Benchmark of GW Methods for Core-Level Binding Energies

Abstract

The GW approximation has recently gained increasing attention as a viable method for the computation of deep core-level binding energies as measured by X-ray photoelectron spectroscopy. We present a comprehensive benchmark study of different GW methodologies (starting point optimized, partial and full eigenvalue-self-consistent, Hedin shift, and renormalized singles) for molecular inner-shell excitations. We demonstrate that all methods yield a unique solution and apply them to the CORE65 benchmark set and ethyl trifluoroacetate. Three GW schemes clearly outperform the other methods for absolute core-level energies with a mean absolute error of 0.3 eV with respect to experiment. These are partial eigenvalue self-consistency, in which the eigenvalues are only updated in the Green’s function, single-shot GW calculations based on an optimized hybrid functional starting point, and a Hedin shift in the Green’s function. While all methods reproduce the experimental relative binding energies well, the eigenvalue self-consistent schemes and the Hedin shift yield with mean absolute errors <0.2 eV the best results.

1. Introduction

X-ray photoelectron spectroscopy (XPS) is a standard characterization tool for materials,¹ liquids,^2,3 and molecules.⁴ In XPS, the core-level binding energies (CLBEs) are measured, which are element-specific but also sensitive to the chemical environment.⁵ However, establishing the link between the measured spectrum and the atomic structure is challenging, in particular for complex materials with heavily convoluted XPS signals.^6,7 Guidance from theory is thus often necessary to interpret XPS spectra.

Theoretical XPS methods can be distinguished into delta and response approaches. In delta-based methods, the CLBE is computed as the total energy difference between the neutral and core-excited system. These calculations can be performed at different levels of theory, for example, with high-level wave function methods, such as delta coupled cluster (ΔCC)^8,9 or with Kohn–Sham density functional theory¹⁰⁻¹² (KS-DFT). The most popular DFT-based approach is the delta self-consistent field (ΔSCF) method,¹³ which has been thoroughly benchmarked.¹⁴⁻²⁰ While high accuracy can be achieved with these approaches, the explicit optimization of a core-ionized wave function leads to conceptual problems, for example, regarding periodicity, constraining spin–orbit coupled states or, in the case of DFT, deteriorating accuracy for larger structures, which was already discussed and demonstrated elsewhere.²¹⁻²⁵

An explicit orbital optimization of core-ionized systems and the related conceptual issues are avoided in response theories, where electron propagators are applied to transform the ground into an excited state. Recently, wave function-based methods, such as linear response and equation-of-motion CC methods²⁶⁻²⁹ and the algebraic diagrammatic construction method,²⁸ were reassessed for absolute CLBEs, yielding partly promising results. Another promising approach in the realm of response methods is the GW approximation³⁰⁻³² to many-body perturbation theory, which is derived from Hedin’s equation³³ by omitting the vertex correction. The GW approximation is considered the “gold standard” for the computation of band structures of materials,^31,34 but it has also been successfully applied to valence excitations of molecules.^31,35−37

Due to its primary application to solids, GW was traditionally implemented in plane wave codes that typically use pseudo-potentials for the deeper states. With the increasing availability of the GW method in localized basis set codes,³⁸⁻⁴⁵ core states moved into focus. CLBE calculations have emerged as a recent trend in GW.^{22,24,44−53} By extension to the Bethe–Salpeter equation (BSE@GW), also K-edge transition energies measured in X-ray absorption spectroscopy can be calculated.⁵⁴ These studies focused primarily on molecules. However, GW is one of the most promising methods for core-level predictions of materials because the scaling with respect to system size is generally smaller than for wave function-based response methods and the method is well established for periodic structures. In addition, GW implementations for localized basis sets are advancing rapidly. Recently, periodic implementations^44,55,56 and low-scaling GW algorithms with complexity emerged in localized basis set formulations.^43,57−60

The application of GW to core states is more difficult than for valence states. We showed that more accurate and computationally more expensive techniques for the frequency integration of the self-energy are required compared to valence excitations.²² Furthermore, we found that the standard single-shot G₀W₀ approach performed on top of DFT calculations with generalized gradient approximations (GGAs) or standard hybrid functionals fails to yield a distinct solution, which is caused by a loss of spectral weight in the quasiparticle (QP) peak.⁴⁹ We demonstrated that eigenvalue self-consistency in G or using a hybrid functional with 45% exact exchange as the starting point for the G₀W₀ calculation restores the QP main excitation. Including also relativistic corrections,⁵⁰ an agreement of 0.3 and 0.2 eV with respect to experiment was reported for absolute and relative CLBEs, respectively.⁴⁹

While G₀W₀ is the computationally least expensive GW flavor, it strongly depends on the density functional approximation (DFA). Tuning the exchange in the hybrid functional to, for example, 45% is conceptually unappealing and introduces undesired small species dependencies, as discussed more in detail in this work. Self-consistency reduces or removes the dependence on the underlying DFT functional but significantly increases the computational cost. The computationally least expensive self-consistent schemes are the so-called eigenvalue self-consistent approaches, where the eigenvalues are iterated in the Green’s function G (evGW₀) or alternatively in G and the screened Coulomb interaction W (evGW).³¹ Higher-level self-consistency schemes, such as fully self-consistent GW(61−63) (scGW) and QP self-consistent GW(64) (qsGW), remove, unlike evGW₀ or evGW, the starting point dependence completely. However, these higher-level self-consistency schemes are much more expensive and not necessarily better because of the inherent underscreening due to the missing vertex correction.^65,66

Recently, we proposed the renormalized singles (RS) Green’s function approach, denoted as G_RSW₀, to reduce the starting point dependence in GW.⁶⁷ The RS concept was developed in the context of the random phase approximation (RPA) for accurate correlation energies^68,69 and termed renormalized single excitation (RSE) correction. Following standard perturbation theory, SEs contribute to the second-order correlation energy. It was shown that their inclusion significantly improves binding energies (BEs).^68,69 The RS Green’s function approach extends the RSE idea from correlation energies to GW QP energies. In the G_RSW₀ scheme, the RS Green’s function is used as a new starting point and the screened Coulomb interaction is calculated with the KS Green’s function. For valence excitations, we found that this renormalization process significantly reduces the starting point dependence and provides improved accuracy over G₀W₀.⁶⁷ The mean absolute errors (MAEs) obtained from the G_RSW₀ approach with different DFAs are smaller than 0.2 eV for predicting ionization potentials of molecules in the GW100 set.⁶⁷ Unlike the self-consistent schemes, the RS Green’s function method hardly increases the computational cost compared to G₀W₀.

Recently, we employed the concept of RS in a multireference DFT approach for strongly correlated systems.⁷⁰ We also used the RS Green’s function in the T-matrix approximation (G_RST_RS).⁷¹ The T-matrix method scales formally as with respect to system size N,^71,72 with reduced scaling possible using effective truncation of the active space.⁷³ In addition to the high computational cost, the performance of G_RST_RS for core levels is not particularly impressive. The error with respect to experiment is 1.5 eV for absolute and 0.3 eV for relative CLBEs.⁷¹ In the present work, we focus thus on RS GW approaches for core levels.

