Exchange-correlation energy for the three-dimensional homogeneous electron gas at arbitrary temperature

We fit finite-temperature path integral Monte Carlo calculations of the exchange-correlation energy of the $3\text{D}$ finite-temperature homogeneous electron gas in the warm-dense regime [$r_s\equiv {(3/4\pi n)}^{1/3}{a}_B^{-1}<40$ and $\Theta \equiv T/T_F>0.0625$]. In doing so, we construct a Padé approximant which collapses to Debye-Hückel theory in the high-temperature, low-density limit. Likewise, the zero-temperature limit matches the numerical results of ground-state quantum Monte Carlo, as well as analytical results in the high-density limit.

Figure 1

Ratio of the exchange-correlation energy $E_{xc}$ at temperature $T$ to that at $T=0$ for the unpolarized $\xi =0$ $3\text{D}$ HEG with $r_s=1.0$, $4.0$, and $10.0$ (respectively). Shown are the results from numerical calculations (RPIMC), the present parametrization (BDHC), and several previous parametrizations. The latter include Debye-Hückel (DH), Hansen (H), Tanaka and Ichimaru (TI), and Perrot and Dharma-wardana (PDW), all of which are discussed in the text.

Figure 2

Ratio of the exchange-correlation energy $E_{xc}$ at temperature $T$ to that at $T=0$ for the polarized $\xi =1$ $3\text{D}$ HEG with $r_s=1.0$, $4.0$, and $10.0$ (respectively). Shown are the results from numerical calculations (RPIMC), the present parametrization (BDHC), and several previous parametrizations. The latter include Debye-Hückel (DH), Hansen (H), Tanaka and Ichimaru (TI), and Perrot and Dharma-wardana (PDW), all of which are discussed in the text.