We report the results of the second chargedparticle 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 chargedparticle transport coefficients relevant to highenergydensity 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 semiempirical models to timedependent 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 firstprinciples 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.
I. INTRODUCTION
Accurate predictions of the properties of highenergydensity (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 properties^{1} (equations of state) of matter over this vast range, but also the response (transport) properties^{2} 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 magnetoradiationhydrodynamic 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 chargedparticle 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 chargedparticle transport coefficient comparison workshop^{3} was held in 2016. Orderofmagnitude 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 frameworks^{4,5} for uncertainty quantification. The second chargedparticle 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 alphaparticle transport (stopping power). We conclude in Sec. VI with a general discussion and a proposal for cases to be considered in a future workshop.
II. TRANSPORT COEFFICIENTS AS CLOSURES TO THE EQUATIONS OF HYDRODYNAMICS
To describe the behavior of currentcarrying 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 largescale 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 chargedparticle transport coefficient comparison workshop.
III. WORKSHOP CASES
The first chargedparticle transport coefficient comparison workshop^{3} 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 highpriority cases with the goal of generating more data at targeted points that could be useful for a broader model comparison. The Priority Level System^{10} 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 multiatom methods based on density functional theory (DFT);

Priority Level 2: direct connections to experiments, additional cases having longitudinal overlap with the first workshop, highvalue cases determined by data analysis, and conditions accessible to multiatom DFTbased methods;

Priority Level 3: data on isochors at densities having longitudinal overlap with the cases of the first workshop and highvalue 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 multiatom DFTbased codes, enabling direct comparisons among different implementations of firstprinciples models.
Priority Level .  Case ID .  Element(s) .  $ n species$ (cm^{–3}) .  $ \rho total$ (g cm^{–3}) .  T (eV) . 

1  H1  H  $ 5.98 \xd7 10 23$  1  2 
1  C1  C  $ 5.01 \xd7 10 23$  10  2 
1  CH1  CH  $ 4.63 \xd7 10 22$  1  2 
1  Al1  Al  $ 6.03 \xd7 10 22$  2.7  1 
1  Cu1  Cu  $ 8.49 \xd7 10 22$  8.96  1 
1  HCu1  HCu  $ 1.68 \xd7 10 22$  1.8  1 
2  Be1  Be  $ 1.23 \xd7 10 23$  1.84  4.4 
2  CH2  CH  $ 4.16 \xd7 10 22$  0.9  7.8 
2  Au1  Au  $ 5.91 \xd7 10 22$  19.32  10 
2  H3  H  $ 5.98 \xd7 10 24$  10  20 
Priority Level .  Case ID .  Element(s) .  $ n species$ (cm^{–3}) .  $ \rho total$ (g cm^{–3}) .  T (eV) . 

1  H1  H  $ 5.98 \xd7 10 23$  1  2 
1  C1  C  $ 5.01 \xd7 10 23$  10  2 
1  CH1  CH  $ 4.63 \xd7 10 22$  1  2 
1  Al1  Al  $ 6.03 \xd7 10 22$  2.7  1 
1  Cu1  Cu  $ 8.49 \xd7 10 22$  8.96  1 
1  HCu1  HCu  $ 1.68 \xd7 10 22$  1.8  1 
2  Be1  Be  $ 1.23 \xd7 10 23$  1.84  4.4 
2  CH2  CH  $ 4.16 \xd7 10 22$  0.9  7.8 
2  Au1  Au  $ 5.91 \xd7 10 22$  19.32  10 
2  H3  H  $ 5.98 \xd7 10 24$  10  20 
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 multiatom DFTbased codes—the current gold standard for simulating most plasmas with degenerate electrons and stronglycoupled 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 averageatom 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}
IV. MODELS AND EQUILIBRIUM PROPERTIES
Transport properties are mediated by interactions among electrons and ions. Approximating these interactions results in modeldependent predictions for atomicscale 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 averageatom models that compute spherically averaged properties, and multiatom DFTbased models including KohnSham molecular dynamics and realtime timedependent 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 exchangecorrelation 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).
Contributor(s) .  Description (code name and version) .  Institution(s) .  Model type . 

G. Röpke  Virial expansion^{12,13}  U. Rostock  Analytic 
L. Stanek  The Lee–More–Desjarlais model,^{14} Yukawa–Gibbs–Bogolyubov model,^{15} Yukawa viscosity model,^{16} and Stanton–Murillo transport model^{11,17}  SNL  Analytic 
L. Babati, S. Baalrud, N. Shaffer  Meanforce kinetic theory^{18–20} using averageatom potentials^{21} with temperaturedependent LDA XC functional^{22} (Scout)  U. Michigan, LLE  KT 
N. Shaffer  quantum Landau–Fokker–Planck kinetic theory with averageatom meanforce electron cross sections^{23} (qLFP)  LLE  KT 
G. Faussurier  Quantum averageatom model^{24–27} (QAAM)  CEA  AA 
S. Hansen  Quantum averageatom model with LDA XC functional^{28–30} (Bemuze)  SNL  AA 
G. Petrov  Averageatom calculation with Dirac exchange^{31} (AAMNRL 4)  NRL  AA 
L. Silvestri, Z. Johnson, M. Murillo, G. Petrov  Molecular dynamics simulation with input from TFDiracvon Weizsäcker averageatom model and integral equation theory^{32} (Sarkas^{33})  MSU, NRL  MD 
L. Stanek, S. Hansen  Molecular dynamics simulations with interatomic potential from an averageatom calculation (LAMMPS^{34})  SNL  MD 
M. Bethkenhagen, M. French, R. Redmer, M. Schörner  Quantum molecular dynamics simulation (VASP^{35–38} 5.4.4)  LULI, U. Rostock  DFTMD 
A. Blanchet, V. Recoules, F. Soubiran, M. Tacu  Quantum molecular dynamics simulation^{39} (Abinit^{40–42} 9.7.4)  CEA  DFTMD 
R. Clay, K. Cochrane, A. Dumi, M. Lentz, C. Melton, J. Townsend  Quantum molecular dynamics simulation (VASP^{35–38} 6.3.2)  SNL  DFTMD 
S. Hu, V. Karasiev  Quantum molecular dynamics simulation with temperaturedependent TSCANL XC functional in combination with thermal hybrid KDT0 for the Kubo–Greenwood calculation for selected cases (VASP^{35–38} 6.2/5.4.4)  LLE  DFTMD 
V. Sharma, L. Collins, A. White  Quantum molecular dynamics simulations with PBE XC functional (SHRED^{43})  LANL  DFTMD 
P. Suryanarayana, S. Kumar  Molecular dynamics using onthefly machine learning force field with DFT simulation using LDA XC functional (SPARC^{44–47})  Georgia Tech.  DFTMD 
A. Kononov, A. Baczewski  TDDFT simulations with LDA XC functional and bare Coulomb or PAW potentials (VASP^{35–38} 5.4.4 extension^{48,49})  SNL  TDDFTMD 
K. Nichols, S. Hu  TDDFT simulations with PBE XC functional and HGH pseudopotentials (SHRED^{43})  LLE  TDDFTMD 
A. White  Mixed stochasticdeterministic TDDFT simulations with PBE XC functional and bare Coulomb potentials (SHRED^{43})  LANL  TDDFTMD 
Contributor(s) .  Description (code name and version) .  Institution(s) .  Model type . 

