Comparing ab initio and quantum-kinetic approaches to electron transport in warm dense matter Open Access

The term “warm dense matter” (WDM) refers to a state of matter that is too dense and cold to be modeled as a classical plasma, yet too hot to be modeled as a liquid metal or solid. In laboratory settings, this roughly corresponds to densities between 1/100 and 100 times a material's ambient solid density and temperatures up to a few tens of electronvolts. Most materials at WDM conditions are electrically conducting, but their electronic properties are exceedingly difficult to measure. Even for basic transport properties like electrical and thermal conductivity, few experimental results exist, and those that do are plagued by large methodological uncertainties.^1–8 For this reason, the understanding of electronic transport in WDM is largely informed by simulations and theoretical models.

Due to the lack of experimental data for WDM, expensive atomic-scale simulations based on density functional theory (DFT) serve as the benchmark for judging the accuracy of simplified theoretical models. However, even the most high-fidelity WDM simulation methods in current use involve approximations that are difficult to quantify. As evidenced by recent workshops,^9,10 different methodological choices in DFT simulations can lead to surprisingly large disagreements in predictions of electronic properties of WDM.

The main purpose of this work is to present what we consider to be the state of the art in both ab initio simulations and theoretical modeling for the electronic transport properties of WDM. For the former, we present DFT-based molecular dynamics simulations using advanced thermal exchange–correlation (XC) functionals and the Kubo–Greenwood (KG) approach for electronic transport. For the latter, we present quantum kinetic theory results based on an average-atom plasma model. The kinetic theory is a quantum Landau–Fokker–Planck (qLFP) equation. In some cases, the agreement is remarkably good, while in others, it is poor. We find that the kinetic theory predictions agree well with the ab initio results whenever the latter predicts that the electronic structure is not especially complex. However, for most cases considered here, this is not so, and in fact the electronic transport properties can be very sensitive to the details of the electronic structure, such as the exchange–correlation potential. Nevertheless, these comparisons indicate clear directions for improvement in both the simulations and quantum kinetic theory. On the simulation side, we argue that advanced DFT functionals designed for the entire range of thermodynamic conditions that provide enhanced accuracy for WDM applications must become the norm in order to continue using their predictions for benchmarking theories. On the theoretical side, it seems that a major limitation of quantum kinetic theory is its reliance on a nearly-free-electron model, so future model development may benefit from relaxing this approximation.

Finally, some attention is given to electron dynamical properties. The mean free path of energetic electrons is controlled mainly by dynamically screened electron–electron interactions. We investigate this process using both time-dependent density functional theory and simpler kinetic theory approaches. Consistent with previous work on the subject,¹¹ we find that even in plasmas that appear to be simple mixtures of ions and free electrons, there are discrepancies between the time-dependent density functional theory results and the kinetic theories. These discrepancies are difficult to explain and indicate a need to better understand dynamic screening and response physics in dense plasmas.

The remainder of the paper is structured as follows: In Sec. II, simulation and theoretical approaches used in this work are reviewed. In Sec. III, some static electronic properties of WDM matter are compared using each method. In Sec. IV, electronic transport properties are compared and connected to relevant static properties. In Sec. V, electron dynamics are compared between time-dependent DFT (TD-DFT) and kinetic theory predictions. In Sec. VI, we offer some concluding remarks as well as outlook and recommendations for making further progress.

The workhorse method for achieving high-fidelity predictions of electronic structure in WDM is to couple a quantum DFT treatment of electrons to classical ion dynamics, a method colloquially known as “quantum molecular dynamics” (QMD). For a fixed arrangement of ions, DFT provides the electronic density and free energy landscape. This free energy landscape determines the electronic contribution to the force on each ion via the Hellman–Feynman theorem. The inter-ionic forces are also calculated and the ion positions and velocities are advanced according to classical dynamics, yielding a new ion arrangement to continue the simulation loop.

