Aufbau principle

Building from low energy upward

For the first elements the sequence is straightforward: H 1s¹, He 1s², Li 1s²2s¹, then 2p begins at boron. In a simple multi-electron ordering the familiar progression starts 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s….

The n + ℓ rule summarizes much of this order: subshells with smaller n + ℓ tend to fill first; if two have the same n + ℓ, the one with smaller n tends to fill first. Thus 4s has n + ℓ = 4 while 3d has 5, explaining why 4s begins filling first in K and Ca.

Changing orbital energies in many-electron atoms

Hydrogen has only one electron, so all orbitals with the same n are degenerate. In many-electron atoms, electron–electron repulsion, shielding and penetration split those energies. Even more importantly, the energies change as the orbitals become occupied.

The 4s orbital penetrates toward the nucleus and can be lower than empty 3d early in period 4. As 3d electrons are added, the balance changes; occupied 3d is stabilized relative to 4s. That is why a single permanent energy ladder is misleading.

Exceptions and transition-metal ions

Chromium and copper are the famous period-4 exceptions, but heavier transition series contain additional irregularities. The important lesson is not to memorize a magical “half-filled stability” rule; it is to recognize that ns and (n−1)d energies are close, so small changes in repulsion and exchange can alter the ground state.

Ions add another layer. Neutral Fe is [Ar]3d⁶4s², yet Fe²⁺ is [Ar]3d⁶: the 4s electrons are removed first. The electron-loss sequence follows the energies of the occupied atom or ion, not the order in which an imaginary empty atom was filled.

Therefore, use Aufbau to propose a configuration, then check known exceptions and apply ionization rules separately.

Ground states, excited states and evidence

Aufbau addresses the ground state. If light or a collision promotes an electron, the atom enters an excited configuration that intentionally violates the lowest-first occupancy pattern. When the electron returns to a lower state, the energy difference can appear as emitted light.

Atomic spectra and ionization measurements are among the experimental tests that constrain orbital energies and configurations. The configuration is not chosen to make the periodic table look neat; it is inferred from quantum theory and evidence.

A good practical workflow is: count electrons, apply the approximate filling order, enforce Pauli and Hund, check the known atom, then treat ions according to the occupied-shell energies.

Beyond chromium and copper

Cr and Cu are useful examples because they occur early, but they are not the only deviations from a simple diagonal filling diagram. Molybdenum, ruthenium, rhodium, palladium, silver and platinum all show configurations influenced by the small energy gaps among neighboring s and d subshells. Palladium is particularly striking: its ground state is commonly written [Kr]4d¹⁰ with no 5s electron.

These cases make the correct lesson clear: the diagonal rule predicts a likely arrangement; the actual ground state is the arrangement of lowest total energy after electron–electron interactions and relativistic effects are included.

The n + ℓ rule is a mnemonic, not a Hamiltonian

The Madelung or n + ℓ ordering works remarkably well for neutral ground-state atoms, but it is not derived as an exact universal theorem. Orbital energies depend on nuclear charge, occupation and the electronic environment. In ions, molecules and solids, the order can differ substantially from the neutral-atom filling chart.

This is why “4s is always below 3d” is incorrect. 4s is filled first for K and Ca, yet occupied 3d is generally lower than 4s through much of the transition series, and 4s electrons are removed first when many transition-metal cations form.

Experimental evidence for orbital ordering

Photoelectron spectra measure electron binding energies and expose subshell structure directly. Atomic emission spectra and magnetic measurements supply additional constraints. These observations are compared with quantum calculations to determine the configuration that best represents the ground state.

Aufbau is therefore best treated as the first pass of a scientific model: predict, check against known low-energy exceptions, and revise when the actual electronic structure requires it.

A practical algorithm for unfamiliar atoms

For an unfamiliar neutral atom, first count Z electrons, write the noble-gas core, then fill the remaining subshells using the approximate n + ℓ order while enforcing Pauli and Hund. After that, check whether the element belongs to a region with known configuration anomalies, especially the d and f blocks.

For a cation, do not simply reverse the filling arrows. Start from the neutral ground-state configuration and remove electrons from the orbitals that are highest in energy in the occupied atom, which usually means the highest ns electrons before (n−1)d in transition metals. This two-stage workflow prevents most common configuration errors.

Exercises

Potassium

Use Aufbau to write the configuration of K, Z = 19.

Chromium

Why is [Ar]3d⁴4s² only a first prediction for Cr?

Fe²⁺

Neutral Fe is [Ar]3d⁶4s². Which electrons are removed first?