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Normalized irreducible tensorial matrices and the Wigner–Eckart theorem for unitary groups: A superposition hamiltonian constructed from octahedral NITM
Author(s) -
Ellzey M. L.
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.560410502
Subject(s) - unitary state , basis (linear algebra) , mathematics , pure mathematics , linear subspace , group (periodic table) , space (punctuation) , quantum mechanics , geometry , physics , linguistics , philosophy , political science , law
The concepts of normalized irreducible tensorial matrices ( NITM ) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M (α 1 × α 2 ). Elements [ X ] α 1 α 2∈ M (α 1 α 2 ) transform under G a ∈ G according to [ G a ] α 1[ X ] α 1 α 2[ G −1 a ] α 2, where [ G a ] α 1and [ G −1 a ] α 2belong to irreducible representations of G . The usual properties of NITM and the Wigner–Eckart theorem follow from these results, which are valid for both finite and compact unitary groups. The NITM span M (α 1 × α 2 ) are orthonormal under the trace and transform irreducibly with respect to G . This NITM basis of M (α 1 × α 2 ) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffith's V coefficients for the octahedral group are tabulated and used to construct a six‐coordinate superposition Hamiltonian.