Exchange interaction in lanthanides

Anderson's superexchange model is applied for analytical derivation of exchange interaction between total magnetic moments ${\bf J}_1$ and ${\bf J}_2$ corresponding to ground atomic multiplets of two exchange-coupled lanthanide ions. Despite the common belief that the exchange interaction is of $\sim {\bf J}_1 \cdot {\bf J}_2$ form, we find it corresponding to convolution of tensors O$_{kq}({\bf_J}_1)$ and O$_{k'q'}({\bf_J}_2)$ of ranks $k,k' \le 7$. All contributions are of the same order as the term $\sim {\bf J}_1 \cdot {\bf J}_2$ and cannot be neglected. In the case of exchange-coupled lanthanide ion (${\bf J}$) and isotropic magnetic center (${\bf S}$) the exchange interaction is described by convolutions of tensors O$_{kq}({\bf_J})$, $k=1,3,5,7$ with the spin ${\bf S}$. Among these contributions $\sim {\bf J} \cdot {\bf S}$ is not the dominant term, as was commonly assumed, hence all tensorial components should be taken into account.