Ab initio-simulated optical response of hot electrons in gold and ruthenium.

Optical femtosecond pump-probe experiments allow to measure the dynamics of ultrafast heating of metals with high accuracy. However, the theoretical analysis of such experiments is often complicated because of the indirect connection of the measured signal and the desired temperature transients. Establishing such a connection requires an accurate model of the optical constants of a metal, depending on both the electron temperature Te and the lattice temperature Tl. In this paper, we present first-principles simulations of the two-temperature scenario with Te ≫ Tl, showing the optical response of hot electrons to laser irradiation in gold and ruthenium. Comparing our simulations with the Kubo-Greenwood approach, we discuss the influence of electron-phonon and electron-electron scattering on the intraband contribution to optical constants. Applying the simulated optical constants to the analysis of ultrafast heating of ruthenium thin films we highlight the importance of the latter scattering channel to understand the measured heating dynamics.

Ultrafast Heating-Induced Suppression of d-Band Dominance in the Electronic Excitation Spectrum of Cuprum.

The combination of isochoric heating of solids by free-electron lasers (FELs) and in situ diagnostics by X-ray Thomson scattering (XRTS) allows for measurements of material properties at warm dense matter (WDM) conditions relevant for astrophysics, inertial confinement fusion, and materials science. In the case of metals, the FEL beam pumps energy directly into electrons with the lattice structure of ions being nearly unaffected. This leads to a unique transient state that gives rise to a set of interesting physical effects, which can serve as a reliable testing platform for WDM theories. In this work, we present extensive linear-response time-dependent density functional theory (TDDFT) results for the electronic dynamic structure factor of isochorically heated copper with a face-centered cubic lattice. At ambient conditions, the plasmon is heavily damped due to the presence of d-band excitations, and its position is independent of the wavenumber. In contrast, the plasmon feature starts to dominate the excitation spectrum and has a Bohm-Gross-type plasmon dispersion for temperatures T ≥ 4 eV, where the quasi-free electrons in the interstitial region are in the WDM regime. In addition, we analyze the thermal changes in the d-band excitations and outline the possibility to use future XRTS measurements of isochorically heated copper as a controlled testbed for WDM theories.

Ab initio path integral Monte Carlo simulations of warm dense two-component systems without fixed nodes: Structural properties.

We present extensive new ab initio path integral Monte Carlo (PIMC) results for a variety of structural properties of warm dense hydrogen and beryllium. To deal with the fermion sign problem-an exponential computational bottleneck due to the antisymmetry of the electronic thermal density matrix-we employ the recently proposed [Y. Xiong and H. Xiong, J. Chem. Phys. 157, 094112 (2022); T. Dornheim et al., J. Chem. Phys. 159, 164113 (2023)] ξ-extrapolation method and find excellent agreement with the exact direct PIMC reference data where available. This opens up the intriguing possibility of studying a gamut of properties of light elements and potentially material mixtures over a substantial part of the warm dense matter regime, with direct relevance for astrophysics, material science, and inertial confinement fusion research.

Ab Initio Path Integral Monte Carlo Simulations of the Uniform Electron Gas on Large Length Scales.

The accurate description of non-ideal quantum many-body systems is of prime importance for a host of applications within physics, quantum chemistry, materials science, and related disciplines. At finite temperatures, the gold standard is given by ab initio path integral Monte Carlo (PIMC) simulations, which do not require any empirical input but exhibit an exponential increase in the required computation time for Fermionic systems with an increase in system size N. Very recently, computing Fermionic properties without this bottleneck based on PIMC simulations of fictitious identical particles has been suggested. In our work, we use this technique to perform very large (N ≤ 1000) PIMC simulations of the warm dense electron gas and demonstrate that it is capable of providing a highly accurate description of the investigated properties, i.e., the static structure factor, the static density response function, and the local field correction, over the entire range of length scales.

Bound-State Breaking and the Importance of Thermal Exchange-Correlation Effects in Warm Dense Hydrogen.

Hydrogen at extreme temperatures and pressures is of key relevance for cutting-edge technological applications, with inertial confinement fusion research being a prime example. In addition, it is ubiquitous throughout our universe and naturally occurs in a variety of astrophysical objects. In the present work, we present exact ab initio path integral Monte Carlo (PIMC) results for the electronic density of warm dense hydrogen along a line of constant degeneracy across a broad range of densities. Using the well-known concept of reduced density gradients, we develop a new framework to identify the breaking of bound states due to pressure ionization in bulk hydrogen. Moreover, we use our PIMC results as a reference to rigorously assess the accuracy of a variety of exchange-correlation (XC) functionals in density functional theory calculations for different density regions. Here, a key finding is the importance of thermal XC effects for the accurate description of density gradients in high-energy-density systems. Our exact PIMC test set is freely available online and can be used to guide the development of new methodologies for the simulation of warm dense matter and beyond.

