Relativistic EOM-CCSD for Core-Excited and Core-Ionized State Energies Based on the Four-Component Dirac-Coulomb(-Gaunt) Hamiltonian

Unveiling the Redox Noninnocence of Metallocorroles: Exploring K-Edge X-ray Absorption Near-Edge Spectroscopy with a Multiconfigurational Wave Function Approach

X-ray absorption near-edge spectroscopy (XANES) is an advanced technique for probing the local electronic structure of catalysts, effectively identifying the noninnocent nature of ligands in transition-metal complexes. Metallocorroles with noninnocent corrole rings exhibit unusual electronic structures that challenge traditional density functional theory (DFT) methods, necessitating more rigorous approaches to describe electron correlation accurately. We explored K-edge XANES spectra of Fe, Mn, and Co metallocorroles using TDDFT and wave function-based methods. This is the first investigation employing multireference methods, specifically RASSCF, RASPT2, and MC-PDFT, to analyze the redox noninnocent nature of metallocorroles reflected in their XANES spectra. We quantified the noninnocent character of the corrole and the oxidation states of the metals, capturing more than singly excited excitations responsible for the pre-edge peak. Our findings demonstrate the importance of these advanced computational techniques for accurately predicting XANES spectra, providing a reliable understanding of the electronic properties of such complexes. This study offers a new strategy for investigating ligand redox noninnocence via integrated experimental and computational XANES.

Unified Implementation of Relativistic Wave Function Methods: 4C-iCIPT2 as a Showcase

In parallel to the unified construction of relativistic Hamiltonians based solely on physical arguments (J. Chem. Phys. 2024, 160, 084111), a unified implementation of relativistic wave function methods is achieved here via programming techniques (e.g., template metaprogramming and polymorphism in C++). That is, once the code for constructing the Hamiltonian matrix is made ready, all the rest can be generated automatically from existing templates used for the nonrelativistic counterparts. This is facilitated by decomposing a second-quantized relativistic Hamiltonian into diagrams that are topologically the same as those required for computing the basic coupling coefficients between spin-free configuration state functions (CSF). Moreover, both time reversal and binary double point group symmetries can readily be incorporated into molecular integrals and Hamiltonian matrix elements. The latter can first be evaluated in the space of (randomly selected) spin-dependent determinants and then transformed to that of spin-dependent CSFs, thanks to simple relations in between. As a showcase, we consider here the no-pair four-component relativistic iterative configuration interaction with selection and perturbation correction (4C-iCIPT2), which is a natural extension of the spin-free iCIPT2 (J. Chem. Theory Comput. 2021, 17, 949), and can provide near-exact numerical results within the manifold of positive energy states (PES), as demonstrated by numerical examples.

Elucidating Anomalous Intensity Ratios in Chlorine L-Edge X-ray Absorption Spectroscopy: Multiplet Effects and Core Rydberg Transitions

A relativistic core-valence-separated equation-of-motion coupled cluster (CVS-EOM-CC) study of chlorine L2,3-edge X-ray absorption near-edge structure (XANES) spectra using CH3Cl and CH2ICl as representative molecules is reported. The nearly identical intensity for the main features in the L2- and L3-edge XANES spectra is attributed to multiplet effects and the overlap between core-valence and core Rydberg transitions. The multiplet effects originating from the interaction between the core hole and the C-Cl σ* orbitals account for around half of the deviation of the L3 and L2 intensity ratio from the 2:1 ratio of the numbers of 2p3/2 and 2p1/2 electrons. The 2p3/2 → 4s core Rydberg transitions are shown to overlap with the 2p1/2 → σ* transitions and contribute to the other half of the intensity anomaly. We demonstrate that triple excitations in CVS-EOM-CC calculations play important roles in accurate simulation of the overlap between the 2p1/2 → σ* and 2p3/2 → 4s transitions.

