The Relationship Between Spin, Orbital and Total Angular Momentum

For an atom in groups 1 or 2, or periods, 2 or 3 of the periodic table, any electron with spin oriented up will be opposed by an electron in the same otherwise identical set of quantum numbers with opposing spin, and any electron with values of angular momentum quantum numbers and magnetic quantum numbers
$l$
and
$m$
and respectively will be similarly opposed, so that spin and angular momentum will sum to zero for electrons in full shells.
The spin and angular momentum of the atoms are due solely (for atoms in the ground state) to electrons in the outermost shell.
If electron in the outermost shell have spins
$s_1, \, s_2,...,s_k$
the total spin in
$s=s_1+s_2+...+s_k$
and the magnitude of the total spin is
$S=\sqrt{s(s+1)} \hbar$
.
If electron in the outermost shell have orbital angular momentum numbers
$l_1, \, l_2,...,s_k$
then the total orbital angular momentum is
$l=l_1+l_2+...+l_k$
and the magnitude of the total orbital angular momentum is
$L=\sqrt{l(l+1)} \hbar$
.
The total angular momentum number is
$j=s+l$
and the magnitude of the total angular momentum is
$\sqrt{j(j+1)} \hbar$
.

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