G. Röpke  Virial expansion^{12,13}  U. Rostock  Analytic 
L. Stanek  The Lee–More–Desjarlais model,^{14} Yukawa–Gibbs–Bogolyubov model,^{15} Yukawa viscosity model,^{16} and Stanton–Murillo transport model^{11,17}  SNL  Analytic 
L. Babati, S. Baalrud, N. Shaffer  Meanforce kinetic theory^{18–20} using averageatom potentials^{21} with temperaturedependent LDA XC functional^{22} (Scout)  U. Michigan, LLE  KT 
N. Shaffer  quantum Landau–Fokker–Planck kinetic theory with averageatom meanforce electron cross sections^{23} (qLFP)  LLE  KT 
G. Faussurier  Quantum averageatom model^{24–27} (QAAM)  CEA  AA 
S. Hansen  Quantum averageatom model with LDA XC functional^{28–30} (Bemuze)  SNL  AA 
G. Petrov  Averageatom calculation with Dirac exchange^{31} (AAMNRL 4)  NRL  AA 
L. Silvestri, Z. Johnson, M. Murillo, G. Petrov  Molecular dynamics simulation with input from TFDiracvon Weizsäcker averageatom model and integral equation theory^{32} (Sarkas^{33})  MSU, NRL  MD 
L. Stanek, S. Hansen  Molecular dynamics simulations with interatomic potential from an averageatom calculation (LAMMPS^{34})  SNL  MD 
M. Bethkenhagen, M. French, R. Redmer, M. Schörner  Quantum molecular dynamics simulation (VASP^{35–38} 5.4.4)  LULI, U. Rostock  DFTMD 
A. Blanchet, V. Recoules, F. Soubiran, M. Tacu  Quantum molecular dynamics simulation^{39} (Abinit^{40–42} 9.7.4)  CEA  DFTMD 
R. Clay, K. Cochrane, A. Dumi, M. Lentz, C. Melton, J. Townsend  Quantum molecular dynamics simulation (VASP^{35–38} 6.3.2)  SNL  DFTMD 
S. Hu, V. Karasiev  Quantum molecular dynamics simulation with temperaturedependent TSCANL XC functional in combination with thermal hybrid KDT0 for the Kubo–Greenwood calculation for selected cases (VASP^{35–38} 6.2/5.4.4)  LLE  DFTMD 
V. Sharma, L. Collins, A. White  Quantum molecular dynamics simulations with PBE XC functional (SHRED^{43})  LANL  DFTMD 
P. Suryanarayana, S. Kumar  Molecular dynamics using onthefly machine learning force field with DFT simulation using LDA XC functional (SPARC^{44–47})  Georgia Tech.  DFTMD 
A. Kononov, A. Baczewski  TDDFT simulations with LDA XC functional and bare Coulomb or PAW potentials (VASP^{35–38} 5.4.4 extension^{48,49})  SNL  TDDFTMD 
K. Nichols, S. Hu  TDDFT simulations with PBE XC functional and HGH pseudopotentials (SHRED^{43})  LLE  TDDFTMD 
A. White  Mixed stochasticdeterministic TDDFT simulations with PBE XC functional and bare Coulomb potentials (SHRED^{43})  LANL  TDDFTMD 
A. Parameterized models
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 electronic^{11} 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 GibbsBogolyubov 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 fit^{51} 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 semianalytic models for plasma transport coefficients can be found in Refs. 52–55.
A second class of parameterized models include those based on kinetic theory (KT) and pairpotential molecular dynamics^{57} (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 input^{23} and meanforce kinetic theory^{18–20} for ionic transport coefficients that uses ion radial distribution functions as input. The MD codes^{33,34} 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., Yukawa^{17,58,59}), pair potentials obtained from averageatom calculations,^{57} or pair potentials from semiclassical methods.^{32} These models are slower than models used for inline evaluation, with fewminute runtimes for KT and tensofminutes 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}
B. Averageatom models
Averageatom models^{21,27,28,60,61} describe the electronic structure surrounding a single, averaged ion suspended in jellium. All of the averageatom models contributed to this workshop use Kohn–Sham orbitals to compute the electronic structure; we will use the term “quantumAA,” or simply AA, to denote averageatom models of this type. Following the pioneering work of Liberman,^{62} these allelectron, fully quantum models find a converged electron–ion potential (including approximate exchange and correlation effects) that supports a selfconsistent set of electronic orbitals. These orbitals provide information on the radial electron density and the energydependent electronic density of states (DOS). The densities of states from two independent AA models are shown in Fig. 4.
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 positiveenergy 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 quantumAA models give widely different predictions for $ Z *$, their underlying electronic densities of state are virtually identical. The choice of $ Z *$ also influences AAmodel predictions for ion–ion interaction potentials, which can be used to determine (spherically symmetric) ionic radial distribution functions^{63} 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 selfconsistent set of electronic and ionic properties, AA models can derive electron–ion collision rates through the Ziman equation^{27,64,65} that inform electronic transport properties. Coupled with a model for stopping numbers or dielectric functions, AA models can also calculate stopping powers.^{27,30}
C. Multiatom DFT models
The final category of models in this workshop are based on a DFT description of a multiatom system, including Kohn–Sham molecular dynamics^{49,66–74} (DFTMD, also known as quantum molecular dynamics or QMD) and realtime timedependent DFT^{75} (TDDFT). These firstprinciples models selfconsistently solve for a threedimensional electron density interacting with an ensemble of ions. DFTMD then simulates psscale 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, TDDFT models fsscale dynamics of electronic excitations in response to an external perturbation. To capture ionic disorder, TDDFT simulations in this workshop began from equilibrated structures obtained from DFTMD and thus we denote this model as TDDFTMD.
Within DFTMD, ionic transport properties are extracted from integrals of autocorrelation functions. Electronic transport properties are extracted from the Kubo–Greenwood^{76,77} formalism, with dynamic properties averaged over ionic configurations and extrapolated to zerofrequency for DC limits. A Kubo relation has also been proposed for electron–ion temperature relaxation rates from DFTMD,^{78,79} but was not applied in this workshop. While TDDFTMD 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 DFTMD and TDDFTMD are fundamentally characterized by an approximate choice of exchange and correlation (XC) potential to describe the meanfield 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 multiatom and threedimensional nature of DFTMD and TDDFTMD 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 DFTbased 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 socalled 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 metaGGAs and hybrid functionals represents an improved approximation for the total energy calculated within DFT.
