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.

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.

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.

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.

Benchmark relativistic delta-coupled-cluster calculations of K-edge core-ionization energies of third-row elements

A benchmark computational study of K-edge core-ionization energies of third-row elements using relativistic delta-coupled-cluster (ΔCC) methods and a revised core-valence separation (CVS) scheme is reported. High-level relativistic (HLR) corrections beyond the spin-free exact two-component theory in its one-electron variant (SFX2C-1e), including the contributions from two-electron picture-change effects, spin-orbit coupling, the Breit term, and quantum electrodynamics effects, have been taken into account and demonstrated to play an important role. Relativistic ΔCC calculations are shown to provide accurate results for core-ionization energies of third-row elements. The SFX2C-1e-CVS-ΔCC results augmented with HLR corrections show a maximum deviation of less than 0.5 eV with respect to experimental values.

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.

Self-Consistent Field Methods for Excited States in Strong Magnetic Fields: a Comparison between Energy- and Variance-Based Approaches

Self-consistent field methods for excited states offer an attractive low-cost route to study not only excitation energies but also properties of excited states. Here, we present the generalization of two self-consistent field methods, the maximum overlap method (MOM) and the σ-SCF method, to calculate excited states in strong magnetic fields and investigate their stability and accuracy in this context. These methods use different strategies to overcome the well-known variational collapse of energy-based optimizations to the lowest solution of a given symmetry. The MOM tackles this problem in the definition of the orbital occupations to constrain the self-consistent field procedure to converge on excited states, while the σ-SCF method is based on the minimization of the variance instead of the energy. To overcome the high computational cost of the variance minimization, we present a new implementation of the σ-SCF method with the resolution of identity approximation, allowing the use of large basis sets, which is an important requirement for calculations in strong magnetic fields. The accuracy of these methods is assessed by comparison with the benchmark literature data for He, H₂, and CH⁺. The results reveal severe limitations of the variance-based scheme, which become more acute in large basis sets. In particular, many states are not accessible using variance optimization. Detailed analysis shows that this is a general feature of variance optimization approaches due to the masking of local minima in the optimization. In contrast, the MOM shows promising performance for computing excited states under these conditions, yielding results consistent with available benchmark data for a diverse range of electronic states.

Density Functional Theory Calculations of Core-Electron Binding Energies at the K-Edge of Heavier Elements

The capability to determine core-electron binding energies (CEBEs) is vital in the analysis of X-ray photoelectron spectroscopy, and the continued development of light sources has made inner shell spectroscopy of heavier elements increasingly accessible. Density functional theory is widely used to determine CEBEs of lighter elements (boron-fluorine). It is shown that good performance of exchange-correlation functionals for these elements does not necessarily translate to the calculation of CEBEs for the heavier elements from the next row of the periodic table, and in general, larger errors are observed. Two strategies are explored that improve the accuracy of the calculated CEBEs. The first is to apply element and functional dependent energy corrections, and the second is a reparametrization of a short-range corrected functional. This functional is able to reproduce experimental phosphorus and sulfur K-edge CEBEs with an average error of 0.15 eV demonstrating the importance of reducing the self-interaction error associated with the core electrons and represents progress toward a density functional theory calculation that performs equally well for ionization at the K-edge of all elements.

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

The availability of new light sources combined with the realization of the unique capabilities of spectroscopy in the X-ray region has driven tremendous advances in the field of X-ray spectroscopy. Currently, these techniques are emerging as powerful analytical tools for the study of a wide range of problems encompassing liquids, materials, and biological systems. Time-resolved measurements add a further dimension to X-ray spectroscopy, opening up the potential to resolve ultrafast chemical processes at an atomic level. X-ray spectroscopy encompasses a range of techniques which provide complementary information, and these include X-ray photoelectron spectroscopy (XPS), X-ray absorption spectroscopy (XAS), X-ray emission spectroscopy (XES), and resonant inelastic X-ray scattering (RIXS). In many studies, the interpretation of the experimental data relies upon calculations to enable the nature of the underlying molecular structure, electronic structure, and bonding to be revealed. Density functional theory (DFT) based methods are some of the most widely used methods for the simulation of X-ray spectra. In this Account, we focus on our recent contributions to the simulation of a range of X-ray spectroscopic techniques using DFT and linear-response time-dependent density functional theory (TDDFT) and show how these methods can provide a computational toolkit for the simulation of X-ray spectroscopy. The importance of the exchange-correlation functional for the calculation of XAS is discussed, and the introduction of short-range corrected functionals is described. The application of these calculations to study large systems through the use of efficient implementations of TDDFT will be highlighted, with the use of these methods illustrated through studies of ionic liquids and transition metal complexes. The extension of TDDFT to calculate XES through the use of a reference determinant for the core-ionized state will be described, and the factors that affect the accuracy of the computed spectra discussed. The application of these approaches will be illustrated through the study of a range of organic molecules and transition metal complexes, which also show how going beyond the dipole approximation in determining the transition intensities can be critical. The application of these approaches to the simulation of the RIXS spectrum of water will also be described, highlighting how ultrafast dynamics on the femtoscale time scale are evident in the measured spectra. In these calculations, the description of the core-ionized and core-excited states becomes increasingly important, and the role of the basis set in accurately describing these states will be explored.