The multifacet graphically contracted function method. II. A general procedure for the parameterization of orthogonal matrices and its application to arc factors
- Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne, Illinois 60439 (United States)
- Department of Chemistry and Biochemistry, Gonzaga University, 502 E. Boone Ave., Spokane, Washington 99258-0102 (United States)
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.
- OSTI ID:
- 22420012
- Journal Information:
- Journal of Chemical Physics, Vol. 141, Issue 6; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9606
- Country of Publication:
- United States
- Language:
- English
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