In the past decade, technical advances in QMD simulation for WDM have progressed along two main axes. The first is improved accuracy in the form of new XC functionals, especially those that take explicit account of temperature and/or exact exchange.^12–16 The second is algorithmic improvements, which circumvent the poor scaling of conventional DFT methods with respect to the number of electrons (⁠ $∼ N 3$⁠) and temperature (⁠ $∼ T 3$⁠).^17–20 

In this work, specific interest is placed on how recently developed advanced thermal XC functionals differ in their predictions of structural and transport properties compared to the widely used PBE functional. XC functionals are often organized into a hierarchy or “Jacob's ladder,” of which the lowest rung contains local density functionals, higher rungs account for gradients in the electronic density, and even higher rungs incorporate the non-local effect of the electron exchange and Coulomb interaction.²¹ PBE is a generalized gradient approximation (GGA) functional, meaning that it depends on both the local value and the gradient of the electron density. It is also a ground-state functional, meaning it is formulated only to be accurate at zero temperature. Below, results will also be presented using the T-SCAN-L and T-r²-SCAN-L functionals, which are both thermal meta-GGA functionals.¹⁶ They go beyond PBE by adding explicit dependence on the electron temperature and the Laplacian of the density. The T-SCAN-L functional is expected to be slightly more accurate than T-r²-SCAN-L, but it is also more expensive to evaluate due to slower convergence, and it is prone to numerical instabilities. For the purpose of this work, both can be considered state-of-the-art functionals for QMD simulations of WDM.

Select calculations were performed using the KDT0 thermal hybrid functional, which combines a thermal GGA functional with the exact finite-temperature Hartree–Fock exchange functional. It is believed to provide the most accurate treatment of exchange in WDM, but it is expensive to evaluate since it is a non-local functional that depends explicitly on all the Kohn–Sham orbitals.¹⁵ For this reason, we do not use KDT0 for the QMD simulations, instead reserving its use for calculations of transport properties in a few cases.

It is important to recognize that QMD as described above has no notion of electron dynamics; the electrons adiabatically follow the ions. This is advantageous for studying static properties such as the equation of state as well as ion dynamic properties like viscosity, since it means that the choice of MD time step is only constrained by the relatively slow ion dynamics. However, the lack of electron dynamics means that electronic transport properties, such as electrical and thermal conductivity, require an alternative method.

In practice, electronic transport properties are almost always obtained from QMD simulations using the Kubo–Greenwood (KG) approach. Kubo derived exact formulas for the transport coefficients, most famously electrical conductivity, by considering the linear response of an interacting many-electron system to external perturbations.²² Greenwood used an independent-electron basis and time-dependent perturbation theory to derive an explicit formula for electrical conductivity.^23,24 One can obtain Greenwood's result when a non-interacting electron model is used to evaluate the Kubo formula for electrical conductivity.²⁵ 

The KG formalism allows the evaluation of electronic transport coefficients using the Kohn–Sham orbitals. In doing so, however, the only scattering process taken into account is that from the ions (“impurities” in condensed matter language). Other scattering mechanisms such as the electron–phonon interaction in solid phases or the electron–electron interaction beyond the static mean-field approximation are not included in QMD+KG predictions of transport coefficients.²⁶ That said, in systems where electron–ion scattering is expected to be dominant, QMD with KG is currently the highest-fidelity method that is practical for WDM applications.

Compared to QMD+KG, quantum kinetic theory has much lower fidelity but greater flexibility. Rather than working in terms of individual electronic states as in DFT-based simulations, one instead takes an interest in the statistical properties of the electrons. In quantum kinetic theory, the object of interest is the electron distribution function, $f ( x , p , t )$⁠, which is the expected number of electrons having momentum $p$ at the space-time point $( x , t )$⁠. More generally, one can work in terms of the (retarded) one-electron Green's function, $G < ( x , p , t , E )$⁠, which contains additional information about the energy spectrum of electrons.²⁷ The simpler kinetic theory considered in this work follows from assuming free-electron dispersion, $G < ( x , p , t , E ) ∝ δ [ E − p 2 / ( 2 m ) ] f ( x , p , t )$⁠, which makes for an easier connection to classical kinetic theory at the cost of fidelity.

