Pauli Exclusion and Hund's Rules

Hund's-rule reasoning is the working language of inorganic and bioinorganic chemistry wherever open-shell metal centers appear. The high-spin versus low-spin question for octahedral $d^6$ complexes (Hund's-rule maximum multiplicity versus a strong ligand field forcing pairing) is precisely the mechanism behind the cooperative oxygen binding of hemoglobin, where iron's spin state changes measurably upon $\mathrm{O_2}$ binding, and behind spin-crossover materials being explored for molecular data storage and switchable pigments. Single-molecule magnets and lanthanide-based qubit candidates are engineered explicitly to maximize $S$ and unquenched orbital angular momentum $L$ — direct exploitation of Hund's rules 1 and 2 to build large, slow-relaxing magnetic moments for quantum information applications. In catalysis, the spin state of a metal center (again a Hund's-rule consequence) controls which reaction pathways are spin-allowed, a central idea in "two-state reactivity" models used to explain unusual barriers in iron- and cobalt-catalyzed C–H activation. Magnetic resonance imaging contrast agents based on high-spin $\mathrm{Gd^{3+}}$ ($4f^7$, the lanthanide analogue of $\mathrm{Fe^{3+}}$'s half-filled-shell stabilization) rely on the very large ground-state spin ($S=7/2$) that Hund's first rule guarantees for a half-filled $f$ shell. Even the periodic table's own shape — why the $d$-block is ten elements wide, why lanthanide contraction and $f$-electron configurations behave the way they do — traces back to counting degenerate orbitals and applying exactly the exclusion-plus-exchange logic developed here, a story picked up directly under Structure of the Periodic Table.

Ask the tutor: How does Hund's rule 1 interact with crystal field splitting to decide whether an octahedral $d^6$ complex is high-spin or low-spin?

Start with two electrons and two boxes. If the boxes represent different quantum states (different $n,\ell,m_\ell,m_s$), quantum mechanics places no restriction on double occupancy of a single state by unlike particles — but electrons are identical fermions, and identical fermions obey a stricter law than "no two in the same box." The full statement of the Pauli exclusion principle is that the total wavefunction of a system of electrons, including spin, must change sign when the coordinates (space and spin) of any two electrons are interchanged. This antisymmetry requirement is more fundamental than the familiar "no two electrons in an atom can have the same four quantum numbers" — that familiar statement is a corollary that emerges once you write the wavefunction as an antisymmetrized product (a Slater determinant) of one-electron orbitals. The consequence for orbital diagrams is immediate: a spatial orbital labeled by $n, \ell, m_\ell$ can hold at most two electrons, and if it holds two, their spin quantum numbers must differ ($m_s = +\tfrac12$ and $-\tfrac12$) — this is the familiar rule that each box gets at most one up-arrow and one down-arrow. But antisymmetry does more than cap orbital occupancy at two; it also correlates the spatial wavefunction with the spin wavefunction whenever there are multiple degenerate orbitals available, which is exactly the $p^2, p^4, d^n$ situation that Hund's rules address. Consider two electrons to be placed in two degenerate $2p$ orbitals, $p_x$ and $p_y$. There are two qualitatively different ways to build an antisymmetric two-electron wavefunction from one-electron spatial orbitals $\phi_x$ and $\phi_y$ and spin states $\alpha$ (up) and $\beta$ (down): A spin-triplet (parallel spin, $S=1$) state must have an antisymmetric spatial part, because the overall wavefunction (spatial × spin) must be antisymmetric, and the triplet spin function is symmetric:

A spin-singlet (paired spin, $S=0$) state must have a symmetric spatial part, because the singlet spin function is antisymmetric:

Notice what the antisymmetric spatial function does at $\mathbf r_1 = \mathbf r_2$: it vanishes identically. Same-spin (triplet) electrons therefore have zero probability of being found at the same point in space — they dig a "Fermi hole" around each other purely from statistics, before you even turn on the Coulomb interaction. The symmetric (singlet) spatial function has no such node; opposite-spin electrons can approach each other freely. Since electrons repel via $e^2/4\pi\varepsilon_0 r_{12}$, the state that keeps electrons farther apart on average has lower Coulomb repulsion energy. That state is the triplet — parallel spins, unpaired, in different orbitals. This is the microscopic origin of Hund's first rule, and it is why the "exchange energy" is not some exotic force but a bookkeeping name for the lowering of Coulomb repulsion caused by the antisymmetry-enforced spatial correlation of same-spin electrons.

