric, and Le Bris, Claude. "On the convergence of SCF algorithms for the Hartree-Fock equations." ESAIM: Mathematical Modelling and Numerical Analysis 34.4 (2010): 749-774. <http://eudml.org/doc/197609>.

@article{Cancès2010,

abstract = {
The present work is a mathematical analysis of two algorithms, namely
the Roothaan and the level-shifting algorithms, commonly used in
practice to solve the Hartree-Fock equations. The level-shifting
algorithm is proved to be well-posed and to converge provided the shift
parameter is large enough. On the contrary, cases when the Roothaan
algorithm is not well defined or fails in converging are
exhibited. These mathematical results are confronted to numerical
experiments performed by chemists.
},

author = {Cancès, Eric, Le Bris, Claude},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Nonlinear eigenvalue problem; Hartree-Fock equations; self-consistent field;
convergence analysis.; convergence analysis.},

language = {eng},

month = {3},

number = {4},

pages = {749-774},

publisher = {EDP Sciences},

title = {On the convergence of SCF algorithms for the Hartree-Fock equations},

url = {http://eudml.org/doc/197609},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Cancès, Eric

AU - Le Bris, Claude

TI - On the convergence of SCF algorithms for the Hartree-Fock equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 4

SP - 749

EP - 774

AB -
The present work is a mathematical analysis of two algorithms, namely
the Roothaan and the level-shifting algorithms, commonly used in
practice to solve the Hartree-Fock equations. The level-shifting
algorithm is proved to be well-posed and to converge provided the shift
parameter is large enough. On the contrary, cases when the Roothaan
algorithm is not well defined or fails in converging are
exhibited. These mathematical results are confronted to numerical
experiments performed by chemists.

LA - eng

KW - Nonlinear eigenvalue problem; Hartree-Fock equations; self-consistent field;
convergence analysis.; convergence analysis.

UR - http://eudml.org/doc/197609

ER -

## References

top- G. Auchmuty and Wenyao Jia, Convergent iterative methods for the Hartree eigenproblem. RAIRO Modél. Math. Anal. Numér. 28 (1994) 575-610.
- V. Bach, E.H. Lieb, M. Loss and J.P. Solovej, There are no unfilled shells in unrestricted Hartree-Fock theory. Phys. Rev. Lett.72 (1994) 2981-2983.
- V. Bonač 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. Acta36 (1975) 163-180.
- J.C. Facelli and R.H. Contreras, 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.
- R. Fletcher, Optimization of SCF LCAO wave functions. Mol. Phys.19 (1970) 55-63.
- 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).
- A. Igawa and H. Fukutome, A new direct minimization algorithm for Hartree-Fock calculations. Progr. Theoret. Phys.54 (1975) 1266-1281.
- J. Koutecký and V. Bonačic, On convergence difficulties in the iterative Hartree-Fock procedure. J. Chem. Phys.55 (1971) 2408-2413.
- E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation. Stud. Appl. Math.57 (1977) 93-105.
- E.H. Lieb, Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A29 (1984) 3018-3028.
- E.H. Lieb and B. Simon, The Hartree-Fock theory for Coulomb systems. Comm. Math. Phys.53 (1977) 185-194.
- P.L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Comm. Math. Phys.109 (1987) 33-97.
- R. McWeeny, The density matrix in self-consistent field theory. I. Iterative construction of the density matrix. Proc. Roy. Soc. London Ser. A235 (1956) 496-509.
- 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.
- P. Pulay, Improved SCF convergence acceleration. J. Comput. Chem.3 (1982) 556-560.
- 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).
- C.C.J. Roothaan, New developments in molecular orbital theory. Rev. Modern Phys.23 (1951) 69-89.
- V.R. Saunders and I.H. Hillier, A ``level-shifting'' method for converging closed shell Hartree-Fock wave functions. Int. J. Quantum Chem.7 (1973) 699-705.
- 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.
- R. Seeger 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.
- R.E. Stanton, The existence and cure of intrinsic divergence in closed shell SCF calculations. J. Chem. Phys.75 (1981) 3426-3432.
- R.E. Stanton, Intrinsic convergence in closed-shell SCF calculations. A general criterion. J. Chem. Phys.75 (1981) 5416-5422.
- M.C. Zerner and M. Hehenberger, A dynamical damping scheme for converging molecular SCF calculations. Chem. Phys. Lett.62 (1979) 550-554.

## Citations in EuDML Documents

top- Konstantin N. Kudin, Gustavo E. Scuseria, Converging self-consistent field equations in quantum chemistry – recent achievements and remaining challenges
- Antoine Levitt, Convergence of gradient-based algorithms for the Hartree-Fock equations
- Antoine Levitt, Convergence of gradient-based algorithms for the Hartree-Fock equations
- Heinz-Jürgen Flad, Reinhold Schneider, s∗-compressibility of the discrete Hartree-Fock equation
- Heinz-Jürgen Flad, Reinhold Schneider, -compressibility of the discrete Hartree-Fock equation
- Antoine Levitt, Convergence of gradient-based algorithms for the Hartree-Fock equations
- Eric Cancès, Gabriel Stoltz, Gustavo E. Scuseria, Viktor N. Staroverov, Ernest R. Davidson, Local Exchange Potentials for Electronic Structure Calculations
- Mathieu Lewin, Séverine Paul, A numerical perspective on Hartree−Fock−Bogoliubov theory

## NotesEmbed ?

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