  In multipole-bound anions, the excess electron is attached by a short-range
multipole potential of a neutral molecule. Such anions are prototypical
marginally-bound open quantum systems. In particular, around the critical
multipole moment required to attach the valence electron, multipole-bound
anions exhibit critical behavior associated with a transition from bound states
dominated by low-$\ell$ partial waves to the electron continuum. In this work,
multipole-bound anions are described using a nonadiabatic electron-plus-rotor
model. The electron-molecule pseudo-potential is represented by a short-range
multipole field with a Gaussian form-factor. The resulting coupled-channel
Schr\"odinger equation is solved by means of the Berggren expansion method, in
which the electron's wave function is decomposed into bound states, narrow
resonances, and the non-resonant scattering continuum. We show that the
Gaussian model predicts the critical transition at the detachment threshold.
Resonant states, including bound states, decaying resonances, subthreshold
resonances, and antibound states are studied, and exceptional points where two
resonant states coalesce are predicted. We discuss the transition of rotational
band structures around the threshold and study the effects of channel coupling
on the decay width of resonant poles.
