Density Functional Theory Based Methods for the Calculation of X-ray Spectroscopy

Validation of Long-Range-Corrected LC2gau Functional for Koopmans' Prediction of Core and Valence Ionization Energies with Diverse Data

Koopmans' theorem, based on Kohn-Sham (KS) orbital energies of current approximate functionals, does not predict well the ionization energies of 1s, 2s, and 2p core electrons in third-period elements due to the self-interaction errors (SIEs). To address this limitation, the LC2gau functional is developed, which we have validated in the present study. With a fixed range-separation parameter (μ) of 0.35 bohr-1, it yields a mean absolute deviation (MAD) of 0.37 eV for 401 valence ionization energies from the Chong-Gritsenko-Baerends (CGB) set and an MAD of 0.20 eV for the highest occupied molecular orbital (HOMO) ionization energies of 34 molecules containing third-period elements. It is less accurate in predicting core electron binding energies (CEBEs) of third-period elements. With μ = 0.35 bohr-1, the MAD of CEBEs is 1.30 eV. We observed that the CEBE increases linearly with μ and tuned it for each element. For 2s and 2p electrons in third-period elements, the optimal μ values are approximately 0.35 and 0.30 bohr-1, respectively. For the corresponding 1s electrons, the optimal μ varies across elements, gradually decreasing from 0.40 bohr-1 for Si to 0.12 bohr-1 for Cl. With the optimized μ, a smaller MAD of 0.64 eV is obtained for CEBEs of the third-period elements.

Revisiting the Half-and-Half Functional

Hybrid density functionals typically provide significantly better accuracy than semilocal functionals. Conventional wisdom holds that incorporating more than 20-25% exact exchange is deleterious to thermochemical properties and should only be used as a last resort, for problems that are dominated by self-interaction error. In such cases, the Becke-Lee-Yang-Parr "half-and-half" functional (BH&H-LYP) has emerged as a go-to choice, especially in time-dependent density functional theory calculations for excitation energies. Here, we examine the assumption that 50% Hartree-Fock exchange sacrifices thermochemical accuracy. Using a sequence of functionals B(α)LYP, with different percentages of exact exchange (0 ≤ α ≤ 100), we find that BH&H-LYP (with α = 50) is nearly optimal and affords accuracy similar to B3LYP for thermochemistry, barrier heights, and excitation energies. Although BH&H-LYP is significantly less accurate than B3LYP for atomization energies, this emerges as the sole rationale for the taboo against values α > 25. Overall, BH&H-LYP is a reasonable choice for problems that are dominated by self-interaction error, including charge-transfer complexes and core-level excitation energies. While B3LYP remains more accurate for valence excitation energies, the use of 50% exact exchange appears to be an acceptable compromise, and BH&H-LYP can be used without undue concern over its diminished accuracy for ground-state properties.

Solvatochromism Observed in the X-ray Absorption Spectrum of Indole Dissolved in Water

Current developments in X-ray absorption spectroscopy (XAS) for liquid samples in the water window demand a rigorous understanding of the interactions between molecules or solute-solvent interactions observed in the spectra. Meanwhile, a theoretical description of such effects, in addition to inner-shell excitations, remains controversial. The controversy is mainly over whether the orbitals should be optimized in the final states or whether the orbital optimizations can be expressed by dynamic electron correlation. In the present work, we measured the XAS spectra of indole in aqueous solution at the carbon and nitrogen K-edges to compare them with those measured in the gas phase. Obvious solvatochromism was observed only in the XAS spectrum measured at the nitrogen K-edge. We then interpreted the observed solvatochromism by simulating spectra with both ΔSCF, where the orbitals were optimized in the final states, and the algebraic-diagrammatic construction through second order [ADC(2)], where the molecular orbitals optimized in the ground state were used throughout. The present results indicate that covalent interactions, such as hydrogen bonds, are the dominant causes of the solvation effects observed in XAS spectra. The present simulations with ΔSCF and ADC(2), in addition to some other reports, highlight the importance of optimizing the orbitals in the final inner-shell excited states for general inner-shell calculations with predictive accuracy.

X-ray Spectroscopy Characterization of Electronic Structure and Metal-Metal Bonding in Dicobalt Complexes

Developing multimetallic complexes with tunable metal-metal interactions has long been a target of synthetic inorganic chemistry efforts due to the unique properties that such compounds can exhibit. However, understanding relationships between metal-metal bonding and chemical properties is challenging due to system-dependent factors that influence metal-metal and metal-ligand interactions, including ligand identity, coordination geometry, and metal-metal distance. In this work, we apply X-ray absorption and emission spectroscopy and quantum chemical calculations to describe electronic structure and bonding in a series of dicobalt complexes. The compounds with silane ligands and pseudo-octahedral coordination geometry exhibit Co-Co σ and multicentered bonding character, which we characterize from both the occupied and vacant perspectives via their contributions to the Co X-ray emission and absorption spectra, respectively. In contrast, the dicobalt complexes with a pseudotetrahedral coordination environment do not exhibit Co-Co bonding due to symmetry constraints on orbital overlap. We extend these insights to the theoretical evaluation of related dicobalt complexes to explain how ligand coordination and symmetry dictate the presence or absence of a Co-Co bond. This work highlights how fundamental insights into electronic structure and bonding through X-ray spectroscopy uncover important factors governing metal-metal interactions and guide the rational design of multimetallic complexes with tunable metal-metal bonds.

Highly Accurate and Robust Constraint-Based Orbital-Optimized Core Excitations

We adapt our recently developed constraint-based orbital-optimized excited-state method (COOX) for the computation of core excitations. COOX is a constrained density functional theory (cDFT) approach based on excitation amplitudes from linear-response time-dependent DFT (LR-TDDFT), and has been shown to provide accurate excitation energies and excited-state properties for valence excitations within a spin-restricted formalism. To extend COOX to core-excited states, we introduce a spin-unrestricted variant which allows us to obtain orbital-optimized core excitations with a single constraint. Using a triplet purification scheme in combination with the constrained unrestricted Hartree-Fock formalism, scalar-relativistic zero-order regular approximation corrections, and a semiempirical treatment of spin-orbit coupling, COOX is shown to produce highly accurate results for K- and L-edge excitations of second- and third-period atoms with subelectronvolt errors despite being based on LR-TDDFT, for which core excitations pose a well-known challenge. L- and M-edge excitations of heavier atoms up to uranium are also computationally feasible and numerically stable, but may require more advanced treatment of relativistic effects. Furthermore, COOX is shown to perform on par with or better than the popular ΔSCF approach while exhibiting more robust convergence, highlighting it as a promising tool for inexpensive and accurate simulations of X-ray absorption spectra.

