Exact conditions on the temperature dependence of density functionals

Universal exact conditions guided the construction of most ground-state density functional approximations in use today. We derive the relation between the entropy and Mermin free energy density functionals for thermal density functional theory. Both the entropy and sum of kinetic and electron-electron repulsion functionals are shown to be monotonically increasing with temperature, while the Mermin functional is concave downwards. Analogous relations are found for both exchange and correlation. The importance of these conditions is illustrated in two extremes: the Hubbard dimer and the uniform gas.

Figure 1

Energy components for the Hubbard dimer in units of $2t$, where $U=2t$ and $\mathrm{\Delta }n=0$: $F^{\tau },F_{\mathrm{I}}^{\tau },S^{\tau }$, both interacting (solid) and noninteracting (dashed).

Figure 2

Temperature dependence of the Mermin functional for spin-unpolarized uniform gas for several values of the Wigner-Seitz radius $r_{\mathrm{S}}$, using the XC parametrization of Ref. [16], where ${\epsilon }_{\mathrm{F}}$ is the Fermi energy.

Figure 3

Correlation entropy in the Hubbard dimer for several values of $\mathrm{\Delta }n$ as a function of temperature, in units of $2t$, where $U=2t$.