# Aufbau principle in modern quantum theory

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.