Exchange interaction between $J$-multiplets

The crystal-field levels in lanthanides and other metal complexes with unquenched orbital momentum originate from the ground atomic $J$-multiplet. It was long believed that the exchange interaction between $J$-multiplets is basically described by Heisenberg form, $\mathbf{J}_1 \cdot \mathbf{J}_2$. In this work, Anderson's superexchange model is applied for analytical derivation of exchange interaction between arbitrary $\mathbf{J}_1$ and $\mathbf{J}_2$ multiplets [1]. The structure and the energy spectrum of the obtained exchange Hamiltonian are significantly different from those of a Heisenberg Hamiltonian. Besides, it is also found that the $1/U$ approximation [2] is not applicable for the description of exchange spectrum, since it gives qualitatively different predictions compared to the present treatment. Similar results are obtained for the exchange interaction between $J$-multiplet ($\mathbf{J}_1$) and isotropic magnetic center ($\mathbf{S}_2$). \\[4pt] [1] N. Iwahara and L. F. Chibotaru, submitted to Phys. Rev. Lett.\\[0pt] [2] P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. Lander, Rev. Mod. Phys. {\bf 81}, 807, (2009).