It is shown that the pseudopotential function, which describes the averaged motion of charged particles with accuracy up to quadratic terms for nonuniform radio-frequency fields, can be replaced by an infinite pseudopotential series for quadrupole radio-frequency electric fields. This replacement provides a more accurate description. It allows us to extend the parameter’s range of the radio-frequency field; in this range,радиоit makes possible to describe the motion of charged particles quantitatively and not just qualitatively. Unfortunately, even this extended concept of pseudopotential is not suitable enough for describing the motion of charged particles when approaching the region of the parametric resonance, where the motion of charged particles loses stability in the quadrupole radio-frequency fields.