Monomial representation of point-group symmetries

Abstract

Most irreducible-matrix representations in point-group symmetries can adopt monomial form. In that case all representational matrices have only one nonzero element in each row and column. The so-obtained standard basis choice contrasts with the conventional Wigner-Racah option. Monomial representations give rise to interesting properties of the corresponding Clebsch-Gordan series: All coupling coefficients are equal in absolute value and a natural intrinsic multiplicity separation is obtained. The concept is also useful in reaching a consistent solution of the multiplicity problem in the reduction of direct products, involving the fourfold U' representation of the octahedral spinor group. Several tables of basis transformations and coupling coefficients in octahedral and icosahedral symmetries are included.

Symmetry relations in the property surfaces of icosahedral molecules

“…the number of times that IV occurs in the product IVi • F# [17]. The Clebsch-Gordan coupling coefficients (FiviiVjYjlFT) may be calculated from the A matrix [18] :…”