In this work, we benchmark GW approaches, which we consider computationally affordable and suitable for large-scale applications. This includes G₀W₀ with tuned starting points, the eigenvalue self-consistent schemes evGW₀ and evGW, and two new methods that we introduce in this work. One is based on the so-called Hedin shift⁷⁴ and can be understood as approximation of the evGW₀ method. We refer to this scheme as G_ΔHW₀. The other is a different flavor of the RS Green’s function approach, where the screened Coulomb interaction is also computed with the RS Green’s function (G_RSW_RS).

The remainder of this article is organized as follows: we introduce the different GW approaches in Section 2 and give the computational details for our calculations in Section 3. The solution behavior of the different methods is discussed in Section 4.1 by comparing self-energy matrix elements and spectral functions. In Section 4.2, results are presented for the CORE65 benchmark set and in Section 4.3 for the ethyl trifluoroacetate (ETFA) molecule. The computational cost for the different GW flavors is discussed in Section 4.4, and we finally draw conclusions in Section 5.

2. Theory

2.1. Single-Shot G₀W₀ Approach

The most popular GW approach is the single-shot G₀W₀ scheme, where the GW QP energies are obtained as corrections to the KS eigenvalues {ϵ_n⁰}

1

where {ψ_n⁰} are the KS molecular orbitals (MOs) and v^xc is the KS exchange–correlation potential. We use i, j for occupied orbitals, a, b for virtual orbitals, and m, n for general orbitals. We omitted the spin index in all equations for simplicity and use the notation and in the following. We can directly obtain the CLBE of state n from the QP energies because they are related by CLBE_n = −ϵ_n. The self-energy Σ is given by

2

where the noninteracting KS Green’s function is denoted G₀ and the screened Coulomb interaction W₀. η is a positive infinitesimal. The KS Green’s function reads

3

where ϵ_F is the Fermi energy. The screened Coulomb interaction is calculated at the level of the RPA as

4

where ε(r,r′,ω) is the dielectric function and v(r,r′) = 1/|r – r′| the bare Coulomb interaction.

The calculation of the self-energy matrix elements Σ_n is split into a correlation part Σ^c and an exchange part Σ^x, that is, Σ_n = Σ_n^c + Σ_n. The HF-like exchange part Σ_n^x is given by

5

The correlation part Σ^c is computed from W₀^c = W₀ – v and is the part that we plot in Section 4.1 to investigate the GW solution behavior. The correlation part in its fully analytic form is given by

6

where Ω_s⁰ are charge neutral excitations at the RPA level and ρ_s are the corresponding transition densities. The fully analytic form of Σ_n^c directly shows the pole structure of the self-energy and is illustrative to understand the solution behavior of GW. However, the evaluation of Ω_s scales with . In practice, the correlation self-energy is usually evaluated with a reduced scaling by using techniques such as analytical continuation or the contour deformation; see ref (31) for an overview of frequency integration techniques.

The QP energies can be obtained by solving eq 1, which is typically the computationally least expensive approach. In this work, we additionally employ two alternative approaches to obtain further insight into the physics and suitability of the different GW approaches. The first is the graphical solution of eq 1, where we plot the self-energy matrix elements Σ_n^c and determine the QP solution by finding the intersections with the straight line ω – ϵ_n + v_n^xc – Σ_n. The presence of several intersections would indicate that more than one solution exists. The second, computationally even more expensive alternative, is the computation of the spectral function,⁴⁹ which is given by

7

The spectral function is related to the photocurrent, which is the experimental observable in photoemission spectroscopy, as discussed in, for example, refs (31) and (75). In eq 7, we include also the imaginary part of the complex self-energy, which gives us direct access to the spectral weights and satellite spectrum.

The GW formalism is primarily applied to closed-shell systems, but the treatment of open shell systems is also possible to some extent. Open shell systems are often multireference problems, while the KS Green’s function G₀ assumes a nondegenerate ground state characterized by a single Slater determinant. If a particular ground state is a sum of several Slater determinants, the GW approach is not applicable and a multiconfiguration method should be used instead. If it is possible to choose among the multiple ground states of the open shell systems one that evolves into a single Slater determinant, for example, the triplet state of O₂, then our definition of G₀ and thus Dyson’s equation are valid. As shown in ref (76), the QP multiplet splittings of valence excitations are correctly predicted by GW if the preceding spin-polarized KS-DFT calculations provide a sufficient approximation of the particular ground states of the open shell system. While the multiplet splittings of open shell valence levels can be rather complex, the 1s excitations give rise to a simple doublet with a weaker line at higher BE and a stronger one at lower BE.⁴ So far, we only investigated the O₂ molecules and found that GW performed very well for the spitting of the O 1s lines.⁴⁹

2.2. Eigenvalue Self-Consistent GW Schemes

Including self-consistency in Hedin’s GW equations is a widely used strategy to go beyond G₀W₀. scGW^31,63 is conceptually the purest approach but also the most expensive one. To reduce the computational demands, different lower-level self-consistent schemes were developed. The simplest approach is an eigenvalue self-consistent scheme, which comes in two different flavors. The first one is to iterate the eigenvalues only in G and keep W fixed at the W₀ level. This scheme is referred to as the evGW₀ approach. In evGW₀, we start by updating the KS eigenvalues in the Green’s function with the G₀W₀ QP energies, re-evaluate the QP equation (see eq 1), and iterate until self-consistency in G is reached. The Green’s function in the eigenvalue self-consistent scheme reads

8

with ϵ_m^QP = ϵ_m + Δϵ_m, where Δϵ_m is the GW correction, see eq 1. The second flavor is evGW, where the KS eigenvalues are updated not only in G but also in the screened Coulomb interaction W. The eigenvalue self-consistent calculations are computationally significantly more expensive than a G₀W₀ calculation, in particular in combination with the accurate self-energy integration techniques that are required for core levels.²² The computational demands are large because G, the screened Coulomb interaction W (in the case of evGW), and the self-energy have to be built repeatedly. In addition, eq 1 must be solved not only for the states of interest but for all states.

2.3. GW with the Hedin Shift

The cost of an evGW₀ scheme can be drastically reduced by using a global shift ΔH instead of an individual shift Δϵ_m for each state m. This scheme was first introduced by Hedin³³ and is referred to as G_ΔHW₀ in the following. The G_ΔHW₀ approach was discussed several times in the literature,^74,77−79 and the effect of evGW₀ and G_ΔHW₀ on the self-energy structure has been discussed for valence states in ref (31). In the Hedin-shift scheme, the Green’s function transforms into

9

where G₀(ω – ΔH) = G_ΔH(ω). The QP equation with the Hedin-shift scheme then becomes

10

Traditionally, ΔH is determined with respect to the Fermi level of G₀ for metals or the valence band maximum for gapped solid-state systems. For the molecular case, ΔH is evaluated with respect to the highest occupied MO (HOMO) by introducing the self-consistency condition ϵ_HOMO^QP = ϵ_HOMO + ΔH, which is inserted in eq 10 and yields

11

As demonstrated in ref (31), evGW₀ and G_ΔHW₀ lead to a shift of the pole structure of the self-energy matrix elements Σ_n to more negative frequencies. For Σ_HOMO(ω), the shift is similar for evGW₀ and G_ΔHW₀, yielding practically the same ϵ_HOMO^QP.

