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 that can distribute the core orbitals over several atoms. We show that the 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. As electron correlation eventually alleviates the issues, but even when using coupled-cluster single-double and perturbative triple, there is a notable discrepancy between core-ionization energies obtained with localized and delocalized core–holes (0.5 eV for the carbon K-edge). Within density functional theory, the discrepancy correlates with 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 a 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 modeled 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.

When modeling x-ray spectroscopies, the question of the core–hole (CH) localization can be highly influential on the integrity of the final spectra. Using a ΔSCF approach, it has long been noted that a localized CH results in ionization energies (IEs) in good agreement with experiments, while a delocalized CH overestimates these IEs.1 Subsequently, when decomposing the terms involved in relaxing a core–hole state,2 it was shown that the relaxation effects associated with a localized core–hole are analogous to relaxation and correlation effects associated with a delocalized core–hole.3 As such, uncorrelated ΔSCF works quite well when the CH is localized but not when it is delocalized. Note that the use of the term correlation in this context can be somewhat misleading, as the deficiency of the delocalized CH has more to do with inadequate orbitals that are then corrected through correlation.4 Nevertheless, these terms are still relatively descriptive and commonly used in the literature, but some care is warranted. Using a localized CH, the symmetry of the molecular system is broken, but it can be retrieved by forming linear combinations of the localized solutions.1 Alternatively, by including electron correlation via various post-Hartree–Fock (HF) methods, theoretical results in good agreement with experimental spectra can be achieved while preserving the full symmetry of the system.5,6

From a more fundamental point of view, the localization of core–holes has been investigated and discussed experimentally, with measurements observing both localized and delocalized CHs, depending on the manner of conducting the measurements. It has thus been seen that CH localization/delocalization can be an effect of the observation (a situation that is not too uncommon in quantum physics).7–11 In certain cases, an entangled system is probed, while under other conditions, the entanglement is broken and an independent particle picture emerges. Care must thus be taken when performing the analysis of these types of measurements, where an overly simplistic single-particle picture should be avoided in lieu of more complete quantum mechanical frameworks.

While the fundamental question of CH localization can be imperative for gaining a deeper insight into the underlying physics, for most practical spectrum calculations, it may not be as vital. Still, it remains important to determine how CH localization/delocalization affects the agreement of calculated spectra to experiment or other computations, and this is the goal of the present study. Using a delocalized CH for core-ionized states results in a significant amount of missing relaxation, which then needs to be accounted for by using, e.g., high-level correlated methods. The amount of missing relaxation has been found to be associated with the number of delocalization sites,12 as it relates to early studies on the core–hole screening effects for atoms.13 The relaxation effects were shown to be quadratic with the change in the shielding constant and, thus, inversely proportional to the number of sites. In this study, we denote the transition or ionization energy difference between a localized CH and a delocalized CH as the delocalization-induced relaxation error (DIRE), which is to be understood as the relaxation that is present when using a localized CH, but not when using a delocalized CH. For fully correlated calculations, this will be close to zero (not exactly zero due to the presence of gerade and ungerade delocalized core–holes), and for approximate methods, it will vary on the level of correlation, probed element, and more. We stress that the DIRE does not arise from the use of delocalized core–holes per se—from a technical point of view, the delocalized CH is the theoretically rigorous approach to preserve the symmetry of the problem. However, it is associated with larger missing relaxation, which can be accounted for by using relatively high-level correlated methods or by enforcing a localized CH.

Studies on the effects of using delocalized core–holes have been conducted for ionization energies,12,14 potential energy surfaces,4 X-ray Raman scattering,15 and resonant inelastic x-ray scattering,16 to name a few. The behavior of DIRE in HF compared to density functional theory (DFT) has been noted, typically yielding the opposite sign.17 A number of measures to remove or minimize DIRE have been proposed and include the above mentioned linear combination of localized solutions,1 the use of high-level wave function theory,5,6 as well as tailoring various specialized methodologies. For example, a Z + Q model has been formulated to replace the Z + 1 (or equivalent core) approximation for cases where the CH is delocalized.18 Furthermore, CH localization has been analyzed for the static-exchange (STEX) approximation description of inner-shell photoionization.19 

The issue of CH localization has also been discussed in the context of double core–hole (DCH) spectroscopy and other multi-electron processes. Here, the creation of core–holes at different atomic sites typically yields superior sensitivity to the local chemical environment when compared to single core–hole approaches. In this context, CH localization/delocalization may have an even stronger influence than in single core–hole spectra, as discussed in a seminal article by Cederbaum et al.20 If a delocalized scheme is used, the correlation effects become substantial and need to be accounted for. A localized approach is thus the more common way of considering DCH processes.21–23 Considering multi-ionized N2, it has been noted that the difference between a localized picture and a delocalized picture depends on the number of holes created—with an even number of holes, the difference is close to zero, but for an uneven number of holes, it becomes substantial.24 

In this work, we investigate the influence of CH delocalization for a number of x-ray spectroscopies, as modeled using several different methods. In recent years, illustrations on the practical impact of using a delocalized CH have been shown for, e.g., single and double core–hole x-ray emission spectroscopy23,25 and core excitation calculations with restricted open-shell DFT.26 We here seek to provide a more systematic overview of where DIRE is significant, consider the influence on spectrum intensities, as well as analyze the energy terms involved. Related, the impact of near-degeneracies for non-equivalent sites when considering x-ray absorption spectroscopy (XAS) is also addressed, with a focus on approaches in which each symmetrically equivalent set of atomic sites is considered individually in order to decrease computational costs. This can be done by considering different sets of core orbitals when using the core–valence separation (CVS) approximation, as it has successfully been done using CVS-algebraic diagrammatic construction (ADC).27,28 We here analyze in more detail the potential pitfalls of this approach.

