We report the results of the second charged-particle transport coefficient code comparison workshop, which was held in Livermore, California on 24–27 July 2023. This workshop gathered theoretical, computational, and experimental scientists to assess the state of computational and experimental techniques for understanding charged-particle transport coefficients relevant to high-energy-density plasma science. Data for electronic and ionic transport coefficients, namely, the direct current electrical conductivity, electron thermal conductivity, ion shear viscosity, and ion thermal conductivity were computed and compared for multiple plasma conditions. Additional comparisons were carried out for electron–ion properties such as the electron–ion equilibration time and alpha particle stopping power. Overall, 39 participants submitted calculated results from 18 independent approaches, spanning methods from parameterized semi-empirical models to time-dependent density functional theory. In the cases studied here, we find significant differences—several orders of magnitude—between approaches, particularly at lower temperatures, and smaller differences—roughly a factor of five—among first-principles models. We investigate the origins of these differences through comparisons of underlying predictions of ionic and electronic structure. The results of this workshop help to identify plasma conditions where computationally inexpensive approaches are accurate, where computationally expensive models are required, and where experimental measurements will have high impact.

Accurate predictions of the properties of high-energy-density (HED) matter are of critical importance in multiple areas of science, including astrophysics and inertial confinement fusion. The HED regime spans enormous ranges of temperatures and densities, from the warm dense matter of planetary interiors to the hot dense plasma at the hearts of stars. Modeling these systems requires understanding not only the equilibrium properties1 (equations of state) of matter over this vast range, but also the response (transport) properties2 of HED matter to gradients in pressure, temperature, and external fields.

Generally, the transport properties of a material are quantified by transport coefficients that inform magneto-radiation-hydrodynamic simulation codes and impact the interpretation of data obtained from experimental diagnostics. These data influence our understanding of the development of hydrodynamic instabilities, the overall energy balance of plasma systems, and the efficacy of alpha heating in burning fusion plasmas. Important transport processes include thermal and electrical conduction, energy exchange between ions and electrons, interdiffusion in ionic mixtures, ion viscosity, and charged-particle stopping. Since it is difficult to create and characterize HED matter in the laboratory, simulation codes rely on model predictions for transport coefficients. However, the systematic uncertainties in transport properties obtained from different models or simulation approaches are not well established. Moreover, the statistical uncertainties from a single model or simulation approach are typically not reported.

To quantify the uncertainties—and understand the capabilities—of our current simulation approaches and theoretical models for generating transport coefficients, the first charged-particle transport coefficient comparison workshop3 was held in 2016. Order-of-magnitude discrepancies were found between different computational approaches, with the largest differences occurring at the lowest temperatures and densities. The second workshop, described here, aimed to (1) add to the data collected in the first workshop, (2) extend the collected quantities for more complete model comparisons to help understand the origin of differences, and (3) select optimal plasma conditions for use in machine learning frameworks4,5 for uncertainty quantification. The second charged-particle transport coefficient code comparison workshop was held at the University of California's Livermore Collaboration Center (UCLCC) in Livermore, California on 24–27 July 2023. Data were submitted by 39 participants, from 14 institutions, using 18 unique models or simulation approaches.

Section II of this paper provides context for the importance of transport coefficients as closures of the magnetohydrodynamic equations that govern simulations of plasmas. Section III defines and justifies the specific elements, temperatures, and densities selected for this workshop. Section IV gives a brief overview of modeling methods along with a comparison of the submitted results for equilibrium properties including ionic radial distribution functions, electronic densities of states, and average ionization. Section V defines the transport coefficients and presents comparisons of submitted results for ionic transport (shear viscosity and thermal diffusivity), electronic transport (electrical and thermal conductivity), and alpha-particle transport (stopping power). We conclude in Sec. VI with a general discussion and a proposal for cases to be considered in a future workshop.

Transport coefficients quantify how various physical properties relax to equilibrium. These properties include momentum, temperature, and concentration distributions. Conservation laws relate these properties over macroscopic length and time scales. For a single species monatomic gas6,7 with no external forces the conservation laws are
ρ t + · ( ρ u ) = 0 ,
(1)
( ρ u ) t + · ( ρ u u ) + · P = 0 ,
(2)
3 2 [ ( n T ) t + · ( u n T ) ] + P : u + · q = 0 ,
(3)
where ρ ρ ( r , t ) = m n is the mass density of the gas defined in terms of the mass m and number density n n ( r , t ) , u u ( r , t ) is the velocity field of the gas, and T T ( r , t ) is the temperature of the gas, for position vector r = x i ̂ + y j ̂ + z k ̂ and time t. Additionally, P denotes the total stress tensor and q is the heat flow vector. Equations (1) and (2) describe the conservation laws of mass and momentum, respectively, and Eq. (3)—which stems from the conservation of energy—describes the temperature evolution of the gas. In this form, Eqs. (1)–(3) are not “closed”—there are more unknowns than equations. If the gas is in local thermodynamic equilibrium, then a commonly employed choice for the closures is
P = p I η [ u + ( u ) T 2 3 ( · u ) I ] ζ ( · u ) I ,
(4)
q = κ T ,
(5)
where p is the scalar pressure and I is the identity matrix. Together, Eqs. (1)–(5) are known as the Navier–Stokes–Fourier equations of hydrodynamics. Now that the stress tensor and heat flow vector have been represented in terms of the gas variables, the last remaining step to close Eqs. (1)–(5) is to quantify the transport coefficients η (the shear viscosity), ζ (the bulk viscosity), and κ (the thermal conductivity). In this workshop, we compare the shear viscosity, η, and the ionic and electronic thermal conductivity—denoted as κi and κe, respectively. We leave comparisons of the bulk viscosity as a topic for future workshops.

To describe the behavior of current-carrying plasmas, Eqs. (1)–(3) are modified to include additional terms from electromagnetic forces; they are referred to as the equations of magnetohydrodynamics.8,9 The equations of magnetohydrodynamics require knowledge of the direct current (DC) electrical conductivity as a closure. In this workshop, we compare values for the electronic DC electrical conductivity, σ.

The equations of magnetohydrodynamics underlie large-scale simulations used to describe phenomena in HED systems ranging from the interior of stars and giant planets to terrestrial fusion plasmas. Their closures require understanding the properties of matter at densities and temperatures that are difficult to create in the laboratory and even more difficult to experimentally constrain and measure. In Secs. III–VI, we will describe modeling approaches for computing material properties in HED conditions and compare predictions from multiple models submitted to the second charged-particle transport coefficient comparison workshop.

The first charged-particle transport coefficient comparison workshop3 studied H, C, and CH mixtures on a regular grid of temperatures and densities. In this second workshop, we extended the materials to more complex atomic systems and expanded the range of both material conditions and requested data. While the cases in the first workshop provided a wide comparison for two elements (i.e., H and C), a subset of the requested temperature and density range was intractable for many codes and simulation tools. Because of this, the comparison in the first workshop at many conditions was limited to results generated from one or two simulation methods. In this second workshop, we developed a Priority Level System to focus on high-priority cases with the goal of generating more data at targeted points that could be useful for a broader model comparison. The Priority Level System10 emphasized the importance of the warm dense matter regime, which is relevant to at least the initial stages of most integrated HED experiments; is theoretically challenging due to its combination of thermal effects, electron degeneracy, and strong ion coupling; and is accessible to computationally expensive models. This focus helped to concentrate the efforts of researchers who would only be able to submit data for a handful of cases. Cases in each Priority Level were chosen with consideration of the following criteria:

  • Priority Level 1: a minimal set of cases having longitudinal overlap with the first workshop and plasma conditions accessible to multi-atom methods based on density functional theory (DFT);

  • Priority Level 2: direct connections to experiments, additional cases having longitudinal overlap with the first workshop, high-value cases determined by data analysis, and conditions accessible to multi-atom DFT-based methods;

  • Priority Level 3: data on isochors at densities having longitudinal overlap with the cases of the first workshop and high-value densities determined by data analysis. This Priority Level was accessible to computationally rapid, more approximate models and provided a backdrop for the cases in Priority Levels 1 and 2.

Each case was assigned a Case ID to facilitate the collection and organization of data, structured as “X#” where the leading letter(s) corresponds to the element(s) and the number corresponds to density and temperature values as specified in Table I. While contributors were encouraged to submit results for as many cases and coefficients as possible, it was understood that not all models can generate every quantity and that many models are limited by computational cost. The Priority Level System was largely successful in focusing the efforts of the contributors, as shown in Fig. 1. Overall, Priority Level 1 cases received more submissions; they also concentrated results from computationally expensive multi-atom DFT-based codes, enabling direct comparisons among different implementations of first-principles models.

TABLE I.

Requested plasma conditions for the workshop. These cases represent Priority Level 1 and a subset of Priority Level 2. Note that for the mixture cases, the ionic number density n species is given in terms of each species, whereas the mass density ρ total is the total mass density of the mixture. As an example, in case CH1, the number density of each species is n H = n C = 4.63 × 10 22 cm−3, and ρ total = ( n H + 12 n C ) / N A = 1 g cm−3, where NA is Avogadro's number. Unless otherwise noted, we assume that T i = T e = T. Recall that in the Case ID column, the number that follows the element does not denote the Priority Level but instead refers to the temperature–density pair for the requested case.

Priority Level Case ID Element(s) n species (cm–3) ρ total (g cm–3) T (eV)
H1  5.98 × 10 23 
C1  5.01 × 10 23  10 
CH1  CH  4.63 × 10 22 
Al1  Al  6.03 × 10 22  2.7 
Cu1  Cu  8.49 × 10 22  8.96 
HCu1  HCu  1.68 × 10 22  1.8 
Be1  Be  1.23 × 10 23  1.84  4.4 
CH2  CH  4.16 × 10 22  0.9  7.8 
Au1  Au  5.91 × 10 22  19.32  10 
H3  5.98 × 10 24  10  20 
Priority Level Case ID Element(s) n species (cm–3) ρ total (g cm–3) T (eV)
H1  5.98 × 10 23 
C1  5.01 × 10 23  10 
CH1  CH  4.63 × 10 22 
Al1  Al  6.03 × 10 22  2.7 
Cu1  Cu  8.49 × 10 22  8.96 
HCu1  HCu  1.68 × 10 22  1.8 
Be1  Be  1.23 × 10 23  1.84  4.4 
CH2  CH  4.16 × 10 22  0.9  7.8 
Au1  Au  5.91 × 10 22  19.32  10 
H3  5.98 × 10 24  10  20 
FIG. 1.

Summary of the quantity of received data for the workshop. Each entry displays the number of received data for different observables over the total number of received submissions for each case (see Table I). For example, out of the 17 total submissions for case H1, 12 included data on the radial distribution function, g(r), and 4 included data on the ionic thermal conductivity, κi. The entries with the darkest color show the cases with the largest percentage of data received. Overall, fewer data were received for ionic mixtures and the correlation function observables [i.e., Z(t) and C(t)]. Computing properties of ionic mixtures—in particular correlation functions—is typically challenging for many of the simulations methods described in Table II.

FIG. 1.

Summary of the quantity of received data for the workshop. Each entry displays the number of received data for different observables over the total number of received submissions for each case (see Table I). For example, out of the 17 total submissions for case H1, 12 included data on the radial distribution function, g(r), and 4 included data on the ionic thermal conductivity, κi. The entries with the darkest color show the cases with the largest percentage of data received. Overall, fewer data were received for ionic mixtures and the correlation function observables [i.e., Z(t) and C(t)]. Computing properties of ionic mixtures—in particular correlation functions—is typically challenging for many of the simulations methods described in Table II.

Close modal
Figure 2 shows where the cases fall in the space of dimensionless plasma parameters Γ, the Coulomb coupling parameter, and θ, the electron degeneracy parameter. The Coulomb coupling parameter is a common heuristic for estimating the validity of models that assume binary collisions between the ions in a plasma:
Γ = ( Z * ) 2 e 2 a ws T i ,
(6)
where Z * is the average ionization of the ions, e is the electron charge, a ws = ( 4 π n i / 3 ) 1 / 3 is the Wigner–Seitz radius corresponding to a uniform ion number density ni, and Ti is the temperature of the ions in energy units. When Γ 1, the plasma is considered “strongly coupled” and the ions cannot be modeled with binary collisions—one of the fundamental assumptions of the Boltzmann equation for classical plasmas.
FIG. 2.

Electron degeneracy θ and ion coupling Γ parameters of the Priority Level 1 and 2 cases reported here. The curves Al1i, C2i, and Cu2i correspond to Priority Level 3 cases. They denote the isochors corresponding to Priority Level 1 cases Al1, C1, and Cu1, respectively; the curve H1i corresponds to the Priority Level 3 case of H at 1.67 g cm−3 (rather than H1’s 1.0 g cm−3). The gray square roughly denotes the warm dense matter regime: a challenging regime for many models.

FIG. 2.

Electron degeneracy θ and ion coupling Γ parameters of the Priority Level 1 and 2 cases reported here. The curves Al1i, C2i, and Cu2i correspond to Priority Level 3 cases. They denote the isochors corresponding to Priority Level 1 cases Al1, C1, and Cu1, respectively; the curve H1i corresponds to the Priority Level 3 case of H at 1.67 g cm−3 (rather than H1’s 1.0 g cm−3). The gray square roughly denotes the warm dense matter regime: a challenging regime for many models.