Fermionic physics from ab initio path integral Monte Carlo simulations of fictitious identical particles.

The ab initio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion sign problem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the sign problem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the sign problem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).

Electronic density response of warm dense hydrogen on the nanoscale.

The properties of hydrogen at warm dense matter (WDM) conditions are of high importance for the understanding of astrophysical objects and technological applications such as inertial confinement fusion. In this work, we present extensive ab initio path integral Monte Carlo results for the electronic properties in the Coulomb potential of a fixed ionic configuration. This gives us unique insights into the complex interplay between the electronic localization around the protons with their density response to an external harmonic perturbation. We find qualitative agreement between our simulation data and a heuristic model based on the assumption of a local uniform electron gas model, but important trends are not captured by this simplification. In addition to being interesting in their own right, we are convinced that our results will be of high value for future projects, such as the rigorous benchmarking of approximate theories for the simulation of WDM, most notably density functional theory.

Averaging over atom snapshots in linear-response TDDFT of disordered systems: A case study of warm dense hydrogen.

Linear-response time-dependent density functional theory (LR-TDDFT) simulations of disordered extended systems require averaging over different snapshots of ion configurations to minimize finite size effects due to the snapshot-dependence of the electronic density response function and related properties. We present a consistent scheme for the computation of the macroscopic Kohn-Sham (KS) density response function connecting an average over snapshot values of charge density perturbations to the averaged values of KS potential variations. This allows us to formulate the LR-TDDFT within the adiabatic (static) approximation for the exchange-correlation (XC) kernel for disordered systems, where the static XC kernel is computed using the direct perturbation method [Moldabekov et al., J. Chem. Theory Comput. 19, 1286 (2023)]. The presented approach allows one to compute the macroscopic dynamic density response function as well as the dielectric function with a static XC kernel generated for any available XC functional. The application of the developed workflow is demonstrated for the example of warm dense hydrogen. The presented approach is applicable for various types of extended disordered systems, such as warm dense matter, liquid metals, and dense plasmas.

Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations.

Rigorous diagnostics of experiments with warm dense matter are notoriously difficult. A key method is X-ray Thomson scattering (XRTS), but the interpretation of XRTS measurements is usually based on theoretical models that entail various approximations. Recently, Dornheim et al. [Nat. Commun. 13, 7911 (2022)] introduced a new framework for temperature diagnostics of XRTS experiments that is based on imaginary-time correlation functions. On the one hand, switching from the frequency to the imaginary-time domain gives one direct access to a number of physical properties, which facilitates the extraction of the temperature of arbitrarily complex materials without relying on any models or approximations. On the other hand, the bulk of theoretical work in dynamic quantum many-body theory is devoted to the frequency domain, and, to the best of our knowledge, the manifestation of physics properties within the imaginary-time density-density correlation function (ITCF) remains poorly understood. In the present work, we aim to fill this gap by introducing a simple, semi-analytical model for the imaginary-time dependence of two-body correlations within the framework of imaginary-time path integrals. As a practical example, we compare our new model to extensive ab initio path integral Monte Carlo results for the ITCF of a uniform electron gas, and find excellent agreement over a broad range of wavenumbers, densities and temperatures. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Energy response and spatial alignment of the perturbed electron gas.

We study the linear energy response of the uniform electron gas to an external harmonic perturbation with a focus on resolving different contributions to the total energy. This has been achieved by carrying out highly accurate ab initio path integral Monte Carlo (PIMC) calculations for a variety of densities and temperatures. We report a number of physical insights into effects such as screening and the relative importance of kinetic and potential energies for different wave numbers. A particularly interesting finding is obtained from the observed non-monotonic behavior of the induced change in the interaction energy, which becomes negative for intermediate wave numbers. This effect is strongly dependent on the coupling strength and constitutes further direct evidence for the spatial alignment of electrons introduced in earlier works [T. Dornheim et al., Commun. Phys. 5, 304 (2022)]. The observed quadratic dependence on the perturbation amplitude in the limit of weak perturbations and the quartic dependence of perturbation amplitude corrections are consistent with linear and nonlinear versions of the density stiffness theorem. All PIMC simulation results are freely available online and can be used to benchmark new methods or as input for other calculations.

Assessing the accuracy of hybrid exchange-correlation functionals for the density response of warm dense electrons.