For core states, we found that the shift ΔH computed as in eq 11 is much smaller than the one from evGW₀. Taking the H₂O molecule as an example, the evGW₀ shift of the self-energy poles with respect to G₀W₀ is in the range of −6 eV for the HOMO, whereas it is in the range of −30 eV for the oxygen 1s state. Generally, we found that the evGW₀ shifts get progressively larger with increasing BE. Therefore, we propose here a modification of the Hedin approach, where we determine an n-specific shift ΔH_n for the (core) state n of interest. ΔH_n is determined as

12

and the QP equation transforms then to

13

ΔH_n is still a global shift, which is, however, specific for the respective (core) state of interest. For example, to obtain ϵ_C1s^QP for the CO molecule, we solve eq 13 with ΔH_C1s, whereas we solve it with ΔH_O1s for ϵ_O1s. In the case of HCOOH, we determine ΔH_n for each O separately.

The flowchart of a G_ΔHW₀ calculation is shown in Figure 1. In a G_ΔHW₀ calculation of state n, we calculate ΔH_n once, insert it in eq 13, and iterate the latter as in a regular G₀W₀ calculation. The shift ΔH_n is kept constant during the iteration of the QP equation. Compared to a G₀W₀ calculation, the computation of ΔH is the only computational overhead that we introduce. The computational cost of a G_ΔHW₀ calculation is thus practically the same as for a G₀W₀ calculation.

Figure 1.

Flowchart for the G_ΔHW₀ scheme starting from a KS-DFT calculation. The additional terms with respect to a G₀W₀ calculation are highlighted in red.

We point out that the Hedin shift is not an arbitrary scissor shift of the KS eigenvalues {ϵ_n⁰}, where the energies of the occupied and virtual states are shifted down and up, respectively. (i) The eigenvalues {ϵ_n} are still used to construct W₀. The shift is only applied in G, and its sign is, unlike in a scissor shift approach, independent on the occupation of state m in eq 9. (ii) More importantly, the expression for ΔH_n is derived by enforcing self-consistency for the energy of state n. The motivation for the Hedin shift and its derivation was comprehensively discussed for the valence case in the literature^74,77,79 and is therein also referred to as “alignment of the chemical potential” or “adjusting the energy scale of G₀”.

The Hedin shift can be understood as the simplest step toward self-consistency and viewed as one-diagonal element correction in the context of the evGW₀ method. In evGW₀, we replace the KS energies ϵ_m⁰ in eq 3 by ϵ_m = ϵ_m⁰ + Δϵ_m, see eq 8, where m is an index that runs over all occupied and virtual states and Δϵ_m is the GW correction. In the G_ΔHW₀ approach that we propose here, we aim to approximate the element Δϵ_m, which has the largest contribution, by ΔH, and neglect the rest. We found that the element Δϵ_m=n, where n is the state of interest, is (by far) the most relevant one. This is the motivation for determining a state-specific shift ΔH_n and using eq 12 instead of eq 11 for our core-level calculations.

2.4. RSEs in RPA

The RS Green’s function approach is based on the same idea as the RSE corrections in RPA. Since we consider the introduction of the RS concept more illustrative for total than QP energies, we will briefly summarize the key equations derived by Ren et al.^68,69,80 in the context of Rayleigh–Schrödinger perturbation theory (RSPT), before proceeding with the RS Green’s function approaches in Section 2.5.

In RSPT, the interacting N-electron Hamiltonian Ĥ is partitioned into a noninteracting mean-field Hamiltonian and an interacting perturbation .

14

15

where is the single-particle Hamiltonian of the mean-field reference. The Hamiltonian includes the kinetic term, the external potential v_ext, and the mean-field potential v^Hxc. The latter can be the single-particle potential from HF or KS-DFT and contains the Hartree and exchange–correlation terms. Following standard perturbation theory, single excitations (SEs) contribute to the second-order correlation energy and are given by^68,69

16

17

where |Ψ₀⟩ is the Slater determinant for the ground-state configuration and for the singly excited configuration. The orbitals ψ_i/a⁰ and corresponding orbital energies ϵ_i/a are the ones of the operator. The derivation of eq 17 from eq 16 is given in detail in the Supporting Information of ref (68). As evident from eq 17, the single correction vanishes if v^Hxc is the HF mean-field potential, which is a consequence of Brillouin’s theorem.⁸¹

The energy E_c^SE is only the second-order correction to the correlation energy, as shown in Figure 2. The infinite summation of the higher-order diagrams yields the RSE correction. The derivation of the RSE correlation energy is given in detail in ref (69). To summarize briefly the procedure, the Fock matrix is evaluated with the KS orbitals. Subsequently, the occupied and unoccupied blocks of this matrix are diagonalized separately (subspace diagonalization), which yields a new set of (RS) eigenvalues and orbitals. Replacing ψ_i/a and ϵ_i/a in eq 17 with the RS eigenvalues and orbitals yields the RSE correlation energy.

Figure 2.

Goldstone diagrammatic for the RSE contributions. Dashed lines, which end with a cross, denote the matrix elements ⟨ψ_p|v^HF – v^Hxc|ψ_q⟩.

2.5. RS Green′s Function GW Approaches

In analogy to the RPA RSE correction, the RS Green’s function G_RS is designed as an effective noninteracting reference system that includes all the single contributions. The RS Green’s function is defined as

18

where is the projection into the occupied orbital space and Q = I – P is the projection into the virtual orbital space. Σ_HF[G₀] refers to a HF-like self-energy constructed with G₀, which is usually the KS Green’s function. Σ_HF is the sum of the Hartree self-energy Σ^H and the exchange self-energy Σ^x, that is, Σ_HF = Σ^H + Σ^x, where and . Note that both are built with the mean-field orbitals ψ_n⁰ provided by, for example, KS-DFT. v^Hxc is the single-particle Hartree-exchange–correlation potential defined in eq 14. If the potential v^Hxc is the one from HF and if G₀ is the HF Green’s function, then the second and third terms on the right-hand side of eq 18 vanish and G_RS corresponds to the HF Green’s function, which is again a consequence of Brillouin’s theorem. G_RS includes the single contributions, which are one source of the starting point dependence. As we showed previously,⁶⁷ the dependence on the DFA is therefore reduced in the G_RSW₀ scheme. As for G₀W₀, the primary use case of the RS approach are closed-shell systems for the same reasons discussed in Section 2.1 and due to the restriction of Brillouin’s theorem to the closed-shell case. While not part of this work, we expect nevertheless that simple spin splittings as observed for 1s excitations can be captured by the RS approach.

The RS Green’s function is given as the solution of the two projected equations in the occupied orbital subspace⁶⁷

19

and the virtual orbital subspace

20

In practice, G_RS is obtained by a similar subspace diagonalization procedure as used for the RSE total energy corrections. The KS density matrix is used to construct the HF Hamiltonian , which defines the RS Hamiltonian . The equations for the occupied

21

and virtual subspace

22

are diagonalized separately.⁶⁷ The subspace diagonalization yields the RS eigenvalues ϵ_n^RS and corresponding eigenvectors ψ_n and is performed only once. The RS Hamiltonian can be reduced to the HF_diag method³⁸ if only diagonal elements in eqs 21 and 22 are used. It has been shown that the G₀W₀(HF_diag) method predicts accurate valence QP energies of molecular systems.³⁸

The RS Green’s function is computed with the RS eigenvalues and orbitals

23

and is diagonal in the occupied subspace and the virtual subspace.