One may derive equations of motion for the distribution function that look essentially like the typical kinetic equation of classical plasma theory, but with corrections to the collision operators to account for the exclusion principle.²⁸ These are illustrated in Fig. 1, with the exact Kadanoff–Baym equations for the Green's function at the top. The Kadanoff–Baym equations require knowledge of the self-energy. While this is unknown in general, there exists a rigorous framework for developing systematic approximations to the self-energy.^27,28 The Markovian T-matrix (or “ladder”) approximation leads to the quantum Boltzmann collision operator (or Uehling–Uhlenbeck operator²⁹) which allows for arbitrarily strong binary collisions. The input to the theory is a scattering T-matrix or collision cross section. Another route, based on the random phase approximation to the self-energy, leads to the quantum Lenard–Balescu equation, with weak but dynamically screened collisions. The input to this theory is the plasma dielectric function. The quantum Landau–Fokker–Planck (qLFP) equation follows from either the weak-scattering limit of Boltzmann or the static-screening limit of Lenard–Balescu. The input to the qLFP equation is a Coulomb logarithm. These are the three threads of traditional plasma kinetic theory, but we also wish to mention work to generalize the Lenard–Balescu framework beyond the random phase approximation^30–32 and efforts to develop a theory of plasma transport coefficients amenable to systematic improvements.^33–35 

In order to apply plasma kinetic equations to WDM in a way that connects with classical theory, an important concession needs to be made: a strict delineation of “free” and “bound” electrons. The bound electrons are associated with the nuclei to define the ions so that the free electrons experience a “bare” ion potential interaction containing both the Coulomb potential of the nucleus and the screening by the bound electrons. There is no unique way to segregate the free and bound electrons, so the results of the kinetic theory are as much a function of the ionization model as the collision model.

The kinetic theory results shown in this work used the average-atom two-component plasma (AA-TCP) model.^36,37 In short, it is a DFT-based model of the approximate spherically symmetric electronic structure around each nucleus. The AA is the “average atom.” The AA density and potential are then used to define which electrons are bound and which are free, which is sufficient to establish a two-component plasma model of ions and free electrons. This model has been shown to give a reasonable description of the effective ionization state and electron–ion interaction for matter with well-defined free-electron-like conduction states, i.e., plasmas and “simple” metals. These are essentially the same requirements for applying plasma kinetic theory in the first place. The AA-TCP model is not suitable for semiconductors or insulators, and it can give surprising results for metals with complex valence structure.

The AA-TCP model can also be used to evaluate the electron–ion and electron–electron potentials of mean force.^38,39 These describe the effective screened interaction between a point ion or point electron and a free electron of the plasma. From these potentials, one may obtain the elastic scattering cross sections needed in a Boltzmann kinetic theory,⁴⁰ which are further reduced to scalar Coulomb logarithms for use in a Landau–Fokker–Planck kinetic theory.⁴¹ Transport coefficients from the qLFP theory are evaluated using a Chapman–Enskog solution of the linearized kinetic equation.^41,42

Time-dependent density-functional theory (TD-DFT) is a third approach to evaluate electronic properties. Unlike QMD, TD-DFT simulations evaluate the dynamic response of electrons to time-dependent perturbations. This can be done either in a linear-response framework atop a time-independent DFT calculation or using explicit time-propagation. In this work, only the time-propagation approach is considered.

In TD-DFT, the XC potential is in principle very complicated and can even depend on the time-history of the density. In most large-scale WDM applications to date, history dependence is neglected by adopting the so-called adiabatic approximation, where $V XC$ is simply evaluated using the instantaneous density. The functional form of $V XC$ determines how exchange and correlation affect the dielectric properties of the plasma, such as the dispersion relation for free-electron density waves.^30,45,46 One would nominally like this to be as accurate as possible; however, the TD-DFT calculations presented here are actually for conditions corresponding to weakly coupled, non-degenerate plasmas, where the details of the XC potential are not expected to be a major factor. Rather, the point is simply to illustrate dynamic electron response physics computed with a high-fidelity method and compare these with simpler kinetic-theory approaches. However, it would appear that in order for TD-DFT to faithfully represent the true dynamic electron–electron interaction, one must eventually eschew the adiabatic approximation.

