Coulomb energies of icosahedral h orbitals

Abstract

Matrix elements of all two-electron and three-electron operators that are scalar with respect to the icosahedral group I have been tabulated for the icosahedral configurations hN. These operators represent the Coulomb interaction between electrons occupying h orbitals, and also the effects (to the lowest orders of perturbation theory) of configuration interaction on the levels of hN. States and operators are labelled by the irreducible representations (irreps) of the continuous groups SO(3) and SO(5) in addition to the irreps of I. An alternative scheme is introduced in which the irreps W of SO(5) are retained, but the orbital angular-momentum quantum numbers L associated with SO(3) are replaced by the irreps of the permutation groups S5 and S6, the latter corresponding to the interchanges (possibly nonfeasible) of the six fivefold axes of an icosahedron among themselves. The kaleidoscope operator 𝒦, which rotates the weight space of SO(5) by π/2, is an element of S5 and S6, and can be used to characterize the operators. The energy matrices in the second scheme are particularly simple, the scalar or pseudoscalar nature of the operators with respect to S5 leading to block forms either on the diagonal or off the diagonal, respectively. Operators of the former kind are invariant under the 𝒦 operation and, in the hypothetical absence of the pseudoscalars, would lead to every level of icosahedral type T1 being degenerate with a level of type T2.