Using the Hohenberg-Kohn theorem, one can prove that there exists a map between the non-interacting electron densities of two molecules to the energy of the interacting system. With this knowledge, in our prior model we aimed to, empirically, find an expression for this map in the form of a sum of fictitious Coulomb-like integral expressions. However, this argument fails for justifying the existence of a map between the non-interacting electron densities to the interacting electron density due to ambiguity issues should the interacting system allow degenerate ground states. Such kind of map is useful to serve as a theoretical foundation of a polarizable model, that is, a model whose, fictitious or not, interacting density depends on the environment the molecules are subjected to.