Hund's Rules and Ground-State Terms

The term symbols of a configuration list the allowed levels but not their order. For chemistry and magnetism the one that matters is the lowest, the ground term. Hund's three rules name it directly from the configuration, without diagonalizing anything, and each rule traces to a piece of physics already assembled: exchange, the anisotropy of the Coulomb repulsion, and the sign of the spin-orbit coupling.

The three rules

For the ground term of a given configuration in LS coupling, apply in order:¹

• Rule 1 — maximum $S$. The ground term has the largest total spin $S$ the Pauli principle permits.
• Rule 2 — maximum $L$. Among the terms of that largest $S$, the ground term has the largest total orbital angular momentum $L$.
• Rule 3 — the $J$ value. For a subshell less than half full, the ground level has $J=|L-S|$; for a subshell more than half full, $J=L+S$. A half-full subshell has $L=0$, so $J=S$ and the question does not arise.

The rules fix the ground level only. They are reliable for the lowest term of a configuration and are not a general ordering of the excited terms.

Why maximum spin and maximum L

Rules 1 and 2 are electrostatic. Aligning spins forces the spatial wave function to be antisymmetric, which keeps the electrons apart and lowers their mutual Coulomb repulsion — the same exchange energy that put the helium triplet below the singlet. A larger $S$ means more parallel pairs, each contributing a favourable exchange term, so the highest-spin term lies lowest.¹

Rule 2 is subtler but the same in spirit. For fixed $S$, a larger $L$ corresponds to the electrons orbiting the nucleus in the same rotational sense, so they meet less often and their average repulsion is smaller. Both rules lower the electrostatic energy by keeping electrons out of each other's way; only the mechanism (spin correlation versus orbital correlation) differs.

Why the J rule flips at half filling

Rule 3 is spin-orbit. Within the ground term, the spin-orbit energy is $A\vec{L}\cdot \vec{S}$ with the level energies

so the sign of the constant $A$ decides which $J$ lies lowest. Summing the one-electron coefficients $\xi (r_i){\vec{\ell }}_i\cdot {\vec{s}}_i$ over the electrons of a subshell gives an $A$ that is positive when the subshell is less than half full and negative when it is more than half full.¹ The reason is the particle-hole symmetry: a more-than-half subshell is better described by its positively contributing holes, whose spin-orbit coupling carries the opposite sign.

• $A>0$ (less than half): $E_J$ increases with $J$, so $J=|L-S|$ is lowest — a normal multiplet.
• $A<0$ (more than half): $E_J$ decreases with $J$, so $J=L+S$ is lowest — an inverted multiplet.

Worked ground terms

The recipe is mechanical. Lay out the subshell's $m_{\ell }$ boxes, fill them to maximize spin (one electron per box, all parallel, before pairing), then read off $S$, $L=|M_L^{\max }|$, and $J$.¹

The two halves of a shell

The pattern across a subshell is symmetric about the half-filled point. Spin rises to a maximum at half filling and falls back; $L$ is zero at both the empty and half-filled shells; and the $J$ rule switches from $J=|L-S|$ to $J=L+S$ as the shell crosses half. A subshell and its hole-complement share the same term set but sit on opposite sides of the $J$ switch, so carbon ($p^2$) and oxygen ($p^4$) both have a ${}^{3}P$ ground term, normal for carbon and inverted for oxygen.

Hund's rules close the many-electron chain that began with the central-field approximation: the central field gives the configuration, the residual electrostatic interaction sorts it into terms, the spin-orbit interaction splits each term into levels, and the three rules pick the ground level. The ground terms they produce are the starting point for an atom's behaviour in an external field, taken up in the Zeeman effect.