You will obtain a very complex interaction potential, which will be significantly simplified if you first expand the potential of the nucleus into a multipole expansion. The ground state also already requires the monopole moment of the nucleus, which is already a multipole moment of the nucleus.

The multipole expansion is only valid if you are far enough away from the source, otherwise the higher order multipole terms will become increasingly more important due to the r^(-n) term.
Perturbation theory is only valid if the energy of the non-perturbed state is significantly larger than the differences in the energy due to the perturbative hamiltonian.
These two requirements are both satisfied in the electron nucleus system if the scale of the deformities in the nucleus is significantly smaller than the typical distance between the electron and the nucleus.