In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners.

Ensemble density functional theory (eDFT) is an exact time-independent alternative to time-dependent DFT (TD-DFT) for the calculation of excitation energies. Despite its formal simplicity and advantages in contrast to TD-DFT (multiple excitations, for example, can be easily taken into account in an ensemble), eDFT is not standard, which is essentially due to the lack of reliable approximate exchange-correlation ($xc$) functionals for ensembles. Following Smith et al. [ Phys. Rev. B 93, 245131 (2016)], we propose in this work to construct an exact eDFT for the nontrivial asymmetric Hubbard dimer, thus providing more insight into the weight dependence of the ensemble $xc$ energy in various correlation regimes. For that purpose, an exact analytical expression for the weight-dependent ensemble exchange energy has been derived. The complementary exact ensemble correlation energy has been computed by means of Legendre-Fenchel transforms. Interesting features like discontinuities in the ensemble $xc$ potential in the strongly correlated limit have been rationalized by means of a generalized adiabatic connection formalism. Finally, functional-driven errors induced by ground-state density-functional approximations have been studied. In the strictly symmetric case or in the weakly correlated regime, combining ensemble exact exchange with ground-state correlation functionals gives better ensemble energies than when calculated with the ground-state exchange-correlation functional. However, when approaching the asymmetric equiensemble in the strongly correlated regime, the former approximation leads to highly curved ensemble energies with negative slope which is unphysical. Using both ground-state exchange and correlation functionals gives much better results in that case. In fact, exact ensemble energies are almost recovered in some density domains. The analysis of density-driven errors is left for future work.

Gross–Oliveira–Kohn density-functional theory (GOK-DFT) for ensembles is the DFT analog of state-averaged wavefunction-based (SA-WF) methods. In GOK-DFT, the SA (so-called ensemble) exchange-correlation (xc) energy is described by a single functional of the density which, for a fixed density, depends on the weights assigned to each state in the ensemble. We show that if a many-weight-dependent xc functional is employed, then it becomes possible to extract, in principle exactly, all individual energy levels from a single GOK-DFT calculation, exactly like in a SA-WF calculation. More precisely, starting from the Kohn–Sham energies, a global Levy–Zahariev-type shift as well as a state-specific (ensemble-based) xc derivative correction must be applied in order to reach the energy level of interest. We illustrate with the asymmetric Hubbard dimer the importance and substantial weight dependence of both corrections. A comparison with more standard extraction procedures, which rely on a sequence of ensemble calculations, is made at the ensemble exact exchange level of approximation.

Gould and Pittalis [ Phys. Rev. Lett. 123, 016401 (2019) recently revealed a density-driven (DD) correlation energy that is specific to many-electron ensembles and must be accounted for by approximations. We derive in this Letter a general and simpler expression in terms of the ensemble weights, the ensemble Kohn-Sham (KS) orbitals, and their linear response to variations in the ensemble weights. As no additional state-driven KS systems are needed, its evaluation is greatly simplified. We confirm the importance of DD effects and introduce a direct and promising route to approximations.

The exact formulation of multi-configuration density-functional theory is discussed in this work. As an alternative to range-separated methods, where electron correlation effects are split in the coordinate space, the combination of configuration interaction methods with orbital occupation functionals is explored at the formal level through the separation of correlation effects in the orbital space. When applied to model Hamiltonians, this approach leads to an exact site-occupation embedding theory (SOET). An adiabatic connection expression is derived for the complementary bath functional and a comparison with density matrix embedding theory is made. Illustrative results are given for the simple two-site Hubbard model. SOET is then applied to a quantum chemical Hamiltonian, thus leading to an exact complete active space site-occupation functional theory (CASSOFT) where active electrons are correlated explicitly within the CAS and the remaining contributions to the correlation energy are described with an orbital occupation functional. The computational implementation of SOET and CASSOFT as well as the development of approximate functionals are left for future work.