Aufbau Principle and Hund's Rules

Chromium has the electron configuration [Ar] 3d⁵4s¹. Naive orbital counting instead predicts [Ar] 3d⁴4s². This single anomaly repeats with variations across the transition series. It is the reason no textbook can teach the Aufbau Principle as pure bookkeeping. Instead, it is a physical approximation, remarkably good but breakable. It is built from real energy competitions inside a many-electron atom. Getting ground-state configurations right matters far beyond a periodic-table party trick. Magnetism, catalytic activity, spectroscopic term symbols, and ionization energies all trace back to orbital occupation. The entire logic of the periodic table depends on which orbitals electrons occupy. It also depends on how their spins align. The big picture is that a many-electron atom has no exact wavefunction. So chemists build one approximately by assigning electrons to hydrogen-like orbitals. These orbitals are ordered by increasing energy — the aufbau ("building-up") procedure. This process obeys the Pauli Exclusion Principle, which follows from antisymmetric Slater Determinants. Within a partially filled subshell, Hund's rules fix the spin and orbital arrangement. This arrangement minimizes electron-electron repulsion. More subtly, it also minimizes the shielding electrons feel from each other. These two ideas together reproduce the entire ground-state electron configuration chart. A short list of well-understood exceptions is also needed. These exceptions are driven by 3d–4s energy near-degeneracy and exchange stabilization. Together they account for only a couple dozen irregularities out of 118 elements. This article builds the filling order from first principles using Effective Nuclear Charge and Shielding. It states Hund's three rules precisely. It derives why maximum multiplicity lowers energy, using exchange integrals from Slater determinants. It works through configurations from carbon to copper to a lanthanide. It closes with the standard exceptions (Cr, Cu, Mo, Ag, Au, Pd, and the lanthanide/actinide irregularities). It also covers the (n+l) Madelung rule as a mnemonic device, with explicit failure modes. Finally, it bridges to Term Symbols and Atomic States. There, the same electron configurations are refined into full spectroscopic states. The reward for taking this seriously is predictive power. From configuration alone you can anticipate paramagnetism and relative ionization energies. You can also anticipate preferred oxidation states. The chemistry of the periodic table is condensed into a filling diagram and three rules.

Key takeaways

The aufbau principle fills orbitals in order of increasing energy, subject to the Pauli exclusion principle from antisymmetrized wavefunctions.
Orbital energies in multi-electron atoms depend on both $n$ and $l$ because of penetration and shielding, unlike the hydrogen atom.
The Madelung $(n+l)$ rule is a mnemonic for filling order, breaking ties by lower $n$ ; it is an approximation, not a law.
Hund's first rule (maximum spin multiplicity) is enforced by exchange stabilization — same-spin electrons avoid each other spatially, lowering Coulomb repulsion.
Hund's second and third rules fix orbital angular momentum and total $J$ , refining degenerate spin arrangements into unique ground terms.
Exceptions like Cr ([Ar] 3d⁵4s¹) and Cu ([Ar] 3d¹⁰4s¹) arise from the near-degeneracy of 3d and 4s and the extra stability of half-filled and filled subshells.

Key idea: Ground-state configurations are the lowest-energy way to distribute electrons over shielded, penetration-split orbitals while maximizing exchange stabilization among parallel spins.

Where this lives today

Ground-state electron configurations, built from aufbau and Hund's rules, are the daily working vocabulary of inorganic and materials chemistry. Crystal-field and ligand-field theory — used to explain the colors, magnetism, and reactivity of transition-metal complexes in everything from hemoglobin's oxygen binding to industrial hydrogenation catalysts — starts from the d-electron count fixed by these rules (high-spin Fe²⁺ in deoxyhemoglobin has four unpaired electrons precisely because Hund's first rule favors maximum spin in the weak-field, high-spin limit). Molecular magnetism and the design of single-molecule magnets for data storage depend on maximizing unpaired-electron count via half-filled or near-half-filled d and f shells, a direct engineering application of Hund's rule. In semiconductor and battery materials — LiMn₂O₄ or LiFePO₄ cathodes, or dilute magnetic semiconductors doped with Mn²⁺ — the oxidation states and spin states accessible to a transition-metal ion trace back to which 3d configuration is energetically preferred, informed by exactly the 3d/4s competition worked through above. In spectroscopy, the ground-state term symbol derived by applying all three Hund's rules in sequence is the starting point for predicting which electronic transitions are spin-allowed versus spin-forbidden (the reason Mn²⁺ salts are pale, not vividly colored), a topic developed fully under Term Symbols and Atomic States. Even computational quantum chemistry codes performing density functional theory or Hartree–Fock calculations use aufbau-filled configurations as their initial guess, and convergence to the wrong spin state (a real, common failure mode in transition-metal DFT calculations) is diagnosed by checking the result against exactly the Hund's-rule expectations developed here.

