Is the atomic shell model valid considering electron-electron interaction? In atomic physics the shell model is motivated by starting looking at the H-atom. Solving Schrödinger's equation leads to atom orbitals labled by quantum numbers n,l,m. Taking Pauli's principle into account it is said that that atom can be 'build up' by filling the shells considering Hund's rules. After that the state of an atom can be written in spectroscopic notation as $^{2S+1}L_J$ and transitions by selection rules. But in this model the electromagnetic interaction between the electrons are neglected cause the shells are filled only taking the quantum numbers into account. My questions is how valid this is because for example a transision is only determined by allowed changes of quantum numbers which are only well defined for a seperation ansatz $\Psi(\vec{r},t)=R_{nl}(r)Y_{lm}(\phi,\theta)$ but this can only be applied for the H-atom.
 A: It depends a bit on what you mean by 'the shell model'.
If you mean that you take the hydrogenic wavefunctions (perhaps scaled to the nuclear charge) and then populate them with the chosen number of electrons, then that is obviously going to be a terrible description of reality. Nobody does that, and nobody would really consider that as a serious model of the atom.
Instead, the shell model really refers to a more sophisticated approach, which is more often called the Hartree-Fock self-consistent-field method.
Here, the idea is that you model the many-electron wavefunction of the atom as a single, fully-antisymmetrized Slater determinant of individual single-electron orbitals $\phi_{1s}$, $\phi_{2s}$, $\phi_{2p}$, $\ldots$, but then to actually solve the Schrödinger equation you reduce it to $n$ separate single-electron Schrödinger equations, where each electron sees the potential produced by the averaged charge density of the other electrons. This is an iterative method: you solve the Schrödinger equation, you get a bunch of orbitals, you put them into the charge density, then you solve the Schrödinger equation again to get new orbitals, and so on and on iteratively until the results stabilize and you get a self-consistent set of orbitals and charge densities. (If it converges, of course $-$ this can be fiddly and is liable to fail.)
So, how good is this model?

*

*On the one hand, we are including the electron-electron electrostatic interaction, so this has a decent shot at reproducing the correct energy levels.

*On the other hand, we are doing a drastic simplification by reducing the problem to an effective single-electron system. This is not what the whole thing should look like: in the full Schrödinger equation, we have a multi-electron wavefunction $\Psi(\mathbf r_1,\ldots,\mathbf r_n)$, and each electron-electron interaction $$\frac{e^2}{|\hat{\mathbf r}_i-\hat{\mathbf r}_j|}$$ is an entangling operator which should induce nontrivial correlations that should break the initial Ansatz that our wavefunction is a (single) Slater determinant (which itself is the "least entangled" wavefunction possible that's still compatible with antisymmetry under exchange).

That second point can actually be quite serious, so the method is in some significant danger. But, surprisingly, Hartree-Fock approaches very often work quite well, at least as an initial approach. They often need to be supplemented by some level of post-Hartree-Fock calculation such as at least some level of configuration interaction (which definitely cannot be described as a "shell model" any more). (For examples of this Hatree-Fock breakdown in real-world systems, see this previous thread.) But on the whole, Hartree-Fock provides a working quantitative model that exactly corresponds to the intuitive idea of the atomic shell model.