Close modal
The electron degeneracy parameter quantifies the extent to which the electrons will be impacted by Fermi–Dirac statistics. Here, we define the electron degeneracy parameter as
θ = T e E F ,
(7)
where Te is the electron temperature (in energy units) and E F = 2 ( 3 π 2 n e ) 2 / 3 / 2 m e is the Fermi energy corresponding to a uniform electron number density ne. When θ 1, the electron species can be approximated with classical mechanics. While the electrons and ions in a plasma can have different temperatures, in this workshop we have assumed that T i = T e = T unless otherwise noted.

The cases in Table I all have fairly large values for the electron degeneracy parameter and moderately to strongly coupled ions. This parameter range is accessible to multi-atom DFT-based codes—the current gold standard for simulating most plasmas with degenerate electrons and strongly-coupled ions. The densities of almost all of the Priority Level 1 cases were included in Priority Level 3 cases for isochoric data, where parameterized and average-atom codes provided a backdrop of data with sufficient temperature resolution to identify important features like conductivity minima and melt transitions. The exception was the H1 case, where the closest corresponding Priority Level 3 case had a density of 1.67 g cm−3 (rather than H1’s 1.0 g cm−3) to facilitate comparisons to published fractional stopping values.11 

Transport properties are mediated by interactions among electrons and ions. Approximating these interactions results in model-dependent predictions for atomic-scale equilibrium properties. This includes the screening of partially ionized nuclear cores by bound electrons, the collective behavior of mobile (unbound) electrons, and the distribution of the nuclear cores (atoms or ions). In this section, we describe three categories of modeling approaches, loosely ordered by increasing computational cost: computationally inexpensive parameterized models that use approximations or external input to describe the electronic and ionic properties, quantum average-atom models that compute spherically averaged properties, and multi-atom DFT-based models including Kohn-Sham molecular dynamics and real-time time-dependent DFT. All of the codes used in this workshop along with corresponding contributors and model categorizations are listed in Table II; this table serves as a reference with information about approximations made in each of the simulation approaches (e.g., the exchange-correlation functional used) and pertinent references to each approach. This information helps to distinguish models from one another within the data submitted for this workshop. We also present comparisons of submitted data for the average ionization Z *, the electronic density of states (DOS), and the ionic radial distribution function g(r).

TABLE II.

Approaches used in this workshop to generate data. The table is organized by model type from lowest computational cost (analytic models) to greatest computational cost (TD-DFT-MD). Note that the organization according to computational cost is only across model types (analytic, KT, AA, DFT-MD, and TD-DFT-MD) and not within each type.

Contributor(s) Description (code name and version) Institution(s) Model type
G. Röpke  Virial expansion12,13  U. Rostock  Analytic 
L. Stanek  The Lee–More–Desjarlais model,14 Yukawa–Gibbs–Bogolyubov model,15 Yukawa viscosity model,16 and Stanton–Murillo transport model11,17  SNL  Analytic 
L. Babati, S. Baalrud, N. Shaffer  Mean-force kinetic theory18–20 using average-atom potentials21 with temperature-dependent LDA XC functional22 (Scout)  U. Michigan, LLE  KT 
N. Shaffer  quantum Landau–Fokker–Planck kinetic theory with average-atom mean-force electron cross sections23 (qLFP)  LLE  KT 
G. Faussurier  Quantum average-atom model24–27 (QAAM)  CEA  AA 
S. Hansen  Quantum average-atom model with LDA XC functional28–30 (Bemuze)  SNL  AA 
G. Petrov  Average-atom calculation with Dirac exchange31 (AAM-NRL 4)  NRL  AA 
L. Silvestri, Z. Johnson, M. Murillo, G. Petrov  Molecular dynamics simulation with input from TF-Dirac-von Weizsäcker average-atom model and integral equation theory32 (Sarkas33 MSU, NRL  MD 
L. Stanek, S. Hansen  Molecular dynamics simulations with interatomic potential from an average-atom calculation (LAMMPS34 SNL  MD 
M. Bethkenhagen, M. French, R. Redmer, M. Schörner  Quantum molecular dynamics simulation (VASP35–38 5.4.4)  LULI, U. Rostock  DFT-MD 
A. Blanchet, V. Recoules, F. Soubiran, M. Tacu  Quantum molecular dynamics simulation39 (Abinit40–42 9.7.4)  CEA  DFT-MD 
R. Clay, K. Cochrane, A. Dumi, M. Lentz, C. Melton, J. Townsend  Quantum molecular dynamics simulation (VASP35–38 6.3.2)  SNL  DFT-MD 
S. Hu, V. Karasiev  Quantum molecular dynamics simulation with temperature-dependent TSCANL XC functional in combination with thermal hybrid KDT0 for the Kubo–Greenwood calculation for selected cases (VASP35–38 6.2/5.4.4)  LLE  DFT-MD 
V. Sharma, L. Collins, A. White  Quantum molecular dynamics simulations with PBE XC functional (SHRED43 LANL  DFT-MD 
P. Suryanarayana, S. Kumar  Molecular dynamics using on-the-fly machine learning force field with DFT simulation using LDA XC functional (SPARC44–47 Georgia Tech.  DFT-MD 
A. Kononov, A. Baczewski  TD-DFT simulations with LDA XC functional and bare Coulomb or PAW potentials (VASP35–38 5.4.4 extension48,49 SNL  TD-DFT-MD 
K. Nichols, S. Hu  TD-DFT simulations with PBE XC functional and HGH pseudopotentials (SHRED43 LLE  TD-DFT-MD 
A. White  Mixed stochastic-deterministic TD-DFT simulations with PBE XC functional and bare Coulomb potentials (SHRED43 LANL  TD-DFT-MD 
Contributor(s) Description (code name and version) Institution(s) Model type
G. Röpke  Virial expansion12,13  U. Rostock  Analytic 
L. Stanek  The Lee–More–Desjarlais model,14 Yukawa–Gibbs–Bogolyubov model,15 Yukawa viscosity model,16 and Stanton–Murillo transport model11,17  SNL  Analytic 
L. Babati, S. Baalrud, N. Shaffer  Mean-force kinetic theory18–20 using average-atom potentials21 with temperature-dependent LDA XC functional22 (Scout)  U. Michigan, LLE  KT 
N. Shaffer  quantum Landau–Fokker–Planck kinetic theory with average-atom mean-force electron cross sections23 (qLFP)  LLE  KT 
G. Faussurier  Quantum average-atom model24–27 (QAAM)  CEA  AA 
S. Hansen  Quantum average-atom model with LDA XC functional28–30 (Bemuze)  SNL  AA 
G. Petrov  Average-atom calculation with Dirac exchange31 (AAM-NRL 4)  NRL  AA 
L. Silvestri, Z. Johnson, M. Murillo, G. Petrov  Molecular dynamics simulation with input from TF-Dirac-von Weizsäcker average-atom model and integral equation theory32 (Sarkas33 MSU, NRL  MD 
L. Stanek, S. Hansen  Molecular dynamics simulations with interatomic potential from an average-atom calculation (LAMMPS34 SNL  MD 
M. Bethkenhagen, M. French, R. Redmer, M. Schörner  Quantum molecular dynamics simulation (VASP35–38 5.4.4)  LULI, U. Rostock  DFT-MD 
A. Blanchet, V. Recoules, F. Soubiran, M. Tacu  Quantum molecular dynamics simulation39 (Abinit40–42 9.7.4)  CEA  DFT-MD 
R. Clay, K. Cochrane, A. Dumi, M. Lentz, C. Melton, J. Townsend  Quantum molecular dynamics simulation (VASP35–38 6.3.2)  SNL  DFT-MD 
S. Hu, V. Karasiev  Quantum molecular dynamics simulation with temperature-dependent TSCANL XC functional in combination with thermal hybrid KDT0 for the Kubo–Greenwood calculation for selected cases (VASP35–38 6.2/5.4.4)  LLE  DFT-MD 
V. Sharma, L. Collins, A. White  Quantum molecular dynamics simulations with PBE XC functional (SHRED43 LANL  DFT-MD 
P. Suryanarayana, S. Kumar  Molecular dynamics using on-the-fly machine learning force field with DFT simulation using LDA XC functional (SPARC44–47 Georgia Tech.  DFT-MD 
A. Kononov, A. Baczewski  TD-DFT simulations with LDA XC functional and bare Coulomb or PAW potentials (VASP35–38 5.4.4 extension48,49 SNL  TD-DFT-MD 
K. Nichols, S. Hu  TD-DFT simulations with PBE XC functional and HGH pseudopotentials (SHRED43 LLE  TD-DFT-MD 
A. White  Mixed stochastic-deterministic TD-DFT simulations with PBE XC functional and bare Coulomb potentials (SHRED43 LANL  TD-DFT-MD 

Many hydrodynamic codes rely on fast, inline evaluations of transport coefficients parameterized by the average ionization Z *. Z * constrains the effective screening of the nuclear charge from highly localized (bound) electrons and it is critical for highly efficient models of partially ionized plasmas, though it is not uniquely defined. Here, we provide a backdrop for more detailed models using the Lee–More–Desjarlais model (labeled as LMD)14,50 for electronic transport coefficients as well as a generalized approach to transport described in Refs. 11 and 17 (labeled as SMT) for both electronic11 and ionic transport coefficients.17 Two additional models for the ion shear viscosity were included as a backdrop: the Yukawa viscosity model (YVM)16 and a model based on a Yukawa plasma that leverages the Gibbs-Bogolyubov inequality (YGBI).15 As part of this workshop the YVM fit has been improved to accurately span hotter temperatures relevant to inertial confinement fusion, and similarly the electronic SMT model has been improved to capture dispersion effects.32 All aforementioned models use Z * values from a fit51 to the Thomas–Fermi (TF) value of Z * based on a fluid approximation to the response of electrons to a central potential embedded in a plasma—a precursor to modern DFT models. The TF Z * is given by the gray lines in Fig. 3. Some additional semi-analytic models for plasma transport coefficients can be found in Refs. 52–55.

FIG. 3.

Average ionization Z * for some of the single-species cases detailed in Table I. Horizontal lines indicate the bare ion charge for reference. Because Z * can be defined in multiple ways,56 there is significant variation across models and even within the same type of model: the quantum-AA models shown here use different definitions for Z *.

FIG. 3.

Average ionization Z * for some of the single-species cases detailed in Table I. Horizontal lines indicate the bare ion charge for reference. Because Z * can be defined in multiple ways,56 there is significant variation across models and even within the same type of model: the quantum-AA models shown here use different definitions for Z *.

Close modal

A second class of parameterized models include those based on kinetic theory (KT) and pair-potential molecular dynamics57 (MD) models that use input from external sources to inform more sophisticated treatments of screening and transport properties. In this workshop, the KT models included a quantum Landau–Fokker–Planck approach to electronic transport properties that uses electron–ion collision cross sections as input23 and mean-force kinetic theory18–20 for ionic transport coefficients that uses ion radial distribution functions as input. The MD codes33,34 for generating data for ionic transport coefficients and correlation functions use interatomic potentials as input. In this workshop, the input quantities are derived either from analytic pair potentials (e.g., Yukawa17,58,59), pair potentials obtained from average-atom calculations,57 or pair potentials from semi-classical methods.32 These models are slower than models used for inline evaluation, with few-minute runtimes for KT and tens-of-minutes runtimes for MD, but they are expected to provide higher fidelity data than fully parameterized models. While these parameterized models can offer a large reduction in computational cost compared to the models discussed in Sec. IV C, their efficacy is heavily dependent on the quality of their inputs and the flexibility of their parametric form.57 

Average-atom models21,27,28,60,61 describe the electronic structure surrounding a single, averaged ion suspended in jellium. All of the average-atom models contributed to this workshop use Kohn–Sham orbitals to compute the electronic structure; we will use the term “quantum-AA,” or simply AA, to denote average-atom models of this type. Following the pioneering work of Liberman,62 these all-electron, fully quantum models find a converged electron–ion potential (including approximate exchange and correlation effects) that supports a self-consistent set of electronic orbitals. These orbitals provide information on the radial electron density and the energy-dependent electronic density of states (DOS). The densities of states from two independent AA models are shown in Fig. 4.

FIG. 4.

Electronic densities of states for some of the single-species cases in Table I. Bound states are labeled for each applicable element. All models have been shifted (to the best of our ability) to the same energy scales, as determined by the binding energies of negative-energy states and the reported chemical potentials (or Fermi energies). Two independent AA models (one model reported both bound and continuum electrons and another reported only the continuum electrons) have virtually identical DsOS, which reproduce the general features of the multi-atom DFT DsOS quite well except for the carbon case. The positive-energy feature in the DFT-MD model of Au near E = 120 eV is believed to be an artifact of the pseudopotential. Multi-atom models tend to predict broader features than AA models due to both variations in the local fields of different atomic centers and, in some implementations, imposed broadening. The positive-energy DsOS from multi-center models artificially drop to zero at different energies, depending on how many continuum states the particular calculation included.

FIG. 4.

Electronic densities of states for some of the single-species cases in Table I. Bound states are labeled for each applicable element. All models have been shifted (to the best of our ability) to the same energy scales, as determined by the binding energies of negative-energy states and the reported chemical potentials (or Fermi energies). Two independent AA models (one model reported both bound and continuum electrons and another reported only the continuum electrons) have virtually identical DsOS, which reproduce the general features of the multi-atom DFT DsOS quite well except for the carbon case. The positive-energy feature in the DFT-MD model of Au near E = 120 eV is believed to be an artifact of the pseudopotential. Multi-atom models tend to predict broader features than AA models due to both variations in the local fields of different atomic centers and, in some implementations, imposed broadening. The positive-energy DsOS from multi-center models artificially drop to zero at different energies, depending on how many continuum states the particular calculation included.