:::quiz[Why is Fe²⁺ written as [Ar]3d⁶ rather than [Ar]3d⁴4s²?]

Because 4s never fills in any transition metal
Because cations lose ns electrons before (n−1)d electrons once the atom is ionized ✓
Because iron violates the Pauli exclusion principle
Because 3d⁶ is always the aufbau-predicted neutral-atom configuration

Ask the tutor: How do Hund's rules change when an atom is placed inside a ligand field, and why does that sometimes favor pairing electrons instead of maximizing spin?

The physical picture

In the hydrogen atom, orbital energy depends only on the principal quantum number $n$ : the 2s and 2p orbitals are exactly degenerate. Add a second electron — as in helium — and this degeneracy shatters. Each electron now feels not only the bare nuclear charge $Z$ but also the repulsive presence of every other electron, which partially screens the nucleus. This screening is not uniform across orbitals: an electron in a 2s orbital has a nonzero probability density at the nucleus (a "penetrating" orbital with a large amplitude near $r=0$ despite its radial node), while a 2p electron is kept away from the nucleus by its angular node. The 2s electron therefore experiences a higher effective nuclear charge $Z_{\text{eff}}$ than the 2p electron of the same shell, using the language of Effective Nuclear Charge and Shielding, and 2s drops below 2p in energy. This same story, repeated shell after shell, is why orbital energies in real atoms split by both $n$ and $l$ , not by $n$ alone. The aufbau principle says: to build the ground-state configuration, pour electrons into the lowest-energy available orbitals one at a time, respecting the Pauli exclusion principle (no two electrons in the atom may share all four quantum numbers, equivalently the many-electron wavefunction must be an antisymmetrized Slater determinant). Because orbital energies depend on the other electrons present, this is a self-consistent, not exactly sequential, problem — but the qualitative energy ordering is stable enough across the periodic table that a single filling diagram works for almost every element. Within a subshell of several degenerate orbitals (say the three 2p orbitals, or the five 3d orbitals), aufbau alone does not say how to distribute the electrons — do two electrons in 2p² go into the same orbital with opposite spins, or into different orbitals with parallel spins? This is where Hund's rules take over. Friedrich Hund's empirical rules, later justified by exchange-integral arguments from quantum mechanics, state that the ground term: 1. Has maximum spin multiplicity $2S+1$ (electrons spread out over degenerate orbitals with parallel spins as much as the Pauli principle allows). 2. Among terms of the same maximum multiplicity, has maximum total orbital angular momentum $L$ . 3. For a subshell that is less than half-filled, the lowest energy has minimum $J = |L-S|$ ; for a subshell more than half-filled, the lowest energy has maximum $J = L+S$ (spin–orbit coupling, relevant to fine structure and covered fully alongside Term Symbols and Atomic States).

Figure

Orbital energy splitting: hydrogen vs. many-electron atom

Figure

Carbon 2p² : Hund's rule vs. paired alternative

The core ideas

Orbital filling order (Madelung / $(n+l)$ rule). Orbitals fill in order of increasing $n + l$ ; for equal $n+l$ , the orbital with smaller $n$ fills first.

\text{fill order: increasing } (n+l), \text{ ties broken by increasing } n

where: $n$ is the principal quantum number, $l$ is the azimuthal (orbital angular momentum) quantum number ( $l = 0,1,2,3 \to s,p,d,f$ ). This reading gives the familiar sequence 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d — the diagonal mnemonic taught with arrows through a periodic-table grid. It is an approximation to the true self-consistent orbital energy ordering, valid because penetration effects (encoded in $Z_{\text{eff}}(n,l)$ ) make $n+l$ track energy fairly well, but it is not derived from a single exact formula and is known to fail for specific elements (worked below).

