Linear interpolation method in ensemble Kohn-Sham and range-separated density-functional approximations for excited states

Gross-Oliveira-Kohn density-functional theory (GOK-DFT) for ensembles is, in principle, very attractive but has been hard to use in practice. A practical model based on GOK-DFT for the calculation of electronic excitation energies is discussed. The model relies on two modifications of GOK-DFT: use of range separation and use of the slope of the linearly interpolated ensemble energy, rather than orbital energies. The range-separated approach is appealing, as it enables the rigorous formulation of a multideterminant state-averaged DFT method. In the exact theory, the short-range density functional, which complements the long-range wave-function-based ensemble energy contribution, should vary with the ensemble weights even when the density is held fixed. This weight dependence ensures that the range-separated ensemble energy varies linearly with the ensemble weights. When the (weight-independent) ground-state short-range exchange-correlation functional is used in this context, curvature appears, thus leading to an approximate weight-dependent excitation energy. In order to obtain unambiguous approximate excitation energies, we propose to interpolate linearly the ensemble energy between equiensembles. It is shown that such a linear interpolation method (LIM) can be rationalized and that it effectively introduces weight dependence effects. As proof of principle, the LIM has been applied to He, Be, and ${\text{H}}_2$ in both equilibrium and stretched geometries as well as the stretched ${\text{HeH}}^{+}$ molecule. Very promising results have been obtained for both single (including charge transfer) and double excitations with spin-independent short-range local and semilocal functionals. Even at the Kohn-Sham ensemble DFT level, which is recovered when the range-separation parameter is set to 0, LIM performs better than standard time-dependent DFT.

Figure 1

Range-separated ensemble energy obtained for He at the WIDFA level while varying the ensemble weight $w$ for $\mu =0$ and $1.0a_0^{-1}$. Comparison is made with the linear interpolation method (LIM) for $\mu =0a_0^{-1}$ and full configuration interaction (FCI). The ensemble contains both $1{}^{1}S$ and $2{}^{1}S$ states. The srLDA functional was used.

Figure 2

Schematic of the linear interpolation method. Top: Ensemble energies. Bottom: Their first-order derivatives. See text for further details.

Figure 3

Effective DD $\left({\mathrm{\Delta }}_{\mathrm{eff}}^{\mu ,w}\right)$, auxiliary $\left(\mathrm{\Delta }{\tilde{\mathcal{E}}}^{\mu ,w}\right)$, and LIM $\left({\omega }_{\mathrm{LIM}}^{\mu }\right)$ excitation energies associated with the excitation $1{}^{1}S\to 2{}^{1}S$ in He. Results are shown for $\mu =0,0.4a_0^{-1}$, and $1.0a_0^{-1}$ with the srLDA (left-hand panels) and srPBE (right-hand panels) functionals while varying the ensemble weight $w$. Comparison is made with the FCI excitation energy ${\omega }_{\mathrm{FCI}}=0.7668E_h$. Open squares show non-self-consistent results.

Figure 4

Effective DD $\left({\mathrm{\Delta }}_{\mathrm{eff}}^{\mu ,w}\right)$, auxiliary $\left(\mathrm{\Delta }{\tilde{\mathcal{E}}}^{\mu ,w}\right)$, and LIM $\left({\omega }_{\mathrm{LIM}}^{\mu }\right)$ excitation energies associated with the excitations $1{}^{1}S\to 2{}^{1}S$ in Be (left-hand panels) and $1{}^1{\mathrm{\Sigma }}^{+}\to 2{}^1{\mathrm{\Sigma }}^{+}$ in the stretched ${\text{HeH}}^{+}$ molecule (right-hand panels). Results are shown for $\mu =0,0.4a_0^{-1}$, and $1.0a_0^{-1}$ with the srLDA functional while varying the ensemble weight $w$. Comparison is made with the FCI excitation energies $({\omega }_{\mathrm{FCI}}=0.2487E_h$ for Be and ${\omega }_{\mathrm{FCI}}=0.4024E_h$ for ${\text{HeH}}^{+})$. Open squares show non-self-consistent results.

Figure 5

Auxiliary excitation energies obtained with $\mu =0a_0^{-1}$ and the srLDA functional (which is equivalent to GOK-LDA) while varying the ensemble weight $w$ in the various systems considered in this work. See text for further details. Excitation energies are shifted by their values at $w=0$ for ease of comparison. Bottom: Zoom-in on the $0\le w\le 0.1$ region.

Figure 6

Effective DD $\left({\mathrm{\Delta }}_{\mathrm{eff}}^{\mu ,w}\right)$, auxiliary $\left(\mathrm{\Delta }{\tilde{\mathcal{E}}}^{\mu ,w}\right)$, and LIM $\left({\omega }_{\mathrm{LIM}}^{\mu }\right)$ excitation energies associated with the excitation $1{}^1{\mathrm{\Sigma }}_g^{+}\to 2{}^1{\mathrm{\Sigma }}_g^{+}$ in ${\text{H}}_2$ at equilibrium (left-hand panels) and in the stretched geometry (right-hand panels). Results are shown for $\mu =0,0.4a_0^{-1}$, and $1.0a_0^{-1}$ with the srLDA functional while varying the ensemble weight $w$. Comparison is made with the FCI excitation energies $({\omega }_{\mathrm{FCI}}=0.4828E_h$ at equilibrium and ${\omega }_{\mathrm{FCI}}=0.3198E_h$ in the stretched geometry). Open squares show non-self-consistent results.

Figure 7

LIM excitation energies obtained for the single excitations discussed in this work with srLDA and srPBE functionals while varying the range-separation parameter $\mu$. Comparison is made with the standard TD-DFT and FCI. For analysis purposes, auxiliary excitation energies obtained from the ground-state density $\left(w=0\right)$ are shown (curves with open circles).

Figure 8

LIM excitation energies calculated for the doubly excited $2{}^1{\mathrm{\Sigma }}_g^{+}$ state in the stretched ${\text{H}}_2$ molecule (top panel) and the $1{}^{1}D$ state in Be (bottom panel) while varying the range-separation parameter $\mu$ with srLDA and srPBE functionals. Comparison is made with the FCI. For ${\text{H}}_2$, auxiliary excitation energies obtained from the ground-state density $\left(w=0\right)$ are shown (curves with open circles) for comparison.