Hund's rules: principles for determining atomic ground terms

The three rules

The rules are usually stated in this order:

1. Maximum multiplicity: Choose the term with the largest total spin S (i.e., maximize the number of unpaired electrons and thus the spin multiplicity 2S+1).
2. Maximum orbital angular momentum: For terms with the same S, choose the term with the largest total orbital angular momentum L.
3. J ordering (spin–orbit consideration): For a given S and L, the total angular momentum J determines the final ordering: if the subshell is less than half-filled, the level with the smallest J lies lowest; if more than half-filled, the level with the largest J lies lowest.

Physical basis and limitations

These rules reflect basic tendencies of electron–electron interactions and the spin–orbit coupling in many-electron systems. Rule 1 reduces Coulomb repulsion by keeping electrons with parallel spins in different orbitals; Rule 2 further lowers energy by arranging orbital angular momenta favorably. Rule 3 derives from the sign of the spin–orbit interaction in a given shell. However, Hund's rules are not exact laws: they are approximate and assume LS coupling. They can fail when spin–orbit coupling is strong (heavy elements), in presence of configuration mixing, or for certain electron counts where exchange and correlation effects alter the simple ordering.

Examples and applications

Hund's rules are routinely used to write ground-state term symbols for atoms and ions and to interpret spectral lines. For example, light p-block atoms follow the rules to give characteristic ground terms (e.g., nitrogen's half-filled p subshell leads to a high-spin term). In chemistry, Hund's first rule underlies the arrangement of electrons in atomic and molecular orbitals, affecting bonding, magnetism, and reactivity. Spectroscopists and chemists use these rules when assigning term symbols and predicting multiplet structures.

History and notable facts

Formulated by Friedrich Hund in the early 20th century, the rules were developed from empirical observations of atomic spectra and later given theoretical justification within quantum mechanics. They remain a standard part of atomic theory courses. For more technical treatments and exceptions, readers can consult specialized texts or reviews in atomic structure literature.

While compact and powerful, Hund's rules are a starting point: detailed calculations or experimental data are needed when precision is required or when systems fall outside the assumptions of LS coupling.