Contents
Introduction The Statement The Madelung Rule Order of Filling Exceptions Periodic Table Misconceptions FAQ Sources Introduction
The Aufbau ("building up") principle is the rule that electrons in a multi-electron atom fill orbitals in order of increasing energy. Combined with Pauli's exclusion principle and Hund's rules, it determines the ground-state electron configuration of any neutral atom and is the foundation of the periodic table's structure. The principle was introduced by Niels Bohr in 1920 as part of his old quantum theory and refined as quantum mechanics developed.
The Statement
The principle states that, in the ground state of an atom, electrons occupy orbitals of lowest energy first. Specifically:
Each orbital holds at most 2 electrons (Pauli exclusion).
Orbitals are filled in order of increasing energy.
For degenerate orbitals, electrons obey Hund's rule (maximum spin first).
The combination of these rules gives the electron configuration that is the chemical "fingerprint" of an element.
The Madelung Rule
The energy ordering of orbitals in multi-electron atoms is not the simple n-ordering of hydrogen. The empirical Madelung rule (or n+ℓ rule) [1 ] states:
Orbitals are filled in order of increasing n + ℓ.
For equal n + ℓ, the orbital with smaller n is filled first.
This produces the order: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, ...
Why It Works
In multi-electron atoms, inner electrons screen the nuclear charge from outer electrons. The 4s orbital (with relatively low ℓ = 0) penetrates closer to the nucleus than 3d (ℓ = 2) and feels more nuclear charge — making 4s lower energy than 3d in many atoms despite having higher n.
Limitations
The Madelung rule is an empirical approximation. Real orbital energies depend on the specific atom and configuration. The rule predicts the gross filling order but has notable exceptions.
Order of Filling
Periods 1–3
Period 1: 1s (H, He).
Period 4 and Beyond
Period 4: 4s, 3d, 4p. The 4s fills before 3d, so K and Ca have 4s electrons before the d block starts at Sc.
Pattern
The Aufbau order reflects orbital energies in neutral atoms. Once an atom is ionized or in a different chemical environment, the energy ordering can differ (4s electrons are usually removed before 3d in transition metal ions).
Exceptions
Several elements deviate from strict Aufbau filling because of subtle energy considerations:
Chromium (Z=24): 4s¹ 3d⁵ (not 4s² 3d⁴). The half-filled 3d shell is energetically favored.Copper (Z=29): 4s¹ 3d¹⁰ (not 4s² 3d⁹). Full d shell favored.Niobium (Z=41), Molybdenum (Z=42): Similar deviations in the second transition row.Palladium (Z=46): 4d¹⁰ 5s⁰. Both s electrons promoted to d.
These exceptions show that "half-filled" and "full" subshells have extra stability — a real but subtle quantum-mechanical effect from exchange interactions and the geometry of orbital filling [2 ].
Heavy Elements
For lanthanides and actinides, relativistic effects on inner electrons modify orbital energies. Predicting exact ground-state configurations for the heaviest elements often requires detailed calculations rather than simple Aufbau filling.
The Periodic Table
The Aufbau principle generates the periodic table's structure:
s block: Groups 1-2 (alkali, alkaline earth metals). Outer ns orbitals.p block: Groups 13-18. Outer np orbitals.d block: Transition metals. Outer (n−1)d orbitals being filled.f block: Lanthanides and actinides. (n−2)f orbitals being filled.
The periodic table's row lengths (2, 8, 8, 18, 18, 32, 32) reflect how many electrons fit in the orbitals filling in each period — direct consequences of Pauli + Aufbau + the angular-momentum structure of atomic orbitals.
Common Misconceptions
"The Aufbau rule is exact"
It's a good first approximation with known exceptions. Real orbital energies require detailed calculations.
"Orbital energies are fixed by the element"
They depend on the electronic configuration. As you remove electrons from a transition metal, the 4s electrons go first because in the ion, 3d is lower in energy than 4s.
"The 4s fills before 3d because 4s is lower in energy"
Partially true. In neutral atoms before transition metals (K, Ca), 4s is lower. But in transition metal atoms themselves, 3d and 4s are very close and the order can shift.
"Half-filled shells have unique stability"
They have somewhat enhanced stability (Hund's rule + exchange interactions), but it's a subtle effect of a few percent in energy. Not a separate fundamental principle.
FAQ
Why does 4s fill before 3d?
In neutral atoms with low atomic number through calcium, 4s orbital energy is lower than 3d (mainly due to penetration of the s orbital closer to the nucleus). After Sc, the ordering shifts and 3d becomes lower than 4s in the ions.
How accurate is the Madelung rule?
It predicts the correct filling order for most elements but has the exceptions noted above. For very heavy elements, relativistic effects can cause additional deviations.
What about excited states?
Aufbau gives ground states. Excited states have one or more electrons in higher orbitals; their configurations follow from how the excitation is generated.
Does Aufbau apply to molecules?
The analogous concept for molecules is filling molecular orbitals (MOs) from lowest to highest. The MO energies must be computed for each molecule; the simple rules of atomic Aufbau don't directly apply.
Sources
Madelung, E. (1936). "Die mathematischen Hilfsmittel des Physikers." Springer.
Levine, I. N. (2014). Quantum Chemistry , 7th ed. Pearson.
Atkins, P. W., de Paula, J. (2014). Physical Chemistry , 10th ed. Oxford University Press.
Cowan, R. D. (1981). The Theory of Atomic Structure and Spectra .
Bohr, N. (1922). "Über die Anwendung der Quantentheorie auf den Atombau." Zeitschrift für Physik , 13(1), 117–165.