We introduce an improved version of our electron densitybased molecular force field, designed to mimic electrostatic polarization effects. This is achieved through the minimization of an energy functional derived from the non-interacting ground states eigendensities of all atoms in the system. Notably, this energy functional shares an identical form with the proposed interaction energy, utilizing distinct optimized parameters tailored specifically to predict the Mulliken charges of a system. To validate the existence of such a functional, we provide a proof using a modified version of the Hohenberg-Kohn theorem, establishing a theoretical foundation for our polarizable molecular force field.