Effective nuclear charge and orbital energy. Within a hydrogen-like approximation dressed by shielding,

E_{n,l} \approx -\frac{Z_{\text{eff}}(n,l)^2}{n^2}\,\text{Ry}

where: Ry $= 13.6$ eV, and $Z_{\text{eff}}(n,l) = Z - S(n,l)$ with $S$ the shielding constant (Slater's rules or numerically fit values), which depends on $l$ because different- $l$ orbitals of the same $n$ penetrate the core to different extents. This equation is why 4s can lie below 3d for neutral K and Ca (large penetration advantage for 4s when the 3d shell is still empty) yet 3d drops below 4s once electrons occupy it and screening rearranges — the famous 3d/4s crossover behind transition-metal exceptions.

Hund's first rule — exchange stabilization. For two electrons in degenerate orbitals $\phi_a, \phi_b$ of the same subshell, the singlet (paired, antisymmetric spatial part) and triplet (parallel spin, symmetric-under-exchange-of-labels-but-antisymmetric-total via Slater determinant) states differ in energy by twice the exchange integral:

E_{\text{singlet}} - E_{\text{triplet}} = 2K_{ab}, \qquad K_{ab} = \iint \phi_a^*(1)\phi_b^*(2)\,\frac{e^2}{4\pi\varepsilon_0 r_{12}}\,\phi_b(1)\phi_a(2)\, d\tau_1\, d\tau_2

where: $K_{ab}$ is the exchange integral (always positive for real orbitals), $r_{12}$ is the inter-electron distance, and the triplet (parallel spin, $S=1$ ) state is lower in energy because its spatial wavefunction vanishes when $\mathbf{r}_1 = \mathbf{r}_2$ (Pauli-driven "Fermi hole"), reducing Coulomb repulsion. This is the microscopic mechanism behind Hund's first rule: maximizing $S$ maximizes the number of same-spin electron pairs, each contributing a stabilizing $-K_{ab}$ term relative to the paired alternative.

Derivation AWhy parallel spins lower energy: exchange integral from a Slater determinant

Goal. Show explicitly, using two electrons in degenerate orbitals $\phi_a$ and $\phi_b$ , that the parallel-spin (triplet) arrangement lies lower in energy than the paired (singlet) arrangement by $2K_{ab}$ , thereby justifying Hund's first rule from Slater-determinant wavefunctions.

Step 1 — Build the antisymmetrized spatial-times-spin wavefunctions. Two electrons occupying distinct orbitals $\phi_a,\phi_b$ can combine spins into a singlet ( $S=0$ ) or triplet ( $S=1$ ). Because the total wavefunction must be antisymmetric under electron exchange (Pauli principle, as encoded in a two-electron Slater determinant), the spatial part must have the opposite symmetry to the spin part:

\Psi_{\text{singlet}} = \frac{1}{\sqrt{2}}\big[\phi_a(1)\phi_b(2)+\phi_a(2)\phi_b(1)\big]\cdot\chi_{S=0}

\Psi_{\text{triplet}} = \frac{1}{\sqrt{2}}\big[\phi_a(1)\phi_b(2)-\phi_a(2)\phi_b(1)\big]\cdot\chi_{S=1}

Here $\chi_{S=0}$ is antisymmetric under spin exchange and $\chi_{S=1}$ is symmetric, forcing the spatial parts shown to compensate.

Step 2 — Compute the electron–electron repulsion energy in each state. The relevant operator is $\hat{V} = e^2/(4\pi\varepsilon_0 r_{12})$ . Expanding $\langle \Psi | \hat{V} | \Psi \rangle$ for the symmetric (singlet) spatial function gives four terms; two are direct Coulomb-type integrals $J_{ab} = \iint |\phi_a(1)|^2 |\phi_b(2)|^2 \hat V \,d\tau_1 d\tau_2$ and two are exchange-type cross terms $K_{ab}$ defined above. Carrying through the normalization $1/\sqrt{2}$ (which contributes a factor of $\tfrac12$ to each of the two identical direct terms and each of the two identical exchange terms):

\langle \hat V \rangle_{\text{singlet}} = J_{ab} + K_{ab}, \qquad \langle \hat V \rangle_{\text{triplet}} = J_{ab} - K_{ab}

The sign flip between the two cases comes directly from the $\pm$ sign in the spatial wavefunctions of Step 1: squaring the sum gives $+2K_{ab}$ (after normalization, $+K_{ab}$ ), while squaring the difference gives $-K_{ab}$ .

