Quantum Physics, when is spin-orbit coupling necessary?

I may try an answer.

The effect is not negligible for hydrogen. The fact is that the classical treatment of hydrogen atom (without spin-orbit coupling etc..) allows one to retrieve the historical spectroscopy results of the hydrogen atom which were not very accurate compared to what we can do nowadays. Actually this is not the only reason. Indeed, since the hydrogen atom has spherical symmetry, spin-orbit coupling first does not affect the ground state and second allow one to see different energy state for different value of the quantum number [tex]l[/tex]. In atoms with more than one electron, there is no spherical symmetry and these particular effects don't appear which implies, if I remember well, that the difference in energy levels for different [tex]l[/tex] is of the order of the spin-orbit coupling.

Practically, the model of an atom structure and its refinement are made using perturbation theory and in order to be coherent with the accuracy you chose to study you have to take into account all the terms of the same order and that is what is done in atomic physics (this is just that there is no equivalent "order zero" structure for polyelectronic atoms that corresponds to the simplest one for the hydrogen atom).