Average Energy of States of Given Multiplicities in Atoms

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 MS. 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.