An examination is made of the effects of spin-spin and spin-other-orbit interactions on the energy levels of $f$-electron configurations. The theory is applied to several rare-earth atoms and ions under a number of simplifying assumptions. Considerable improvements in the fits between experiment and theory are obtained, particularly for the sextets of Gd iv. Electrostatically correlated spin-orbit interaction is studied and found not to be susceptible to parametric absorption into the ordinary spin-orbit and spin-other-orbit interactions. The various contributions to the effective Hamiltonian for two electrons are decomposed into parts having well-defined group-theoretical properties, in preparation for their calculation for any configuration $f^N$.

• Received 15 December 1967

DOI:https://doi.org/10.1103/PhysRev.169.130