Polyelectronic atoms : Quantum mechanics

The total electronic spin angular momentum (S) of an atom of N electrons, $\vec{S}=\sum_{i=1}^N\vec{s}_i$, is the vector sum of the angular momenta of spin of individual electrons. The operators $\hat{S}^2$ and $\hat{S}_z$ commute with each other, with the operators of the total orbital angular momentum and with the Hamiltonian of the many-electron atom, which allows to know simultaneously the observables associated with these operators in each quantum state of the polyelectronic atom. Let's see what are the eigenvalues of the operators $\hat{S}^2$ and $\hat{S}_z$: \begin{equation} \hat{S}^2\;\;\rightarrow\;\; \hbar^2 S(S+1) \end{equation} \begin{equation} \hat{S}_z\;\;\rightarrow\;\;M_s\hbar \end{equation} with $M_s$ taking values between -S,........,+S