Developments in the Theory of Complex Spectra

The use of orthogonal operators to fit atomic energy levels often requires the calculation of matrix elements of multi-electron operators defined by the irreducible representations (irreps) of Lie groups. One way to do this is to imagine the 2^{4/ +2} states of an atomic l shell as arising from the coupling in SO(2l + 1) of four inequivalent quasi-particles, each of which is labeled by the elementary spinor irrep (½ ½... ½) (of dimension 2^l) of SO(2l + 1). Two parity labels (specifying the oddness or evenness of the numbers of electrons with spins up and spins down) complete the classification scheme, which leads to N (electron number) and S (total spin) as good quantum numbers. A unique correspondence (to within a phase) is established with the classic states of Racah. The Coulomb operators e₂ and e₃ for f electrons are examined, and the selection rule ΔS = 0 for e₃ reappears as symmetry conditions on the 9-W symbols, where W is an irrep of SO(7). Work underway (with Shaozhong Li) is outlined. In keeping with a presentation to a general EGAS audience, these matters are described broadly and through examples.