These DFTMD and TDDFTMD models represent the state of the art for HED materials properties calculations. While these models still have difficulttoquantify 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 xray diffraction measurements; we illustrate radial distribution functions, g(r), in Fig. 5 for some of the cases of Table I. However, multiatom 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 manyelectron ions. To reduce computational expense, the number of ions may be restricted (potentially leading to finitesize effects^{57}), the electrons within ion cores may be included in psuedopotentials (approximating their behavior), and/or subcubic scaling algorithms that rely on sparsity or stochastic sampling may be applied.^{43,45,84–86}
V. TRANSPORT COEFFICIENTS
In this section, we describe how transport coefficients are computed within the atomicscale 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).
A. Electronic transport coefficients
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 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 generically^{11,17} or computed on a casebycase 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 approach^{92,93} 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 approach^{27,64,65} to calculate electron–ion collision rates, integrating over both ionic and electronic structure. The DFTMD codes obtain the DC conductivity by extrapolating the frequencydependent conductivity obtained by Kubo^{76} and Greenwood^{77} relations to the zero frequency limit.^{94,95} This extrapolation is often a significant source of uncertainty for evaluating the DC conductivity from multiatom DFTMDbased models. A description of the extrapolation procedure and other considerations for computing the Onsager coefficients, and thus the electronic and thermal conductivity, from multiatom DFTMD codes can be found in Refs. 96 and 97. Another source of uncertainty for the evaluation of the electronic transport coefficients is the meanfield treatment of electron–electron interactions.^{12,96} The method of DFTMD 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 κ_{e} 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 DFTMD 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.
Case ID/Submitter .  $ \sigma \u2009 ( 1 \Omega \u2009 cm 10 4 )$ .  $ \kappa e \u2009 ( erg s \u2009 cm \u2009 K 10 7 )$ .  $ \eta ( g cm \u2009 s 10 \u2212 3 )$ .  $ \kappa i \u2009 ( erg s \u2009 cm \u2009 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  DFTMD 
M. French  1.05 ± 0.02  6.4 ± 0.13  ⋯  ⋯  ⋯  HSE  DFTMD 
S. Hansen  $ 1 \u2212 0.42 + 1.50$  $ 7 \u2212 2.14 + 7.69$  ⋯  ⋯  10.23 ± 1.12  LDA  AA 
S. Hu  1.4 ± 0.10  7.5 ± 0.31  11.48  ⋯  ⋯  PBE  DFTMD 
S. Hu  1.25 ± 0.08  6.7 ± 0.26  13.36  ⋯  ⋯  TSCANL  DFTMD 
N. Shaffer  2.01  9.61  ⋯  ⋯  11.17  LDA  KT 
V. Sharma  1.4 ± 0.15  7.2 ± 0.41  ⋯  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
L. Stanek  ⋯  ⋯  12 ± 3.18  0.5 ± 0.11  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  12 ± 0.91  0.168 ± 0.002  ⋯  LDA  DFTMD 
J. Townsend  1.4 ± 0.004  8.05 ± 0.01  8 ± 1.64  ⋯  ⋯  LDA  DFTMD 
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  DFTMD 
G. Faussurier  1.14  6.44  ⋯  ⋯  10.55  LDA  AA 
S. Hansen  $ 1 \u2212 0.36 + 1.06$  $ 6 \u2212 2.02 + 5.99$  ⋯  ⋯  8 ± 3.52  LDA  AA 
V. Karasiev  1.69 ± 0.06  9.5 ± 0.16  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Karasiev  1.04 ± 0.06  6.9 ± 0.28  ⋯  ⋯  ⋯  Tr^{2}SCANL^{*}  DFTMD 
C. Melton  1.60 ± 0.01  8.80  28.2 ± 0.3  ⋯  ⋯  LDA  DFTMD 
N. Shaffer  1.87  9.39  ⋯  ⋯  11.40  LDA  KT 
V. Sharma  1.26 ± 0.06  7.7 ± 0.24  ⋯  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
L. Stanek  ⋯  ⋯  64 ± 14.57  0.14 ± 0.04  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  29 ± 1.61  1.02 ± 0.02  ⋯  LDA  DFTMD 
CH1 (1 g cm^{–3}, 2 eV)  
M. Bethkenhagen  0.174 ± 0.003  0.71 ± 0.02  ⋯  ⋯  ⋯  PBE  DFTMD 
R. Clay  0.168 ± 0.004  0.72 ± 0.01  6.5 ± 0.9  ⋯  ⋯  LDA  DFTMD 
V. Karasiev  0.19 ± 0.03  0.80 ± 0.06  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Karasiev  0.14 ± 0.02  0.74 ± 0.05  ⋯  ⋯  ⋯  TSCANL^{*}  DFTMD 
V. Sharma  0.188 ± 0.009  0.78 ± 0.02  ⋯  ⋯  ⋯  PBE  DFTMD 
F. Soubiran  0.18 ± 0.02  0.67 ± 0.02  8 ± 3.2  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
G. Faussurier  4.18  11.5  ⋯  ⋯  0.0218  LDA  AA 
S. Hansen  $ 4 \u2212 1.30 + 2.25$  $ 7 \u2212 2.36 + 4.06$  ⋯  ⋯  0.016 ± 0.002  LDA  AA 
V. Karasiev  2.03 ± 0.05  5.69 ± 0.15  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Karasiev  1.38 ± 0.08  4.75 ± 0.22  ⋯  ⋯  ⋯  TSCANL^{*}  DFTMD 
G. Petrov  5.65  15.3  ⋯  ⋯  0.0101  LDA  AA 
M. Schörner  2.6 ± 0.13  7.33 ± 0.53  ⋯  ⋯  ⋯  PBE  DFTMD 
N. Shaffer  2.15  4.09  ⋯  ⋯  0.036  LDA  KT 
V. Sharma  2.38 ± 0.07  6.46 ± 0.19  ⋯  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
L. Stanek  ⋯  ⋯  8 ± 3.13  0.014 ± 0.003  ⋯  LDA  MD 