Step 3 — Take the difference. Subtracting,

E_{\text{singlet}} - E_{\text{triplet}} = (J_{ab}+K_{ab}) - (J_{ab}-K_{ab}) = 2K_{ab}

Since $K_{ab} = \iint \phi_a^*(1)\phi_b^*(2)\,\hat V\,\phi_b(1)\phi_a(2)\,d\tau_1 d\tau_2 > 0$ for any real, non-orthogonal-in-space overlap density (it is a self-repulsion-like integral of the overlap density $\phi_a\phi_b$ with itself through $\hat V$ ), the singlet state is always higher in energy. The triplet, with maximum spin multiplicity $2S+1=3$ , is the ground state — exactly Hund's first rule.

Step 4 — Physical interpretation. The triplet spatial wavefunction has a node when $\mathbf{r}_1=\mathbf{r}_2$ (the "Fermi hole"): same-spin electrons are automatically kept apart by antisymmetry, lowering their mutual Coulomb repulsion below the value $J_{ab}$ that opposite-spin, spatially unrestricted electrons would experience. This is a purely quantum statistical effect — no classical force pulls parallel-spin electrons apart, but the wavefunction's required symmetry does the job for free.

Go deeper · Derivation B · 2 min3d/4s crossover: why the Aufbau order is not fixed, using effective nuclear charge

Derivation B3d/4s crossover: why the Aufbau order is not fixed, using effective nuclear charge

Goal. Show quantitatively, using Slater's shielding rules, why 4s fills before 3d in neutral atoms (K, Ca) yet 3d lies below 4s once several 3d electrons are present — the mechanism behind transition-metal aufbau exceptions.

Step 1 — Slater's rules for $Z_{\text{eff}}$ . For an electron in orbital $(n,l)$ , $Z_{\text{eff}} = Z - S$ , where $S$ is built from contributions of other electrons: electrons in the same group contribute $0.35$ each (or $0.30$ for 1s); electrons one shell lower contribute $0.85$ (for s,p groups) or $1.00$ (if the electron of interest is in a d or f orbital, all electrons in lower $n$ contribute $1.00$ , and same- $n$ s/p electrons contribute $1.00$ too since d/f orbitals do not penetrate through inner shells as efficiently); electrons two or more shells lower contribute $1.00$ .

Step 2 — Apply to potassium, $Z=19$ , configuration [Ar]4s¹. For the 4s electron, the shielding comes from all 18 core electrons (each contributing effectively close to $1.00$ per Slater's grouping of $(4s,4p)$ together, with lower shells contributing $1.00$ ):

S_{4s} \approx 18 \times 1.00 = 18.0 \quad(\text{approximately, before same-group correction})

More precisely using the standard Slater grouping $(1s)(2s,2p)(3s,3p)(3d)(4s,4p)(\dots)$ , the 4s electron in K sees 2 electrons in 1s, 8 in $n=2$ , 8 in $n=3(s,p)$ all contributing $0.85$ each (since they are one shell below), plus zero same-group partners:

S_{4s} = 2(1.00) + 8(0.85) + 8(0.85) = 2 + 6.8 + 6.8 = 15.6

Z_{\text{eff},4s} = 19 - 15.6 = 3.4

Step 3 — Compare to a hypothetical 3d electron in the same atom. If the 19th electron were instead placed in 3d, it would belong to the $(3d)$ group by itself (Slater places 3d in its own group), and all 18 electrons below it (through $n=3\,s,p$ ) count as "lower shell," each contributing $1.00$ (Slater's rule: for d and f electrons, anything in a lower group contributes fully, no partial 0.85 discount):

S_{3d} = 18 \times 1.00 = 18.0, \qquad Z_{\text{eff},3d} = 19 - 18.0 = 1.0

Step 4 — Compare orbital energies with the hydrogenic estimate $E_{n,l}\approx -Z_{\text{eff}}^2/n^2$ Ry.

