Virial coefficients of the uniform electron gas from path-integral Monte Carlo simulations

The properties of plasmas in the low-density limit are described by virial expansions. Analytical expressions are known from Green's function approaches only for the first three virial coefficients. Accurate path-integral Monte Carlo (PIMC) simulations have recently been performed for the uniform electron gas, allowing the virial expansions to be analyzed and interpolation formulas to be derived. The exact expression for the second virial coefficient is used to test the accuracy of the PIMC simulations and the range of validity of the interpolation formula of Groth et al. [Phys. Rev. Lett. 119, 135001 (2017)], and we discuss the fourth virial coefficient, which is not exactly known yet. Combining PIMC simulations with benchmarks from exact virial expansion results would allow us to obtain more accurate representations of the equation of state for parameter ranges of conditions which are of interest, e.g., for helioseismology.

Figure 1

Extrapolating PIMC results for the interaction energy of the UEG to the thermodynamic limit (TDL) at $r_s=2$ and $\mathrm{\Theta }=217.204$ ($T_{H}a=100$). Left: Raw PIMC results for the interaction energy per particle $V/N$ (green crosses) and finite-size corrected values (red circles) as a function of system size. Right: Magnified segment around the finite-size corrected results; the solid black lines show empirical linear and constant fits, and the blue cross depicts the extrapolated result in the limit of $N\to \infty$.

Figure 2

Isotherms for $T_{\mathrm{Ha}}=0.589$ (16.03 eV) and 0.295 (8.03 eV). In both subfigures the effective second virial coefficients $v_2^{\mathrm{eff}}(T,n)$, Eq. (8), are plotted as function of $n_B^{1/2}ln(T_{\mathrm{Ha}}^2/4\pi n_B)$. The blue full line shows the isotherms according to the interpolation formula (13) for the respective two temperatures, taking $v_{\mathrm{XC}}^{\mathrm{GDSMFB}}(T,n)$ instead of $v(T,n)$. The exact contribution of $v_2\left(T\right)$ and $v_3\left(T\right)$ is given by the dashed line [virial $2+3$: Eq. (9)]. The red squares corresponds to $v_2^{\mathrm{eff},\mathrm{PIMC}}(T,n)$. See Tables 1 and 2. (Atomic units used. Left panel: Abscissa 0.1 corresponds $n=5.4\times {10}^{21}{\mathrm{cm}}^{-3}$, right panel: Abscissa 0.06 corresponds $n=4.1\times {10}^{21}{\mathrm{cm}}^{-3}$.)

Figure 3

Effective second virial coefficient $v_2^{\mathrm{eff},\mathrm{GDSMFB}}(T,n)$, Eq. (8), plotted as function of $n_B^{1/2}ln(T_{\mathrm{Ha}}^2/4\pi n_B)$, shown in blue full line. The exact value $v_2\left(100\right)=-0.0114705$ is also shown [virial $2+3$: Eq. (9)]. The slope according to $v_3\left(T\right)$, Eq. (7), becomes very small at high temperatures. In addition, PIMC simulations according Table 2, No. 3, are presented. A linear extrapolation $-0.0128+0.000289\sqrt{n_B}log(T_{\mathrm{Ha}}^2/4\pi n_B)$ is also shown (dash-dotted line). (Atomic units used. $T_{\mathrm{Ha}}=100$ corresponds 2730 eV, the abscissa 7 corresponds $n=7.7\times {10}^{24}{\mathrm{cm}}^{-3}$.)

Figure 4

Isotherms for $T_{\mathrm{Ha}}=100$. (a) The effective third virial coefficient $v_3^{\mathrm{eff},\mathrm{GDSMFB}}(T,n)$, Eq. (15), plotted as a function of $1/ln(T_{\mathrm{Ha}}^2/4\pi n_B)$. The slope of $v_3^{\mathrm{eff}}(T,n)$ determines $v_4\left(100\right)$. The linear relation Eq. (16) with $v_3\left(100\right)=-8.352\times {10}^{-7}$ is denoted as virial $3+4$. The dash-dotted curve corresponds to $v_4\left(100\right)=0.00129$, Eq. (14).  (b) The effective second virial coefficient $v_2^{\mathrm{eff},\mathrm{GDSMFB}}(T,n)$, Eq. (8), plotted as function of $n_B^{1/2}$. From analytical approaches [Eqs. (7) and (14)] follows $v_2^{\mathrm{eff}}\left(100\right)\approx -0.01147+0.00129n_B^{1/2}$ (shown as a dash-dotted line). The dashed line represents a linear fit. (Atomic units used.)