Aufbau Principle, Hund’s Rule, and Pauli Exclusion

The Aufbau principle, Hund’s rule and the Pauli exclusion principle are the three rules that determine how electrons fill atomic orbitals: electrons occupy the lowest-energy orbitals first, they spread out over orbitals of equal energy with parallel spins before pairing up, and no orbital can hold more than two electrons, which must have opposite spins. In brief: for nitrogen (7 electrons), the configuration is 1s² 2s² 2p³, with the three 2p electrons in three separate orbitals, all spinning in the same direction.

Together, these rules explain the arrangement of electrons in every atom and the structure of the periodic table. They tell us which elements are magnetic, why the 4s orbital fills before 3d and why certain elements, such as chromium and copper, behave differently. This article explains each rule, gives the filling order, works through examples including unpaired electrons and exceptions and gives the history of each principle.

What is the Aufbau principle?

“Aufbau” is a German word meaning “building up”. The principle states that electrons occupy the available orbital of lowest energy first, and move to higher energy orbitals only when the lower ones are full. This gives the ground state, the most stable arrangement of electrons for an atom.

What is the order of filling?

Orbital energies increase with n + l, where n is the principal quantum number and l the subshell number (s = 0, p = 1, d = 2, f = 3). When two subshells have the same value of n + l, the one with the lower n is filled first. This is called the Madelung rule, and it gives the order:

1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p

The 3p and 4s subshells both have n + l = 4, but 3p (n = 3) has the lower n and fills first. Note also that 4s (n + l = 4) fills before 3d (n + l = 5), and that 3d fills before 4p because they both have n + l = 5 and 3d has the lower n.

What is the Pauli exclusion principle?

Formulated by Wolfgang Pauli in 1925, the exclusion principle states that no two electrons in the same atom can have the same set of four quantum numbers. Since two electrons in the same orbital share n, l and m_l, they must differ in spin: one has m_s = +½ and the other −½. An orbital can therefore hold a maximum of two electrons, which are said to be paired.

What is Hund’s rule?

Friedrich Hund’s rule (1925 to 1927) applies when electrons fill orbitals of equal energy (degenerate orbitals), such as the three 2p orbitals. It states that the lowest energy arrangement has the maximum number of unpaired electrons with parallel spins. Electrons first occupy separate orbitals, each with the same spin, before any pairing begins. The reason is that electrons repel each other, and electrons in different orbitals are on average farther apart than electrons in the same orbital. Parallel spins also reduce the repulsion through a quantum effect called exchange energy.

Orbital filling in the second period

An orbital diagram shows each orbital as a box and electrons as arrows (↑ and ↓).

Worked examples

1. Carbon (6 electrons). The first two go in 1s (paired), the next two in 2s (paired) and the last two into two different 2p orbitals with parallel spins, not in the same orbital. 1s² 2s² 2p² with 2 unpaired electrons.
2. Oxygen (8 electrons). After 1s² 2s², there are 4 electrons for the 2p subshell. Three go in separate orbitals (↑ ↑ ↑) and the fourth pairs up with one of them (↑↓ ↑ ↑). Oxygen has 2 unpaired electrons, which explains its tendency to form two bonds.

3. Magnetism. Unpaired electrons make atoms paramagnetic, which means they are attracted into a magnetic field. Nitrogen atoms with 3 unpaired electrons are paramagnetic, while neon, with all electrons paired, is diamagnetic (very weakly repelled).
4. Iron (26 electrons). Fill in order: 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d⁶. The six 3d electrons occupy five orbitals: one orbital has a pair and four have a single electron (↑↓ ↑ ↑ ↑ ↑). Iron has 4 unpaired electrons, which is the basis of its strong magnetism.

5. Check the total. 2 + 2 + 6 + 2 + 6 + 2 + 6 = 26 ✓.
6. Violation example. The configuration 1s³ is impossible, because the 1s orbital can hold only 2 electrons (Pauli). The configuration 2p² written as ↑↓ _ _ is not the ground state, because it breaks Hund’s rule, since the electrons should be in separate orbitals.

What are the exceptions?

The Aufbau order predicts most elements correctly but not all. A half-filled or completely filled d subshell is especially stable, so some transition metals promote an electron from the 4s orbital to the 3d.

Chromium has 6 unpaired electrons (one in 4s and five in 3d), the most of any atom in the fourth period. The reasons for exceptions are subtle, and involve the small energy differences between the 4s and 3d orbitals.

Where do these rules come from?

Niels Bohr used the term Aufbauprinzip in the early 1920s to explain how the periodic table is built up. Wolfgang Pauli proposed the exclusion principle in 1925 to explain the structure of electron shells and received the Nobel Prize in Physics in 1945. Friedrich Hund formulated his rules in 1925 to 1927 from the study of atomic spectra. Erwin Madelung proposed the n + l rule in 1936, and Vsevolod Klechkovsky explained it theoretically in the 1950s, so it is also called the Klechkowski rule, especially in France and Russia.

Which mistakes are common?

Mistake 1: Pairing electrons too early

In a subshell with several orbitals, put one electron in each orbital before pairing any.

Mistake 2: Filling 3d before 4s

In the neutral atom, 4s fills first. Remember that the order is 4s, then 3d. (In cations, the 4s electrons are lost first.)

Mistake 3: Giving the paired electrons the same spin

Two electrons in the same orbital have opposite spins, shown by arrows pointing up and down.

Mistake 4: Thinking that the rules are exact laws for all atoms

The filling order is a very good guide, but exceptions exist, as in chromium and copper.