E_{4s} \approx -\frac{(3.4)^2}{4^2}\,\text{Ry} = -\frac{11.56}{16}\,\text{Ry} = -0.723\ \text{Ry}

E_{3d} \approx -\frac{(1.0)^2}{3^2}\,\text{Ry} = -\frac{1.00}{9}\,\text{Ry} = -0.111\ \text{Ry}

Since $-0.723 < -0.111$ , the 4s orbital is far lower in energy than 3d for the 19th electron of potassium — 4s fills first, exactly as observed ([Ar]4s¹, not [Ar]3d¹).

Step 5 — Why the order flips once 3d starts filling. As electrons are added to 3d in the transition series, they shield each other poorly (the 0.35 same-group Slater constant, smaller than the 0.85 an s/p electron would receive from an inner shell), so $Z_{\text{eff}}$ for 3d rises faster with atomic number than $Z_{\text{eff}}$ for 4s. By scandium and beyond, the 3d orbital — though it started higher in K/Ca — drops below 4s. This is why, once ionized, transition metals lose 4s electrons before 3d electrons (Fe²⁺ is [Ar]3d⁶, not [Ar]3d⁴4s²): the filled-atom ordering of 3d below 4s applies to cations, even though 4s filled first during neutral-atom aufbau. The lesson: aufbau order and removal (ionization) order are not simply the reverse of each other, because $Z_{\text{eff}}$ itself depends on which electrons are already present.

Worked examples

Example 1 — Foundational: build the ground-state configuration and term for carbon (Z = 6). Fill orbitals by the $(n+l)$ order: 1s (2 electrons), 2s (2 electrons), then 2p (2 electrons remain), giving

\text{C}:\ 1s^2\,2s^2\,2p^2

For the 2p² subshell (three degenerate orbitals, $m_l = +1,0,-1$ ), Hund's first rule demands maximum $S$ : place the two electrons in different $m_l$ orbitals with parallel spin,

S = \tfrac12+\tfrac12 = 1,\qquad \text{multiplicity } 2S+1 = 3

Hund's second rule then picks maximum $L$ consistent with the Pauli principle: putting electrons in $m_l=+1$ and $m_l=0$ (rather than $+1,-1$ , which is also allowed by spin but gives smaller $M_L$ possibilities once properly antisymmetrized) yields the maximum $M_L = 1$ , and careful term analysis (standard for $p^2$ ) shows the ground term is $^3P$ ( $L=1$ ). Since the 2p subshell is less than half filled (2 of 6 electrons), Hund's third rule gives minimum $J = |L-S| = |1-1| = 0$ . The full ground-state term symbol is

{}^{3}P_0

This matches the experimentally observed ground state of atomic carbon exactly, and previews the systematic term-symbol machinery developed alongside Term Symbols and Atomic States.

Example 2 — Applied: predict the ground-state configuration and magnetic behavior of manganese (Z = 25), an "on-schedule" transition metal used in steelmaking and battery cathodes (LiMn₂O₄). Following the $(n+l)$ filling order through argon (18 electrons: [Ar] = $1s^22s^22p^63s^23p^6$ ), the next orbitals are 4s then 3d:

\text{Mn} = [\text{Ar}]\,4s^2\,3d^5

This uses $18 + 2 + 5 = 25$ electrons — matches $Z=25$ , no shortcut assumed. The 3d⁵ subshell is exactly half-filled across five degenerate orbitals; Hund's first rule places one electron in each $d$ orbital with parallel spins:

S = 5\times\tfrac12 = \tfrac52,\qquad 2S+1 = 6

giving a sextet ground term ( $^6S_{5/2}$ , since a half-filled subshell has $L=0$ by symmetry — every $m_l$ is singly occupied so $M_L=0$ is the only possibility). This is why Mn²⁺ (which loses the two 4s electrons to give [Ar]3d⁵, keeping the same half-filled sensitivity) is a textbook high-spin, strongly paramagnetic ion with five unpaired electrons — the basis of its characteristic weak, spin-forbidden d–d transitions (pale pink Mn²⁺ solutions) and its utility as an EPR-active probe ion in bioinorganic chemistry. Note that manganese, unlike chromium and copper, follows the naive aufbau prediction exactly — no exception needed here, since [Ar]4s²3d⁵ (half-filled d, full s) is already the electronically favorable arrangement without any 4s→3d electron transfer.

