Formation of Stable Molecules within the Statistical Theory of Atoms

We show that no theory can describe a stable molecule or molecular ion if it expresses the charge density as a function of the electric potential alone. Such theories comprise the Thomas-Fermi, Thomas-Fermi-Dirac, Thomas-Fermi-Gomb\'as theories. The Thomas-Fermi theory with the Fermi-Amaldi correction cannot give rise to molecular binding for homonuclear, diatomic, neutral molecules if the total number of electrons in the molecule equals or exceeds 4. The Thomas-Fermi-Weizs\"acker theory gives stable homonuclear, diatomic molecules.