Case ID/Submitter .  $ \sigma \u2009 ( 1 \Omega \u2009 cm 10 4 )$ .  $ \kappa e \u2009 ( erg s \u2009 cm \u2009 K 10 7 )$ .  $ \eta ( g cm \u2009 s 10 \u2212 3 )$ .  $ \kappa i \u2009 ( erg s \u2009 cm \u2009 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  DFTMD 
M. French  1.05 ± 0.02  6.4 ± 0.13  ⋯  ⋯  ⋯  HSE  DFTMD 
S. Hansen  $ 1 \u2212 0.42 + 1.50$  $ 7 \u2212 2.14 + 7.69$  ⋯  ⋯  10.23 ± 1.12  LDA  AA 
S. Hu  1.4 ± 0.10  7.5 ± 0.31  11.48  ⋯  ⋯  PBE  DFTMD 
S. Hu  1.25 ± 0.08  6.7 ± 0.26  13.36  ⋯  ⋯  TSCANL  DFTMD 
N. Shaffer  2.01  9.61  ⋯  ⋯  11.17  LDA  KT 
V. Sharma  1.4 ± 0.15  7.2 ± 0.41  ⋯  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
L. Stanek  ⋯  ⋯  12 ± 3.18  0.5 ± 0.11  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  12 ± 0.91  0.168 ± 0.002  ⋯  LDA  DFTMD 
J. Townsend  1.4 ± 0.004  8.05 ± 0.01  8 ± 1.64  ⋯  ⋯  LDA  DFTMD 
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  DFTMD 
G. Faussurier  1.14  6.44  ⋯  ⋯  10.55  LDA  AA 
S. Hansen  $ 1 \u2212 0.36 + 1.06$  $ 6 \u2212 2.02 + 5.99$  ⋯  ⋯  8 ± 3.52  LDA  AA 
V. Karasiev  1.69 ± 0.06  9.5 ± 0.16  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Karasiev  1.04 ± 0.06  6.9 ± 0.28  ⋯  ⋯  ⋯  Tr^{2}SCANL^{*}  DFTMD 
C. Melton  1.60 ± 0.01  8.80  28.2 ± 0.3  ⋯  ⋯  LDA  DFTMD 
N. Shaffer  1.87  9.39  ⋯  ⋯  11.40  LDA  KT 
V. Sharma  1.26 ± 0.06  7.7 ± 0.24  ⋯  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
L. Stanek  ⋯  ⋯  64 ± 14.57  0.14 ± 0.04  ⋯  LDA  MD 
P. Suryanarayana  ⋯  ⋯  29 ± 1.61  1.02 ± 0.02  ⋯  LDA  DFTMD 
CH1 (1 g cm^{–3}, 2 eV)  
M. Bethkenhagen  0.174 ± 0.003  0.71 ± 0.02  ⋯  ⋯  ⋯  PBE  DFTMD 
R. Clay  0.168 ± 0.004  0.72 ± 0.01  6.5 ± 0.9  ⋯  ⋯  LDA  DFTMD 
V. Karasiev  0.19 ± 0.03  0.80 ± 0.06  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Karasiev  0.14 ± 0.02  0.74 ± 0.05  ⋯  ⋯  ⋯  TSCANL^{*}  DFTMD 
V. Sharma  0.188 ± 0.009  0.78 ± 0.02  ⋯  ⋯  ⋯  PBE  DFTMD 
F. Soubiran  0.18 ± 0.02  0.67 ± 0.02  8 ± 3.2  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
G. Faussurier  4.18  11.5  ⋯  ⋯  0.0218  LDA  AA 
S. Hansen  $ 4 \u2212 1.30 + 2.25$  $ 7 \u2212 2.36 + 4.06$  ⋯  ⋯  0.016 ± 0.002  LDA  AA 
V. Karasiev  2.03 ± 0.05  5.69 ± 0.15  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Karasiev  1.38 ± 0.08  4.75 ± 0.22  ⋯  ⋯  ⋯  TSCANL^{*}  DFTMD 
G. Petrov  5.65  15.3  ⋯  ⋯  0.0101  LDA  AA 
M. Schörner  2.6 ± 0.13  7.33 ± 0.53  ⋯  ⋯  ⋯  PBE  DFTMD 
N. Shaffer  2.15  4.09  ⋯  ⋯  0.036  LDA  KT 
V. Sharma  2.38 ± 0.07  6.46 ± 0.19  ⋯  ⋯  ⋯  PBE  DFTMD 
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  DFTMD 
L. Stanek  ⋯  ⋯  8 ± 3.13  0.014 ± 0.003  ⋯  LDA  MD 
Case ID/Submitter .  $ \sigma \u2009 ( 1 \Omega \u2009 cm 10 4 )$ .  $ \kappa e \u2009 ( erg s \u2009 cm \u2009 K 10 7 )$ .  $ \eta ( g cm \u2009 s 10 \u2212 3 )$ .  $ \kappa i \u2009 ( erg s \u2009 cm \u2009 K 10 7 )$ .  g $ ( erg cm 3 K 10 20 )$ .  XC .  Model . 

Cu1 (8.96 g cm^{−3}, 1 eV)  
K. Cochrane  2  4.25  14 ± 0.5  ⋯  ⋯  PBE  DFTMD 
G. Faussurier  2.75  7.62  ⋯  ⋯  0.0133  LDA  AA 
S. Hansen  $ 6 \u2212 2.70 + 32.20$  $ 10 \u2212 4.68 + 55.93$  ⋯  ⋯  0.01 ± 0.015  LDA  AA 
S. Hu  0.90 ± 0.02  2.50 ± 0.03  16.87  ⋯  ⋯  PBE  DFTMD 
S. Hu  1.20 ± 0.01  3.75 ± 0.08  16.42  ⋯  ⋯  TSCANL  DFTMD 
G. Petrov  3.13  18.5  ⋯  ⋯  0.0116  LDA  AA 
F. Soubiran  1.80 ± 0.02  3.90 ± 0.03  19 ± 8.00  ⋯  ⋯  LDA  DFTMD 
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  DFTMD 
S. Hu  0.11 ± 0.004  0.463 ± 0.005  5.135  ⋯  ⋯  TSCANL  DFTMD 
M. Lentz  0.09 ± 0.002  0.39 ± 0.007  1.8 ± 0.58  ⋯  ⋯  LDA  DFTMD 
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 \u2212 0.04 + 0.23$  $ 5.55 \u2212 0.34 + 2.16$  ⋯  ⋯  0.4 ± 0.14  LDA  AA 
S. Hu  0.65 ± 0.02  7.2 ± 0.11  12.84  ⋯  ⋯  PBE  DFTMD 
S. Hu  0.65 ± 0.03  7.4 ± 0.18  6.33  ⋯  ⋯  TSCANL  DFTMD 
M. Schörner  0.66 ± 0.01  7.3 ± 0.17  ⋯  ⋯  ⋯  PBE  DFTMD 
N. Shaffer  0.81  9.62  ⋯  ⋯  0.749  LDA  KT 
V. Sharma  0.58 ± 0.01  6.3 ± 0.11  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Sharma  0.57 ± 0.01  6.1 ± 0.27  ⋯  ⋯  ⋯  PBE  (mix)DFTMD 
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  DFTMD 
V. Sharma  0.23 ± 0.015  7.52 ± 0.69  ⋯  ⋯  ⋯  PBE  (mix)DFTMD 
F. Soubiran  0.21 ± 0.002  6.5 ± 0.1  15.00  ⋯  ⋯  PBE  (ext)DFTMD 
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 \u2212 0.067 + 0.001$  $ 15.50 \u2212 1.45 + 0.01$  ⋯  ⋯  0.024 ± 0.0072  LDA  AA 
S. Hu  1.07 ± 0.03  26.2 ± 0.3  44 ± 1.00  ⋯  ⋯  TSCANL  DFTMD 
V. Karasiev  0.99 ± 0.02  24.7 ± 0.3  ⋯  ⋯  ⋯  TSCANL  DFTMD 
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)DFTMD 
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  DFTMD 
M. Bethkenhagen  14.8 ± 0.6  850.0 ± 34.0  ⋯  ⋯  ⋯  PBE  DFTMD 
Case ID/Submitter .  $ \sigma \u2009 ( 1 \Omega \u2009 cm 10 4 )$ .  $ \kappa e \u2009 ( erg s \u2009 cm \u2009 K 10 7 )$ .  $ \eta ( g cm \u2009 s 10 \u2212 3 )$ .  $ \kappa i \u2009 ( erg s \u2009 cm \u2009 K 10 7 )$ .  g $ ( erg cm 3 K 10 20 )$ .  XC .  Model . 

