Normalized irreducible tensorial matrices and the Wigner–Eckart theorem for unitary groups: A superposition hamiltonian constructed from octahedral NITM

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.