Fig. 1. Schematic illustration of the PIMC representation, showing a configuration of N = 3 hydrogen atoms in the x–τ plane for P = 6 imaginary-time slices. Each electron (filled circles) and each proton (empty circles) is represented by a path along the imaginary-time axis τ of P coordinates, which are effectively connected by harmonic spring potentials (red and green connections); the extension of the paths is directly related to the respective thermal wavelength λTa=2πβ/ma. The yellow horizontal lines illustrate the evaluation of the density of species a and b in reciprocal space at an imaginary-time distance τ to estimate the ITCF F_ab(q, τ) [Eq. (3)].

Fig. 2. Ab initio PIMC results for the partial imaginary-time density–density correlation functions of warm dense hydrogen for N = 32 hydrogen atoms at the electronic Fermi temperature Θ = 1 and a metallic density r_s = 2 in the τ–q plane: (a) electron–electron ITCF F_ee(q, τ); (b) proton–proton ITCF F_pp(q, τ); (c) electron–proton ITCF F_ep(q, τ).

Fig. 3. Ab initio PIMC results for partial hydrogen ITCFs at r_s = 2 and Θ = 1 for (a) q = 1.53 Å⁻¹ (or q = 0.84q_F) and (b) q = 7.65 Å⁻¹ (or q = 4.21q_F): solid red line, F_ee(q, τ); dashed green line, F_pp(q, τ); dotted blue line, F_ep(q, τ); double-dashed yellow line, UEG model.¹⁰⁸ The shaded intervals correspond to 1σ error bars. The insets in (b) show magnified segments around F_ep(q, τ) and F_pp(q, τ).

Fig. 4. Ab initio PIMC results for the partial static density responses of hydrogen at r_s = 2 and Θ = 1: red, green, and blue symbols, χ_ee(q), χ_pp(q), and χ_ep(q), respectively, for full hydrogen evaluated from the ITCF via Eq. (9); gray squares, χ_ee(q), χ_pp(q), and χ_ep(q) for full hydrogen evaluated from the direct perturbation approach [Eq. (7)]; black circles, electronic density response of fixed ion snapshot;⁷⁴ solid yellow line, UEG;⁷¹ dashed black lines, ideal density responses χee(0)(q) and χpp(0)(q).

Fig. 5. Partial induced density for q = 1.53 Å⁻¹ as a function of the perturbation strength A [cf. Eq. (4)] at r_s = 2 and Θ = 1 for N = 14 H atoms. The dashed blue lines show the linear-response limit computed from the ITCF via Eq. (9), and the solid green lines the cubic polynomial fits via Eq. (7). (a) Induced electronic density ρ_e as function of A_e (red stars, A_p = 0) and A_p (yellow crosses, A_e = 0), and induced proton density ρ_p as a function of A_p (yellow squares, A_e = 0) and A_e (red crosses, A_p = 0). (b) Induced electronic density ρ_e as a function of the electronic perturbation strength A_e for full hydrogen (red diamonds, A_p = 0), a fixed proton snapshot⁷⁴ (black crosses), and the UEG¹²⁹ (yellow squares). Additional results are shown in Appendix B.

Fig. 6. Induced density change Δn(z)/n₀ for an electronic perturbation amplitude of A_e = 0.05 Ha (A_e = 1.36 eV) [cf. Eq. (4)] for r_s = 2 and Θ = 1: solid red line, electron density n_e(z) for full hydrogen (with A_p = 0); dotted blue line, corresponding linear-response prediction [Eq. (25)]; solid green line, proton density n_p(z) for full hydrogen (with A_p = 0); solid yellow line, electron density n_e(z) for the UEG model;¹²⁹ dashed black line, electron density n_e(z) of a fixed proton snapshot.⁷⁴

Fig. 7. Ab initio PIMC results for the partial local field factors θ_ab(q) for r_s = 2 and Θ = 1: red, green, and blue lines, θ_ee(q), θ_ep(q), and θ_pp(q), respectively, of full hydrogen [Eqs. (14)–(16)]; yellow line, electron–electron local field factor of the UEG model.⁷¹

Fig. 8. Ab initio PIMC results for the partial imaginary-time density–density correlation functions of warm dense hydrogen for N = 32 hydrogen atoms at the electronic Fermi temperature Θ = 1 and a solid density r_s = 3.23 in the τ–q plane: (a) electron–electron ITCF F_ee(q, τ); (b) proton–proton ITCF F_pp(q, τ); (c) electron–proton ITCF F_ep(q, τ).

