Exact thermal density functional theory for a model system: Correlation components and accuracy of the zero-temperature exchange-correlation approximation

Thermal density functional theory calculations often use the Mermin-Kohn-Sham scheme, but employ ground-state approximations to the exchange-correlation (XC) free energy. In the simplest solvable nontrivial model, an asymmetric Hubbard dimer, we calculate the exact many-body energies and the exact Mermin-Kohn-Sham functionals for this system and extract the exact XC free energy. For moderate temperatures and weak correlation, we find this approximation to be excellent. We extract various exact free-energy correlation components and the exact adiabatic connection formula.

Figure 1

Free energy for different values of $\mathrm{\Delta }v$. Solid lines are exact; dashed lines are the zero-temperature XC approximation (ZTA), evaluated on the self-consistent thermal density.

Figure 2

Exact entropy (solid) and self-consistent Kohn-Sham entropy (dashed) for different values of $\mathrm{\Delta }v$. All curves approach $4log2$.

Figure 3

Densities as a function of temperature for the system of Fig. 1. Solid lines are exact; dashed lines are self-consistent KS using the ZTA.

Figure 4

Panel 1: Correlation free-energy functional for various temperatures. Panel 2: Sum of kinetic- and potential-energy functional for various temperatures. Panel 3: Entropic correlation functional for various temperatures.

Figure 5

Adiabatic connection integrand for the symmetric dimer at several different temperatures.

Figure 6

Same as Fig. 5, except $\mathrm{\Delta }n=1$.

Figure 7

Correlation free energy for the symmetric case with increasing values of $U$ ranging from weak to strong correlation.

Figure 8

Error in ZTA densities of Fig. 3, density from self-consistent MKS subtracted from exact density.