M{\o}ller-Plesset and density-fixed adiabatic connections for a model diatomic system at different correlation regimes

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

Excitations and benchmark ensemble density functional theory for two electrons

A new method for extracting ensemble Kohn-Sham potentials from accurate excited state densities is applied to a variety of two electron systems, exploring the behavior of exact ensemble density functional theory. The issue of separating the Hartree energy and the choice of degenerate eigenstates is explored. A new approximation, spin eigenstate Hartree-exchange (SEHX), is derived. Exact conditions that are proven include the signs of the correlation energy components, the virial theorem for both exchange and correlation, and the asymptotic behavior of the potential for small weights of the excited states. Many energy components are given as a function of the weights for two electrons in a one-dimensional flat box, in a box with a large barrier to create charge transfer excitations, in a three-dimensional harmonic well (Hooke's atom), and for the He atom singlet-triplet ensemble, singlet-triplet-singlet ensemble, and triplet bi-ensemble.Comment: 15 pages, supplemental material pd