Long-range Corrected Density Functional Theory Including a Two-Gaussian Hartree-Fock Operator for High Accuracy Core-excitation Energy Calculations of Both the Second- and Third-Row Atoms (LC2gau-core-BOP)

In the previous work, LCgau-core-BOP, which includes the short-range interelectronic Gaussian attenuating Hartree-Fock (HF) exchange to the long-range HF exchange, showed high accuracy core-excitation energies from 1s orbitals of the second-row atoms (1s → π*, 1s → σ*, 1s → n*, and 1s → Rydberg), but underestimates the core-excitation energies from 1s orbitals of the third-row atoms. To improve this, we added one more Gaussian attenuating HF exchange to LCgau-core-BOP. We named it LC2gau-core-BOP, which achieves a mean absolute error (MAE) of 0.6 and 0.3 eV for core excitation energies of the second- and third-row atoms of the tested small molecules, respectively. We found that the inclusion of the short-range interelectronic HF exchange at a distance ranging from 0.2 to 0.6 a.u. contributes to the increase of performances on 1s orbital energy calculations of the second-row atoms, while the inclusion of more short-range interelectronic HF exchange at a distance ranging from 0 to 0.2 a.u. does to the increase of performance on 1s orbital energy calculations of the third-row atoms. It is notable that all of these improvements were accomplished using flexible Gaussian attenuating HF exchange inclusion. LC2gau-core-BOP shows deviations of less than 0.8 eV from experimental values for all of the core-excitation energies of the tested medium-size molecules consisting of thymine, oxazole, glycine, and dibenzothiophene sulfone. Moreover, by optimizing one parameter of the OP correlation functional, LC2gau-core-BOP provides atomization energies over the G3 test set with an accuracy comparable to that of B3LYP.

Efficient State-Specific Natural Orbital Based Equation of Motion Coupled Cluster Method for Core-Ionization Energies: Theory, Implementation, and Benchmark

We have implemented a reduced-cost partial triples correction scheme to the equation of motion coupled cluster method for core-ionization energy based on state-specific natural orbitals. The second-order Algebraic Diagrammatic Construction (ADC) method is used to generate the state-specific natural orbital, which provides quicker convergence of the core-IP value with respect to the size of the virtual space than that observed in standard MP2-based natural orbitals. The error due to truncation of the virtual orbital can be reduced by using a perturbative correction. The accuracy of the method can be controlled by a single threshold, and there is a black box to use. The inclusion of the partial triples correction in the natural orbital based EOM-CCSD method greatly improves the agreement of the results with the experiment. The efficiency of the present implementation is demonstrated by calculating the core-ionization energy of a molecule containing 60 atoms and more than 2000 basis functions.

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.

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.

Unified construction of relativistic Hamiltonians

It is shown that the four-component (4C), quasi-four-component (Q4C), and exact two-component (X2C) relativistic Hartree-Fock equations can be implemented in a unified manner by making use of the atomic nature of the small components of molecular 4-spinors. A model density matrix approximation can first be invoked for the small-component charge/current density functions, which gives rise to a static, pre-molecular mean field to be combined with the one-electron term. As a result, only the nonrelativistic-like two-electron term of the 4C/Q4C/X2C Fock matrix needs to be updated during the iterations. A "one-center small-component" approximation can then be invoked in the evaluation of relativistic integrals, that is, all atom-centered small-component basis functions are regarded as extremely localized near the position of the atom to which they belong such that they have vanishing overlaps with all small- or large-component functions centered at other nuclei. Under these approximations, the 4C, Q4C, and X2C mean-field and many-electron Hamiltonians share precisely the same structure and accuracy. Beyond these is the effective quantum electrodynamics Hamiltonian that can be constructed in the same way. Such approximations lead to errors that are orders of magnitude smaller than other sources of errors (e.g., truncation errors in the one- and many-particle bases as well as uncertainties of experimental measurements) and are, hence, safe to use for whatever purposes. The quaternion forms of the 4C, Q4C, and X2C equations are also presented in the most general way, based on which the corresponding Kramers-restricted open-shell variants are formulated for "high-spin" open-shell systems.

