XABOOM: An X-ray Absorption Benchmark of Organic Molecules Based on Carbon, Nitrogen, and Oxygen 1s → π* Transitions

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.

Benchmark Study of Core-Ionization Energies with the Generalized Active Space-Driven Similarity Renormalization Group

X-ray photoelectron spectroscopy (XPS) is a powerful experimental technique for probing the electronic structure of molecules and materials; however, interpreting XPS data requires accurate computational methods to model core-ionized states. This work proposes and benchmarks a new approach based on the generalized active space-driven similarity renormalization group (GAS-DSRG) for calculating core-ionization energies and treating correlation effects at the perturbative and nonperturbative levels. We tested the GAS-DSRG across three data sets. First, the vertical core-ionization energies of small molecules containing first-row elements are evaluated. GAS-DSRG achieves mean absolute errors below 0.3 eV, which is comparable to high-level coupled cluster methods. Next, the accuracy of GAS-DSRG is evaluated for larger organic molecules using the CORE65 data set, with the DSRG-MRPT3 level yielding a mean absolute error of only 0.34 eV for 65 core-ionization transitions. Insights are provided into the treatment of static and dynamic correlation, the importance of high-order perturbation theory, and notable differences from density functional theory in the predicted energy ordering of core-ionized states for specific molecules. Finally, vibrationally resolved XPS spectra of diatomic molecules (CO, N2, and O2) are simulated, showing excellent agreement with experimental data.

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.

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.

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.

Efficient and Parallel Implementation of Real and Complex Response Functions Employing the Second-Order Algebraic-Diagrammatic Construction Scheme for the Polarization Propagator

We present the implementation of an efficient matrix-folded formalism for the evaluation of complex response functions and the calculation of transition properties at the level of the second-order algebraic-diagrammatic construction (ADC(2)) scheme. The underlying algorithms, in combination with the adopted hybrid MPI/OpenMP parallelization strategy, enabled calculations of the UV/vis spectra of a guanine oligomer series ranging up to 1032 contracted basis functions, thereby utilizing vast computational resources from up to 32,768 CPU cores. Further analysis of the convergence behavior of the involved iterative subspace algorithms revealed the superiority of a frequency-separated treatment of response equations even for a large spectral window, including 101 frequencies. We demonstrate the applicability to general quantum mechanical operators by the first reported electronic circular dichroism spectrum calculated with a complex polarization propagator approach at the ADC(2) level of theory.

Toward a correct treatment of core properties with local hybrid functionals

In local hybrid functionals (LHs), a local mixing function (LMF) determines the position-dependent exact-exchange admixture. We report new LHs that focus on an improvement of the LMF in the core region while retaining or partly improving upon the high accuracy in the valence region exhibited by the LH20t functional. The suggested new pt-LMFs are based on a Padé form and modify the previously used ratio between von Weizsäcker and Kohn-Sham local kinetic energies by different powers of the density to enable flexibly improved approximations to the correct high-density and iso-orbital limits relevant for the innermost core region. Using TDDFT calculations for a set of K-shell core excitations of second- and third-period systems including accurate state-of-the-art relativistic orbital corrections, the core part of the LMF is optimized, while the valence part is optimized as previously reported for test sets of atomization energies and reaction barriers (Haasler et al., J Chem Theory Comput 2020, 16, 5645). The LHs are completed by a calibration function that minimizes spurious nondynamical correlation effects caused by the gauge ambiguities of exchange-energy densities, as well as by B95c meta-GGA correlation. The resulting LH23pt functional relates to the previous LH20t functional but specifically improves upon the core region.

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.

Unraveling the variational breakdown of core valence separation calculations: Diagnostic and cure to the over relaxation error of double core hole states

The core valence separation (CVS) approximation is the most employed strategy to prevent the variational collapse of standard wave function optimization when attempting to compute electronic states bearing one or more electronic vacancies in core orbitals. Here, we explore the spurious consequences of this approximation on the properties of the computed core hole states. We especially focus on the less studied case of double core hole (DCH) states, whose spectroscopic interest has recently been rapidly growing. We show that the CVS error leads to a systematic underestimation of DCH energies, a property in stark contrast with the case of single core hole states. We highlight that the CVS error can then be interpreted as an over relaxation effect and design a new correction strategy adapted to these specificities.

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.

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.