The concept of a mean ionization state has no rigorous first-principles meaning. However, it is useful for connecting with theories of electron transport in plasmas and metals, which often presume that some fraction of the electrons are bound to nuclei to form ions, and the remainder are the free electrons which participate in conduction. Indeed, as previously mentioned, this kind of decomposition is necessary to apply plasma kinetic theory to WDM.

The AA-TCP model uses an alternative definition of ionization, $Z ¯ TCP$ based on the concept of a neutral pseudoatom, whose density distribution extends beyond the ion-sphere radius.^36,37 In practice, $Z ¯ TCP$ is usually close in value to $Z ¯ AA$⁠, except at conditions where near-threshold bound states or continuum resonances occur, as in the gold case shown below. For the qLFP transport theory calculations presented in here, $Z ¯ TCP$ is the ionization model used.

Finally, there have been some recent attempts to define a mean ionization from QMD simulation results.^49,50 It is important to note that QMD simulations need not (and cannot) differentiate between bound and scattering states in the way that average-atom models do. Thus, computing an effective ionization from QMD is useful mainly as a way to understand the approximations in AA models, where such a definition is necessary for evaluating transport coefficients.

The approaches described above are summarized in Table I. For both aluminum and carbon, the AA, TCP, and QMD estimates of $Z ¯$ are identical. This indicates that across the models, there is a consistent qualitative description of aluminum and carbon as having well-defined bound-state structure with closed shells (⁠ $1 s 2 2 s 2 2 p 6$ and $1 s 2$⁠, respectively) with the remaining electrons being in conducting states. However, these conduction electrons are not perfectly free, as evidenced by the fact that the AA $Z *$ value based on a free-electron DOS is systematically lower than the $Z ¯$ value based on the actual DOS. For gold, there is some variability due to the occurrence of a 5d-like valence electron whose classification as bound vs free is model-dependent.

The DOS is a much richer way to understand difference in electronic structure between different methods. Densities of states are plotted in Fig. 2. In addition to QMD results, two types of DOS relevant to the AA-TCP model are shown. The curves labeled “AA” are the DOS obtained directly from an ion–sphere average-atom model, which is the DOS used to evaluate $Z ¯ AA$⁠. The curves labeled “TCP” is the free-electron DOS that is implicitly assumed in the qLFP kinetic theory model based on the AA-TCP model. Since there is not a common zero-point for the energy in each method, the energy scale in Fig. 2 is zeroed according to the chemical potential relevant to each method, given in Table II. In the case of AA and QMD, this is simply the chemical potential that ensures the Fermi–Dirac distribution is consistent with the total number of electrons. For AA, this is all electrons, bound plus free. For QMD, this is all active electrons not contained in the pseudopotential. The TCP chemical potential is determined by requiring that there are $Z ¯ TCP$ free electrons per atom. Further analysis of the DOS is deferred until Sec. IV, where salient features are discussed in the context of understanding conductivity predictions.

While the ion–ion radial distribution function (RDF), g(r), is not directly used in any of the electronic transport calculations here, it is useful to determine at a glance how strongly coupled the plasma is and whether it is in a gas-like, liquid-like, or solid-like phase. In the QMD simulations, the RDF is determined directly from the nuclear positions at each time step and involves no approximations beyond those intrinsic to QMD, e.g., the choice of XC functional or finite-size effects. In the AA-TCP model, the RDF is determined by solving the quantum Ornstein–Zernike equations. These extend the classical Ornstein–Zernike theory for liquids to the case of a two-component plasma of ions and free electrons.^36,37

Comparing the RDFs between the AA-TCP model and the QMD simulations in Fig. 3 is a useful check on several assumptions made by the AA-TCP model. Notably, the AA-TCP model assumes that the effective ion–ion interaction is spherically symmetric, behaves like $( Z e ) 2 / ( 4 π ϵ 0 r )$ at small r, and has a frozen-in screening cloud that depends on $Z ¯$⁠. Differences in peak height are an indication that the AA-TCP model's ion–ion interaction is inaccurate, which is largely a function of $Z ¯$⁠.

