This workshop serves as a platform to discuss methods of density functional theory (DFT) that are used to study the behavior of matter under extreme conditions as found in planets and stars.
As the most stable element in the periodic table and a final fusion product of stars, iron is ubiquitous in the universe, and its presence likely leads to the formation of cores in planetary bodies that form from refractory oxides, similar to the terrestrial bodies in our solar system, termed super-Earths. Physical and chemical properties of iron at high pressure are therefore of great interest in the study of planetary interiors, beyond the Earth and our solar system. At the same time, iron has played a critical role in the development of approximations to the exchange and correlation potential in Kohn-Sham density functional theory (KS-DFT). The failure to properly describe the ground-state of iron (ferromagnetic bcc) in the local density approximation provided an early impetus to the development of generalized gradient formulations. The magnetic structure of the high-temperature phase fcc led to early non-collinear magnetic formulations of the charge density. At pressures of the Earth’s core (~300 GPa, with a relative compression of ~0.6 with respect to the ambient pressure volume), hybridization effects for 3d and 4s electrons (typically considered valence at ambient conditions) with 3s and 3p states (core) makes choices on the treatment of electronic bands a tricky business. At higher pressure that prevail in the interior (and therefore hypothetical cores) of super-Earths (exceeding TPa), choices on the treatment of electronic states become even more complex. Here I review high-pressure results using KS-DFT for iron and dilute iron alloys that are of importance in a planetary context, and discuss apparent and potential limitations. Properties discussed range rom pure compression behavior to atomic and electronic transport properties.
The interiors of icy planets are thought to comprise H/He-rich atmospheres followed by vast ‘hot ice’ mantle regions and, possibly, small rocky cores. The ‘hot ice’ layers, the largest water reservoirs in planetary systems, are chemically very complex and while individual constituents (water, methane, ammonia) have been studied in great detail both computationally and experimentally, their mixtures are much less explored. However, molecular mixtures allow for chemical changes that can drastically alter their properties at specific conditions, with consequences for internal stratification, depth-density profiles, thermal conduction, convection, magnetism, etc.
In this talk I will discuss some of the recent computational progress in studies of complex molecular mixtures at extreme conditions, including ‘hot ice’ mixtures (such as ammonia-water), ad-mixtures of atmospheric materials (such as methane-hydrogen), and the structure of the full H-C-N-O quaternary chemical space near Neptune’s core pressure conditions, as explored via unbiased crystal structure searching. I will show that planetary ice mixtures can exhibit properties not found in their constituting compounds, and argue that the full complexity of these mixtures needs to be considered to fully capture their behaviour.
The detection of more than 4000 exoplanets, most of them having no equivalent in the Solar Systems, stimulates numerous fields of research in order to better understand the structure, the formation, the evolution and the habitability of these new worlds. In order to build realistic models, astrophysicists are in need for the physical properties (equations of state, phase diagrams, transport properties,…) of various materials on a wide range of thermodynamic conditions. In this regard, ab initio numerical simulations have proven to be a wonderful tool to accurately characterize the properties of matter under extreme conditions. These advances lead to paradigm shifts regarding the structure of planets, such as Jupiter’s core erosion for instance or the origin of Super-Earth magnetic field. In this presentation, I will show on a few examples how the concurrent use of ab initio simulations and laser experiments can help to better comprehend exoplanets as well as to provide constraints on the selection of priority targets for followup observations.
The question of whether hydrogen and water mixtures are miscible at planetary temperatures and pressures deep in the interior of planets like Uranus and Neptune remains unresolved. In 2021, Bergermann et al  used the Gibbs-ensemble Monte Carlo simulation method and found a huge miscibility gap which shows reasonable agreement with the experimental data of Bali et al . Soubiran and Militzer  conducted simulations based on density functional theory (DFT-MD) and predicted hydrogen-water mixtures to be completely miscible under those conditions. To resolve this problem, we performed extensive DFT-MD simulations for water-hydrogen mixtures at temperatures of 1000 K < T < 2000 K and pressures of 40 kbar < p < 300 kbar. By using the ideal entropy and calculating the free enthalpy we found hydrogen-water to be partially immiscible at temperatures of T = 1000 K and pressures of 40 kbar < p < 120 kbar. Despite the fact, that the non-ideal entropy is still neglected, our simulations provide evidence, that hydrogen-water mixtures might be immiscible under temperature and pressure conditions deep in the interior of planets like Uranus and Neptune. These results are important for interior and evolution models for ice giant planets because H2-H20 demixing would induce compositional gradients which could inhibit convection and, therefore, the cooling of those planets [4, 5].