This section includes a brief overview of the methods considered in this article, whereas more extensive discussions on modeling x-ray spectroscopies can be found in, e.g., Ref. 29.

When constructing a CH reference state, some constraint has to be imposed in order to ensure that the SCF optimization does not collapse into a valence-ionized state. One of the approaches, which can be used to achieve this, is the maximum overlap method (MOM).30–33 In MOM, the wave function is optimized with overlap to previous iterations in mind, rather than from energetic arguments. With this non-Aufbau procedure, core–holes and other energetically unstable wave functions can be constructed. Alternative approaches of forming a CH reference state are available, including the initial maximum overlap method (IMOM),34 state-targeted energy projection (STEP),35 and square gradient minimization (SGM).36 Core–hole states constructed this way can be used to estimate ionization energies (IEs) via the ΔSCF approach, i.e., from the difference in energy between the ground state and a core-ionized state. Using post-HF methods, such as Møller–Plesset (MP) perturbation theory or coupled cluster (CC) theory, ΔMP and ΔCC IEs can be obtained by using these neutral and core-ionized states as the reference for a MP or CC calculation.

In addition to the calculation of core-ionization energies and as an initial state of x-ray emission calculations, core–holes can also be used to include the relaxation effects involved in x-ray absorption spectroscopy (XAS). One example is Slater’s transition state method that models relaxation by considering a half-empty core orbital in combination with the addition of half an electron to each probed unoccupied MO.37,38 The total spectrum is then constructed by calculating orbital energy differences and transition matrix elements, but the approach requires one explicit calculation for each final state. A further simplification of the method is to relax the electronic structure in the presence of a partial core–hole, constructing the full spectrum from a single Hamiltonian. Within a DFT framework, this approach constitutes the transition potential DFT (TP-DFT) method,39,40 which commonly employs a half core–hole (HCH), although other occupations are also used.41,42 By introducing a shift such that the eigenvalue of the core level is equal to the calculated IE, TP-DFT provides XAS spectra that compare well to experiment in many cases,39,40,43 albeit with some occasional difficulties in sufficiently capturing relaxation effects,41 and relatively large spread in relative energies.28 More recent developments related to this method include the combination of the TP-DFT philosophy with coupled cluster, yielding the transition potential coupled cluster (TP-CC) method.44 

Alternatively, an approach using the energy difference and transition dipole moments between the ground state molecular orbitals has been used, with good relative agreement to x-ray emission experiments.45,46

Despite a successful application to large molecules47–49 and extended systems,50–52 TP-DFT is essentially a ground state theory for XAS. A different approach is to determine the response of the electron system to an applied electromagnetic field.53 In time-dependent HF (TDHF) and time-dependent DFT (TDDFT), this is achieved through the random phase approximation (RPA) equation.54,55 Computational costs can be lowered by using the Tamm–Dancoff approximation (TDA),56 which for TDHF becomes equivalent to configuration interaction singles (CIS).55 In TDDFT, the approximate nature of the exchange–correlation (xc) functional leads to self-interaction errors (SIE)29,57–59 that are exacerbated in the case of core excitations (due to the high densities in the core region). This has spurred the design of a plethora of tailored xc-functionals,29,58,60 which can yield improved absolute energies, but not necessarily improved relative energies.28 

Among correlated ab initio methods for excited states, of interest here are the algebraic diagrammatic construction (ADC) scheme for the polarization propagator and coupled cluster (CC) theory. In ADC, the polarization propagator is expanded in a perturbation series, where the poles and residues of its spectral representation correspond to excitation energies and transition amplitudes, respectively.61–65 The hierarchy of ADC methods is obtained by truncating the perturbation expansion at desired order, with efficient implementations available for ADC(1), ADC(2), ADC(2)-x, and ADC(3/2). Note that ADC(1) is equivalent to CIS in terms of transition energies. An alternative hierarchy of post-HF methods is available through the coupled cluster (CC) approach, which can be considered either in an equation-of-motion (EOM)66,67 or linear response (LR)68–70 formalism. For XAS, ADC(2)-x and (EOM-) coupled-cluster single-double (CCSD) have been shown to yield good agreement to experiment,28,71,72 while for x-ray emission spectra, good absolute agreement with experiments is provided by ADC(2) and EOM-CCSD [with ADC(2)-x yielding good relative features].58,73–75

When calculating x-ray absorption spectra, a flavor of the core–valence separation (CVS) approximation must typically be applied.29,71,72,76–80 CVS avoids the convergence of a great manifold of valence-excited and valence-ionized states by utilizing the fact that the coupling between core- and valence-excited states is small, motivating the decoupling of core excitations from the total manifold. The precise implementation of this scheme varies between electronic structure methods and software packages. For the K-edge, the error associated with the CVS approximation in TDHF and TDDFT has been shown to be negligible, while in post-HF methods, it has been shown to be small and stable over different compounds.78,79,81,82

When it comes to x-ray emission spectra, they involve the use of a core–hole reference state for which the valence-to-core transitions occur as the first (negative) eigenvalues.58,73,74 X-ray emission spectrum (XES) calculations have successfully been performed using TDDFT,58,83,84 EOM-CCSD,58,73,83 and ADC,74,75 where we note that the relaxation (and thus the performance of different methods) is markedly different compared to that of core excitation processes.74 