Cu1 (8.96 g cm^{−3}, 1 eV)  
K. Cochrane  2  4.25  14 ± 0.5  ⋯  ⋯  PBE  DFTMD 
G. Faussurier  2.75  7.62  ⋯  ⋯  0.0133  LDA  AA 
S. Hansen  $ 6 \u2212 2.70 + 32.20$  $ 10 \u2212 4.68 + 55.93$  ⋯  ⋯  0.01 ± 0.015  LDA  AA 
S. Hu  0.90 ± 0.02  2.50 ± 0.03  16.87  ⋯  ⋯  PBE  DFTMD 
S. Hu  1.20 ± 0.01  3.75 ± 0.08  16.42  ⋯  ⋯  TSCANL  DFTMD 
G. Petrov  3.13  18.5  ⋯  ⋯  0.0116  LDA  AA 
F. Soubiran  1.80 ± 0.02  3.90 ± 0.03  19 ± 8.00  ⋯  ⋯  LDA  DFTMD 
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  DFTMD 
S. Hu  0.11 ± 0.004  0.463 ± 0.005  5.135  ⋯  ⋯  TSCANL  DFTMD 
M. Lentz  0.09 ± 0.002  0.39 ± 0.007  1.8 ± 0.58  ⋯  ⋯  LDA  DFTMD 
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 \u2212 0.04 + 0.23$  $ 5.55 \u2212 0.34 + 2.16$  ⋯  ⋯  0.4 ± 0.14  LDA  AA 
S. Hu  0.65 ± 0.02  7.2 ± 0.11  12.84  ⋯  ⋯  PBE  DFTMD 
S. Hu  0.65 ± 0.03  7.4 ± 0.18  6.33  ⋯  ⋯  TSCANL  DFTMD 
M. Schörner  0.66 ± 0.01  7.3 ± 0.17  ⋯  ⋯  ⋯  PBE  DFTMD 
N. Shaffer  0.81  9.62  ⋯  ⋯  0.749  LDA  KT 
V. Sharma  0.58 ± 0.01  6.3 ± 0.11  ⋯  ⋯  ⋯  PBE  DFTMD 
V. Sharma  0.57 ± 0.01  6.1 ± 0.27  ⋯  ⋯  ⋯  PBE  (mix)DFTMD 
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  DFTMD 
V. Sharma  0.23 ± 0.015  7.52 ± 0.69  ⋯  ⋯  ⋯  PBE  (mix)DFTMD 
F. Soubiran  0.21 ± 0.002  6.5 ± 0.1  15.00  ⋯  ⋯  PBE  (ext)DFTMD 
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 \u2212 0.067 + 0.001$  $ 15.50 \u2212 1.45 + 0.01$  ⋯  ⋯  0.024 ± 0.0072  LDA  AA 
S. Hu  1.07 ± 0.03  26.2 ± 0.3  44 ± 1.00  ⋯  ⋯  TSCANL  DFTMD 
V. Karasiev  0.99 ± 0.02  24.7 ± 0.3  ⋯  ⋯  ⋯  TSCANL  DFTMD 
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)DFTMD 
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  DFTMD 
M. Bethkenhagen  14.8 ± 0.6  850.0 ± 34.0  ⋯  ⋯  ⋯  PBE  DFTMD 
The modest disagreement among DFTMD models may be attributable to several factors arising from the finite size of the simulations, which produces a discrete spectrum of allowable lowenergy 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 DFTMD simulations are typically carried out at increasing particle number. In this workshop, the typical number of particles that participants employed in their DFTMD calculations ranged from tens to hundreds. Despite all these choices, the DFTMD models tend to agree better with one another than with AA or parameterized models.
The larger disagreement among AA models, which perform an allelectron 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 quantumAA and DFTMD models and especially at low temperatures, the AAinformed 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 codependence on the equation of state^{101,103–107} and/or probe the optical response far from the DC limit.^{108,109} Recent advances with THz probes coupled with xray diffraction measurements at xray freeelectron laser facilities^{110} offer a promising approach to experimental validation.
B. Ionic transport coefficients
Ionic transport coefficients are governed by collisions among ions. In partially ionized systems, these collisions are mediated by electron distributions that screen longrange forces between ions. If binary collisions are assumed between ions, KT codes that use an effective Boltzmann approach^{17,18} (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.
Case ID .  $ \rho i c p eff$ $ ( erg Kcm 3 )$ .  $ \rho i c p mon$ $ ( erg Kcm 3 )$ .  $ \rho i c p eff \rho i c p mon$ . 

H1  1.91 $ \xd7 10 8$  2.06 $ \xd7 10 8$  0.93 
C1  6.35 $ \xd7 10 8$  1.73 $ \xd7 10 8$  3.67 
Al1  4.14 $ \xd7 10 7$  2.08 $ \xd7 10 7$  1.99 
Cu1  ⋯  2.93 $ \xd7 10 7$  ⋯ 
Be1  ⋯  4.24 $ \xd7 10 7$  ⋯ 
Au1  ⋯  2.04 $ \xd7 10 7$  ⋯ 
H3  ⋯  2.06 $ \xd7 10 9$  ⋯ 
Case ID .  $ \rho i c p eff$ $ ( erg Kcm 3 )$ .  $ \rho i c p mon$ $ ( erg Kcm 3 )$ .  $ \rho i c p eff \rho i c p mon$ . 

H1  1.91 $ \xd7 10 8$  2.06 $ \xd7 10 8$  0.93 
C1  6.35 $ \xd7 10 8$  1.73 $ \xd7 10 8$  3.67 
Al1  4.14 $ \xd7 10 7$  2.08 $ \xd7 10 7$  1.99 
Cu1  ⋯  2.93 $ \xd7 10 7$  ⋯ 
Be1  ⋯  4.24 $ \xd7 10 7$  ⋯ 
Au1  ⋯  2.04 $ \xd7 10 7$  ⋯ 
H3  ⋯  2.06 $ \xd7 10 9$  ⋯ 
Here, we compare the shear viscosity, η, and the ion thermal conductivity, κ_{i}. 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 strongcoupling 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 DFTMD models.