Robust relativistic many-body Green's function based approaches for assessing core ionized and excited states

A two-component contour deformation (CD) based GW method that employs frequency sampling to drastically reduce the computational effort when assessing quasiparticle states far away from the Fermi level is outlined. Compared to the canonical CD-GW method, computational scaling is reduced by an order of magnitude without sacrificing accuracy. This allows for an efficient calculation of core ionization energies. The improved computational efficiency is used to provide benchmarks for core ionized states, comparing the performance of 15 density functional approximations as Kohn-Sham starting points for GW calculations on a set of 65 core ionization energies of 32 small molecules. Contrary to valence states, GW calculations on core states prefer functionals with only a moderate amount of Hartree-Fock exchange. Moreover, modern ab initio local hybrid functionals are also shown to provide excellent generalized Kohn-Sham references for core GW calculations. Furthermore, the core-valence separated Bethe-Salpeter equation (CVS-BSE) is outlined. CVS-BSE is a convenient tool to probe core excited states. The latter is tested on a set of 40 core excitations of eight small inorganic molecules. Results from the CVS-BSE method for excitation energies and the corresponding absorption cross sections are found to be in excellent agreement with those of reference damped response BSE calculations.

Ab Initio Investigation of the Auger Spectra of Methane, Ethane, Ethylene, and Acetylene

We present an ab initio computational study of the Auger spectra of methane, ethane, ethylene, and acetylene. Auger spectroscopy is an established technique to probe the electronic structure of molecules and exploits the Auger-Meitner effect that core-ionized states undergo. We compute partial decay widths using coupled-cluster theory with single and double substitutions (CCSD) and equation-of-motion CCSD theory combined with complex-scaled basis functions and Feshbach-Fano projection. We generate Auger spectra from these partial widths and draw conclusions about the strength of particular decay channels and trends among the four molecules. A connection to experimental results about fragmentation pathways of the electronic states produced by Auger decay is also made.

Resonant Inner-Shell Photofragmentation of Adamantane (C10H16)

Adamantane, the smallest diamondoid molecule with a symmetrical cage, contains two distinct carbon sites, CH and CH2. The ionization/excitation of the molecule leads to the cage opening and strong structural reorganization. While theoretical predictions suggest that the carbon site CH primarily causes the cage opening, the role of the other CH2 site remains unclear. In this study, we used advanced experimental Auger electron-ion coincidence techniques and theoretical calculations to investigate the fragmentation dynamics of adamantane after resonant inner-shell photoexcitation. Our results demonstrate that some fragmentation channels exhibit site-sensitivity of the initial core-hole location, indicating that different carbon site excitations could lead to unique cage opening mechanisms.

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.

A theoretical study of the time-resolved x-ray absorption spectrum of the photoionized BT-1T cation

The time-resolved x-ray absorption spectrum of the BT-1T cation (BT-1T+) is theoretically simulated in order to investigate the charge transfer reaction of the system. We employ both trajectory surface hopping and quantum dynamics to simulate the structural evolution over time and the changes in the state populations. To compute the static x-ray absorption spectra (XAS) of the ground and excited states, we apply both the time-dependent density functional theory and the coupled cluster singles and doubles method. The results obtained are in good agreement between the methods. It is, furthermore, found that the small structural changes that occur during the reaction have little effect on the static XAS. Hence, the tr-XAS can be computed based on the state populations determined from a nuclear dynamics simulation and one set of static XAS calculations, utilizing the ground state optimized geometry. This approach can save considerable computational resources, as the static spectra need not to be calculated for all geometries. As BT-1T is a relatively rigid molecule, the outlined approach should only be considered when investigating non-radiative decay processes in the vicinity of the Franck-Condon point.

A study of core-excited states of organic molecules computed with the generalized active space driven similarity renormalization group

This work examines the accuracy and precision of x-ray absorption spectra computed with a multireference approach that combines generalized active space (GAS) references with the driven similarity renormalization group (DSRG). We employ the x-ray absorption benchmark of organic molecule (XABOOM) set, consisting of 116 transitions from mostly organic molecules [Fransson et al., J. Chem. Theory Comput. 17, 1618 (2021)]. Several approximations to a full-valence active space are examined and benchmarked. Absolute excitation energies and intensities computed with the GAS-DSRG truncated to second-order in perturbation theory are found to systematically underestimate experimental and reference theoretical values. Third-order perturbative corrections significantly improve the accuracy of GAS-DSRG absolute excitation energies, bringing the mean absolute deviation from experimental values down to 0.32 eV. The ozone molecule and glyoxylic acid are particularly challenging for second-order perturbation theory and are examined in detail to assess the importance of active space truncation and intruder states.