The molecular geometries were optimized at the frozen-core MP285/cc-pVTZ86 level of theory using the Q-Chem 5.2 program.87 Property calculations were performed using cc-pVDZ for hydrogen atoms and aug-cc-pCVTZ for the remaining elements. Core–holes were localized by applying effective core potentials (ECPs) of the Stuttgart–Cologne type88 for all save one atomic site of an investigated element, or in some cases (Figs. 5 and 7) by using a distorted structure. For the delocalized core–hole calculations, the maximum overlap method (MOM)30–33 was used to obtain a core–hole with the same symmetry as that of the corresponding ground state core orbital. The smallest absolute values of the MP2 energy denominator for the ethene core–holes were seen to be ∼1 hartree, and numerical instabilities due to near-singularities are thus not present for these calculations.75 The Perdew–Burke–Ernzerhof (PBE)89,90 exchange–correlation functional was used for the DFT calculations unless otherwise stated. Convolution of the calculated energies and intensities was performed using a Lorentzian function of 0.25 eV half-width at half maximum (HWHM) to facilitate the analysis and comparison of the spectra.

The TDDFT and ADC calculations were carried out in Q-Chem 5.2,87 with the restricted energy-window and CVS77,91 approaches being applied when computing x-ray absorption spectra. Calculations using tailored CVS spaces were run using the adcc software package,27 with SCF results obtained from pyscf.92,93 The TDA was applied for TDDFT calculations of emission spectra.

An in-house modified version of the psiXAS plugin module94 of psi495 was used for TP-DFT and ground state DFT calculations. For XAS calculations in the TP-DFT framework, the half core–hole (HCH) approximation was used, without any additional shifts in energy. In order to perform XES calculations, we have modified the psiXAS module to also allow the computation of transition dipole moment integrals between the core–hole state and occupied orbitals (rather than unoccupied orbitals as required by core excitation calculations). This modification was also used to determine the transition dipole moments between ground state orbitals, which together with MO energy differences yields spectra within ground state (GS) DFT.

We first illustrate the effects of core–hole delocalization in XAS, XES, and IE/XPS calculations. The influence of various xc-functionals, probed element, and system size is then discussed, followed by a decomposition of energy contributions in HF and DFT. The impact of electron correlation in a post-HF framework is then considered, together with some general recommendations for calculations involving explicit core–holes. Finally, we consider the use of a tailored CVS space, including cases of fully symmetric atomic sites, near-degeneracy, and localized core orbitals.

The effects of using localized or delocalized CHs are illustrated in Fig. 1, considering x-ray absorption and emission spectra using five different methods. In the case of XAS, a delocalized CH yields the same spectra for all methods save TP-DFT for which the delocalized CH spectrum is 8.39 eV lower in photon energy due to DIRE (as compared to the localized case). The delocalized TP-DFT x-ray absorption spectrum has thus been shifted by this amount. Only minor differences in relative spectrum features are noted following this energy shift. The difference in transition energies is due to the use of an explicit (fractional) core–hole in TP-DFT, something which is not present for the other approaches. Compared to experimental measurements, the 1sπ* peak is situated at 284.7 eV (the lower left panel of Fig. 1),96 which is close to the localized results.

FIG. 1.

X-ray absorption (left) and x-ray emission (right) spectra of ethene calculated at different levels of theory using a localized (blue) or delocalized (red) core–hole, as compared to experimental measurements.96,97 Where necessary, spectra calculated with a delocalized CH have been shifted to align with the corresponding localized CH spectrum using rigid shifts in energy.

FIG. 1.

X-ray absorption (left) and x-ray emission (right) spectra of ethene calculated at different levels of theory using a localized (blue) or delocalized (red) core–hole, as compared to experimental measurements.96,97 Where necessary, spectra calculated with a delocalized CH have been shifted to align with the corresponding localized CH spectrum using rigid shifts in energy.

Close modal

For XES, significant differences in both absolute energies and relative features are noted for all methods except using DFT ground state MOs, which is the only approach not using an explicit CH reference state. This approach also compares reasonably to the experiment97 (lower right panel in Fig. 1) in terms of relative peak positions and intensities, albeit with a large rigid shift of ∼20 eV. Comparing localized and delocalized calculations, the largest differences in both absolute and relative features are observed for TDDFT and ADC(1), as these methods do not include the highly influential relaxation effects involved in core properties—the difference observed between a localized and a delocalized CH is ultimately a consequence of the improper treatment of electron correlation and orbital relaxation.2 

The distinction between electron correlation and relaxation is somewhat arbitrary,2,3 with relaxation for core properties often being included through electron correlation.28,29,98 Still, the distinction can be helpful for analysis, and in Fig. 2, we use these designations in order to better illustrate various contributions for calculations using HF and DFT. Included here are results for IEs, and one feature each of XES and XAS, as compared to reference calculations using coupled-cluster single-double and perturbative triple [CCSD(T)]. Note that the PBE0 absolute energies are not directly comparable to the CCSD(T) absolute energies, as the Hamiltonians are different, but relative energies can still be examined. The results are thus given on the same absolute energy scale, where the closed-shell ground state energies of PBE0 and CCSD(T) happen to take very similar values. The CCSD(T) calculations are all performed considering different HF reference states (neutral ground state, core-ionized, neutral core-, and valence-excited), while the XES/XAS results for HF and DFT have been obtained within the Tamm–Dancoff approximation. For the CCSD(T) reference calculations, it is clear that the use of a localized or delocalized CH yields very similar results, with ionization or transition energies varying by 0.40–0.54 eV (larger for a delocalized CH, where DIRE is present). This is primarily a result of CCSD(T) still lacking some electron correlation, with minor contributions also stemming from the use of ECPs when localizing the CH. Note that the CCSD(T) calculations of the final state in XAS are not spin-pure, but the resulting energies are similar to those obtained by, e.g., CVS-ADC(2)-x or CVS-EOM-CCSD, and are thus expected to be reasonably good.

