Many-electron atoms and Hund’s rules

A subshell of orbital quantum number l has $2(2l+1)$ spin-orbitals. Choosing N occupied spin-orbitals gives a binomial number of Slater determinants, with each complete spin-orbital occupied at most once. A closed subshell has total L and S zero, so open subshells carry the angular-momentum structure. Configuration parity is $(-1)^{\sum_i l_i}$, which is unchanged by coupling the same occupied orbitals into different terms. Parity is separate from spin multiplicity.

For two equivalent p electrons there are six available spin-orbitals and fifteen determinants. Their allowed terms are $^1S$, $^1D$, and $^3P$, with dimensions one, five, and nine. The sum is fifteen. A tempting $^3D$ term from naive addition is forbidden by electron antisymmetry, rather than merely high in energy. This check is stronger than drawing two arrows in boxes: it verifies the entire angular-momentum state count and exposes terms that cannot occur.

For two electrons in distinct orthonormal spatial orbitals, symmetric and antisymmetric spatial combinations have different Coulomb expectation values. To make the complete fermionic wavefunction antisymmetric, the spin singlet uses the symmetric spatial combination and the spin triplet uses the antisymmetric one. The direct integral J is shared, while the exchange integral K enters with opposite signs. For a repulsive Coulomb interaction in this simple fixed-orbital comparison, positive K lowers the triplet relative to the singlet.

This energy difference does not arise from a new force that recognizes parallel arrows. The same Coulomb interaction acts in both states; antisymmetry changes their spatial pair probability. Also, a singlet is not synonymous with two opposite definite spins, because its spin wavefunction is a coherent antisymmetric superposition. Orbital relaxation and correlation change quantitative energies. An exchange explanation is a useful mechanism, but claiming that one fixed-orbital argument proves every Hund ordering would extend it beyond its assumptions.

For a given configuration, the first Hund rule usually favors the largest allowed total spin. Among terms with that spin, the second favors the largest allowed L. For the resulting ground term, the third favors the smallest J for a subshell less than half full and the largest J for one more than half full. At half filling the maximum-spin term often has L zero. These rules concern energy ordering, whereas triangle rules and Pauli restrictions concern existence.

The p-squared example therefore favors $^3P$ and, below half filling, its J equal to zero level. The p-to-the-fourth configuration has two holes and the same term types, but the ground $^3P$ level has J equal to two under the third rule. Strong spin-orbit coupling, configuration mixing, or accidental near degeneracy can weaken this picture. A simple two-state phase instrument can distinguish singlet-like and triplet-like spin combinations in a restricted subspace, but it cannot enforce orbital antisymmetry or calculate exchange energies.