In the beryllium case shown, there is excellent agreement between the AA-TCP and QMD RDFs. This is largely due to the fact that at these conditions, the 2s shell of beryllium is unambiguously ionized, while the 1s shell is unambiguously bound. Indeed, the AA-TCP model predicts $Z ¯ = 2$ in this case, which appears to be a good description of the beryllium ions at these conditions.

Two other examples of good agreement are the aluminum and gold cases. Here, the AA-TCP model and QMD are reasonably close, but not as tight as for beryllium. Cross-referencing these RDFs with their corresponding DOSs in Fig. 2, it is seen that the aluminum and gold DOS largely follows that of the TCP free-electron gas, except for some non-trivial valence structure near threshold. While these electrons do have an outsize influence on the screening between ions, it is not too dramatic here. One might imagine that a more sophisticated prescription for the TCP ionization state might close these gaps.

The remaining two cases, carbon and hydrogen, highlight important physics that is present in QMD but which any average-atom model will struggle to predict. In the case of carbon, the AA-TCP model has a simple oscillatory structure, which is characteristic of isotropic fluids. The QMD RDFs, however, have a distinct break from this behavior near $r = 3.5 a B$⁠, which is indicative of angular effects due to the interaction between p-like valence electrons. Since the AA-TCP model is spherically symmetric, it cannot treat this anisotropic valence-electron interaction accurately.⁵¹ In the case of hydrogen, the AA-TCP model predicts $Z ¯ = 1$⁠. This appears to be an over-estimate of the actual degree of ionization, judging by the more pronounced oscillations in the RDF compared to QMD. Instead of a clear peak, the QMD RDFs show a “shoulder,” which could result from the coexistence of two hydrogen populations: ionized and atomic. These two structural signatures indicate that hydrogen is not fully ionized at these conditions. If so, there would likely be significant hybridization of each atom's electron, which is neither unambiguously free nor bound. Such latent bonding physics is likewise not able to be modeled in a spherically symmetric average-atom approach.

The electrical and thermal conductivity were computed from the AA-TCP model using the qLFP kinetic theory with Coulomb logarithms computed using the mean-force potential method.^39,41 Both electron–electron and electron–ion collisions were included, and the Chapman–Enskog expansion was carried out to the third order, which is adequate for convergence to the 1% level.⁴² From QMD simulations, the electrical and thermal conductivity were evaluated using the Kubo–Greenwood method, which was applied on several independent QMD ion configurations, between 10 and 25. The ion configurations were obtained using the same MD trajectories as the static properties already considered, and for most cases, direct comparisons between the ground-state PBE GGA functional and the thermal meta-GGA T-(r²)-SCAN-L functional were made. For the calculations using thermal XC functionals, in some cases, the more sophisticated (and expensive) KDT0 hybrid functional was used to evaluate the transport coefficients. The results are presented in Tables III and IV.⁵² The reported values and uncertainties are the sample mean and standard deviation across the sampled MD configurations.

Focusing at first on just the QMD results, there are a few remarkable trends. For gold, beryllium, and hydrogen, calculations using both thermal and ground-state semi-local functionals are in reasonable agreement, though the differences are still outside the estimated uncertainty range for gold and hydrogen. Of these three cases, hydrogen shows the largest sensitivity to XC functional for both electrical and thermal conductivity (about 10% and 14%, respectively). This was not necessarily obvious based on expectations from the static properties, where it was seen that T-SCAN-L and PBE gave largely identical RDFs. Such a result underpins that electronic transport properties can be more sensitive to XC effects than ionic ones.

Continuing the analysis of the QMD results, we now focus on the cases of aluminum and carbon, where more pronounced differences are seen between the thermal KG calculations vs PBE calculations. In these two cases, the thermal KG calculations used KDT0 as the XC functional. Since KDT0 is a hybrid functional, it incorporates the exact (orbital-dependent) non-local electron exchange interaction. In both cases, the use of KDT0 leads to reductions in both thermal and electrical conductivities on the order of several tens of percent. Such differences are striking but consistent with past work on the optical properties of warm dense polystyrene.⁵³ Evidently, the treatment of the exchange interaction is a significant lever on KG predictions of electronic transport properties of warm dense matter.