Now build the full picture for a $p$ subshell across a period. As electrons are added to a set of three degenerate $2p$ orbitals ($p_x, p_y, p_z$), Hund's first rule says: place one electron in each orbital, all spins parallel, before pairing any. Only after all three are singly occupied does the fourth electron pair up. Carbon ($2p^2$), nitrogen ($2p^3$), and oxygen ($2p^4$) illustrate the three regimes — building, maximally half-filled, and beginning to pair.

Why does oxygen pair up at all, if parallel spins are always favored? Because there is no fourth degenerate $2p$ orbital to put the fourth electron into — only three $2p$ orbitals exist ($\ell=1 \Rightarrow m_\ell = -1,0,+1$). Once all three are singly occupied (nitrogen's half-filled, maximally-symmetric configuration), the next electron must enter an already-occupied orbital, paying a pairing energy $P$ (partly the loss of exchange stabilization, partly the extra direct Coulomb repulsion of two electrons sharing the same spatial orbital) rather than being promoted to a higher shell, which would cost far more. Hund's rule operates strictly within a single, degenerate set of orbitals; it does not compete against intershell promotion once that set is full.

The full statement usually taught as "Hund's rules" is a three-part hierarchy for finding the ground term ($^{2S+1}L_J$) of a free atom or ion, used after the electron configuration is fixed by Electron Configurations and the Aufbau Principle:

where: $S$ is total spin angular momentum quantum number, obtained by aligning as many electron spins as the Pauli principle allows within the degenerate subshell; $2S+1$ is the number of spin states (the term's multiplicity, e.g. singlet, doublet, triplet).

where: $L$ is total orbital angular momentum quantum number; $m_{\ell,i}$ are the individual orbital magnetic quantum numbers of the occupied orbitals, summed algebraically over all electrons and then maximized subject to $S$ already fixed at its maximum.

where: $J$ is the total angular momentum quantum number labeling the ground-state fine-structure level; the two branches reflect that spin-orbit coupling has opposite sign effects for shells filled from empty versus filled from full (a shell exactly half-full has $L=0$ so $J=S$ trivially, with no splitting ambiguity). The exchange energy that underlies rule 1 can be written for a pair of electrons $i,j$ with parallel spin in different orbitals of the same subshell as a genuine two-electron quantum-mechanical integral:

where: $\phi_i, \phi_j$ are one-electron orbitals in the degenerate subshell; $K_{ij} > 0$ always, and it enters the total energy subtracted for parallel-spin pairs (lowering their energy) and not at all for antiparallel pairs (they only feel the ordinary Coulomb/"direct" integral $J_{ij}$, which is always present regardless of spin). The number of parallel-spin electron pairs available in a given configuration, multiplied by $K$, gives the total exchange stabilization, which is why a half-filled subshell (maximum number of same-spin pairs) is particularly stabilized — this is the microscopic reason behind the well-known "extra stability of half-filled and filled subshells" invoked for configurations like $\mathrm{Cr}: [\mathrm{Ar}]3d^5 4s^1$.

Reading the equations: Rule 1 is not a separate law of nature; it is what maximizing exchange stabilization $nK$ (where $n$ is the number of parallel-spin pairs) demands, subject to the hard constraint of Pauli exclusion. Rule 2 is a subtler electron-correlation effect: for a fixed spin arrangement, different ways of distributing the $m_\ell$ values among electrons still give different Coulomb repulsions, and the arrangement with electrons "orbiting in the same sense" (large $|L|$) keeps them apart in an azimuthal sense, again lowering the direct Coulomb integral. Rule 3 is a genuinely different physical effect — spin-orbit coupling, a relativistic magnetic interaction between an electron's spin and its own orbital motion — layered on top of the electrostatically-determined $S,L$ manifold.