Example 3 — Exam/competition level: rationalize the two real 3d/4s exceptions among the first-row transition metals (Cr and Cu), and predict the ground-state configuration of niobium (Z = 41), which behaves anomalously in the second row. Chromium, Z = 24. Naive aufbau, filling strictly by $(n+l)$ then breaking ties by $n$ , would give [Ar]4s²3d⁴ ( $18+2+4=24$ ). The actual ground state is

\text{Cr} = [\text{Ar}]\,4s^1\,3d^5

The energy gain comes from two compounding effects. First, promoting one electron from 4s to 3d converts a 3d⁴ configuration (four electrons in five orbitals, so one orbital must be doubly occupied, paying a Coulomb penalty $J$ and losing one potential exchange stabilization) into 3d⁵ (five electrons, one in each of the five d orbitals, all spins parallel — maximum possible number of exchange pairs, $\binom{5}{2}=10$ pairs each worth $-K$ ). Second, the resulting configuration has two half-filled subshells (4s¹ and 3d⁵) rather than one full and one partially filled — extra stability empirically associated with half-filled and fully filled shells (symmetric charge distribution, spherical symmetry, and maximized exchange energy). The 4s–3d gap in Cr is small enough (of order the exchange stabilization gained) that this rearrangement is thermodynamically favorable. Cr's ground term is $^7S_3$ — six unpaired electrons (5 in 3d, 1 in 4s), all parallel: $S=3$ , $2S+1=7$ . Copper, Z = 29. Naive aufbau gives [Ar]4s²3d⁹ ( $18+2+9=29$ ), but the true ground state is

\text{Cu} = [\text{Ar}]\,4s^1\,3d^{10}

Here promoting the electron completes the 3d shell entirely: 3d¹⁰ is a closed, spherically symmetric, maximally exchange-stabilized subshell (all ten electrons paired but every possible same-spin pairing already exhausted at 3d⁵ parallel-spin capacity, and the filled shell has zero net angular momentum and maximal orbital symmetry). The energetic reward for completing the d-shell outweighs the cost of half-emptying 4s. This is why Cu(I), with configuration [Ar]3d¹⁰, is exceptionally stable (closed-shell, diamagnetic) — the chemical basis of Cu(I)'s preference over Cu(III) in many complexes, and part of why coinage metals (Cu, Ag, Au) share the ns¹(n−1)d¹⁰ pattern. Niobium, Z = 41 (second-row anomaly). Following the same logic one row down: naive aufbau predicts [Kr]5s²4d³ ( $36+2+3=41$ ). The observed ground state is

\text{Nb} = [\text{Kr}]\,5s^1\,4d^4

Unlike Cr, Nb does not reach a half-filled d⁵ configuration by borrowing only one s electron — 4d⁴ is not half-filled. The anomaly here is driven by an even smaller 5s–4d energy gap in the second transition row (4d orbitals are more diffuse and penetrate less effectively than 3d relative to 5s, shifting the crossover point earlier in the row), so even a non-half-filled d configuration with one fewer paired s electron is favored. This illustrates the important caveat: while "half-filled/filled shell stability" rationalizes Cr and Cu cleanly, the deeper cause is the size of the ns/(n−1)d gap, which varies row to row and metal to metal — half-filled-shell language is a useful heuristic, not the fundamental mechanism, and it does not predict every exception (Nb, Mo, Ru, Rh, Pd all deviate from naive aufbau in the second transition row, each for gap-size reasons that must be checked against experiment rather than assumed from a rule of thumb).

