Electronic Structure

The lattice vibrations (phonon modes) of crystals underpin a large number of material properties. The harmonic phonon spectrum of a solid is the simplest description of its structural dynamics and can be straightforwardly derived from the Hellman–Feynman forces obtained in a ground-state electronic structure calculation. The presence of imaginary harmonic modes in the spectrum indicates that a structure is not a local minimum on the structural potential-energy surface and is instead a saddle point or a hilltop, for example. This can in turn yield important insight into the fundamental nature and physical properties of a material. In this review article, we discuss the physical significance of imaginary harmonic modes and distinguish between cases where imaginary modes are indicative of such phenomena, and those where they reflect technical problems in the calculations. We outline basic approaches for exploring and renormalising imaginary modes, and demonstrate their utility through a set of three case studies in the materials sciences.

This Roadmap article provides a succinct, comprehensive overview of the state of electronic structure (ES) methods and software for molecular and materials simulations. Seventeen distinct sections collect insights by 51 leading scientists in the field. Each contribution addresses the status of a particular area, as well as current challenges and anticipated future advances, with a particular eye towards software related aspects and providing key references for further reading. Foundational sections cover density functional theory and its implementation in real-world simulation frameworks, Green's function based many-body perturbation theory, wave-function based and stochastic ES approaches, relativistic effects and semiempirical ES theory approaches. Subsequent sections cover nuclear quantum effects, real-time propagation of the ES, challenges for computational spectroscopy simulations, and exploration of complex potential energy surfaces. The final sections summarize practical aspects, including computational workflows for complex simulation tasks, the impact of current and future high-performance computing architectures, software engineering practices, education and training to maintain and broaden the community, as well as the status of and needs for ES based modeling from the vantage point of industry environments. Overall, the field of ES software and method development continues to unlock immense opportunities for future scientific discovery, based on the growing ability of computations to reveal complex phenomena, processes and properties that are determined by the make-up of matter at the atomic scale, with high precision.

In recent years, we have been witnessing a paradigm shift in computational materials science. In fact, traditional methods, mostly developed in the second half of the XXth century, are being complemented, extended, and sometimes even completely replaced by faster, simpler, and often more accurate approaches. The new approaches, that we collectively label by machine learning, have their origins in the fields of informatics and artificial intelligence, but are making rapid inroads in all other branches of science. With this in mind, this Roadmap article, consisting of multiple contributions from experts across the field, discusses the use of machine learning in materials science, and share perspectives on current and future challenges in problems as diverse as the prediction of materials properties, the construction of force-fields, the development of exchange correlation functionals for density-functional theory, the solution of the many-body problem, and more. In spite of the already numerous and exciting success stories, we are just at the beginning of a long path that will reshape materials science for the many challenges of the XXIth century.

Materials design based on density functional theory (DFT) calculations is an emergent field of great potential to accelerate the development and employment of novel materials. Magnetic materials play an essential role in green energy applications as they provide efficient ways of harvesting, converting, and utilizing energy. In this review, after a brief introduction to the major functionalities of magnetic materials, we demonstrated how the fundamental properties can be tackled via high-throughput DFT calculations, with a particular focus on the current challenges and feasible solutions. Successful case studies are summarized on several classes of magnetic materials, followed by bird-view perspectives.

The development of novel materials for vacuum electron sources in particle accelerators is an active field of research that can greatly benefit from the results of ab initio calculations for the characterization of the electronic structure of target systems. As state-of-the-art many-body perturbation theory calculations are too expensive for large-scale material screening, density functional theory offers the best compromise between accuracy and computational feasibility. The quality of the obtained results, however, crucially depends on the choice of the exchange–correlation potential, v_xc. To address this essential point, we systematically analyze the performance of three popular approximations of v_xc [PBE, strongly constrained and appropriately normed (SCAN), and HSE06] on the structural and electronic properties of bulk Cs₃Sb and Cs₂Te as representative materials of Cs-based semiconductors employed in photocathode applications. Among the adopted approximations, PBE shows expectedly the largest discrepancies from the target: the unit cell volume is overestimated compared to the experimental value, while the band gap is severely underestimated. On the other hand, both SCAN and HSE06 perform remarkably well in reproducing both structural and electronic properties. Spin–orbit coupling, which mainly impacts the valence region of both materials inducing a band splitting and, consequently, a band-gap reduction of the order of 0.2 eV, is equally captured by all functionals. Our results indicate SCAN as the best trade-off between accuracy and computational costs, outperforming the considerably more expensive HSE06.

