Exact and approximate Kohn-Sham potentials in ensemble density-functional theory

We construct exact Kohn-Sham potentials for the ensemble density-functional theory (EDFT) from the ground and excited states of helium. The exchange-correlation (XC) potential is compared with the quasi-local-density approximation and both single-determinant and symmetry-eigenstate ghost-corrected exact exchange approximations. Symmetry-eigenstate Hartree exchange recovers distinctive features of the exact XC potential and is used to calculate the correlation potential. Unlike the exact case, excitation energies calculated from these approximations depend on ensemble weight, and it is shown that only the symmetry-eigenstate method produces an ensemble derivative discontinuity. Differences in asymptotic and near-ground-state behavior of exact and approximate XC potentials are discussed in the context of producing accurate optical gaps.

Figure 1

Radial densities for the three lowest eigenstates of helium.

Figure 2

Radial densities and KS potentials for helium in singlet EDFT. The black solid lines are equiensemble properties. The red dashed line in the top panel shows an equiensemble density constructed from orbitals of the ground-state KS potential; the red dashed line in the bottom panel shows the exact ground-state KS potential.

Figure 3

XC potentials for the helium ground state, biensemble, and symmetry-projected singlet ensemble, produced by inverting ensemble densities constructed from the states shown in Fig. 1.

Figure 4

The exact XC potential for the helium singlet ensemble at various ensemble weights.

Figure 5

The exact potential jump $\Delta v_{\mathrm{XC}}$ as $w\to 0$. The location of the step depends logarithmically on $w$. As $w\to 0$, the drop off to the $w=0$ value moves infinitely far from the origin.

Figure 6

The exact potential jump $\Delta v_{\mathrm{XC}}$, showing the shoulder in the XC potential developing from the small-$w$ step as $w$ increases. Since $w$ is no longer near zero, the asymptotic formula for the position of the drop off no longer holds.

Figure 7

Equation (10) applied to the exact helium singlet ensemble, demonstrating the exact cancellation of all $w$ dependence in KS gaps (red bottom line for small $w$) and corrections to the KS gap (green top line for small $w$), leading to no $w$ dependence in the calculated optical gap (blue middle line). Gaps are shifted by the true optical gap $\omega$ for ease of comparison.

Figure 8

The exact and approximated $v_{\mathrm{XC}}\left(\mathbf{r}\right)$ for the helium singlet equiensemble. The approximated $v_{\mathrm{XC}}$'s are evaluated using the exact ensemble density as input.

Figure 9

The exact $v_{\mathrm{C}}\left(r\right)$ for the helium singlet equiensemble shows two upward bumps and does not depend on the definition of the Hartree potential used. These are obtained by subtracting the SEHX $v_{\mathrm{X},w}\left(r\right)$ of the exact ensemble density from the exact $v_{\mathrm{XC},w}\left(r\right)$.

Figure 10

Self-consistent $\Delta v_{\mathrm{XC}}\left(r\right)$ of various approximations at $w=0.0001$. Only SEHX (long-dashed red line) replicates a shift similar to that of the exact curve (solid black line).

Figure 11

Equation (10) applied to self-consistent quasi-LDA results. The correction to the quasi-LDA KS gap (dashed green line) is not zero, but it is too small to be noticed on this scale. This correction is inadequate to cancel the $w$ dependence in the qLDA KS gap (dot-dashed red line), resulting in inaccurate, $w$-dependent calculated optical gaps (dotted blue line). The gaps have been shifted by the optical gap $\omega$ for easier comparison, and the exact results of Fig. 7 are also shown for context.

Figure 12

Equation (10) applied to self-consistent SD results. The spin-up SD KS gap (dot-dashed red line) is insufficiently corrected by the SD corrections to the KS gap (dashed green line), yielding calculated optical gaps that are too small (dotted blue line). Although the $w$ dependence is less severe than for qLDA, it is still non-negligible. The gaps have been shifted by the optical gap $\omega$ for easier comparison, and the exact results of Fig. 7 are also shown for context.

Figure 13

Equation (10) applied to approximate self-consistent SEHX results. SEHX produces far less variation in calculated excitation energies with $w$ (dotted blue line), which appears to be the result of its ensemble derivative discontinuity. This produces approximate KS gaps (dot-dashed red line) and KS gap corrections (dashed green line) that most closely resemble the exact curves in overall shape. The exact results (as in Fig. 7) are also shown for context. The gaps have been shifted by the optical gap $\omega$ for easier comparison, and the exact results of Fig. 7 are also shown for context.