Most realistic calculations of moderately correlated materials begin with a
ground-state density functional theory (DFT) calculation. While Kohn-Sham DFT
is used in about 40,000 scientific papers each year, the fundamental
underpinnings are not widely appreciated. In this chapter, we analyze the
inherent characteristics of DFT in their simplest form, using the asymmetric
Hubbard dimer as an illustrative model. We begin by working through the core
tenets of DFT, explaining what the exact ground-state density functional yields
and does not yield. Given the relative simplicity of the system, almost all
properties of the exact exchange-correlation functional are readily visualized
and plotted. Key concepts include the Kohn-Sham scheme, the behavior of the XC
potential as correlations become very strong, the derivative discontinuity and
the difference between KS gaps and true charge gaps, and how to extract optical
excitations using time-dependent DFT. By the end of this text and accompanying
exercises, the reader will improve their ability to both explain and visualize
the concepts of DFT, as well as better understand where others may go wrong.Comment: This chapter appears in the book "Autumn School on Correlated
Electrons: Simulating Correlations with Computers" (2021) prepared by
Forschungszentrum J\"ulich; see http://hdl.handle.net/2128/28665 for
published chapter. Updated versions correct citation