Dielectric function and thermodynamic properties of jellium in the $GW$ approximation

The fully self-consistent $\mathit{GW}$ approximation is an established method for electronic structure calculations. Its most serious deficiency is known to be an incorrect prediction of the dielectric response. In this work, we examine the $\mathit{GW}$ approximation for the homogeneous electron gas and find that problems with the dielectric response are drastically improved by enforcing the particle-number conservation law in the polarization function. We also find that previously reported data for the ground-state energy contradict each other well outside of reported error bounds. Some of these results created a false impression of how accurate the fully self-consistent $\mathit{GW}$ approximation is. Our two independent implementations of the $GW$ method agree with the data plotted in X.-Z. Yan [Phys. Rev. E 84, 016706 (2011)], thus confirming only that data set. We also present values for other key Fermi-liquid properties.

Figure 1

Energy per electron (in Hartrees) as a function of ${(T/{\epsilon }_F)}^2$ revealing the Fermi-liquid behavior. The solid line is a linear fit giving a ground-state energy of $E/N=0.5783\left(2\right)$ Ha.

Figure 2

Imaginary part of the dielectric function within the $\mathit{GW}$ approximation at $r_s=1, k/k_F=0.1$, and $T/{\epsilon }_F=0.02$. The red dashed curve is the original $\mathit{GW}$ result and the solid black line is the corrected $\mathit{GW}$ spectrum. The crucial difference at frequencies $\omega >k{v}_F$ is clearly seen in the inset.

Figure 3

Real part of the dielectric function within the $\mathit{GW}$ approximation at $r_s=1, k/k_F=0.1$, and $T/{\epsilon }_F=0.02$. The original $\mathit{GW}$ result (red dashed curve) completely misses the plasmon zero and predicts unrealistically large response at frequencies above ${\epsilon }_F$. The corrected result (solid black line) crosses zero within $10\%$ of ${\omega }_p$ and saturates to unity at $\omega >{\epsilon }_F$.