  In recent years, Adiabatic Connection Interpolations developed within Density
Functional Theory (DFT) have been found to provide satisfactory performances in
the calculation of interaction energies when used with Hartree-Fock (HF)
ingredients. The physical and mathematical reasons for such unanticipated
performance have been clarified, to some extent, by studying the
strong-interaction limit of the M\o ller-Plesset (MP) adiabatic connection. In
this work, we calculate both the MP and the DFT adiabatic connection (AC)
integrand for the asymmetric Hubbard dimer, which allows for a systematic
investigation at different correlation regimes by varying two simple parameters
in the Hamiltonian: the external potential, $\Delta v$, and the interaction
strength, $U$. Noticeably, we find that, while the DFT AC integrand appears to
be convex in the full parameter space, the MP integrand may change curvature
twice. Furthermore, we discuss different aspects of the second-order expansion
of the correlation energy in each adiabatic connection and we demonstrate that
the derivative of the $\lambda$-dependent density in the MP adiabatic
connection at $\lambda=0$ (i.e., at the HF density) is zero. Concerning the
strong-interaction limit of both adiabatic connections, we show that while, for
a given density, the asymptotic value of the MP adiabatic connection,
$W_\infty^\text{HF}$, is lower (or equal) than its DFT analogue,
$W_\infty^\text{KS}$, this is not always the case for a given external
potential.