Full Breit Hamiltonian in the Multiwavelets Framework

New techniques in core-electron spectroscopy are necessary to resolve the structures of oxides of f-elements and other strongly correlated materials that are present only as powders and not as single crystals. Thus, accurate quantum chemical methods must be developed to calculate core spectroscopic properties in such materials. In this contribution, we present an important development in this direction, extending our fully adaptive real-space multiwavelet basis framework to tackle the four-component Dirac-Coulomb-Breit Hamiltonian. We show that multiwavelets can reproduce one-dimensional grid-based approaches. They are however a fully three-dimensional approach which can later be extended to molecules and materials. Our multiwavelet implementation attained precise results irrespective of the chosen nuclear model, provided that the error threshold is tight enough and that the chosen polynomial basis is sufficiently large. Furthermore, our results confirmed that in two-electron species, the magnetic and Gauge contributions from s-orbitals are identical in magnitude and can account for the experimental evidence from K and L edges.

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.

Core Excitations of Uranyl in Cs2UO2Cl4 from Relativistic Embedded Damped Response Time-Dependent Density Functional Theory Calculations

X-ray spectroscopies, by their high selectivity and sensitivity to the chemical environment around the atoms probed, provide significant insights into the electronic structures of molecules and materials. Interpreting experimental results requires reliable theoretical models, accounting for environmental, relativistic, electron correlation, and orbital relaxation effects in a balanced manner. In this work, we present a protocol for the simulation of core excited spectra with damped response time-dependent density functional theory based on the Dirac-Coulomb Hamiltonian (4c-DR-TD-DFT), in which environmental effects are accounted for through the frozen density embedding (FDE) method. We showcase this approach for the uranium M4- and L3-edges and oxygen K-edge of the uranyl tetrachloride (UO2Cl42-) unit as found in a host Cs2UO2Cl4 crystal. We have found that the 4c-DR-TD-DFT simulations yield excitation spectra that very closely match the experiment for the uranium M4-edge and the oxygen K-edge, with good agreement for the broad experimental spectra for the L3-edge. By decomposing the complex polarizability in terms of its components, we have been able to correlate our results with angle-resolved spectra. We have observed that for all edges, but in particular the uranium M4-edge, an embedded model in which the chloride ligands are replaced by an embedding potential reproduces rather well the spectral profile obtained for UO2Cl42-. Our results underscore the importance of the equatorial ligands to simulating core spectra at both uranium and oxygen edges.

Quantum self-consistent equation-of-motion method for computing molecular excitation energies, ionization potentials, and electron affinities on a quantum computer

Near-term quantum computers are expected to facilitate material and chemical research through accurate molecular simulations. Several developments have already shown that accurate ground-state energies for small molecules can be evaluated on present-day quantum devices. Although electronically excited states play a vital role in chemical processes and applications, the search for a reliable and practical approach for routine excited-state calculations on near-term quantum devices is ongoing. Inspired by excited-state methods developed for the unitary coupled-cluster theory in quantum chemistry, we present an equation-of-motion-based method to compute excitation energies following the variational quantum eigensolver algorithm for ground-state calculations on a quantum computer. We perform numerical simulations on H2, H4, H2O, and LiH molecules to test our quantum self-consistent equation-of-motion (q-sc-EOM) method and compare it to other current state-of-the-art methods. q-sc-EOM makes use of self-consistent operators to satisfy the vacuum annihilation condition, a critical property for accurate calculations. It provides real and size-intensive energy differences corresponding to vertical excitation energies, ionization potentials and electron affinities. We also find that q-sc-EOM is more suitable for implementation on NISQ devices as it is expected to be more resilient to noise compared with the currently available methods.

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.