Fig. 9. Ab initio PIMC results for partial hydrogen ITCFs at r_s = 3.23 and Θ = 1 for (a) q = 0.95 Å⁻¹ (or q = 0.84q_F) and (b) q = 4.73 Å⁻¹ (or q = 4.21q_F): solid red line, F_ee(q, τ); dashed green line, F_pp(q, τ); dotted blue line, F_ep(q, τ); double-dashed yellow line, UEG model.¹⁰⁸ The shaded intervals correspond to 1σ error bars. The insets in (b) show magnified segments around F_ep(q, τ) and F_pp(q, τ).

Fig. 10. Ab initio PIMC results for the partial static density responses of hydrogen at r_s = 3.23 and Θ = 1: red, green, and blue symbols, χ_ee(q), χ_pp(q), and χ_ep(q), respectively, for full hydrogen evaluated from the ITCF via Eq. (9); solid yellow line, UEG model;⁷¹ dashed black lines, ideal density responses χee(0)(q) and χpp(0)(q).

Fig. 11. Ab initio PIMC results for the partial local field factors θ_ab(q) for r_s = 3.23 and Θ = 1: red, green, and blue lines, θ_ee(q), θ_ep(q), and θ_pp(q), respectively, of full hydrogen [Eqs. (14)–(16)]; yellow line, local field factor of the UEG model.⁷¹

Fig. 12. Ab initio PIMC results for the partial imaginary-time density–density correlation functions of warm dense hydrogen for N = 32 hydrogen atoms at the electronic Fermi temperature Θ = 1 and a compressed density r_s = 1 in the τ–q plane: (a) electron–electron ITCF F_ee(q, τ); (b) proton–proton ITCF F_pp(q, τ); (c) electron–proton ITCF F_ep(q, τ).

Fig. 13. Ab initio PIMC results for partial hydrogen ITCFs at r_s = 1 and Θ = 1 for (a) q = 3.06 Å⁻¹ (or q = 0.84q_F) and (b) q = 15.3 Å⁻¹ (or q = 4.21q_F): solid red line, F_ee(q, τ); dashed green line, F_pp(q, τ); dotted blue line, F_ep(q, τ); double-dashed yellow line, UEG model.¹⁰⁸ The shaded intervals correspond to 1σ error bars. The insets in (b) show magnified segments around F_ep(q, τ) and F_pp(q, τ).

Fig. 14. Ab initio PIMC results for the partial static density responses of hydrogen at r_s = 1 and Θ = 1: red, green, and blue symbols, χ_ee(q), χ_pp(q), and χ_ep(q), respectively, for full hydrogen evaluated from the ITCF via Eq. (9); solid yellow line, UEG model;⁷¹ dashed black lines, ideal density responses χee(0)(q) and χpp(0)(q).

Fig. 15. Ab initio PIMC results for the partial local field factors θ_ab(q) for r_s = 1 and Θ = 1: red, green, and blue lines, θ_ee(q), θ_ep(q), and θ_pp(q), respectively, of full hydrogen [Eqs. (14)–(16)]; yellow line, local field factor of the UEG model.⁷¹

Fig. 16. Comparison of the electronic density response function χ_ee(q) (multiplied by the density parameter r_s) at the electronic Fermi temperature, Θ = 1, for the three considered values of r_s. The symbols show our new PIMC results for hydrogen, and the lines correspond to the UEG model.⁷¹

Fig. 17. Application of the ξ-extrapolation method for the electron–electron density response χ_ee(q) of hydrogen at Θ = 1 and (a) r_s = 3.23, (b) r_s = 2, and (c) r_s = 1: green crosses, direct fermionic PIMC results for ξ = −1; red circles, extrapolation from the sign-problem-free domain of ξ ∈ [0, 1] (shaded gray area) via Eq. (A1).

Fig. 18. (a) and (b) Partial induced electronic ρ_e(q) and proton densities ρ_p(q), respectively, as functions of the electronic perturbation amplitude A_e [cf. Eq. (4)] at r_s = 2 and Θ = 1 for N = 14 hydrogen atoms: dashed blue line, linear response limit evaluated from the ITCF [Eq. (9)]; solid green line, cubic fits via Eq. (7); the different symbols distinguish different perturbation wavenumbers q.