When does the separation of variables work

When studying the hydrogen atom, given that the potential depends only on the distance and not an any angle, we can do a separation of variables of the wavefunction as the product between a function depending only on the distance between particles (protons and electrons) and a spherical harmonic. However I saw this done even when the potential does depend on angle, for example when having the interaction between 2 electric dipoles. What is the criterion based on which I know if I can just a factorizable wavefunction or not? Thank you!

Two answers:

(1) You just have to try it. Sometimes it helps and sometimes it just makes a mess.

(2) There is a deep relation between separation of variables and symmetries. The hydrogen atom can be solved by separation of variables in two different coordinate systems, spherical and parabolic. This is why you have the degeneracy of radial and angular energy levels.

Separation variables works when it works. If it doesn't work, it can often lead to a series solution like the Legendre polynomial expansion.