To understand the commonly quoted magnetic values of coordination complexes (central ion) we use $$m_l=\sqrt{n(n+2)} \text{BM where BM}=\frac{e\hbar}{2m_e}\text{JT}^{-1}$$ $n$=number of unpaired electrons.
How did we derive this equation?
I know the orbital angular momentum of electron is given by $\sqrt{l(l+1)}\hbar$, spin angular momentum is given by $\sqrt{s(s+1)}\hbar$ and the gyromagnetic ratio for electron is $$\vec \mu_B=\frac{-e}{2m_e}\vec L$$. There is also the "total" angular momentum given by $n_0\hbar$.
I fail to see how these magnetic moments and angular momentums are related, and how they can be added or related to get the corresponding values for an entire atom or a complex?.
I have also heard that the Lande g-factor is relevant here.
Key:-
$n_0$=principal quantum number, $l$=azimuthal quantum number,$s$=spin (half for an electron), $\hbar=\frac{h}{2\pi}$, $S=\Sigma s+1$(spin multiplicity), $\vec L$=angular momentum, $\vec \mu_B$=Magnetic moment.
