Physics:Racah seniority number

The Racah seniority number (seniority quantum number) [math]\displaystyle{ \nu }[/math] 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 [math]\displaystyle{ L }[/math] and total spin [math]\displaystyle{ S }[/math], which gives the degree of unpaired particles. A spin-independent interaction [math]\displaystyle{ \hat{V} }[/math] is assumed with the property

[math]\displaystyle{ \langle l^2;LM_L|\hat{V}|l^2;LM_L \rangle = g(2l+1)\delta_{L0} }[/math],

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