Intermediate coupling

For some rare earth ions and for transition metals or actinides it is necessary to include more single-ion ion states with different L, S into the calculation. Two-ion intermediate interaction can be written as

  $$\hat{\mathcal{H}}_{IC} = - \frac{1}{2} \sum_{nn^{\prime},... ..._{n}) \hat{\mathcal{I}}_{\alpha}^n \hat{\mathcal{I}}_{\beta}^{n^{\prime}}$$ (9)

where the operators are understood as $\hat{\mathcal{I}}_{1} \leftrightarrow \hat{S}_x$, $\hat{\mathcal{I}}_{2} \leftrightarrow \hat{S}_y$, $\hat{\mathcal{I}}_{3} \leftrightarrow \hat{S}_z$, $\hat{\mathcal{I}}_{4} \leftrightarrow \hat{L}_x$, $\hat{\mathcal{I}}_{5} \leftrightarrow \hat{L}_y$, $\hat{\mathcal{I}}_{6} \leftrightarrow \hat{L}_z$.