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

We show that the molecular relativistic Thomas-Fermi-Weizsäcker 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 2.56Z. This gives lower and upper bounds on the maximal ionization of heavy atoms.