Close modal

Several definitions of Z *56 can be extracted from AA models and used to parameterize the methods described in Sec. IV A. In Fig. 3, for example, some of the AA models define Z * to include all occupied positive-energy orbitals while others count only the plane wave (ideal) portion of occupied orbitals as free electrons. The first definition can result in discontinuities in Z * as orbitals move from negative (bound) to positive energies, while the second definition gives smoother behavior more similar to the TF fit. It is notable that while several independent quantum-AA models give widely different predictions for Z *, their underlying electronic densities of state are virtually identical. The choice of Z * also influences AA-model predictions for ion–ion interaction potentials, which can be used to determine (spherically symmetric) ionic radial distribution functions63 and ion–ion transport coefficients.57 

While AA models are relatively efficient, with runtimes of several minutes, and they are expected to be more accurate than Z *-parameterized models for plasmas and liquids, they cannot account for bonding or crystalline effects that are important at low temperatures. They also do not calculate transport properties directly without appealing to additional models. For example, from their self-consistent set of electronic and ionic properties, AA models can derive electron–ion collision rates through the Ziman equation27,64,65 that inform electronic transport properties. Coupled with a model for stopping numbers or dielectric functions, AA models can also calculate stopping powers.27,30

The final category of models in this workshop are based on a DFT description of a multi-atom system, including Kohn–Sham molecular dynamics49,66–74 (DFT-MD, also known as quantum molecular dynamics or QMD) and real-time time-dependent DFT75 (TD-DFT). These first-principles models self-consistently solve for a three-dimensional electron density interacting with an ensemble of ions. DFT-MD then simulates ps-scale ion dynamics with interatomic forces determined by the equilibrium electronic structure at each time step. This separation of electronic and ionic time scales is known as the Born–Oppenheimer approximation. In contrast, TD-DFT models fs-scale dynamics of electronic excitations in response to an external perturbation. To capture ionic disorder, TD-DFT simulations in this workshop began from equilibrated structures obtained from DFT-MD and thus we denote this model as TD-DFT-MD.

Within DFT-MD, ionic transport properties are extracted from integrals of autocorrelation functions. Electronic transport properties are extracted from the Kubo–Greenwood76,77 formalism, with dynamic properties averaged over ionic configurations and extrapolated to zero-frequency for DC limits. A Kubo relation has also been proposed for electron–ion temperature relaxation rates from DFT-MD,78,79 but was not applied in this workshop. While TD-DFT-MD can offer an alternative framework for accessing electronic transport properties,80,81 contributions to this workshop only use this method to predict electronic stopping powers by simulating electron dynamics as an alpha particle traverses the plasma. The stopping power is determined by the average force exerted on the alpha particle by the electronic system, or equivalently, the average rate at which the alpha particle deposits energy into electronic excitations.82 

Both DFT-MD and TD-DFT-MD are fundamentally characterized by an approximate choice of exchange and correlation (XC) potential to describe the mean-field electron–electron interaction, an approximation that also enters AA models.12 Additionally, these methods typically employ pseudopotentials to avoid large computational costs associated with highly oscillatory wavefunctions near ion cores. Nonetheless, the multi-atom and three-dimensional nature of DFT-MD and TD-DFT-MD allows these methods to describe interatomic bonding and anisotropic behavior. They also treat bound and free electrons on the same footing without relying on a state partitioning or Z * definition. In practice, rigorous application of these DFT-based models requires preliminary calculations to ensure convergence with respect to numerical parameters and quantify sensitivity to model choices like XC potentials and pseudopotentials.

Systematic improvements of XC functionals have been performed along the so-called Jacob's ladder.83 Its lowest rung is the local density approximation (LDA) followed by the generalized gradient approximation (GGA). Each higher rung of the ladder such as meta-GGAs and hybrid functionals represents an improved approximation for the total energy calculated within DFT.

These DFT-MD and TD-DFT-MD models represent the state of the art for HED materials properties calculations. While these models still have difficult-to-quantify systematic errors, they are computationally viable over a wide range of conditions from ambient materials to warm dense matter, and strong empirical evidence indicates that their accuracy suffices for many relevant applications. Their predictions for ionic radial distribution functions generally agree well with directly observable x-ray diffraction measurements; we illustrate radial distribution functions, g(r), in Fig. 5 for some of the cases of Table I. However, multi-atom DFT calculations are computationally expensive, consuming many hours—or weeks—of CPU time, and they can become intractable or require further approximation for high temperatures and many-electron ions. To reduce computational expense, the number of ions may be restricted (potentially leading to finite-size effects57), the electrons within ion cores may be included in psuedopotentials (approximating their behavior), and/or sub-cubic scaling algorithms that rely on sparsity or stochastic sampling may be applied.43,45,84–86

FIG. 5.

Ionic radial distribution function, g(r), for some of the single-species cases in Table I. g(r)s from AA models are obtained by solving the quantum Ornstein–Zernike equations and employing the hyper-netted chain approximation while g(r)s obtained from DFT-MD and MD models are obtained directly from ensemble-averaged particle positions.

FIG. 5.

Ionic radial distribution function, g(r), for some of the single-species cases in Table I. g(r)s from AA models are obtained by solving the quantum Ornstein–Zernike equations and employing the hyper-netted chain approximation while g(r)s obtained from DFT-MD and MD models are obtained directly from ensemble-averaged particle positions.

Close modal

In this section, we describe how transport coefficients are computed within the atomic-scale modeling approaches described above and show comparisons of the results submitted to this workshop. For clarity and discussion of the results, we divide this section into four parts: electronic transport coefficients (electrical and thermal conductivities; Sec. V A), ionic transport coefficients (shear viscosity and thermal diffusivity; Sec. V B), electron–ion temperature relaxation rate (Sec. V C), and stopping power (Sec. V D).

In the warm dense regime, electronic transport coefficients—such as DC electrical conductivity and the electronic thermal conductivity—are governed by collisions between mobile electrons and collisions between mobile electrons and static ions. In classical and fully ionized plasmas, collisions are often treated using Coulomb logarithms based on minimum and maximum approach distances. For the partially ionized and partially degenerate plasma cases considered in this workshop, the calculations require a more complete treatment of collisions.

In general, the DC electrical conductivity can be computed via
σ = L 11 ,
(8)
where Lij denote the Onsager coefficients,87–89 which are related to detailed balance among collisional rates. The electronic thermal conductivity is also represented by Onsager coefficients and has the form
κ e = 1 T ( L 22 L 12 2 L 11 ) .
(9)

In practice, the codes contributing to this workshop treat electronic transport properties in quite distinct ways. Most of the codes designated as KT in Table II use an effective Boltzmann approach that requires an effective interaction potential; the potentials may be specified generically11,17 or computed on a case-by-case basis.18,90,91 Once an effective potential is obtained, the KT codes numerically evaluate collision integrals—avoiding the need for a Coulomb logarithm. The KT codes then utilize the Chapman–Enskog approach92,93 to determine the relevant transport coefficients. A primary source of uncertainty in KT codes of this kind results from the choice of the effective interaction potential.

Most of the codes designated as AA use modifications of the Ziman approach27,64,65 to calculate electron–ion collision rates, integrating over both ionic and electronic structure. The DFT-MD codes obtain the DC conductivity by extrapolating the frequency-dependent conductivity obtained by Kubo76 and Greenwood77 relations to the zero frequency limit.94,95 This extrapolation is often a significant source of uncertainty for evaluating the DC conductivity from multi-atom DFT-MD-based models. A description of the extrapolation procedure and other considerations for computing the Onsager coefficients, and thus the electronic and thermal conductivity, from multi-atom DFT-MD codes can be found in Refs. 96 and 97. Another source of uncertainty for the evaluation of the electronic transport coefficients is the mean-field treatment of electron–electron interactions.12,96 The method of DFT-MD plus Kubo–Greenwood has been used to calculate thermal conductivities of materials relevant for inertial confinement fusion (e.g., D and CH) in a wide range of densities and temperatures.95,98–100 The results are generally larger than what traditional plasma models predict in the WDM regime, which was aligned with AA prediction and recent experiment.101 A thorough review on this is given by Ref. 102 in this special issue.

Comparisons of the submitted data for the DC electrical conductivity σ and electronic thermal conductivity κe are given in Figs. 6 and 7, respectively; the corresponding data are provided in Tables V and VI. Of the submitted data in Priority Level 1 and 2 cases, the greatest difference appears in the cases of Al1 and Cu1. The maximum difference between all models for the DC electrical conductivity is roughly a factor of seven for case Cu1. This difference decreases if we consider the differences within a single model type. For a single model type, the maximum difference in the DC electrical conductivity is roughly a factor of two—which occurs between the AA models of case Cu1. For the electron thermal conductivity, the maximum difference between all models—which occurs for case Al1—is roughly one order of magnitude; the maximum difference within a single model type is roughly a factor of five which occurs between DFT-MD models for case Al1. These differences are much larger than those observed in the electronic densities of states or even Z * values (among models reliant on a Z * definition), which should together largely determine the electron–ion interaction.

FIG. 6.

Electrical conductivities for some of the single-species cases in Table I. For one of the AA models (upward pointing triangles), error bars are given indicating the possible range for different choices of ion structure and average ionization Z *. Another AA model (downward pointing triangles) shows discontinuities from unconverged cases. The DFT-MD results are concentrated at the cases defined in Table I.

FIG. 6.

Electrical conductivities for some of the single-species cases in Table I. For one of the AA models (upward pointing triangles), error bars are given indicating the possible range for different choices of ion structure and average ionization Z *. Another AA model (downward pointing triangles) shows discontinuities from unconverged cases. The DFT-MD results are concentrated at the cases defined in Table I.

Close modal
FIG. 7.

Same as Fig. 6, but for electron thermal conductivities.

FIG. 7.

Same as Fig. 6, but for electron thermal conductivities.

Close modal
TABLE V.

Computed transport coefficients for plasma conditions detailed in Table I. For brevity, entries in the Case ID/Submitter column have been labeled by a single submitter determined by how the data were received; please refer to Table II for a full list of the collaborations and institutions. Exchange-correlation functionals used include LDA,118 PBE,119 and Tr2SCANL.120 Kubo–Greenwood calculations marked with (*) were performed with employment of the thermal hybrid KDT0 XC functional121 on top of the Tr2SCANL snapshots (see further details in Ref. 122).

