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Title: The multifacet graphically contracted function method. II. A general procedure for the parameterization of orthogonal matrices and its application to arc factors

Practical algorithms are presented for the parameterization of orthogonal matrices Q ∈ R {sup m×n} in terms of the minimal number of essential parameters (φ). Both square n = m and rectangular n < m situations are examined. Two separate kinds of parameterizations are considered, one in which the individual columns of Q are distinct, and the other in which only Span(Q) is significant. The latter is relevant to chemical applications such as the representation of the arc factors in the multifacet graphically contracted function method and the representation of orbital coefficients in SCF and DFT methods. The parameterizations are represented formally using products of elementary Householder reflector matrices. Standard mathematical libraries, such as LAPACK, may be used to perform the basic low-level factorization, reduction, and other algebraic operations. Some care must be taken with the choice of phase factors in order to ensure stability and continuity. The transformation of gradient arrays between the Q and (φ) parameterizations is also considered. Operation counts for all factorizations and transformations are determined. Numerical results are presented which demonstrate the robustness, stability, and accuracy of these algorithms.
Authors:
;  [1] ;  [2]
  1. Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439 (United States)
  2. Department of Chemistry and Biochemistry, Gonzaga University, 502 E. Boone Ave., Spokane, Washington 99258-0102 (United States)
Publication Date:
OSTI Identifier:
22420012
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Chemical Physics; Journal Volume: 141; Journal Issue: 6; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; ACCURACY; ALGORITHMS; STABILITY