Density functional theory with exact XC potentials: Lessons from DMRG studies and exactly solvable models

Density functional theory in the Kohn-Sham formulation is the most successful method in chemistry and physics when it comes to predicting crystal and molecular structures. Very frequently it is also used with benefit for estimating the corresponding electronic dispersion relations. Therefore, it is not surprising that computations have been attempted of observables that, strictly speaking, are beyond those protected by the Hohenberg-Kohn theorems. The most infamous one of them is the conductance of single molecules. In this article we offer a brief overview of recent developments that justify such transport studies in situations where transport is dominated by a single orbital. In such situations the Friedel sum rule establishes a relation between conductance and ground-state density that is free of microscopic parameters, such as the interaction strength, and therefore can provide a rigorous link between Kohn-Sham transport and the physical observables. What happens in more general situations is also discussed.