A DFT/MRCI Hamiltonian parameterized using only ab initio data. II. Core-excited states

A newly parameterized combined density functional theory and multi-reference configuration interaction (DFT/MRCI) Hamiltonian, termed core-valence separation (CVS)-QE12, is defined for the computation of K-shell core-excitation and core-ionization energies. This CVS counterpart to the recently reported QE8 Hamiltonian [Costain et al., J. Chem. Phys, 160, 224106 (2024)] is parameterized by fitting to benchmark quality ab initio data. The definition of the CVS-QE12 and QE8 Hamiltonians differ from previous CVS-DFT/MRCI parameterizations in three primary ways: (i) the replacement of the BHLYP exchange-correlation functional with QTP17 to yield a balanced description of both core and valence excitation energies, (ii) the adoption of a new, three-parameter damping function, and (iii) the introduction of separate scaling of the core-valence and valence-valence Coulombic interactions. Crucially, the parameters of the CVS-QE12 Hamiltonian are obtained via fitting exclusively to highly accurate ab initio vertical core-excitation and ionization energies computed at the CVS-EOM-CCSDT level of theory. The CVS-QE12 Hamiltonian is validated against further benchmark computations and is found to furnish K-edge core vertical excitation and ionization energies exhibiting absolute errors ≤0.5 eV at low computational cost.

Metal-ligand delocalization of iron and cobalt porphyrin complexes in aqueous solutions probed by soft X-ray absorption spectroscopy

Metal-ligand delocalization of metal porphyrin complexes in aqueous solutions was investigated by analyzing the electronic structure of both the metal and ligand sides using soft X-ray absorption spectroscopy (XAS) at the metal L2,3-edges and nitrogen K-edge, respectively. In the N K-edge XAS spectra of the ligands, the energies of the CN π* peaks of cobalt protoporphyrin IX (CoPPIX) are higher than iron protoporphyrin IX (FePPIX). The energy difference between the two lowest peaks in the XAS spectrum of CoPPIX is also larger than that of FePPIX. Nitrogen K-edge inner-shell calculations of metalloporphyrins with different central metals indicate that the energy differences between these peaks reflect the electronic configurations and spin multiplicities of metalloporphyrins. We also investigated the hydration structure of CoPPIX in aqueous solution by analyzing the electronic structure of the ligand and revealed that CoPPIX maintains its five-coordination geometry in aqueous solution. The present study shows high performance of N K-edge XAS of ligands for studying the coordination structures of metalloporphyrins in solutions rather than the metal L2,3-edges of central metals.

Exploiting the Correlation between the 1s, 2s, and 2p Energies for the Prediction of Core-Level Binding Energies of Si, P, S, and Cl species

The binding energies (BEs) of the 1s, 2s, and 2p core electrons of third-period elements (Si, P, S, Cl) were calculated using Hartree-Fock (HF) and B3LYP, BH&HLYP, and LC-BOP ΔSCF, and the shifted KS ΔSCF methods. Linear relationships between two BEs were derived and compared with the Auger parameter. The derived lines are essentially parallel, with only the intercepts differing. The difference in intercepts is due to the lack of electron correlation effects in HF and the self-interaction errors (SIEs) of the functional. The slope is the slope of the linear relationship between the chemical shifts. The straight lines between the 2s and 2p BEs also coincided with the Auger parameter lines, which have a slope of 1 by definition and an intercept being the difference between the 2s and 2p BEs. The shifted KS ΔSCF scheme compensates for SIEs, yielding equations that are approximately invariant. The calculated average gaps for the 2s and 2p BEs are 51.21 eV for Si, 57.48 eV for P, 63.85 eV for S, and 70.48 eV for Cl. The straight lines representing the relationships between the BEs of the 1s and 2s and 1s and 2p electrons are also parallel to each other in ΔSCF and converge into a single line in the shifted ΔSCF scheme. However, these lines are steeper than the Auger parameter line. The derived relationships can be used to predict unknown BEs, which we have applied to many molecules. The results are highly accurate, with mean absolute errors (MAEs) of less than 0.2 eV compared to experimental values.

A new parameterization of the DFT/CIS method with applications to core-level spectroscopy

Time-dependent density functional theory (TD-DFT) within a restricted excitation space is an efficient means to compute core-level excitation energies using only a small subset of the occupied orbitals. However, core-to-valence excitation energies are significantly underestimated when standard exchange-correlation functionals are used, which is partly traceable to systemic issues with TD-DFT's description of Rydberg and charge-transfer excited states. To mitigate this, we have implemented an empirically modified combination of configuration interaction with single substitutions (CIS) based on Kohn-Sham orbitals, which is known as "DFT/CIS." This semi-empirical approach is well-suited for simulating x-ray near-edge spectra, as it contains sufficient exact exchange to model charge-transfer excitations yet retains DFT's low-cost description of dynamical electron correlation. Empirical corrections to the matrix elements enable semi-quantitative simulation of near-edge x-ray spectra without the need for significant a posteriori shifts; this should be useful in complex molecules and materials with multiple overlapping x-ray edges. Parameter optimization for use with a specific range-separated hybrid functional makes this a black-box method intended for both core and valence spectroscopy. Results herein demonstrate that realistic K-edge absorption and emission spectra can be obtained for second- and third-row elements and 3d transition metals, with promising results for L-edge spectra as well. DFT/CIS calculations require absolute shifts that are considerably smaller than what is typical in TD-DFT.

Toward Consistent Predictions of Core/Valence Ionization Potentials and Valence Excitation Energies by MRSF-TDDFT

Optimizing exchange-correlation functionals for both core/valence ionization potentials (cIPs/vIPs) and valence excitation energies (VEEs) at the same time in the framework of MRSF-TDDFT is self-contradictory. To overcome the challenge, within the previous "adaptive exact exchange" or double-tuning strategy on Coulomb-attenuating XC functionals (CAM), a new XC functional specifically for cIPs and vIPs was first developed by enhancing exact exchange to both short- and long-range regions. The resulting DTCAM-XI functional achieved remarkably high accuracy in its predictions with errors of less than half eV. An additional concept of "valence attenuation", where the amount of exact exchange for the frontier orbital regions is selectively suppressed, was introduced to consistently predict both VEEs and IPs at the same time. The second functional, DTCAM-XIV, exhibits consistent overall prediction accuracy at ∼0.64 eV. By preferentially optimizing VEEs within the same "valence attenuation" concept, a third functional, DTCAM-VAEE, was obtained, which exhibits improved performance as compared to that of the previous DTCAM-VEE and DTCAM-AEE in the prediction of VEEs, making it an attractive alternative to BH&HLYP. As the combination of "adaptive exchange" and "valence attenuation" is operative, it would be exciting to explore its potential with a more tunable framework in the future.

