must also exist. This argument can be used on a per atom basis, which serves to justify separating the entire interaction energy on a per atom pair sum of contributions. Mapping the non-interacting ground states eigendensities of each atom in our system to the ground states eigendensity of the whole is advantageous since it allows us to exploit the spherical symmetry of their ground states eigendensities. However, just like when Hohenberg and Kohn's original proof is used instead, this generalization only holds for the ground states eigendensities of both interacting and non-interacting system.