Ground state of an ion fluctuating between two magnetic valence states

As a model for mixed-valence thulium which fluctuates between the configurations $f^{13}$, S=(1/2), and $f^{12}$, 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. If we confine ourselves to the simplest variational functions, i.e., those that do not include electron-hole excitations of the Fermi sea, then the splitting between singlet and triplet is very small, of the order of a few percent of the mixing width. (Under the same approximation, this splitting for the case where one of the valence states is nonmagnetic, is of the order of the mixing width.) When excited states of the Fermi sea are included, we find that the 1/N expansion, which shows that in $f^0$ to $f^1$ fluctuating valence the corrections due to these states are small for large N, is not applicable to the present case because the degeneracies of $f^1$ and $f^2$ are comparable, and so N is of order unity. We do a perturbation calculation to all orders, keeping only the dominant logarithmic terms. This has the effect of enhancing the energy separation between singlet and magnetic states by a factor of order 3. This separation is still very small however; we conclude that the magnetism of TmSe is probably caused by interactions between ions.