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10^-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnF_n, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Ultrafast Charge Relocation Dynamics in Enol-Keto Tautomerization Monitored with a Local Soft-X-ray Probe

Proton-coupled electron transfer (PCET) is the underlying mechanism governing important reactions ranging from water splitting in photosynthesis to oxygen reduction in hydrogen fuel cells. The interplay of proton and electronic charge distribution motions can vary from sequential to concerted schemes, with elementary steps occurring on ultrafast time scales. We demonstrate with a simulation study that femtosecond soft-X-ray spectroscopy provides key insights into the PCET mechanism of a photoinduced intramolecular enol* → keto* tautomerization reaction. A full quantum treatment of the electronic and nuclear dynamics of 2-(2'-hydroxyphenyl)benzothiazole upon electronic excitation reveals how spectral signatures of local excitations from core to frontier orbitals display the distinctly different stages of charge relocation for the H atom, donating, and accepting sites. Our findings indicate that ultraviolet/X-ray pump-probe spectroscopy provides a unique way to probe ultrafast electronic structure rearrangements in photoinduced chemical reactions essential to understanding the mechanism of PCET.

Basis Set Selection for Molecular Core-Level GW Calculations

The GW approximation has been recently gaining popularity among the methods for simulating molecular core-level X-ray photoemission spectra. Traditionally, Gaussian-type orbital GW core-level binding energies have been computed using either the cc-pVnZ or def2-nZVP basis set families, extrapolating the obtained results to the complete basis set limit, followed by an element-specific relativistic correction. Despite achieving rather good accuracy, it has been previously stated that these binding energies are chronically underestimated. In the present work, we show that those previous studies obtained results that were not well-converged with respect to the basis set size and that, once basis set convergence is achieved, there seems to be no such underestimation. Standard techniques known to offer a good cost-accuracy ratio in other theories demonstrate that the cc-pVnZ and def2-nZVP families exhibit contraction errors and might lead to unreliable complete basis set extrapolations for absolute binding energies, often deviating about 200-500 meV from the putative basis set limit found in this work. On the other hand, uncontracted versions of these basis sets offer vastly improved convergence. Even faster convergence can be obtained using core-rich property-optimized basis set families like pcSseg-n, pcJ-n, and ccX-nZ. Finally, we also show that the improvement observed for core properties using these specialized basis sets does not degrade their description of valence excitations: vertical ionization potentials and electron affinities computed with these basis sets converge as fast as the ones obtained with the aug-cc-pVnZ family, thus offering a balanced description of both core and valence regions.

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.

Assessing MP2 frozen natural orbitals in relativistic correlated electronic structure calculations

The high computational scaling with the basis set size and the number of correlated electrons is a bottleneck limiting applications of coupled cluster algorithms, in particular for calculations based on two- or four-component relativistic Hamiltonians, which often employ uncontracted basis sets. This problem may be alleviated by replacing canonical Hartree-Fock virtual orbitals by natural orbitals (NOs). In this paper, we describe the implementation of a module for generating NOs for correlated wavefunctions and, in particular, second order Møller-Plesset perturbation frozen natural orbitals (MP2FNOs) as a component of our novel implementation of relativistic coupled cluster theory for massively parallel architectures [Pototschnig et al. J. Chem. Theory Comput. 17, 5509, (2021)]. Our implementation can manipulate complex or quaternion density matrices, thus allowing for the generation of both Kramers-restricted and Kramers-unrestricted MP2FNOs. Furthermore, NOs are re-expressed in the parent atomic orbital (AO) basis, allowing for generating coupled cluster singles and doubles NOs in the AO basis for further analysis. By investigating the truncation errors of MP2FNOs for both the correlation energy and molecular properties-electric field gradients at the nuclei, electric dipole and quadrupole moments for hydrogen halides HX (X = F-Ts), and parity-violating energy differences for H₂Z₂ (Z = O-Se)-we find MP2FNOs accelerate the convergence of the correlation energy in a roughly uniform manner across the Periodic Table. It is possible to obtain reliable estimates for both energies and the molecular properties considered with virtual molecular orbital spaces truncated to about half the size of the full spaces.