In the G_RSW₀ approach,⁶⁷ the RS Green’s function is used as a new starting point and the screened interaction is calculated with the KS Green’s function. The correlation part of the G_RSW₀ self-energy is⁶⁷

24

where ρ_s⁰ and Ω_s are the transition densities and the RPA excitation energies calculated with the KS Green’s function. Then, the QP equation for G_RSW₀ is

25

Note that eqs 24 and 25 are slightly different from the original QP equation for G_RSW₀,⁶⁷ where we used the RS eigenvectors ψ_n^RS. Here, we use for simplicity the KS instead of the RS orbitals because we found that the difference induced upon using RS orbitals is marginal.⁶⁷ Since we use the KS orbitals, the exchange part of the G_RSW₀ self-energy is the same as in G₀W₀, see eq 5.

In this work, we introduce a new approach that uses the RS Green’s function as a new starting point and also calculates the screened Coulomb interaction with the RS Green’s function, denoted as G_RSW_RS. This means that W_RS is obtained by inserting the RS Green’s function into the RPA equation. Similar to G_RSW₀, the exchange part of the G_RSW_RS self-energy is also the same as for G₀W₀, but the correlation part of the G_RSW_RS self-energy becomes

26

where ρ_s^RS and Ω_s are the transition densities and the RPA excitation energies calculated with the RS Green’s function. Then, the QP equation for the G_RSW_RS approach follows as

27

The construction of the RS Green’s function scales only as with respect to system size N. The overall scaling of G_RSW_RS depends on the frequency integration technique that is used, which will be discussed more in detail in Section 4.4.

3. Computational Details

Core-level calculations were performed at the G₀W₀, evGW₀, evGW, G_ΔHW₀, and G_RSW_RS levels for the CORE65 benchmark set,⁴⁹ which contains 65 1s excitation energies of 32 small molecules (30 × C 1s, 21 × O 1s, 11 × N 1s, and 3 × F 1s). Geometries and experimental reference values were taken from ref (49). Additionally, we studied the ETFA molecule. The structure of the latter was obtained upon request from the authors of ref (18) and is available in the Supporting Information.

The G₀W₀, evGW₀, evGW, and G_ΔHW₀ calculations were carried out with the FHI-aims program package,^82,83 which is based on numeric atom-centered orbitals (NAOs). The G₀W₀ and evGW₀ data were extracted from our previous work,⁴⁹ while the evGW and G_ΔHW₀ data were generated for this benchmark study. In the FHI-aims calculations, the contour deformation technique^22,31,84 is used to evaluate the self-energy using a modified Gauss–Legendre grid³⁹ with 200 grid points for the imaginary frequency integral part. The G_RSW_RS calculations were performed with the QM4D program.⁸⁵ In QM4D, the GW self-energy integral is calculated fully analytically, see eq 6. In FHI-aims and also in QM4D, the QP equation is always solved iteratively.

For evGW₀, evGW, and G_ΔHW₀, we used the Perdew–Burke–Ernzerhof (PBE)⁸⁶ functional for the underlying DFT calculation, while the G₀W₀ calculations employ the PBEh(α) hybrid functional⁸⁷ with 45% exact exchange (α = 0.45). The G_RSW_RS calculations were performed with PBE and three different hybrid functionals, namely, PBE0,^88,89 PBEh(α = 0.45), and B3LYP.^90,91 Note that PBE0 corresponds to PBEh(α = 0.25).

All GW results are extrapolated to the complete basis set limit using the Dunning basis set family cc-pVnZ.^92,93 For a discussion of the basis set choice, we refer the reader to ref (52) and theSupporting Information of ref (50). Following ref (49), the extrapolation is performed by a linear regression with respect to the inverse of the total number of basis functions. A four-point extrapolation with n = 3–6 is performed for G₀W₀, evGW₀, evGW, and G_ΔHW₀. For G_RSW_RS, we use only two points (n = 3, 4) due to computational limitations. We verified that this two-point extrapolation deviates only by 0.1 eV from the four-point scheme on average. The cc-pVnZ family consists of contracted Gaussian-type orbitals (GTOs), which can be considered as a special case of an NAO and are treated numerically in FHI-aims. Note that the cc-pVnZ basis sets are treated as spherical GTOs in FHI-aims, whereas in QM4D, they are processed as pure Cartesian GTOs. Both codes use the resolution-of-the-identity (RI) approach with the Coulomb metric (RI-V).⁹⁴ In FHI-aims, the RI auxiliary basis sets are generated on-the-fly as described in ref (39). For the QM4D calculations, the corresponding RI basis sets for cc-pVTZ and cc-pVQZ from ref (95) were used.

Relativistic effects were included for all calculations as post-processing step following the approach in refs (49) and (50); that is, we performed a nonrelativistic GW calculation on top of a nonrelativistic KS-DFT calculation and added the corrective term derived in ref (50) to the GW QP energies. The magnitude of the corrections increases with the atomic number and ranges from 0.12 eV for C 1s to 0.71 eV for F 1s. The relativistic corrections were derived for a free neutral atom at the PBE level and were obtained by evaluating the difference between the 1s eigenvalues from the radial KS and the 4-component Dirac-KS equation.

4. Results and Discussion

4.1. Solution Behavior

In our previous work,⁴⁹ we showed that standard G₀W₀ calculations starting from a GGA functional, which are routinely applied to valence states, lead to an erroneous multi-solution behavior for deep core states. It is thus important to confirm that the respective GW flavors yield indeed a unique solution. Only looking at the QP energies obtained by iterating eq 1 is typically not enough to verify the latter. Detailed insight into the solution behavior of GW-based methods can be obtained by (i) plotting the real part of the correlation self-energy Σ^c and (ii) plotting the spectral function A(ω) as defined in eq 7. In Figure 3, we investigate A(ω) and the diagonal matrix elements Σ_n^c(ω) for the 1s oxygen orbital of a single water molecule. Results are shown for G₀W₀ and G_RSW_RS with different starting points [PBE, PBE0, B3LYP, PBEh(α = 0.45)] as well as partial self-consistent schemes, namely, evGW₀, evGW, and G_ΔHW₀, using PBE for the underlying DFT calculation.

Figure 3.

O 1s excitation for a single water molecule from G₀W₀, evGW₀, G_ΔHW₀, evGW, and G_RSW_RS computed at the cc-pVQZ level. (a) Spectral function A(ω) (eq 7) from G₀W₀ using starting points with no exact exchange or a low amount. (b,c) Real part of the correlation self-energy Σ^c(ω) in evGW₀@PBE, G_ΔHW₀, and evGW@PBE. Diagonal matrix elements for the oxygen 1s orbital. The intersection with the red line is the graphical solution of eq 1. The vertical gray-dashed line indicates the QP solution of this graphical solution, and Δ indicates the shift with respect to G₀W₀@PBE (gray). (d) Spectral functions in evGW₀@PBE, G_ΔHW₀@PBE, evGW@PBE, and G₀W₀@PBEh(α = 0.45). (e) Self-energy matrix elements ReΣ_n^c(ω) in G_RSW_RS@PBE (green) compared to G₀W₀@PBE (gray). (f) Spectral functions A(ω) in G_RSW_RS with different starting points.

