How does one determine orbital configurations in multi-electron atoms? When we measure absorption/emission spectra for hydrogen-like atoms, we can point to a particular line/energy level and say "Aha! That's almost exactly the $A\rightarrow B$ transition that we predicted, so we have good reason to  hereby label that spectral line the line that corresponds to the $A\rightarrow B$ transition." 
How is this done for complex multi-electron atoms? Are specific transitions between resonant states able to be predicted/calculated for such complicated systems? I would imagine that even the notion of electron orbitals (in terms of those for the hydrogen atom) would get mixed up for more complicated atoms.
 A: This is closely related to the questions:


*

*Why do non-hydrogen atomic orbitals have the same degeneracy structure as hydrogen orbitals?

*Which electron is first ionized $n=2,\ell=1, m=?$
Strictly speaking polyelectronic atoms do not have atomic orbitals because atomic orbitals exist only for a central force. In polyelectronic atoms the repulsion between electrons means the force is not central, and the result is that the atomic orbitals are mixed up with each other. Rather than a set of seperate atomic orbitals, each containing two electrons, we have a single wavefunction containing all the electrons that can't be factored into separate orbitals.
So when we calculate the energy for a transition we are comparing the energies of two $n$-electron wavefunctions. That is we calculate the $n$-electron wavefunction for one state then the $n$-electron wavefunction for the other state and compare the difference.
But for the vast majority of atoms it's actually a pretty good approximation to ignore the electron correlation and treat the wavefunction of the atom as a product of atomic orbitals. In fact when we do the calculation we usually start by calculating atomic orbitals using a self consistent field approach, then do a configuration interaction calculation to mix up the atomic orbitals and get the wavefunction for the whole atom.
