Matter and Radiation at Extremes, Volume. 9, Issue 5, 057401(2024)

Ab initio density response and local field factor of warm dense hydrogen

Tobias Dornheim1...2,a), Sebastian Schwalbe1,2, Panagiotis Tolias3, Maximilian P. Böhme1,2,4, Zhandos A. Moldabekov1,2 and Jan Vorberger2 |Show fewer author(s)
Author Affiliations
  • 1Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
  • 2Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
  • 3Space and Plasma Physics, Royal Institute of Technology (KTH), Stockholm SE-100 44, Sweden
  • 4Technische Universität Dresden, D-01062 Dresden, Germany
  • show less
    Figures & Tables(18)
    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 Fab(q, τ) [Eq. (3)].
    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 rs = 2 in the τ–q plane: (a) electron–electron ITCF Fee(q, τ); (b) proton–proton ITCF Fpp(q, τ); (c) electron–proton ITCF Fep(q, τ).
    Ab initio PIMC results for partial hydrogen ITCFs at rs = 2 and Θ = 1 for (a) q = 1.53 Å−1 (or q = 0.84qF) and (b) q = 7.65 Å−1 (or q = 4.21qF): solid red line, Fee(q, τ); dashed green line, Fpp(q, τ); dotted blue line, Fep(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 Fep(q, τ) and Fpp(q, τ).
    Ab initio PIMC results for the partial static density responses of hydrogen at rs = 2 and Θ = 1: red, green, and blue symbols, χee(q), χpp(q), and χep(q), respectively, for full hydrogen evaluated from the ITCF via Eq. (9); gray squares, χee(q), χpp(q), and χep(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).
    Partial induced density for q = 1.53 Å−1 as a function of the perturbation strength A [cf. Eq. (4)] at rs = 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 ρe as function of Ae (red stars, Ap = 0) and Ap (yellow crosses, Ae = 0), and induced proton density ρp as a function of Ap (yellow squares, Ae = 0) and Ae (red crosses, Ap = 0). (b) Induced electronic density ρe as a function of the electronic perturbation strength Ae for full hydrogen (red diamonds, Ap = 0), a fixed proton snapshot74 (black crosses), and the UEG129 (yellow squares). Additional results are shown in Appendix B.
    Induced density change Δn(z)/n0 for an electronic perturbation amplitude of Ae = 0.05 Ha (Ae = 1.36 eV) [cf. Eq. (4)] for rs = 2 and Θ = 1: solid red line, electron density ne(z) for full hydrogen (with Ap = 0); dotted blue line, corresponding linear-response prediction [Eq. (25)]; solid green line, proton density np(z) for full hydrogen (with Ap = 0); solid yellow line, electron density ne(z) for the UEG model;129 dashed black line, electron density ne(z) of a fixed proton snapshot.74
    Ab initio PIMC results for the partial local field factors θab(q) for rs = 2 and Θ = 1: red, green, and blue lines, θee(q), θep(q), and θpp(q), respectively, of full hydrogen [Eqs. (14)–(16)]; yellow line, electron–electron local field factor of the UEG model.71
    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 rs = 3.23 in the τ–q plane: (a) electron–electron ITCF Fee(q, τ); (b) proton–proton ITCF Fpp(q, τ); (c) electron–proton ITCF Fep(q, τ).
    Ab initio PIMC results for partial hydrogen ITCFs at rs = 3.23 and Θ = 1 for (a) q = 0.95 Å−1 (or q = 0.84qF) and (b) q = 4.73 Å−1 (or q = 4.21qF): solid red line, Fee(q, τ); dashed green line, Fpp(q, τ); dotted blue line, Fep(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 Fep(q, τ) and Fpp(q, τ).
    Ab initio PIMC results for the partial static density responses of hydrogen at rs = 3.23 and Θ = 1: red, green, and blue symbols, χee(q), χpp(q), and χep(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).
    Ab initio PIMC results for the partial local field factors θab(q) for rs = 3.23 and Θ = 1: red, green, and blue lines, θee(q), θep(q), and θpp(q), respectively, of full hydrogen [Eqs. (14)–(16)]; yellow line, local field factor of the UEG model.71
    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 rs = 1 in the τ–q plane: (a) electron–electron ITCF Fee(q, τ); (b) proton–proton ITCF Fpp(q, τ); (c) electron–proton ITCF Fep(q, τ).
    Ab initio PIMC results for partial hydrogen ITCFs at rs = 1 and Θ = 1 for (a) q = 3.06 Å−1 (or q = 0.84qF) and (b) q = 15.3 Å−1 (or q = 4.21qF): solid red line, Fee(q, τ); dashed green line, Fpp(q, τ); dotted blue line, Fep(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 Fep(q, τ) and Fpp(q, τ).
    Ab initio PIMC results for the partial static density responses of hydrogen at rs = 1 and Θ = 1: red, green, and blue symbols, χee(q), χpp(q), and χep(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).
    Ab initio PIMC results for the partial local field factors θab(q) for rs = 1 and Θ = 1: red, green, and blue lines, θee(q), θep(q), and θpp(q), respectively, of full hydrogen [Eqs. (14)–(16)]; yellow line, local field factor of the UEG model.71
    Comparison of the electronic density response function χee(q) (multiplied by the density parameter rs) at the electronic Fermi temperature, Θ = 1, for the three considered values of rs. The symbols show our new PIMC results for hydrogen, and the lines correspond to the UEG model.71
    Application of the ξ-extrapolation method for the electron–electron density response χee(q) of hydrogen at Θ = 1 and (a) rs = 3.23, (b) rs = 2, and (c) rs = 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).
    (a) and (b) Partial induced electronic ρe(q) and proton densities ρp(q), respectively, as functions of the electronic perturbation amplitude Ae [cf. Eq. (4)] at rs = 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.
    Tools

    Get Citation

    Copy Citation Text

    Tobias Dornheim, Sebastian Schwalbe, Panagiotis Tolias, Maximilian P. Böhme, Zhandos A. Moldabekov, Jan Vorberger. Ab initio density response and local field factor of warm dense hydrogen[J]. Matter and Radiation at Extremes, 2024, 9(5): 057401

    Download Citation

    EndNote(RIS)BibTexPlain Text
    Save article for my favorites
    Paper Information

    Category:

    Received: Mar. 30, 2024

    Accepted: Jul. 8, 2024

    Published Online: Oct. 14, 2024

    The Author Email: Dornheim Tobias (t.dornheim@hzdr.de)

    DOI:10.1063/5.0211407

    Topics