We start our discussion with the G₀W₀ spectral functions and self-energy elements displayed in Figure 3a,b,d, where we reproduced for convenience the G₀W₀@PBE, G₀W₀@PBE0, and G₀W₀@PBEh(α = 0.45) results, which are also presented in ref (49). Figure 3b shows the self-energy from a G₀W₀@PBE calculation (gray line), which exhibits many poles. The poles are broadened by the η-term in eq 3 and thus appear as spikes in the self-energy. For G₀W₀@PBE, we find that the poles are located in the frequency region where the QP solution is expected (around −540 eV). As already outlined in Section 2.1, we can obtain the graphical solution of eq 1 by finding the intersections with the straight line ω – ε_n⁰ + v_n – Σ_n^x. For G₀W₀, we observe many intersections, which are all valid solutions of the QP equation. The corresponding spectral function in Figure 3a shows many peaks with equal spectral weight but no clear main peak that could be assigned to the QP excitation in contrast to the experiment, where a sharp peak at 539.7 eV⁴ is observed. A main peak starts to emerge for hybrid functional starting points, such as PBE0 (α = 0.25) and B3LYP (α = 0.20). However, G₀W₀@PBE0 and G₀W₀@B3LYP still yield an unphysical second peak, which carries a large fraction of the spectral weight.

As already discussed in ref (49), the reason for this unphysical behavior is the overlap of the satellite spectrum with the QP peak. Satellites are due to multielectron excitations that accompany the photoemission process, for example, shake-up satellites, which are produced when the core photoelectron scatters a valence shell electron to a higher unoccupied energy level.^96,97 These peaks appear as series of smaller peaks at higher BEs than the QP energy. For molecules, the spectral weight of these peaks is orders of magnitudes smaller than for the main excitation.⁹⁷ Satellites occur in frequency regions where the real part of the self-energy has poles. As demonstrated in, for example, ref (31), the imaginary part of the self-energy exhibits complementary peaks at these frequencies (Kramers–Kronig relation). According to eq 7, large imaginary parts lead to peaks with small spectral weight, that is, peaks with a satellite character.

The occurrence of pole features around the QP excitations for deep core states can best be understood by analyzing the denominator of the fully analytic expression of the self-energy given in eq 6. Σ_n^c(ω) has poles on the real axis for η → 0 at ϵ_i – Ω_s (occupied states i) and ϵ_a⁰ + Ω_s (virtual states a). For occupied states, the eigenvalues ϵ_i are too large (too positive), and the charge neutral excitations Ω_s are underestimated at the PBE level. As a result, the poles ϵ_i⁰ – Ω_s are at too positive frequencies and the satellite thus too close to the QP peak. For virtual states, the same reasoning holds for the poles ϵ_a + Ω_s but with reversed sign, that is, the poles are at too small frequencies. The separation between satellites and QP peak is also too small for valence excitations. However, the problem gets progressively worse further away from the Fermi level since the absolute differences between PBE eigenvalues ϵ_i⁰ and the QP excitation increase. We demonstrated this for semicore states,⁵⁸ for which a distinct QP peak is still obtained. However, for deep core states, the separation becomes finally so small that the satellites merge with the QP peak.

The proper separation between the QP excitation and satellites can be restored by using an evGW₀ scheme. The evGW₀@PBE self-energy is shown in Figure 3b (reproduced from ref (49)). Iterating the eigenvalues in G shifts the on-set of the pole structure too more negative frequencies. The overall pole structure is very similar to G₀W₀@PBE but shifted by a constant value of Δ = −28.7 eV. The G_ΔHW₀@PBE self-energy displayed in Figure 3b is almost identical to evGW₀@PBE. The shift of the pole structure compared to G₀W₀@PBE is with Δ = −28.8 eV only slightly larger than for evGW₀. The rigid Δ shift of the pole features can be understood as follows: in evGW₀ and G_ΔHW₀, the KS eigenvalue ϵ_m⁰ is replaced with ϵ_m + Δϵ_m and ϵ_m⁰ + ΔH_1s, respectively, where Δϵ_m is the self-consistent GW correction for state m and ΔH_1s its non-self-consistent approximation for the O 1s state (see eq 12). The poles are consequently located at ϵ_1s + Δϵ_1s – Ω_s and ϵ_1s⁰ + ΔH_1s – Ω_s. Since both corrections, Δϵ_1s and ΔH_1s, are negative for PBE starting points, the poles, that is, satellites, move to more negative frequencies and are properly separated from the main excitation. The spectral function now exhibits a distinct QP peak as evidenced by Figure 3d.

While the main effect of evGW₀ and G_ΔHW₀@PBE for deep core levels is the correction of the spurious transfer of spectral weight to the satellites, it affects also the location of the QP peak. This has been graphically demonstrated for molecular valence states in Figure 25 of ref (31): by shifting the pole structure to more negative frequencies, the slope of ReΣ_n^c will be flatter in the region where the QP excitation is expected. As a consequence, the intersection with the straight line ω – ϵ_n + v_n^xc – Σ_n (= QP solution) will be at more negative frequencies compared to G₀W₀@PBE.

As discussed in detail previously,⁴⁹ the effect of eigenvalue self-consistency can be mimicked in a G₀W₀ calculation by using a hybrid functional with a high amount of exact exchange α. We showed that increasing α progressively shifts the pole features to more negative frequencies. For α = 0.45, the evGW₀@PBE self-energy is approximately reproduced, and the spectral function shows a distinct peak as displayed in Figure 3d. We note here again that values of α < 0.3 do not yield a clear main peak⁴⁹ and thus no unique solution, which is also demonstrated in Figure 3a.

The evGW approach and the G_RSW_RS schemes lead to a significantly stronger shift of the pole features than evGW₀@PBE or G_ΔHW₀@PBE, as shown in Figure 3c,e. The spectral functions displayed in Figure 3d,f confirm that evGW and G_RSW_RS yield a distinct peak in the spectrum. The RS eigenvalues of the occupied orbitals are more negative, and the ones of the virtual orbitals are more positive than the KS eigenvalues. In addition, RPA evaluated with RS fundamental gaps provides larger excitation energies Ω_s. In G_RSW_RS, the poles at ϵ_a + Ω_s are shifted in the positive direction and the poles at ϵ_i – Ω_s are shifted in the negative direction. Therefore, satellites from these poles are separated from the main peak. For G_RSW_RS, a unique solution is obtained for all starting points. As shown in Figure S1 (see Supporting Information), the G_RSW₀ approach suffers from a multi-solution behavior in the deep core region and cannot be applied for core-level calculations.

4.2. CORE65 Benchmark

In the following, we discuss the CORE65 results for the GW schemes for which a physical solution behavior was confirmed in Section 4.1, namely, evGW₀@PBE, evGW@PBE, G₀W₀@PBEh(α = 0.45), G_ΔHW₀@PBE, and G_RSW_RS with four different starting points [PBE, PBE0, B3LYP, and PBEh(α = 0.45)]. The distribution of the errors with respect to experiment is shown in Figures 4 and 5 for the absolute CLBEs and the relative CLBEs, respectively. The corresponding MAE and the mean errors (MEs) are given in Table 1. The error of excitation i is defined as error_i = CLBE_i^theory – CLBE_i.

