  This review explains the relationship between density functional theory and
strongly correlated models using the simplest possible example, the two-site
Hubbard model. The relationship to traditional quantum chemistry is included.
Even in this elementary example, where the exact ground-state energy and site
occupations can be found analytically, there is much to be explained in terms
of the underlying logic and aims of Density Functional Theory. Although the
usual solution is analytic, the density functional is given only implicitly. We
overcome this difficulty using the Levy-Lieb construction to create a
parametrization of the exact function with negligible errors. The symmetric
case is most commonly studied, but we find a rich variation in behavior by
including asymmetry, as strong correlation physics vies with charge-transfer
effects. We explore the behavior of the gap and the many-body Green's function,
demonstrating the `failure' of the Kohn-Sham method to reproduce the
fundamental gap. We perform benchmark calculations of the occupation and
components of the KS potentials, the correlation kinetic energies, and the
adiabatic connection. We test several approximate functionals (restricted and
unrestricted Hartree-Fock and Bethe Ansatz Local Density Approximation) to show
their successes and limitations. We also discuss and illustrate the concept of
the derivative discontinuity. Useful appendices include analytic expressions
for Density Functional energy components, several limits of the exact
functional (weak- and strong-coupling, symmetric and asymmetric), the Kohn-Sham
hopping energy functional for 3 sites, various adiabatic connection results,
proofs of exact conditions for this model, and the origin of the Hubbard model
from a minimal basis model for stretched H$_2$.