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.

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.

Self-Adaptive Real-Time Time-Dependent Density Functional Theory for X-ray Absorptions

Real-time time-dependent density functional theory (RT-TDDFT) can in principle access the whole absorption spectrum of a many-electron system exposed to a narrow pulse. However, this requires an accurate and efficient propagator for the numerical integration of the time-dependent Kohn-Sham equation. While a low-order time propagator is already sufficient for the low-lying valence absorption spectra, it is no longer the case for the X-ray absorption spectra (XAS) of systems composed even only of light elements, for which the use of a high-order propagator is indispensable. It is then crucial to choose a largest possible time step and a shortest possible simulation time, so as to minimize the computational cost. To this end, we propose here a robust AutoPST approach to determine automatically (Auto) the propagator (P), step (S), and time (T) for relativistic RT-TDDFT simulations of XAS.

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.

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.

Using core-hole reference states for calculating X-ray photoelectron and emission spectra

For the calculation of core-ionization energies (IEs), X-ray photoelectron spectra (XPS), and X-ray emission spectra (XES), a commonly applied approach is to use non-Aufbau reference states with a core-hole as either final (IE and XPS) or initial (XES) state. However, such reference states can introduce numerical instabilities in post-HF methods, relating to the denominator of the energy corrections involved. This may become arbitrarily close to zero if a negative virtual orbital is present, e.g. a core-hole, leading to near-singularities. The resulting instabilities lead to severe convergence issues of the calculation schemes and, in addition, can strongly affect both energies and intensities, with oscillator strengths seen to reach values up to 4 × 10⁷. For the K-edge we propose freezing the highest-energy virtual orbitals which contribute to any denominator below a threshold of 0.1 Hartree. Stable and reliable spectra are then produced, with minimal influence due to freezing energetically high-lying virtual orbitals (typically removing <5% of the total number of MOs). The developed protocol is here tested for Møller-Plesset perturbation theory and for the algebraic diagrammatic construction scheme for the polarization propagator, and it is also relevant for coupled cluster theory and other related methods.

Characterization of Electrochemical Processes in Metal-Organic Batteries by X-ray Raman Spectroscopy

X-ray Raman spectroscopy (XRS) is an emerging spectroscopic technique that utilizes inelastic scattering of hard X-rays to study X-ray absorption edges of low Z elements in bulk material. It was used to identify and quantify the amount of carbonyl bonds in a cathode sample, in order to track the redox reaction inside metal-organic batteries during the charge/discharge cycle. XRS was used to record the oxygen K-edge absorption spectra of organic polymer cathodes from different multivalent metal-organic batteries. The amount of carbonyl bond in each sample was determined by modeling the oxygen K-edge XRS spectra with the linear combination of two reference compounds that mimicked the fully charged and the fully discharged phases of the battery. To interpret experimental XRS spectra, theoretical calculations of oxygen K-edge absorption spectra based on density functional theory were performed. Overall, a good agreement between the amount of carbonyl bond present during different stages of battery cycle, calculated from linear combination of standards, and the amount obtained from electrochemical characterization based on measured capacity was achieved. The electrochemical mechanism in all studied batteries was confirmed to be a reduction of double carbonyl bond and the intermediate anion was identified with the help of theoretical calculations. X-ray Raman spectroscopy of the oxygen K-edge was shown to be a viable characterization technique for accurate tracking of the redox reaction inside metal-organic batteries.

Higher-order transition state approximation

We generalize Slater’s transition state concept by deriving systematic higher-order transition state approximations. Numerical validation is performed by the calculation of transition energies for various excitations, including core, valence, and charge-transfer excitations, at Hartree–Fock and Kohn–Sham density functional theory levels. All higher-order transition state approximations introduced in this study accurately reproduce the results from delta self-consistent-field calculations. In particular, we demonstrate that the third-order generalized transition state (GTS3) approximation is a promising alternative to the original, owing to a good balance between the accuracy and computational cost. We also demonstrate that accurate and reliable results can be obtained with a low computational cost by combining the GTS3 approximation with the transition potential scheme.

