Electronic Excited States in Extreme Limits via Ensemble Density Functionals

Density functional theory (DFT) has greatly expanded our ability to affordably compute and understand electronic ground states, by replacing intractable ab initio calculations by models based on paradigmatic physics from high- and low-density limits. But, a comparable treatment of excited states lags behind. Here, we solve this outstanding problem by employing a generalization of density functional theory to ensemble states (EDFT). We thus address important paradigmatic cases of all electronic systems in strongly (low-density) and weakly (high-density) correlated regimes. We show that the high-density limit connects to recent, exactly solvable EDFT results. The low-density limit reveals an unnoticed and most unexpected result—density functionals for strictly correlated ground states can be reused directly for excited states. Nontrivial dependence on excitation structure only shows up at third leading order. Overall, our results provide foundations for effective models of excited states that interpolate between exact low- and high-density limits, which we illustrate on the cases of singlet-singlet excitations in ${\mathrm{H}}_2$ and a ring of quantum wells.

Figure 1

Ratio of quantum and classical interaction energies for two electrons in a Harmonic well. Six triplet (top) and singlet (bottom) energies are shown. Scaled densities, $4\pi r^{2}n(r)$, of the states are also shown for the case $\lambda =50$.

Figure 2

Excitation energies, for dissociating ${\mathrm{H}}_2$ (top); and seven random runs and the mean of 25 runs for a ring of wells with disorder (bottom). Ensemblized EXX [dotted line (green)], GL2 [dashed line (red)], ISI [dash-dotted line (blue)], and exact [solid lines (maroon)] energies are shown. GL2 errors in the ring are so large they cannot be shown in the lower panel.