 A. Bergermann, M. French, R. Redmer, Phys. Chem. Chem. Phys., 23 (2021)  E. Bali, A. Audétat and H. Keppler, Nature, 495, 7440 (2013)
 F. Soubiran and B. Militzer, The Astrophysical Journal, 806 (2015)
 R. Helled, N. Nettelmann and T. Guillot, Space Sci. Rev., 216 (2020)
 E. Bailey and D. J. Stevenson, Planet. Sci. J., 64 (2020)
An earlier study  benchmarked Density Functional Theory (DFT) coupled with classical Molecular Dynamics (MD) with all available experimental data on dense helium in recent years. A subsequent study  calculated the helium melting line with DFT-MD. These two studies allows for the examination of the metallization of fluid helium consistently with DFTMD .
We study the insulator-to-metal transition at densities between 1 and 22 g/cm and
temperatures between 10 000 and 50 000 K. We calculate the equation of state, the band gap dependent on density and temperature by using different definitions [4-7], the DC conductivity, the reflectivity, and the ionization degree for which a novel method has been proposed recently [see M. Bethkenhagen et al., Phys. Rev. Res. 2, 023260 (2020)]. We find no indication of a first-order phase transition in any of the properties studied here and conclude that the metallization of fluid helium is continuous. For instance, we do not observe jumps in the DC conductivity and/or the reflectivity when the band gap closes.
However, the ionization degree increases from below 10% at the lowest to over 99% at the highest densities which reflects the continuous insulator-to-metal transition. The increase is almost exclusively driven by pressure ionization and shows only a weak temperature dependence.
We discuss the high-pressure phase diagram of helium and the implications of our results on the structure of astrophysical objects like gas giant planets and brown dwarfs.
 Preising et al., “Equation of state and optical properties of warm dense helium”, Phys. Plasmas 25, 012706 (2018).
 Preising et al., “High-pressure melting line of helium from ab initio calculations”, Phys. Rev. B. 100, 184107 (2019).
 Preising et al., “Metallization of dense fluid helium from ab initio simulations”, Phys. Rev. B. 102, 224107 (2020).
 Kowalski et al. “Equation of state and optical properties of warm dense helium”, Proc. Rev. B 76, 11071 (2007).
 Stixrude et al. “Fluid helium at conditions of giant planetary interiors”, Proc. Natl. Acad. Sci. USA 32, 11071 (2008).
 Monserrat et al. “Fluid helium at conditions of giant planetary interiors”, Phys. Rev. Lett. 112, 055504 (2014).
 W. Zhang et al., “Revisiting metallization boundary of warm dense helium in a wide ρ-T regime from ab initio study”, Sci. Rep. 7, 41885 (2017).
Stochastic vector computational approaches for the electronic structure of extended condensed matter systems help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. The electronic density is estimated by a stochastic process that samples the Kohn-Sham eigenstate contribution according to the Fermi Dirac occupation, completely avoiding the actual calculation of eigenstates. The method is expected to be especially useful for finite-temperature density functional-based molecular dynamics calculations. We also discuss the use of stochastic approaches for estimating quasiparticle energies within the G0W0 approximaiton.
We developed a stochastic density functional theory (sDFT) approach under a nonorthogonal , atom-centered basis set representation. The method is a highly parallelizable linear-scaling approach in which the reduced scaling is achieved without imposing (or relying on) a sparse structure to the Kohn-Sham density matrix, and as such may be applicable to a wide variety of systems in biology and material science. Observables in sDFT are calculated in a trace-based formalism using the stochastic trace formula and can therefore be regarded as random variables, with an expected value and fluctuation. Due to the non-linear nature of the SCF iterations, sDFT observables are also characterized by a systematic bias error, whose magnitude can be controlled by increasing sampling as well as by employing an embedded-fragments-based, variance-reducing technique. We developed and implemented a new formalism for the calculation of forces using sDFT in the nonorthogonal, atom-centered basis set, which also includes the treatment of Pulay force terms acting on the nuclei. This is a key step towards the goal of using the “noisy” forces calculated from sDFT for geometry optimization when implemented in a Langevin dynamics framework. For this approach to be useful it is key to make sure that the systematic bias errors in the forces are sufficiently small. We present a statistical analysis of the sDFT errors in the forces acting on a Tryptophan Zipper 2 peptide solvated in water and results indicate that the sDFT bias in the forces is small and independent of system size, paving the way for future Langevin dynamics structural studies of peptides in solution.