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 \u2009 order \u2009 o f \u2009 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 selfdiffusion coefficient calculated from a variety of pairinteraction potentials used in MD simulations of dense plasmas^{57} where the maximum differences were on the order of ten between pairpotential 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 finitesize effects^{57}) 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 finitesize 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 functions^{112} and by exploiting stationarity of the autocorrelation function.^{111}
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 employed^{111} 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 DFTMD 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 DFTMD 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 selfdiffusion 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)].
From Figs. 10 and 11, three things are clear. First, due to the significant computational cost, the DFTMD based simulation approaches are often plagued by finitesize 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.
C. Electron–ion temperature relaxation rate
D. Stopping power
Finally, we compare electronic stopping powers for alpha particles traversing some singlespecies 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 lowvelocity regime, likely due to different parameterizations of the stopping number in a uniform electron gas. TDDFTMD is generally expected to be more accurate and can serve as a valuable benchmark for AA models.^{30} However, the high computational cost of TDDFTMD calculations limited this method to a subset of the Priority Level 1 cases illustrated in Figs. 12(a)–12(c).
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 $ \xd7 10 8$  9.13 $ \xd7 10 9$  LDA  AA  1.7 $ \xd7 10 2$ 
S. Hansen  4.90 $ \xd7 10 8$  8.36 $ \xd7 10 9$  LDA  AA  ⋯ 
A. Kononov  5.47 $ \xd7 10 8$  6.39 $ \xd7 10 9$  LDA  TDDFTMD  3.8 $ \xd7 10 8$ 
K. Nichols  4.38 $ \xd7 10 8$  6.21 $ \xd7 10 9$  PBE  TDDFTMD  1.5 $ \xd7 10 8$ 
A. White  5.47 $ \xd7 10 8$  6.68 $ \xd7 10 9$  PBE  (mix)TDDFTMD  1.7 $ \xd7 10 9$ 
C1 (10 g cm^{–3}, 2 eV)  
G. Faussurier  6.19 $ \xd7 10 8$  1.68 $ \xd7 10 10$  LDA  AA  1.9 $ \xd7 10 2$ 
S. Hansen  6.92 $ \xd7 10 8$  1.54 $ \xd7 10 10$  LDA  AA  ⋯ 
A. Kononov  6.56 $ \xd7 10 8$  1.26 $ \xd7 10 10$  LDAa  TDDFTMD  5.2 $ \xd7 10 6$ 
A. Kononov  6.56 $ \xd7 10 8$  1.06 $ \xd7 10 10$  LDAb  TDDFTMD  9.6 $ \xd7 10 7$ 
CH1 (1 g cm^{–3}, 2 eV)  
A. Kononov  4.38 $ \xd7 10 8$  2.32 $ \xd7 10 9$  LDA  TDDFTMD  2.4 $ \xd7 10 8$ 
Al1 (2.7 g cm^{–3}, 1 eV)  
G. Faussurier  3.10 $ \xd7 10 8$  6.43 $ \xd7 10 9$  LDA  AA  1.1 $ \xd7 10 2$ 
S. Hansen  3.47 $ \xd7 10 8$  5.66 $ \xd7 10 9$  LDA  AA  ⋯ 
A. Kononov  3.83 $ \xd7 10 8$  3.40 $ \xd7 10 9$  LDA  TDDFTMD  5.2 $ \xd7 10 9$ 
Cu1 (8.96 g cm^{−3}, 1 eV)  
G. Faussurier  3.90 $ \xd7 10 8$  7.78 $ \xd7 10 9$  LDA  AA  6.4 $ \xd7 10 2$ 
Be1 (1.84 g cm^{–3}, 4.4 eV)  
G. Faussurier  3.48 $ \xd7 10 8$  6.32 $ \xd7 10 9$  LDA  AA  2.3 $ \xd7 10 2$ 
S. Hansen  3.89 $ \xd7 10 8$  5.96 $ \xd7 10 9$  LDA  AA  ⋯ 
Au1 (19.32 g cm^{–3}, 10 eV)  
G. Faussurier  4.92 $ \xd7 10 8$  8.13 $ \xd7 10 9$  LDA  AA  2.4 $ \xd7 10 2$ 
S. Hansen  4.90 $ \xd7 10 8$  7.45 $ \xd7 10 9$  LDA  AA  ⋯ 
H3 (10 g cm^{–3}, 20 eV)  
G. Faussurier  8.74 $ \xd7 10 8$  2.57 $ \xd7 10 10$  LDA  AA  2.3 $ \xd7 10 2$ 
S. Hansen  9.78 $ \xd7 10 8$  2.38 $ \xd7 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 $ \xd7 10 8$  9.13 $ \xd7 10 9$  LDA  AA  1.7 $ \xd7 10 2$ 
S. Hansen  4.90 $ \xd7 10 8$  8.36 $ \xd7 10 9$  LDA  AA  ⋯ 
A. Kononov  5.47 $ \xd7 10 8$  6.39 $ \xd7 10 9$  LDA  TDDFTMD  3.8 $ \xd7 10 8$ 
K. Nichols  4.38 $ \xd7 10 8$  6.21 $ \xd7 10 9$  PBE  TDDFTMD  1.5 $ \xd7 10 8$ 
A. White  5.47 $ \xd7 10 8$  6.68 $ \xd7 10 9$  PBE  (mix)TDDFTMD  1.7 $ \xd7 10 9$ 
C1 (10 g cm^{–3}, 2 eV)  
G. Faussurier  6.19 $ \xd7 10 8$  1.68 $ \xd7 10 10$  LDA  AA  1.9 $ \xd7 10 2$ 
S. Hansen  6.92 $ \xd7 10 8$  1.54 $ \xd7 10 10$  LDA  AA  ⋯ 
A. Kononov  6.56 $ \xd7 10 8$  1.26 $ \xd7 10 10$  LDAa  TDDFTMD  5.2 $ \xd7 10 6$ 
A. Kononov  6.56 $ \xd7 10 8$  1.06 $ \xd7 10 10$  LDAb  TDDFTMD  9.6 $ \xd7 10 7$ 
CH1 (1 g cm^{–3}, 2 eV)  
A. Kononov  4.38 $ \xd7 10 8$  2.32 $ \xd7 10 9$  LDA  TDDFTMD  2.4 $ \xd7 10 8$ 
Al1 (2.7 g cm^{–3}, 1 eV)  
G. Faussurier  3.10 $ \xd7 10 8$  6.43 $ \xd7 10 9$  LDA  AA  1.1 $ \xd7 10 2$ 
S. Hansen  3.47 $ \xd7 10 8$  5.66 $ \xd7 10 9$  LDA  AA  ⋯ 
A. Kononov  3.83 $ \xd7 10 8$  3.40 $ \xd7 10 9$  LDA  TDDFTMD  5.2 $ \xd7 10 9$ 
Cu1 (8.96 g cm^{−3}, 1 eV)  
G. Faussurier  3.90 $ \xd7 10 8$  7.78 $ \xd7 10 9$  LDA  AA  6.4 $ \xd7 10 2$ 
Be1 (1.84 g cm^{–3}, 4.4 eV)  
G. Faussurier  3.48 $ \xd7 10 8$  6.32 $ \xd7 10 9$  LDA  AA  2.3 $ \xd7 10 2$ 
S. Hansen  3.89 $ \xd7 10 8$  5.96 $ \xd7 10 9$  LDA  AA  ⋯ 
Au1 (19.32 g cm^{–3}, 10 eV)  
G. Faussurier  4.92 $ \xd7 10 8$  8.13 $ \xd7 10 9$  LDA  AA  2.4 $ \xd7 10 2$ 
S. Hansen  4.90 $ \xd7 10 8$  7.45 $ \xd7 10 9$  LDA  AA  ⋯ 
H3 (10 g cm^{–3}, 20 eV)  
G. Faussurier  8.74 $ \xd7 10 8$  2.57 $ \xd7 10 10$  LDA  AA  2.3 $ \xd7 10 2$ 
S. Hansen  9.78 $ \xd7 10 8$  2.38 $ \xd7 10 10$  LDA  AA  ⋯ 
For this case allelectron calculations with bare Coulomb potentials were carried out.