TDDFT and the x-ray absorption spectrum of liquid water: Finding the "best" functional

We investigate the performance of time-dependent density functional theory (TDDFT) for reproducing high-level reference x-ray absorption spectra of liquid water and water clusters. For this, we apply the integrated absolute difference (IAD) metric, previously used for x-ray emission spectra of liquid water [T. Fransson and L. G. M. Pettersson, J. Chem. Theory Comput. 19, 7333-7342 (2023)], in order to investigate which exchange-correlation (xc) functionals yield TDDFT spectra in best agreement to reference, as well as to investigate the suitability of IAD for x-ray absorption spectroscopy spectrum calculations. Considering highly asymmetric and symmetric six-molecule clusters, it is seen that long-range corrected xc-functionals are required to yield good agreement with the reference coupled cluster (CC) and algebraic-diagrammatic construction spectra, with 100% asymptotic Hartree-Fock exchange resulting in the lowest IADs. The xc-functionals with best agreement to reference have been adopted for larger water clusters, yielding results in line with recently published CC theory, but which still show some discrepancies in the relative intensity of the features compared to experiment.

Core Binding Energy Calculations: A Scalable Approach with the Quantum Embedding-Based Equation-of-Motion Coupled-Cluster Method

We investigated the use of density matrix embedding theory to facilitate the computation of core ionization energies (IPs) of large molecules at the equation-of-motion coupled-cluster singles doubles with perturbative triples (EOM-CCSD*) level in combination with the core-valence separation (CVS) approximation. The unembedded IP-CVS-EOM-CCSD* method with a triple-ζ basis set produced ionization energies within 1 eV of experiment with a standard deviation of ∼0.2 eV for the core65 data set. The embedded variant contributed very little systematic error relative to the unembedded method, with a mean unsigned error of 0.07 eV and a standard deviation of ∼0.1 eV, in exchange for accelerating the calculations by many orders of magnitude. By employing embedded EOM-CC methods, we computed the core ionization energies of the uracil hexamer, doped fullerene, and chlorophyll molecule, utilizing up to ∼4000 basis functions within 1 eV from experimental values. Such calculations are not currently possible with the unembedded EOM-CC method.

Simulating transient X-ray photoelectron spectra of Fe(CO)5 and its photodissociation products with multireference algebraic diagrammatic construction theory

Accurate simulations of transient X-ray photoelectron spectra (XPS) provide unique opportunities to bridge the gap between theory and experiment in understanding the photoactivated dynamics in molecules and materials. However, simulating X-ray photoelectron spectra along a photochemical reaction pathway is challenging as it requires accurate description of electronic structure incorporating core-hole screening, orbital relaxation, electron correlation, and spin-orbit coupling in excited states or at nonequilibrium ground-state geometries. In this work, we employ the recently developed multireference algebraic diagrammatic construction theory (MR-ADC) to investigate the core-ionized states and X-ray photoelectron spectra of Fe(CO)5 and its photodissociation products (Fe(CO)4, Fe(CO)3) following excitation with 266 nm light. The simulated transient Fe 3p and CO 3σ XPS spectra incorporating spin-orbit coupling and high-order electron correlation effects are shown to be in a good agreement with the experimental measurements by Leitner et al. [J. Chem. Phys., 2018, 149, 044307]. Our calculations suggest that core-hole screening, spin-orbit coupling, and ligand-field splitting effects are similarly important in reproducing the experimentally observed chemical shifts in transient Fe 3p XPS spectra of iron carbonyl complexes. Our results also demonstrate that the MR-ADC methods can be very useful in interpreting the transient XPS spectra of transition metal compounds.

Simulating X-ray Absorption Spectroscopy in Challenging Environments: Methodological Insights from Water-Solvated Ammonia and Ammonium Systems

Core-electron excitations in solvated systems, influenced by solvent geometry and hydrogen bonding, make X-ray absorption spectroscopy (XAS) a valuable tool for assessing solvent-solute interactions. However, calculating XAS spectra with electronic-structure methods has proven challenging due to a delicate interplay between correlation and solvation effects. This study provides a computational procedure for XAS modeling in solvated systems, with water-solvated ammonia and ammonium systems serving as probes. Exploring methodological challenges, we investigate explicit embedding models, specifically the polarizable embedding family, including polarizable density embedding and extended polarizable density embedding. Our linear-response time-dependent density functional theory (LR-TDDFT) XAS calculations reveal the efficiency of this approach, with extended polarizable density embedding emerging as a robust improvement over polarizable density embedding. Contrary to some recent literature, our study challenges the belief that LR-TDDFT cannot accurately describe XAS spectra of ammonia and ammonium solvated in water.

Understanding X-ray absorption in liquid water using triple excitations in multilevel coupled cluster theory

X-ray absorption (XA) spectroscopy is an essential experimental tool to investigate the local structure of liquid water. Interpretation of the experiment poses a significant challenge and requires a quantitative theoretical description. High-quality theoretical XA spectra require reliable molecular dynamics simulations and accurate electronic structure calculations. Here, we present the first successful application of coupled cluster theory to model the XA spectrum of liquid water. We overcome the computational limitations on system size by employing a multilevel coupled cluster framework for large molecular systems. Excellent agreement with the experimental spectrum is achieved by including triple excitations in the wave function and using molecular structures from state-of-the-art path-integral molecular dynamics. We demonstrate that an accurate description of the electronic structure within the first solvation shell is sufficient to successfully model the XA spectrum of liquid water within the multilevel framework. Furthermore, we present a rigorous charge transfer analysis of the XA spectrum, which is reliable due to the accuracy and robustness of the electronic structure methodology. This analysis aligns with previous studies regarding the character of the prominent features of the XA spectrum of liquid water.

