  We make a multipole expansion of the atomic/molecular electrostatic charge
distribution as seen by the valence electron up to the quadrupole term. The
Tridiagonal Representation Approach (TRA) is used to obtain an exact bound
state solution associated with an effective quadrupole moment and assuming that
the electron-molecule interaction is screened by suborbital electrons. We show
that the number of states available for binding the electron is finite forcing
an energy jump in its transition to the continuum that could be detected
experimentally in some favorable settings. We expect that our solution gives an
alternative, viable and simple description of the binding of valence
electron(s) in atoms/molecules with electric dipole and quadrupole moments. We
also ascertain that our model implies that a pure quadrupole-bound anion cannot
exist in such a charge-screening environment.
