For $N$ electrons in $N$ orbitals — each orbital a spatial function together with a spin state — there is one antisymmetric state, and Slater wrote it as a determinant: the orbitals along one direction, the electrons along the other. Swapping two electrons swaps two rows, which changes the sign of a determinant: the state is antisymmetric. Putting two electrons in the same spin-orbital makes two columns equal, and the determinant vanishes: no such state exists. That is Pauli’s exclusion principle, and it builds the periodic table. An orbital with given $n$, $l$ and $m$ holds two electrons, one of each spin; a subshell $nl$ holds $2(2l+1)$; a shell $n$ holds $2n^{2}$. When electrons fill an atom’s orbitals from the bottom, they must stack upward instead of all sinking to the lowest.

A car park with numbered spaces, one car to a space: the tenth car cannot squeeze in beside the first, however much it would like the space nearest the door. Electrons fill orbitals the same way, which is why atoms have size and chemistry.

Sodium’s eleven electrons in the ground configuration $1s^{2}2s^{2}2p^{6}3s$. Each orbital slot holds at most two electrons, one with spin up and one with spin down, so ten electrons fill the first two shells and the eleventh has to start the third.

$N$ electrons in the orthonormal spin-orbitals $\psi_{k_{1}},\dots,\psi_{k_{N}}$, with the position and spin of electron , have the antisymmetric state

• The Pauli principle: no two electrons occupy the same spin-orbital; in an atom, no two share all of $n$, $l$, $m_{l}$ and $m_{s}$.

Written out, the determinant is a signed sum over the ways of assigning the orbitals to the electrons. Swapping two electrons swaps two rows, so the sign flips. A repeated orbital gives two equal columns, so it vanishes. For the norm, only the assignments that agree survive the orthogonality of the orbitals, and there are $N!$ of them.

The determinant written out, a sum over the permutations $\sigma$ of the electrons.

Swapping two electrons swaps two rows, which changes the sign of a determinant.

Two equal columns make a determinant vanish.

With orthonormal orbitals, a product of overlaps vanishes unless the two permutations agree; each of the $N!$ that do gives 1.

$N$ electrons in $N$ spin-orbitals have one antisymmetric state: the determinant of the orbitals evaluated at each electron, divided by $\sqrt{N!}$.

Each orbital $nlm$ holds two electrons, so a subshell $nl$ holds $2(2l+1)$ and a shell $n$ holds $$.

An orbital fixed by $n$, $l$ and $m$ leaves only the spin free, so it holds at most two electrons.

Swapping two electrons swaps two rows of the matrix, and a determinant changes sign when two rows are swapped.

Great work! You now know how to:

• write the antisymmetric state of $N$ electrons as a Slater determinant
• derive the exclusion principle from it
• count the electrons that fit in an orbital, a subshell and a shell