Introductory lecture: when the density of the noninteracting reference system is not the density of the physical system in density functional theory.

A major challenge in density functional theory (DFT) is the development of density functional approximations (DFAs) to overcome errors in existing DFAs, leading to more complex functionals. For such functionals, we consider roles of the noninteracting reference systems. The electron density of the Kohn-Sham (KS) reference with a local potential has been traditionally defined as being equal to the electron density of the physical system. This key idea has been applied in two ways: the inverse calculation of such a local KS potential for the reference from a given density and the direct calculation of density and energy based on given DFAs. By construction, the inverse calculation can yield a KS reference with the density equal to the input density of the physical system. In application of DFT, however, it is the direct calculation of density and energy from a DFA that plays a central role. For direct calculations, we find that the self-consistent density of the KS reference defined by the optimized effective potential (OEP), is not the density of the physical system, when the DFA is dependent on the external potential. This inequality holds also for the density of generalized KS (GKS) or generalized OEP reference, which allows a nonlocal potential, when the DFA is dependent on the external potential. Instead, the density of the physical system, consistent with a given DFA, is given by the linear response of the total energy with respect to the variation of the external potential. This is a paradigm shift in DFT on the use of noninteracting references: the noninteracting KS or GKS references represent the explicit computational variables for energy minimization, but not the density of the physical system for external potential-dependent DFAs. We develop the expressions for the electron density so defined through the linear response for general DFAs, demonstrate the results for orbital functionals and for many-body perturbation theory within the second-order and the random-phase approximation, and explore the connections to developments in DFT.