This review outlines fundamental principles, instrumentation, and capabilities of angle-resolved photoemission spectroscopy (ARPES) and microscopy. We will present how high-resolution ARPES enables to investigate fine structures of electronic band dispersions, Fermi surfaces, gap structures, and many-body interactions, and how angle-resolved photoemission microscopy (spatially-resolved ARPES) utilizing micro/nano-focused light allows to extract spatially localized electronic information at small dimensions. This work is focused on specific results obtained by the author from strongly correlated copper and ruthenium oxides, to help readers to understand consistently how these techniques can provide essential electronic information of materials, which can, in principle, apply to a wide variety of systems.

Organic-inorganic metal-halide perovskite semiconductors have outstanding and widely tunable optoelectronic properties suited for a broad variety of applications. First-principles numerical modelling techniques are playing a key role in unravelling structure-property relationships of this structurally and chemically diverse family of materials, and for predicting new materials and properties. Herein we review first-principles calculations of the photophysics of halide perovskites with a focus on the band structures, optical absorption spectra and excitons, and the effects of electron- and exciton-phonon coupling and temperature on these properties. We focus on first-principles approaches based on density functional theory and Green's function-based many-body perturbation theory and provide an overview of these approaches. While a large proportion of first-principles studies have been focusing on the prototypical ABX₃ single perovskites based on Pb and Sn, recent years have witnessed significant efforts to further functionalize halide perovskites, broadening this family of materials to include double perovskites, quasi-low-dimensional structures, and other organic-inorganic materials, interfaces and heterostructures. While this enormous chemical space of perovskite and perovskite-like materials has only begun to be tapped experimentally, recent advances in theoretical and computational methods, as well as in computing infrastructure, have led to the possibility of understanding the photophysics of ever more complex systems. We illustrate this progress in our review by summarizing representative studies of first-principles calculations of halide perovskites with various degrees of complexity.

By summarizing the constraint-based development of orbital-free free-energy density functional approximations, we provide a perspective on progress over the last 15 years, the limitations of existing functionals, and the challenges awaiting resolution. We outline the chronology of the development of noninteracting and exchange-correlation free-energy orbital-free functionals and summarize the theoretical basis of existing local density approximation, second-order approximation, generalized gradient approximation (GGA), and meta-GGAs. We discuss limitations and challenges such as problems with thermodynamic derivatives, free-energy nonadditivity and the closely related issue of all-electron versus valence-only local pseudo-potential performance.

Work function is a fundamental property of metals and is related to many surface-related phenomena of metals. Theoretically, it can be calculated with a metal slab supercell in density functional theory (DFT) calculations. In this paper, we discuss how the commensurability of atomic structure with the underlying fast Fourier transform (FFT) grid affects the accuracy of work function obtained from plane-wave pseudopotential DFT calculations. We show that the macroscopic average potential, which is an important property in work function calculations under the 'bulk reference' method, is more numerically stable when it is calculated with commensurate FFT grids than with incommensurate FFT grids. Due to the stability of the macroscopic average potential, work function calculated with commensurate FFT grids shows better convergence with respect to basis set size, vacuum length and slab thickness of a slab supercell. After we control the FFT grid commensurability issue in our work function calculations, we obtain well-converged work functions for Al, Pd, Au and Pt of (100), (110) and (111) surface orientations. For all the metals considered, the ordering of our calculated work functions of the three surface orientations agrees with experiment. Our findings reveal the importance of the FFT grid commensurability issue, which is usually neglected in practice, in obtaining accurate metal work functions, and are also meaningful to other DFT calculations which can be affected by the FFT grid commensurability issue.

