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A study of the electronic and ionic structure, for competing states of fully and partially ionized hydrogen, using the neutral pseudo-atom method as well as a classical map for the electron subsystem

M. W. C. Dharma-wardana

TL;DR

This work addresses hydrogen under warm-dense conditions where conventional perturbative methods fail, by combining the neutral-pseudo-atom (NPA) average-atom approach with a classical-map (CMap) representation of the electron subsystem. It demonstrates the existence of two competing plasma states—fully ionized and partially ionized hydrogen—at the same density and temperature, characterized by distinct mean ionizations $\bar{Z}$ and captured via free energies, structure factors, and transport properties. The authors validate CMap against path-integral Monte Carlo (PIMC) and QMD data where available, while highlighting the ability of NPA to access the small-$k$ limit of structure factors and to provide self-consistent estimates of $\bar{Z}$ through Friedel-sum-rule minimization. The study further shows that the classical-map framework can efficiently extend quantum structure data to low-$k$ regimes and that $\bar{Z}$ acts as a robust indicator of phase miscibility, with implications for X-ray Thomson scattering diagnostics and planetary interiors modeling. Together, these methods offer fast, first-principles tools to predict equations of state and transport in hydrogen plasmas across relevant warm-dense regimes, guiding experiments and planetary science applications.

Abstract

Prof. Bonitz and his collaborators have made seminal contributions to the study of the uniform electron fluid and the electron-proton fluid, viz., hydrogen, in using {\it ab initio} simulations, as reflected in this festschrift. Here we review the theoretical methods available for these systems where traditional small-parameter methods fail. We use the neutral-pseudo atom (NPA) method, and a classical map for quantum electrons to study hydrogen plasmas. We show that {\it both} fully-ionized and partially-ionized hydrogen phases can exist with the same nominal density and temperature, at pressures and temperatures of interest to planetary physics. The mean ionization $\bar{Z}$, pair-distribution functions, free energies, pressures and conductivities are calculated for the competing phases. Here $\bar{Z}$ is also a measure of the miscibility of fully ionized and un-ionized hydrogen. Recent studies using path-integral Monte Carlo methods, and $N$-atom Density Functional Theory (DFT) simulations have provided essential structure data including the electron-electron structure factor $S_{ee}(k)$ that enters into interpretation of X-ray Thomson scattering and other diagnostics. We show that these structure data can be inexpensively evaluated using classical-map schemes for fully ionized plasmas, and more generally, using one-atom (average-atom) DFT methods for partially ionized systems.

A study of the electronic and ionic structure, for competing states of fully and partially ionized hydrogen, using the neutral pseudo-atom method as well as a classical map for the electron subsystem

TL;DR

This work addresses hydrogen under warm-dense conditions where conventional perturbative methods fail, by combining the neutral-pseudo-atom (NPA) average-atom approach with a classical-map (CMap) representation of the electron subsystem. It demonstrates the existence of two competing plasma states—fully ionized and partially ionized hydrogen—at the same density and temperature, characterized by distinct mean ionizations and captured via free energies, structure factors, and transport properties. The authors validate CMap against path-integral Monte Carlo (PIMC) and QMD data where available, while highlighting the ability of NPA to access the small- limit of structure factors and to provide self-consistent estimates of through Friedel-sum-rule minimization. The study further shows that the classical-map framework can efficiently extend quantum structure data to low- regimes and that acts as a robust indicator of phase miscibility, with implications for X-ray Thomson scattering diagnostics and planetary interiors modeling. Together, these methods offer fast, first-principles tools to predict equations of state and transport in hydrogen plasmas across relevant warm-dense regimes, guiding experiments and planetary science applications.

Abstract

Prof. Bonitz and his collaborators have made seminal contributions to the study of the uniform electron fluid and the electron-proton fluid, viz., hydrogen, in using {\it ab initio} simulations, as reflected in this festschrift. Here we review the theoretical methods available for these systems where traditional small-parameter methods fail. We use the neutral-pseudo atom (NPA) method, and a classical map for quantum electrons to study hydrogen plasmas. We show that {\it both} fully-ionized and partially-ionized hydrogen phases can exist with the same nominal density and temperature, at pressures and temperatures of interest to planetary physics. The mean ionization , pair-distribution functions, free energies, pressures and conductivities are calculated for the competing phases. Here is also a measure of the miscibility of fully ionized and un-ionized hydrogen. Recent studies using path-integral Monte Carlo methods, and -atom Density Functional Theory (DFT) simulations have provided essential structure data including the electron-electron structure factor that enters into interpretation of X-ray Thomson scattering and other diagnostics. We show that these structure data can be inexpensively evaluated using classical-map schemes for fully ionized plasmas, and more generally, using one-atom (average-atom) DFT methods for partially ionized systems.

Paper Structure

This paper contains 18 sections, 10 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Review of theoretical methods
  3. $N$-atom DFT simulations with MD evolution of ions
  4. Path-integral and Monte-Carlo methods
  5. Limitations of PIMC and QMD
  6. Average-atom models
  7. Accessing the small-$k$ limit
  8. Short-comings of the NPA model
  9. The classical-map approach to WDM calculations
  10. The degree of ionization $\bar{Z}$ for Warm dense Hydrogen
  11. Metastable states and plasma-phase transitions in hydrogen
  12. Classical Map calculations for the UEF and H-plasmas
  13. Calculations using the Classical Map for the UEF
  14. Classical Map and NPA calculations for H-plasmas
  15. Results for $r_s = 2$, $\theta = 1$
  16. ...and 3 more sections

Figures (8)

  • Figure 1: (online color) The competition and possible co-existence of a partially ionized phase with $\bar{Z} < 1$, with a fully ionized phase with $\bar{Z} = 1$ for the hydrogen isochore, $\bar{\rho} = 0.1$ g/cc, i.e., $r_{\rm ws} \simeq 3$.
  • Figure 2: (online color) The proton-proton pair distribution function $g_{\rm pp}(r)$ and the corresponding $S_{\rm pp}(k)$ for a fully-ionized plasma ($\bar{Z} = 1$), and for a partially ionized plasma ($\bar{Z} = 0.3$) from within the metastable region, for the density $\bar{\rho} = 0.1$ g/cc, i.e., $r_{ws}\simeq3$. The PIMC $g(r)$ is from Ref. BonitzPOP24, Fig. 18, where the conditions $r_s$ = 3, $T$ = 62500K are specified, with $\bar{Z}$ unspecified.
  • Figure 3: (online color) (a) The Helmholtz free energy $F$ per atom (Hartrees) for the partially-ionized plasma (labeled partial-H$^+$), compared with the fully ionized plasma (labeled full-H$^+$). (b) The electrical conductivities of the partially-ionized and fully-ionized plasmas, both of density 0.1 g/cc are displayed, calculated using a T-matrix approach suitable for strong scattering. The QMD values of $\sigma$ are from the calculation by Holst at al. Holst11 using the $T=0$ PBE-XC functional.
  • Figure 4: (online color) The total pressures along the 0.1 g/cc isochores of the two phases, calculated from the Helmholtz free energy $F$ are displayed. The UEF contributions are also displayed. The PIMC data are from Militzer et al. MilitzerCataldo21.
  • Figure 5: (online color) The UEF components of the total Helmholtz free energy $F$ per atom (Hartrees) for the partially-ionized plasma (labeled partial-H$^+$), and for the fully ionized plasma (labeled fully-H$^+$) at 0.1 g/cc, calculated using the classical map (i.e., PDW-XC) PDWXC, and via the parametrization given by Groth et al. (GDS-XC) GDS17 are displayed.
  • ...and 3 more figures