Racah seniority number

The Racah seniority number (seniority quantum number) $\nu$ was introduced by Giulio Racah for the classification of electrons in an atomic configuration. The "seniority number", in a loosing statement, is quantum number additional to the total angular momentum $L$ and total spin $S$, which gives the degree of unpaired particles.

A spin-independent interaction $\hat{V}$ is assumed with the property

$⟨l^2;L{M}_L|\hat{V}|l^2;L{M}_L⟩=g(2l+1){\delta }_{L0}$,

where $L$ is the combined angular momentum, $M_L$ magnetic quantum number, $l$ is electrons' orbital angular momenta, and $g$ is the dimensionless magnetic moment. The equation above shows there is no interaction unless the two electrons' orbital angular momenta are coupled to $L=0$. The eigenvalue is the "seniority number" $\nu$.