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[en] 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 then used to obtain exact bound state solutions associated with an effective quadrupole moment and assuming that the electron-molecule interaction is screened by sub-orbital 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. Graphical abstract: .