We present the first calculations of the inelastic part of the dynamical structure factor (DSF) for warm dense matter (WDM) using time-dependent orbital-free density functional theory (TD-OF-DFT) and mixed-stochastic-deterministic (mixed) Kohn Sham TD-DFT (KS TD-DFT). WDM is an intermediate phase of matter found in planetary cores and laser-driven experiments, where the accurate calculation of the DSF is critical for interpreting x-ray Thomson scattering measurements. Traditional TD-DFT methods, while highly accurate, are computationally expensive, motivating the exploration of TD-OF-DFT and mixed TD-KS-DFT as more efficient alternatives. We applied these methods to experimentally measured WDM systems, including solid-density aluminum and beryllium, compressed beryllium, and carbon–hydrogen mixtures. Our results show that TD-OF-DFT requires a dynamical kinetic energy potential in order to qualitatively capture the plasmon response. Additionally, it struggles with capturing bound electron contributions. In contrast, mixed TD-KS-DFT offers greater accuracy in distinguishing bound and free electron effects, aligning well with experimental data, though at a higher computational cost. This study highlights the trade-offs between computational efficiency and accuracy, demonstrating that TD-OF-DFT remains a valuable tool for rapid scans of parameter space, while mixed TD-KS-DFT should be preferred for high-fidelity simulations. Our findings provide insight into the future development of DFT methods for WDM and suggest potential improvements for TD-OF-DFT.

The environment induced shift of any property derived from the embedded wavefunctions obtained from any method based on Frozen Density Embedding Theory depend on (a) the approximation made for the universal bi-functional $v^\mathrm {nad}_\mathrm {xct}[\rho_A,\rho_B]$, and (b) the modeller-decided density $\rho$_B used as the only quantum descriptor for N_B electrons associated with the environment. In several previous works, a particular choice for (a) and (b) was tested and shown to yield very accurate complexation induced shifts of vertical excitation energies for chromophores non-covalently bound to their environment for such excitations which do not involve large redistribution of the electron density upon excitation (errors in the range of 0.04 eV). The present work extend the previous studies to transition moments and oscillator strengths and demonstrated the limits of the predictive power of the protocol used for transition energies. For the errors in the oscillator strengths, lie below 0.06. The role of the environment polarisation and the intermolecular Pauli repulsion is analysed in detail.

The kinetic energy (KE) kernel, which is defined as the second order functional derivative of the KE functional with respect to density, is the key ingredient to the construction of KE models for orbital free density functional theory applications. For solids, KE kernels are usually approximated using the uniform electron gas (UEG) model or the UEG-with-gap model. These kernels do not have knowledge about the core electrons since there are no orbitals directly available to couple with nonlocal pseudopotentials (NLPs). To illuminate this aspect, we provide a methodology for computing KE kernels from pseudopotential Kohn–Sham DFT and apply them to the valence electrons in bulk aluminum (Al) with a face-centered cubic lattice and in bulk silicon (Si) in a semiconducting crystal diamond state. We find that bulk-derived local pseudopotentials provide accurate KE kernels in the interstitial region. However, the effect of using NLPs manifests at short wavelengths, roughly defined by the cutoff radius of the nonlocal part of the Kohn–Sham DFT pseudopotential. In this region, we record significant deviations between KE kernels and the von Weizsäcker result.

The increased availability of computing time, in recent years, allows for systematic high-throughput studies of material classes. Such studies serve the purpose of both screening for materials with remarkable properties and understanding how structural configuration and material composition affect macroscopic attributes manifestation. However, when conducting systematic high-throughput studies, the individual ab initio calculations' success depends on the quality of the chosen input quantities. On a large scale, improving input parameters by trial and error is neither efficient nor systematic. We present a systematic, high-throughput compatible, and machine learning (ML)-based approach to improve the input parameters optimized during a density functional theory computation or workflow. This approach of integrating ML into a typical high-throughput workflow demonstrates the advantages and necessary considerations for a systematic study of magnetic multilayers of 3d transition metal layers on FCC noble metal substrates. For 6660 film systems, we were able to improve the overall success rate of our high-throughput FLAPW-based structural relaxations from 64.8% to 94.3% while at the same time requiring 17% less computational time for each successful relaxation.