We will present a massively parallel DFT approach which doesn’t rely on electron localization and is formally quadratic scaling yet enables highly efficient linear wall-time complexity in the weak scalability regime. The method extends from the stochastic DFT approach described in Fabian et al. WIRES: Comp. Mol. Science, e1412 2019 but is entirely deterministic and is well suited for the warm dense matter regime since its computational effort is inversely proportional to the system's temperature. The algorithm is based on standard quantum chemical atom-centered Gaussian basis sets to represent the electronic wave functions combined with Cartesian real-space grids for some operators and enables a fast solver for the Poisson equation. Our main conclusion is that when a processor-abundant high-performance computing (HPC) infrastructure is available, this type of approach has the potential to allow the study of large systems in regimes where quantum confinement or electron delocalization prevents linear-scaling.
Accurately modeling dense plasmas over wide ranging conditions of pressures and temperatures is a grand challenge problem critically important to our understanding of inertial confinement fusion (ICF), stellar physics, exoplanets, and planetary formation. Over the last few years planewave- based Kohn-Sham Density Functional Theory Molecular Dynamics (DFT-MD) has proven highly successful in describing the properties of many materials in the condensed and warm dense matter regimes from first principles. However, the tractable number of particles is rather limited due to the method’s O(N^3) scaling and it becomes challenging to employ at high temperatures, where the number of partially occupied states increases significantly. Both problems are addressed with the novel SQDFT code , which is a large-scale implementation of the Spectral Quadrature (SQ) method for O(N) Kohn-Sham DFT calculations [2,3].
This talk discusses the new capabilities of SQDFT by comparing to the widely-used planewave Kohn-Sham DFT codes as well as other methods such as Path Integral Monte Carlo. Extensive benchmark calculations for the thermodynamic and structural properties of carbon are presented. In particular, the Hugoniot curve for carbon was calculated for conditions spanning the condensed matter regime, the warm dense matter regime and the plasma regime with temperatures up to 10 million Kelvin all within the framework of many particles full Kohn-Sham DFT-MD .
 P. Suryanarayana, P.P. Pratapa, A. Sharma, and J.E. Pask, Computer Physics Communications 224, 288 (2018).
 P.P. Pratapa, P. Suryanarayana, and J.E. Pask, Computer Physics Communications 200, 96 (2016).
 P. Suryanarayana, Physics Letters 584, 182 (2013).
 M. Bethkenhagen, A. Sharma, P. Suryanarayana, J.E. Pask, B. Sadigh, S. Hamel, arXiv:2110.01034
We present an extensive description of the application of a generalized collective modes model to ab initio simulations in the warm dense matter regime. We calculate the intermediate scattering function for warm dense aluminum by using density functional theory molecular dynamics simulations. From this data set we derive the static and dynamic ion-ion structure factors. Applying a generalized collective modes model, we can fit the excitation spectra of the ion system and thereby extract the dispersion for the ion acoustic modes, as well as the decay coefficients for the diffusive and collective modes. The results are discussed and compared with experimental data if available. We show that computational limitations prevent sufficient access to the hydrodynamic limit and demonstrate that this can be circumvented using machine learning.
Average-atom (AA) models are an important tool in the modelling of warm dense matter, being both a computationally cheap and conceptually straightforward alternative to full DFT MD simulations. AA models are typically based on a common premise - namely, an atom immersed in a plasma environment - but use a range of different assumptions and approximations, which can cause inconsistent predictions for various properties. In this talk, I will compare results across several models, differing for example in their choice of boundary conditions and exchange-correlation functional. I will focus on the mean ionization state (MIS), an important property in WDM. I will compare different methods for computing the MIS, including methods which are historically popular and still widely-used in AA codes, and also consider more novel approaches using the electron localization function and Kubo-Greenwood formalism. If time permits, these results with also be compared with results from full DFT-MD simulations.
