Exact ensemble density functional theory for excited states in a model system: Investigating the weight dependence of the correlation energy

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.

Figure 1

Variation of the ground- $n^0$ and first-excited-state $n^1$ densities with the local potential $\mathrm{\Delta }v$ in the Hubbard dimer for various $U$ values.

Figure 2

Derivative discontinuity obtained for the Hubbard dimer with different $\mathrm{\Delta }{v}_{\mathrm{ext}}$ and $U$ values. Results obtained by numerical differentiation are shown for $\mathrm{\Delta }{v}_{\mathrm{ext}}=1$ and $U=1$ (see the black dots on the right-hand top panel). See text for further details.

Figure 3

Derivative discontinuity plotted as a function of $\mathrm{\Delta }{v}_{\mathrm{ext}}$ for various $w$ and $U$ values.

Figure 4

Derivative discontinuity plotted as a function of $U$ for various $w$ and $\mathrm{\Delta }{v}_{\mathrm{ext}}$ values.

Figure 5

Solution of Eq. (88) plotted with respect to the potential $\mathrm{\Delta }{v}_{\mathrm{ext}}$ for different $U$ values. Physical values should be lower than 1/2 (i.e., below the dashed line).

Figure 6

Variation of the exchange-correlation GACE integrand with the ensemble weight $\xi$ for various $U$ values and densities. Comparison is made with the exact exchange-only contribution ${\mathrm{\Delta }}_x^{\xi }\left(n\right)$ (dashed lines).

Figure 7

Exact ensemble density-functional exchange-correlation energies for various $U$ and $w$ values. The mixed ensemble exchange/ground-state correlation energy $E_x^w\left(n\right)+E_c^{w=0}\left(n\right)$ is shown (dashed lines) for analysis purposes.

Figure 8

Exact ensemble exchange-correlation potential for various $U$ and $w$ values. The exact ensemble exchange potential (dashed lines) is shown for analysis purposes.

Figure 9

Comparing exact with approximate ensemble energies (left panels) and first-order derivatives (right panels) for various $\mathrm{\Delta }{v}_{\mathrm{ext}}$ and $U$ values. See text for further details.