  The asymmetric Hubbard dimer is a model that allows for explicit expressions
of the Hartree-Fock (HF) and Kohn-Sham (KS) states as analytical functions of
the external potential, $\Delta v$, and of the interaction strength, $U$. We
use this unique circumstance to establish a rigorous comparison between the
individual contributions to the correlation energies stemming from the two
theories in the $\{U,\Delta v\}$ parameter space. Within this analysis of the
Hubbard dimer, we observe a change in the sign of the HF kinetic correlation
energy, compare the indirect repulsion energies, and derive an expression for
the 'traditional' correlation energy, i.e. the one that corrects the HF
estimate, in a pure site-occupation function theory spirit [Eq. (43)]. Next, we
test the performances of the Liu-Burke and the Seidl-Perdew-Levy functionals,
which model the correlation energy based on its weak- and strong-interaction
limit expansions and can be used for both the traditional and the KS
correlation energies. Our results show that, in the Hubbard dimer setting, they
typically work better for the HF reference, despite having been originally
devised for KS. These conclusions are somewhat in line with prior assessments
of these functionals on various chemical data sets. However, the Hubbard dimer
model allows us to show the extent of the error that may occur in using the
strong-interaction ingredient for the KS reference in place of the one for the
HF reference, as has been carried out in most of the prior assessments.

Chemical Physics