Figure 4.

Distribution of errors with respect to the experiment for absolute 1s BEs of the CORE65 benchmark set, where the error is defined as error_i = CLBE_i^theory – CLBE_i. The histogram is stacked. Eight GW approaches are compared: (a) evGW₀@PBE, (b) G₀W₀@PBEh(α = 0.45), (c) evGW@PBE, (d) G_ΔHW₀@PBE, (e) G_RSW_RS@PBE, (f) G_RSW_RS@PBE0, (g) G_RSW_RS@B3LYP, and (h) G_RSW_RS@PBEh(α = 0.45).

Figure 5.

Distribution of errors with respect to the experiment for relative 1s BEs of the CORE65 benchmark set. The histogram is stacked. CH₄, NH₃, H₂O, and CH₃F have been used as reference molecules for C 1s, N 1s, O 1s, and F 1s, respectively. Eight GW approaches are compared: (a) evGW₀@PBE, (b) G₀W₀@PBEh(α = 0.45), (c) evGW@PBE, (d) G_ΔHW₀@PBE, (e) G_RSW_RS@PBE, (f) G_RSW_RS@PBE0, (g) G_RSW_RS@B3LYP, and (h) G_RSW_RS@PBEh(α = 0.45).

Table 1. MAE and ME in eV with Respect to Experiment for Absolute and Relative CLBEs of the CORE65 Benchmark Set^a.

	evGW0@		G0W0@		evGW@		GΔHW0@		GRSWRS@
	PBE		PBEh		PBE		PBE		PBE		PBE0		B3LYP		PBEh
core level	MAE	ME	MAE	ME	MAE	ME	MAE	ME	MAE	ME	MAE	ME	MAE	ME	MAE	ME
absolute CLBEs
all	0.30	–0.29	0.33	–0.08	1.53	1.53	0.25	–0.20	4.88	4.88	5.32	5.32	4.94	4.94	5.72	5.72
C 1s	0.27	–0.27	0.24	0.19	1.37	1.37	0.20	–0.13	3.97	3.97	4.50	4.50	4.24	4.24	5.03	5.03
N 1s	0.30	–0.30	0.16	–0.01	1.58	1.58	0.14	–0.13	5.13	5.13	5.59	5.59	5.12	5.12	6.00	6.00
O 1s	0.32	–0.28	0.48	–0.40	1.70	1.70	0.35	–0.31	5.91	5.91	6.24	6.24	5.71	5.71	7.48	7.48
F 1s	0.44	–0.44	0.83	–0.83	1.65	1.65	0.54	–0.54	6.56	6.56	6.75	6.75	6.32	6.32	6.88	6.88
Relative CLBEs
all	0.18	0.02	0.26	0.23	0.18	–0.03	0.19	–0.02	0.40	0.39	0.43	0.43	0.37	0.36	0.48	0.46
C 1s	0.18	–0.05	0.29	0.25	0.19	–0.01	0.20	0.07	0.36	0.36	0.33	0.33	0.30	0.30	0.41	0.41
N 1s	0.14	0.14	0.23	0.21	0.14	–0.13	0.16	–0.15	0.29	0.29	0.39	0.39	0.30	0.30	0.40	0.40
O 1s	0.22	0.08	0.25	0.24	0.18	–0.03	0.20	–0.06	0.56	0.56	0.66	0.66	0.56	0.56	0.68	0.65
F 1s	0.05	–0.05	0.11	0.10	0.11	0.02	0.16	–0.16	0.16	–0.10	0.16	0.00	0.20	0.02	0.22	0.00

^aThe error for excitation i is defined as error_i = CLBE_i^theory – CLBE_i. The relative CLBEs are the shifts with respect to a reference molecule, ΔCLBE = CLBE – CLBE_{ref_mol}. CH₄, NH₃, H₂O, and CH₃F have been used as reference molecules for C 1s, N 1s, O 1s, and F 1s, respectively.

Starting the discussion with the absolute CLBEs, we find that evGW₀@PBE, G₀W₀@PBEh(α = 0.45), and G_ΔHW₀@PBE yield the best results with error distributions close to zero and MAEs of ≈0.3 eV. The smallest overall MAE of 0.25 eV is obtained with G_ΔHW₀@PBE. Figure 4a,d shows that the errors from evGW₀@PBE and G_ΔHW₀@PBE are tightly distributed but mostly negative; that is, the computed CLBEs are slightly underestimated. Generally, we find that the G_ΔHW₀@PBE scheme reproduces the evGW₀@PBE results almost perfectly. The slight underestimation of the absolute CLBEs by evGW₀@PBE and G_ΔHW₀@PBE might be due to insufficiencies in the cc-pVnZ basis sets, which are not captured by the extrapolation procedure. A very recent study⁵² with G₀W₀@PBEh showed that increasing the number of core functions by, for example, uncontracting the cc-pVnZ basis sets yields larger absolute CLBEs. The reported increase is in the range of 0.25 to maximal 0.5 eV, indicating that the CLBEs from evGW₀ and G_ΔHW₀@PBE might be even closer to experiment with core-rich basis sets.

The error distribution of G₀W₀@PBEh(α = 0.45), which is displayed in Figure 4b, is centered around zero, yielding also the smallest overall ME, see Table 1. Compared to evGW₀@PBE and G_ΔHW₀@PBE, the spread of the G₀W₀@PBEh errors is larger and a clustering by species can be observed. The C 1s BEs are overestimated, while N 1s, O 1s, and F 1s are increasingly underestimated. This is due to the species dependence of the α parameter, which we discussed in refs (49) and (50). As we showed in ref (50), including relativistic effects reduces the species dependence of α but does not remove it completely. The optimal α value, α_opt, increases from 0.44 (C 1s) to 0.49 (F 1s), after including relativistic corrections. For α < α_opt, the CLBEs are too small and for α > α_opt too large. As a result, the C 1s BEs are overestimated for α = 0.45, and O 1s and F 1s BEs are underestimated.

evGW@PBE systematically overestimates the absolute CLBEs by 1–2 eV since iterating also the eigenvalues in W effectively leads to an underscreening. This underscreening is also expected for the higher-level self-consistency schemes, such as scGW⁶³ and qsGW,⁶⁴ and is due to the missing vertex correction. The performance of evGW@PBE for deep core levels seems to be comparable to qsGW. An exploratory study by van Setten et al.⁴⁷ reported that qsGW overestimates the absolute 1s BEs of small molecules by 2 eV. This indicates that qsGW suffers from similar underscreening effects and that the orbitals inserted in the GW scheme have a minor effect on the core-level QP energies. The success of evGW₀@PBE on the contrary is based on a fortuitous error cancellation effect, which can be explained as follows: at the PBE level, the fundamental gap is underestimated and inserting the PBE eigenvalues in W consequently yields an overscreened potential. However, the overscreening in W compensates the underscreening introduced by the missing vertex corrections. Our observation for deep core levels agrees with previous work on valence states. Underscreening effects have been previously discussed for scGW and qsGW for the GW100 benchmark set.⁶⁵ Comparing evGW₀@PBE and evGW@PBE for valence excitations, it was found that evGW₀ improves upon G₀W₀,^98,99 while evGW yields too large band gaps⁹⁸ and overly stretched spectra.¹⁰⁰ We acknowledge that the performance difference between evGW₀@PBE and evGW@PBE is less pronounced for valence states (mostly <0.5 eV) and might be partly superimposed by a system dependence and numerical effects, for example, basis set convergence, which are in a similar range. For CLBEs on the contrary, we operate on energy scales which are an order of magnitude larger leading also to errors/performance differences which are an order of magnitude larger.

