Importance of finite-temperature exchange correlation for warm dense matter calculations

The effects of an explicit temperature dependence in the exchange correlation (XC) free-energy functional upon calculated properties of matter in the warm dense regime are investigated. The comparison is between the Karasiev-Sjostrom-Dufty-Trickey (KSDT) finite-temperature local-density approximation (TLDA) XC functional [Karasiev et al., Phys. Rev. Lett. 112, 076403 (2014)] parametrized from restricted path-integral Monte Carlo data on the homogeneous electron gas (HEG) and the conventional Monte Carlo parametrization ground-state LDA XC [Perdew-Zunger (PZ)] functional evaluated with $T$-dependent densities. Both Kohn-Sham (KS) and orbital-free density-functional theories are used, depending upon computational resource demands. Compared to the PZ functional, the KSDT functional generally lowers the dc electrical conductivity of low-density Al, yielding improved agreement with experiment. The greatest lowering is about 15% for $T=15$ kK. Correspondingly, the KS band structure of low-density fcc Al from the KSDT functional exhibits a clear increase in interband separation above the Fermi level compared to the PZ bands. In some density-temperature regimes, the deuterium equations of state obtained from the two XC functionals exhibit pressure differences as large as 4% and a 6% range of differences. However, the hydrogen principal Hugoniot is insensitive to the explicit XC $T$ dependence because of cancellation between the energy and pressure-volume work difference terms in the Rankine-Hugoniot equation. Finally, the temperature at which the HEG becomes unstable is $T\ge 7200$ K for the $T$-dependent XC, a result that the ground-state XC underestimates by about 1000 K.

Figure 1

Map in the $(r_{\mathrm{s}},T)$ plane that shows the relative importance of the explicit $T$ dependence in the exchange correlation free-energy functional for the HEG measured as ${log}_{10}\left\{\right|f_{\mathrm{xc}}(r_{\mathrm{s}},T)-e_{\mathrm{xc}}\left(r_{\mathrm{s}}\right)|/[|f_{\mathrm{s}}(r_{\mathrm{s}},T)|+|e_{\mathrm{xc}}\left(r_{\mathrm{s}}\right)\left|\right]\}$.

Figure 2

Total free-energy, internal energy, internal XC energy, and entropic component (per particle) magnitudes for the spin-unpolarized HEG for $r_{\mathrm{s}}=20$ bohr calculated with the KSDT XC free-energy parametrization [14].

Figure 3

Aluminum dc conductivity as a function of density from calculations with $T$-dependent KSDT (dot-dashed line) and ground-state PZ (dashed line) XC functionals for five isotherms. From bottom to top $T=5$ kK (circles), 10 kK (triangles up), 15 kK (diamonds), 20 kK (squares), and 30 kK (triangles down). Experimental data [51] correspond to 10 kK (triangles up) and 30 kK (triangles down).

Figure 4

Relative error in dc conductivity for Al as a function of density for five different temperatures.

Figure 5

Comparison of fcc Al KS band structures for $\rho =0.2\mathrm{g}/{\mathrm{cm}}^3$ from KSDT $T$-dependent XC and PZ ground-state XC for $T=5$, 10, and $20\mathrm{kK}$ (bottom to top). The Fermi level ${\epsilon }_F$ is set to zero. The right-hand panels display the density of states $D\left(\epsilon \right)$ and Fermi-Dirac occupation $f\left(\epsilon \right)$.

Figure 6

Deuterium electronic pressure as a function of $T$ from KS and OFDFT calculations with the finite-$T$ KSDT and ground-state PZ XC functionals. The inset shows the relative difference between the total pressure from the calculations with the PZ and KSDT XC; see Eq. (8). The system density ${\rho }_{\mathrm{D}}=0.20\mathrm{g}/{\mathrm{cm}}^3$ ($r_{\mathrm{s}}=3$ bohr). The PIMC results are shown for comparison.

Figure 7

Same as in Fig. 6 for deuterium, with ${\rho }_{\mathrm{D}}=0.506\mathrm{g}/{\mathrm{cm}}^3$ ($r_{\mathrm{s}}=2.2$ bohr).

Figure 8

Same as in Fig. 6 for deuterium, with ${\rho }_{\mathrm{D}}=1.964\mathrm{g}/{\mathrm{cm}}^3$ ($r_{\mathrm{s}}=1.4$ bohr).

Figure 9

Same as in Fig. 6 for deuterium, with ${\rho }_{\mathrm{D}}=4.04819\mathrm{g}/{\mathrm{cm}}^3$ ($r_{\mathrm{s}}=1.10$ bohr).

Figure 10

Same as in Fig. 6 for deuterium, with ${\rho }_{\mathrm{D}}=10.0\mathrm{g}/{\mathrm{cm}}^3$ ($r_{\mathrm{s}}=0.81373$ bohr).

Figure 11

Hydrogen principal Hugoniot. The initial density is ${\rho }_0=0.0855\mathrm{g}/{\mathrm{cm}}^3$.

Figure 12

Pressure (top) and specific internal energy difference (bottom) along the hydrogen Hugoniot as functions of $T$.

Figure 13

The HEG total free energy per electron as a function of $r_{\mathrm{s}}$ for selected temperatures calculated with the KSDT XC functional.

Figure 14

Shown on top is the value of equilibrium $r_{\mathrm{s}}$ corresponding to the minimum of the total free energy per electron for the HEG as a function of $T$. The bottom shows the HEG barrier height (binding energy; see Fig. 13 and the text) as a function of $T$.

Figure 15

Equilibrium $r_{\mathrm{s}}$ as a function $T$ for scH. See the text.

Figure 16

Electronic heat capacity at constant volume $C_V^{\text{el}}$ as a function of electronic temperature for scH at material density ${\rho }_{\mathrm{H}}$ = 0.60 g/${\mathrm{cm}}^3$ with the finite-$T$ KSDT and ground-state PZ XC functionals. The inset shows the difference $\mathrm{\Delta }{C}_V^{\text{el}}=C_V^{\text{el},\mathrm{PZ}}-C_V^{\text{el},\mathrm{KSDT}}$.