Intramolecular charge transfer enhanced optical limiting in novel hydrazone derivatives with a D1-D-Ai-π-A structure

The present paper investigates one of the hydrazone derivatives (BTH with a D-π-A structure) based on density functional theory. With the computation results of ground state absorption (GSA), excited-state absorption (ESA) and multi-photon absorption (MPA), the optical limiting effect observed in the experiment for the BTH molecule can be well predicted and elucidated by the MPA-ESA mechanism. The analysis of the hole-electron and the electron density differences between two transition states reveal that the main transitions involved in the GSA and ESA of BTH could be recognized as local excitation. Based on these observations, four novel hydrazone derivatives based on the BTH unit with a D1-D-Ai-π-A structure were designed to promote intramolecular charge transfer (ICT). It shows that the ICT effect is well improved by adding the D1 and Ai units. Compared with the original BTH molecule, the main bands of GSA and ESA of D1-D-Ai-π-A molecules are both red-shifted. In addition, GSA, ESA and MPA probabilities are all improved because the obvious charge transfer character results in the transition dipole moment change from localized to delocalized. Accordingly, the optical limiting effect in these hydrazone derivatives is well enhanced. These observations provide guidance for designing novel optical limiting materials based on the hydrazone derivatives.

Intercalated PtCo Electrocatalyst of Vanadium Metal Oxide Increases Charge Density to Facilitate Hydrogen Evolution

Efforts to develop high-performance electrocatalysts for the hydrogen evolution reaction (HER) are of utmost importance in ensuring sustainable hydrogen production. The controllable fabrication of inexpensive, durable, and high-efficient HER catalysts still remains a great challenge. Herein, we introduce a universal strategy aiming to achieve rapid synthesis of highly active hydrogen evolution catalysts using a controllable hydrogen insertion method and solvothermal process. Hydrogen vanadium bronze HxV2O5 was obtained through controlling the ethanol reaction rate in the oxidization process of hydrogen peroxide. Subsequently, the intermetallic PtCoVO supported on two-dimensional graphitic carbon nitride (g-C3N4) nanosheets was prepared by a solvothermal method at the oil/water interface. In terms of HER performance, PtCoVO/g-C3N4 demonstrates superior characteristics compared to PtCo/g-C3N4 and PtCoV/g-C3N4. This superiority can be attributed to the notable influence of oxygen vacancies in HxV2O5 on the electrical properties of the catalyst. By adjusting the relative proportions of metal atoms in the PtCoVO/g-C3N4 nanomaterials, the PtCoVO/g-C3N4 nanocomposites show significant HER overpotential of η10 = 92 mV, a Tafel slope of 65.21 mV dec-1, and outstanding stability (a continuous test lasting 48 h). The nanoarchitecture of a g-C3N4-supported PtCoVO nanoalloy catalyst exhibits exceptional resistance to nanoparticle migration and corrosion, owing to the strong interaction between the metal nanoparticles and the g-C3N4 support. Pt, Co, and V simultaneous doping has been shown by Density Functional Theory (DFT) calculations to enhance the density of states (DOS) at the Fermi level. This augmentation leads to a higher charge density and a reduction in the adsorption energy of intermediates.

Evaluating the Impact of the Tamm-Dancoff Approximation on X-ray Spectrum Calculations

The impact of the Tamm-Dancoff approximation (TDA) for time-dependent density functional theory (TDDFT) calculations of X-ray absorption and X-ray emission spectra (XAS and XES) is investigated, showing small discrepancies in the excitation energies and intensities. Through explicit diagonalization of the TDDFT Hessian, XES was considered by using full TDDFT with a core-hole reference state. This has previously not been possible with most TDDFT implementations as a result of the presence of negative eigenvalues. Furthermore, a core-valence separation (CVS) scheme for XES is presented, in which only elements including the core-hole are considered, resulting in a small Hessian with the dimension of the number of remaining occupied orbitals of the same spin as the core-hole (CH). The resulting spectra are in surprisingly good agreement with the full-space counterpart, illustrating the weak coupling between the valence-valence and valence-CH transitions. Complications resulting from contributions from the discretized continuum are discussed, which can occur for TDDFT calculations of XAS and XES and for TDA calculations of XAS. In conclusion, we recommend that TDA be used when calculating X-ray emission spectra, and either CVS-TDA or CVS-TDDFT can be used for X-ray absorption spectra.

Solvent Effects and pH Dependence of the X-ray Absorption Spectra of Proline from Electrostatic Embedding Quantum Mechanics/Molecular Mechanics and Mixed-Reference Spin-Flip Time-dependent Density-Functional Theory

The accurate description of solvent effects on X-ray absorption spectra (XAS) is fundamental for comparing the simulated spectra with experiments in solution. Currently, few protocols exist that can efficiently reproduce the effects of the solute/solvent interactions on XAS. Here, we develop an efficient and accurate theoretical protocol for simulating the solvent effects on XAS. The protocol combines electrostatic embedding QM/MM based on electrostatic potential fitted operators for describing the solute/solvent interactions and mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT) for simulating accurate XAS spectra. To demonstrate the capabilities of our protocol, we compute the X-ray absorption of neutral proline in the gas phase and ionic proline in water in all relevant K-edges, showing excellent agreement with experiments. We show that states represented by core to π* transitions are almost unaffected by the interaction with water, whereas the core to σ* transitions are more impacted by the fluctuation of proline structure and the electrostatic interaction with the solvent. Finally, we reconstruct the pH-dependent XAS of proline in solution, determining that the N K-edge can be used to distinguish its three protonation states.

Calibrating TDDFT Calculations of the X-ray Emission Spectrum of Liquid Water: The Effects of Hartree-Fock Exchange

The structure and dynamics of liquid water continue to be debated, with insight provided by, among others, X-ray emission spectroscopy (XES), which shows a split in the high-energy 1b1 feature. This split is yet to be reproduced by theory, and it remains unclear if these difficulties are related to inaccuracies in dynamics simulations, spectrum calculations, or both. We investigate the performance of different methods for calculating XES of liquid water, focusing on the ability of time-dependent density functional theory (TDDFT) to reproduce reference spectra obtained by high-level coupled cluster and algebraic-diagrammatic construction scheme calculations. A metric for evaluating the agreement between theoretical spectra termed the integrated absolute difference (IAD), which considers the integral of shifted difference spectra, is introduced and used to investigate the performance of different exchange-correlation functionals. We find that computed spectra of symmetric and asymmetric model water structures are strongly and differently influenced by the amount of Hartree-Fock exchange, with best agreement to reference spectra for ∼40-50%. Lower percentages tend to yield high density of contributing states, resulting in too broad features. The method introduced here is useful also for other spectrum calculations, in particular where the performance for ensembles of structures are evaluated.

