Ground State of an Ion Fluctuating Between two Magnetic Valence States

As a model for mixed-valence thulium which fluctuates between the configurations the inverse of the function aup 13, S = 1 over 2, and the inverse of the function sup 12, S = 1, we consider an ion with orbital degeneracy embedded in a free-electron continuum. We construct variational states of different total-spin multiplicity, namely singlet, doublet, and triplet. We find that the ground state is a singlet, with the triplet and doublet lying higher in energy and in that order.