The simulation of correlated fermions is important for various phenomena in warm dense matter, plasmonics, and ultracold atoms. In order to enable simulations at larger length and longer time scales, there is a need to develop quantum hydrodynamics (QHD) as a complementary method to commonly used first-principles methods. The key difference of the QHD from classical fluid equations is the inclusion of the quantum non-locality. This is usually done by using the Bohm potential. We performed the very first investigation of the Bohm potential for a correlated many-fermion system based on the data from KS-DFT. Despite its long history in quantum mechanics since its derivation by Bohm in 1952 and its importance for QHD, this has not been done before. Our key result shows the very limited applicability of the standard Bohm potential which is used in virtually all previous works of QHD. We showed that it is only valid for a very weakly perturbed electron gas. We illustrate that the many-fermion quantum Bohm potential is needed to model nonlinear phenomena in quantum plasmas and WDM [1, 2].
 Z. A. Moldabekov, T. Dornheim, G. Gregori, F. Graziani, M. Bonitz, A. Cangi, Towards a Quantum Fluid Theory of Correlated Many-Fermion Systems from First Principles, SciPost Physics [accepted for publication], scipost_202106_00020v3
 F. Graziani, Z. Moldabekov, B. Olson, M. Bonitz, Shock Physics in Warm Dense Matter--a quantum hydrodynamics perspective, arXiv:2109.09081
Dinner at the conference venue
The successful characterization of high energy density (HED) phenomena in laboratories using pulsed power facilities and coherent light sources is possible only with numerical modeling for design, diagnostic development, and data interpretation. The persistence of electron correlation in HED matter is one of the greatest challenges for accurate numerical modeling and has hitherto impeded our ability to model HED phenomena across multiple length and time scales at sufficient accuracy. Standard methods from electronic structure theory capture electron correlation at high accuracy, but are limited to small scales due to their high computational cost. In this talk, I will summarize our recent efforts on devising a data-driven and physics-informed workflow to tackle this challenge . Based on first-principles data we generate machine-learning surrogate models that replace traditional density functional theory calculations. Our surrogates predict the electronic structure and related properties of matter under extreme conditions highly efficiently while maintaining the accuracy of traditional methods.
 J. A. Ellis, L. Fiedler, G. A. Popoola, N. A. Modine, J. A. Stephens, A. P. Thompson, A. Cangi, and S. Rajamanickam, Phys. Rev. B 104, 035120 (2021).
While the high efficiency of Density Functional Theory (DFT) calculations has enabled many important materials science application over the past decades, modern scientific problems require accurate electronic structure data beyond the scales attainable with DFT. For instance, the modeling of materials at extreme conditions across multiple length and time scales, which is important for the understanding for physical phenomena such as radiation damages in fusion reactor walls, evades ab-initio treatment.
One possible method to obtain such models at near ab-initio accuracy are DFT surrogate models, that, based on machine learning (ML) algorithms, reproduce DFT results at a fraction of the cost. One drawback of the ML workflow is the need for hyperparameter optimization, i.e., the need to tune the employed ML algorithm in order to best perform on the given dataset. Manually performing this optimization becomes prohibitive if a wide range of materials and conditions is eventually to be treated. Here, we present results of an hyperparameter study in an effort to find optimal surrogate models for aluminium at ambient conditions 1, that investigates how modern hyperparameter optimization techniques can be used to automate large parts of the model selection process and eventually move towards automated surrogate model creation. The models are based upon the Materials Learning Algorithms (MALA) package 2 and the therein implemented LDOS based machine learning workflow 3.
The physical and chemical state of Earth’s core is challenging to match using only pure iron. An additional amount of light element(s) is required to account for its geophysical features (e.g., core density deficit), and among various candidates proposed in the Earth’s core , hydrogen attracts special attention: (i) It is the most abundant element in the universe and, due to pressure stabilization of iron hydrides FeHx (where x denotes nonstoichiometric H), substantial hydrogen might have been sequestered into the core during planetary accretion. (ii) FeHx reproduces not only the density and compressional velocity of the outer core, but also the anomalously high Poisson’s ratio of the inner core . (iii) Low solidus temperatures of pyrolite at core–mantle boundary pressure conditions require a substantial reduction of the melting temperature of the outer core by impurities ; hydrogen is known to depress the melting temperature of iron more efficiently than other alloying elements. Given that a substantial amount of hydrogen likely is present during planetary differentiation, whether, and to what extent, hydrogen was stripped from the bulk silicate Earth into the core hinge primarily on the hydrogen partitioning behavior between metallic and silicate melts. In this talk, I will present our recent efforts to constrain the hydrogen partition coefficients between silicate and iron, and between liquid and solid iron determined by (i) Gibbs energy calculations using Kohn-Sham (KS) density functional theory (DFT)-based thermodynamic integration methods, and (ii) large-scale two-phase coexisting simulations. In the latter approach, to bypass the computational cost of solving the KS equations while representing potential energy surfaces at KS-DFT accuracy, we build a neural network potential that directly uses relative atomic coordinates to describe local atomic environments to obtain atomic energies and forces. With these simulation results, I will discuss the hydrogen compositions of the bulk, inner and outer core.