We assess the accuracy of common hybrid exchange-correlation (XC) functionals (PBE0, PBE0-1/3, HSE06, HSE03, and B3LYP) within the Kohn-Sham density functional theory for the harmonically perturbed electron gas at parameters relevant for the challenging conditions of the warm dense matter. Generated by laser-induced compression and heating in the laboratory, the warm dense matter is a state of matter that also occurs in white dwarfs and planetary interiors. We consider both weak and strong degrees of density inhomogeneity induced by the external field at various wavenumbers. We perform an error analysis by comparing with the exact quantum Monte Carlo results. In the case of a weak perturbation, we report the static linear density response function and the static XC kernel at a metallic density for both the degenerate ground-state limit and for partial degeneracy at the electronic Fermi temperature. Overall, we observe an improvement in the density response when the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are used, compared with the previously reported results for the PBE, PBEsol, local-density approximation, and AM05 functionals; B3LYP, on the other hand, does not perform well for the considered system. Additionally, the PBE0, PBE0-1/3, HSE06, and HSE03 functionals are more accurate for the density response properties than SCAN in the regime of partial degeneracy.

Ab Initio Static Exchange-Correlation Kernel across Jacob's Ladder without Functional Derivatives.

The electronic exchange─correlation (XC) kernel constitutes a fundamental input for the estimation of a gamut of properties such as the dielectric characteristics, the thermal and electrical conductivity, or the response to an external perturbation. In this work, we present a formally exact methodology for the computation of the system specific static XC kernel exclusively within the framework of density functional theory (DFT) and without employing functional derivatives─no external input apart from the usual XC-functional is required. We compare our new results with exact quantum Monte Carlo (QMC) data for the archetypical uniform electron gas model under both ambient and warm dense matter conditions. This gives us unprecedented insights into the performance of different XC functionals, and it has important implications for the development of new functionals that are designed for the application at extreme temperatures. In addition, we obtain new DFT results for the XC kernel of warm dense hydrogen as it occurs in fusion applications and astrophysical objects. The observed excellent agreement to the QMC reference data demonstrates that presented framework is capable to capture nontrivial effects such as XC-induced isotropy breaking in the density response of hydrogen at large wave numbers.

Non-empirical Mixing Coefficient for Hybrid XC Functionals from Analysis of the XC Kernel.

We present an analysis of the static exchange-correlation (XC) kernel computed from hybrid functionals with a single mixing coefficient such as PBE0 and PBE0-1/3. We break down the hybrid XC kernels into the exchange and correlation parts using the Hartree-Fock functional, the exchange-only PBE, and the correlation-only PBE. This decomposition is combined with exact data for the static XC kernel of the uniform electron gas and an Airy gas model within a subsystem functional approach. This gives us a tool for the non-empirical choice of the mixing coefficient under ambient and extreme conditions. Our analysis provides physical insights into the effect of the variation of the mixing coefficient in hybrid functionals, which is of immense practical value. The presented approach is general and can be used for other types of functionals like screened hybrids.

Ab initio path integral Monte Carlo simulations of hydrogen snapshots at warm dense matter conditions.

We combine ab initio path integral Monte Carlo (PIMC) simulations with fixed ion configurations from density functional theory molecular dynamics (DFT-MD) simulations to solve the electronic problem for hydrogen under warm dense matter conditions [Böhme et al., Phys. Rev. Lett. 129, 066402 (2022)0031-900710.1103/PhysRevLett.129.066402]. The problem of path collapse due to the Coulomb attraction is avoided by utilizing the pair approximation, which is compared against the simpler Kelbg pair potential. We find very favorable convergence behavior towards the former. Since we do not impose any nodal restrictions, our PIMC simulations are afflicted with the notorious fermion sign problem, which we analyze in detail. While computationally demanding, our results constitute an exact benchmark for other methods and approximations within DFT. Our setup gives us the unique capability to study important properties of warm dense hydrogen such as the electronic static density response and exchange-correlation kernel without any model assumptions, which will be very valuable for a variety of applications such as the interpretation of experiments and the development of new XC functionals.

Accurate temperature diagnostics for matter under extreme conditions.

The experimental investigation of matter under extreme densities and temperatures, as in astrophysical objects and nuclear fusion applications, constitutes one of the most active frontiers at the interface of material science, plasma physics, and engineering. The central obstacle is given by the rigorous interpretation of the experimental results, as even the diagnosis of basic parameters like the temperature T is rendered difficult at these extreme conditions. Here, we present a simple, approximation-free method to extract the temperature of arbitrarily complex materials in thermal equilibrium from X-ray Thomson scattering experiments, without the need for any simulations or an explicit deconvolution. Our paradigm can be readily implemented at modern facilities and corresponding experiments will have a profound impact on our understanding of warm dense matter and beyond, and open up a variety of appealing possibilities in the context of thermonuclear fusion, laboratory astrophysics, and related disciplines.

Diamond formation kinetics in shock-compressed C─H─O samples recorded by small-angle x-ray scattering and x-ray diffraction.

