Aufbau principle in modern quantum theory

Question

What is the rigorous definition of the Aufbau principle and the mathematical model used for its description?

From Wikipedia, we have that the principle postulates a hypothetical process in which an atom is "built up" by progressively adding electrons. For modeling the postulate, we use the Madelung rule, Pauli exclusion and "if two same energy orbitals are available, choose the less occupied one)". So, $n+\ell$ ordering with constraints. And I also know that the Aufbau principle has its exceptions.

Answer 1

The Aufbau principle isn't rigorous because it's based upon the approximation that the electron-electron interaction can be averaged into a mean field. This is called the Hartree–Fock or self-consistent field method. The centrally-symmetric mean field results in a set of atomic orbitals that you can populate 2 electrons at a time.

The trouble is that the electron correlations mix up the atomic orbitals so that distinct atomic orbitals no longer exist. Instead you have a single wavefunction that describes all the electrons and does not factor into parts for each electron. For example this is explicitly done in the configuration interaction technique for improving the accuracy of Hartree–Fock calculations.