For this case pseudopotentials were employed with 4 valence electrons per C ion.
Figure 12(a) includes several TDDFTMD datasets for alphaparticle stopping in case H1 that were computed using different codes and methodological details, including deterministic vs mixed stochasticdeterministic TDDFT variants, HGH pseudopotentials vs bare Coulomb potentials, and LDA vs PBE XC functionals. Additionally, some of the TDDFT calculations optimized the alphaparticle trajectory to representatively sample a cubic simulation cell and mitigate finitesize effects,^{73} whereas others used an elongated simulation cell with the alpha particle traveling along the long direction.
Nonetheless, the different TDDFTMD stopping power datasets in Fig. 12(a) agree amongst each other quite well, and the minor discrepancies reflect the sensitivity that even this firstprinciples model can have to methodological choices. Furthermore, high computational costs can make it challenging to assess convergence and quantify uncertainties in TDDFTMD stopping powers. A separate article within this special issue scrutinizes the sources of these small discrepancies and finds highest sensitivities to the pseudopotential approximation, finitesize errors, and alphaparticle trajectory choice.^{116}
In Fig. 12(a), all of the TDDFTMD datasets agree quite well with both averageatom models beyond the stopping power peak. Good agreement is also obtained for case C1 shown in Fig. 12(b), where one TDDFTMD 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 TDDFTMD predictions. To reduce computational costs, these TDDFTMD calculations neglected contributions from Al core electrons, which become increasingly significant for fast projectiles^{73} 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 uniformelectrongas approximation used by the AA models may not accurately capture lowvelocity 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 formula^{117} that becomes increasingly important for slow projectiles.
VI. CONCLUSIONS AND FUTURE WORKSHOPS
The results of this workshop have quantified differences between stateoftheart 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 DFTMD models.
In this second iteration of the chargedparticle 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 costbenefit 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 indepth 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 highpriority cases for the next workshop that explore more extreme conditions. We have chosen a single temperature of 3 eV, which remains within reach for multiatom DFTbased 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 semilongitudinal study that will explore the persistence of multiatom effects on ionic and electronic structure. To facilitate further exploration of mixtures, we have pressurematched 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.
Element(s) .  $ n species$ (cm^{−3}) .  $ \rho total$ (g cm^{–3}) .  T (eV) . 

H  $ 1.8 \xd7 10 24$  3  3 
C  $ 5.0 \xd7 10 23$  10  3 
CH  $ 3.9 \xd7 10 23$  8.4  3 
Be  $ 3.2 \xd7 10 23$  4.7  3 
Al  $ 1.4 \xd7 10 23$  6.4  3 
Cu  $ 1.7 \xd7 10 23$  18.3  3 
Element(s) .  $ n species$ (cm^{−3}) .  $ \rho total$ (g cm^{–3}) .  T (eV) . 

H  $ 1.8 \xd7 10 24$  3  3 
C  $ 5.0 \xd7 10 23$  10  3 
CH  $ 3.9 \xd7 10 23$  8.4  3 
Be  $ 3.2 \xd7 10 23$  4.7  3 
Al  $ 1.4 \xd7 10 23$  6.4  3 
Cu  $ 1.7 \xd7 10 23$  18.3  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 largescale HED simulations of both astrophysical objects and inertial confinement fusion experiments.
ACKNOWLEDGMENTS
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. MendozaOlivera. 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 multimission 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. DENA0003525. 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. DEAC5207NA27344. 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. DENA0004128 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. DENA0004146). 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. DENA0004144). 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.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
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).
DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.
APPENDIX: ADDITIONAL DATA
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.