FIG. 2.

Initial and final state energies for carbon 1s ionization energy (IE), the highest feature of the x-ray emission spectrum (XES), and lowest feature of the X-ray absorption spectrum (XAS), as obtained for ethene using a localized or delocalized core–hole. Reference CCSD(T) results are shown in black, and HF and PBE0 results are in blue and red, respectively, including resulting ionization and transition energies. For IEs, the binding energy from Koopmans’ theorem is also shown (dashed lines). Note that the PBE0 and CCSD(T) absolute energies are not directly comparable, but energy differences are (see the main text). The right side of horizontal lines shows a discrepancy of HF results compared to CCSD(T) values (or HF values for binding energies from Koopmans’ theorem), which is associated with lacking correlation (Ec), relaxation (Er), partial relaxation (Er), and combined relaxation and correlation (Erc). All energies are expressed in eV.

FIG. 2.

Initial and final state energies for carbon 1s ionization energy (IE), the highest feature of the x-ray emission spectrum (XES), and lowest feature of the X-ray absorption spectrum (XAS), as obtained for ethene using a localized or delocalized core–hole. Reference CCSD(T) results are shown in black, and HF and PBE0 results are in blue and red, respectively, including resulting ionization and transition energies. For IEs, the binding energy from Koopmans’ theorem is also shown (dashed lines). Note that the PBE0 and CCSD(T) absolute energies are not directly comparable, but energy differences are (see the main text). The right side of horizontal lines shows a discrepancy of HF results compared to CCSD(T) values (or HF values for binding energies from Koopmans’ theorem), which is associated with lacking correlation (Ec), relaxation (Er), partial relaxation (Er), and combined relaxation and correlation (Erc). All energies are expressed in eV.

Close modal

For HF, the missing correlation and relaxation in the initial and final state can be identified by comparison to CCSD(T) total energies (see energy differences marked as Ec, Er, Er, and Erc in Fig. 2). Thus, ∼11 eV of correlation energy (Ec) is lacking for the initial states of IE and XAS and for the localized description of XES. For the final states, there is generally a lack of both relaxation and correlation (Erc), although for the ΔHF calculations of IEs, the relaxation is largely accounted for through the explicit optimization of a core-ionized state (yielding Er). This is seen for the localized CH, where a lowering in final state energy of 15.26 eV is achieved when compared to Koopmans’ theorem (dashed lines). Here, the missing correlation in the final state largely compensates the missing correlation in the initial state, and the resulting IE is thus quite close to the CCSD(T) reference. In the delocalized CH case, only approximately half of the relaxation energy (7.38 eV) is retrieved, yielding a remaining 19.77 eV difference with respect to CSSD(T), and an overestimation of the final IE of about 7 eV. This partial relaxation (Er′) is in line with the previous observation, where the factor one-half (compared to the localized case) comes from the relation of relaxation for a localized CH being analogous to correlation and relaxation of a delocalized CH, and the introduced discrepancy being inversely proportional to the number of delocalization sites.3,12,13 For XES, the final state lacks both relaxation and correlation (when using TDHF—for ΔHF the situation would be closer to that of IEs) and is thus further from the CCSD(T) reference than the ∼11 eV Ec of most initial states. For a localized CH, the energy difference between the initial and final state is thus too low, and the emission energy is ∼10 eV lower compared to CCSD(T). For a delocalized CH, the situation is quite different: the initial state has only partial relaxation accounted for (Er′) and is thus 19.77 eV above the reference energy. The final state again lacks Erc, but this is in total smaller than for a localized CH due to the use of a poorer (and thus more GS-like) initial state. The resulting emission energy, therefore, overestimates the reference value by almost 6 eV, yielding a large discrepancy when compared to the localized description. For XAS, no explicit core–hole is used, and the localized and delocalized results are thus very similar, both featuring an overestimation in transition energy of ∼9 eV due to the unrelaxed (and thus high-energy) final state.

In the case of DFT, relaxation and correlation cannot be determined by comparison to CCSD(T) since the Hamiltonians are different, but the resulting relative energies are possible to compare. The case of DFT also differs by (partially) including correlation in the functional, as well as by the presence of self-interaction errors (SIEs).29,57–59 The SIEs will generally decontract high-density orbitals and will partially cancel out the effects of lacking relaxation. This is most clearly seen for XAS, where TDDFT yields virtually identical results with a localized or delocalized CH, both underestimating the CCSD(T) experimental excitation energy by almost 10 eV. For IEs, the localized CH leads to results well in line with the reference, while the delocalized CH lowers the total IE by about 2 eV. The difference, when compared to HF, will be discussed more below. Finally, for XES, TDDFT yields a too large energy difference between the initial and final states in both the localized and delocalized cases, thus resulting in emission energies that are too high. Here, the emission energies from using a delocalized CH case are actually a bit closer to the CCSD(T) reference value.

