Essentially, what we have shown is that it is possible to map the electronic densities of the separated molecules to the interaction energy since using the HK theorem it follows that the maps ρ A → E A and ρ B → E B exist since ρ A and ρ B can be mapped to their respective external potentials they can be mapped to the cartesian coordinates and atomic numbers, therefore, the pair can also be mapped to the Hamiltonian operator of the interacting system: