Stability of half filled and completely filled orbitals

Chromium's real configuration, $[\text{Ar}]3d^5\,4s^1$ rather than the "expected" $[\text{Ar}]3d^4\,4s^2$, is not an isolated quirk - it is one instance of a general rule: exactly half-filled and exactly completely-filled sets of degenerate orbitals are more stable than any other, partially-filled arrangement of the same electrons. Two separate, mutually reinforcing effects explain why.

1. Symmetrical distribution of electrons. A half-filled configuration ($p^3$, $d^5$, $f^7$ - one electron in every orbital of the set, all with parallel spin) or a completely-filled one ($p^6$, $d^{10}$, $f^{14}$ - every orbital fully paired) is, in both cases, perfectly symmetrical: every orbital in the set is treated identically (Figure 2.13). Symmetry of this kind translates directly into extra stability. The reasoning traces back to how the degenerate orbitals are actually laid out in space - the three p orbitals, for instance, point along three mutually perpendicular directions (Figure 2.14) rather than overlapping each other. Because of this spatial separation, an electron sitting in one of these orbitals only weakly shields (screens) an electron sitting in a different orbital of the same degenerate set from the nucleus's pull. Less mutual shielding means each electron feels a stronger net attraction from the nucleus, and a stronger nuclear attraction means a more stable, lower-energy configuration overall.

2. Exchange energy. The second, generally larger effect is more subtle. Whenever two or more electrons of the same spin occupy different orbitals within one degenerate set, quantum mechanics allows them to effectively "exchange" places with each other - and each such possible exchange releases a small amount of stabilising energy, called exchange energy. The more such same-spin exchanges are possible, the more exchange energy is released, and the more stable the configuration becomes. Crucially, the number of possible pairwise exchanges among $k$ same-spin electrons works out to $\binom{k}{2}=\tfrac{k(k-1)}{2}$, and this count is largest precisely when as many electrons as possible share the same spin - which happens exactly in the half-filled and fully-filled cases.

Chromium makes this concrete (Figure 2.15). In the real configuration $[\text{Ar}]3d^5\,4s^1$, all five 3d electrons occupy separate orbitals with parallel spin, giving $\binom{5}{2}=10$ possible pairwise exchanges (visualised in the figure as 4 + 3 + 2 + 1 = 10, counting down as each electron is compared against the ones "left" to exchange with). In the hypothetical $[\text{Ar}]3d^4\,4s^2$ configuration, only four d electrons carry parallel spin, giving just $\binom{4}{2}=6$ possible exchanges (3 + 2 + 1 = 6). Ten exchanges release noticeably more stabilising exchange energy than six, and this extra energy is more than enough to offset the cost of promoting one electron out of the otherwise-lower-energy 4s orbital - which is exactly why chromium's true ground state has the "irregular" $3d^5\,4s^1$ configuration rather than the Aufbau-predicted $3d^4\,4s^2$. …