Ab initio Exchange-Correlation Free Energy of the Uniform Electron Gas at Warm Dense Matter Conditions

In a recent Letter [T. Dornheim et al., Phys. Rev. Lett. 117, 156403 (2016)], we presented the first quantum Monte Carlo (QMC) results for the warm dense electron gas in the thermodynamic limit. However, a complete parametrization of the exchange-correlation free energy with respect to density, temperature, and spin polarization remained out of reach due to the absence of (i) accurate QMC results below $\theta =k_{B}T/E_F=0.5$ and (ii) QMC results for spin polarizations different from the paramagnetic case. Here we overcome both remaining limitations. By closing the gap to the ground state and by performing extensive QMC simulations for different spin polarizations, we are able to obtain the first completely ab initio exchange-correlation free energy functional; the accuracy achieved is an unprecedented $\sim 0.3\%$. This also allows us to quantify the accuracy and systematic errors of various previous approximate functionals.

Figure 1

Temperature dependence of the XC free energy and potential energy—the top row shows $f_{\text{xc}}$ (dashed lines) from this work (red), KSDT (blue [53]), IIT (black [48, 49]), Tanaka (green [47]), and Perrot–Dharma-wardana (yellow dashed line and triangles, PDW [44]), as well as the corresponding interaction energy $v$ (solid lines) from this work, KSDT, and the restricted PIMC results by Brown et al. (blue dots [52]). The red rhombs correspond to ground-state QMC results plus a temperature correction function obtained from the STLS theory. The inset corresponds to an enlargement of the gray box. The bottom row displays the relative deviations of the different models of $f_{\text{xc}}$ with respect to our new parametrization.

Figure 2

Cross-check of our parametrization ($\xi =1$, $r_s=1$). The XC energy per electron (red line), as calculated from our Padé function for $f_{\text{xc}}$ (dashed line), is compared to new, independent finite-size-corrected QMC data (red dots) [68]. While our functional has been constructed solely using the interaction energy $v$ [cf. Eq. (3)], the KSDT curve [53] (solid blue) was fitted to the restricted PIMC data [52] for $e_{\text{xc}}$ (blue circles, BCDC).

Figure 3

Dependence of the XC free energy on spin polarization at $r_s=1$. The ab initio functional derived here (red curve) is compared to the parametrizations of KSDT (blue curve) [53], IIT (black curve) [48, 49], and Tanaka (green curve) [47].