Although metallic elements favor three-dimensional (3D) geometries due to their isotropic, metallic bonding, experiments have reported metals also with two-dimensional allotropes, the so-called metallenes. And while bulk metals' electronic and structural properties are well known, the corresponding knowledge for atomically thin metallenes remains scattered. Therefore, in this work, we use density-functional theory to investigate the electronic and structural properties of 45 elemental metals with honeycomb, square, and hexagonal lattices, along with their buckled counterparts, resulting in a comprehensive catalog of 270 metallenes with their properties. We systematically present their structural, energetic, and electronic structure properties and discuss similarities and differences compared to their 3D counterparts. As a result, simple and noble metals exhibit similar characteristics and lack buckled hexagonal lattice. Apart from scattered exceptions, the trends in several properties, such as bond lengths, cohesion energies, and projected densities of states, are governed by coordination numbers and exhibit systematic patterns. This systematic reporting provides a necessary reference for the selection and categorization of metallenes for further experimental efforts to develop them for catalytic, sensing, plasmonic, and nanoelectronics applications.

Superalkalis with low ionization energies have properties mimicking those of alkali atoms, therefore serving as potentially useful elements for constructing innovative nanostructured materials. Ab initio techniques were successfully utilized for investigating entirely new classes of metallic superalkali cationic clusters, F_nM_n+1⁺ cations (M = Li, Na and K and n = 1–7) using second-order Møller–Plesset perturbation theory (MP2). We have demonstrated a connection between the core atoms and the structural characteristics of such clusters by analyzing the NBO charge, bond distances, and vertical electron affinity (EA_v). The F_nLi_n+1⁺, F_nNa_n+1⁺, and F_nK_n+1⁺ (n = 1–7) cations have exceptionally low vertical electron affinities under the range of 3.61–2.64 eV, 3.26–2.23 eV, and 2.84–2.08 eV, respectively, and consequently, they ought to be characterized as novel superalkali clusters.

By summarizing the constraint-based development of orbital-free free-energy density functional approximations, we provide a perspective on progress over the last 15 years, the limitations of existing functionals, and the challenges awaiting resolution. We outline the chronology of the development of noninteracting and exchange-correlation free-energy orbital-free functionals and summarize the theoretical basis of existing local density approximation, second-order approximation, generalized gradient approximation (GGA), and meta-GGAs. We discuss limitations and challenges such as problems with thermodynamic derivatives, free-energy nonadditivity and the closely related issue of all-electron versus valence-only local pseudo-potential performance.

This Roadmap article provides a succinct, comprehensive overview of the state of electronic structure (ES) methods and software for molecular and materials simulations. Seventeen distinct sections collect insights by 51 leading scientists in the field. Each contribution addresses the status of a particular area, as well as current challenges and anticipated future advances, with a particular eye towards software related aspects and providing key references for further reading. Foundational sections cover density functional theory and its implementation in real-world simulation frameworks, Green's function based many-body perturbation theory, wave-function based and stochastic ES approaches, relativistic effects and semiempirical ES theory approaches. Subsequent sections cover nuclear quantum effects, real-time propagation of the ES, challenges for computational spectroscopy simulations, and exploration of complex potential energy surfaces. The final sections summarize practical aspects, including computational workflows for complex simulation tasks, the impact of current and future high-performance computing architectures, software engineering practices, education and training to maintain and broaden the community, as well as the status of and needs for ES based modeling from the vantage point of industry environments. Overall, the field of ES software and method development continues to unlock immense opportunities for future scientific discovery, based on the growing ability of computations to reveal complex phenomena, processes and properties that are determined by the make-up of matter at the atomic scale, with high precision.

