= Solution
{c}
For a fixed open-shell <electron configuration> in the <LS coupling> regime, <Hund's rules> select the free-<ion> ground term in the following order:
* Maximize the total <spin> $S$, subject to the <Pauli exclusion principle>.
* Among terms with that $S$, maximize the total <orbital angular momentum> $L$.
* For a less-than-half-filled subshell choose the smallest total <angular momentum>, $J=|L-S|$; for a more-than-half-filled subshell choose the largest, $J=L+S$. A half-filled subshell has $L=0$ and $J=S$.
The first two rules come from the <Coulomb interaction> between <electrons> within the partially filled subshell, including its <exchange interaction>. Parallel <spins> require an antisymmetric spatial <wavefunction>, reducing close approach and often lowering the repulsive <Coulomb interaction> <energy>. Among the allowed high-<spin> terms, the angular dependence of the <Coulomb interaction> generally favours the largest $L$; a simple magnetic alignment argument alone does not derive this second ordering rule.
The weaker <spin-orbit coupling> then splits a selected $L,S$ term. In <LS coupling>, its term-level <energy> is proportional to
$$
\langle\mathbf L\cdot\mathbf S\rangle
=\frac{\hbar^2}{2}\left[J(J+1)-L(L+1)-S(S+1)\right].
$$
The effective sign appropriate to less-than-half versus more-than-half filling gives the third rule. These rules assume the electrostatic term separation dominates <spin-orbit coupling>; strong <crystal field splitting> or other coupling regimes require further analysis.
Back to article page