Simulating core electron binding energies of halogenated species adsorbed on ice surfaces and in solution via relativistic quantum embedding calculations

In this work, we investigate the effects of the environment on the X-ray photoelectron spectra of hydrogen chloride and chloride ions adsorbed on ice surfaces, as well as of chloride ions in water droplets. In our approach, we combine a density functional theory (DFT) description of the ice surface with that of halogen species using the recently developed relativistic core-valence separation equation of motion coupled cluster (CVS-EOM-IP-CCSD) via the frozen density embedding formalism (FDE), to determine the K and L_1,2,3 edges of chlorine. Our calculations, which incorporate temperature effects through snapshots from classical molecular dynamics simulations, are shown to reproduce experimental trends in the change of the core binding energies of Cl^- upon moving from a liquid (water droplets) to an interfacial (ice quasi-liquid layer) environment. Our simulations yield water valence band binding energies in good agreement with experiment, which vary little between the droplets and the ice surface. For halide core binding energies there is an overall trend for overestimating experimental values, though good agreement between theory and experiment is found for Cl^- in water droplets and on ice. For HCl on the other hand there are significant discrepancies between experimental and calculated core binding energies when we consider structural models that maintain the H-Cl bond more or less intact. An analysis of models that allow for pre-dissociated and dissociated structures suggests that experimentally observed chemical shifts in binding energies between Cl^- and HCl would require that H⁺ (in the form of H₃O⁺) and Cl^- are separated by roughly 4-6 Å.

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.

SOiCI and iCISO: combining iterative configuration interaction with spin-orbit coupling in two ways

The near-exact iCIPT2 approach for strongly correlated systems of electrons, which stems from the combination of iterative configuration interaction (iCI, an exact solver of full CI) with configuration selection for static correlation and second-order perturbation theory (PT2) for dynamic correlation, is extended to the relativistic domain. In the spirit of spin separation, relativistic effects are treated in two steps: scalar relativity is treated by the infinite-order, spin-free part of the exact two-component (X2C) relativistic Hamiltonian, whereas spin-orbit coupling (SOC) is treated by the first-order, Douglas-Kroll-Hess-like SOC operator derived from the same X2C Hamiltonian. Two possible combinations of iCIPT2 with SOC are considered, i.e., SOiCI and iCISO. The former treats SOC and electron correlation on an equal footing, whereas the latter treats SOC in the spirit of state interaction, by constructing and diagonalizing an effective spin-orbit Hamiltonian matrix in a small number of correlated scalar states. Both double group and time reversal symmetries are incorporated to simplify the computation. Pilot applications reveal that SOiCI is very accurate for the spin-orbit splitting (SOS) of heavy atoms, whereas the computationally very cheap iCISO can safely be applied to the SOS of light atoms and even of systems containing heavy atoms when SOC is largely quenched by ligand fields.

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.

Two-Component Multireference Restricted Active Space Configuration Interaction for the Computation of L-Edge X-ray Absorption Spectra

X-ray absorption spectroscopy is a powerful probe of local electronic and nuclear structures, providing insights into chemical processes. The theoretical prediction and interpretation of metal L-edge X-ray absorption spectra are complicated by both relativistic effects, including spin-orbit coupling and the multiconfigurational nature of the states involved. This work details an exact two-component multireference restricted active space (RAS) configuration interaction scheme that uses an exact two-component state-averaged complete active space self-consistent-field method, which includes the spin-orbit coupling in a variational manner, for the accurate description of the electronic structure before using a RAS configuration interaction method to describe the core excited states of the X-ray spectrum. Benchmark calculations are presented for a series of iron-containing complexes, with results showing key features of the spectrum being reproduced, including ligand-to-metal charge transfer and shake-up excitations.

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.

