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Algebrants in many‐electron quantum mechanics: Applications of generalized determinants or matrix functions
Author(s) -
Poshusta R. D.,
Kinghorn D. B.
Publication year - 1992
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560410106
Subject(s) - gaussian elimination , multilinear map , generalization , matrix (chemical analysis) , quantum computer , computation , gaussian , algebra over a field , analogy , mathematics , quantum , computer science , pure mathematics , statistical physics , quantum mechanics , physics , algorithm , chemistry , mathematical analysis , linguistics , philosophy , chromatography
We define the algebrant , a mathematical generalization of the determinant, the immanant, the permanent, and the Schur functions. Algebrants are classified as multilinear matrix functions or multicomponent symmetrized tensors. In applications, such as N ‐electron quantum mechanics, where extensive computation is required, it is vital to reduce computational effort, e.g., the well‐known N ‐factorial problem. We derive certain mathematical properties that can be incorporated in efficient computing algorithms for algebrants. Foremost is our “elimination theorem,” which allows (in important special cases) zeros to be introduced into an algebrant in close analogy with Gaussian elimination for determinants. Savings accruing from such elimination can be substantial. We show examples from Matsen's spin‐free quantum chemistry where elimination effectively removes the N ‐factorial problem that has hitherto stifled possible applications.