Organic-inorganic metal-halide perovskite semiconductors have outstanding and widely tunable optoelectronic properties suited for a broad variety of applications. First-principles numerical modelling techniques are playing a key role in unravelling structure-property relationships of this structurally and chemically diverse family of materials, and for predicting new materials and properties. Herein we review first-principles calculations of the photophysics of halide perovskites with a focus on the band structures, optical absorption spectra and excitons, and the effects of electron- and exciton-phonon coupling and temperature on these properties. We focus on first-principles approaches based on density functional theory and Green's function-based many-body perturbation theory and provide an overview of these approaches. While a large proportion of first-principles studies have been focusing on the prototypical ABX₃ single perovskites based on Pb and Sn, recent years have witnessed significant efforts to further functionalize halide perovskites, broadening this family of materials to include double perovskites, quasi-low-dimensional structures, and other organic-inorganic materials, interfaces and heterostructures. While this enormous chemical space of perovskite and perovskite-like materials has only begun to be tapped experimentally, recent advances in theoretical and computational methods, as well as in computing infrastructure, have led to the possibility of understanding the photophysics of ever more complex systems. We illustrate this progress in our review by summarizing representative studies of first-principles calculations of halide perovskites with various degrees of complexity.

Topotactic chemical reaction (TCR) is a chemical process that transforms one crystalline phase to another while maintaining one or more of the original structural frameworks, typically induced by the local insertion, removal, or replacement of atoms in a crystal. The utilization of TCR in atomic-layer materials and surfaces of bulk crystals leads to exotic quantum phases, as highlighted by the control of topological phases, the emergence of two-dimensional (2D) superconductivity, and the realization of 2D ferromagnetism. Advanced surface-sensitive spectroscopies such as angle-resolved photoemission spectroscopy and scanning tunneling microscopy are leading techniques to visualize the electronic structure of such exotic states and provide us a guide to further functionalize material properties. In this review article, we summarize the recent progress in this field, with particular emphasis on intriguing results obtained by combining spectroscopies and TCR in thin films.

Solid state theory, density functional theory and its generalizations for correlated systems together with numerical simulations on supercomputers allow nowadays to model magnetic systems realistically and in detail and can be even used to predict new materials, paving the way for more rapid material development for applications in energy storage and conversion, information technologies, sensors, actuators etc. Modeling magnets on different length scales (between a few $\mathrm{\unicode{x00C5}}$ngström and several micrometers) requires, however, approaches with very different mathematical formulations. Parameters defining the material in each formulation can be determined either by fitting experimental data or from theoretical calculations and there exists a well-established approach for obtaining model parameters for each length scale using the information from the smaller length scale. In this review, this approach will be explained step-by-step in textbook style with examples of successful scale-bridging modeling of different classes of magnetic materials from the research literature as well as based on results newly obtained for this review.

Ab initio calculations based on density functional theory (DFT) have been performed to investigate the role of trivalent atoms
substituting silicon atom in the 2D zeolite model. The effects of the B and Ga atoms on the stability, structural and electronic
properties of the 2D zeolite model are explored. Our DFT calculations reveal that the introduction of B atom is exothermic whereas
that one of Ga atom is endothermic. The structural analysis shows that the incorporation of B and Ga atoms affects the bond lengths
of the system, however it does not lead to a significant deformation of the structure. The Fermi level of the doped systems is shifted
towards the valence band, indicating that the incorporation of these trivalent atoms leads to p−type materials. The second purpose
of this study is to find the suitable charge compensations among hydrogen and alkali metals as well as their site preference (either
on the surface or in the cages of the silica bilayer). The calculated formation energy values are similar, suggesting both configura-
tions could co-exist. Hydrogen has the lowest formation energy and the proton affinity analysis predicts low acid strength of H-B- compared to H-Ga-doped 2D zeolite, a similar trend to that of bulk zeolite. Among the alkali elements, we found that Na and K atoms are the most stable ones. The density of states analysis shows that the Fermi level is lying within the gap, and defect states are observed near the band edges narrowing the band gap of the system. This work provides detailed and valuable information about the atomic-level
properties of the relatively recent 2D zeolite model, which is beneficial for its industrial applications.

We report on results from an experiment at the European XFEL where we measured the x-ray Thomson scattering (XRTS) spectrum of single crystal silicon with ultrahigh resolution. Compared to similar previous experiments, we consider a more complex scattering setup, in which the scattering vector changes orientation through the crystal lattice. In doing so, we are able to observe strong geometric dependencies in the inelastic scattering spectrum of silicon at low scattering angles. Furthermore, the high quality of the experimental data allows us to benchmark state-of-the-art TDDFT calculations, and demonstrate TDDFT's ability to accurately predict these geometric dependencies. Finally, we note that this experimental data was collected at a much faster rate than another recently reported dataset using the same setup, demonstrating that ultrahigh resolution XRTS data can be collected in more general experimental scenarios.