The error distributions for the absolute CLBEs from G_RSW_RS are shown in Figure 4e–h for four different starting points. G_RSW_RS overestimates the absolute CLBEs by 3–8 eV with an MAE between 5 and 6 eV. The reason for the large overestimation is that the RS fundamental gap is too large, which then leads, similarly as in G₀W₀@HF,¹⁰⁰ to an underscreening in W. One way to reduce the underscreening is to include corrections for the electron correlation in the RS Hamiltonian, which is dominated by exchange interactions. An alternative strategy is to compensate the underscreening by including vertex corrections, for example, to use the T-matrix formalism,^30,71,72 where the two-point screened interaction W is replaced with a four-point effective interaction T. However, methods such as the T-matrix are computationally much more expensive than GW due to their higher complexity. We recently applied the G_RST_RS scheme to the CORE65 benchmark set.⁷¹ Comparing G_RSW_RS and G_RST_RS, the overestimation is indeed reduced by G_RST_RS, which yields an ME of ≈1.5 eV. However, the errors for the absolute CLBEs are still an order of magnitude larger than for computationally cheaper schemes such as evGW₀ or G_ΔHW₀, which rely on a very fortunate error cancellation effect that leads to a balanced screening.

Furthermore, we find that the overestimation with G_RSW_RS increases with the atomic number, that is, from the C 1s to the F 1s excitations. This species dependence is inherited from the KS-DFT calculation. As shown in Table S8 in the Supporting Information, the deviations of the CLBEs obtained from KS-DFT eigenvalues generally increase from C 1s to F 1s for all DFAs. We expect that, for example, adding correlation contributions to the RS Hamiltonian would also reduce this undesired species dependence.

The motivation of the RS approach is the reduction of the starting point dependence. As evident from Figure 4e–h, the large overestimation is observed for all DFAs. Based on the MAEs and MEs in Table 1, it can be shown that the starting point dependence is on average <1 eV for α = 0.0–0.45. As shown in Figure S2 (Supporting Information) for a single water molecule, evGW₀ seems to reduce the starting point dependence less. For the same α range, the CLBE changes by ≈2 eV. The direct comparison with G₀W₀ is difficult because a unique solution is only obtained for α > 0.3. However, we can study the change of the CLBEs for α = 0.3–1.0, see Figure S2 (Supporting Information), which shows that the starting point dependence is 10 eV with G₀W₀ compared to 2.8 eV with evGW₀.

Moving now to the relative CLBEs, we observe that evGW₀@PBE, G₀W₀@PBEh, evGW@PBE, and G_ΔHW₀@PBE yield MAEs of ≈0.2–0.3 eV, and G_RSW_RS MAEs of 0.4–0.5 eV, see Table 1. The errors of the relative CLBEs are centered and tightly distributed around zero for evGW₀@PBE, evGW@PBE, and G_ΔHW₀@PBE. The perturbative schemes G₀W₀@PBEh and G_RSW_RS slightly overestimate the relative CLBEs. The latter is evident from the positive MEs and the error distributions in Figure 4b,e–h, which are not centered at zero but exhibit a small offset toward positive values. The RS results show a larger spread compared to G₀W₀@PBEh and the partial self-consistent schemes. Furthermore, outliers with errors >1 eV are observed for G₀W₀@PBEh and in particular for G_RSW_RS. The largest outliers are primarily O 1s excitations, which originate from the underlying DFT calculation, as evident from Table S9 (see Supporting Information), which shows the MAEs of the relative CLBEs from the KS-DFT eigenvalues. For all four functionals (PBE, PBE0, B3LYP, and PBEh), we obtained the largest MAE at the KS-DFT level for the O 1s excitations. These errors are inherited in the one-shot G₀W₀ and G_RSW_RS approaches because of their perturbative nature.

The chemical shifts between CLBEs of the same atomic type can be smaller than 0.5 eV for second row elements⁴ and even as small as 0.1 eV for C 1s.¹⁰¹ Therefore, the errors for absolute CLBEs from evGW and G_RSW_RS are too large to align or resolve experimental XPS spectra, for which reference data are not available. The most promising methods are evGW₀@PBE, G₀W₀@PBEh, and G_ΔHW₀@PBE. With MAEs between 0.2 and 0.3 eV for absolute and relative CLBEs, the accuracy is well within the chemical resolution required to interpret most XPS data. The disadvantage of the G₀W₀@PBEh(α) scheme is the need for tuning the α parameter. In addition, the species dependence of α_opt for C–F cannot be completely removed and is expected to increase for heavier elements. For example, we found α_opt = 0.61 for sulfur 1s excitations.⁵⁴ Conversely, the accuracy of evGW₀@PBE and G_ΔHW₀@PBE is species independent. In addition, the already very good agreement of evGW₀@PBE and G_ΔHW₀@PBE with experiment might further improve with core-rich basis sets, as already mentioned before.

4.3. ETFA Molecule

We further examine the performance of the eight GW approaches, which we applied to the CORE65 benchmark set in Section 4.2, for C 1s excitations of the ETFA molecule, which is also referred to as the “ESCA molecule” in the literature.¹⁰² The ETFA molecule was synthesized to demonstrate the potential of XPS for chemical analysis in the late 1960s. It contains four carbon atoms in various chemical environments, see the inset of Figure 6a. The ETFA molecule presents a challenge because of the extreme variations of the chemical shifts, which range up to 8.0 eV and decrease from the CF₃ to the CH₃ end. The four C 1s signals are separated by several electronvolts due to the widely different electronegativities of the substituents on the carbon atoms. ETFA is thus an important reference system and was very recently used to benchmark the performance of different functionals in ΔSCF calculations^18,104 and GW approaches.⁴⁵

Figure 6.

Comparison of the C 1s XPS spectrum of ETFA (a) for absolute CLBEs and (b) relative CLBEs obtained from evGW₀@PBE, G₀W₀@PBEh(α = 0.45), evGW@PBE, G_ΔHW₀@PBE, G_RSW_RS@PBE, G_RSW_RS@PBE0, G_RSW_RS@B3LYP, and G_RSW_RS@PBEh(α = 0.45). Dashed lines indicate the experimental ref (102).

In equilibrium, ETFA exists in two dominating conformations (anti-anti and anti-gauche); that is, each peak in the spectrum is a superposition of the signals from both conformers.¹⁰² However, including the different conformations is primarily important when resolving the vibrational profiles of the peaks, which is not the scope of our benchmarking effort. The experimental conformational shifts are <0.1 eV.¹⁰² To ensure direct comparability with the computational data from ref (18), we include thus only the anti-anti conformer.