Electronic spectra of ytterbium fluoride from relativistic electronic structure calculations

We report an investigation of the low-lying excited states of the YbF molecule-a candidate molecule for experimental measurements of the electron electric dipole moment-with 2-component based multi-reference configuration interaction (MRCI), equation of motion coupled cluster (EOM-CCSD) and the extrapolated intermediate Hamiltonian Fock-space coupled cluster (XIHFS-CCSD). Specifically, we address the question of the nature of these low-lying states in terms of configurations containing filled or partially-filled Yb 4f shells. We show that while it does not appear possible to carry out calculations with both kinds of configurations contained in the same active space, reliable information can be extracted from different sectors of Fock space-that is, by performing electron attachment and detachment IHFS-CCSD and EOM-CCSD calculation on the closed-shell YbF+ and YbF- species, respectively. From these calculations we predict Ω = 1/2, 3/2 states, arising from the 4f13σ26s, 4f145d1/6p1, and 4f135d1σ16s configurations to be able to interact as they appear in the same energy range around the ground-state equilibrium geometry. As these states are generated from different sectors of Fock space, they are almost orthogonal and provide complementary descriptions of parts of the excited state manifold. To obtain a comprehensive picture, we introduce a simple adiabatization model to extract energies of interacting Ω = 1/2, 3/2 states that can be compared to experimental observations.

Short Iterative Lanczos Integration in Time-Dependent Equation-of-Motion Coupled-Cluster Theory

A time-dependent (TD) formulation of equation-of-motion coupled-cluster (EOM-CC) theory can provide excited-state information over an arbitrarily wide energy window with a reduced memory footprint relative to conventional, frequency-domain EOM-CC theory. However, the floating-point costs of the time-integration required by TD-EOM-CC are generally far larger than those of the frequency-domain form of the approach. This work considers the potential of the short iterative Lanczos (SIL) integration scheme [J. Chem. Phys. 1986, 85, 5870-5876] to reduce the floating-point costs of TD-EOM-CC simulations. Low-energy and K-edge absorption features for small molecules are evaluated using TD-EOM-CC with single and double excitations, with the time-integrations carried out via SIL and fourth-order Runge-Kutta (RK4) schemes. Spectra derived from SIL- and RK4-driven simulations are nearly indistinguishable, and with an appropriately chosen subspace dimension, the SIL requires far fewer floating-point operations than are required by RK4. For K-edge spectra, SIL is the more efficient scheme by an average factor of 7.2.

Relativistic EOM-CCSD for Core-Excited and Core-Ionized State Energies Based on the Four-Component Dirac-Coulomb(-Gaunt) Hamiltonian

We report an implementation of the core-valence separation approach to the four-component relativistic Hamiltonian-based equation-of-motion coupled-cluster with singles and doubles theory (CVS-EOM-CCSD) for the calculation of relativistic core-ionization potentials and core-excitation energies. With this implementation, which is capable of exploiting double group symmetry, we investigate the effects of the different CVS-EOM-CCSD variants and the use of different Hamiltonians based on the exact two-component (X2C) framework on the energies of different core-ionized and -excited states in halogen- (CH₃I, HX, and X^-, X = Cl-At) and xenon-containing (Xe, XeF₂) species. Our results show that the X2C molecular mean-field approach [Sikkema, J.; J. Chem. Phys. 2009, 131, 124116], based on four-component Dirac-Coulomb mean-field calculations (²DC^M), is capable of providing core excitations and ionization energies that are nearly indistinguishable from the reference four-component energies for up to and including fifth-row elements. We observe that two-electron integrals over the small-component basis sets lead to non-negligible contributions to core binding energies for the K and L edges for atoms such as iodine or astatine and that the approach based on Dirac-Coulomb-Gaunt mean-field calculations (²DCG^M) are significantly more accurate than X2C calculations for which screened two-electron spin-orbit interactions are included via atomic mean-field integrals.