Path-Integral Monte Carlo Simulation of the Warm Dense Homogeneous Electron Gas

We perform calculations of the 3D finite-temperature homogeneous electron gas in the warm-dense regime ($r_s\equiv (3/4\pi n{)}^{1/3}{a}_0^{-1}=1.0–40.0$ and $\Theta \equiv T/T_F=0.0625–8.0$) using restricted path-integral Monte Carlo simulations. Precise energies, pair correlation functions, and structure factors are obtained. For all densities, we find a significant discrepancy between the ground state parametrized local density approximation and our results around $T_F$. These results can be used as a benchmark for developing finite-temperature density functionals, as well as input for orbital-free density function theory formulations.

Figure 1

Temperature-density points considered in the current study (dots). Several values of the Coulomb coupling parameter $\Gamma$ (dashed lines) and the electron degeneracy parameter $\Theta$ (dotted lines) are also shown.

Figure 2

Static structure factors for $r_s=1.0$ and $r_s=10.0$ in the unpolarized state. At $\Theta =0.0$ we plot the ground state structure factor from Ref. 25. Also shown is the small $k$ part of $S_{\mathrm{DH}}(k)$ at $\Theta =8.0$; see Eq. (3).

Figure 3

Excess energies for $r_s=4.0$ (top) and $r_s=40.0$ (bottom) for the polarized state. For both densities, the high temperature results fall smoothly on top of previous Monte Carlo energies for the classical electron gas [10] (solid line). Differences from the classical Coulomb gas occur for $\Theta <2.0$ for $r_s=4.0$ and $\Theta <4.0$ for $r_s=40.0$. Simulations with the Fermion sign (squares) confirm the fixed-node results at $\Theta =1.0$ and 8.0. The zero-temperature limit (dotted line) smoothly extrapolates to the ground state QMC results of Ceperley-Alder [4] (dashed line).

Figure 4

Pair correlation functions for $r_s=1.0$ and $r_s=10.0$ in the unpolarized state. At $\Theta =0.0$ is shown the ground state correlation function from Ref. 25. Deviation from RPIMC is seen at small $r$, but this is most likely due to poor ground state QMC data [29]. Also shown is the small $r$ part of $g_{\mathrm{DH}}(r)$ at $\Theta =8.0$; see Eq. (4). The Debye-Huckel limit is not yet reached at $\Theta =8.0$ for the lower density $r_s=10.0$.

Figure 5

Correlation energy $e_c(T)$ of the 3D HEG at several temperatures and densities for the unpolarized (top) and fully spin-polarized (bottom) states. Exact (signful) calculations (squares) confirm the fixed-node results where possible ($\Theta =8.0$ for $\xi =0$ and $\Theta =4.0$, 8.0 for $\xi =1$). For comparison, we plot the $\Theta =0.0$ correlation energy used in local density approximation DFT calculations.