Case ID/Submitter σ ( 1 Ω cm 10 4 ) κ e ( erg s cm K 10 7 ) η ( g cm s 10 3 ) κ i ( erg s cm K 10 7 ) g ( erg cm 3 K 10 20 ) XC Model
H1 (1 g cm–3, 2 eV)               
L. Babati  ⋯  ⋯  7.88  0.24  ⋯  LDA  KT 
G. Faussurier  1.47  8.03  ⋯  ⋯  11.81  LDA  AA 
M. French  1.29 ± 0.03  7.5 ± 0.15  ⋯  ⋯  ⋯  PBE  DFT-MD 
M. French  1.05 ± 0.02  6.4 ± 0.13  ⋯  ⋯  ⋯  HSE  DFT-MD 
S. Hansen  1 0.42 + 1.50  7 2.14 + 7.69  ⋯  ⋯  10.23 ± 1.12  LDA  AA 
S. Hu  1.4 ± 0.10  7.5 ± 0.31  11.48  ⋯  ⋯  PBE  DFT-MD 
S. Hu  1.25 ± 0.08  6.7 ± 0.26  13.36  ⋯  ⋯  TSCANL  DFT-MD 
N. Shaffer  2.01  9.61  ⋯  ⋯  11.17  LDA  KT 
V. Sharma  1.4 ± 0.15  7.2 ± 0.41  ⋯  ⋯  ⋯  PBE  DFT-MD 
L. Silvestri  ⋯  ⋯  7.0 ± 0.03  0.845 ± 0.004  ⋯  LDA  MD 
F. Soubiran  1.31 ± 0.04  7.58 ± 0.01  11 ± 4.00  ⋯  ⋯  PBE  DFT-MD 
L. Stanek  ⋯  ⋯  12 ± 3.18  0.5 ± 0.11  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  12 ± 0.91  0.168 ± 0.002  ⋯  LDA  DFT-MD 
J. Townsend  1.4 ± 0.004  8.05 ± 0.01  8 ± 1.64  ⋯  ⋯  LDA  DFT-MD 
C1 (10 g cm–3, 2 eV)               
L. Babati  ⋯  ⋯  5.80  0.015  ⋯  LDA  KT 
M. Bethkenhagen  1.63 ± 0.03  9 ± 0.18  ⋯  ⋯  ⋯  PBE  DFT-MD 
G. Faussurier  1.14  6.44  ⋯  ⋯  10.55  LDA  AA 
S. Hansen  1 0.36 + 1.06  6 2.02 + 5.99  ⋯  ⋯  8 ± 3.52  LDA  AA 
V. Karasiev  1.69 ± 0.06  9.5 ± 0.16  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Karasiev  1.04 ± 0.06  6.9 ± 0.28  ⋯  ⋯  ⋯  Tr2SCANL*  DFT-MD 
C. Melton  1.60 ± 0.01  8.80  28.2 ± 0.3  ⋯  ⋯  LDA  DFT-MD 
N. Shaffer  1.87  9.39  ⋯  ⋯  11.40  LDA  KT 
V. Sharma  1.26 ± 0.06  7.7 ± 0.24  ⋯  ⋯  ⋯  PBE  DFT-MD 
L. Silvestri  ⋯  ⋯  104 ± 0.5  0.942 ± 0.005  ⋯  LDA  MD 
F. Soubiran  1.58 ± 0.04  8.8 ± 0.15  32 ± 21.00  ⋯  ⋯  PBE  DFT-MD 
L. Stanek  ⋯  ⋯  64 ± 14.57  0.14 ± 0.04  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  29 ± 1.61  1.02 ± 0.02  ⋯  LDA  DFT-MD 
CH1 (1 g cm–3, 2 eV)               
M. Bethkenhagen  0.174 ± 0.003  0.71 ± 0.02  ⋯  ⋯  ⋯  PBE  DFT-MD 
R. Clay  0.168 ± 0.004  0.72 ± 0.01  6.5 ± 0.9  ⋯  ⋯  LDA  DFT-MD 
V. Karasiev  0.19 ± 0.03  0.80 ± 0.06  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Karasiev  0.14 ± 0.02  0.74 ± 0.05  ⋯  ⋯  ⋯  TSCANL*  DFT-MD 
V. Sharma  0.188 ± 0.009  0.78 ± 0.02  ⋯  ⋯  ⋯  PBE  DFT-MD 
F. Soubiran  0.18 ± 0.02  0.67 ± 0.02  8 ± 3.2  ⋯  ⋯  PBE  DFT-MD 
Al1 (2.7 g cm–3, 1 eV)               
L. Babati  ⋯  ⋯  1.51  0.0018  ⋯  LDA  KT 
A. Dumi  2.461 ± 0.004  1.45 ± 0.02  4 ± 0.82  ⋯  ⋯  LDA  DFT-MD 
G. Faussurier  4.18  11.5  ⋯  ⋯  0.0218  LDA  AA 
S. Hansen  4 1.30 + 2.25  7 2.36 + 4.06  ⋯  ⋯  0.016 ± 0.002  LDA  AA 
V. Karasiev  2.03 ± 0.05  5.69 ± 0.15  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Karasiev  1.38 ± 0.08  4.75 ± 0.22  ⋯  ⋯  ⋯  TSCANL*  DFT-MD 
G. Petrov  5.65  15.3  ⋯  ⋯  0.0101  LDA  AA 
M. Schörner  2.6 ± 0.13  7.33 ± 0.53  ⋯  ⋯  ⋯  PBE  DFT-MD 
N. Shaffer  2.15  4.09  ⋯  ⋯  0.036  LDA  KT 
V. Sharma  2.38 ± 0.07  6.46 ± 0.19  ⋯  ⋯  ⋯  PBE  DFT-MD 
L. Silvestri  ⋯  ⋯  7.25 ± 0.04  0.0436 ± 0.0002  ⋯  LDA  MD 
F. Soubiran  2.44 ± 0.03  6.8 ± 0.01  8 ± 4.40  ⋯  ⋯  GGA  DFT-MD 
L. Stanek  ⋯  ⋯  8 ± 3.13  0.014 ± 0.003  ⋯  LDA  MD 
Case ID/Submitter σ ( 1 Ω cm 10 4 ) κ e ( erg s cm K 10 7 ) η ( g cm s 10 3 ) κ i ( erg s cm K 10 7 ) g ( erg cm 3 K 10 20 ) XC Model
H1 (1 g cm–3, 2 eV)               
L. Babati  ⋯  ⋯  7.88  0.24  ⋯  LDA  KT 
G. Faussurier  1.47  8.03  ⋯  ⋯  11.81  LDA  AA 
M. French  1.29 ± 0.03  7.5 ± 0.15  ⋯  ⋯  ⋯  PBE  DFT-MD 
M. French  1.05 ± 0.02  6.4 ± 0.13  ⋯  ⋯  ⋯  HSE  DFT-MD 
S. Hansen  1 0.42 + 1.50  7 2.14 + 7.69  ⋯  ⋯  10.23 ± 1.12  LDA  AA 
S. Hu  1.4 ± 0.10  7.5 ± 0.31  11.48  ⋯  ⋯  PBE  DFT-MD 
S. Hu  1.25 ± 0.08  6.7 ± 0.26  13.36  ⋯  ⋯  TSCANL  DFT-MD 
N. Shaffer  2.01  9.61  ⋯  ⋯  11.17  LDA  KT 
V. Sharma  1.4 ± 0.15  7.2 ± 0.41  ⋯  ⋯  ⋯  PBE  DFT-MD 
L. Silvestri  ⋯  ⋯  7.0 ± 0.03  0.845 ± 0.004  ⋯  LDA  MD 
F. Soubiran  1.31 ± 0.04  7.58 ± 0.01  11 ± 4.00  ⋯  ⋯  PBE  DFT-MD 
L. Stanek  ⋯  ⋯  12 ± 3.18  0.5 ± 0.11  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  12 ± 0.91  0.168 ± 0.002  ⋯  LDA  DFT-MD 
J. Townsend  1.4 ± 0.004  8.05 ± 0.01  8 ± 1.64  ⋯  ⋯  LDA  DFT-MD 
C1 (10 g cm–3, 2 eV)               
L. Babati  ⋯  ⋯  5.80  0.015  ⋯  LDA  KT 
M. Bethkenhagen  1.63 ± 0.03  9 ± 0.18  ⋯  ⋯  ⋯  PBE  DFT-MD 
G. Faussurier  1.14  6.44  ⋯  ⋯  10.55  LDA  AA 
S. Hansen  1 0.36 + 1.06  6 2.02 + 5.99  ⋯  ⋯  8 ± 3.52  LDA  AA 
V. Karasiev  1.69 ± 0.06  9.5 ± 0.16  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Karasiev  1.04 ± 0.06  6.9 ± 0.28  ⋯  ⋯  ⋯  Tr2SCANL*  DFT-MD 
C. Melton  1.60 ± 0.01  8.80  28.2 ± 0.3  ⋯  ⋯  LDA  DFT-MD 
N. Shaffer  1.87  9.39  ⋯  ⋯  11.40  LDA  KT 
V. Sharma  1.26 ± 0.06  7.7 ± 0.24  ⋯  ⋯  ⋯  PBE  DFT-MD 
L. Silvestri  ⋯  ⋯  104 ± 0.5  0.942 ± 0.005  ⋯  LDA  MD 
F. Soubiran  1.58 ± 0.04  8.8 ± 0.15  32 ± 21.00  ⋯  ⋯  PBE  DFT-MD 
L. Stanek  ⋯  ⋯  64 ± 14.57  0.14 ± 0.04  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  29 ± 1.61  1.02 ± 0.02  ⋯  LDA  DFT-MD 
CH1 (1 g cm–3, 2 eV)               
M. Bethkenhagen  0.174 ± 0.003  0.71 ± 0.02  ⋯  ⋯  ⋯  PBE  DFT-MD 
R. Clay  0.168 ± 0.004  0.72 ± 0.01  6.5 ± 0.9  ⋯  ⋯  LDA  DFT-MD 
V. Karasiev  0.19 ± 0.03  0.80 ± 0.06  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Karasiev  0.14 ± 0.02  0.74 ± 0.05  ⋯  ⋯  ⋯  TSCANL*  DFT-MD 
V. Sharma  0.188 ± 0.009  0.78 ± 0.02  ⋯  ⋯  ⋯  PBE  DFT-MD 
F. Soubiran  0.18 ± 0.02  0.67 ± 0.02  8 ± 3.2  ⋯  ⋯  PBE  DFT-MD 
Al1 (2.7 g cm–3, 1 eV)               
L. Babati  ⋯  ⋯  1.51  0.0018  ⋯  LDA  KT 
A. Dumi  2.461 ± 0.004  1.45 ± 0.02  4 ± 0.82  ⋯  ⋯  LDA  DFT-MD 
G. Faussurier  4.18  11.5  ⋯  ⋯  0.0218  LDA  AA 
S. Hansen  4 1.30 + 2.25  7 2.36 + 4.06  ⋯  ⋯  0.016 ± 0.002  LDA  AA 
V. Karasiev  2.03 ± 0.05  5.69 ± 0.15  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Karasiev  1.38 ± 0.08  4.75 ± 0.22  ⋯  ⋯  ⋯  TSCANL*  DFT-MD 
G. Petrov  5.65  15.3  ⋯  ⋯  0.0101  LDA  AA 
M. Schörner  2.6 ± 0.13  7.33 ± 0.53  ⋯  ⋯  ⋯  PBE  DFT-MD 
N. Shaffer  2.15  4.09  ⋯  ⋯  0.036  LDA  KT 
V. Sharma  2.38 ± 0.07  6.46 ± 0.19  ⋯  ⋯  ⋯  PBE  DFT-MD 
L. Silvestri  ⋯  ⋯  7.25 ± 0.04  0.0436 ± 0.0002  ⋯  LDA  MD 
F. Soubiran  2.44 ± 0.03  6.8 ± 0.01  8 ± 4.40  ⋯  ⋯  GGA  DFT-MD 
L. Stanek  ⋯  ⋯  8 ± 3.13  0.014 ± 0.003  ⋯  LDA  MD 
TABLE VI.

Computed transport coefficients for plasma conditions detailed in Table I. For brevity, entries in the Case ID/Submitter column have been labeled by a single submitter determined by how the data were received; please refer to Table II for a full list of the collaborations and institutions. Note that (ext)-DFT-MD denotes that extended DFT-MD was used—see Ref. 42 for details. Also note that (mix)-DFT-MD denotes that mixed stochastic-deterministic DFT-MD was used—see Ref. 43.