Time-Dependent Density Functional Theory for X-ray Absorption Spectra: Comparing the Real-Time Approach to Linear Response

We simulate X-ray absorption spectra at elemental K-edges using time-dependent density functional theory (TDDFT) in both its conventional linear-response implementation and its explicitly time-dependent or "real-time" formulation. Real-time TDDFT simulations enable broadband spectra calculations without the need to invoke frozen occupied orbitals ("core/valence separation"), but we find that these spectra are often contaminated by transitions to the continuum that originate from lower-energy core and semicore orbitals. This problem becomes acute in triple-ζ basis sets, although it is sometimes sidestepped in double-ζ basis sets. Transitions to the continuum acquire surprisingly large dipole oscillator strengths, leading to spectra that are difficult to interpret. Meaningful spectra can be recovered by means of a filtering technique that decomposes the spectrum into contributions from individual occupied orbitals, and the same procedure can be used to separate L- and K-edge spectra arising from different elements within a given molecule. In contrast, conventional linear-response TDDFT requires core/valence separation but is free of these artifacts. It is also significantly more efficient than the real-time approach, even when hundreds of individual states are needed to reproduce near-edge absorption features and even when Padé approximants are used to reduce the real-time simulations to just 2-4 fs of time propagation. Despite the cost, the real-time approach may be useful to examine the validity of the core/valence separation approximation.

Experimentally revealed and theoretically certified synergistic electronic interaction of V-doped CoS for facilitating the oxygen evolution reaction

Since electrocatalytic oxygen evolution (OER) is a four-electron transfer reaction with very slow kinetics, there is great competition to develop cheap, durable and efficient catalysts for oxygen evolution. A molecular model is designed for density functional theory (DFT) simulation calculations to guide the experiment, and this hypothesis is fully supported by the experimental data. Herein, regulating the composition and morphology of the bimetallic VCo and MoCo sulfide and monometallic sulfide nanoparticles (NPs) at the oil-water interface was achieved via a one-step hydrothermal method for efficient and durable OER electrocatalysts. Compared to CoS and MoCoS, the VCoS NPs show superior OER performance. By adjusting the atomic composition ratio of the VCoS nanoparticles, the VCoS NPs (1 : 2 : 1.5 mole ratio) showed a significant OER overpotential η = 255 mV (10 mA cm-2), and their outstanding stability was demonstrated after 48 h of continuous testing. The CoS and MoCoS NPs were also tested for comparison. Density functional theory (DFT) calculations showed that appropriate V doping (VCoS) can heighten the density of states (DOS) of the Fermi level, which generates more charge density and reduces the intermediate adsorption energy. This study not only provides efficient and powerful integrated catalysts, but also details a DFT calculation model guided by experiments to regulate the oxygen insertion technology, thus leading to the design of ideal materials and enabling more far-reaching applications in electrocatalysis.

NEXAFS spectra of model sulfide chains: implications for sulfur networks obtained from inverse vulcanization

Inverse vulcanization is a promising route to stabilize sulfur in lithium-sulfur batteries, but the resulting sulfur strand lengths in the materials are elusive. We address the strand length by characterization via sulfur near edge X-ray absorption fine structure (NEXAFS) spectroscopy. Theoretical predictions of NEXAFS spectra for model molecules containing strands with up to three sulfur atoms are verified by experiment. The near perfect agreement between simulation and experiment on the absolute energy scale allows for the predictions for larger chain lengths also. Inspection and interpretation of NEXAFS spectra from real battery materials on this basis reveals the appearance of single connecting sulfur atoms for very low sulfur content, and of longer strands when the sulfur fraction increases.

Fractional-Electron and Transition-Potential Methods for Core-to-Valence Excitation Energies Using Density Functional Theory

Methods for computing X-ray absorption spectra based on a constrained core hole (possibly containing a fractional electron) are examined. These methods are based on Slater's transition concept and its generalizations, wherein core-to-valence excitation energies are determined using Kohn-Sham orbital energies. Methods examined here avoid promoting electrons beyond the lowest unoccupied molecular orbital, facilitating robust convergence. Variants of these ideas are systematically tested, revealing a best-case accuracy of 0.3-0.4 eV (with respect to experiment) for K-edge transition energies. Absolute errors are much larger for higher-lying near-edge transitions but can be reduced below 1 eV by introducing an empirical shift based on a charge-neutral transition-potential method, in conjunction with functionals such as SCAN, SCAN0, or B3LYP. This procedure affords an entire excitation spectrum from a single fractional-electron calculation, at the cost of ground-state density functional theory and without the need for state-by-state calculations. This shifted transition-potential approach may be especially useful for simulating transient spectroscopies or in complex systems where excited-state Kohn-Sham calculations are challenging.

Sulfur X-ray Absorption and Emission Spectroscopy of Organic Sulfones

The sulfones are a widespread group of organo-sulfur compounds, which contain the sulfonyl SO₂ group attached to two carbons and have a formal sulfur oxidation state of +2. We have examined the sulfur K near-edge X-ray absorption spectroscopy (XAS) of a range of different sulfones and find substantial spectroscopic variability depending upon the nature of the coordination to the sulfonyl group. We have also examined the sulfur Kβ X-ray emission spectroscopy (XES) of selected representative sulfones. Density functional theory simulations show satisfactory reproduction of both absorption and emission spectra while enabling assignment of the various transitions comprising the spectra. The correspondence between observed and simulated spectra shows promise for ab initio prediction of sulfur X-ray absorption and emission spectra of sulfones of any substituent. The absorption spectra and, to a lesser extent, the emission spectra are sensitive to the nature of the organic groups bound to the sulfonyl (SO₂) moiety, clearly showing the potential of X-ray spectroscopy as an in situ probe of sulfone chemistry.

