Polyelectronic atoms : Quantum mechanics

The total orbital angular momentum (L) of an atom of N electrons, $\vec{L}=\sum_{i=1}^N\vec{l}_i$, is the vector sum of the orbital angular momentum of the electrons individual. The operators $\hat{L}^2$ and $\hat{L}_z$ commute with each other and with $\hat{H}$, their eigenvalues being: \begin{equation} \hat{L}^2 \;\;\rightarrow \hbar^2L(L+1) \end{equation} \begin{equation} \hat{L}_z\;\;\;\rightarrow M_L\hbar \end{equation} with $M_L $ taking values between -L,........,+L. Orbital angular momentum is designated by an uppercase letter, while lowercase is for the orbital angular momentum of individual electrons.