I was reconsidering the Madelung’s rule and now I am not sure whether I understand well why the exeptions of the rule are possible. Does anybody still know the answer?
In question 1.3 of of the LCAO model for example is called that Cupper, Palladium (Pd), silver and gold have exceptions. I think this is due to the large amount of electrons between the outer electrons and the nucleus in those electrons. The attractive force of the nucleus on the outer electrons is not that large due to the distance to the nucleus and due to the repulsive force of the electrons in the lower shells. Therefore it is sometimes energetically profitable to form an electron couple in the next shell. The exception atoms of question 1.3 are in the end all in group 11 (or 10 for Pd) of the periodic system. Therefore all those atoms have a lot of electrons in the lower shells and in all of those atoms it is possible to form a electron couple when an electron is moved from the second highest shell to the highest shell.
I was reconsidering the Madelung’s rule and now I am not sure whether I understand well why the exceptions of the rule are possible. Does anybody still know the answer?
Madelung’s rule / Aufbau principle are simplistic models built on a set assumptions about a fixed nature and fixed energies of individual orbitals. However, the electron configuration is a quantum many-body phenomenon where various interactions can play decisive role in determining the total energy of an electron in an orbital that then disrupts the ordering principle. An analytical solution of the Schrödinger Equation for a ground state of interacting-electrons has not yet been possible. Therefore, the nature of some of those interactions has been traditionally captured heuristically in some regular exceptions to those models that are quite intuitive, like the preference for electron pairing and half-filling, and some interactions can be incorporated into models aimed at solving the quantum many-body problem approximately, like Hartree-Fock method. For the heavier elements, the effects of special relativity may have to be taken into account.
From this perspective, Madelung’s rule / Aufbau principle are imperfect models that capture the essence of the noninteracting-electron configuration and serve as a useful and a sufficient tool for many branches of science and engineering, but a deeper understanding of the exceptions in the electron configuration order requires considering for example; electronic correlations and spin-orbit interaction.
The attractive force of the nucleus on the outer electrons is not that large due to the distance to the nucleus and due to the repulsive force of the electrons in the lower shells. Therefore it is sometimes energetically profitable to form an electron couple in the next shell. The exception atoms of question 1.3 are in the end all in group 11 (or 10 for Pd) of the periodic system. Therefore all those atoms have a lot of electrons in the lower shells and in all of those atoms it is possible to form a electron couple when an electron is moved from the second highest shell to the highest shell.
In view of what I wrote above, your explanation builds on a working and popular principle, but to my knowledge, I doubt it is truly a satisfying explanation from the perspective of the current state of physics, but that depends of course on how precisely you would like to describe the cause of, for example, Palladium’s exception.
I invite a discussion with @team if anyone knows something more
