How does adding electrons break the angular momentum degeneracy? In the hydrogen atom, the energy does not depend on l. This degeneracy is sometimes called "accidental" (because it does not correspond to some symmetry?). However, there is l dependence in the energy multi-electron atoms. Since the original degeneracy was "accidental", is it no longer correct to talk about what breaks this degeneracy? Physically, I understand this is a result of the lower l-value states being concentrated closer to the nucleus, but is there a way to see that this would be the case in advance, without finding the eigenvalues of the Hamiltonian?
 A: You could say that the degeneracy for the H-atom is accidential, but this unique property that the energy does not depend on l is because of the fact that you have a pure Coulomb-potential. For this pure Coulomb potential there exists an operator:
$\textbf{R} = \frac{1}{2\mu}(\textbf{p}\times\textbf{l}-\textbf{l}\times\textbf{p})-\frac{e^2}{r}\textbf{r}$
which is derived from the Lenz-Rungevector of classical mechanics and has the properties:
$[\mathcal{H},\textbf{R}] = 0 = [\textbf{l},\textbf{R}]$
So actually it isn't an accidential symmetry!
The correct way to predict what symmetries are present is the machinery of group-theory (or if you want to go to practical calculations representation theory), this provides a systematical way to reduce your system to it's most fundamental symmetry-groups (irreducible representations).
Atomic physics always works the same scheme:

*

*Look for the operators $\mathbf{A}_i$ which belong to the symmetry group of the Hamiltonian (so $[\mathcal{H},\mathbf{A}_i] = 0$) so that they all commute pair-wise, so $[\mathbf{A}_i,\mathbf{A}_j] = 0 \,\forall i,j$.

*Determine the corresponding quantumnumbers and representations of the states, with the right theorems of group theory you can now determine what dependancies your system has.

To go to more numerical results you'll need some approximations ans perturbation theory:


*Determine the assumptions in which you are working (when having a magnetic field for example)

*Apply perturbation-theory (most of the time degenerate perturbation theory)