Example 1 — Foundational: build the ground term of carbon ($2p^2$) from the box diagram up. Carbon's configuration is $[\mathrm{He}]2s^2 2p^2$; only the two $2p$ electrons are open-shell. By Hund's rule 1, place them in different orbitals with parallel spins, e.g. spin-up electrons in $p_x$ and $p_y$:

For rule 2, we must check which pair of $m_\ell$ values, among all ways to keep $S=1$, gives the largest $|L|$. With two parallel-spin electrons restricted to different orbitals (Pauli forbids two same-spin electrons in one orbital), the possible unordered pairs of $m_\ell \in \{-1,0,+1\}$ and their sums are:

The maximum available is $L=1$ (a "P" term), achieved e.g. by occupying $m_\ell=+1$ and $m_\ell=0$ — physically the real $p_x,p_y,p_z$ real-orbital combinations are linear combinations of these complex $m_\ell$ orbitals, but the term symbol bookkeeping is done in the $m_\ell$ basis. So the maximum-multiplicity, maximum-$L$ manifold is $^3P$. For rule 3, the $2p$ subshell is less than half full (2 of 6 electrons), so

Ground term: $^3P_0$. This matches the tabulated ground term of atomic carbon exactly, and it is the textbook example used to introduce term symbols precisely because both non-trivial rules (1 and 2) are needed and the arithmetic stays small. Limiting-case check: if carbon's two $2p$ electrons were forced (hypothetically, violating Hund's rule) to pair in a single orbital, $S=0$, multiplicity $=1$, and the state would be an excited singlet, not the ground state — consistent with the experimental fact that atomic carbon's lowest excited term, $^1D_2$, lies about $121\,\text{kJ mol}^{-1}$ (≈1.26 eV) above the $^3P$ ground term, a gap that is entirely exchange energy in origin (same orbital occupancy pattern reshuffled in spin/orbital coupling, not a different electron configuration).

Example 2 — Applied: why is $\mathrm{O_2}$ paramagnetic, and how much unpaired spin does the box diagram predict? Molecular oxygen's paramagnetism is a molecular-orbital phenomenon, but it is seeded by the same exclusion + Hund logic applied to molecular orbitals instead of atomic ones. The $\mathrm{O_2}$ MO diagram places the last four valence electrons in the degenerate antibonding pair $\pi_{2p}^{*}$ (two components, $\pi_x^*$ and $\pi_y^*$, from the two perpendicular $2p_\perp$ pairs). With four electrons to distribute between $\sigma_{2p}$ (fully bonding, filled) and the $\pi^*$ pair, the relevant open-shell count is two electrons in two degenerate $\pi^*$ orbitals — structurally identical to the carbon $p^2$ problem above. Applying Hund's rule 1 to these two $\pi^*$ electrons: they occupy $\pi_x^*$ and $\pi_y^*$ singly, with parallel spins, rather than pairing in one $\pi^*$ orbital.

This predicts a triplet ground state, $^3\Sigma_g^-$ in molecular term notation, which is exactly the experimentally observed ground state of $\mathrm{O_2}$ — and the reason liquid oxygen is drawn into the poles of a magnet (paramagnetism requires unpaired spins; a naive Lewis structure with all electrons paired, $:\!\!\ddot O\!=\!\ddot O\!\!:$, predicts zero unpaired electrons and would wrongly predict $\mathrm{O_2}$ to be diamagnetic). Limiting-case check: contrast with $\mathrm{N_2}$, whose valence electron count leaves the $\pi^*$ orbitals empty (bond order 3, all bonding MOs filled, no antibonding electrons at all) — there is no open shell to apply Hund's rule to, correctly predicting $\mathrm{N_2}$ is diamagnetic. And for $\mathrm{O_2^{2-}}$ (peroxide), four electrons fill both $\pi^*$ orbitals completely ($\pi_x^{*2}\pi_y^{*2}$), pairing is forced by simple orbital exhaustion, $S=0$, and peroxide is diamagnetic — showing that the rule's prediction changes correctly as electron count changes, not by fiat but by re-running the same Pauli/Hund logic on the new electron count.

Example 3 — Exam/competition level: ground term and effective magnetic moment of high-spin $\mathrm{Co^{2+}}$ ($d^7$), with exchange-energy comparison against a hypothetical low-spin alternative. $\mathrm{Co^{2+}}$ has configuration $3d^7$: seven electrons in five degenerate $3d$ orbitals (free-ion, no ligand field splitting considered yet — that refinement belongs to crystal field theory). Apply the full hierarchy. Rule 1. To maximize $S$, fill all five orbitals singly first (5 electrons, $S=5/2$ contribution), then the remaining 2 electrons must pair up in already-occupied orbitals (Pauli forbids a sixth or seventh unpaired slot — only five orbitals exist). Maximum spin achievable:

