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

Reinhold Schneider, Thorsten Rohwedder, Alexey Neelov & Johannes Blauert

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 steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the admissible manifold, i.e., the invariance with respect to unitary transformations, to reformulate the problem on the Grassmann manifold as the admissible set. We then prove asymptotical linear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the Grassmann manifold at the minimizer.

  • Keywords

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

  • AMS Subject Headings

65Z05, 58E50, 49R50.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-360, author = {Reinhold Schneider, Thorsten Rohwedder, Alexey Neelov and Johannes Blauert}, 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 steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the admissible manifold, i.e., the invariance with respect to unitary transformations, to reformulate the problem on the Grassmann manifold as the admissible set. We then prove asymptotical linear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the 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 AU - Reinhold Schneider, Thorsten Rohwedder, Alexey Neelov & Johannes Blauert 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, Grassmann manifolds, Optimization, Orthogonality constraints, Hartree-Fock theory, Density functional theory, PINVIT. AB -

In this article, we analyse three related preconditioned steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the admissible manifold, i.e., the invariance with respect to unitary transformations, to reformulate the problem on the Grassmann manifold as the admissible set. We then prove asymptotical linear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the Grassmann manifold at the minimizer.

Reinhold Schneider, Thorsten Rohwedder, Alexey Neelov and Johannes Blauert. (2009). 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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