Considering the dependence of DIRE with probed atom type and xc-functional, Fig. 3 shows the size of this difference for various elements and functionals. The left panel shows the IE discrepancies as a function of Z, considering ethene, N2, O2, and F2. HF and the PBE functional with varying levels of HF exchange both show an increasing IE difference as Z increases. This simply reflects the increasing relaxation effect and SIE of heavier elements. Focusing on the IE of ethene, the right panel shows the DIRE of a number of different xc-functionals, grouped into four main categories. For global and range-separated generalized gradient approximation (GGA) functionals, the relation between the DIRE and the amount of HF exchange is clear, showing a linear behavior with increasing HF exchange. This points to the HF exchange parameter being an influential factor to consider when analyzing the performance of DFT for CH delocalization, as will be discussed further below. By comparison, for global and range-separated meta-GGAs, there are some clear outliers, although the general trend is in line with the GGAs. The outliers are primarily meta-GGAs with exchange coefficients that sum up to more than one, with M06-HF, MN15, and MN12-SX deviating the most from the otherwise linear behavior.

FIG. 3.

Left: DIRE of ionization energies as a function of atomic number obtained using the ΔSCF approach. DFT results obtained using the PBE functional with different amounts of HF exchange. Right: DIRE of the IE of ethene, shown as a function of the amount of (short-range) HF exchange. Results obtained using hybrid GGA (light blue), meta-GGA (dark blue), range-separated GGA (orange), and range-separated meta-GGA (red) xc-functionals, with HF result shown as a black star.

FIG. 3.

Left: DIRE of ionization energies as a function of atomic number obtained using the ΔSCF approach. DFT results obtained using the PBE functional with different amounts of HF exchange. Right: DIRE of the IE of ethene, shown as a function of the amount of (short-range) HF exchange. Results obtained using hybrid GGA (light blue), meta-GGA (dark blue), range-separated GGA (orange), and range-separated meta-GGA (red) xc-functionals, with HF result shown as a black star.

Close modal

Another parameter that plays a role in the absolute value of DIRE is the size of the system over which the CH is delocalized. This is illustrated in Fig. 4, where the error in IE for ethene and cycloalkanes of increasing size is shown as a function of the inverse number of atomic sites over which the CH is delocalized. The IEs were calculated for the core MOs that are delocalized over the entire cycloalkanes. Irrespective of the amount of HF exchange included in the functional, the DIRE increases in absolute value with increasing the number of sites (N). Represented as a function of 1/N, the behavior is almost linear, as expected.12,13 Note that what is important is not the actual system size, but rather the number of involved atomic sites.

FIG. 4.

DIRE of ionization energies as a function of 1/N, where N is the number of sites over which the core–hole is delocalized. The delocalized core–hole corresponds to the fully delocalized C 1s bonding orbital in cycloalkanes of increasing size from ethene to cyclohexane. Differences obtained using various amounts of HF exchange included in a PBE-based functional are shown in different colors. The dotted lines are obtained by linear regression, with corresponding coefficients of determination.

FIG. 4.

DIRE of ionization energies as a function of 1/N, where N is the number of sites over which the core–hole is delocalized. The delocalized core–hole corresponds to the fully delocalized C 1s bonding orbital in cycloalkanes of increasing size from ethene to cyclohexane. Differences obtained using various amounts of HF exchange included in a PBE-based functional are shown in different colors. The dotted lines are obtained by linear regression, with corresponding coefficients of determination.

Close modal

To further analyze the behavior of the DIRE in HF and DFT, we decompose different terms involved in calculating the IE, as a function of amount of HF exchange. In ΔSCF, the IE is calculated as the total energy difference between a core-ionized (ECH) state and the ground state (EGS),

IE=ECHEGS,
(1)

or decomposed into the one-electron contribution stemming from the core–Hamiltonian (EH), Coulomb (EJ), HF exchange (EK), and DFT exchange and correlation (EDFT),

IE=(EHCHEHGS)+(EJCHEJGS)+(EKCHEKGS)+(EDFTCHEDFTGS),
(2)

where individual components are defined as

EH=ii|ĥ|i,
(3)
EJ=12i,jij|ij,
(4)
EK=α12i,jij|ji,
(5)
EDFT=(1α)f(n(r),n(r))dr.
(6)

Here, i, j are occupied orbital indices, |i⟩ are spin–orbitals, ĥ is the core–Hamiltonian operator, and the summation is over all occupied orbitals. The amount of HF exchange is given as α, f is a generic GGA functional of the electron density (n), and its gradient (∇n). The electron density, n(r), is obtained by representing the density matrix (P = {Pμν}) on a grid,

Pμν=iCμiCνi,
(7)

where {Cμi} are the MO coefficients. By restricting the summations in Eqs. (3)(7) to core orbitals, core–electron contributions can be separated out

EHcore=II|ĥ|I,
(8)
EJcore=I,jIj|Ij,
(9)
EKcore=αI,jIj|jI,
(10)

where I are occupied core orbitals and j are generic occupied orbitals (including core). In the case of DFT exchange–correlation, the core orbital contribution can be computed by determining the electron density required in Eq. (6) based on a density matrix constructed using only core orbitals,

Pμνcore=ICμICνI.
(11)

For the exchange energy, we also calculated the core–core contribution of the β spin channel alone (where β is the spin channel with the core–hole),