 K. Hirose, B. Wood, and L. Vočadlo, Nat. Rev. Earth Environ. 2, 645 (2021).
 W. Wang, Y. Li, J. P. Brodholt, L. Vočadlo, M. J. Walter, and Z. Wu, Earth Planet. Sci. Lett. 568, 117014 (2021).
 R. Nomura, K. Hirose, K. Uesugi, Y. Ohishi, A. Tsuchiyama, A. Miyake, and Y. Ueno, Science 343, 522 (2014).
Improving the predictive capability of molecular properties in ab initio simulations is essential for advanced material discovery. Despite recent progress making use of machine learning, utilizing deep neural networks to improve quantum chemistry modelling remains severely limited by the scarcity and heterogeneity of appropriate experimental data. Here we show how training a neural network to replace the exchange-correlation functional within a fully-differentiable three-dimensional Kohn-Sham density functional theory (DFT) framework can greatly improve simulation accuracy. Using only eight experimental data points on diatomic molecules, our trained exchange-correlation networks enable improved prediction accuracy of atomization energies across a collection of 104 molecules containing new bonds and atoms that are not present in the training dataset.
Warm dense matter is of high current interest for many applications, including astrophysics, material science, and fusion research. Yet, the accurate description of electronic correlation effects at these conditions is most difficult, and often computationally intensive ab-initio methods have to be used . The most accurate approach is given by the quantum Monte Carlo (QMC) technique, which is, in principle, capable to give one exact results for the full quantum many-body problem of interest without any empirical input. Consequently, parametrizations of accurate QMC data constitute the basis for a gamut of applications, such as the construction of XC-functionals for density functional theory (DFT).
In this talk, I focus on the electronic density response of WDM to an external perturbation, which is of central interest for WDM theory, such as the interpretation of X-ray Thomson scattering (XRTS) experiments and the construction of advanced XC-functionals for DFT. In particular, I will show how we can use the ab-initio path integral Monte Carlo (PIMC) method to estimate the exact density response to an external harmonic perturbation. First and foremost, this allows us to compute the electronic XC-kernel (also known as local field correction in the context of dielectric theory), which has recently become available as a neural-net representation  for a uniform electron gas. Secondly, I will show how our imaginary-time PIMC data can be used as a starting point for an analytic continuation . This gives us access to the dynamic structure factor, which is the key property in XRTS experiments. Lastly, I will talk about nonlinear effects in WDM , which cannot be neglected in many situations of experimental relevance.
 M. Bonitz, T. Dornheim, Z.A. Moldabekov, S. Zhang, P. Hamann, H. Kählert, A. Filinov, K. Ramakrishna, and J. Vorberger, Phys. Plasmas 27, 042710 (2020)
 T. Dornheim, J. Vorberger, S. Groth, N. Hoffmann, Z. Moldabekov, and M. Bonitz, J. Chem. Phys. 151, 194104 (2019)
 T. Dornheim, S. Groth, J. Vorberger, and M. Bonitz, Phys. Rev. Lett. 121, 255001 (2018)
 T. Dornheim, J. Vorberger, and M. Bonitz, Phys. Rev. Lett. 125, 085001 (2020)
Libxc, one of a few available libraries for exchange-correlation functionals, currently contains well above 100 different LDA, GGA, hybrid GGA, meta-GGA, and hybrid meta-GGA functionals. The aim of this talk is to show the results of trying quite a few of these for different warm dense matter states and different properties. When possible, quantum Monte Carlo data will be used to benchmark the DFT results.
Density functional theory (DFT) is an in-principle exact theoretical framework for any many-body system. It is particularly relevant for the warm dense matter (WDM) regime, where large systems have to be addressed at the electron-electron level, at relatively high temperatures and pressures. The high-temperature regime requires addressing not only the ground state, but (many) excited states of the system. It is well known, however, that the relation between excitation energies and Kohn-Sham energy-level differences is by no means straightforward, and usually involves discontinuities in the exact exchange-correlation potential , which are very difficult to model. The natural question that arises in this context is whether we can construct an xc functional that is suitable to predict excitation energies, preferably at a moderate computational cost. In my talk I present a thorough study of atomic systems throughout the periodic table  calculated with ensemble-generalized xc functionals [2,3] and examine, as a first step, their IP and fundamental gap, obtained from the Kohn-Sham spectrum. Analysis of the s-, p- and d-block systems vs experiment allows us to make a connection between the accuracy of the method and the position of the system in the Periodic Table and to suggest further steps to remove the remaining discrepancy. Generalization of the method to higher excitations and inclusion of temperature will be discussed.
