By use of the diagonal sum rule, formulas are derived for the weighted mean energy of the multiplets with a given $S$ value in a partly filled shell of equivalent electrons with a given azimuthal quantum number $l$. As an incidental result, a formula is also derived for the mean energy of multiplet states with a given $M_S$. The results can be applied to spectroscopic and magnetic problems. They lead to a straightforward proof that the average energy of multiplets increases with the value of $S$, one of Hund's rules.

• Received 7 August 1967

DOI:https://doi.org/10.1103/PhysRev.165.655