Ab Initio Quantum Monte Carlo Simulation of the Warm Dense Electron Gas in the Thermodynamic Limit

We perform ab initio quantum Monte Carlo (QMC) simulations of the warm dense uniform electron gas in the thermodynamic limit. By combining QMC data with the linear response theory, we are able to remove finite-size errors from the potential energy over the substantial parts of the warm dense regime, overcoming the deficiencies of the existing finite-size corrections by Brown et al. [Phys. Rev. Lett. 110, 146405 (2013)]. Extensive new QMC results for up to $N=1000$ electrons enable us to compute the potential energy $V$ and the exchange-correlation free energy $F_{\mathrm{xc}}$ of the macroscopic electron gas with an unprecedented accuracy of $|\mathrm{\Delta }V|/|V|,|\mathrm{\Delta }{F}_{\mathrm{xc}}|/|F{|}_{\mathrm{xc}}\sim {10}^{-3}$. A comparison of our new data to the recent parametrization of $F_{\mathrm{xc}}$ by Karasiev et al. [Phys. Rev. Lett. 112, 076403 (2014)] reveals significant deviations to the latter.

Figure 1

Potential energy per particle of the unpolarized UEG at $\theta =2$ and $r_s=0.5$. The exact CPIMC results for different system sizes are indicated by green crosses; the yellow asterisks show these results after the $\mathrm{\Delta }{V}_{\mathrm{BCDC}}$ finite-size correction from Eq. (4) has been applied. The horizontal arrows refer to many-body theories (RPA, STLS [21], MW, and e4 [45]; see the text). The black lines are two different, equally plausible, extrapolations of the QMC data to infinite system size [44].

Figure 2

Static structure factors for $\theta =2$, $r_s=0.5$, and three values of $N$. In (a), the discrete QMC $k$ points are plotted as vertical lines for $N=100$; the minimum $k$ values for $N=66$ and $N=38$ are indicated by the green and yellow line, respectively. The colored horizontal bars indicate the $k$ ranges where $S^{\mathrm{STLS}}$ (red), $S^{\mathrm{RPA}}$ RPA (gray), and $S_0^{\mathrm{RPA}}$ (light blue) are accurate. (b) shows that the QMC results for $S(k)$ converge rapidly with $N$ (see the colored symbols in the inset). The black curve shows $S_{\mathrm{comb}}$ connecting $S^{\mathrm{STLS}}(k)$ at small $k$ with the QMC data for $N=100$ which yields accurate results for all $k$.

Figure 3

(a) Finite-size corrected QMC data for the potential energy for $\theta =2$ and $r_s=0.5$. The yellow asterisks are obtained using Eq. (4); the red diamonds use the combined SF $S_{\mathrm{comb}}$ (cf. Fig. 2) to evaluate the discretization error, Eq. (5). (b) Magnified part of (a) including an extrapolation of the residual finite-size error to the TDL (the red cross). Results obtained using only the full RPA (blue) and STLS structure factors (black) in Eq. (5) are also shown.

Figure 4

Potential energy of the UEG in the TDL. (a) Our new FS-corrected QMC data, the fits to our data [see Eq. (S.2) of Ref. 49], and the RPIMC results of Brown et al. [27], which include BCDC FSCs. (b) Relative deviations of our data (for $\mathrm{\Theta }=8$) and Brown’s BCDC-corrected data from the corresponding fit. (c) Relative deviation of our exchange-correlation free energies from the fit of Ref. [52] for five temperatures. For details, see Ref. [49].