Electronic Structure and Core Spectroscopy of Scandium Fluoride Polymorphs

Microscopic knowledge of the structural, energetic, and electronic properties of scandium fluoride is still incomplete despite the relevance of this material as an intermediate for the manufacturing of Al-Sc alloys. In a work based on first-principles calculations and X-ray spectroscopy, we assess the stability and electronic structure of six computationally predicted ScF₃ polymorphs, two of which correspond to experimentally resolved single-crystal phases. In the theoretical analysis based on density functional theory (DFT), we identify similarities among the polymorphs based on their formation energies, charge-density distribution, and electronic properties (band gaps and density of states). We find striking analogies between the results obtained for the low- and high-temperature phases of the material, indirectly confirming that the transition occurring between them mainly consists of a rigid rotation of the lattice. With this knowledge, we examine the X-ray absorption spectra from the Sc and F K-edge contrasting first-principles results obtained from the solution of the Bethe-Salpeter equation on top of all-electron DFT with high-energy-resolution fluorescence detection measurements. Analysis of the computational results sheds light on the electronic origin of the absorption maxima and provides information on the prominent excitonic effects that characterize all spectra. A comparison with measurements confirms that the sample is mainly composed of the high- and low-temperature polymorphs of ScF₃. However, some fine details in the experimental results suggest that the probed powder sample may contain defects and/or residual traces of metastable polymorphs.

Slater transition methods for core-level electron binding energies

Methods for computing core-level ionization energies using self-consistent field (SCF) calculations are evaluated and benchmarked. These include a "full core hole" (or "ΔSCF") approach that fully accounts for orbital relaxation upon ionization, but also methods based on Slater's transition concept in which the binding energy is estimated from an orbital energy level that is obtained from a fractional-occupancy SCF calculation. A generalization that uses two different fractional-occupancy SCF calculations is also considered. The best of the Slater-type methods afford mean errors of 0.3-0.4 eV with respect to experiment for a dataset of K-shell ionization energies, a level of accuracy that is competitive with more expensive many-body techniques. An empirical shifting procedure with one adjustable parameter reduces the average error below 0.2 eV. This shifted Slater transition method is a simple and practical way to compute core-level binding energies using only initial-state Kohn-Sham eigenvalues. It requires no more computational effort than ΔSCF and may be especially useful for simulating transient x-ray experiments where core-level spectroscopy is used to probe an excited electronic state, for which the ΔSCF approach requires a tedious state-by-state calculation of the spectrum. As an example, we use Slater-type methods to model x-ray emission spectroscopy.

Projected Hybrid Density Functionals: Method and Application to Core Electron Ionization

This work introduces a new class of hybrid density functional theory (DFT) approximations, which incorporate different fractions of nonlocal exact exchange in predefined states such as core atomic orbitals (AOs). These projected hybrid density functionals are related to range-separated hybrid functionals, which incorporate different fractions of nonlocal exchange at different electron-electron separations. This work derives projected hybrids using the Adiabatic Projection formalism. One projects the electron-electron interaction operator onto the chosen predefined states, introduces the projected operator into the noninteracting Kohn-Sham reference system, and employs a formally exact density functional to model the remaining electron-electron interactions. Projected hybrids, like range-separated hybrids, approximate the partially interacting reference system's ground-state wave function as a single Slater determinant. Projected hybrids are readily implemented into existing density functional codes, requiring only a projection of the one-particle density matrices and exchange operators entering existing routines. This work also presents an application to core electron ionization. Projecting onto core atomic orbitals allows us to introduce additional nonlocal exchange into atomic core regions. This reduces the impact of self-interaction error on computed core electron properties. Benchmark studies are reported for PBE0c70, a core-projected variant of the Perdew-Burke-Ernzerhof global hybrid PBE0, in which the fraction of nonlocal exchange is increased from 25% to 70% in atomic core regions. PBE0c70-predicted core orbital energies accurately recover nonrelativistic core-electron binding energies of second-period elements Li-Ne and third-period elements Na-Ar, without degrading the good performance of PBE0 for atomization energies and valence ionization potentials.

Exact Two-Component TDDFT with Simple Two-Electron Picture-Change Corrections: X-ray Absorption Spectra Near L- and M-Edges of Four-Component Quality at Two-Component Cost

X-ray absorption spectroscopy (XAS) has gained popularity in recent years as it probes matter with high spatial and elemental sensitivities. However, the theoretical modeling of XAS is a challenging task since XAS spectra feature a fine structure due to scalar (SC) and spin-orbit (SO) relativistic effects, in particular near L and M absorption edges. While full four-component (4c) calculations of XAS are nowadays feasible, there is still interest in developing approximate relativistic methods that enable XAS calculations at the two-component (2c) level while maintaining the accuracy of the parent 4c approach. In this article we present theoretical and numerical insights into two simple yet accurate 2c approaches based on an (extended) atomic mean-field exact two-component Hamiltonian framework, (e)amfX2C, for the calculation of XAS using linear eigenvalue and damped response time-dependent density functional theory (TDDFT). In contrast to the commonly used one-electron X2C (1eX2C) Hamiltonian, both amfX2C and eamfX2C account for the SC and SO two-electron and exchange-correlation picture-change (PC) effects that arise from the X2C transformation. As we demonstrate on L- and M-edge XAS spectra of transition metal and actinide compounds, the absence of PC corrections in the 1eX2C approximation results in a substantial overestimation of SO splittings, whereas (e)amfX2C Hamiltonians reproduce all essential spectral features such as shape, position, and SO splitting of the 4c references in excellent agreement, while offering significant computational savings. Therefore, the (e)amfX2C PC correction models presented here constitute reliable relativistic 2c quantum-chemical approaches for modeling XAS.

Electron and ion spectroscopy of azobenzene in the valence and core shells

Azobenzene is a prototype and a building block of a class of molecules of extreme technological interest as molecular photo-switches. We present a joint experimental and theoretical study of its response to irradiation with light across the UV to x-ray spectrum. The study of valence and inner shell photo-ionization and excitation processes combined with measurement of valence photoelectron-photoion coincidence and mass spectra across the core thresholds provides a detailed insight into the site- and state-selected photo-induced processes. Photo-ionization and excitation measurements are interpreted via the multi-configurational restricted active space self-consistent field method corrected by second order perturbation theory. Using static modeling, we demonstrate that the carbon and nitrogen K edges of azobenzene are suitable candidates for exploring its photoinduced dynamics thanks to the transient signals appearing in background-free regions of the NEXAFS and XPS.

