The Madelung ordering rule says that as you increase the atomic number of a neutral atom, the electron orbitals fill up in order of increasing value of $n+l$, and in increasing order of $n$ within a given total value $n + l$, where $n$ is the principal quantum number and $l$ is the angular momentum quantum number. This rule qualitatively explains most of the patterns of regularity contained within the periodic table.
The Wikipedia article also claims that there are certain exceptions to the Madelung ordering rule, and that for some elements, "experiment" gives a different pattern of electron orbital filling.
But electron orbitals are only well-defined for hydrogenic atoms, as discussed here. For multi-electron atoms, the notion of electron orbitals only makes sense within the Hartree-Fock approximation, as Slater determinants of single-electron orbitals do not exactly solve the Schrodinger equation because they do not include correlation effects. So what it exactly does it mean to "experimentally" determine which electron orbitals are filled? Does the claim that an element "violates" the Madelung ordering rule just mean that in a rough sense, it is less chemically similar to the element directly above it on the periodic table than it is to another element in the period above it? Or is the claim referring to a more precise experimental result?