Since warm dense aluminum is widely studied, we further mention a few other reported values of the electrical and thermal conductivity of aluminum at the same conditions considered here. Witte et al. performed KG calculations using the PBE (GGA) and HSE (range-separated hybrid) functionals.⁵⁴ For both the electrical and thermal conductivities, their PBE values are in excellent agreement with ours, as they should be. Their calculations using the ground-state HSE functional obtained very similar values as their PBE calculations, in contrast to the pronounced drop in conductivity reported here for KDT0 relative to PBE. There is also the experimental work by Milchberg et al., who measured the resistivity of aluminum. They obtained a value corresponding to an electrical conductivity of about $5.3 ± 2.0 MS / m$⁠,^1,55 which is far larger than any of the Kubo–Greenwood or kinetic theory predictions considered here and does not help much to clarify the question of XC effects on the conductivity. It would seem that the treatment of exchange for KG calculations of WMD is far from a settled point, and further development is necessary to understand the relative importance of finite-temperature vs exact-exchange effects.

We next consider the predictions of the qLFP mean-force kinetic theory. The main piece of physics that qLFP accounts for which is absent from the KG calculations is electron–electron scattering. However, for the cases considered here, electron–electron scattering is expected not to be important for two reasons. First, with the exception of gold, all the cases presented here involve rather degenerate electrons, as evidenced by the fact that $μ / ( k B T ) > 1$⁠. This is important because the exclusion principle effectively suppresses electron–electron scattering in degenerate matter.⁵⁶ Second, electron–electron scattering becomes unimportant relative to electron–ion scattering when the effective ionization is large. The only non-degenerate case considered here is gold, which the AA-TCP model predicts to have an ionization state of about 5. Thus, electron–electron scattering, which is qLFP's distinguishing strength, is not expected to be relevant for any of the cases for which QMD results are available.

Since electron–electron scattering is not significant in these comparisons, the main physics being elucidated is how well the electron–ion interaction is treated in qLFP. By and large, the qLFP results are in better agreement with the PBE QMD predictions, the only exceptions being the thermal conductivity of aluminum and gold. It is difficult to draw a clear inference from this, since the effect of different XC functionals in qLFP is very indirect and generally not a major lever on the final results. That is, even though an XC functional is chosen for the average atom model, the construction of the two-component plasma model introduces more severe approximations. In other words, the intrinsic approximations of qLFP are more important than the choice of XC functional.

Another general trend appearing is that the qLFP conductivities are somewhat higher than the QMD results. A plausible explanation for this is the nearly-free-electron approximation implicit in qLFP. The electronic transport properties of degenerate plasmas are determined largely by those electrons with energies near the Fermi energy. In terms of the DOS plots in Fig. 2, this corresponds to the origin. In both aluminum and carbon, the PBE QMD DOS is in reasonable agreement with the TCP free-electron DOS near the Fermi energy, while in the same energy range, there are marked differences from the QMD results using thermal functionals. In the case of gold, the electrons are not quite so degenerate, so electrons with a wider range of energies participate in conduction, up to about $E − μ ∼ k B T = 10 eV$⁠. Again, the continuum DOS in the case is largely free-electron-like, which may explain the remarkably good agreement between qLFP and KG predictions.

The first is a parameterized fit to TD-DFT results of hot dense CH.¹¹ In brief, that work simulated the passage of a test electron through a CH plasma using TD-DFT. The stopping power was evaluated by computing the Feynman–Hellmann force on the electron by the background plasma ions and electrons. This was done for several electron energies so that the stopping power could be interpolated for evaluating Eq. (9).

The third approach is the so-called “modified Lee–More” (MLM) method. It is an amalgam of several known limits from binary scattering theory and stopping power theory. A complete description and justification of the MLM model is given in Appendix E of Ref. 58.

