Contents
Introduction The Three Rules Rule 1: Maximum Multiplicity Rule 2: Maximum Angular Momentum Rule 3: J Value Worked Examples Misconceptions FAQ Sources Introduction
Friedrich Hund (1925) [1 ] formulated three empirical rules describing how electrons fill atomic orbitals to give the ground-state configuration of an atom. The rules predict the ground-state term symbol of an atom from its electron configuration, accurately for most atoms in the periodic table. They originated as empirical observations from atomic spectroscopy and were later derived from quantum mechanics combined with electron-electron interactions.
This article walks through the three rules, the physics behind each, worked examples, and the misconceptions that arise from over-applying them.
The Three Rules
For atoms with partially filled subshells, Hund's rules determine the ground state:
Maximize the total spin S.
For given S, maximize the total orbital angular momentum L.
For less-than-half-filled subshells, J = |L − S| (regular); for more-than-half-filled, J = L + S (inverted).
The result is the term symbol ²ˢ⁺¹LJ for the ground state.
Rule 1: Maximum Multiplicity
The ground state has the maximum possible total spin S. Equivalently, electrons singly occupy degenerate orbitals before pairing up.
Why
Electrons with parallel spins have antisymmetric spatial wave functions (by Pauli's exclusion). Antisymmetric spatial wave functions keep electrons farther apart on average, reducing the Coulomb repulsion energy. So maximum-spin configurations have lower energy [2 ].
Example: Nitrogen
N has electron configuration 1s² 2s² 2p³. The three 2p electrons go into separate px , py , pz orbitals with parallel spins. Total S = 3/2; multiplicity 2S+1 = 4 (quartet). Ground term: ⁴S3/2 .
Rule 2: Maximum Angular Momentum
For configurations with the same S, the one with maximum L has lowest energy.
Why
Higher L means electrons circulate in the same rotational sense; this also keeps them apart on average, reducing Coulomb repulsion. The argument is more subtle than for Rule 1 but produces consistent predictions [3 ].
Example: Carbon
C has configuration 1s² 2s² 2p². The two 2p electrons have parallel spins (Rule 1) and align m values to give maximum L. m₁ = 1, m₂ = 0 gives M_L = 1, so L = 1 (P term). Ground term: ³P0 .
Rule 3: J Value
For less-than-half-filled subshells: J = |L − S| has lowest energy (regular multiplet).
Why
Spin-orbit coupling. The interaction ξL·S has opposite sign for electron-like vs hole-like systems. For less than half filled, electrons (negative-energy interaction) prefer low J. For more than half filled, holes flip the sign, preferring high J [4 ].
Examples
Carbon (2p², less than half): ³P0 ground state (J = |1−1| = 0).2 ground state (J = 1+1 = 2).
Worked Examples
Hydrogen (1s¹)
Single electron: S = ½, L = 0, J = ½. Ground state: ²S½ .
Carbon (2p²)
Rule 1: maximum S = 1. Rule 2: maximum L = 1. Rule 3: less than half-filled, J = 0. Ground state: ³P0 .
Nitrogen (2p³)
Half-filled. Maximum S = 3/2; L = 0. Ground state: ⁴S3/2 .
Oxygen (2p⁴)
More than half-filled. S = 1, L = 1, J = 2. Ground state: ³P2 .
Iron (3d⁶)
More than half-filled. Maximum S = 2, L = 2, J = 4. Ground state: ⁵D4 .
Manganese (3d⁵)
Half-filled. S = 5/2, L = 0, J = 5/2. Ground state: ⁶S5/2 .
Common Misconceptions
"Hund's rules are exact"
They are approximate, working well for ground states of atoms with weakly interacting electron shells. For some heavy atoms with strong spin-orbit coupling, jj-coupling rather than LS-coupling is more appropriate, and Hund's rules in LS form may not apply.
"Rule 1 is about spin alignment due to some mysterious force"
It's about Coulomb repulsion. Parallel spins force spatial wave functions to be antisymmetric, keeping electrons apart. The "spin alignment" is statistical, not a force.
"The rules apply to all atoms"
For neutral atoms in their ground states, they work well. Excited states, ions, and certain heavy atoms can deviate.
"You can use Hund's rules for molecules"
In molecules, the orbital structure is very different and Hund's rules in their atomic form don't directly apply. Molecular spin states follow related but different principles.
FAQ
Why is Manganese ⁶S?
Half-filled d shell with all spins parallel gives S = 5/2 and L = 0 (each orbital singly occupied, contributions sum to zero). The pure spin state ⁶S5/2 is the result.
How accurate are Hund's rules for transition metals?
Generally good for 3d transition metals. For 4d, 5d, and especially lanthanides/actinides with significant spin-orbit coupling, deviations occur.
Do Hund's rules predict chemical reactivity?
They predict ground-state term symbols, which influence chemistry. But the rules don't directly give reactivity; that depends on molecular orbital structure, bond strengths, etc.
What is jj-coupling?
For very heavy atoms, spin-orbit coupling dominates over residual electron-electron interactions. Individual electron j values couple first; then total J results. This is jj-coupling. LS coupling (and Hund's rules) is the limit where spin-orbit is weak.
Sources
Hund, F. (1925). "Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel." Zeitschrift für Physik , 33(1), 345–371.
Slater, J. C. (1929). "The theory of complex spectra." Physical Review , 34(10), 1293–1322.
Cowan, R. D. (1981). The Theory of Atomic Structure and Spectra . University of California Press.
Bethe, H. A., Salpeter, E. E. (1957). Quantum Mechanics of One- and Two-Electron Atoms .
Levine, I. N. (2014). Quantum Chemistry , 7th ed. Pearson.