Filling of Orbitals: Aufbau, Pauli and Hund's Rule

Three rules, applied together, determine exactly how the electrons of a many-electron atom are distributed among its orbitals in the ground state.

The Aufbau principle. ('Aufbau' is German for 'building up'.) In the ground state of an atom, orbitals are filled in order of increasing energy -- the lowest-energy orbital available is filled first, and only once it is full does the next electron go into the next-lowest-energy orbital. For a many-electron atom, the relative energy of a subshell is decided by the sum $(n + l)$: a subshell with a lower value of $(n+l)$ is filled before one with a higher value; if two subshells happen to have the same value of $(n+l)$, the one with the lower $n$ is filled first. Applying this $(n+l)$ rule systematically gives the standard filling order:

The most commonly misjudged step in this order is that $4s$ (with $n+l = 4+0 = 4$) is filled before $3d$ (with $n+l = 3+2 = 5$), even though $3d$ belongs to a lower shell -- because the Aufbau principle ranks subshells by $(n+l)$, not by $n$ alone.

The Pauli exclusion principle. Wolfgang Pauli (1925) stated that no two electrons in the same atom can have an identical set of all four quantum numbers. Since $n$, $l$ and $m_l$ together already specify one particular orbital, and $m_s$ can only be $+\tfrac{1}{2}$ or $-\tfrac{1}{2}$, this principle has a direct and very useful consequence: any one orbital can hold a maximum of two electrons, and if it holds two, their spins must be paired (opposite) -- one $+\tfrac{1}{2}$ and one $-\tfrac{1}{2}$. This is exactly what fixes the maximum electron capacity of every subshell and shell (2 for an s subshell, 6 for p, 10 for d, 14 for f; and $2n^2$ for a shell of principal quantum number $n$). …