Bound states of dipole-bound anions are studied by using a nonadiabatic pseudopotential method and the Berggren expansion involving bound states, decaying resonant states, and nonresonant scattering continuum. The method is benchmarked by using the traditional technique of direct integration of coupled-channel equations. A good agreement between the two methods has been found for well-bound states. For weakly bound subthreshold states with binding energies comparable to rotational energies of the anion, the direct integration approach breaks down and the Berggren expansion method becomes the tool of choice.