Electronic Configuration of Atoms, Exchange Energy and Orbital Stability

Applying the Aufbau order, the Pauli exclusion principle and Hund's rule in sequence gives the ground-state electronic configuration of an atom: the number of electrons in each subshell, written as a superscript after that subshell's symbol, in Aufbau order (for example, sodium, $Z = 11$: $1s^2\, 2s^2\, 2p^6\, 3s^1$, often abbreviated using the previous noble gas core as $[\text{Ne}]\, 3s^1$). For most elements this procedure gives exactly the experimentally observed configuration.

Exceptions: chromium, copper, and the reason for them. A small number of elements -- most notably chromium ($Z=24$) and copper ($Z=29$) in the first transition series -- do not follow the simple predicted pattern. The Aufbau order predicts chromium to be $[\text{Ar}]\,3d^4\,4s^2$, but its actual, experimentally observed configuration is $[\text{Ar}]\,3d^5\,4s^1$; similarly, the Aufbau order predicts copper to be $[\text{Ar}]\,3d^9\,4s^2$, but its actual configuration is $[\text{Ar}]\,3d^{10}\,4s^1$. In both cases, one electron that the simple rule would have kept in $4s$ instead moves into $3d$, giving an exactly half-filled ($3d^5$, chromium) or exactly completely filled ($3d^{10}$, copper) $d$ subshell, at the cost of an exactly half-filled ($4s^1$) $s$ subshell in place of a filled one. This rearrangement happens because half-filled and completely filled subshells are unusually stable, for two reinforcing reasons: (i) a symmetrical distribution of electrons, one in each orbital of a degenerate set (half-filled) or two in each (completely filled), produces a more even, more symmetric distribution of charge around the nucleus, which lowers electron-electron repulsion; and (ii), more importantly in size, exchange energy -- electrons of parallel spin occupying different orbitals of the same subshell can exchange positions with each other without violating the Pauli principle, and quantum mechanics shows that every such possible exchange between two parallel-spin electrons lowers the atom's total energy by a fixed amount. A half-filled or completely filled subshell has the maximum possible number of electrons with parallel spin arranged this way, and therefore the maximum number of stabilising exchanges, making that configuration lower in energy (hence more stable) than the one the simple $(n+l)$ rule alone would predict. The same pattern recurs, for exactly the same reason, in a few heavier transition and inner-transition elements (for instance molybdenum, directly below chromium, is $[\text{Kr}]\,4d^5\,5s^1$ rather than $4d^4\,5s^2$). …