Atomic Structure

A configuration is written as a list of subshells, each with a coefficient and a superscript: the coefficient names the shell, the letter names the subshell, the superscript counts the electrons in it. Sodium is $1s^22s^22p^63s^1$, and the $3s^1$ at the end is where the interesting decisions were made. Three rules decide them. The Aufbau principle, the building up principle, says subshells fill in increasing order of energy; since that energy depends on the principal quantum number $n$ and the azimuthal quantum number $\ell$, the order comes off the sum $n + \ell$, the lower sum filling first and the lower $n$ winning a tie. The Pauli exclusion principle caps an orbital at two electrons, because no two electrons in an atom may share the same set of four quantum numbers, so a second electron in an orbital must have the opposite spin. Hund's rule governs degenerate orbitals within one subshell: each is singly occupied with parallel spins before any pairing happens, which keeps repulsion down.

The order the rules produce is $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s$, and $4s$ precedes $3d$ by arithmetic: $4s$ has $n + \ell = 4 + 0 = 4$ against $3d$ at $3 + 2 = 5$. A repeated inner block is compressed into a noble gas core, so calcium is $[\mathrm{Ar}],4s^2$. Two elements break the ladder outright: chromium is $[\mathrm{Ar}],3d^54s^1$ and copper is $[\mathrm{Ar}],3d^{10}4s^1$, because an exactly half-filled and a completely filled $d$ subshell both carry extra stability. Across the first transition series the $d$ electrons occupy their orbitals singly and in parallel before pairing, so manganese, at $3d^5$, has five unpaired electrons.