A non-Euclidean crystal field for f electrons

Abstract

To honor the memory of Clyde A. Morrison, some aspects are discussed of the two-electron crystal field that exhibits the symmetry of a regular didodecahedron, a non-Euclidean figure in which 24 regular heptagons meet in threes at 56 vertices. The group D, which comprises the 168 elements representing the rotations that send the didodecahedron into itself, is a subgroup of the group G 2 that Racah used for f electrons, but it is not a subgroup of SO(3). Reflections extend D to D h. The matrix elements of four two-electron operators scalar with respect to D h, which have been previously evaluated in f 2 and f 3 within an SO(3) basis, are studied to find some isoscalar factors involving the irreducible representations of G 2 and D h. Selection rules and reciprocity relations are examined, and two unexpectedly null matrix elements of an operator scalar with respect to D but odd under reflection are accounted for. The effect of a perturbation possessing didodecahedral symmetry on the energy levels of a free ion is shown in a correlation diagram for the singlet terms of f 2.