The three approaches are compared in Fig. 4, which shows that by and large, the qLFP and MLM approaches are in good agreement, being within order-unity factors of one another. This is not surprising since they are both binary collision models being evaluated in a weakly coupled, non-degenerate plasma. At such conditions, the account of Fermi–Dirac statistics in the qLFP model is irrelevant, so the differences mainly come down to minor differences in Coulomb logarithms, particularly at high energy. The MLM attempts to take into account dynamic screening effects through a velocity-dependent Coulomb logarithm, whereas the qLFP Coulomb logarithm is a thermally averaged one. Since the velocity-dependent Coulomb logarithm is a monotonically increasing function of velocity, thermal averaging under-estimates its value for electrons with $E ≫ k B T$⁠. This leads to qLFP predicting a slightly lower mean free path compared to MLM, but the difference is not dramatic.

Compared with the TD-DFT-fitted model, however, there are significant differences as previously observed in Ref. 11. For high-energy electrons, the TD-DFT fit predicts a much larger mean free path than either kinetic theory model, which results from the fact that the stopping power in this energy range was predicted to be much lower than predictions from linear response and binary collision theories.¹¹ The ultimate cause of this remains unknown and is being actively investigated. The leading hypotheses for the lower than expected stopping power relative to traditional theories is that nonlinear dynamic response physics is relevant, even at high electron energies. As evidence for this, it was previously shown that TD-DFT simulations of charged-particle stopping exhibit a non-negligible Barkas effect, which is a sensitivity to the sign of the projectile's charge.^11,61 Such an effect absent in both Born-approximation collision theories as well as linear-response treatments of projectile stopping; however, the Barkas effect alone cannot account for the factor-of-two differences observed at large electron energy.¹¹ We take this as an indication that electron dynamic response may be more complex than expected based on simple free-electron and Coulomb collision models and that further developments in understanding nonlinear dynamic response properties of plasmas would be valuable in the context of non-local electron transport.

In summary, we have investigated connections between electronic structure and transport in warm dense matter through the lenses of both high-fidelity DFT-based molecular dynamics and the Kubo–Greenwood method as well as a quantum kinetic theory based on a Fokker–Planck collision model. The two approaches are in surprisingly good agreement for materials and conditions were a nearly-free-electron model prevails. However, this does not necessarily hold for warm dense matter, and there were many cases considered here where kinetic theory approaches would likely benefit from relaxing the free-electron assumption. It was also observed that in degenerate warm dense matter, even the QMD-based approaches can be sensitive to the inclusion of thermal XC effects and treatment of electron exchange. Kubo–Greenwood calculations using the KDT0 thermal hybrid functionals predict considerably lower electrical and thermal conductivity compared to PBE, and this difference is much larger than those between T-(r²-)SCAN-L and ground-state GGA. This would indicate that even within ab initio approaches, there is progress yet to be made toward establishing high-quality benchmark values of electronic transport coefficients. Finally, we looked at dynamic electron screening physics using TD-DFT and kinetic theory to compute the mean free path of energetic electrons in hot dense plasmas. These results indicate that the dynamically screened electron–electron interaction is quite complex, and there remain serious unresolved issues either with TD-DFT or kinetic theory in accurately describing the transient response of electrons in dense plasmas.

This material is based upon work supported by the Department of Energy (National Nuclear Security Administration) University of Rochester “National Inertial Confinement Fusion Program” under Award No. DE-NA0004144. S.X.H. and V.V.K. also acknowledge support by the U.S. National Science Foundation PHY Grant No. 2205521.

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.

LANL is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 89233218NCA000001.

The authors have no conflicts to disclose.

N. R. Shaffer: Investigation (equal); Methodology (equal); Visualization (lead); Writing – original draft (lead). S. X. Hu: Investigation (equal); Methodology (equal); Resources (lead); Writing – review & editing (equal). V. V. Karasiev: Investigation (equal); Methodology (equal); Writing – review & editing (equal). K. A. Nichols: Investigation (equal); Methodology (equal); Writing – review & editing (equal). C. E. Starrett: Methodology (supporting); Writing – review & editing (equal). A. J. White: Methodology (supporting); Writing – review & editing (equal).

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