Hund's Rule of Maximum Multiplicity

Hund's Rule of Maximum Multiplicity is one of the three fundamental rules governing how electrons occupy atomic orbitals, alongside the Aufbau principle and the Pauli Exclusion Principle. Formulated by the German physicist Friedrich Hund in 1927, it states that when electrons occupy orbitals of equal energy — known as degenerate orbitals — each orbital must receive one electron before any orbital receives a second. Furthermore, all singly occupied orbitals must contain electrons with the same spin quantum number, either all spin-up (ms = +½) or all spin-down (ms = −½). This maximizes the total spin multiplicity of the atom, which is defined as 2S + 1, where S is the total spin quantum number. The physical basis for Hund's Rule lies in electron–electron repulsion. When two electrons are forced to share the same orbital, they occupy the same region of space and repel each other strongly due to their like charges. By spreading out across different degenerate orbitals, electrons minimize this repulsion and lower the overall energy of the atom, producing a more stable ground-state configuration. The requirement for identical spin directions in singly occupied orbitals further reduces repulsion through a quantum mechanical effect called exchange energy: parallel-spin electrons avoid each other more effectively because the Pauli Exclusion Principle prevents them from ever being in the same quantum state simultaneously. Hund's Rule is especially important when filling p, d, and f subshells, which contain 3, 5, and 7 degenerate orbitals respectively. Classic examples include carbon (Z = 6) with two electrons in its 2p subshell occupying separate orbitals with parallel spins, nitrogen (Z = 7) with three 2p electrons each in separate orbitals, and chromium (Z = 24) whose half-filled 3d subshell exemplifies the special stability associated with maximum multiplicity. Understanding this rule is essential for correctly writing electron configurations, predicting magnetic properties such as paramagnetism and diamagnetism, and interpreting atomic spectra.

Key takeaways
  • Identify the Subshell Being Filled
  • Singly Occupy Each Orbital First
  • Assign Parallel Spins to All Singly Occupied Orbitals
  • Begin Pairing Only After All Orbitals Are Half-Filled
  • Determine Magnetic Properties from Unpaired Electrons

Identify the Subshell Being Filled

Determine which subshell the valence electrons enter. Use the Aufbau filling order (1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p …). Hund's Rule applies when more than one degenerate orbital is available — that is, within a p (3 orbitals), d (5 orbitals), or f (7 orbitals) subshell.

Singly Occupy Each Orbital First

Place one electron in each degenerate orbital before doubling up. For the 2p subshell of nitrogen (three electrons to place), electron 1 enters 2p_x, electron 2 enters 2p_y, and electron 3 enters 2p_z. No pairing occurs until all orbitals have one electron each.

Assign Parallel Spins to All Singly Occupied Orbitals

By convention, the first electron in each orbital is assigned spin-up (↑, ms = +½). All singly occupied orbitals in the same subshell must have electrons of the same spin. For nitrogen's 2p³, the configuration is 2p_x(↑) 2p_y(↑) 2p_z(↑), giving a total spin S = 3/2 and a spin multiplicity of 2S + 1 = 4 (a quartet state).

Begin Pairing Only After All Orbitals Are Half-Filled

Once every degenerate orbital contains one electron, the next electron must pair with an existing electron, adopting the opposite spin (↓, ms = −½) in accordance with the Pauli Exclusion Principle. For oxygen (four 2p electrons): 2p_x(↑↓) 2p_y(↑) 2p_z(↑). The two unpaired electrons make oxygen paramagnetic — it is weakly attracted to magnetic fields.

Determine Magnetic Properties from Unpaired Electrons

Count unpaired electrons to predict magnetic behavior. Atoms with one or more unpaired electrons are paramagnetic (attracted to magnetic fields). Atoms with all electrons paired are diamagnetic (weakly repelled by magnetic fields). Manganese in its ground state [Ar] 3d⁵ 4s² has five unpaired 3d electrons — one in each of the five degenerate 3d orbitals — making it strongly paramagnetic with a multiplicity of 2(5/2)+1 = 6.

The Key Equation

M=2S+1M = 2S + 1
  • M — Spin Multiplicity (dimensionless): A measure of the number of possible orientations of the total spin angular momentum; equals the number of unpaired electrons plus one for a simple case.
  • S — Total Spin Quantum Number (dimensionless): The sum of individual electron spin quantum numbers ms for unpaired electrons; each unpaired electron contributes +½, so S = (number of unpaired electrons) × ½.
  • 2S + 1 — Multiplicity Term (dimensionless): For a single unpaired electron S = ½, giving M = 2; for two unpaired electrons S = 1, giving M = 3 (triplet state); for three unpaired electrons S = 3/2, giving M = 4 (quartet state).

Worked Example

Problem. Write the full ground-state electron configuration of iron (Fe, Z = 26) using orbital box notation for the 3d subshell. Determine the number of unpaired electrons and the spin multiplicity. State whether iron is paramagnetic or diamagnetic.

Fe: [Ar] 3d6 4s2,S=2,M=2(2)+1=5\text{Fe: } [\text{Ar}]\, 3d^6 \, 4s^2, \quad S = 2, \quad M = 2(2)+1 = 5

Answer. Iron has 4 unpaired electrons, a spin multiplicity of 5, and is paramagnetic.

Figure
Diagram
Hund's Rule: Orbital Box Diagrams for C, N, O, FElement2p subshell (3 degenerate orbitals)Unpaired e⁻C (Z=6)2p²↑↑2 (paramagnetic)N (Z=7)2p³↑↑↑3 (paramagnetic)O (Z=8)2p⁴↑↓↑↑2 (paramagnetic)F (Z=9)2p⁵↑↓↑↓↑1 (paramagnetic)2p_x2p_y2p_z↑ = spin-up↓ = spin-down
Origins · 1 minFriedrich Hund and how this was discovered
FH
Friedrich Hund
1896–1997 · German

Friedrich Hund formulated the rule bearing his name in 1927 while working on atomic spectra and quantum mechanics. He made major contributions to molecular orbital theory, quantum tunneling, and the interpretation of spectroscopic data. The 'Hund coupling cases' used in spectroscopy also bear his name. He worked closely with Werner Heisenberg and Erwin Schrödinger during the golden age of quantum mechanics. Friedrich Hund lived to the remarkable age of 101 years, witnessing nearly a century of physics — from the early days of quantum mechanics all the way to modern computational chemistry. He continued working well into his nineties and was honored with the Max Planck Medal in 1943.

  • Aufbau Principle — Hund's Rule works in conjunction with the Aufbau Principle, which dictates the order in which orbitals are filled from lowest to highest energy before Hund's Rule determines how electrons distribute within degenerate orbitals.
  • Pauli Exclusion Principle — The Pauli Exclusion Principle enforces a maximum of two electrons per orbital and requires opposite spins when pairing occurs — this is precisely when Hund's Rule has already been satisfied and pairing becomes necessary.
  • Quantum Numbers — Understanding the spin quantum number ms (+½ or −½) and the magnetic quantum number m_l (which distinguishes degenerate orbitals within a subshell) is essential for applying Hund's Rule correctly.
  • Paramagnetism and Diamagnetism — The number of unpaired electrons predicted by Hund's Rule directly determines whether an element is paramagnetic (unpaired electrons present) or diamagnetic (all electrons paired), linking electronic configuration to observable physical properties.
  • Electronic Configuration — Hund's Rule is one of the three core rules used to write accurate ground-state electron configurations for multi-electron atoms, and exceptions such as Cr and Cu illustrate the stabilizing effect of maximum multiplicity.