The "egg-box effect" is a known challenge in Density Functional Theory (DFT) calculations which arises from the discretization of continuous quantities, e.g. the electron density. This effect is observed when the system is moved relative to the underlying computational grid and causes an unphysical change in the system's total energy, violating translational invariance. The pattern of energy change with translation is reminiscent of an egg-box. This effect can cause unphysical results such as geometry relaxations finding incorrect crystal symmetries or imaginary phonon modes in vibrational calculations. The egg-box effect can be mitigated by using finer grids for all continuous quantities, but this greatly increases computational cost. For plane-wave DFT the effect's origin is the evaluation of the Exchange and Correlation (XC) energy (E_XC ). We present a novel technique for estimating the violation of translational invariance in E_XC , using a Fourier interpolation scheme, providing an estimate of the uncertainty. Our results also show that the numerical behaviour of the XC approximations is strongly linked to the magnitude of the violation of the translation invariance. The more numerically ill behaved XC functionals, such as the more advanced meta-GGAs functionals, exhibit changes in E_XC (due to violation of translation invariance) that are orders of magnitude worse than less advanced functionals. Performing this analysis at an early stage of a workflow can inform the user about the expected accuracy of subsequent calculations. Further, our results demonstrate that by selectively computing E_XC and its corresponding potential (V_XC ) on a finer grid, the egg-box effect can be significantly reduced. Coupled with our uncertainty quantification method, egg-box related inaccuracies can be avoided more conveniently and efficiently than just increasing grid resolution until inaccuracies appear suppressed. This work offers a promising pathway towards mitigating the egg-box effect in a diverse range of materials modelling applications.

The environment induced shift of any property derived from the embedded wavefunctions obtained from any method based on Frozen Density Embedding Theory depend on (a) the approximation made for the universal bi-functional $v^\mathrm {nad}_\mathrm {xct}[\rho_A,\rho_B]$, and (b) the modeller-decided density $\rho$_B used as the only quantum descriptor for N_B electrons associated with the environment. In several previous works, a particular choice for (a) and (b) was tested and shown to yield very accurate complexation induced shifts of vertical excitation energies for chromophores non-covalently bound to their environment for such excitations which do not involve large redistribution of the electron density upon excitation (errors in the range of 0.04 eV). The present work extend the previous studies to transition moments and oscillator strengths and demonstrated the limits of the predictive power of the protocol used for transition energies. For the errors in the oscillator strengths, lie below 0.06. The role of the environment polarisation and the intermolecular Pauli repulsion is analysed in detail.

The kinetic energy (KE) kernel, which is defined as the second order functional derivative of the KE functional with respect to density, is the key ingredient to the construction of KE models for orbital free density functional theory applications. For solids, KE kernels are usually approximated using the uniform electron gas (UEG) model or the UEG-with-gap model. These kernels do not have knowledge about the core electrons since there are no orbitals directly available to couple with nonlocal pseudopotentials (NLPs). To illuminate this aspect, we provide a methodology for computing KE kernels from pseudopotential Kohn–Sham DFT and apply them to the valence electrons in bulk aluminum (Al) with a face-centered cubic lattice and in bulk silicon (Si) in a semiconducting crystal diamond state. We find that bulk-derived local pseudopotentials provide accurate KE kernels in the interstitial region. However, the effect of using NLPs manifests at short wavelengths, roughly defined by the cutoff radius of the nonlocal part of the Kohn–Sham DFT pseudopotential. In this region, we record significant deviations between KE kernels and the von Weizsäcker result.