Case ID/Submitter σ ( 1 Ω cm 10 4 ) κ e ( erg s cm K 10 7 ) η ( g cm s 10 3 ) κ i ( erg s cm K 10 7 ) g ( erg cm 3 K 10 20 ) XC Model
Cu1 (8.96 g cm−3, 1 eV)               
K. Cochrane  4.25  14 ± 0.5  ⋯  ⋯  PBE  DFT-MD 
G. Faussurier  2.75  7.62  ⋯  ⋯  0.0133  LDA  AA 
S. Hansen  6 2.70 + 32.20  10 4.68 + 55.93  ⋯  ⋯  0.01 ± 0.015  LDA  AA 
S. Hu  0.90 ± 0.02  2.50 ± 0.03  16.87  ⋯  ⋯  PBE  DFT-MD 
S. Hu  1.20 ± 0.01  3.75 ± 0.08  16.42  ⋯  ⋯  TSCANL  DFT-MD 
G. Petrov  3.13  18.5  ⋯  ⋯  0.0116  LDA  AA 
F. Soubiran  1.80 ± 0.02  3.90 ± 0.03  19 ± 8.00  ⋯  ⋯  LDA  DFT-MD 
L. Stanek  ⋯  ⋯  19 ± 5.76  0.013 ± 0.003  ⋯  LDA  MD 
HCu1 (1.8 g cm–3, 1 eV)               
S. Hu  0.09 ± 0.008  0.332 ± 0.023  0.796  ⋯  ⋯  PBE  DFT-MD 
S. Hu  0.11 ± 0.004  0.463 ± 0.005  5.135  ⋯  ⋯  TSCANL  DFT-MD 
M. Lentz  0.09 ± 0.002  0.39 ± 0.007  1.8 ± 0.58  ⋯  ⋯  LDA  DFT-MD 
Be1 (1.84 g cm–3, 4.4 eV)               
L. Babati  ⋯  ⋯  8.81  0.031  ⋯  LDA  KT 
G. Faussurier  0.83  8.01  ⋯  ⋯  0.470  LDA  AA 
S. Hansen  0.6 0.04 + 0.23  5.55 0.34 + 2.16  ⋯  ⋯  0.4 ± 0.14  LDA  AA 
S. Hu  0.65 ± 0.02  7.2 ± 0.11  12.84  ⋯  ⋯  PBE  DFT-MD 
S. Hu  0.65 ± 0.03  7.4 ± 0.18  6.33  ⋯  ⋯  TSCANL  DFT-MD 
M. Schörner  0.66 ± 0.01  7.3 ± 0.17  ⋯  ⋯  ⋯  PBE  DFT-MD 
N. Shaffer  0.81  9.62  ⋯  ⋯  0.749  LDA  KT 
V. Sharma  0.58 ± 0.01  6.3 ± 0.11  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Sharma  0.57 ± 0.01  6.1 ± 0.27  ⋯  ⋯  ⋯  PBE  (mix)-DFT-MD 
L. Stanek  ⋯  ⋯  11 ± 2.21  0.06 ± 0.015  ⋯  LDA  MD 
CH2 (0.9 g cm–3, 7.8 eV)               
L. Babati  ⋯  ⋯  8.61  0.0257  ⋯  LDA  KT 
N. Shaffer  ⋯  ⋯  ⋯  ⋯  1.72  LDA  KT 
V. Sharma  0.23 ± 0.010  7.38 ± 0.24  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Sharma  0.23 ± 0.015  7.52 ± 0.69  ⋯  ⋯  ⋯  PBE  (mix)-DFT-MD 
F. Soubiran  0.21 ± 0.002  6.5 ± 0.1  15.00  ⋯  ⋯  PBE  (ext)-DFT-MD 
Au1 (19.32 g cm–3, 10 eV)               
L. Babati  ⋯  ⋯  24.98  0.0040  ⋯  LDA  KT 
G. Faussurier  1.07  22.8  ⋯  ⋯  0.026  LDA  AA 
S. Hansen  0.73 0.067 + 0.001  15.50 1.45 + 0.01  ⋯  ⋯  0.024 ± 0.0072  LDA  AA 
S. Hu  1.07 ± 0.03  26.2 ± 0.3  44 ± 1.00  ⋯  ⋯  TSCANL  DFT-MD 
V. Karasiev  0.99 ± 0.02  24.7 ± 0.3  ⋯  ⋯  ⋯  TSCANL  DFT-MD 
N. Shaffer  1.18  28.3  ⋯  ⋯  0.036  LDA  KT 
F. Soubiran  1.024 ± 0.001  27.34 ± 0.03  51 ± 1.00  ⋯  ⋯  LDA  (ext)-DFT-MD 
L. Stanek  ⋯  ⋯  42 ± 12.04  0.011 ± 0.003  ⋯  LDA  MD 
H3 (10 g cm–3, 20 eV)               
L. Babati  ⋯  ⋯  209.9  0.651  ⋯  LDA  KT 
G. Faussurier  23.8  1206.0  ⋯  ⋯  87.3  LDA  AA 
S. Hansen  24.7  730.9  ⋯  ⋯  ⋯  LDA  AA 
N. Shaffer  17.8  867  ⋯  ⋯  129  LDA  KT 
V. Sharma  12 ± 1.58  757 ± 80.4  ⋯  ⋯  ⋯  PBE  DFT-MD 
M. Bethkenhagen  14.8 ± 0.6  850.0 ± 34.0  ⋯  ⋯  ⋯  PBE  DFT-MD 
Case ID/Submitter σ ( 1 Ω cm 10 4 ) κ e ( erg s cm K 10 7 ) η ( g cm s 10 3 ) κ i ( erg s cm K 10 7 ) g ( erg cm 3 K 10 20 ) XC Model
Cu1 (8.96 g cm−3, 1 eV)               
K. Cochrane  4.25  14 ± 0.5  ⋯  ⋯  PBE  DFT-MD 
G. Faussurier  2.75  7.62  ⋯  ⋯  0.0133  LDA  AA 
S. Hansen  6 2.70 + 32.20  10 4.68 + 55.93  ⋯  ⋯  0.01 ± 0.015  LDA  AA 
S. Hu  0.90 ± 0.02  2.50 ± 0.03  16.87  ⋯  ⋯  PBE  DFT-MD 
S. Hu  1.20 ± 0.01  3.75 ± 0.08  16.42  ⋯  ⋯  TSCANL  DFT-MD 
G. Petrov  3.13  18.5  ⋯  ⋯  0.0116  LDA  AA 
F. Soubiran  1.80 ± 0.02  3.90 ± 0.03  19 ± 8.00  ⋯  ⋯  LDA  DFT-MD 
L. Stanek  ⋯  ⋯  19 ± 5.76  0.013 ± 0.003  ⋯  LDA  MD 
HCu1 (1.8 g cm–3, 1 eV)               
S. Hu  0.09 ± 0.008  0.332 ± 0.023  0.796  ⋯  ⋯  PBE  DFT-MD 
S. Hu  0.11 ± 0.004  0.463 ± 0.005  5.135  ⋯  ⋯  TSCANL  DFT-MD 
M. Lentz  0.09 ± 0.002  0.39 ± 0.007  1.8 ± 0.58  ⋯  ⋯  LDA  DFT-MD 
Be1 (1.84 g cm–3, 4.4 eV)               
L. Babati  ⋯  ⋯  8.81  0.031  ⋯  LDA  KT 
G. Faussurier  0.83  8.01  ⋯  ⋯  0.470  LDA  AA 
S. Hansen  0.6 0.04 + 0.23  5.55 0.34 + 2.16  ⋯  ⋯  0.4 ± 0.14  LDA  AA 
S. Hu  0.65 ± 0.02  7.2 ± 0.11  12.84  ⋯  ⋯  PBE  DFT-MD 
S. Hu  0.65 ± 0.03  7.4 ± 0.18  6.33  ⋯  ⋯  TSCANL  DFT-MD 
M. Schörner  0.66 ± 0.01  7.3 ± 0.17  ⋯  ⋯  ⋯  PBE  DFT-MD 
N. Shaffer  0.81  9.62  ⋯  ⋯  0.749  LDA  KT 
V. Sharma  0.58 ± 0.01  6.3 ± 0.11  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Sharma  0.57 ± 0.01  6.1 ± 0.27  ⋯  ⋯  ⋯  PBE  (mix)-DFT-MD 
L. Stanek  ⋯  ⋯  11 ± 2.21  0.06 ± 0.015  ⋯  LDA  MD 
CH2 (0.9 g cm–3, 7.8 eV)               
L. Babati  ⋯  ⋯  8.61  0.0257  ⋯  LDA  KT 
N. Shaffer  ⋯  ⋯  ⋯  ⋯  1.72  LDA  KT 
V. Sharma  0.23 ± 0.010  7.38 ± 0.24  ⋯  ⋯  ⋯  PBE  DFT-MD 
V. Sharma  0.23 ± 0.015  7.52 ± 0.69  ⋯  ⋯  ⋯  PBE  (mix)-DFT-MD 
F. Soubiran  0.21 ± 0.002  6.5 ± 0.1  15.00  ⋯  ⋯  PBE  (ext)-DFT-MD 
Au1 (19.32 g cm–3, 10 eV)               
L. Babati  ⋯  ⋯  24.98  0.0040  ⋯  LDA  KT 
G. Faussurier  1.07  22.8  ⋯  ⋯  0.026  LDA  AA 
S. Hansen  0.73 0.067 + 0.001  15.50 1.45 + 0.01  ⋯  ⋯  0.024 ± 0.0072  LDA  AA 
S. Hu  1.07 ± 0.03  26.2 ± 0.3  44 ± 1.00  ⋯  ⋯  TSCANL  DFT-MD 
V. Karasiev  0.99 ± 0.02  24.7 ± 0.3  ⋯  ⋯  ⋯  TSCANL  DFT-MD 
N. Shaffer  1.18  28.3  ⋯  ⋯  0.036  LDA  KT 
F. Soubiran  1.024 ± 0.001  27.34 ± 0.03  51 ± 1.00  ⋯  ⋯  LDA  (ext)-DFT-MD 
L. Stanek  ⋯  ⋯  42 ± 12.04  0.011 ± 0.003  ⋯  LDA  MD 
H3 (10 g cm–3, 20 eV)               
L. Babati  ⋯  ⋯  209.9  0.651  ⋯  LDA  KT 
G. Faussurier  23.8  1206.0  ⋯  ⋯  87.3  LDA  AA 
S. Hansen  24.7  730.9  ⋯  ⋯  ⋯  LDA  AA 
N. Shaffer  17.8  867  ⋯  ⋯  129  LDA  KT 
V. Sharma  12 ± 1.58  757 ± 80.4  ⋯  ⋯  ⋯  PBE  DFT-MD 
M. Bethkenhagen  14.8 ± 0.6  850.0 ± 34.0  ⋯  ⋯  ⋯  PBE  DFT-MD 

The modest disagreement among DFT-MD models may be attributable to several factors arising from the finite size of the simulations, which produces a discrete spectrum of allowable low-energy electronic transitions. A broadening procedure—as discussed in Ref. 96—recovers continuous optical conductivities, but these are only accurate above the minimum captured transition energy. Thus, the DC limit requires extrapolation using fit functions of known forms (e.g., the Drude model), which may not be accurate for all materials. Also, the extrapolated values converge slowly with the number of particles in the simulation, and multiple DFT-MD simulations are typically carried out at increasing particle number. In this workshop, the typical number of particles that participants employed in their DFT-MD calculations ranged from tens to hundreds. Despite all these choices, the DFT-MD models tend to agree better with one another than with AA or parameterized models.

The larger disagreement among AA models, which perform an all-electron calculation for a single atom, is also attributable to a combination of factors. These models are heavily dependent on choices for Z * and the ionic structure factor. Variations due to different possible choices of these quantities are illustrated by the error bars on the points from one submitted AA model in Figs. 6 and 7. Most (but not all) of the independent AA models fall within the range defined by these variations. While the fully parameterized SMT and LMD models have yet larger differences, both among themselves and with the quantum-AA and DFT-MD models and especially at low temperatures, the AA-informed kinetic theory models are in encouraging agreement with AA models at high temperatures.

There is very little experimental data to provide guidance on electrical and thermal conductivities in warm and hot dense matter due to the difficulty of creating sufficiently uniform states of matter in these extreme conditions, independently characterizing their temperature and density, and measuring or inferring the conductivities. Many experimental approaches to measuring these transport coefficients have a significant co-dependence on the equation of state101,103–107 and/or probe the optical response far from the DC limit.108,109 Recent advances with THz probes coupled with x-ray diffraction measurements at x-ray free-electron laser facilities110 offer a promising approach to experimental validation.

Ionic transport coefficients are governed by collisions among ions. In partially ionized systems, these collisions are mediated by electron distributions that screen long-range forces between ions. If binary collisions are assumed between ions, KT codes that use an effective Boltzmann approach17,18 (see Sec. V A) can be employed to estimate the ionic transport coefficients. Because these methods are based on a Boltzmann kinetic theory framework, KT codes are most accurate for weakly coupled plasmas. In contrast, one could simulate a system of interacting particles using MD to go beyond the binary collision approximation. Approaches for estimating ionic transport coefficients using particle trajectories from MD are discussed in the paragraphs that follow.

The shear viscosity—also referred to as the dynamic viscosity—quantifies the magnitude of a fluid's response due to gradients in fluid velocity. The shear viscosity η may be computed by the Green–Kubo relation
η = 1 V k B T 0 d t P γ β ( t ) P γ β ( 0 ) ,
(10)
where V is the volume of the system, kB is the Boltzmann's constant, T is the system temperature, γ , β { x , y , z } are Cartesian indices, and P γ β denotes an off diagonal component of the pressure tensor,111 i.e., γ β. The brackets · denote an ensemble average over thermodynamic configurations. The averaging may be done over multiple independent MD simulations or by partitioning a single MD simulation into uncorrelated samples. The ion thermal conductivity may also be defined in terms of a Green–Kubo relation, namely,
κ i = 1 V k B T 2 0 d t j γ e ( t ) j γ e ( 0 ) ,
(11)
where j γ e denotes the γ-component of the energy current.111 By exploiting isotropy of the systems considered here, we can average the results of the integrals appearing in Eqs. (10) and (11) for all γ , β, yielding a more statistically accurate result.
From Eq. (11), the ion thermal diffusivity is defined by
α κ i ρ i c p ,
(12)
where cp denotes the specific heat capacity at constant pressure. For monatomic gasses, an approximation for the specific heat capacity is c p 20.785 J K−1 g 1 / A, where A denotes the relative atomic mass. In practice, values for cp are obtained from a numerical derivative of the enthalpy generated from “lookup” tables of energy and pressure values. In Table III, we compare the monatomic approximation to that generated from a “lookup” table approach with pressures and energies derived from effective ionic potentials. We see that for case C1, the monatomic gas approximation and MD results differ by nearly a factor of four.
TABLE III.

Mass density times ionic specific heat at constant pressure, ρ i c p. The column denoted ρ i c p eff indicates that cp has been computed from an effective ionic potential approach—submitted by L. Silvestri (see Table II for the full list of contributors); the column denoted ρ i c p mon indicates that cp has been computed from the relation for the monatomic gas as mentioned in the main text.

Case ID ρ i c p eff ( erg Kcm 3 ) ρ i c p mon ( erg Kcm 3 ) ρ i c p eff ρ i c p mon
H1  1.91  × 10 8  2.06  × 10 8  0.93 
C1  6.35  × 10 8  1.73  × 10 8  3.67 
Al1  4.14  × 10 7  2.08  × 10 7  1.99 
Cu1  ⋯  2.93  × 10 7  ⋯ 
Be1  ⋯  4.24  × 10 7  ⋯ 
Au1  ⋯  2.04  × 10 7  ⋯ 
H3  ⋯  2.06  × 10 9  ⋯ 
Case ID ρ i c p eff ( erg Kcm 3 ) ρ i c p mon ( erg Kcm 3 ) ρ i c p eff ρ i c p mon
H1  1.91  × 10 8  2.06  × 10 8  0.93 
C1  6.35  × 10 8  1.73  × 10 8  3.67 
Al1  4.14  × 10 7  2.08  × 10 7  1.99 
Cu1  ⋯  2.93  × 10 7  ⋯ 
Be1  ⋯  4.24  × 10 7  ⋯ 
Au1  ⋯  2.04  × 10 7  ⋯ 
H3  ⋯  2.06  × 10 9  ⋯ 

Here, we compare the shear viscosity, η, and the ion thermal conductivity, κi. The data received for the shear viscosity are presented in Fig. 8 and data for the thermal conductivity—which is cast in terms of the thermal diffusivity using Eq. (12)—are displayed in Fig. 9; the data of these coefficients are given in Tables V and VI. Of the data received for the shear viscosity, we find that the maximum difference between all data submitted for a given Priority Level 1 or 2 case is roughly a factor of twenty; the maximum difference occurs for the case C1. While this difference is substantial, it is not unexpected if we consider the disparate approaches and models used for computing the shear viscosity. In particular, results for obtaining the shear viscosity from Eq. (10) are expected to be more accurate in the strong-coupling regime in contrast to expressions that rely on numerically evaluated collision integrals or from approaches that only require the collision rate as input. If instead we consider the difference between similar models within a given case, we find that the maximum difference is roughly a factor of six for case HCu1 which occurs between DFT-MD models.

FIG. 8.

Ion shear viscosity η for some of the single-species cases in Table I.

FIG. 8.

Ion shear viscosity η for some of the single-species cases in Table I.

Close modal
FIG. 9.

Thermal diffusivity α for some of the single-species cases in Table I.

FIG. 9.

Thermal diffusivity α for some of the single-species cases in Table I.

