Exact conditions for ensemble density functional theory

Ensemble density functional theory (EDFT) is a promising alternative to time-dependent density functional theory for computing electronic excitation energies. Using coordinate scaling, we prove several fundamental exact conditions in EDFT and illustrate them on the exact singlet bi-ensemble of the Hubbard dimer. Several approximations violate these conditions, and some ground-state conditions from quantum chemistry do not generalize to EDFT. The strong correlation limit is derived for the dimer, revealing weight-dependent derivative discontinuities in EDFT.

Figure 1

The Hubbard dimer singlet bi-ensemble correlation energies (negative values) and kinetic contribution (positive values) for $U=4$ (light blue) and $U=5$ (black) as a function of site occupation and different weights. Red curves deduced from $U=4$ constrain the $U=5$ curve via Eq. (22).

Figure 2

Absolute value of the density of the Hubbard dimer bi-ensemble, plotted for various weight values as a function of $\mathrm{\Delta }v$. Here we set $U=0$ (left), $U=1$ (center), and $U=5$ (right).

Figure 3

$F_w/U$ of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }v$ for various $w$ values. Here we set $U=1$ (top) and $U=5$ (bottom).

Figure 4

$F_w/U$ of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }{n}_w$ for various $w$ values. Here we set $U=1$ (top) and $U=5$ (bottom).

Figure 5

$E_{\mathrm{HXC},w}/U$ of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }{n}_w$ for various $w$ values. Here we set $U=1$ (top) and $U=5$ (bottom).

Figure 6

Total energy of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }v$ for various $w$ values. Here we set $U=1$ (top) and $U=5$ (bottom).

Figure 7

Total energy of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }{n}_w$ for various $w$ values. Here we set $U=1$ (top) and $U=5$ (bottom).

Figure 8

Variation of the potential (blue), kinetic (red), and total (black) correlation energies in the Hubbard dimer bi-ensemble, plotted as functions of site-occupation of the first site for various weights. We set $U=1$ on the left and $U=5$ on the right.

Figure 9

Correlation inequalities [Eq. (22)] for the total (top), kinetic (middle), and potential (bottom) correlation energies, depicted by varying $\lambda$ in the Hubbard dimer bi-ensemble with $U=1$.

Figure 10

Correlation inequalities [Eq. (22)] for the total (top), kinetic (middle), and potential (bottom) correlation energies, depicted by varying $\lambda$ in the Hubbard dimer bi-ensemble with $U=5$.

Figure 11

Ensemble adiabatic connection for $U=5$ and various $\mathrm{\Delta }{n}_w$; circles represent the weight-dependent HX energy, which the HXC expression approaches as $\lambda \to 0$.

Figure 12

The second derivative of the ensemble adiabatic connection with respect to $\lambda$ for $U=5$ and $\mathrm{\Delta }{n}_w=0.9$.

Figure 13

The second derivative of the Hubbard dimer bi-ensemble correlation energy with respect to $U$ for all values of the reduced variable $\tilde{u}=U/\sqrt{1+U^2}$. The concavity condition is satisfied exactly, but is violated by the $\delta$-PT2 approximation in certain regimes (red denotes positive). $U$-PT2 automatically satisfies the concavity condition by construction.

Figure 14

Exact correlation energy (black), leading-order expansion in large $U$ (red), and the expansion in the symmetric limit (blue) for the correlation energy are all plotted as a function of the exact density. Small arrows indicate the region between $2w$ and $-2w$ where the symmetric expansion matches the exact.

Figure 15

Smooth approximation for the density Eq. (42) (light red) and the exact density (black) Eq. (32) plotted against $\mathrm{\Delta }v$.

Figure 16

The exact (black) correlation energy as a function of the exact density and our higher-order expansion in large $U$ (light purple) for the correlation energy Eq. (44), plotted as a function of the smooth approximation for the density Eq. (42).

Figure 17

$T_{c,w}$, Eq. (49), $U_{c,w}$, Eq. (50), $T_w$, Eq. (51), and $V_{\mathrm{ee},w}$, Eq. (52) for various values of $w$ and $U=100$. The exact values in the strongly correlated limit are represented by the black curves, which are exact as $U\to \infty$.

Figure 18

Absolute value of the density of the Hubbard dimer bi-ensemble, plotted for various weight values as a function of $\mathrm{\Delta }v$. Here we set $U=0$ (left), $U=1$ (center), and $U=5$ (right). Dashed curves represent the ensemble HF approximation, and the solid curves are exact.

Figure 19

Total energy of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }v/U$ for various $w$ values. Dashed is HF and solid is exact.

Figure 20

Total correlation energy of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }v$ for various $w$ values. Dashed is HF and solid is exact.

Figure 21

Kinetic correlation energy of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }v$ for various $w$ values. Dashed is HF and solid is exact.

Figure 22

Potential correlation energy of the Hubbard dimer bi-ensemble plotted as a function of $\mathrm{\Delta }v$ for various $w$ values. Dashed is HF and solid is exact.