Electronic Configuration and Pauli/Hund Rules

Three rules decide the ground-state configuration of every atom:

• Aufbau principle — orbitals fill in order of increasing energy, lowest first. The order follows the (n + l) rule: lower $(n+l)$ fills first, and for a tie the lower $n$ fills first. This gives $1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, \dots$
• Pauli's exclusion principle — no two electrons in an atom can have the same set of all four quantum numbers. Consequence: an orbital holds at most 2 electrons, and they must have opposite spins.
• Hund's rule of maximum multiplicity — electrons occupy degenerate orbitals (same subshell) singly first, all with parallel spin, and pairing begins only after every such orbital has one electron.

Writing a ground-state configuration, and the two anomalies:

• Fill in Aufbau order, obeying Pauli (2 per orbital) and Hund (singly first): e.g. nitrogen $(Z=7)$ is $1s^2\,2s^2\,2p^3$.
• Extra-stability rule: exactly half-filled $(p^3, d^5, f^7)$ and fully-filled $(p^6, d^{10}, f^{14})$ subshells are especially stable (symmetric distribution + favourable exchange energy).
• Chromium $(Z=24)$ is $[\text{Ar}]\,4s^1\,3d^5$, not $4s^2\,3d^4$ — one 4s electron shifts to give a half-filled $3d^5$.
• Copper $(Z=29)$ is $[\text{Ar}]\,4s^1\,3d^{10}$, not $4s^2\,3d^9$ — the shift gives a fully-filled $3d^{10}$.

The counting procedure:

• Write the ground-state configuration; only the incompletely filled subshell(s) can carry unpaired electrons.
• Fill that subshell's degenerate orbitals by Hund's rule (singly, parallel spin, before pairing) and count the singly-occupied boxes.
• For a subshell holding $k$ electrons in $N$ orbitals: if $k \le N$, all $k$ are unpaired; if $k > N$, the number unpaired is $2N - k$.
• A fully-filled subshell $(p^6, d^{10}, s^2)$ has zero unpaired electrons — e.g. zinc's $3d^{10}4s^2$ is entirely paired.