Pauli exclusion principle

The precise statement

An orbital is specified by n, ℓ and mℓ. If two electrons share that orbital, those first three quantum numbers are necessarily identical. The only remaining quantum number is the spin projection mₛ, which can be +1/2 or −1/2. The two electrons must therefore have opposite spin projections.

This is not mainly a rule about electrostatic repulsion. Even hypothetical non-repelling electrons would still obey Pauli because electrons are identical fermions. The many-electron wavefunction changes sign when two identical fermions are exchanged; two electrons cannot occupy the same one-electron state.

From Pauli to the periodic table

A subshell with angular quantum number ℓ contains 2ℓ + 1 orbitals. Multiplying by two electrons per orbital gives 2, 6, 10 and 14 electrons for s, p, d and f. Those capacities are visible directly in the widths of the periodic-table blocks.

The maximum number of electrons in a shell n is 2n². For n = 2, the 2s orbital plus three 2p orbitals give four orbitals × two = 8 electrons. For n = 3, 3s + 3p + 3d give 1 + 3 + 5 = 9 orbitals, hence 18 electrons.

Pauli and Hund are complementary

Pauli says what cannot happen inside one orbital; Hund says how electrons arrange among several equal-energy orbitals. For carbon 2p², Pauli would allow a pair in one p orbital, but Hund tells us the lower-energy ground state places the two electrons in different p orbitals with parallel spin projections.

Together with Aufbau, these rules generate the familiar ground-state orbital diagrams used throughout chemistry.

Consequences beyond electron bookkeeping

The exclusion principle is responsible for much of the structure of matter. Because electrons cannot all collapse into the same lowest state, atoms develop shells and subshells. The resulting valence structure underlies periodicity and chemical bonding.

Pauli exclusion also enters molecular-orbital filling and the electronic structure of solids. In a metal, many electrons fill an enormous number of closely spaced quantum states up to a characteristic Fermi energy rather than all occupying the same lowest state.

The principle does not say that two electrons can never be in the same place. Opposite-spin electrons in the same orbital have overlapping spatial probability distributions; what is excluded is identity of the complete quantum state.

Antisymmetry and the fermion origin of exclusion

Electrons are indistinguishable particles with half-integer spin. Quantum mechanics requires the total wavefunction of identical fermions to change sign when two particles are exchanged. If two electrons attempted to occupy exactly the same one-electron state, exchanging them would change nothing physically but would also require the wavefunction to equal its own negative. The only solution is zero probability for that forbidden state.

This antisymmetry is the deeper origin of the exclusion rule. It is independent of the ordinary electrostatic repulsion between two negatively charged electrons.

Pauli exclusion and chemical bonds

Two electrons can occupy the same bonding molecular orbital when their spin projections are opposite. That paired occupation is central to the ordinary two-electron covalent bond. A third electron cannot enter the identical bonding state; it must occupy a different molecular orbital, often one of higher energy.

Thus Pauli simultaneously permits paired covalent bonds and prevents unlimited accumulation of electrons into one low-energy orbital.

A macroscopic consequence: degeneracy pressure

The same exclusion principle that organizes atomic shells also resists extreme compression of electron matter. When matter is compressed so strongly that ordinary atomic structure breaks down, electrons must occupy successively higher momentum states because the lower states are already filled. The resulting electron degeneracy pressure supports white-dwarf stars against gravity.

This astrophysical example is far beyond ordinary chemistry, but it shows that Pauli exclusion is a fundamental property of matter rather than a classroom rule invented for orbital boxes.

Pauli is why closed shells are special

Once every orbital in a subshell is occupied by two opposite-spin electrons, adding another electron requires a different, usually higher-energy orbital. This creates energetic shell closures at 1s², 2p⁶, 3p⁶ and related configurations and contributes to the exceptional stability of noble-gas electron structures.

The next electron after Ne cannot enter the already filled 2p set; sodium begins a new 3s shell. That single exclusion-driven step is visible macroscopically as the start of a new period of the periodic table.

Exercises

1s orbital

Can a 1s orbital contain three electrons if their spins are arranged differently?

3d capacity

Use Pauli to determine the maximum population of a 3d subshell.

Same position?

Does Pauli forbid two opposite-spin electrons from occupying the same orbital region in space?