Double-Hybrid Density Functional Theory for Core Excitations: Theory and Benchmark Calculations

The double-hybrid (DH) time-dependent density functional theory is extended to core excitations. Two different DH formalisms are presented utilizing the core-valence separation (CVS) approximation. First, a CVS-DH variant is introduced relying on the genuine perturbative second-order correction, while an iterative analogue is also presented using our second-order algebraic-diagrammatic construction [ADC(2)]-based DH ansatz. The performance of the new approaches is tested for the most popular DH functionals using the recently proposed XABOOM [J. Chem. Theory Comput.2021, 17, 1618] benchmark set. In order to make a careful comparison, the accuracy and precision of the methods are also inspected. Our results show that the genuine approaches are highly competitive with the more advanced CVS-ADC(2)-based methods if only excitation energies are required. In contrast, as expected, significant differences are observed in oscillator strengths; however, the precision is acceptable for the genuine functionals as well. Concerning the performance of the CVS-DH approaches, the PBE0-2/CVS-ADC(2) functional is superior, while its spin-opposite-scaled variant is also recommended as a cost-effective alternative. For these approaches, significant improvements are realized in the error measures compared with the popular CVS-ADC(2) method.

Modeling Nonresonant X-ray Emission of Second- and Third-Period Elements without Core-Hole Reference States and Empirical Parameters

Nonresonant X-ray emission (XE) energies and oscillator strengths are obtained using the effective potential of the generalized Kohn-Sham semi-canonical projected random phase approximation (GKS-spRPA) method. XE energies are estimated as a difference between the valence and core ionization eigenvalues, while the oscillator strengths are obtained within a frozen orbital approximation. This straightforward approach provides accurate XE energies without any need for core-hole reference states, empirical shifting parameters, or tuning of density functionals. To account for relativistic corrections to the core orbitals, we have formulated a scalar relativistic (sr) GKS-spRPA approach based on the spin-free X2C one-electron Hamiltonian. The sr-GKS-spRPA method provides highly reliable XE energies using uncontracted basis-sets on atoms where the core-hole is created prior to emission. For the largest basis-sets used in our study, using completely uncontracted polarized core-valence Dunning basis-sets, the mean absolute errors (MAEs) are within 0.7 eV compared to experimental reference values for a test-set consisting of 27 valence-to-core XE energies of molecules with second- and third-period elements. Considering a balance of accuracy and computational effort, we recommend the use of s-uncontracted def2-TZVP for second-period and all-uncontracted def2-TZVP for third-period elements. For this recommended basis-set, the MAE is 0.2 eV. The analytically continued sr-GKS-spRPA approach, with an O(N4) computational cost, enables efficient computation of XE spectra of molecules such as S₈ and C₆₀ with several core-hole states.

Electron-Affinity Time-Dependent Density Functional Theory: Formalism and Applications to Core-Excited States

The lack of particle-hole attraction and orbital relaxation within time-dependent density functional theory (TDDFT) lead to extreme errors in the prediction of K-edge X-ray absorption spectra (XAS). We derive a linear-response formalism that uses optimized orbitals of the n - 1-electron system as the reference, building orbital relaxation and a proper hole into the initial density. Our approach is an exact generalization of the static-exchange approximation that ameliorates the particle-hole interaction error associated with the adiabatic approximation and reduces errors in TDDFT XAS by orders of magnitude. With a statistical performance of just 0.5 eV root-mean-square error and the same computational scaling as TDDFT under the core-valence separation approximation, we anticipate that this approach will be of great utility in XAS calculations of large systems.

Modeling X-ray Photoelectron Spectroscopy of Macromolecules Using GW

We propose a simple additive approach to simulate X-ray photoelectron spectra (XPS) of macromolecules based on the GW method. Single-shot GW (G₀W₀) is a promising technique to compute accurate core-electron binding energies (BEs). However, its application to large molecules is still unfeasible. To circumvent the computational cost of G₀W₀, we break the macromolecule into tractable building blocks, such as isolated monomers, and sum up the theoretical spectra of each component, weighted by their molar ratio. In this work, we provide a first proof of concept by applying the method to four test polymers and one copolymer and show that it leads to an excellent agreement with experiments. The method could be used to retrieve the composition of unknown materials and study chemical reactions, by comparing the simulated spectra with experimental ones.

Pair Natural Orbital Equation-of-Motion Coupled-Cluster Method for Core Binding Energies: Theory, Implementation, and Benchmark

We present the theory and implementation of a lower scaling core-valence separated equation-of-motion coupled-cluster approach based on domain-based local pair natural orbitals for core binding energies. The accuracy of the new method has been compared with that of the standard equation-of-motion coupled-cluster method and experimentally measured results. The use of pair natural orbitals significantly reduces the computation cost and can be applied to large molecules.

Computational approaches for XANES, VtC-XES, and RIXS using linear-response time-dependent density functional theory based methods

The emergence of state-of-the-art X-ray light sources has paved the way for novel spectroscopies that take advantage of their atomic specificity to shed light on fundamental physical, chemical, and biological processes both in the static and time domains. The success of these experiments hinges on the ability to interpret and predict core-level spectra, which has opened avenues for theory to play a key role. Over the last two decades, linear-response time-dependent density functional theory (LR-TDDFT), despite various theoretical challenges, has become a computationally attractive and versatile framework to study excited-state spectra including X-ray spectroscopies. In this context, we focus our discussion on LR-TDDFT approaches for the computation of X-ray Near-Edge Structure (XANES), Valence-to-Core X-ray Emission (VtC-XES), and Resonant Inelastic X-ray Scattering (RIXS) spectroscopies in molecular systems with an emphasis on Gaussian basis set implementations. We illustrate these approaches with applications and provide a brief outlook of possible new directions.

Core-hole delocalization for modeling X-ray spectroscopies: A cautionary tale

The influence of core-hole delocalization for X-ray photoelectron, X-ray absorption, and X-ray emission spectrum calculations is investigated in detail, using approaches including response theory, transition-potential methods, and ground state schemes. The question of a localized/delocalized vacancy is relevant for systems with symmetrically equivalent atoms, as well as near-degeneracies which can distribute the core-orbitals over several atoms. We show that issues relating to core-hole delocalization are present for calculations considering explicit core-hole states, e.g. when using a core-excited or core-ionized reference state, or for fractional occupation numbers. Including electron correlation eventually alleviates the issues, but even using CCSD(T) there is a noticable discrepancy between core-ionization energies obtained with a localized and delocalized core-hole (0.5 eV for the carbon K-edge). Within density functional theory, the discrepancy correlates to the exchange interaction involving the core orbitals of the same spin symmetry as the delocalized core-hole. The use of a localized core-hole allows for a reasonably good inclusion of relaxation at lower level of theory, whereas the proper symmetry solution involving a delocalized core-hole requires higher levels of theory to account for the correlation effects involved in orbital relaxation. For linear response methods, we further show that if X-ray absorption spectra are modelled by considering symmetry-unique sets of atoms, care has to be taken such that there are no delocalizations of the core orbitals, which would otherwise introduce shifts in absolute energies and relative features.

