Motivation
The question "Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?" asks for an explanation of the sequence 2, 8, 8, 18, 18, 32, … of row lengths in the periodic table. The question currently has two very interesting answers (Carlo Beenakker's and Aaron Bergman's) providing good insight on the question. However, both of those answers revolve around analyzing eigenstate degeneracies in the single-electron atom quantum system (aka the hydrogen atom), and then explaining how the sequence 2, 8, 8, … can be understood in terms of dimensions of irreducible representations of the underlying symmetry group of that quantum system.
While this is very nice and somewhat satisfying, I feel that these explanations still leave an annoying gap in the understanding of where electron shells come from. The point is that the hydrogen atom is not the correct quantum system that describes multi-electron atoms. The correct system (in non-relativistic quantum mechanics) involves a Hamiltonian that acts on a much larger Hilbert space, with a much larger symmetry group. So the missing piece in the story seems to be giving a convincing mathematical explanation for why it is sufficient to analyze the single-electron quantum system to understand electron shells and their sizes. What I'm able to understand from doing some cursory reading on the subject is that this has to do with certain approximation schemes to the multi-electron atom quantum system (some relevant technical terms are configuration interaction, Hartree–Fock method, and Slater determinant). However, the mathematical details of why such approximations produce acceptable results and what is the relationship of the approximate solutions to the solutions of the original system remain murky.
Questions
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What is the mathematical justification for analyzing the multi-electron atom quantum system in terms of solutions to the hydrogen atom system? Is there some rigorous analysis that justifies this step, or is it something that gives accurate enough results in sufficiently many situations that physicists are happy to use it regardless? (If it is not rigorous, is there at least a good clean heuristic that makes it seem plausible?)
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In the original multi-electron atom system without any approximations, is there a well-defined notion of "electron shells"? Or is the electron shell picture only meaningful to speak of after taking the step of replacing the solutions to the original system with various approximations based on single-electron wave functions?
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Related to question 2, has there been any attempt to explain electron shell sizes through a representation-theoretic analysis of the symmetry group of the original multi-electron atom quantum system, in the spirit of the answers given to the earlier linked question? This would skip the approximation step so in my eyes would give a more satisfying explanation than the hydrogen atom-based representation-theoretic analysis.
Best Answer
Let me expand a bit on my comment, focusing on points 1 and 2. (I have no meaningful response to 3.) It may be instructive to consider the simplest multi-electron atom, Helium, with two electrons. In the shell model one says these occupy two hydrogenic 1s states with opposite spin. This is a qualitative, approximate statement, the exact two-electron wave function has only about 90% weight on the antisymmetric product of two 1s states.
So one might think that the missing 10% is from other hydrogenic bound states, but even if all higher hydrogenic shells are included one does not reach the exact ground state energy of Helium, –79 eV, one ends up about 1 eV too high. This is because the hydrogenic bound states are not a complete basis set (I mistakenly stated that in a comment), for a complete basis set also the continuum states are needed. They are needed for quantitative agreement, but also to satisfy rigorous sum rules.
All of this is explained clearly in The Spectral Decomposition of the Helium atom two-electron configuration in terms of Hydrogenic orbitals (see also this comment).
So the answer to 1 and 2 would be: No, there is no mathematical justification for a description of a multi-electron atom in terms of hydrogenic shells, these have no well-defined meaning. There are more accurate representations of the multi-electron wave function (the cited paper discusses these), but these are less intuitive.