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  DFTMD 
M. French  4.75  HSE  DFTMD 
S. Hansen  5.46  LDA  AA 
S. Hu  4.82 ± 0.07  PBE  DFTMD 
S. Hu  4.81 ± 0.06  TSCANL  DFTMD 
P. Suryanarayana  4.79 ± 0.01  LDA  DFTMD 
F. Soubiran  4.82  PBE  DFTMD 
J. Townsend  4.73 ± 0.004  LDA  DFTMD 
C1 (10 g cm^{–3}, 2 eV)  
M. Bethkenhagen  28.6  PBE  DFTMD 
G. Faussurier  43.5  LDA  AA 
S. Hansen  34.7  LDA  AA 
V. Karasiev  28.5  PBE  DFTMD 
V. Karasiev  28.7  Tr^{2}SCANL  DFTMD 
C. Melton  28.1  LDA  DFTMD 
P. Suryanarayana  28.93 ± 0.02  LDA  DFTMD 
F. Soubiran  28.5 ± 0.002  PBE  DFTMD 
CH1 (1 g cm^{–3}, 2 eV)  
M. Bethkenhagen  0.370  PBE  DFTMD 
R. Clay  0.338  LDA  DFTMD 
V. Karasiev  0.370  PBE  DFTMD 
V. Karasiev  0.330  TSCANL  DFTMD 
F. Soubiran  0.368 ± 0.001  PBE  DFTMD 
Al1 (2.7 g cm^{–3}, 1 eV)  
A. Dumi  0.513 ± 0.001  LDA  DFTMD 
G. Faussurier  1.01  LDA  AA 
S. Hansen  0.808  LDA  AA 
V. Karasiev  0.443  PBE  DFTMD 
V. Karasiev  0.425  TSCANL  DFTMD 
M. Schörner  0.441 ± 0.001  PBE  DFTMD 
F. Soubiran  0.440 ± 0.001  GGA  DFTMD 
Cu1 (8.96 g cm^{–3}, 1 eV)  
K. Cochrane  0.766  PBE  DFTMD 
G. Faussurier  1.67  LDA  AA 
S. Hansen  1.04  LDA  AA 
S. Hu  0.377 ± 0.053  PBE  DFTMD 
S. Hu  0.340 ± 0.053  TSCANL  DFTMD 
F. Soubiran  0.651 ± 0.003  LDA  DFTMD 
HCu1 (1.8 g cm^{–3}, 1 eV)  
S. Hu  0.02 ± 0.0158  PBE  DFTMD 
S. Hu  0.01 ± 0.0139  TSCANL  DFTMD 
M. Lentz  0.0311 ± 0.0001  LDA  DFTMD 
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  DFTMD 
S. Hu  2.29 ± 0.04  TSCANL  DFTMD 
M. Schörner  2.37 ± 0.003  PBE  DFTMD 
CH2 (0.9 g cm^{–3}, 7.8 eV)  
F. Soubiran  2.04 ± 0.002  PBE  (ext)DFTMD 
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  DFTMD 
V. Karasiev  5.76  TSCANL  DFTMD 
F. Soubiran  6.01 ± 0.01  LDA  (ext)DFTMD 
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  DFTMD 
M. French  4.75  HSE  DFTMD 
S. Hansen  5.46  LDA  AA 
S. Hu  4.82 ± 0.07  PBE  DFTMD 
S. Hu  4.81 ± 0.06  TSCANL  DFTMD 
P. Suryanarayana  4.79 ± 0.01  LDA  DFTMD 
F. Soubiran  4.82  PBE  DFTMD 
J. Townsend  4.73 ± 0.004  LDA  DFTMD 
C1 (10 g cm^{–3}, 2 eV)  
M. Bethkenhagen  28.6  PBE  DFTMD 
G. Faussurier  43.5  LDA  AA 
S. Hansen  34.7  LDA  AA 
V. Karasiev  28.5  PBE  DFTMD 
V. Karasiev  28.7  Tr^{2}SCANL  DFTMD 
C. Melton  28.1  LDA  DFTMD 
P. Suryanarayana  28.93 ± 0.02  LDA  DFTMD 
F. Soubiran  28.5 ± 0.002  PBE  DFTMD 
CH1 (1 g cm^{–3}, 2 eV)  
M. Bethkenhagen  0.370  PBE  DFTMD 
R. Clay  0.338  LDA  DFTMD 
V. Karasiev  0.370  PBE  DFTMD 
V. Karasiev  0.330  TSCANL  DFTMD 
F. Soubiran  0.368 ± 0.001  PBE  DFTMD 
Al1 (2.7 g cm^{–3}, 1 eV)  
A. Dumi  0.513 ± 0.001  LDA  DFTMD 
G. Faussurier  1.01  LDA  AA 
S. Hansen  0.808  LDA  AA 
V. Karasiev  0.443  PBE  DFTMD 
V. Karasiev  0.425  TSCANL  DFTMD 
M. Schörner  0.441 ± 0.001  PBE  DFTMD 
F. Soubiran  0.440 ± 0.001  GGA  DFTMD 
Cu1 (8.96 g cm^{–3}, 1 eV)  
K. Cochrane  0.766  PBE  DFTMD 
G. Faussurier  1.67  LDA  AA 
S. Hansen  1.04  LDA  AA 
S. Hu  0.377 ± 0.053  PBE  DFTMD 
S. Hu  0.340 ± 0.053  TSCANL  DFTMD 
F. Soubiran  0.651 ± 0.003  LDA  DFTMD 
HCu1 (1.8 g cm^{–3}, 1 eV)  
S. Hu  0.02 ± 0.0158  PBE  DFTMD 
S. Hu  0.01 ± 0.0139  TSCANL  DFTMD 
M. Lentz  0.0311 ± 0.0001  LDA  DFTMD 
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  DFTMD 
S. Hu  2.29 ± 0.04  TSCANL  DFTMD 
M. Schörner  2.37 ± 0.003  PBE  DFTMD 
CH2 (0.9 g cm^{–3}, 7.8 eV)  
F. Soubiran  2.04 ± 0.002  PBE  (ext)DFTMD 
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  DFTMD 
V. Karasiev  5.76  TSCANL  DFTMD 
F. Soubiran  6.01 ± 0.01  LDA  (ext)DFTMD 
Case/Submitter .  T (eV) .  $ \sigma \u2009 ( 1 \Omega \u2009 cm 10 4 )$ .  η $ ( g cm \u2009 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  2  ⋯  0.00901  LDA  KT 
L. Babati  5  ⋯  0.0233  LDA  KT 
L. Babati  8  ⋯  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  DFTMD 
L. Babati  50  ⋯  0.491  LDA  KT 
P. Suryanarayana  64.6  ⋯  2.0 ± 0.2  LDA  DFTMD 
L. Babati  80  ⋯  1.11  LDA  KT 
P. Suryanarayana  86.2  ⋯  3.1 ± 0.3  LDA  DFTMD 
L. Babati  100  ⋯  1.67  LDA  KT 
G. Röpke  100  13 ± 0.25  ⋯  ⋯  Analytic 
P. Suryanarayana  107.7  ⋯  4.5 ± 0.5  LDA  DFTMD 
P. Suryanarayana  129.3  ⋯  6.1 ± 0.7  LDA  DFTMD 
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  8  ⋯  0.0715  LDA  KT 
N. Shaffer  8  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) .  $ \sigma \u2009 ( 1 \Omega \u2009 cm 10 4 )$ .  η $ ( g cm \u2009 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  2  ⋯  0.00901  LDA  KT 
L. Babati  5  ⋯  0.0233  LDA  KT 
L. Babati  8  ⋯  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  DFTMD 
L. Babati  50  ⋯  0.491  LDA  KT 
P. Suryanarayana  64.6  ⋯  2.0 ± 0.2  LDA  DFTMD 
L. Babati  80  ⋯  1.11  LDA  KT 
P. Suryanarayana  86.2  ⋯  3.1 ± 0.3  LDA  DFTMD 
L. Babati  100  ⋯  1.67  LDA  KT 
G. Röpke  100  13 ± 0.25  ⋯  ⋯  Analytic 
P. Suryanarayana  107.7  ⋯  4.5 ± 0.5  LDA  DFTMD 
P. Suryanarayana  129.3  ⋯  6.1 ± 0.7  LDA  DFTMD 
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  8  ⋯  0.0715  LDA  KT 
N. Shaffer  8  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 