Relativistic Orbital-Optimized Density Functional Theory for Accurate Core-Level Spectroscopy

Core-level spectra of 1s electrons of elements heavier than Ne show significant relativistic effects. We combine advances in orbital-optimized density functional theory (OO-DFT) with the spin-free exact two-component (X2C) model for scalar relativistic effects to study K-edge spectra of third period elements. OO-DFT/X2C is found to be quite accurate at predicting energies, yielding a ∼0.5 eV root-mean-square error versus experiment with the modern SCAN (and related) functionals. This marks a significant improvement over the >50 eV deviations that are typical for the popular time-dependent DFT (TDDFT) approach. Consequently, experimental spectra are quite well reproduced by OO-DFT/X2C, sans empirical shifts for alignment. OO-DFT/X2C combines high accuracy with ground state DFT cost and is thus a promising route for computing core-level spectra of third period elements. We also explored K and L edges of 3d transition metals to identify limitations of the OO-DFT/X2C approach in modeling the spectra of heavier atoms.

Insights into the thermal degradation mechanisms of polyethylene terephthalate dimer using DFT method

The thermal degradation mechanisms of polyethylene terephthalate (PET) dimer were studied by the B3P86 density functional theory (DFT) approach at 6-31++G (d, p) base set in this paper. Seven possible reaction paths were designed and analyzed, and the thermodynamic parameters for all reactions were computed. The calculated results indicate that the bond dissociation energy values (BDEs) of C-C bonds on the main-chain are the smallest, followed by those of C-O bonds. The kinetics analysis indicates that the concerted reactions are obviously liable to occur rather than radical reactions in the initial thermal decomposition process. In the processes of initial reactions, all concerted reactions occurred by six-membered cyclic transition states (TSs) are more prone to carry out than those happened by four-membered cyclic transition states. The research results show that the primary products of PET dimer pyrolysis are terephthalic acid, vinyl terephthalate, CH₃CHO and divinyl terephthalate. CH₃CHO is mainly formed by a concerted reaction in the initial degradation process, and CO₂ is mainly produced by the decarboxylation via a concerted reaction and CO is mainly produced by the decarbonylation of a radical in secondary degradation.

Frontier orbital stability of nitroxyl organic radicals probed by means of inner shell resonantly enhanced valence band photoelectron spectroscopy

We have investigated the frontier orbitals of persistent organic radicals known as nitroxyls by resonant photoelectron spectroscopy (ResPES) under inner shell excitation. By means of this site-specific technique, we were able to disentangle the different atomic contributions to the outer valence molecular orbitals and examine several core-hole relaxation pathways involving the singly occupied molecular orbital (SOMO) localized on the nitroxyl group. To interpret the ResPES intensity trends, especially the strong enhancement of the SOMO ionized state at the N K-edge, we computed the Dyson spin orbitals (DSOs) pertaining to the transitions between the core-excited initial states and several of the singly ionized valence final states. We found that the computed vertical valence ionization potentials and norms of the DSOs are reasonably reliable when based on the long-range corrected CAM-B3LYP density functional. Thanks to their unpaired electrons, nitroxyls have recently found application in technological fields implying a spin control, such as spintronics and quantum computing. The present findings on the electronic structure of nitroxyl persistent radicals furnish important hints for their implementation in technological devices and, more in general, for the synthesis of new and stable organic radicals with tailored properties.

Accurate X-ray Absorption Spectra near L- and M-Edges from Relativistic Four-Component Damped Response Time-Dependent Density Functional Theory

The simulation of X-ray absorption spectra requires both scalar and spin-orbit (SO) relativistic effects to be taken into account, particularly near L- and M-edges where the SO splitting of core p and d orbitals dominates. Four-component Dirac-Coulomb Hamiltonian-based linear damped response time-dependent density functional theory (4c-DR-TDDFT) calculates spectra directly for a selected frequency region while including the relativistic effects variationally, making the method well suited for X-ray applications. In this work, we show that accurate X-ray absorption spectra near L2,3- and M4,5-edges of closed-shell transition metal and actinide compounds with different central atoms, ligands, and oxidation states can be obtained by means of 4c-DR-TDDFT. While the main absorption lines do not change noticeably with the basis set and geometry, the exchange-correlation functional has a strong influence with hybrid functionals performing the best. The energy shift compared to the experiment is shown to depend linearly on the amount of Hartee-Fock exchange with the optimal value being 60% for spectral regions above 1000 eV, providing relative errors below 0.2% and 2% for edge energies and SO splittings, respectively. Finally, the methodology calibrated in this work is used to reproduce the experimental L2,3-edge X-ray absorption spectra of [RuCl2(DMSO)2(Im)2] and [WCl4(PMePh2)2], and resolve the broad bands into separated lines, allowing an interpretation based on ligand field theory and double point groups. These results support 4c-DR-TDDFT as a reliable method for calculating and analyzing X-ray absorption spectra of chemically interesting systems, advance the accuracy of state-of-the art relativistic DFT approaches, and provide a reference for benchmarking more approximate techniques.

Simulating X-ray photoelectron spectra with strong electron correlation using multireference algebraic diagrammatic construction theory

A new theoretical approach for the simulations of X-ray photoelectron spectra of strongly correlated molecular systems that combines multireference algebraic diagrammatic construction theory (MR-ADC) with a core–valence separation (CVS) technique.

On the choice of reference orbitals for linear-response calculations of solution-phase K-edge X-ray absorption spectra

X-ray absorption spectra of liquids calculated with linear-response theories like TDDFT and CIS are dramatically improved with core-ion reference orbitals.

Core-shell excitation of isoxazole at the C, N, and O K-edges – an experimental NEXAFS and theoretical TD-DFT study

The near-edge X-ray absorption fine structure (NEXAFS) spectra of the gas-phase isoxazole molecule have been measured by collecting total ion yields at the C, N, and O K-edges. The spectral...

Extended quasiparticle approach to non-resonant and resonant X-ray emission spectroscopy

The initial state of X-ray emission spectroscopy (XES) and resonant inelastic X-ray emission spectroscopy (RIXS) is a highly excited eigenstate with a deep core hole after a X-ray photoelectron spectroscopy...