Ab Initio Thermodynamic Results for the Degenerate Electron Gas at Finite Temperature

The uniform electron gas at finite temperature is of key relevance for many applications in dense plasmas, warm dense matter, laser excited solids, and much more. Accurate thermodynamic data for the uniform electron gas are an essential ingredient for many-body theories, in particular, density-functional theory. Recently, first-principles restricted path integral Monte Carlo results became available, which, however, had to be restricted to moderate degeneracy, i.e., low to moderate densities with $r_s=\overline{r}/a_B\gtrsim 1$. Here we present novel first-principles configuration path integral Monte Carlo results for electrons for $r_s\le 4$. We also present quantum statistical data within the $e^4$ approximation that are in good agreement with the simulations at small to moderate $r_s$.

Figure 1

Density-temperature plain in the warm dense matter range. Typical inertial confinement fusion (ICF) parameters [8]. Quantum (classical) behavior dominates below (above) the line $\mathrm{\Theta }=1$. $\mathrm{\Gamma }=e^2/\overline{r}{k}_{B}T$ is the classical coupling parameter. Red dots, available finite-temperature RPIMC [27] and DPIMC [30] data for the UEG; blue dots, ground state data of Ref. [31]; green crosses, CPIMC and analytical results of this work.

Figure 2

Typical closed path in Slater determinant (Fock) space. The state with three occupied orbitals $|{\overrightarrow{k}}_0⟩,|{\overrightarrow{k}}_1⟩,|{\overrightarrow{k}}_3⟩$ undergoes a two-particle excitation $(s_1,{\tau }_1)$ that replaces the occupied orbitals $|{\overrightarrow{k}}_0⟩,|{\overrightarrow{k}}_3⟩$ by $|{\overrightarrow{k}}_2⟩,|{\overrightarrow{k}}_5⟩$. Two further excitations occur at ${\tau }_2$ and ${\tau }_3$. The states at the “imaginary times” $\tau =0$ and $\tau =\beta$ coincide. All possible paths contribute to the partition function $Z$, Eq. (4).

Figure 3

Convergence of the CPIMC simulations. (a) Convergence with the single-particle basis size $N_B$ for $r_s=0.4$. The expected scaling with $N_B^{5/3}$ [48] is well reproduced. (b) Convergence with respect to the kink potential parameter $\kappa$ (see text) and extrapolation to $1/\kappa \to 0$, corresponding to $K\to \infty$, for $r_s=1.0$ and $\theta =0.0625$. The asymptotic value is enclosed between the red line and the blue line.

Figure 4

Exchange-correlation energy (times $r_s$) for 33 spin-polarized electrons and four temperatures. Comparison of our CPIMC results (full symbols with error bars [53]) and RPIMC results of Ref. [27] (open symbols). The dotted line is an interpolation between the CPIMC and RPIMC data for $\mathrm{\Theta }=0.0625$. Also shown is the data point of DuBois et al. [34] for $\mathrm{\Theta }=0.125$.

Figure 5

Exchange correlation energy of the macroscopic polarized UEG at $\mathrm{\Theta }=0.0625$ (blue) and $\mathrm{\Theta }=0.5$ (red). Open symbols, RPIMC results [27]. CPIMC (a), our results with FSC from Ref. [54]. CPIMC (b), our data (3 points) with numerical extrapolation $N\to \infty$ [57]. Dashes, analytical $e^4$ approximation [49]; dots, fit of Ref. [32]. For better visibility, the curves for $\mathrm{\Theta }=0.5$ are up-shifted by 0.2.