Extreme conditions inside ice giants such as Uranus and Neptune can result in peculiar chemistry and structural transitions, e.g., the precipitation of diamonds or superionic water, as so far experimentally observed only for pure C─H and H2O systems, respectively. Here, we investigate a stoichiometric mixture of C and H2O by shock-compressing polyethylene terephthalate (PET) plastics and performing in situ x-ray probing. We observe diamond formation at pressures between 72 ± 7 and 125 ± 13 GPa at temperatures ranging from ~3500 to ~6000 K. Combining x-ray diffraction and small-angle x-ray scattering, we access the kinetics of this exotic reaction. The observed demixing of C and H2O suggests that diamond precipitation inside the ice giants is enhanced by oxygen, which can lead to isolated water and thus the formation of superionic structures relevant to the planets' magnetic fields. Moreover, our measurements indicate a way of producing nanodiamonds by simple laser-driven shock compression of cheap PET plastics.

Static Electronic Density Response of Warm Dense Hydrogen: Ab Initio Path Integral Monte Carlo Simulations.

The properties of hydrogen under extreme conditions are important for many applications, including inertial confinement fusion and astrophysical models. A key quantity is given by the electronic density response to an external perturbation, which is probed in x-ray Thomson scattering experiments-the state of the art diagnostics from which system parameters like the free electron density n_{e}, the electronic temperature T_{e}, and the charge state Z can be inferred. In this work, we present highly accurate path integral Monte Carlo results for the static electronic density response of hydrogen. We obtain the static exchange-correlation (XC) kernel K_{XC}, which is of central relevance for many applications, such as time-dependent density functional theory. This gives us a first unbiased look into the electronic density response of hydrogen in the warm-dense matter regime, thereby opening up a gamut of avenues for future research.

Effective electronic forces and potentials from ab initio path integral Monte Carlo simulations.

The rigorous description of correlated quantum many-body systems constitutes one of the most challenging tasks in contemporary physics and related disciplines. In this context, a particularly useful tool is the concept of effective pair potentials that take into account the effects of the complex many-body medium consistently. In this work, we present extensive, highly accurate ab initio path integral Monte Carlo (PIMC) results for the effective interaction and the effective force between two electrons in the presence of the uniform electron gas. This gives us a direct insight into finite-size effects, thereby, opening up the possibility for novel domain decompositions and methodological advances. In addition, we present unassailable numerical proof for an effective attraction between two electrons under moderate coupling conditions, without the mediation of an underlying ionic structure. Finally, we compare our exact PIMC results to effective potentials from linear-response theory, and we demonstrate their usefulness for the description of the dynamic structure factor. All PIMC results are made freely available online and can be used as a thorough benchmark for new developments and approximations.

Density Functional Theory Perspective on the Nonlinear Response of Correlated Electrons across Temperature Regimes.

We explore a new formalism to study the nonlinear electronic density response based on Kohn-Sham density functional theory (KS-DFT) at partially and strongly quantum degenerate regimes. It is demonstrated that the KS-DFT calculations are able to accurately reproduce the available path integral Monte Carlo simulation results at temperatures relevant for warm dense matter research. The existing analytical results for the quadratic and cubic response functions are rigorously tested. It is demonstrated that the analytical results for the quadratic response function closely agree with the KS-DFT data. Furthermore, the performed analysis reveals that currently available analytical formulas for the cubic response function are not able to describe simulation results, neither qualitatively nor quantitatively, at small wavenumbers q < 2qF, with qF being the Fermi wavenumber. The results show that KS-DFT can be used to describe warm dense matter that is strongly perturbed by an external field with remarkable accuracy. Furthermore, it is demonstrated that KS-DFT constitutes a valuable tool to guide the development of the nonlinear response theory of correlated quantum electrons from ambient to extreme conditions. This opens up new avenues to study nonlinear effects in a gamut of different contexts at conditions that cannot be accessed with previously used path integral Monte Carlo methods.

Energy Relaxation and Electron-Phonon Coupling in Laser-Excited Metals.

The rate of energy transfer between electrons and phonons is investigated by a first-principles framework for electron temperatures up to Te = 50,000 K while considering the lattice at ground state. Two typical but differently complex metals are investigated: aluminum and copper. In order to reasonably take the electronic excitation effect into account, we adopt finite temperature density functional theory and linear response to determine the electron temperature-dependent Eliashberg function and electron density of states. Of the three branch-dependent electron-phonon coupling strengths, the longitudinal acoustic mode plays a dominant role in the electron-phonon coupling for aluminum for all temperatures considered here, but for copper it only dominates above an electron temperature of Te = 40,000 K. The second moment of the Eliashberg function and the electron phonon coupling constant at room temperature Te=315 K show good agreement with other results. For increasing electron temperatures, we show the limits of the T=0 approximation for the Eliashberg function. Our present work provides a rich perspective on the phonon dynamics and this will help to improve insight into the underlying mechanism of energy flow in ultra-fast laser-metal interaction.