Anomalous Electronic Configurations of Chromium and Copper

Two elements among the first thirty break the straightforward Aufbau prediction: chromium (Z=24) and copper (Z=29). Simple $(n+l)$-order filling predicts chromium's configuration as $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^4$ — but a 3d subshell holding 4 electrons is neither half-filled nor fully filled, and is therefore less stable than it could be. Because of the repulsion between electrons, one of the two 4s electrons shifts into the empty 3d orbital, giving both 4s and 3d subshells a HALF-filled arrangement ($4s^1\,3d^5$) and a genuine gain in stability; chromium's actual configuration is therefore $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^1\,3d^5$. The same logic applies to copper: the naive prediction is $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^2\,3d^9$, where 3d, holding 9 electrons, is neither half-filled nor fully filled. Here interelectronic repulsion pushes one 4s electron into 3d, completely FILLING the 3d subshell ($3d^{10}$) while leaving 4s half-filled ($4s^1$), again gaining extra stability; copper's actual configuration is $1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^1\,3d^{10}$. Both cases illustrate the same underlying principle: a half-filled or fully …