Molecular Auger Decay Rates from Complex-Variable Coupled-Cluster Theory

The emission of an Auger electron is the predominant relaxation mechanism of core-vacant states in molecules composed of light nuclei. In this non-radiative decay process, one valence electron fills the core vacancy while a second valence electron is emitted into the ionization continuum. Because of this coupling to the continuum, core-vacant states represent electronic resonances that can be tackled with standard quantum-chemical methods only if they are approximated as bound states, meaning that Auger decay is neglected. Here, we present an approach to compute Auger decay rates of core-vacant states from coupled-cluster and equation-of-motion coupled-cluster wave functions combined with complex scaling of the Hamiltonian or, alternatively, complex-scaled basis functions. Through energy decomposition analysis, we illustrate how complex-scaled methods are capable of describing the coupling to the ionization continuum without the need to model the wave function of the Auger electron explicitly. In addition, we introduce in this work several approaches for the determination of partial decay widths and Auger branching ratios from complex-scaled coupled-cluster wave functions. We demonstrate the capabilities of our new approach by computations on core-ionized states of neon, water, dinitrogen, and benzene. Coupled-cluster and equation-of-motion coupled-cluster theory in the singles and doubles approximation both deliver excellent results for total decay widths, whereas we find partial widths more straightforward to evaluate with the former method. We also observe that the requirements towards the basis set are less arduous for Auger decay than for other types of resonances so that extensions to larger molecules are readily possible.

Soft X-ray Spectroscopy Simulations with Multiconfigurational Wave Function Theory: Spectrum Completeness, Sub-eV Accuracy, and Quantitative Reproduction of Line Shapes

Multireference methods are known for their ability to accurately treat states of very different nature in many molecular systems, facilitating high-quality simulations of a large variety of spectroscopic techniques. Here, we couple the multiconfigurational restricted active space self-consistent field RASSCF/RASPT2 method (of the CASSCF/CASPT2 methods family) to the displaced harmonic oscillator (DHO) model, to simulate soft X-ray spectroscopy. We applied such an RASSCF/RASPT2+DHO approach at the K-edges of various second-row elements for a set of small organic molecules that have been recently investigated at other levels of theory. X-ray absorption near-edge structure (XANES) and X-ray photoelectron spectroscopy (XPS) are simulated with a sub-eV accuracy and a correct description of the spectral line shapes. The method is extremely sensitive to the observed spectral shifts on a series of differently fluorinated ethylene systems, provides spectral fingerprints to distinguish between stable conformers of the glycine molecule, and accurately captures the vibrationally resolved carbon K-edge spectrum of formaldehyde. Differences with other theoretical methods are demonstrated, which show the advantages of employing a multireference/multiconfigurational approach. A protocol to systematically increase the number of core-excited states considered while maintaining a contained computational cost is presented. Insight is eventually provided for the effects caused by removing core-electrons from a given atom in terms of bond rearrangement and influence on the resulting spectral shapes within a unitary orbital-based framework for both XPS and XANES spectra.

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.

Simulations of x-ray absorption spectra for CO desorbing from Ru(0001) with transition-potential and time-dependent density functional theory approaches

The desorption of a carbon monoxide molecule from a Ru(0001) surface was studied by means of X-ray Absorption Spectra (XAS) computed with Transition Potential (TP-DFT) and Time Dependent (TD-DFT) DFT methods. By unraveling the evolution of the CO electronic structure upon desorption, we observed that at 2.3 Å from the surface, the CO molecule has already predominantly gas-phase character. While C 1s XAS is quite insensitive to changes in the C-O bond length, the O 1s excitation is very sensitive with the π* coming down in energy upon CO bond stretching, which competes with the increase in orbital energy due to the repulsive interaction with the metallic surface. We show in a systematic way that the TP-DFT method can describe the XAS rather well at the endpoints (chemisorbed and gas phase) but is affected by artificial charge transfer and/or incorrect spin treatment in the transition region in cases like CO, where there are low-lying π* orbitals and large exchange interactions between the core 1s and valence-acceptor π* orbitals. As an alternative, we demonstrate by comparing with experimental data that a linear response approach using TD-DFT employing common exchange-correlation functionals and finite-size clusters can yield a good description of the spectral evolution of the 1s → π* transition with correct spin and gas-to-chemisorbed chemical shifts in good agreement with experiment.