(Five singly-occupied orbitals would need only 5 electrons; with 7 electrons, 2 orbitals must become doubly-occupied, each pairing removing one electron's worth of net spin contribution but not two, since one electron of each pair still contributes $+\tfrac12$ — net unpaired count is $5 - 2 = 3$.) Rule 2. Maximize $L$ subject to $S=3/2$. Distribute 3 unpaired electrons (one each in three of the five orbitals) plus 2 pairs (each pair contributing $m_\ell$ twice, net contribution $2m_\ell$ per paired orbital since both spins occupy the same $m_\ell$). To maximize $\sum m_\ell$, put the paired electrons in the orbitals with the largest $|m_\ell|$ (since they contribute double), and the unpaired electrons in the remaining slots, again choosing large $|m_\ell|$ first. Using $m_\ell = +2,+1,0,-1,-2$:

• Pair in $m_\ell=+2$: contributes $+4$
• Pair in $m_\ell=+1$: contributes $+2$
• Unpaired singles in $m_\ell = 0, -1, -2$: contributes $0-1-2=-3$

giving an "F" term (in agreement with the standard tabulated result for $d^7$). Rule 3. The $3d$ subshell has 7 electrons out of 10 — more than half-filled — so use $J=L+S$:

Ground term: $^4F_{9/2}$, matching tables exactly. Exchange-energy comparison. Now quantify why this high-spin arrangement ($S=3/2$) beats a hypothetical lower-spin alternative, e.g. $S=1/2$ (six electrons force-paired into three orbitals, one unpaired electron left). Counting same-spin pairs in each: High-spin ($3$ unpaired, spins all parallel; the 2 paired orbitals contribute no net new same-spin pairs among themselves but each unpaired electron can pair in spin with same-spin members of the doubly-occupied orbitals depending on which spin is "majority"): treating the standard case of 5 majority-spin electrons (one per orbital, all $\uparrow$) and 2 minority-spin electrons ($\downarrow$, doubling into two of those orbitals), the number of $\uparrow\uparrow$ pairs is

Low-spin alternative (hypothetically only 1 net unpaired electron, meaning at most 4 majority-spin electrons arranged with heavy pairing): the majority-spin count drops sharply, e.g. 4 up-spin electrons singly placed plus 3 forced pairs, giving $\binom{4}{2}=6$ same-spin pairs — a full $4K$ less exchange stabilization, on top of the extra direct-Coulomb pairing penalties $P$ from cramming more electrons into shared orbitals. Both effects push decisively toward the experimentally correct high-spin, $^4F_{9/2}$, free-ion ground term. Magnetic moment check (spin-only formula, an application downstream of $S$):

where $n=3$ is the number of unpaired electrons found above. This is close to (though not identical to, owing to orbital contributions from the non-zero $L=3$) the experimentally observed magnetic moments of octahedral high-spin $\mathrm{Co^{2+}}$ complexes, typically $4.3$–$5.2\,\mu_B$ — the enhancement above the spin-only value is itself a diagnostic fingerprint of unquenched orbital angular momentum, consistent with $L=3\neq0$ found above.

Friedrich Hund arrived at his empirical rules for atomic spectral term ordering in 1925, working within the young discipline of quantum spectroscopy at Göttingen, in close intellectual contact with Max Born, Werner Heisenberg, and later with Robert Mulliken (with whom Hund co-developed molecular orbital theory itself). Hund's rules were originally an empirical distillation of atomic emission-spectrum data — he noticed that observed ground terms of atoms consistently obeyed the maximum-multiplicity pattern before quantum mechanics could fully justify why. Wolfgang Pauli's exclusion principle, published in 1925 as well, initially took the form "no two electrons in an atom can have all four quantum numbers identical," derived from spectroscopic anomalies (the anomalous Zeeman effect and the puzzle of why atomic electrons don't all collapse into the $1s$ orbital). The deeper justification — that exclusion follows from the requirement of total wavefunction antisymmetry for identical fermions — came slightly later, cemented by Heisenberg's and Dirac's treatments of identical particles and by the spin-statistics theorem (fully general only with relativistic quantum field theory, associated with Pauli's later 1940 work). The two results, Pauli's principle and Hund's rules, were thus born in the same short burst of mid-1920s quantum theory but answer different kinds of question: one is exact and universal, the other is an emergent energetic ranking that quantum mechanics later explained in terms of exchange integrals.