On the excess charge of a relativistic statistical model of molecules with an inhomogeneity correction

We show that the molecular relativistic Thomas-Fermi-Weizsacker functional consisting of atoms of atomic numbers $Z_1,...,Z_k$ has a minimizer, if the particle number $N$ is constrained to a number less or equal to the total nuclear charge $Z:=Z_1+...+Z_K$. Moreover, there is no minimizer, if the particle number exceeds $2.56 Z$. This gives lower and upper bounds on the maximal ionization of heavy atoms.