We report on results from an experiment at the European XFEL where we measured the x-ray Thomson scattering (XRTS) spectrum of single crystal silicon with ultrahigh resolution. Compared to similar previous experiments, we consider a more complex scattering setup, in which the scattering vector changes orientation through the crystal lattice. In doing so, we are able to observe strong geometric dependencies in the inelastic scattering spectrum of silicon at low scattering angles. Furthermore, the high quality of the experimental data allows us to benchmark state-of-the-art TDDFT calculations, and demonstrate TDDFT's ability to accurately predict these geometric dependencies. Finally, we note that this experimental data was collected at a much faster rate than another recently reported dataset using the same setup, demonstrating that ultrahigh resolution XRTS data can be collected in more general experimental scenarios.

The "egg-box effect" is a known challenge in Density Functional Theory (DFT) calculations which arises from the discretization of continuous quantities, e.g. the electron density. This effect is observed when the system is moved relative to the underlying computational grid and causes an unphysical change in the system's total energy, violating translational invariance. The pattern of energy change with translation is reminiscent of an egg-box. This effect can cause unphysical results such as geometry relaxations finding incorrect crystal symmetries or imaginary phonon modes in vibrational calculations. The egg-box effect can be mitigated by using finer grids for all continuous quantities, but this greatly increases computational cost. For plane-wave DFT the effect's origin is the evaluation of the Exchange and Correlation (XC) energy (E_XC ). We present a novel technique for estimating the violation of translational invariance in E_XC , using a Fourier interpolation scheme, providing an estimate of the uncertainty. Our results also show that the numerical behaviour of the XC approximations is strongly linked to the magnitude of the violation of the translation invariance. The more numerically ill behaved XC functionals, such as the more advanced meta-GGAs functionals, exhibit changes in E_XC (due to violation of translation invariance) that are orders of magnitude worse than less advanced functionals. Performing this analysis at an early stage of a workflow can inform the user about the expected accuracy of subsequent calculations. Further, our results demonstrate that by selectively computing E_XC and its corresponding potential (V_XC ) on a finer grid, the egg-box effect can be significantly reduced. Coupled with our uncertainty quantification method, egg-box related inaccuracies can be avoided more conveniently and efficiently than just increasing grid resolution until inaccuracies appear suppressed. This work offers a promising pathway towards mitigating the egg-box effect in a diverse range of materials modelling applications.

The increased availability of computing time, in recent years, allows for systematic high-throughput studies of material classes. Such studies serve the purpose of both screening for materials with remarkable properties and understanding how structural configuration and material composition affect macroscopic attributes manifestation. However, when conducting systematic high-throughput studies, the individual ab initio calculations' success depends on the quality of the chosen input quantities. On a large scale, improving input parameters by trial and error is neither efficient nor systematic. We present a systematic, high-throughput compatible, and machine learning (ML)-based approach to improve the input parameters optimized during a density functional theory computation or workflow. This approach of integrating ML into a typical high-throughput workflow demonstrates the advantages and necessary considerations for a systematic study of magnetic multilayers of 3d transition metal layers on FCC noble metal substrates. For 6660 film systems, we were able to improve the overall success rate of our high-throughput FLAPW-based structural relaxations from 64.8% to 94.3% while at the same time requiring 17% less computational time for each successful relaxation.

By summarizing the constraint-based development of orbital-free free-energy density functional approximations, we provide a perspective on progress over the last 15 years, the limitations of existing functionals, and the challenges awaiting resolution. We outline the chronology of the development of noninteracting and exchange-correlation free-energy orbital-free functionals and summarize the theoretical basis of existing local density approximation, second-order approximation, generalized gradient approximation (GGA), and meta-GGAs. We discuss limitations and challenges such as problems with thermodynamic derivatives, free-energy nonadditivity and the closely related issue of all-electron versus valence-only local pseudo-potential performance.

