×

Preconditioned gradient flows for nonlinear eigenvalue problems and application to the Hartree-Fock functional. (English) Zbl 1166.65039

An exponentially convergent numerical method for the eigenvalue problem for a Hermitian operator is discussed. This method reduces the eigenvalue problem to the solution of an initial value problem for a system of the first order differential equations (gradient flow) under some constraints. Two modifications of the method leading to a reduction of the computational costs and to numerical stability are proposed. Numerical examples including the Hartree-Fock problem are given.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
34L30 Nonlinear ordinary differential operators
65L20 Stability and convergence of numerical methods for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Szabo, Modern quantum chemistry: introduction to advanced electronic structure theory (1996)
[2] Cancès, On the convergence of SCF algorithms for the Hartree-Fock equations, Math Model Numer Anal 34 pp 749– (2000) · Zbl 1090.65548
[3] Kresse, Efficient iteratives schemes for ab initio total-energy calculations using a plane-wave basis set, Phys Rev B 54 pp 11169– (1996)
[4] Pulay, Convergence acceleration of iterative sequences. The case of SCF iteration, Chem Phys Lett 73 pp 393– (1980)
[5] C. Audouze, Méthodes performantes d’approximations de solutions en chimie quantique moléculaire, Orsay University, 2004.
[6] Helmke, Optimization and dynamical systems (1994) · doi:10.1007/978-1-4471-3467-1
[7] Parr, Density functional theory of atoms and molecules (1989)
[8] Lieb, On solutions to the Hartree-Fock problem for atoms and molecules, J Chem Phys 61 pp 735– (1974)
[9] Lieb, The Hartree-Fock theory for coulomb systems, Commun Math Phys 53 pp 185– (1977)
[10] Lions, Solutions of Hartree-Fock equations for coulomb systems, Commun Math Phys 109 pp 33– (1987) · Zbl 0618.35111
[11] Alouges, A new algorithm for computing liquid crystal stable configurations: the Harmonic mapping case, SIAM J Numer Anal 34 pp 1708– (1997) · Zbl 0886.35010
[12] Payne, Iterative minimization techniques for Ab-initio total-energy calculations: molecular dynamics and conjugate gradients, Rev Modern Phys 64 pp 1045– (1992)
[13] http://www.abinit.org.
[14] Crouzeix, Analyse numérique des équations différentielles (1992)
[15] Nougier 2 (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.