EKcore=αI,JIβJβ|JβIβ.
(12)

In order to determine these terms for a localized and delocalized CH, localization using ECPs will not be useful, as this will yield very different total energies and number of electrons. Instead, we localize the CH by using an asymmetric ethene molecule where one of the CH2 groups has CH bonds stretched by 0.05 Å from the equilibrium value. This distortion is large enough to lift the degeneracy of the core MOs, allowing full localization on the atomic sites, but small enough not to affect the IEs significantly. With this, energy differences (ΔE) of the contributions defined in Eqs. (3)(6), (8)(10), and (12) are determined as

ΔE=(EdelocCHEdelocGS)(ElocCHElocGS).
(13)

The resulting decomposition in contributions is illustrated in Fig. 5, as a function of the amount of HF exchange mixed in a hybrid PBE-based xc-functional. The left panel shows total components (full lines), as well as those obtained using only the C 1s core orbitals (dotted lines). All contributions display a close to linear dependence with respect to the amount of HF exchange, and we see that the main variation in the IE error is correlated with the variation in HF exchange and DFT exchange–correlation. The difference in one-electron energy is quite small, and the Coulomb energy contributes mainly by a shift. Importantly, the large variations observed for HF exchange and DFT exchange–correlation stem mainly from the core orbitals, as the corresponding dotted lines lie almost perfectly over the total variations. The difference in Coulomb is also primarily involving the core, although a larger contribution from the other MOs is present there.

FIG. 5.

Difference in IE energy terms using a localized or delocalized core–hole on ethene, as a function of the amounts of HF exchange included in a PBE-based hybrid functional. Left: IE difference decomposed into: one-electron contribution (gray), Coulomb (blue), HF exchange (pink), and DFT exchange and correlation (yellow). The contributions to these energy differences from only the core electrons are also shown (dotted lines). Right: the IE difference and the main contributions: the HF exchange contribution from the core β electrons (spin channel with a CH), DFT exchange and correlation of the core levels, and the Coulomb interaction for the core levels.

FIG. 5.

Difference in IE energy terms using a localized or delocalized core–hole on ethene, as a function of the amounts of HF exchange included in a PBE-based hybrid functional. Left: IE difference decomposed into: one-electron contribution (gray), Coulomb (blue), HF exchange (pink), and DFT exchange and correlation (yellow). The contributions to these energy differences from only the core electrons are also shown (dotted lines). Right: the IE difference and the main contributions: the HF exchange contribution from the core β electrons (spin channel with a CH), DFT exchange and correlation of the core levels, and the Coulomb interaction for the core levels.

Close modal

The right panel of Fig. 5 shows the variation in core Coulomb energy and core DFT exchange–correlation, alongside the HF exchange contribution of the core β spin channel (the channel with the CH). It is evident that the behavior of DIRE correlates mainly with energy contributions from the core β channel. Furthermore, if these three energy components are summed together (the black dotted line), the trend in IE error is almost perfectly captured. As such, the numerical difference in DIRE for HF and DFT is associated with differences in pure HF core β exchange (20.6 eV) and pure DFT core β exchange–correlation (6.9 eV), shifted by a relatively constant discrepancy in core Coulomb interaction. It is important to note that what we have observed here is a correlation and not necessarily a cause of DIRE. The analysis of the origin of this error in DFT is complicated by the fact that DFT is affected by self-interaction errors (SIE). Disentangling the contribution of SIE to the IE error is not easy, but the density-driven part of SIE may be alleviated through the use of a self-interaction free density (e.g., from a self-consistent HF calculation).99,100 With this approach, we estimated that density-driven self-interaction produces an error of ∼0.5–0.7 eV in the PBE total energies of our systems. However, since in ΔKohn-Sham  (ΔKS) the IEs are calculated as total energy differences between the ground and core-ionized state, these density-driven SIEs cancel out to a large extent for both the localized and delocalized cases. In addition, functionals of different types (right panel of Fig. 3), which attempt to minimize SIE in various ways, produce ΔKS IEs comparably close to the experimental ionization energy of 290.7 eV101 if the localized CH scheme is used. It can be noted that the inadequacy of single-reference methods for describing dynamic correlation of highly symmetrical systems has also been discussed in other areas, such as for strongly correlated magnetic systems.102,103

To investigate the influence of electron correlation on DIRE, Fig. 6 reports XPS spectra for ethene at various post-HF levels of theory, as obtained from total energy differences. With a localized CH, the resulting IEs are all close to the experimental value of 290.7 eV,101 with CCSD(T) yielding an IE of 290.9 eV (accounting for a relativistic shift of about 0.1 eV). Using a delocalized CH, much larger deviations are visible, with HF results being about 8 eV too high in energy. For comparison, the IE determined with the GGA functional PBE and the localized CH approach is 290.0 eV, while the delocalized CH underestimates the IE by about 7 eV. The trends in IE error for different types of functionals can be seen in Fig. 3 (right panel). As previously discussed, the HF IE error is due to a lack of relaxation (and, to a smaller extent, correlation), and the agreement improves significantly once electron correlation is included. Even so, the variations remain noticeable, with a difference of 0.5 eV remaining at the CCSD(T) level of theory. As such, relatively high order in perturbation theory is needed to yield identical results, and we note that this increases the likelihood of numerical instabilities due to the use of a non-Aufbau reference state.75 

FIG. 6.

