  We study the ionization problem in the Thomas-Fermi-Dirac-von Weizs\"acker
theory for atoms and molecules. We prove the nonexistence of minimizers for the
energy functional when the number of electrons is large and the total nuclear
charge is small. This nonexistence result also applies to external potentials
decaying faster than the Coulomb potential. In the case of arbitrary nuclear
charges, we obtain the nonexistence of stable minimizers and radial minimizers.
