In quantum physics there is a special case known as a particle in a spherically symmetric potential. I have a problem which is similar to the case of a hydrogen atom in that there is one free electron, but the "nucleus" is actually a group of several atoms. So I'm not sure how to find the radial wavefunction for this situation? If I could say that it's a point charge I could use the same radial wavefunction as for the hydrogen atom (I think). But since the group of atoms takes up space, how does that affect the potential/wavefunction?
You can solve the radial equation piecewise. Here it is an example done in perturbation theory, but because the radial and angular parts of the Hamiltonian separate out, you can probably find a way to get an analytic solution. The first approximation here is to treat the "ball o' atoms" as a uniformly charged sphere.
In effect you're trying to calculate the wavefunction for a molecule i.e. the molecular orbitals. There is no simple solution for this. Three decades ago when I was doing this sort of thing we used a Hartree-Fock self consistent field calculation to compute the molecular orbitals on an IBM mainframe. I imagine that these days the calculation is trivial and you'd be able to do it on your PC (or phone :-).
The highly excited states, where the electron has a high probability of being a long way from the nuclei, will resemble the hydrogen atomic orbitals. However the ground state will be very different.