X-ray photoelectron spectra of ethene calculated at different levels of theory, as obtained using a delocalized (red) and localized (blue) core–hole and compared to experimental peak position (dotted vertical line).101 

FIG. 6.

X-ray photoelectron spectra of ethene calculated at different levels of theory, as obtained using a delocalized (red) and localized (blue) core–hole and compared to experimental peak position (dotted vertical line).101 

Close modal

In order to better understand the influence of electron correlation, Fig. 7 shows a decomposition of the MP2 energy corrections for localized and delocalized CHs. We again use symmetric and asymmetric structures of ethene, and thus, note that there will be a small error due to this use of different structures. The total MP2 energy correction is the sum of the same-spin (ESS) and opposite-spin (EOS) terms, which are calculated as

ESS=i,j,a,b,σiσjσ|aσbσiσjσaσbσεa+εbεiεj,
(14)
EOS=i,j,a,b,σ,σiσjσ|aσbσiσjσ|aσbσεa+εbεiεj,
(15)

where σ indicates the spin, i and j denote occupied orbitals, and a and b denote virtual orbitals. By restricting these summations over different occupied spaces, the total MP2 correlation energy can be divided into separate components involving only valence-occupied orbitals, only core-occupied orbitals, and mixed core- and valence-occupied orbitals. This can also be done using frozen core or valence orbitals, with core–valence contribution being the remaining MP2 energy of the full-space calculation.104 

FIG. 7.

MP2 energy corrections for the ground state (GS) and core–hole (CH) state of symmetric (delocalized CH) and asymmetric (localized CH) ethene. The energy corrections have been decomposed into different contributions, categorized by the involved MO levels and spin components. The opposite spin MP2 correction of the core–valence term has been further divided into contributions involving core α and the remaining core β orbitals (CH placed in β).

FIG. 7.

MP2 energy corrections for the ground state (GS) and core–hole (CH) state of symmetric (delocalized CH) and asymmetric (localized CH) ethene. The energy corrections have been decomposed into different contributions, categorized by the involved MO levels and spin components. The opposite spin MP2 correction of the core–valence term has been further divided into contributions involving core α and the remaining core β orbitals (CH placed in β).

Close modal

As shown in Fig. 7, the largest difference lies in the core–valence correlation contribution of the delocalized CH, which is about 9 eV larger than the near-zero core–valence contributions of the ground state and localized CH. Looking more closely at the precise terms, this discrepancy is dominated by same-spin β and the opposite-spin contribution involving a core-β orbital. Returning to the results presented and discussed in connection to Fig. 5, we see that the inconsistent exchange energy that is achieved when creating a delocalized CH is primarily accounted for by correlating the remaining β orbitals with the valence region, although we note that the difference in ΔMP2 IE remains relatively large at around 2 eV. Small additional differences are seen in the valence–valence and core–core contributions, where in particular the valence–valence contribution to the delocalized CH is shifted by about 0.8 eV. The core–core contribution of the neutral system is about 2 eV, which is lowered to 1 eV for a localized CH or 1.5 eV for a delocalized CH. For a localized CH, this decrease by one-half is easy to understand, as the same-spin contribution is close to zero due to the near-zero spatial overlap, and one out of two (close to equal) opposite-spin contributions disappears when a core–hole is created.

In terms of x-ray emission spectrum calculations, the spectra obtained with a localized and delocalized CH at different levels of theory are reported in Fig. 8. As seen above, XES is the spectroscopy most affected by CH localization discussed here, with both absolute energies and relative energies and intensities being significantly affected. We here see that ADC(1) shows a particularly poor comparison between the two CH localization schemes, as this method lacks in both relaxation and correlation. For higher order ADC and EOM-CCSD, the absolute energy discrepancy is much smaller (0.3–4.1 eV, with either sign), and the relative features are in better agreement. Still, the difference remains large, and we recommend using the localized CH approach for practical calculations.

FIG. 8.

X-ray emission spectra of ethene calculated with localized (blue) and delocalized (red) core–holes, as obtained using EOM-CCSD and with the ADC hierarchy. Delocalized CH results are shifted in energy such that the high-energy features overlap.

FIG. 8.

X-ray emission spectra of ethene calculated with localized (blue) and delocalized (red) core–holes, as obtained using EOM-CCSD and with the ADC hierarchy. Delocalized CH results are shifted in energy such that the high-energy features overlap.

Close modal

A different context in which core MO localization becomes important is in the selection of a CVS space for calculating x-ray absorption spectra. Generally, the core–valence separation is justified from the observation that the core and valence orbitals are well-separated both spatially and energetically. Still, there are cases where a CVS space containing only some of the core MOs and leaving out others of the same atomic type may be useful. For example, such a tailored CVS space can be used to lower computational cost, as well as to avoid state mixing that can make analysis more difficult.27,28 Doing so breaks the energetic argument in CVS, so this tailoring relies on the coupling between orbitals still being small from a spatial perspective. This is clearly not the case if MOs delocalized over different atomic sites are set in separated CVS spaces, as will now be discussed.

Figure 9 shows the x-ray absorption spectrum of ethene, as obtained when using a CVS space consisting of both core MOs, or tailored spaces containing one MO at a time. Furthermore, results obtained when localizing the core orbitals with ECPs are also shown, yielding practically identical results as when using the full CVS space. It is clear that using a CVS space tailored to each core MO at a time introduces an error in both absolute and relative energies and features. For ADC(1), the difference in absolute energy is the smallest, as neither the full nor the tailored calculations include relaxation. By comparison, the relative features are quite different. The closest agreement in relative features is present for ADC(2)-x, which also has the largest absolute energy difference, as well as being the method that has been shown to perform best for XAS.28,71

FIG. 9.

