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Orbital Filling Order Exceptions

In assembling the electron configurations for many-electron atoms, one tool that students find valuable is the diagonal rule. This rule provides a guideline that is readily remembered and easily followed to produce accurate electron configurations for even complicated \begin{align*}d-\end{align*}d− and \begin{align*}f-\end{align*}f− block atoms. One confusing consequence of the diagonal rule is the order of filling the \begin{align*}4s\end{align*}4s and \begin{align*}3d\end{align*}3d subshells.

When these orbitals are filled, they are very close in energy. Though as the electrons begin to occupy the empty orbitals, the 4s level is slightly lower in energy than the \begin{align*}3d\end{align*}3d, thus it is filled first. On the other hand, when both are occupied with electrons, the \begin{align*}4d\end{align*}4d orbital becomes higher in energy. Thus, in the case that both of these filled levels are composed of valence electrons, the \begin{align*}4s\end{align*}4s level loses its valence electrons before the \begin{align*}3d\end{align*}3d level.

The preferential filling of the \begin{align*}4s\end{align*}4s orbital can also be explained by means of the electron penetration effect. Due to the spherical shape of the \begin{align*}s\end{align*}s orbital probability density distribution, the likelihood that an electron is found closer to the nucleus is greater than the multi-lobed \begin{align*}3d\end{align*}3d orbitals.

The similarity in the energy levels of the \begin{align*}4s\end{align*}4s and \begin{align*}3d\end{align*}3d orbitals also leads to another interesting consequence. In the electron configuration of the neutral Chromium atom with \begin{align*}24\end{align*}24 electrons, the diagonal rule suggests an electron configuration of \begin{align*}1s^22s^22p^63s^23p^64s^23d^4\end{align*}1s22s22p63s23p64s23d4. The actual electron configuration is \begin{align*}1s^22s^22p^63s^23p^64s^13d^5\end{align*}1s22s22p63s23p64s13d5, where due to the similarity in energy between the \begin{align*}4s\end{align*}4s and \begin{align*}3d\end{align*}3d orbitals, one electron transfers from the \begin{align*}4s\end{align*}4s to the \begin{align*}3d\end{align*}3d orbital. The net effect of this exchange yields half-filled \begin{align*}4s\end{align*}4s and \begin{align*}3d\end{align*}3d orbitals, and therefore can be justified in terms of generating additional stability. This is also the case for neutral copper atoms, with \begin{align*}29\end{align*}29 electrons and a putative electron configuration of \begin{align*}1s^22s^22p^63s^23p^64s^23d^9\end{align*}1s22s22p63s23p64s23d9. Again as in the example of chromium, an electron transfer occurs, shifting one electron from the \begin{align*}4s\end{align*}4s orbital to the \begin{align*}3d\end{align*}3d orbital. For copper, the \begin{align*}4s\end{align*}4s orbital is now half-filled but added stability is attained by completing the 3d subshell.

The stability afforded to half-filled orbitals is also noted among the \begin{align*}f-\end{align*}block elements. For example, the electron configuration for Europium (atomic number \begin{align*}63\end{align*}) is \begin{align*}1s^22s^22p^63s^23p^64s^23d^{10}\end{align*} \begin{align*}4p^65s^24d^{10}5p^66s^24f^7\end{align*} whereas the next atom, Gadolinium, with atomic number \begin{align*}64\end{align*}, has the additional electron added to the \begin{align*}5d\end{align*} orbital in order to maintain the half-filled stability of the \begin{align*}4f^7\end{align*} configuration. The electron configuration for Gadolinium is therefore \begin{align*}1s^22s^22p^63s^23p^64s^23d^{10}4p^65s^24d^{10}5p^6\end{align*} \begin{align*}6s^24f^75d^1\end{align*}.

The unusual stability of half-filled orbitals can be explained in terms of the disruption afforded by the addition of another electron to this configuration. After the orbital is half-filled, the next additional electron must pair up with another electron, increasing the spin-spin interaction energy and destabilizing the configuration.