The direct product structure of atomic orbital manifolds: extension to non-Euclidean permutation groups

Abstract

Atomic orbital manifolds are described as Γ × Γ direct products of degenerate irreducible representations (irreps) of odd dimension in suitable groups, which may be either the polyhedral symmetry point groups (T, O, I) or larger simple non-Euclidean permutation groups containing the polyhedral symmetry point groups as subgroups. Such larger non-Euclidean groups include the simple pollakispolyhedral groups 7 O [≈L 2(7)] and 11 I [≈L 2(11)] of orders 168 and 660, respectively, and the simple alternating groups A 6 and A 7 of orders 360 and 2520, respectively. Thus the 9-dimensional (9D) direct product T × T of a 3D irrep of the polyhedral or larger point groups leads to the nine-orbital sp3d5 manifold. In icosahedral symmetry I this 9D direct product splits into separate irreps for the s, p and d orbitals, whereas in the larger non-Euclidean 7 O group this direct product splits only into a 6D irrep for the gerade orbitals (s + d) and a 3D irrep for the ungerade p orbitals. A similar method can be used to study the 25-orbital sp3d5f7g9 manifold as H × H direct products in the icosahedral group I and the two larger simple non-Euclidean groups A 6 and 11 I, in which I is a subgroup of index 6 and 11, respectively. The mathematics implicit in these direct products appears to have some connections with a few areas of chemical physics including icosahedral quasicrystals and unusual degeneracies in the Coulomb energies of terms in transition metal atomic spectra.