Limiting cases, scope & common traps

Regime / check	What happens	Why
Light main-group atoms (Z ≤ 20)	$(n+l)$ rule and Hund's rules reproduce all configurations exactly	4s/3d gap not yet relevant; $Z_{\text{eff}}$ splitting by $l$ dominates cleanly
First-row transition metals (Sc–Zn)	Cr and Cu deviate; all others match [Ar]4sⁿ3dᵐ pattern	Half-filled (3d⁵) and filled (3d¹⁰) exchange stabilization exceeds small 4s–3d gap only at those two points
Second/third-row transition metals	More frequent exceptions (Nb, Mo, Ru, Rh, Pd, Ag; La, Pt, Au)	4d/5d–5s/6s gaps are smaller and less regular; case-by-case experimental configurations needed
Lanthanides/actinides	4f vs 5d (and 5f vs 6d) near-degeneracy causes irregular filling (La is [Xe]5d¹6s², not 4f¹6s²)	4f orbitals are extremely contracted and poorly shielded by $(n+l)$ heuristics; relativistic effects also matter for heavy actinides
Ions (transition-metal cations)	Electrons are removed from ns before (n−1)d, even though ns filled first	Once populated, 3d drops below 4s in energy for the same electron count — filling order ≠ removal order
$Z\to\infty$ (superheavy, hypothetical)	Relativistic contraction/expansion of s,p₁/₂ vs d,f orbitals scrambles simple $(n+l)$ predictions	Relativistic mass-velocity and spin-orbit terms become comparable to shielding effects, outside the non-relativistic aufbau framework
Half-filled subshell, general $l$	Term is always $^{(2l+2)}S$ ( $L=0$ ), maximal $S=(2l+1)/2$	Every $m_l$ singly occupied, parallel spins — no orbital angular momentum possible, maximum possible exchange pairs

The chemist behind it

Friedrich Hund derived his rules in 1925 from careful analysis of atomic spectra — the arrangement of energy levels revealed by line spectra of multi-electron atoms — years before Slater determinants gave a clean theoretical justification for why parallel-spin arrangements should be favored. Hund worked in Göttingen alongside Max Born and was part of the generation that translated the new quantum mechanics into working rules for real atoms and molecules; he later made foundational contributions to molecular orbital theory (the Hund–Mulliken approach) and to understanding diatomic molecular spectra (Hund's coupling cases). The aufbau principle itself is usually credited to Niels Bohr, who in the early 1920s used it, alongside spectroscopic data, to rationalize the structure of the periodic table before the Pauli exclusion principle or full quantum mechanics existed to justify it rigorously — Bohr's building-up scheme was a bold, largely correct guess that quantum mechanics later placed on firm footing.

Origins · 1 minFriedrich Hund and how this was discovered

Friedrich Hund

1896–1997 · Göttingen school, atomic and molecular spectroscopy, 1920s–1930s

Hund's 1925 empirical rules for atomic ground-state term symbols, derived from spectroscopic term analysis rather than first-principles calculation, were later shown to follow directly from the exchange integral in antisymmetrized many-electron wavefunctions — one of the cleanest examples in atomic physics of an empirical rule preceding, and then being explained by, the underlying quantum theory.

History

- 1913 — Bohr's early atomic model introduces shell-filling ideas for the periodic table.

- 1922–1925 — Bohr formalizes the aufbau ("building-up") principle to rationalize element-by-element periodicity.

- 1925 — Hund publishes his empirical rules for ground-state term symbols from atomic spectra.

- 1925–1926 — Pauli's exclusion principle and Uhlenbeck–Goudsmit electron spin give aufbau a rigorous quantum basis.

- 1929 — Slater's determinant formalism provides the exchange-integral justification for Hund's first rule.

Is the aufbau principle exactly true, or just a rule of thumb?

It is an approximation, though a very good one. It assumes each electron moves in an averaged, effective potential created by the nucleus and all other electrons, and that orbitals retain a fixed energy ordering as they fill — both assumptions break down modestly in real atoms, since adding electrons changes shielding and can reorder orbital energies (the 3d/4s crossover is the clearest example). It correctly predicts roughly 95% of ground-state configurations across the periodic table, with well-documented exceptions in the d-block, f-block, and a few p-block anomalies, all traceable to specific near-degeneracies rather than to any failure of quantum mechanics itself.

Why doesn't Hund's first rule apply to filled subshells?

