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

Abstract

AbstractThe 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(α1 × α2). Elements [X] ∈ M(α1 α2) transform under Ga ∈ G according to [Ga] [X][G−1a], where [Ga] and [G−1a] belong 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(α1 × α2) are orthonormal under the trace and transform irreducibly with respect to G. This NITM basis of M(α1 × α2) 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.