Static Electronic Density Response of Warm Dense Hydrogen: Ab Initio Path Integral Monte Carlo Simulations

The properties of hydrogen under extreme conditions are important for many applications, including inertial confinement fusion and astrophysical models. A key quantity is given by the electronic density response to an external perturbation, which is probed in x-ray Thomson scattering experiments—the state of the art diagnostics from which system parameters like the free electron density $n_e$, the electronic temperature $T_e$, and the charge state $Z$ can be inferred. In this work, we present highly accurate path integral Monte Carlo results for the static electronic density response of hydrogen. We obtain the static exchange-correlation (XC) kernel $K_{\mathrm{XC}}$, which is of central relevance for many applications, such as time-dependent density functional theory. This gives us a first unbiased look into the electronic density response of hydrogen in the warm-dense matter regime, thereby opening up a gamut of avenues for future research.

Figure 1

Perturbation strength dependence of the electronic density response of hydrogen at $\mathbf{q}=2\pi L^{-1}(0,0,2{)}^T$ ($q\approx 1.5q_F$) at the electronic Fermi temperature ($\theta =1$) for (a) $r_s=2$, (b) $r_s=4$, and (c) $r_s=6$. The green crosses show our new PIMC data for hydrogen, and the red circles corresponding results for a uniform electron gas partly taken from Ref. [40]; note that the statistical error bars do not exceed the symbol size. The dashed black line shows a cubic fit to the PIMC data for $\mathrm{\Delta }\rho (A)$, and the solid blue horizontal lines the linear-response limit.

Figure 2

Electronic density response of warm dense hydrogen at $\theta =1$ with (a) $r_s=2$ and (b) $r_s=4$ (bottom). Green symbols: new PIMC results for different $N$; blue dots: Kohn-Sham response function within LDA ${\chi }_0(\mathbf{q})$; black stars: corresponding RPA; dotted red (dash-dotted gray): UEG model with LFC [52] (in RPA); dashed black: Lindhard function. Also shown for $r_s=4$ are results for the Kohn-Sham response function without XC effects (gray circles), the corresponding RPA (purple triangles), the UEG model assuming a free-electron fraction of $\alpha =0.6$ (dash-dotted yellow), and the corresponding Lindhard function (solid gray).

Figure 3

Electronic local-field correction $G(q)=G(q,0)$ [cf. Eq. (2)] of warm dense hydrogen for $r_s=2$ (a) and $r_s=4$ (b). Dotted red lines: UEG model [52]; blue (green) symbols: solving Eq. (2) for the LFC using for ${\chi }_0(\mathbf{q})$ the Kohn-Sham response within LDA (the response function of an ideal uniform Fermi gas [78]). The black dashed line gives the ALDA kernel. Also shown are results for the UEG model under the assumption of a free-electron fraction of $\alpha =0.6$ (dash-dotted yellow) and for the LFC using as input the KS response computed without any XC effects (purple symbols) for $r_s=4$.