Volume 27, Issue 2-3
Direct Minimization for Calculating Invariant Subspaces in Density Functional Computations of the Electronic Structure
DOI:

J. Comp. Math., 27 (2009), pp. 360-387.

Published online: 2009-04

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• Abstract

In this article, we analyse three related preconditioned steepestdescent algorithms, which are partially popular in Hartree-Fock andKohn-Sham theory as well as invariant subspace computations, fromthe viewpoint of minimization of the corresponding functionals,constrained by orthogonality conditions. We exploit the geometry ofthe admissible manifold, i.e., the invariance with respect tounitary transformations, to reformulate the problem on the Grassmannmanifold as the admissible set. We then prove asymptotical linearconvergence of the algorithms under the condition that the Hessianof the corresponding Lagrangian is elliptic on the tangent space ofthe Grassmann manifold at the minimizer.

• Keywords

Eigenvalue computation Grassmann manifolds Optimization Orthogonality constraints Hartree-Fock theory Density functional theory PINVIT

65Z05 58E50 49R50.

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@Article{JCM-27-360, author = {}, title = {Direct Minimization for Calculating Invariant Subspaces in Density Functional Computations of the Electronic Structure}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {2-3}, pages = {360--387}, abstract = {

In this article, we analyse three related preconditioned steepestdescent algorithms, which are partially popular in Hartree-Fock andKohn-Sham theory as well as invariant subspace computations, fromthe viewpoint of minimization of the corresponding functionals,constrained by orthogonality conditions. We exploit the geometry ofthe admissible manifold, i.e., the invariance with respect tounitary transformations, to reformulate the problem on the Grassmannmanifold as the admissible set. We then prove asymptotical linearconvergence of the algorithms under the condition that the Hessianof the corresponding Lagrangian is elliptic on the tangent space ofthe Grassmann manifold at the minimizer.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8577.html} }
TY - JOUR T1 - Direct Minimization for Calculating Invariant Subspaces in Density Functional Computations of the Electronic Structure JO - Journal of Computational Mathematics VL - 2-3 SP - 360 EP - 387 PY - 2009 DA - 2009/04 SN - 27 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8577.html KW - Eigenvalue computation KW - Grassmann manifolds KW - Optimization KW - Orthogonality constraints KW - Hartree-Fock theory KW - Density functional theory KW - PINVIT AB -

In this article, we analyse three related preconditioned steepestdescent algorithms, which are partially popular in Hartree-Fock andKohn-Sham theory as well as invariant subspace computations, fromthe viewpoint of minimization of the corresponding functionals,constrained by orthogonality conditions. We exploit the geometry ofthe admissible manifold, i.e., the invariance with respect tounitary transformations, to reformulate the problem on the Grassmannmanifold as the admissible set. We then prove asymptotical linearconvergence of the algorithms under the condition that the Hessianof the corresponding Lagrangian is elliptic on the tangent space ofthe Grassmann manifold at the minimizer.

Reinhold Schneider, Thorsten Rohwedder, Alexey Neelov & Johannes Blauert. (2019). Direct Minimization for Calculating Invariant Subspaces in Density Functional Computations of the Electronic Structure. Journal of Computational Mathematics. 27 (2-3). 360-387. doi:
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