A Core-Valence Separated Similarity Transformed EOM-CCSD Method for Core-Excitation Spectra

We present the theory and implementation of a core-valence separated similarity transformed EOM-CCSD (STEOM-CCSD) method for K-edge core excitation spectra. The method can select an appropriate active space using CIS natural orbitals and near "black box" to use. The second similarity transformed Hamiltonian is diagonalized in the space of single excitation. Therefore, the final diagonalization step is free from the convergence problem arising due to the coupling of the core-excited states with the continuum of doubly excited states. Convergence trouble can appear for the preceding core-ionized state calculation in STEOM-CCSD. A core-valence separation (CVS) scheme compatible with the natural orbital based active space selection (CVS-STEOM-CCSD-NO) is implemented to overcome the problem. The CVS-STEOM-CCSD-NO has a similar accuracy to that of the standard CVS-EOM-CCSD method but comes with a lower computational cost. The modification required in the CVS scheme to make use of the CIS natural orbital is highlighted. The suitability of the CVS-STEOM-CCSD-NO method for chemical application is demonstrated by simulating the K-edge spectra of glycine and thymine.

The porphyrin center as a regulator for metal-ligand covalency and π hybridization in the entire molecule

The central moiety of porphyrins is shown to control the charge state of the inner complex and links it by covalent interaction to the peripheral substituents. This link, which enables the versatile functions of porphyrins, is not picked up in the established, reduced four orbital picture [Gouterman, J. Mol. Spectrosc., 1961, 6, 138]. X-ray absorption spectroscopy at the N K-edge with density functional theory approaches gives access to the full electronic structure, in particular the π* manifold beyond the Gouterman orbitals. Systematic variation of the central moiety highlights two linked, governing trends: The ionicity of the porphyrin center increases from the aminic N-H to N-Cu to N-Zn to N-Mg to the iminic N:. At the same time covalency with peripheral substituents increases and compensates the buildup of high charge density at the coordinated nitrogen sites.

Nitrogen K-Edge X-ray Absorption Spectra of Ammonium and Ammonia in Water Solution: Assessing the Performance of Polarizable Embedding Coupled Cluster Methods

The recent development of liquid jet and liquid leaf sample delivery systems allows for accurate measurements of soft X-ray absorption spectra in transmission mode of solutes in a liquid environment. As this type of measurement becomes increasingly accessible, there is a strong need for reliable theoretical methods for assisting in the interpretation of the experimental data. Coupled cluster methods have been extensively developed over the past decade to simulate X-ray absorption in the gas phase. Their performance for solvated species, on the contrary, remains largely unexplored. Here, we investigate the current state of the art of coupled cluster modeling of nitrogen K-edge X-ray absorption of aqueous ammonia and ammonium based on quantum mechanics/molecular mechanics, where both the level of coupled cluster calculations and polarizable embedding are scrutinized. The results are compared to existing experimental data as well as simulations based on transition potential density functional theory.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

Dynamic-then-Static Approach for Core Excitations of Open-Shell Molecules

Delta self-consistent-field methods are widely used in studies of electronically excited states. However, the nonaufbau determinants are generally spin-contaminated. Here, we describe a general approach for spin-coupling interactions of open-shell molecules, making use of multistate density functional theory (MSDFT). In particular, the effective exchange integrals that determine spin coupling are obtained by enforcing the multiplet degeneracy of the S+1 state in the M_S = S manifold. Consequently, they are consistent with the energy of the high-spin state that is adequately treated by Kohn-Sham density functional theory (DFT) and, thereby, free of double counting of correlation. The method was applied to core excitations of open-shell molecules and compared with those by spin-adapted time-dependent DFT. An excellent agreement with experiment was found employing the BLYP functional and aug-cc-pCVQZ basis set. Overall, MSDFT provides an effective combination of the strengths of DFT and wave function theory to achieve efficiency and accuracy.

Analytical gradients for core-excited states in the algebraic diagrammatic construction (ADC) framework

Expressions for analytical molecular gradients of core-excited states have been derived and implemented for the hierarchy of algebraic diagrammatic construction (ADC) methods up to extended second-order within the core-valence separation (CVS) approximation. We illustrate the use of CVS-ADC gradients by determining relaxed core-excited state potential energy surfaces and optimized geometries for water, formic acid, and benzene. For water, our results show that in the dissociative lowest core-excited state, a linear configuration is preferred. For formic acid, we find that the O K-edge lowest core-excited state is non-planar, a fact that is not captured by the equivalent core approximation where the core-excited atom with its hole is replaced by the "Z + 1" neighboring atom in the periodic table. For benzene, the core-excited state gradients are presented along the Jahn-Teller distorted geometry of the 1s → π* excited state. Our development may pave a new path to studying the dynamics of molecules in their core-excited states.