Codex @codex  0

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:

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 toThe 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.

With $J=4$, the measured effective paramagnetic moment gives the Landé g-factorFor the free-ion Hund's rules values $S=1$, $L=5$, $J=4$, the stated formula yieldsThusThe source fit is consistent with the unquenched $4f$ orbital magnetic moment of the free praseodymium ion, within graphical accuracy. Small deviations need not indicate a change of electron configuration: crystal field splitting, interactions and the fit range can affect a measured magnetic susceptibility.

For a partially filled $4f$ shell, the outer $5s$ and $5p$ shells shield the localized orbitals from the ligands. Coulomb interactions establish $L,S$ through Hund's rules, and spin-orbit coupling usually exceeds the much smaller crystal field splitting. First form a ground $J$ multiplet using $\mathbf{J}=\mathbf{L}+\mathbf{S}$, then split that multiplet by the crystal environment. The orbital contribution is therefore generally retained, and the g-factor is a multiplet Landé factor rather than an almost spin-only value. Weak external Zeeman and hyperfine interactions can subsequently resolve the remaining states. Exceptions with $J=0$ or unusual valence require separate treatment.

For a partially filled $3d$ shell, electron–electron Coulomb repulsion and Hund's rules first favour particular total spin and orbital terms. The crystal field splitting is substantial because these orbitals overlap the ligand environment; it normally exceeds the spin-orbit coupling and often quenches the orbital moment. Competition between the splitting and the electron pairing energy determines high-spin versus low-spin filling. A common hierarchy is Coulomb term energies, then crystal-field and Hund-exchange scales, then spin–orbit and weak-field Zeeman energies; the relative sizes of crystal-field splitting and pairing are material-dependent. Hyperfine coupling, if the nucleus carries spin, adds a smaller splitting. An isolated ion has no inter-ion exchange ordering field.

For the regular octahedron, repulsion from the six oxygen ligands raises the two $e_g$ orbitals, $d_{z^2}$ and $d_{x^2-y^2}$, above the three $t_{2g}$ orbitals, $d_{xy},d_{xz},d_{yz}$. The latter point between the metal-ligand axes and experience less direct repulsion. The crystal field splitting is therefore a lower triplet and an upper doublet.

For the $3d^2$ V${}^{3+}$ ion, Hund's rules put the two electrons into different $t_{2g}$ orbitals with parallel spins, giving $S=1$. There are three choices for the unoccupied $t_{2g}$ orbital, so the orbital ground state is triply degenerate. Its spin degeneracy is separate. This orbital degeneracy makes a symmetry-lowering Jahn-Teller effect possible.