The trouble with the aufbau principle

The use of the aufbau principle to predict electronic configurations of atoms, and therefore explain the layout of the periodic table, is a key point when teaching chemistry. However, the version of this method that has been taught to generations of students is actually deeply flawed. The error is rather subtle and may well have arisen from an attempt to simplify matters. 

The orbitals

The different atomic orbitals come in various kinds that are distinguished by labels such as s, p, d and f. Each shell of electrons can be broken down into various orbitals and as we move away from the nucleus each shell contains a progressively larger number of types of orbital: the first shell only contains a 1s orbital, the second shell 2s and 2p orbitals, the third shell 3s, 3p and 3d orbitals, the fourth shell 4s, 4p, 4d and 4f orbitals and so on.

Next, we need to know how many of these orbitals occur in each shell. The answer is provided by the formula 2l + 1, where l takes different values depending on whether we are speaking of s, p, d or f orbitals. For s orbitals l = 0, for p orbitals l = 1, for d orbitals l = 2 and so on. As a result there is potentially one s orbital, three p orbitals, five d orbitals, seven f orbitals and so on for each shell.

The flaw

So far, so good. Now comes the magic ingredient in the sloppy version of this principle that claims to predict the order in which these orbitals fill (and here is where the fallacy lurks): rather than filling the shells around the nucleus in a simple sequence, where each shell must fill completely before moving onto the next shell, we are told that the correct procedure is more complicated. But we are also reassured that there is a nice simple pattern that governs the order of shell and consequently of orbital filling. This is demonstrated using the aufbau diagram, which lies at the heart of the trouble...