Close modal

Similarly, we find that of the data received for the ion thermal conductivity, the maximum difference between all models is on the order of one order o f magnitude (for cases C1 and Al1), and that the maximum difference within a given model type is roughly a factor of seven for case C1 between MD models. These results are consistent with the differences in the self-diffusion coefficient calculated from a variety of pair-interaction potentials used in MD simulations of dense plasmas57 where the maximum differences were on the order of ten between pair-potential models.

The discussion of results up to this point has been on the uncertainty between models. We now switch our discussion to the statistical uncertainty incurred when using a single model. Depending on the model used to generate the transport coefficient data, there are multiple ways statistical uncertainties manifest. In particular, for MD simulations, statistical uncertainties occur from the inability to simulate an infinite number of particles (often referred to as uncertainty from finite-size effects57) the truncation of the upper bound on the integrals defined in Eqs. (10), (11), and (14), and uncertainties due to the incomplete sampling of the thermodynamic ensemble during the MD simulation. By carrying out multiple MD simulations—at the same plasma conditions—but with an increasing number of particles, one can estimate uncertainties due to finite-size effects (see Ref. 57). Statistical noise in the integration of the expressions in Eqs. (10), (11), and (14) can be mitigated by utilizing appropriate fit functions112 and by exploiting stationarity of the autocorrelation function.111 

Equations (10) and (11) are integrals over autocorrelation functions, e.g., the stress autocorrelation function for the shear viscosity. Because transport coefficients are integrated quantities, it is useful to compare the autocorrelation functions generated by the various simulation approaches. In this workshop, we requested data for two autocorrelation functions: the velocity autocorrelation function and the stress autocorrelation function. The normalized velocity autocorrelation function is defined as
Z ( t ) v ( t ) · v ( 0 ) v ( 0 ) · v ( 0 ) ,
(13)
where v ( t ) denotes the velocity vector of a particle at time t. Equation (13) is the integrand of the Green–Kubo relation for the self-diffusion coefficient D, where
D = T m 0 d t Z ( t ) .
(14)
We note that the self-diffusion coefficient does not appear in the single-species hydrodynamics equations (1)–(5), nor does it appear in the multi-species case. However, the velocity autocorrelation function is a useful metric for comparing dynamical properties of the system by providing insight into differences in integrated observables (e.g., transport coefficients). Similarly, the normalized stress autocorrelation function is given by
C ( t ) P ¯ ( t ) P ¯ ( 0 ) P ¯ ( 0 ) P ¯ ( 0 ) ,
(15)
where P ¯ ( t ) [ P x y ( t ) + P y z ( t ) + P x z ( t ) ] / 3.

As concrete examples of how the autocorrelation functions were computed in this workshop, we turn to the data received from the two models denoted MD (see Table II). For these data, on the order of thousands of particles were used in each simulation. Then, on the order of tens of MD simulations were carried out with differing initial conditions (particle placement and velocity) for each case. The autocorrelation functions from each simulation were averaged to obtain an average autocorrelation function which is used to estimate the pertinent transport coefficients. A block averaging scheme may also be employed111 in each MD run to reduce statistical noise.

In Fig. 10, we show a comparison of the normalized velocity autocorrelation function [Eq. (13)] for the models described in Table II. The only methods in Table II that can directly compute the velocity autocorrelation function are DFT-MD and MD as they are the only approaches that directly simulate the motion of particles over sufficiently long time scales. In Fig. 10, we observe that in most cases, the DFT-MD simulation methods generally agree—most notably in terms of the autocorrelation time. In contrast, the MD simulation methods appear to agree in some cases, but disagree in other cases; the differences in the representation of Z(t) are directly related to the choice of interatomic potential used in the MD simulations. We caution that even though Z(t) can vary between models, the self-diffusion coefficient obtained by integrating Z(t) using Eq. (14) can be approximately the same.57 A similar comparison is displayed in Fig. 11 where we show the stress autocorrelation function [Eq. (15)].

FIG. 10.

Normalized velocity autocorrelation function Z(t) [Eq. (13)] vs time for some of the single-species cases in Table I.

FIG. 10.

Normalized velocity autocorrelation function Z(t) [Eq. (13)] vs time for some of the single-species cases in Table I.

Close modal
FIG. 11.

Normalized stress autocorrelation function C(t) [Eq. (15)] vs time for some of the single-species cases in Table I.

FIG. 11.

Normalized stress autocorrelation function C(t) [Eq. (15)] vs time for some of the single-species cases in Table I.

Close modal

From Figs. 10 and 11, three things are clear. First, due to the significant computational cost, the DFT-MD based simulation approaches are often plagued by finite-size effects—appearing as noise in the autocorrelation functions. Second, in contrast to Z(t), C(t) requires a longer simulation time to converge to a meaningful result. Third, the autocorrelation time varies between models.

While conductivity is governed by momentum scattering frequencies, temperature relaxation in non-equilibrium plasmas is governed by the energy exchange between ions and electrons. For example, in the two-temperature model, the energy exchange is described by the coupled equations
C e d T e ( t ) d t = g [ T e ( t ) T i ( t ) ] ,
(16)
C i d T i ( t ) d t = g [ T e ( t ) T i ( t ) ] ,
(17)
where Ce and Ci are the heat capacity of the electrons and ions, respectively, and g is the electron–ion temperature relaxation rate. Note that Eqs. (16) and (17) are a simplified form of the general two-temperature model.113 Here, we have assumed there is no external source term, there is no heat flux, and that the heat capacity terms are independent of temperature. Finally, we note that calculations of dT/dt that include heat capacities native to a given model could enable more direct comparisons with experimental data than calculations of g, since experiments measure temperature as a function of time.114,115 Most contributors of this workshop generate g from collision rates which would represent equilibration rates at a small perturbation from Te = Ti—much like collisions for conductivity are determined in the presence of a weak electric field. In practice, these calculations are relatively insensitive to Ti. Submitted values of g are given in Tables V and VI.

Finally, we compare electronic stopping powers for alpha particles traversing some single-species plasma cases as specified in Table I. Data for the electronic stopping powers are displayed in Fig. 12, and provided in Table VII, which additionally includes data for the CH1 and H3 cases. The two AA models differ mainly in the low-velocity regime, likely due to different parameterizations of the stopping number in a uniform electron gas. TD-DFT-MD is generally expected to be more accurate and can serve as a valuable benchmark for AA models.30 However, the high computational cost of TD-DFT-MD calculations limited this method to a subset of the Priority Level 1 cases illustrated in Figs. 12(a)–12(c).

FIG. 12.

Electronic contribution to the alpha-particle stopping power for some of the single-species cases in Table I as a function of alpha particle velocity vp.

FIG. 12.

Electronic contribution to the alpha-particle stopping power for some of the single-species cases in Table I as a function of alpha particle velocity vp.

Close modal
TABLE VII.

Submitted data for the electronic contribution to the alpha-particle stopping power. The column denoted “peak position” gives the alpha particle velocity that corresponds to the largest stopping power—denoted as “peak height.” The computation time reported denotes the total simulation time. For example, the total computation time for a TD-DFT-MD simulation is determined from t total = p t p , sim N p , CPUs, where p is the number of projectiles simulated (i.e., the number of alpha particle velocities for a case), t p , sim is the duration of the simulation for projectile p, and N p , CPUs is the number of CPU-cores used for the simulation of projectile p. Note that (mix)-TD-DFT-MD denotes that mixed stochastic-deterministic TD-DFT-MD was used—see Ref. 43.

Case ID/Submitter Peak position (cm/s) Peak height (eV/cm) XC Model Computation time (s)
H1 (1 g cm–3, 2 eV)           
G. Faussurier  4.38  × 10 8  9.13  × 10 9  LDA  AA  1.7  × 10 2 
S. Hansen  4.90  × 10 8  8.36  × 10 9  LDA  AA  ⋯ 
A. Kononov  5.47  × 10 8  6.39  × 10 9  LDA  TD-DFT-MD  3.8  × 10 8 
K. Nichols  4.38  × 10 8  6.21  × 10 9  PBE  TD-DFT-MD  1.5  × 10 8 
A. White  5.47  × 10 8  6.68  × 10 9  PBE  (mix)-TD-DFT-MD  1.7  × 10 9 
C1 (10 g cm–3, 2 eV)           
G. Faussurier  6.19  × 10 8  1.68  × 10 10  LDA  AA  1.9  × 10 2 
S. Hansen  6.92  × 10 8  1.54  × 10 10  LDA  AA  ⋯ 
A. Kononov  6.56  × 10 8  1.26  × 10 10  LDAa  TD-DFT-MD  5.2  × 10 6 
A. Kononov  6.56  × 10 8  1.06  × 10 10  LDAb  TD-DFT-MD  9.6  × 10 7 
CH1 (1 g cm–3, 2 eV)           
A. Kononov  4.38  × 10 8  2.32  × 10 9  LDA  TD-DFT-MD  2.4  × 10 8 
Al1 (2.7 g cm–3, 1 eV)           
G. Faussurier  3.10  × 10 8  6.43  × 10 9  LDA  AA  1.1  × 10 2 
S. Hansen  3.47  × 10 8  5.66  × 10 9  LDA  AA  ⋯ 
A. Kononov  3.83  × 10 8  3.40  × 10 9  LDA  TD-DFT-MD  5.2  × 10 9 
Cu1 (8.96 g cm−3, 1 eV)           
G. Faussurier  3.90  × 10 8  7.78  × 10 9  LDA  AA  6.4  × 10 2 
Be1 (1.84 g cm–3, 4.4 eV)           
G. Faussurier  3.48  × 10 8  6.32  × 10 9  LDA  AA  2.3  × 10 2 
S. Hansen  3.89  × 10 8  5.96  × 10 9  LDA  AA  ⋯ 
Au1 (19.32 g cm–3, 10 eV)           
G. Faussurier  4.92  × 10 8  8.13  × 10 9  LDA  AA  2.4  × 10 2 
S. Hansen  4.90  × 10 8  7.45  × 10 9  LDA  AA  ⋯ 
H3 (10 g cm–3, 20 eV)           
G. Faussurier  8.74  × 10 8  2.57  × 10 10  LDA  AA  2.3  × 10 2 
S. Hansen  9.78  × 10 8  2.38  × 10 10  LDA  AA  ⋯ 
Case ID/Submitter Peak position (cm/s) Peak height (eV/cm) XC Model Computation time (s)
H1 (1 g cm–3, 2 eV)           
G. Faussurier  4.38  × 10 8  9.13  × 10 9  LDA  AA  1.7  × 10 2 
S. Hansen  4.90  × 10 8  8.36  × 10 9  LDA  AA  ⋯ 
A. Kononov  5.47  × 10 8  6.39  × 10 9  LDA  TD-DFT-MD  3.8  × 10 8 
K. Nichols  4.38  × 10 8  6.21  × 10 9  PBE  TD-DFT-MD  1.5  × 10 8 
A. White  5.47  × 10 8  6.68  × 10 9  PBE  (mix)-TD-DFT-MD  1.7  × 10 9 
C1 (10 g cm–3, 2 eV)           
G. Faussurier  6.19  × 10 8  1.68  × 10 10  LDA  AA  1.9  × 10 2 
S. Hansen  6.92  × 10 8  1.54  × 10 10  LDA  AA  ⋯ 
A. Kononov  6.56  × 10 8  1.26  × 10 10  LDAa  TD-DFT-MD  5.2  × 10 6 
A. Kononov  6.56  × 10 8  1.06  × 10 10  LDAb  TD-DFT-MD  9.6  × 10 7 
CH1 (1 g cm–3, 2 eV)           
A. Kononov  4.38  × 10 8  2.32  × 10 9  LDA  TD-DFT-MD  2.4  × 10 8 
Al1 (2.7 g cm–3, 1 eV)           
G. Faussurier  3.10  × 10 8  6.43  × 10 9  LDA  AA  1.1  × 10 2 
S. Hansen  3.47  × 10 8  5.66  × 10 9  LDA  AA  ⋯ 
A. Kononov  3.83  × 10 8  3.40  × 10 9  LDA  TD-DFT-MD  5.2  × 10 9 
Cu1 (8.96 g cm−3, 1 eV)           
G. Faussurier  3.90  × 10 8  7.78  × 10 9  LDA  AA  6.4  × 10 2 
Be1 (1.84 g cm–3, 4.4 eV)           
G. Faussurier  3.48  × 10 8  6.32  × 10 9  LDA  AA  2.3  × 10 2 
S. Hansen  3.89  × 10 8  5.96  × 10 9  LDA  AA  ⋯ 
Au1 (19.32 g cm–3, 10 eV)           
G. Faussurier  4.92  × 10 8  8.13  × 10 9  LDA  AA  2.4  × 10 2 
S. Hansen  4.90  × 10 8  7.45  × 10 9  LDA  AA  ⋯ 
H3 (10 g cm–3, 20 eV)           
G. Faussurier  8.74  × 10 8  2.57  × 10 10  LDA  AA  2.3  × 10 2 
S. Hansen  9.78  × 10 8  2.38  × 10 10  LDA  AA  ⋯ 
a

For this case all-electron calculations with bare Coulomb potentials were carried out.

b

For this case pseudopotentials were employed with 4 valence electrons per C ion.

Figure 12(a) includes several TD-DFT-MD datasets for alpha-particle stopping in case H1 that were computed using different codes and methodological details, including deterministic vs mixed stochastic-deterministic TD-DFT variants, HGH pseudopotentials vs bare Coulomb potentials, and LDA vs PBE XC functionals. Additionally, some of the TD-DFT calculations optimized the alpha-particle trajectory to representatively sample a cubic simulation cell and mitigate finite-size effects,73 whereas others used an elongated simulation cell with the alpha particle traveling along the long direction.