X-ray absorption spectrum of ethene calculated using several orders of CVS-ADC with a full CVS space (blue), ECPs (blue dotted), or by considering the two C 1s core MOs in separate CVS spaces (red). The tailored CVS spectrum is the sum of these two individual contributions and has been shifted in energy to overlap with the first feature of the full CVS calculation.

FIG. 9.

X-ray absorption spectrum of ethene calculated using several orders of CVS-ADC with a full CVS space (blue), ECPs (blue dotted), or by considering the two C 1s core MOs in separate CVS spaces (red). The tailored CVS spectrum is the sum of these two individual contributions and has been shifted in energy to overlap with the first feature of the full CVS calculation.

Close modal

Moving to the more interesting case of non-equivalent atomic sites, the use of full or tailored CVS spaces for cyclopentadiene and furan is illustrated in Fig. 10. Here, the tailored spectra are constructed by considering separately each set of (what can be thought to be) MOs of chemically unique atoms. For furan, the tailored CVS approach works very well, while for cyclopentadiene, this approach clearly does not work. This is because the C1 and C2 sets of chemically nonequivalent atoms in cyclopentadiene have closer C 1s binding energies, with the corresponding MOs partially delocalized over all atomic sites.

FIG. 10.

X-ray absorption spectra of cyclopentadiene (top) and furan (bottom) calculated at the CVS-ADC(2) level of theory. Results obtained using the full CVS space encompassing all carbon atoms (blue) or tailored CVS spaces including the chemically nonequivalent C atoms separately (red). The tailored CVS spectrum of cyclopentadiene has been shifted such that the peak max approximately overlaps with that of the full CVS calculation.

FIG. 10.

X-ray absorption spectra of cyclopentadiene (top) and furan (bottom) calculated at the CVS-ADC(2) level of theory. Results obtained using the full CVS space encompassing all carbon atoms (blue) or tailored CVS spaces including the chemically nonequivalent C atoms separately (red). The tailored CVS spectrum of cyclopentadiene has been shifted such that the peak max approximately overlaps with that of the full CVS calculation.

Close modal

As such, in order to reduce the computational costs by considering separate CVS spaces, delocalization over different sets of symmetry-unique atoms needs to be avoided. This can be ensured by checking the delocalization of the ground state MOs or by using ECPs and considering each different site at a time. While using a full CVS space is the approach most consistent with the original formulation of the core–valence separation approximation, the use of tailored CVS spaces can lower computational cost by reducing matrix sizes, as well as the number of eigenstates needed at a time. The approach thus has benefits, but some care must be observed.

In summary, we have analyzed the influence of core–hole (CH) delocalization on x-ray photoionization, x-ray absorption, and x-ray emission spectra calculated at various levels of theory, including TP-DFT, TDDFT, and the post-HF correlated methods ADC and CC. In cases where an explicit CH is required in the calculation of a highly symmetric molecule with chemically equivalent atoms, namely, for IE/XPS and XAS (TP-DFT), or XES (TDDFT, TP-DFT, ADC, CC), the use of a delocalized core–hole can produce a large discrepancy to experiment and localized core–holes. This is due to a lack of orbital relaxation, which then needs to be accounted for (typically by higher levels of theory). Not including this relaxation leads to a larger discrepancy in energy,1–3 which we here term the delocalization-induced relaxation error (DIRE). This is to be understood as a result of lacking relaxation, or electron correlation, rather than a fundamental error in using delocalized core–holes per se. In the case of IE/XPS and XAS, DIRE manifests mainly in terms of an absolute energy shift. By comparison, XES calculations are sensitive both in terms of absolute energies, as well as for relative peak positions and intensities. The size of DIRE depends on the element (the larger the atomic number, the larger the differences), the number of sites over which the CH is delocalized (the more sites, the larger the DIRE), and the description of exchange and correlation. In the case of DFT, DIRE is directly related to the amount of HF exchange included in the xc-functional, where the largest correlation is found with the exchange interaction between the core orbitals with the same spin as the delocalized core–hole.

As a best practice, for state-specific calculations that require an explicit core–hole, the recommendation is to use a localized core–hole to minimize the relaxation error, especially if the system under investigation has a high degree of symmetry. This can be done by, e.g., using ECPs for all atoms of the same type, save the probed one. This is a practical solution, as the use of delocalized core–holes preserves the symmetry of the molecular system, while localizing the core–hole artificially breaks the symmetries. Moreover, care must be given to calculations that involve tailored CVS spaces. In this case, all nearly degenerate core orbitals must be included in the same CVS space to avoid errors, or ECPs can be used again.

T.F. acknowledges financial support of the Air Force Office of Scientific Research (Grant No. FA8655-20-1-7010). I.E.B. acknowledges support provided by Professor Y. M. Rhee through a grant from the National Research Foundation (NRF) of Korea (Grant No. 2020R1A5A1019141) funded by the Ministry of Science and ICT (MSIT). I.E.B. also acknowledges support from the Swedish Research Council (Grant No. 2017-06419). We thank Patrick Norman for constructive feedback on our manuscript, and we are grateful to Szymon Śmiga and Igor Di Marco for insightful discussions on the self-interaction error. The calculations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC, partially funded by the Swedish Research Council through Grant Agreement No. 2018–05973.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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