Ghost-interaction correction in ensemble density-functional theory for excited states with and without range separation

Ensemble density-functional theory (eDFT) suffers from the so-called “ghost-interaction” error when approximate exchange-correlation functionals are used. In this work, we present a rigorous ghost-interaction correction (GIC) scheme in the context of range-separated eDFT. The method relies on an exact decomposition of the ensemble short-range exchange-correlation energy into a multideterminantal exact exchange term, which involves the long-range interacting ensemble density matrix, instead of the Kohn-Sham (KS) one, and a complementary density-functional correlation energy. A generalized adiabatic connection formula is derived for the latter. In order to perform practical calculations, the complementary correlation functional is simply modeled by its ground-state local density approximation (LDA), while long-range interacting ground- and excited-state wave functions are obtained self-consistently by combining a long-range configuration-interaction calculation with a short-range LDA potential. We show that the GIC reduces the curvature of approximate range-separated ensemble energies drastically while providing considerably more accurate excitation energies, even for charge-transfer and double excitations. Interestingly, the method performs well also in the context of standard KS-eDFT, which is recovered when the range-separation parameter is set to 0.

Figure 1

Relative error with respect to the FCI obtained for the auxiliary (solid lines) and FOC (dash-dotted lines) excitation energies with $\mu =0$ (top) and $\mu =0.4a_0^{-1}$ (bottom). The stretched ${\mathrm{H}}_2$ molecule is represented as $\mathrm{H}\cdots \mathrm{H}$. Relative errors are calculated as $\frac{\tilde{\omega }-{\omega }_{\mathrm{FCI}}}{{\omega }_{\mathrm{FCI}}}$, where $\tilde{\omega }$ is the approximate excitation energy.

Figure 2

Weight dependence of the WIDFA and GIC ensemble energies in ${\mathrm{HeH}}^{+}$ for $\mu =0$ and $\mu =0.4a_0^{-1}$. The FCI and LIM (dashed lines) are also shown. Energies are shifted by their values at $w=0$ for ease of comparison.

Figure 3

Relative error with respect to the FCI obtained for the LIM (dash-dotted lines) and GIC-LIM (solid lines) excitation energies when varying $\mu$. Inset: Zoom-in on the range $0\le \mu \le 2.0a_0^{-1}$. Excitations in the stretched ${\mathrm{HeH}}^{+}$ ($1^1{\mathrm{\Sigma }}^{+}\to 2^1{\mathrm{\Sigma }}^{+}$) and $\mathrm{H}...\mathrm{H}$ ($1^1{\mathrm{\Sigma }}_g^{+}\to 2^1{\mathrm{\Sigma }}_g^{+}$) molecules correspond to a charge transfer and a double excitation, respectively.

Figure 4

Relative error with respect to the FCI obtained for the GIC-LIM (solid lines) and FOC $[w=0]$ (dash-dotted lines) excitation energies in He and the stretched ${\mathrm{HeH}}^{+}$ molecule. Inset: The convergence towards FCI when $\mu \to +\infty$.

Figure 5

Relative error with respect to the FCI obtained for the GIC-LIM (solid lines) and FOC $[w=0.5]$ (dash-dotted lines) excitation energies in He and the stretched ${\mathrm{HeH}}^{+}$ molecule. Inset: The convergence towards FCI when $\mu \to +\infty$.

Figure 6

LIM and GIC-LIM $1^{1}S\to 2^{1}S$ excitation energies obtained in He with and without extrapolation corrections when varying $\mu$. Comparison is made with the FCI. See text for further details.

Figure 7

Convergence towards the FCI (solid horizontal lines) of the LIM (dash-dotted lines with symbols), ELIM (dotted lines with symbols), GIC-LIM (dashed lines with symbols) and EGIC-LIM (solid lines with symbols) excitation energies obtained for the singly excited $1^{1}S\to 2^{1}S$ (bottom, red curves) and doubly-excited $1^{1}S\to 1^{1}D$ (top, blue curves) transitions in Be with increasing $\mu$.