Nonetheless, the different TD-DFT-MD stopping power datasets in Fig. 12(a) agree amongst each other quite well, and the minor discrepancies reflect the sensitivity that even this first-principles model can have to methodological choices. Furthermore, high computational costs can make it challenging to assess convergence and quantify uncertainties in TD-DFT-MD stopping powers. A separate article within this special issue scrutinizes the sources of these small discrepancies and finds highest sensitivities to the pseudopotential approximation, finite-size errors, and alpha-particle trajectory choice.116 

In Fig. 12(a), all of the TD-DFT-MD datasets agree quite well with both average-atom models beyond the stopping power peak. Good agreement is also obtained for case C1 shown in Fig. 12(b), where one TD-DFT-MD dataset included contributions from C 1s electrons and the other excluded them through the use of pseudopotentials. For case Al1 of Fig. 12(c), however, both AA models significantly exceed the TD-DFT-MD predictions. To reduce computational costs, these TD-DFT-MD calculations neglected contributions from Al core electrons, which become increasingly significant for fast projectiles73 and likely explain the discrepancy beyond the stopping power peak, since the AA models do include contributions from core electrons. On the other hand, the local uniform-electron-gas approximation used by the AA models may not accurately capture low-velocity stopping power.30 Furthermore, the AA models do not account for partial neutralization of the alpha particle as it captures electrons from the plasma—a nonlinear effect beyond the standard Lindhard stopping formula117 that becomes increasingly important for slow projectiles.

The results of this workshop have quantified differences between state-of-the-art approaches for computing fundamental material properties of plasmas. Namely, we have compared ionic and electronic transport coefficients, correlation functions, and scalar quantities that characterize the system. Of the data received, we found significant differences between the shear viscosity and thermal conductivity of the ions. For the shear viscosity the difference was at worst one order of magnitude between all models and a factor of six between similar models. For the ion thermal conductivity, the difference was at worst one order of magnitude between all models and a factor of seven between similar models.

We also found significant differences in the DC electrical conductivity and electron thermal conductivity—where the difference in the DC electrical conductivity was at worst a factor of seven between all models and a factor of two between similar models. For the electron thermal conductivity, the difference was at worst one order of magnitude between all models and a factor of five between similar models. Disagreement was generally larger at lower temperatures and smaller among the most sophisticated DFT-MD models.

In this second iteration of the charged-particle transport coefficient code comparison workshop, we built upon the first workshop by requesting more detailed quantities to characterize the ionic structure, electronic structure, and compare particle trajectories from autocorrelation functions. As a result, additional insight was provided beyond the comparison of integrated quantities; this insight may provide a path forward for improving upon extant models.

Through a comparison with analytic models that estimate transport coefficients within fractions of a second, we have shown the plasma conditions for which these more approximate approaches are viable—primarily in the weakly coupled regime. Additionally, the results of this workshop highlight some of the inherent difficulties in computing transport coefficients—which often constitute cost-benefit tradeoffs of model and statistical accuracy.

In future workshops, we aim to continue exploring disparate plasma conditions relevant to inertial confinement fusion. While the Priority Level System was largely successful in guiding participants to the cases with maximal impact, fewer cases would allow for a more in-depth model comparison, including additional quantities like ion–ion potentials and optical properties (e.g., dynamic conductivities or dielectric functions). In Table IV, we propose a set of six high-priority cases for the next workshop that explore more extreme conditions. We have chosen a single temperature of 3 eV, which remains within reach for multi-atom DFT-based methods. We have also selected higher densities for every element except carbon, whose density remains the same as the present workshop's C1 case as a semi-longitudinal study that will explore the persistence of multi-atom effects on ionic and electronic structure. To facilitate further exploration of mixtures, we have pressure-matched the C, H, and CH cases using an isothermal–isobaric mixing rule, setting the CH molecular volume to the sum of the atomic volumes of the pure C and H cases. The cases of Be, Al, and Cu were chosen to lie along the principal Hugoniot, allowing for access from experimental platforms.

TABLE IV.

Suggested cases for the next workshop. For the CH mixture, the ionic number density n species is given in terms of each species, whereas the mass density ρ total is the total mass density of the mixture. We will also request data along these isobars from computationally inexpensive codes.

Element(s) n species (cm−3) ρ total (g cm–3) T (eV)
1.8 × 10 24 
5.0 × 10 23  10 
CH  3.9 × 10 23  8.4 
Be  3.2 × 10 23  4.7 
Al  1.4 × 10 23  6.4 
Cu  1.7 × 10 23  18.3 
Element(s) n species (cm−3) ρ total (g cm–3) T (eV)
1.8 × 10 24 
5.0 × 10 23  10 
CH  3.9 × 10 23  8.4 
Be  3.2 × 10 23  4.7 
Al  1.4 × 10 23  6.4 
Cu  1.7 × 10 23  18.3 

Continued efforts to compare transport coefficients predicted by various models will further characterize regimes of accuracy for different approximations, offer insight into underlying physical processes, and inspire improvements to efficient models suitable for tabulating material properties over the wide range of conditions accessed by hydrodynamic simulations. This line of research also helps estimate and reduce uncertainties in transport coefficients, which contribute to uncertainties in hydrodynamic simulations. Ultimately, this work is an important step toward improving the predictive power of large-scale HED simulations of both astrophysical objects and inertial confinement fusion experiments.

We are grateful for the hospitality of the University of California's Livermore Collaboration Center along with the help of A. Cuevas, G. Weiss, C. Bibeau, and A. Mendoza-Olivera. The participants of the workshop included N. Acharya. T. Chuna, P. Efthimion, S. Glenzner, F. Graziani, T. Griffin, T. Haxhimali, F. Kraus, S. Malko, I. Martinez, O. Schilling, J. Shang, and T. White. The authors would like to thank W. Lewis and G. Shipley for helpful feedback and L. Shulenburger for careful proofreading of the manuscript and useful conversations. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International, Inc., for DOE's National Nuclear Security Administration under Contract No. DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. Los Alamos National Laboratory is managed by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy (Contract No. 89233218CNA000001). P.S. and S.K. gratefully acknowledge support from Grant No. DE-NA0004128 funded by the U.S. Department of Energy (DOE), National Nuclear Security Administration (NNSA). L.J.S., J.P.T., C.A.M., A.E.D., R.C.C., K.R.C., M.K.L., A.K., and A.D.B. were supported by the Laboratory Directed Research and Development program (Project Nos. 229428, 230332, and 233196) at Sandia National Laboratories. A.K. and A.D.B. were also partially supported by the U.S. Department of Energy Science Campaign 1. This work was supported in part by NNSA Stewardship Science Academic Programs (DOE Cooperative Agreement No. DE-NA0004146). This material is based upon work supported by the Department of Energy (National Nuclear Security Administration) University of Rochester “National Inertial Confinement Fusion Program” (Award No. DE-NA0004144). This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof. A.J.W., L.A.C., and V.S. were supported by Science Campaign 4 and Laboratory Directed Research and Development of LANL (Project Nos. 20210233ER and 20230322ER). We gratefully acknowledge the support of the Center for Nonlinear Studies (CNLS). This research used computing resources provided by the LANL Institutional Computing and Advanced Scientific Computing programs.

The authors have no conflicts to disclose.

Lucas James Stanek: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Supervision (lead); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Scott David Baalrud: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Lucas Babati: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Andrew Baczewski: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Mandy Bethkenhagen: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Augustin Blanchet: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Raymond C. Clay: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Kyle R. Cochrane: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Lee Collins: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Amanda Dumi: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Gerald Faussurier: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Alina Kononov: Data curation (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Martin French: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Zachary A. Johnson: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Valentin V. Karasiev: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Shashikant Kumar: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Meghan K. Lentz: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Cody Allen Melton: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Katarina A. Nichols: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). George M. Petrov: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Vanina Recoules: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Ronald Redmer: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Stephanie B. Hansen: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Gerd Roepke: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Maximilian Schörner: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Nathaniel R. Shaffer: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Vidushi Sharma: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Luciano Germano Silvestri: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). François Soubiran: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Phanish Suryanarayana: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Mikael Tacu: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Joshua Townsend: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Alexander James White: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Brian Michael Haines: Conceptualization (equal); Supervision (equal); Writing – review & editing (supporting). Suxing Hu: Conceptualization (equal); Data curation (supporting); Supervision (equal); Writing – review & editing (supporting). Patrick Francis Knapp: Conceptualization (equal); Supervision (equal); Writing – review & editing (supporting). Michael S. Murillo: Conceptualization (equal); Data curation (supporting); Methodology (equal); Supervision (equal); Validation (supporting); Writing – review & editing (equal). Liam Stanton: Conceptualization (equal); Project administration (supporting); Writing – review & editing (supporting). Heather D. Whitley: Conceptualization (equal); Funding acquisition (equal); Project administration (lead); Resources (equal); Supervision (equal); Writing – review & editing (supporting).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Due to the quantity of the received data, not all data were suitable for a broad comparison between models. In this section, we present additional data that participants submitted. These data include the total pressure for the cases given in Table I, which is displayed in Table VIII. The DC electrical conductivity and shear viscosity for hydrogen at various densities are also provided in Table IX. Finally, we display g(r)s for the mixture cases of Table I in Fig. 13.

TABLE VIII.

Computed total pressure for plasma conditions detailed in Table I. For brevity, entries in the Case ID/Submitter column have been labeled by a single submitter determined by how the data were received; see Table II for a full list of the collaborations and institutions. Note that (ext)-DFT-MD denotes that extended DFT-MD was used–see Ref. 42 for details.

Case ID/Submitter P total (Mbar) XC Model
H1 (1 g cm–3, 2 eV)       
G. Faussurier  6.70  LDA  AA 
M. French  4.81  PBE  DFT-MD 
M. French  4.75  HSE  DFT-MD 
S. Hansen  5.46  LDA  AA 
S. Hu  4.82 ± 0.07  PBE  DFT-MD 
S. Hu  4.81 ± 0.06  TSCANL  DFT-MD 
P. Suryanarayana  4.79 ± 0.01  LDA  DFT-MD 
F. Soubiran  4.82  PBE  DFT-MD 
J. Townsend  4.73 ± 0.004  LDA  DFT-MD 
C1 (10 g cm–3, 2 eV)       
M. Bethkenhagen  28.6  PBE  DFT-MD 
G. Faussurier  43.5  LDA  AA 
S. Hansen  34.7  LDA  AA 
V. Karasiev  28.5  PBE  DFT-MD 
V. Karasiev  28.7  Tr2SCANL  DFT-MD 
C. Melton  28.1  LDA  DFT-MD 
P. Suryanarayana  28.93 ± 0.02  LDA  DFT-MD 
F. Soubiran  28.5 ± 0.002  PBE  DFT-MD 
CH1 (1 g cm–3, 2 eV)       
M. Bethkenhagen  0.370  PBE  DFT-MD 
R. Clay  0.338  LDA  DFT-MD 
V. Karasiev  0.370  PBE  DFT-MD 
V. Karasiev  0.330  TSCANL  DFT-MD 
F. Soubiran  0.368 ± 0.001  PBE  DFT-MD 
Al1 (2.7 g cm–3, 1 eV)       
A. Dumi  0.513 ± 0.001  LDA  DFT-MD 
G. Faussurier  1.01  LDA  AA 
S. Hansen  0.808  LDA  AA 
V. Karasiev  0.443  PBE  DFT-MD 
V. Karasiev  0.425  TSCANL  DFT-MD 
M. Schörner  0.441 ± 0.001  PBE  DFT-MD 
F. Soubiran  0.440 ± 0.001  GGA  DFT-MD 
Cu1 (8.96 g cm–3, 1 eV)       
K. Cochrane  0.766  PBE  DFT-MD 
G. Faussurier  1.67  LDA  AA 
S. Hansen  1.04  LDA  AA 
S. Hu  0.377 ± 0.053  PBE  DFT-MD 
S. Hu  0.340 ± 0.053  TSCANL  DFT-MD 
F. Soubiran  0.651 ± 0.003  LDA  DFT-MD 
HCu1 (1.8 g cm–3, 1 eV)       
S. Hu  0.02 ± 0.0158  PBE  DFT-MD 
S. Hu  0.01 ± 0.0139  TSCANL  DFT-MD 
M. Lentz  0.0311 ± 0.0001  LDA  DFT-MD 
Be1 (1.84 g cm–3, 4.4 eV)       
G. Faussurier  3.10  LDA  AA 
S. Hansen  2.78  LDA  AA 
S. Hu  2.36 ± 0.04  PBE  DFT-MD 
S. Hu  2.29 ± 0.04  TSCANL  DFT-MD 
M. Schörner  2.37 ± 0.003  PBE  DFT-MD 
CH2 (0.9 g cm–3, 7.8 eV)       
F. Soubiran  2.04 ± 0.002  PBE  (ext)-DFT-MD 
Au1 (19.32 g cm–3, 10 eV)       
G. Faussurier  6.09  LDA  AA 
S. Hansen  6.51  LDA  AA 
S. Hu  5.87 ± 0.2  TSCANL  DFT-MD 
V. Karasiev  5.76  TSCANL  DFT-MD 
F. Soubiran  6.01 ± 0.01  LDA  (ext)-DFT-MD 
Case ID/Submitter P total (Mbar) XC Model
H1 (1 g cm–3, 2 eV)       
G. Faussurier  6.70  LDA  AA 
M. French  4.81  PBE  DFT-MD 
M. French  4.75  HSE  DFT-MD 
S. Hansen  5.46  LDA  AA 
S. Hu  4.82 ± 0.07  PBE  DFT-MD 
S. Hu  4.81 ± 0.06  TSCANL  DFT-MD 
P. Suryanarayana  4.79 ± 0.01  LDA  DFT-MD 
F. Soubiran  4.82  PBE  DFT-MD 
J. Townsend  4.73 ± 0.004  LDA  DFT-MD 
C1 (10 g cm–3, 2 eV)       
M. Bethkenhagen  28.6  PBE  DFT-MD 
G. Faussurier  43.5  LDA  AA 
S. Hansen  34.7  LDA  AA 
V. Karasiev  28.5  PBE  DFT-MD 
V. Karasiev  28.7  Tr2SCANL  DFT-MD 
C. Melton  28.1  LDA  DFT-MD 
P. Suryanarayana  28.93 ± 0.02  LDA  DFT-MD 
F. Soubiran  28.5 ± 0.002  PBE  DFT-MD 
CH1 (1 g cm–3, 2 eV)       
M. Bethkenhagen  0.370  PBE  DFT-MD 
R. Clay  0.338  LDA  DFT-MD 
V. Karasiev  0.370  PBE  DFT-MD 
V. Karasiev  0.330  TSCANL  DFT-MD 
F. Soubiran  0.368 ± 0.001  PBE  DFT-MD 
Al1 (2.7 g cm–3, 1 eV)       
A. Dumi  0.513 ± 0.001  LDA  DFT-MD 
G. Faussurier  1.01  LDA  AA 
S. Hansen  0.808  LDA  AA 
V. Karasiev  0.443  PBE  DFT-MD 
V. Karasiev  0.425  TSCANL  DFT-MD 
M. Schörner  0.441 ± 0.001  PBE  DFT-MD 
F. Soubiran  0.440 ± 0.001  GGA  DFT-MD 
Cu1 (8.96 g cm–3, 1 eV)       
K. Cochrane  0.766  PBE  DFT-MD 
G. Faussurier  1.67  LDA  AA 
S. Hansen  1.04  LDA  AA 
S. Hu  0.377 ± 0.053  PBE  DFT-MD 
S. Hu  0.340 ± 0.053  TSCANL  DFT-MD 
F. Soubiran  0.651 ± 0.003  LDA  DFT-MD 
HCu1 (1.8 g cm–3, 1 eV)       
S. Hu  0.02 ± 0.0158  PBE  DFT-MD 
S. Hu  0.01 ± 0.0139  TSCANL  DFT-MD 
M. Lentz  0.0311 ± 0.0001  LDA  DFT-MD 
Be1 (1.84 g cm–3, 4.4 eV)       
G. Faussurier  3.10  LDA  AA 
S. Hansen  2.78  LDA  AA 
S. Hu  2.36 ± 0.04  PBE  DFT-MD 
S. Hu  2.29 ± 0.04  TSCANL  DFT-MD 
M. Schörner  2.37 ± 0.003  PBE  DFT-MD 
CH2 (0.9 g cm–3, 7.8 eV)       
F. Soubiran  2.04 ± 0.002  PBE  (ext)-DFT-MD 
Au1 (19.32 g cm–3, 10 eV)       
G. Faussurier  6.09  LDA  AA 
S. Hansen  6.51  LDA  AA 
S. Hu  5.87 ± 0.2  TSCANL  DFT-MD 
V. Karasiev  5.76  TSCANL  DFT-MD 
F. Soubiran  6.01 ± 0.01  LDA  (ext)-DFT-MD 
TABLE IX.

