Hund's Rules

Hund's rules are a set of three empirical quantum-mechanical principles formulated between 1925 and 1927 by the German physicist Friedrich Hund that determine which Russell–Saunders (LS-coupled) spectroscopic term symbol (^2*S*+1L_J) corresponds to the lowest-energy electronic ground state of a multi-electron atom or ion containing partially filled degenerate subshells (pⁿ, dⁿ, or fⁿ). In a hydrogenic one-electron atom, orbital energy depends solely on the principal quantum number n; in a multi-electron atom, however, the non-central inter-electronic Coulombic repulsion operator (Σ_*i*<*j* e² / [4*π ε*₀ r_ij]) and relativistic spin–orbit coupling split a single electron configuration (such as carbon 1*s*²2*s*²2*p*², which encompasses 15 distinct microstates) into multiple spectroscopic energy terms. Hund's three rules rank these terms in hierarchical order of energy: first by total spin angular momentum (S), second by total orbital angular momentum (L), and third by total electronic angular momentum (J).

Historically, Hund derived these rules while analyzing complex atomic emission line spectra at the University of Göttingen with Max Born and James Franck shortly after Wolfgang Pauli formulated the exclusion principle (1925) and Samuel Goudsmit and George Uhlenbeck introduced electron spin. Although all three rules are essential in atomic spectroscopy and transition metal magnetochemistry, Hund's first rule (the rule of maximum spin multiplicity) is the most widely applied across general and physical chemistry: it dictates that electrons occupy degenerate orbitals (p_x, p_y, p_z or the five d orbitals) singly with parallel spins before any orbital is doubly occupied.

Beyond isolated atoms, Hund's rules govern the electronic structure and magnetic susceptibility of free radicals, diradicals, dioxygen (O₂), and coordination complexes. In molecular orbital theory and orbital hybridization, the competition between Hund's first-rule spin-pairing energy (P) and the orbital splitting gap (Δ) determines whether open-shell molecules and transition metal ions adopt paramagnetic high-spin or diamagnetic low-spin ground states.

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For light and medium-weight atoms (typically atomic number Z ≤ 30, including the first-row d-block transition metals), electrostatic repulsion between electrons is substantially larger than relativistic spin–orbit interaction. Under this Russell–Saunders (LS) coupling approximation (developed by Henry Norris Russell and Frederick Saunders in 1925), the individual orbital angular momenta l_i of all valence electrons vector-couple into a total orbital angular momentum L = Σ l_i (specified by the quantum number L, with projection M_L = Σ m_l,*i*), and the individual spin angular momenta s_i vector-couple into a total spin angular momentum S = Σ s_i (specified by the quantum number S, with projection M_S = Σ m_s,*i*).

Subsequently, L and S couple weakly via spin–orbit interaction (Ĥ_SO = ζ(r) L · S) to yield the total electronic angular momentum J = L + S, where the quantum number J takes integer or half-integer values spaced by unity:

J = |*L* − *S*|, |*L* − *S*| + 1, ..., L + S − 1, L + S

Each resulting energy level is denoted by a term symbol:

^2*S*+1L_J

where:

• 2*S* + 1 is the spin multiplicity (1 = singlet, 2 = doublet, 3 = triplet, 4 = quartet, 5 = quintet, 6 = sextet, 7 = septet), equal to the number of degenerate spin orientations (M_S = −*S* to +*S*).
• L is designated by an uppercase spectroscopic letter corresponding to the numerical value of L:

L = 0 → S, L = 1 → P, L = 2 → D, L = 3 → F, L = 4 → G, L = 5 → H, L = 6 → I.

• J is the total angular momentum subscript, with each J-level possessing a (2*J* + 1)-fold spatial degeneracy in the absence of an external magnetic field. Closed shells and subshells (s², p⁶, d¹⁰, f¹⁴) always have L = 0, S = 0, and J = 0 (¹S₀) and can be ignored when computing the term symbol of open-shell configurations.

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1. Hund's First Rule: Maximum Spin Multiplicity (2*S* + 1)

Rule 1: For a given electron configuration, the spectroscopic term with the largest value of total spin quantum number S—and therefore the highest spin multiplicity (2*S* + 1)—lies lowest in energy.