We present the first calculations of the inelastic part of the dynamical structure factor (DSF) for warm dense matter (WDM) using time-dependent orbital-free density functional theory (TD-OF-DFT) and mixed-stochastic-deterministic (mixed) Kohn Sham TD-DFT (KS TD-DFT). WDM is an intermediate phase of matter found in planetary cores and laser-driven experiments, where the accurate calculation of the DSF is critical for interpreting x-ray Thomson scattering measurements. Traditional TD-DFT methods, while highly accurate, are computationally expensive, motivating the exploration of TD-OF-DFT and mixed TD-KS-DFT as more efficient alternatives. We applied these methods to experimentally measured WDM systems, including solid-density aluminum and beryllium, compressed beryllium, and carbon–hydrogen mixtures. Our results show that TD-OF-DFT requires a dynamical kinetic energy potential in order to qualitatively capture the plasmon response. Additionally, it struggles with capturing bound electron contributions. In contrast, mixed TD-KS-DFT offers greater accuracy in distinguishing bound and free electron effects, aligning well with experimental data, though at a higher computational cost. This study highlights the trade-offs between computational efficiency and accuracy, demonstrating that TD-OF-DFT remains a valuable tool for rapid scans of parameter space, while mixed TD-KS-DFT should be preferred for high-fidelity simulations. Our findings provide insight into the future development of DFT methods for WDM and suggest potential improvements for TD-OF-DFT.

Thanks to their favorable electronic and optical properties, sodium–potassium-antimonides are an emerging class of crystals used as photocathodes in particle accelerators. The persisting challenges related to the synthesis and characterization of these materials demand support from theory and make the study of computationally predicted polymorphs particularly relevant to identifying the structure, composition, and properties of the samples. Using first-principles methods based on density-functional theory and many-body perturbation theory, the electronic and optical properties of cubic NaK₂Sb and hexagonal Na₂KSb are studied. Both systems, experimentally reported in the hexagonal and cubic phase, respectively, exhibit an indirect fundamental gap that is energetically very close to the direct band gap at Γ of magnitude ${0.81}\,\textrm{eV}$ for NaK₂Sb and ${0.70}\,\textrm{eV}$ for Na₂KSb. In the band structure of both materials, Sb p-states dominate the valence region with minor contributions from the alkali p-states, while the alkali s-states mainly contribute at lower energies. The optical spectra of both crystals are not subject to sizeable excitonic effects, except for a redshift of the excitation energies of 50–100 meV and some redistribution of the oscillator strength beyond the lowest-energy peak in the near-infrared region. Our results indicate that computationally predicted cubic NaK₂Sb and hexagonal Na₂KSb have favorable characteristics as photocathodes and, as such, their presence in polycrystalline samples is not detrimental for this application.

Understanding the electrical conductivity of warm dense hydrogen is critical for both fundamental physics and applications in planetary science and inertial confinement fusion. We demonstrate how to calculate the electrical conductivity using the continuum form of Ohm's law, with the current density obtained from real-time time-dependent density functional theory. This approach simulates the dynamic response of hydrogen under warm dense matter conditions, with temperatures around 30 000 K and mass densities ranging from 0.02 to 0.98 g cm⁻³. We systematically address finite-size errors in real-time time-dependent density functional theory, demonstrating that our calculations are both numerically feasible and reliable. Our results show good agreement with other approaches, highlighting the effectiveness of this method for modeling electronic transport properties from ambient to extreme conditions.

This Roadmap article provides a succinct, comprehensive overview of the state of electronic structure (ES) methods and software for molecular and materials simulations. Seventeen distinct sections collect insights by 51 leading scientists in the field. Each contribution addresses the status of a particular area, as well as current challenges and anticipated future advances, with a particular eye towards software related aspects and providing key references for further reading. Foundational sections cover density functional theory and its implementation in real-world simulation frameworks, Green's function based many-body perturbation theory, wave-function based and stochastic ES approaches, relativistic effects and semiempirical ES theory approaches. Subsequent sections cover nuclear quantum effects, real-time propagation of the ES, challenges for computational spectroscopy simulations, and exploration of complex potential energy surfaces. The final sections summarize practical aspects, including computational workflows for complex simulation tasks, the impact of current and future high-performance computing architectures, software engineering practices, education and training to maintain and broaden the community, as well as the status of and needs for ES based modeling from the vantage point of industry environments. Overall, the field of ES software and method development continues to unlock immense opportunities for future scientific discovery, based on the growing ability of computations to reveal complex phenomena, processes and properties that are determined by the make-up of matter at the atomic scale, with high precision.