New Perspectives on the Fundamental Theorem of Density Functional Theory

The fundamental theorem of time-dependent/time-independent DFT due to Runge-Gross(RG)/Hohenberg-Kohn(HK) proves the bijectivity between the density $ \rho({\bf r} t)/ \rho({\bf r})$ and the Hamiltonian ${\hat H}(t)/{\hat H}$ to within a function $C(t)$/constant $C$, and wave function $\Psi(t) /\Psi$. (Implicit in the RG theorem is that the initial condition $\Psi(t_{0})$ is fixed.) As such in DFT the wave function is considered solely a functional of the density. Since the density is gauge invariant, the wave function as a functional of the density is also gauge invariant. However, it is well known that the Hamiltonian and wave function are gauge variant. There is, therefore, an inherent inconsistency in the RG/HK theorem. We resolve this inconsistency of the theorem via a unitary or equivalently a gauge transformation. As a consequence we generalize the theorem to external potentials that include the momentum operator and a curl-free vector potential operator. The RG/HK theorems each then constitute a special case of this generalization.