Because the Pauli exclusion principle requires the total many-electron wavefunction Ψ(1, 2, ..., N) to be antisymmetric with respect to the exchange of any two electrons (meaning two electrons in the same spatial orbital must have opposite spins, m_s = +½ and −½), maximizing S requires placing electrons into different degenerate spatial orbitals (m_l) with parallel spins (↑, ↑, ↑) before pairing any two electrons in the same orbital.

Physical Origin — Exchange Energy vs. Reduced Nuclear Screening: Historically, textbook explanations attributed Hund's first rule solely to reduced electron–electron repulsion (V_ee). When two electrons have parallel spins (triplet state, symmetric spin function), their spatial wavefunction must be antisymmetric:

Ψ_spatial(1, 2) = (1 / √2) [ φ_a(1)*φ*_b(2) − φ_b(1)*φ*_a(2) ]

Because Ψ_spatial(1, 2) vanishes identically whenever r₁ = r₂, each electron is surrounded by a Fermi hole (exchange hole) that forbids another electron of the same spin from occupying the same point in space. In standard first-order perturbation theory using fixed atomic orbitals, the total electrostatic energy of two parallel-spin electrons is J_ab − K_ab (where J_ab is the classical Coulomb integral and K_ab > 0 is the quantum-mechanical exchange integral), whereas antiparallel spins give J_ab + K_ab, stabilizing the high-spin triplet by 2*K*_ab.

However, high-level self-consistent field (Hartree–Fock) and configuration-interaction (CI) calculations by Robert S. Mulliken, ** Ernest R. Davidson, and Russell J. Boyd** revealed a critical subtlety when orbitals are allowed to relax variationally: in the true ground state of carbon (³P) and other atoms, electron–electron repulsion V_ee is actually slightly larger in the high-spin state than in the low-spin state! The reason is that the Fermi hole reduces inter-electronic screening of the nucleus (σ). Experiencing a higher effective nuclear charge (Z_eff), the radial wavefunctions of the parallel-spin electrons contract closer to the nucleus. Although this radial contraction slightly increases V_ee and electronic translational kinetic energy (½*mv*², by the quantum virial theorem T = −*E*), it produces a much larger stabilization in electron–nucleus attraction potential energy (V_ne), which is the true dominant physical driver of Hund's first rule in neutral atoms.

2. Hund's Second Rule: Maximum Orbital Angular Momentum (L)

Rule 2: For a given spin multiplicity (2*S* + 1), the term with the largest value of the total orbital angular momentum quantum number L lies lowest in energy.

If two terms share the same maximum spin S, the one with higher L (e.g., a ³F term [L = 3] over a ³P term [L = 1] in a d² ion) is more stable. Semi-classically, high L means the valence electrons are orbiting the nucleus in the same rotational direction (m_l values of the same sign), allowing them to stay farther apart on average than electrons orbiting in opposite directions, which pass each other twice per revolution.

3. Hund's Third Rule: Spin–Orbit Coupling (J)

Rule 3: Within a given multiplet term (fixed S and L):
* If the open subshell is less than half-filled (n < 2*l* + 1), the level with the lowest value of J = |*L* − *S*| lies lowest in energy (regular multiplet).
* If the open subshell is more than half-filled (n > 2*l* + 1), the level with the highest value of J = L + S lies lowest in energy (inverted multiplet).
* If the subshell is exactly half-filled (n = 2*l* + 1, such as p³, d⁵, or f⁷), then M_L = Σ m_l = 0 so L = 0 (S term), leaving a single unique level with J = S.

The energy shift of a spin–orbit level J relative to the term barycenter is given by the Landé interval rule:

Δ*E*_SO(J) = ½ A(L, S) [ J(J + 1) − L(L + 1) − S(S + 1) ]

where the multiplet spin–orbit coupling constant A(L, S) is positive (A > 0) for less-than-half-filled shells (favoring minimum J when L and S are antiparallel) and negative (A < 0) for more-than-half-filled shells (because holes in an almost-full shell behave like positively charged positrons, inverting the sign of the magnetic dipole interaction and favoring maximum J).

