Interpretation of higher order magnetic effects in the spectra of transition metal ions in terms of SO(5) and Sp(10)

Small discrepancies in the fitted energy levels of the configurations 3d(N) of transition metal ions are ascribed to effective three-electron magnetic operators y(i). Surprisingly it has been found that, of the 16 possible operators with ranks 1 in both spin and orbital spaces, four operators labeled by the irreducible representation (irrep) (11) of SO(5) are sufficient to obtain results which appear to be limited by the errors in the experimental energy levels. An interpretation is given involving products of operators labeled by the irreps of SO(5) and the symplectic group Sp(10).