Bound on the maximum negative ionization of atoms and molecules

It is proved that $N_c$, the number of negative particles that can be bound to an atom of nuclear charge $z$, satisfies $N_c<2z+1\mathrm{}$. For a molecule of $K$ atoms, $N_c<2Z+K$ where $Z$ is the total nuclear charge. As an example, for hydrogen $N_c=2\mathrm{}$, and thus ${\mathrm{H}}^{--}$ is not stable, which is a result not proved before. The bound particles can be a mixture of different species, e.g., electrons and $\pi$ mesons; statistics plays no role. The theorem is proved in the static-nucleus approximation, but if the nuclei are dynamical, a related, weaker result is obtained. The kinetic energy operator for the particles can be either $\frac{{\left[\frac{p-eA\left(x\right)\mathrm{}}{c}\right]}^2}{2m}$ (nonrelativistic with magnetic field) or ${\{{\mathrm{}[pc-eA(x\left)\right]}^2+m^2{c}^4\}}^{\frac{1}{2}}-m{c}^2$ (relativistic with magnetic field). This result is not only stronger than that obtained before, but the proof (at least in the atomic case) is simple enough to be given in an elementary quantum-mechanics course.