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To find the ground-state term symbol of any lⁿ configuration using a Hund box diagram, draw the 2*l* + 1 boxes labeled from m_l = +*l* down to −*l*, fill electrons singly with spin-up (↑, m_s = +½) starting from the highest m_l box (+*l*) to maximize both M_S and M_L, and then apply Rule 3 for J:

Half-Filled Subshell Stability and d-Block Anomalies

Hund's first rule explains two classic periodic patterns:

1. Ionization Energy Inversions at Group 15: The first ionization energy of Nitrogen (2*p*³, ⁴S_3/2, IE₁ = 1,402 kJ mol⁻¹) is higher than that of Oxygen (2*p*⁴, ³P₂, IE₁ = 1,314 kJ mol⁻¹) because nitrogen's three parallel-spin 2*p* electrons enjoy maximum exchange stabilization (3*K*_2*p*,2*p* pairs), whereas oxygen's fourth 2*p* electron is forced to pair with an opposite spin in an already occupied 2*p* orbital, incurring inter-electronic Coulombic pairing repulsion (Π_c).
2. Aufbau Exceptions in Chromium and Molybdenum: Neutral Chromium (Z = 24) adopts the ground-state electron configuration [Ar] 3*d*⁵ 4*s*¹ (⁷S₃) rather than [Ar] 3*d*⁴ 4*s*² (⁵D₀) because avoiding double-occupancy Coulomb repulsion in the compact 4*s* orbital while gaining 15 parallel-spin exchange interactions across six singly occupied valence orbitals lowers the total energy.

Hund's Rule in Molecules: Triplet Dioxygen (O₂) and Organic Diradicals

Hund's first rule applies equally to degenerate molecular orbitals. In ground-state molecular oxygen (O₂, 12 valence electrons), the highest occupied molecular orbital (HOMO) is the doubly degenerate pair of π* antibonding orbitals (1π_gx, 1π_gy) containing 2 electrons. By Hund's first rule, these two electrons occupy the orthogonal π*_x and π*_y orbitals singly with parallel spins (S = 1), giving O₂ a paramagnetic triplet diradical ground state (³Σ_g⁻) that lies 94.3 kJ mol⁻¹ below the lowest excited singlet state (¹Δ_g). Because nearly all organic biomolecules have closed-shell singlet ground states (S = 0), direct concerted reaction with triplet O₂ is spin-forbidden by angular momentum conservation, preventing spontaneous combustion of living matter at ambient temperature.

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Why do electrons occupy degenerate orbitals singly with parallel spins before pairing?

When two electrons have parallel spins, the Pauli exclusion principle requires their spatial wavefunction to be antisymmetric, creating a Fermi hole where the probability of finding both electrons at the same point in space is zero. In atomic Hartree–Fock theory, this reduced inter-electronic screening allows the orbitals of high-spin states to contract slightly closer to the positively charged nucleus, lowering electron–nucleus potential energy (V_ne) while avoiding the Coulombic repulsion of sharing the same spatial orbital.

Do Hund's second and third rules apply to excited states as well as the ground state?

Strictly speaking, Hund's three rules are guaranteed to identify the single lowest-energy ground-state term of a given electron configuration, and Hund's third rule accurately orders the spin–orbit J levels within that ground multiplet. Among higher-energy excited terms arising from the same configuration (for example, ordering excited singlet vs. triplet terms of different L, or in conjugated organic π systems), configuration interaction with higher shells can occasionally invert the energetic ordering of excited states.

Why does Hund's third rule reverse for subshells that are more than half full?

Spin–orbit coupling arises from the magnetic interaction between an electron's intrinsic spin magnetic moment and the internal magnetic field generated by its orbital motion around the nucleus. When a subshell is less than half full, antiparallel alignment of total L and S (J = |*L* − *S*|) minimizes magnetic energy. When a subshell is more than half full, the missing electrons behave mathematically like positively charged "holes" relative to a closed spherical shell, reversing the sign of the spin–orbit coupling constant A and stabilizing J = L + S.

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1. Hund, F. (1925). "Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel." Zeitschrift für Physik, 33(1), 345–371. https://doi.org/10.1007/BF01328319
2. Russell, H. N., & Saunders, F. A. (1925). "New regularities in the spectra of the alkaline earths." The Astrophysical Journal, 61, 38–69. https://doi.org/10.1086/142872
3. Boyd, R. J. (1984). "A quantum mechanical explanation for Hund's multiplicity rule." Nature, 310(5977), 480–481. https://doi.org/10.1038/310480a0
4. Kutzelnigg, W., & Morgan, J. D., III. (1996). "Hund's rules." Zeitschrift für Physik D Atoms, Molecules and Clusters, 36(3–4), 197–214. https://doi.org/10.1007/BF01426405