Calculating atomic spectra means to me that one takes a Hamiltonian neglecting some effects known to be present (any such Hamiltonian will always neglect something, be it relativistic effects or even quantum gravity) and tries to approximate its (math.) spectrum.
I've only every seen this scheme applied to Hydroge: The most naive, exactly solvable Hamiltonian gives rise to the Bohr formula of energy levels. One can then add to this "relativistic corrections to the dispersion relation" while keeping the theory non-relativistic, as well as couplings between the electron spin (or rather magnetic moment) and angular momentum. These are referred to as Fine Structure and provide corrections that are smaller than the Bohr energies by a factor of roughly $\alpha^2$ (Fine Structure Constant). There is then the Lamb shift, smaller by another factor of $\alpha$, that in a sense incorporates the fact, that the em-field is quantized without actually quantizing it, and the Hyper-Fine Structure from the magnetic interaction between dipole moments of electron and proton.
Alternatively, one can consider the Dirac equation (and Dirac Hamiltonian), which is fully relativistic and should be phenomenologically consistent in the regime where no charged particles are created, despite being not bounded below. There is also the Fierz-Pauli model, which considers non-relativistic particles in external potentials inside a fully quantized, non-self interacting electromagnetic field. I'm not sure what the respective predictions are (I think it's hard to work out the spectra of these models) and how to evaluate the precisions of each of these descriptions. If anyone knows sources that compare all of these calculations to good experimental data, I would be very interested even for Hydrogen.
In 1951 Kato proved that the Coulomb Hamiltonian for an arbitrary (finite) number of moving positive and negative charges, with instantanous Coulomb potentials acting between them, is self-adjoint. In principle the same calculations should be applicable to other atoms, even if it may be harder to work out the spectra of those Hamiltonians, perhaps by some approximations or numerical methods. I imagine that similar schemes to the ones above should be possible to apply to more complicated atoms than Hydrogen, without too much conceptual difficulty. My question is:
Which of these models have been applied to different atoms and how good is the agreement with experiment? What effects are more important in larger atoms, that may be neglected for Hydrogen?