 E. Kraisler, M. J. P. Hodgson, E. K. U. Gross, From Kohn–Sham to Many-Electron Energies via Step Structures in the Exchange-Correlation Potential, J. Chem. Theory Comput. 17, 1390 (2021)
 E. Kraisler, L. Kronik, Piecewise linearity of approximate density functionals revisited: Implications for frontier orbital energies, Phys. Rev. Lett. 110, 126403 (2013)
 E. Kraisler, L. Kronik, Fundamental gaps with approximate density functionals: The derivative discontinuity revealed from ensemble considerations, J. Chem Phys. 140, 18A540 (2014)
 S. Lavie, E. Kraisler, Ionization potentials and Fundamental Gaps in Atomic Systems from the Ensemble-DFT Approach, in preparation.
Real-time time-dependent density functional theory (TDDFT) has been wildly successful in many contexts, but it is likely to remain far too expensive to be used in tabulating materials properties for the plasma physics codes used to design inertial fusion experiments or other systems with many orders of magnitude of variation in density and temperature. For that task, average atom (AA) models are orders of magnitude more efficient but their accuracy depends on choices concerning the treatment of electron-ion collisions and the partitioning of electrons into bound and free populations. I will compare TDDFT and AA for two observables - the dynamic structure factor and electron stopping power - focusing on aluminum and iron in the warm dense regime. TDDFT corroborates the use of improved collisional models in AA, as well as the adoption of a scheme for treating bound and free electrons consistently as they change their identities under the influence of thermodynamic conditions (e.g., pressure ionization).
Understanding electronic transport properties of materials under high temperatures and pressures is essential for constraining geophysical processes and provides indispensable insights useful for novel materials discovery. The difficulty of measuring the electrical conductivity of iron under Earth-core conditions reliably in experiments  calls for sophisticated theoretical methods that can support diagnostics. We present results of the electrical conductivity in iron within the pressure and temperature ranges found in Earth's core from simulating microscopic Ohm's law using time-dependent density functional theory. Our predictions are independent of previous studies, which primarily used the Kubo-Greenwood formula, and therefore provide a new perspective on resolving discrepancies in recent experiments [2,3].  D. Dobson, Nature 534, 45 (2016)  K. Ohta, Nature 534, 95 (2016)  Z. Konopkova, Nature 534, 99 (2016)
The determination of thermoelectric transport coefficients of dense, partially ionized plasmas is a great challenge for both experiment and theory. In the past two decades, density functional theory (DFT) has evolved to an efficient tool for making theoretical predictions of properties of dense plasmas. Many of these are of high relevance for modelling the interior states, evolution, and magnetic field dynamics of stellar and planetary objects.
Here I will give an overview on the generalized Kubo-Greenwood formalism  that is frequently used in calculations of electronic transport properties using the Kohn-Sham states from DFT. Several examples for successful application of this technique to various solid and fluid metals will be presented. Furthermore, a comparison of optical reflectivities of molecular fluids with shock compression experiments  will be made.
Finally, the limits of the Kubo-Greenwood formalism with respect to its capability of describing electron-electron collisions will be discussed by comparing the thermopower and Lorenz number of weakly degenerate hydrogen plasmas with the known Spitzer results.
This work is supported by the DFG within the FOR 2440 "Matter under Planetary Interior
Conditions - High Pressure, Planetary, and Plasma Physics."
 B. Holst, M. French and R. Redmer, "Electronic transport coefficients from ab initio simulations and application to dense liquid hydrogen", Phys. Rev. B 83, 235120 (2011).
 A. Ravasio, M. Bethkenhagen, J.-A. Hernandez, A. Benuzzi-Mounaix, F. Datchi, M. French, M. Guarguaglini, F. Lefevre, S. Ninet, R. Redmer, and T. Vinci, "Metallization of Shock-Compressed Liquid Ammonia", Phys. Rev. Lett. 126, 025003 (2019).