Potential errors in density functional theory

Kohn-Sham density functional theory (KS-DFT) is routinely used to compute ground-state properties, but despite its success, better approximate energy functionals are still sought out. Many modern DFT approaches, such as van der Waal's corrections and DFT+U, amongst others, introduce some form of explicit dependence on atomic information. In principle, the underlying KS equations only work if the exchange-correlation functional is expressed purely as a density functional. These mixed approaches directly alter the density-potential mapping invoked in the standard ground-state theory, and the consequences have not been studied. While we expect these new errors to be generally small, can this be stated definitively? Moreover, are there scenarios where mixed functional approaches result in large errors? Using the two-site Hubbard model, we produce exact solutions for these mixed functional theories and invoke plausible approximations to study the emergence of errors and address these questions.