Additional DC electrical conductivity and shear viscosity data for hydrogen. For brevity, entries in the Case ID/Submitter column have been labeled by a single submitter determined by how the data were received; see Table II for a full list of the collaborations and institutions.

Case/Submitter T (eV) σ ( 1 Ω cm 10 4 ) η ( g cm s ) XC Model
H (1 g cm–3          
G. Röpke  70  7.75  ⋯  ⋯  Analytic 
G. Röpke  100  10.7  ⋯  ⋯  Analytic 
G. Röpke  200  21.9  ⋯  ⋯  Analytic 
G. Röpke  400  48.3  ⋯  ⋯  Analytic 
G. Röpke  700  95.1  ⋯  ⋯  Analytic 
G. Röpke  1000  148  ⋯  ⋯  Analytic 
H (1.67 g cm–3          
L. Babati  ⋯  0.00901  LDA  KT 
L. Babati  ⋯  0.0233  LDA  KT 
L. Babati  ⋯  0.0396  LDA  KT 
L. Babati  10  ⋯  0.0513  LDA  KT 
L. Babati  20  ⋯  0.124  LDA  KT 
P. Suryanarayana  43.1  ⋯  1.0 ± 0.17  LDA  DFT-MD 
L. Babati  50  ⋯  0.491  LDA  KT 
P. Suryanarayana  64.6  ⋯  2.0 ± 0.2  LDA  DFT-MD 
L. Babati  80  ⋯  1.11  LDA  KT 
P. Suryanarayana  86.2  ⋯  3.1 ± 0.3  LDA  DFT-MD 
L. Babati  100  ⋯  1.67  LDA  KT 
G. Röpke  100  13 ± 0.25  ⋯  ⋯  Analytic 
P. Suryanarayana  107.7  ⋯  4.5 ± 0.5  LDA  DFT-MD 
P. Suryanarayana  129.3  ⋯  6.1 ± 0.7  LDA  DFT-MD 
L. Babati  200  ⋯  6.43  LDA  KT 
L. Babati  500  ⋯  43.4  LDA  KT 
L. Babati  800  ⋯  121  LDA  KT 
L. Babati  1000  ⋯  197  LDA  KT 
G. Röpke  1000  159 ± 3.2  ⋯  ⋯  Analytic 
G. Röpke  10 000  3130 ± 60  ⋯  ⋯  Analytic 
G. Röpke  20 000  7950 ± 160  ⋯  ⋯  Analytic 
H (10 g cm–3          
L. Babati  ⋯  0.0715  LDA  KT 
N. Shaffer  22.1  ⋯  LDA  KT 
L. Babati  10  ⋯  0.0912  LDA  KT 
N. Shaffer  10  20.8  ⋯  LDA  KT 
L. Babati  20  ⋯  0.210  LDA  KT 
N. Shaffer  20  17.8  ⋯  LDA  KT 
L. Babati  50  ⋯  0.776  LDA  KT 
N. Shaffer  50  18.4  ⋯  LDA  KT 
L. Babati  80  ⋯  1.66  LDA  KT 
N. Shaffer  80  21.6  ⋯  LDA  KT 
L. Babati  100  ⋯  2.44  LDA  KT 
G. Röpke  100  26.6 ± 0.5  ⋯  ⋯  Analytic 
N. Shaffer  100  23.7  ⋯  LDA  KT 
L. Babati  200  ⋯  8.74  LDA  KT 
N. Shaffer  200  33.9  ⋯  LDA  KT 
L. Babati  500  ⋯  54.7  LDA  KT 
N. Shaffer  500  87.1  ⋯  LDA  KT 
L. Babati  800  ⋯  147  LDA  KT 
N. Shaffer  800  148.9  ⋯  LDA  KT 
L. Babati  1000  ⋯  238  LDA  KT 
G. Röpke  1000  207 ± 4  ⋯  ⋯  Analytic 
N. Shaffer  1000  193.8  ⋯  LDA  KT 
H (100 g cm–3          
L. Babati  50  ⋯  1.49  LDA  KT 
N. Shaffer  50  182  ⋯  LDA  KT 
L. Babati  80  ⋯  2.92  LDA  KT 
N. Shaffer  80  159  ⋯  LDA  KT 
L. Babati  100  ⋯  4.14  LDA  KT 
N. Shaffer  100  153  ⋯  LDA  KT 
L. Babati  200  ⋯  13.5  LDA  KT 
N. Shaffer  200  153  ⋯  LDA  KT 
L. Babati  500  ⋯  77.4  LDA  KT 
N. Shaffer  500  213  ⋯  LDA  KT 
L. Babati  800  ⋯  200  LDA  KT 
N. Shaffer  800  269  ⋯  LDA  KT 
L. Babati  1000  ⋯  317  LDA  KT 
G. Röpke  1000  340 ± 7  ⋯  ⋯  Analytic 
N. Shaffer  1000  320  ⋯  LDA  KT 
Case/Submitter T (eV) σ ( 1 Ω cm 10 4 ) η ( g cm s ) XC Model
H (1 g cm–3          
G. Röpke  70  7.75  ⋯  ⋯  Analytic 
G. Röpke  100  10.7  ⋯  ⋯  Analytic 
G. Röpke  200  21.9  ⋯  ⋯  Analytic 
G. Röpke  400  48.3  ⋯  ⋯  Analytic 
G. Röpke  700  95.1  ⋯  ⋯  Analytic 
G. Röpke  1000  148  ⋯  ⋯  Analytic 
H (1.67 g cm–3          
L. Babati  ⋯  0.00901  LDA  KT 
L. Babati  ⋯  0.0233  LDA  KT 
L. Babati  ⋯  0.0396  LDA  KT 
L. Babati  10  ⋯  0.0513  LDA  KT 
L. Babati  20  ⋯  0.124  LDA  KT 
P. Suryanarayana  43.1  ⋯  1.0 ± 0.17  LDA  DFT-MD 
L. Babati  50  ⋯  0.491  LDA  KT 
P. Suryanarayana  64.6  ⋯  2.0 ± 0.2  LDA  DFT-MD 
L. Babati  80  ⋯  1.11  LDA  KT 
P. Suryanarayana  86.2  ⋯  3.1 ± 0.3  LDA  DFT-MD 
L. Babati  100  ⋯  1.67  LDA  KT 
G. Röpke  100  13 ± 0.25  ⋯  ⋯  Analytic 
P. Suryanarayana  107.7  ⋯  4.5 ± 0.5  LDA  DFT-MD 
P. Suryanarayana  129.3  ⋯  6.1 ± 0.7  LDA  DFT-MD 
L. Babati  200  ⋯  6.43  LDA  KT 
L. Babati  500  ⋯  43.4  LDA  KT 
L. Babati  800  ⋯  121  LDA  KT 
L. Babati  1000  ⋯  197  LDA  KT 
G. Röpke  1000  159 ± 3.2  ⋯  ⋯  Analytic 
G. Röpke  10 000  3130 ± 60  ⋯  ⋯  Analytic 
G. Röpke  20 000  7950 ± 160  ⋯  ⋯  Analytic 
H (10 g cm–3          
L. Babati  ⋯  0.0715  LDA  KT 
N. Shaffer  22.1  ⋯  LDA  KT 
L. Babati  10  ⋯  0.0912  LDA  KT 
N. Shaffer  10  20.8  ⋯  LDA  KT 
L. Babati  20  ⋯  0.210  LDA  KT 
N. Shaffer  20  17.8  ⋯  LDA  KT 
L. Babati  50  ⋯  0.776  LDA  KT 
N. Shaffer  50  18.4  ⋯  LDA  KT 
L. Babati  80  ⋯  1.66  LDA  KT 
N. Shaffer  80  21.6  ⋯  LDA  KT 
L. Babati  100  ⋯  2.44  LDA  KT 
G. Röpke  100  26.6 ± 0.5  ⋯  ⋯  Analytic 
N. Shaffer  100  23.7  ⋯  LDA  KT 
L. Babati  200  ⋯  8.74  LDA  KT 
N. Shaffer  200  33.9  ⋯  LDA  KT 
L. Babati  500  ⋯  54.7  LDA  KT 
N. Shaffer  500  87.1  ⋯  LDA  KT 
L. Babati  800  ⋯  147  LDA  KT 
N. Shaffer  800  148.9  ⋯  LDA  KT 
L. Babati  1000  ⋯  238  LDA  KT 
G. Röpke  1000  207 ± 4  ⋯  ⋯  Analytic 
N. Shaffer  1000  193.8  ⋯  LDA  KT 
H (100 g cm–3          
L. Babati  50  ⋯  1.49  LDA  KT 
N. Shaffer  50  182  ⋯  LDA  KT 
L. Babati  80  ⋯  2.92  LDA  KT 
N. Shaffer  80  159  ⋯  LDA  KT 
L. Babati  100  ⋯  4.14  LDA  KT 
N. Shaffer  100  153  ⋯  LDA  KT 
L. Babati  200  ⋯  13.5  LDA  KT 
N. Shaffer  200  153  ⋯  LDA  KT 
L. Babati  500  ⋯  77.4  LDA  KT 
N. Shaffer  500  213  ⋯  LDA  KT 
L. Babati  800  ⋯  200  LDA  KT 
N. Shaffer  800  269  ⋯  LDA  KT 
L. Babati  1000  ⋯  317  LDA  KT 
G. Röpke  1000  340 ± 7  ⋯  ⋯  Analytic 
N. Shaffer  1000  320  ⋯  LDA  KT 
FIG. 13.

Ionic radial distribution function, g(r), for the mixture cases in Table I. g(r)s from AA models are obtained by solving the quantum Ornstein–Zernike equations and employing the hyper-netted chain approximation while g(r)s obtained from DFT-MD models are obtained directly from ensemble-averaged particle positions. Note that the g(r)s for each pair of species have been shifted vertically for visual clarity.

FIG. 13.

Ionic radial distribution function, g(r), for the mixture cases in Table I. g(r)s from AA models are obtained by solving the quantum Ornstein–Zernike equations and employing the hyper-netted chain approximation while g(r)s obtained from DFT-MD models are obtained directly from ensemble-averaged particle positions. Note that the g(r)s for each pair of species have been shifted vertically for visual clarity.

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