An efficient method for calculating maxima of symmetric homogeneous functions of orthogonal matrices: Applications to localized occupied orbitals

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The authors present here three new algorithms (one purely iterative and two DIIS-like) to compute maxima of symmetric homogeneous functions of orthogonal matrices. These algorithms revolve around the mathematical lemma that, given an invertible matrix A, the function f(U) = Tr(AU) has exactly one local (and global) maximum for U special orthogonal (i.e. UU{sup T} = 1 and det(U) = 1). This is proved in the appendix. One application of these algorithms is the computation of localized orbitals, including, for example, Boys and Edmiston-Reudenberg (ER) orbitals. The Boys orbitals are defined as the set of orthonormal orbitals which, for a ... continued below

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32 pages

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Subotnik, Joseph E.; Shao, Yihan; Liang, WanZhen & Head-Gordon, Martin June 4, 2004.

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The authors present here three new algorithms (one purely iterative and two DIIS-like) to compute maxima of symmetric homogeneous functions of orthogonal matrices. These algorithms revolve around the mathematical lemma that, given an invertible matrix A, the function f(U) = Tr(AU) has exactly one local (and global) maximum for U special orthogonal (i.e. UU{sup T} = 1 and det(U) = 1). This is proved in the appendix. One application of these algorithms is the computation of localized orbitals, including, for example, Boys and Edmiston-Reudenberg (ER) orbitals. The Boys orbitals are defined as the set of orthonormal orbitals which, for a given vector space of orbitals, maximize the sum of the distances between orbital centers. The ER orbitals maximize total self-interaction energy. The algorithm presented here computes Boys orbitals roughly as fast as the traditional method (Jacobi sweeps), while, for large systems, it finds ER orbitals potentially much more quickly than traditional Jacobi sweeps. In fact, the required time for convergence of the algorithm scales quadratically in the region of a few hundred basis functions (though cubicly asymptotically), while Jacobi sweeps for the ER orbitals traditionally scale as the number of occupied orbitals to the fifth power. As an example of the utility of the method, they provide the ER orbitals of nitrated and nitrosated benzene, and they discuss the chemical implications.

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32 pages

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OSTI as DE00836400

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  • Journal Name: Journal of Chemical Physics; Journal Volume: 121; Journal Issue: 19; Other Information: Submitted to Journal of Chemical Physics: Volume 121, No.19; Journal Publication Date: 11/15/2004

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  • Report No.: LBNL--55997
  • Grant Number: AC03-76SF00098
  • DOI: 10.1063/1.1790971 | External Link
  • Office of Scientific & Technical Information Report Number: 836400
  • Archival Resource Key: ark:/67531/metadc786512

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  • June 4, 2004

Added to The UNT Digital Library

  • Dec. 3, 2015, 9:30 a.m.

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  • Sept. 21, 2017, 6 p.m.

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Subotnik, Joseph E.; Shao, Yihan; Liang, WanZhen & Head-Gordon, Martin. An efficient method for calculating maxima of symmetric homogeneous functions of orthogonal matrices: Applications to localized occupied orbitals, article, June 4, 2004; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc786512/: accessed September 19, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.