Accurate Exchange-Correlation Energies for the Warm Dense Electron Gas

The density matrix quantum Monte Carlo (DMQMC) method is used to sample exact-on-average $N$-body density matrices for uniform electron gas systems of up to ${10}^{124}$ matrix elements via a stochastic solution of the Bloch equation. The results of these calculations resolve a current debate over the accuracy of the data used to parametrize finite-temperature density functionals. Exchange-correlation energies calculated using the real-space restricted path-integral formalism and the $k$-space configuration path-integral formalism disagree by up to $\sim 10\%$ at certain reduced temperatures $T/T_F\le 0.5$ and densities $r_s\le 1$. Our calculations confirm the accuracy of the configuration path-integral Monte Carlo results available at high density and bridge the gap to lower densities, providing trustworthy data in the regime typical of planetary interiors and solids subject to laser irradiation. We demonstrate that the DMQMC method can calculate free energies directly and present exact free energies for $T/T_F\ge 1$ and $r_s\le 2$.

Figure 1

Exchange-correlation energy per particle (times $r_s$) as a function of $r_s$, showing excellent agreement between the $i$-DMQMC results and configuration PIMC (CPIMC) for $r_s\le 1$ and differences between the $i$-DMQMC results and restricted PIMC (RPIMC) for $1\le r_s\le 4$. For the electron gas, $u_{\mathrm{xc}}=(U-U_0)/N$, where $U_0$ is the internal energy of the $N=33$ noninteracting electron gas in the canonical ensemble. The lines are weighted third-order polynomial interpolations [58] between the $i$-DMQMC data and the restricted PIMC data for $r_s>4$ and are meant as guides to the eye. The $i$-DMQMC results at $r_s=3$ and $r_s=4$ were obtained using a basis set extrapolation not required at lower $r_s$. The error bars include estimates of the remaining initiator and basis set errors [43].

Figure 2

Extrapolation [59] of the internal energy to $\mathrm{\Theta }=0$ for the $N=33$, $\zeta =1$, $r_s=1$ system. The restricted PIMC energies are systematically too low and extrapolate to a value considerably below the $i$-FCIQMC ground-state energy. This discrepancy cannot be explained by finite-size effects. The $i$-DMQMC (and by extension configuration PIMC) results fare significantly better. Also shown are two Hartree-Fock-like mean-field estimates (labeled HF0 and THF) of the internal energy in the canonical ensemble [43].

Figure 3

Top panel: exchange-correlation free energies for the $N=33$, $\zeta =1$ electron gas calculated using the DMQMC method. The sign problem prohibits calculations below $\mathrm{\Theta }=4$ at $r_s=2$. Also plotted is $f_{\mathit{x}}^0=N^{-1}⟨\widehat{V}{⟩}_0$, the first-order exchange contribution to the free energy evaluated in the canonical ensemble. Bottom panel: the exchange-correlation entropy for the same system. Additional data and more details of the calculation procedures can be found in the Supplemental Material [43].