Reformulation of density functional theory for N-representable densities and the resolution of the v-representability problem

• We reformulate the density functional theory using parametric functional derivatives defined with the delta function.
• A mathematical procedure is constructed to enable the determination whether a density is v-representable.
• Generalized Hohenberg–Kohn theorems are derived for N-representable densities.

Density functional theory for the case of general, N-representable densities is reformulated in terms of density functional derivatives of expectation values of operators evaluated with wave functions leading to a density, making no reference to the concept of potential. The developments provide proof of existence of a mathematical procedure that determines whether a density is v-representable and in the case of an affirmative answer determines the potential (within an additive constant) as a derivative with respect to the density of a constrained search functional. It also establishes the existence of an energy functional of the density that, for v-representable densities, assumes its minimum value at the density describing the ground state of an interacting many-particle system. The theorems of Hohenberg and Kohn emerge as special cases of the formalism. Numerical results for one-dimensional non-interacting systems illustrate the formalism. Some direct formal and practical implications of the present reformulation of DFT are also discussed.