Shape of orbitals in atoms with multiple electrons I found this statement when browsing the Wikipedia article for atomic orbitals:
"Orbitals of multi-electron atoms are qualitatively similar to those of hydrogen."
Is this true? Googling around I could only found this article where in page 50 it seems to address how to obtain the wave function of atoms with multiple electrons, but I don't have the necessary background to understand if it proves the statement or not.
Please include academic sources or a brief proof if possible.
I find it surprising that adding electrons wouldn't change the shape of the orbitals substantially, but that's what is implied when I've studied chemistry.
 A: An approximation that seems to work well for the multi-electron case is the Hartree-Fock method. 
In Hartree-Fock, we assume the mean-field approximation. Each electron feels the repulsion from other electrons based on their average, not instantaneous, positions. (This assumption prevents Hartree-Fock from predicting van der Waals forces.)
We thus modify the hydrogen Hamiltonian by introducing two new operators. One is the average Coulombic repulsion between electrons, and the other is the exchange interaction. However, because we're using the average position of the electrons, then for our spherical atom these operators don't have an angular dependence. Thus the spherical harmonics are still separable as in the hydrogen case, so roughly the shape of the orbitals must remain the same. The only part that can change is the radial part of the wavefunction. Doing the calculations, you'll see that the radial part of the wavefunctions are squeezed or stretched a little bit due to Coulombic repulsion and the exchange interaction between electrons, and the increased Coulombic attraction to the nucleus. But as Wikipedia says, qualitatively they don't change much until you introduce multiple atoms.  
Without the mean-field approximation, I suppose even the angular shape would change, but that's beyond me.
A: What is meant is that the quantum numbers of the hydrogen solutions are still relevant for multi-electron orbitals. You still have shells 1s, 2sp, 3spd, etc.
There is not a "proof". What is there is excellent agreement between quantum chemical calculation and experiment. 
