  In quadrupole-bound anions, an extra electron is attached at a sufficiently
large quadrupole moment of a neutral molecule, which is lacking a permanent
dipole moment. The nature of the bound states and low-lying resonances of such
anions is of interest for understanding the threshold behavior of open quantum
systems in general. In this work, we investigate the properties of quadrupolar
anions as extreme halo systems, the formation of rotational bands, and the
transition from a subcritical to supercritical electric quadrupole moment. We
solve the electron-plus-molecule problem using a non-adiabatic coupled-channel
formalism by employing the Berggren ensemble, which explicitly contains bound
states, narrow resonances, and the scattering continuum. We demonstrate that
binding energies and radii of quadrupolar anions strictly follow the scaling
laws for two-body halo systems. Contrary to the case of dipolar anions,
ground-state band of quadrupolar anions smoothly extend into the continuum, and
many rotational bands could be identified above the detachment threshold. We
study the evolution of a bound state of an anion as to dives into the continuum
at a critical quadrupole moment and we show that the associated critical
exponent is consistent with the second-order phase transition. Everything
considered, quadrupolar anions represent a perfect laboratory for the studies
of marginally bound open quantum systems.