Hund's rules only distinguish between different ways of arranging electrons within a partially filled degenerate subshell; a filled subshell (like 2p⁶ or 3d¹⁰) has only one possible arrangement — every orbital singly then doubly occupied, spins all paired — so there is no competition to resolve. Filled and empty subshells always have $S=0$ , $L=0$ , contributing nothing to the atom's overall spin or orbital angular momentum, which is why term symbol analysis (see Term Symbols and Atomic States) only needs to consider the open (partially filled) subshells.

Does Hund's rule mean electrons "repel" each other into different orbitals like charged billiard balls?

Not quite — the stabilization is a quantum statistical effect, not an extra classical force. Same-spin electrons are already spatially correlated by the antisymmetry requirement of the total wavefunction (built from Slater Determinants): the spatial part of a triplet wavefunction is forced to vanish whenever two electrons coincide in space, creating a "Fermi hole" that keeps them apart automatically. This reduces their average Coulomb repulsion below what unrestricted (opposite-spin) electrons would experience, but no new force is invoked — it is entirely a consequence of the Pauli exclusion principle acting through the symmetry of the wavefunction.

Why do lanthanum and cerium not follow the (n+l) rule for 4f filling?

Lanthanum is [Xe]5d¹6s² rather than the "expected" [Xe]4f¹6s², because the 4f orbitals are extraordinarily contracted and poorly shielded, making their energy highly sensitive to the exact electron count — a subtlety the simple $(n+l)$ heuristic cannot capture. Once a few more protons are added (cerium onward), 4f does drop below 5d in energy and begins filling in earnest, though even then several lanthanides (Ce, Gd, and others) show irregular 4f/5d electron distributions. This region is one of the least "rule-following" parts of the periodic table and is generally learned from tabulated experimental configurations rather than derived from first principles.

How do Hund's rules connect to paramagnetism?

The number of unpaired electrons predicted by Hund's first rule directly determines whether an atom or ion is paramagnetic (attracted into a magnetic field, due to unpaired electron spins) or diamagnetic (weakly repelled, all spins paired). High-spin Mn²⁺, with five unpaired 3d electrons from a half-filled subshell, is strongly paramagnetic, while Zn²⁺, with a filled 3d¹⁰ subshell and $S=0$ , is diamagnetic. This link is the practical basis of techniques like electron paramagnetic resonance (EPR) spectroscopy and magnetic susceptibility measurements used to determine oxidation states and spin states of transition-metal complexes in coordination chemistry and bioinorganic chemistry.

Why is chromium's exception explained by "half-filled shell stability" when other elements near it are not exceptions?

Half-filled (and filled) shell stability is a real, quantifiable effect — it maximizes the number of parallel-spin exchange pairs, each worth a stabilizing $-K_{ab}$ — but whether it actually flips the ground-state configuration depends on how that stabilization energy compares to the specific ns–(n−1)d orbital gap in that particular atom. In chromium the 4s–3d gap happens to be small enough that the exchange gain wins; in manganese, the "naive" configuration [Ar]4s²3d⁵ is already half-filled without needing any electron transfer, so there is no exception to make; in vanadium (3d³) or iron (3d⁶4s²), reaching a half-filled or filled state would require moving more than one electron, which costs more energy than it recovers. The rule of thumb is real but must be checked case by case against the actual orbital energy gap.

Do Hund's rules ever fail, and how would you know?

Hund's first rule (maximum multiplicity) is very robust for atomic ground states because exchange stabilization is a large effect, but the second and third rules (maximum $L$ , and the $J$ ordering) are weaker and can occasionally be reordered by more subtle correlation and relativistic effects in heavy atoms, particularly where spin-orbit coupling is strong. In molecules and especially in transition-metal complexes with ligand fields, "Hund's rule violations" are common and expected — ligand-field splitting can override the free-atom exchange preference, giving low-spin complexes (e.g., low-spin Fe²⁺ in many octahedral complexes) that pair electrons despite the loss of exchange stabilization, because the crystal-field stabilization energy is larger. This is not a failure of quantum mechanics, just a reminder that Hund's rules describe free atoms and ions, not atoms embedded in a ligand environment.