The first high-quality experimental spectrum of the free ETFA molecule in gas phase was reported in 1973,¹⁰³ while new results were published by Travnikova et al. in 2012,¹⁰² see Table 2. Both results were referenced in the most recent studies.^18,45 The newer results have a higher resolution and are vibrationally resolved. More relevant for this work is that the chemical shifts of the more “descreened” carbon atoms are significantly smaller than in the older spectrum. The difference between the experimental spectra was attributed to missing correction techniques in early multi-channel plate detectors. We follow here the reasoning of ref (45), pointing out that coupled-cluster results⁸ are significantly closer to the new experimental data (in particular the chemical shifts). We use thus the data by Travnikova et al. as the experimental reference.

Table 2. Absolute C 1s CLBEs for the ETFA Molecule^a.

	C1	C2	C3	C4	ME	MAE
experiment103	299.45	296.01	293.07	291.20
experiment102	298.93	295.80	293.19	291.47
evGW0@PBE	–0.70	–0.54	–0.19	–0.12	–0.36	0.36
G0W0@PBEh	0.56	0.54	0.30	0.16	0.39	0.39
GΔHW0@PBE	–0.53	–0.44	–0.10	–0.02	–0.27	0.27
evGW@PBE	0.93	1.01	1.35	1.40	1.17	1.17
GRSWRS@PBE	3.56	3.70	4.13	4.14	3.88	3.88
GRSWRS@PBE0	4.41	4.53	4.65	4.51	4.52	4.52
GRSWRS@B3LYP	4.20	4.29	4.33	4.23	4.26	4.26
GRSWRS@PBEh	5.02	5.13	5.02	4.82	5.00	5.00
evGW0@PBE45	–0.41	–0.18	–0.04	–0.09	–0.18	0.18
ΔSCAN18	–0.15	–0.08	0.05	0.17	0.00	0.11
ΔCCSD(T)8	–0.35	–0.24	–0.31	–0.23	–0.28	0.28

^aThe deviation Δ_exp with respect to experiment and the corresponding ME and MAE are computed with the experimental data by Travnikova et al.¹⁰² (in bold), where error_i = CLBE_i^theory – CLBE_i. The labels of the C atoms are given in the inset of Figure 6a. For the @PBEh calculations, we use α = 0.45.

The comparison between the experimental spectrum and calculated spectra is shown in Figure 6. The differences to the experimental peak positions are reported in Tables 2 and 3. For the absolute CLBEs, the ETFA predictions are in line with the CORE65 benchmark results. evGW₀@PBE and G_ΔHW₀@PBE slightly underestimate the CLBEs, while evGW@PBE and the RS schemes severely overestimate them, see Figure 6. G₀W₀@PBEh(α = 0.45) also overestimates the C 1s energies slightly. As discussed in Section 4.2, this is due to the fact that α_opt is slightly smaller than 0.45 for C 1s excitation. C 1s excitations are consequently slightly overestimated in G₀W₀@PBEh. G_ΔHW₀@PBE provides the best accuracy followed by evGW₀@PBE with an MAE of 0.27 and 0.36 eV, respectively, which is consistent with the conclusion for the CORE65 set benchmark.

Table 3. Relative C 1s CLBEs for the ETFA Molecule^a.

	C1	C2	C3	C4	ME	MAE
experiment103	8.25	4.81	1.87	0.00
experiment102	7.46	4.33	1.72	0.00
evGW0@PBE	–0.58	–0.42	–0.07	0.00	–0.36	0.36
G0W0@PBEh	0.40	0.38	0.14	0.00	0.31	0.31
GΔHW0@PBE	–0.50	–0.42	–0.07	0.00	–0.33	0.33
evGW@PBE	–0.47	–0.39	–0.05	0.00	–0.30	0.30
GRSWRS@PBE	–0.58	–0.44	–0.01	0.00	–0.34	0.34
GRSWRS@PBE0	–0.10	0.02	0.14	0.00	0.02	0.09
GRSWRS@B3LYP	–0.03	0.05	0.10	0.00	0.04	0.06
GRSWRS@PBEh	0.21	0.31	0.20	0.00	0.24	0.24
evGW0@PBE45	–0.32	–0.09	0.05	0.00	–0.12	0.15
ΔSCAN18	–0.32	–0.25	–0.12	0.00	–0.23	0.23
ΔCCSD(T)8	–0.12	–0.01	–0.08	0.00	–0.07	0.07

^aThe deviation Δ_exp with respect to experiment and the corresponding ME and MAE are computed with the experimental data by Travnikova et al.¹⁰² (in bold), where Δ_exp = ΔCLBE_i^theory – ΔCLBE_i. The relative energies ΔCLBE are computed with respect to C4. The labels of the C atoms are given in the inset of Figure 6a. For the @PBEh calculations, we use α = 0.45.

Turning to the relative CLBEs, the three partial self-consistent schemes and G₀W₀@PBEh yield MAEs around 0.3 eV, which is only slightly worse than the CORE65 MAEs for C 1s. evGW₀, evGW, and G_ΔHW₀ underestimate all shifts, while G₀W₀@PBEh overestimates the shifts of the more “descreened” C atoms (C1 and C2), see Figure 6b. G_RSW_RS also provides good shifts for ETFA. The RS results are again not completely independent on the starting point. With PBE, the C 1s shifts are similarly underestimated as for the self-consistent schemes, while they are slightly too large with PBEh. The RS schemes with the conventional hybrid functionals PBE0 and B3LYP yield, with MAEs below 0.1 eV, the best overall result of the 8 investigated schemes. Furthermore, except for G_RSW_RS with hybrid starting points, we find that our predictions are worse progressively with the electronegativity of the substituents at the C atoms. The relative CLBEs of C1 (CF₃) and C2 (carbonyl) seem to be the ones that are more difficult to predict, while the predictions of the C3 (CH₂) shifts are mostly within 0.1 eV of the experimental references. However, this trend can be also found in the deviations between the experimental references, see Table 3, that is, deviations between the experiment values increase with the “descreening” of the C atoms.

Finally, we compare our results to previously published computational XPS data.^{8,18,45,102,104−106} We focus here on the results from (i) the same method (evGW₀), (ii) one of the most successful functionals for ΔSCF (ΔSCAN), and (iii) from a higher-level method (coupled cluster), see Tables 2 and 3. All three literature results include relativistic effects. The previous evGW₀@PBE study⁴⁵ uses the same correction scheme⁵⁰ as employed in this work. The ΔSCAN results were obtained with the atomic zeroth-order regular approximation.⁸² For the delta coupled-cluster double results with triple correction (ΔCCSD(T)),⁸ scalar relativistic effects were taken into account by an exact two-component theory. The ΔSCAN and coupled-cluster results were computed for the anti-anti conformer, whereas the evGW₀ literature data are actually an average of the CLBEs of both conformers.

The absolute evGW₀@PBE CLBEs from ref (45) are less underestimated with respect to experiment in comparison with our results. The largest differences are observed for the descreened C1 and C2 atoms. The previously published evGW₀ results⁴⁵ are not extrapolated but obtained with a basis set with more core functions. As already discussed for the CORE65 results, substantially increasing the amount of core functions seems to cure the (small) systematic underestimation of evGW₀ for absolute CLBEs. The predictions upon including more core functions should change on average by 0.25 eV for C atoms,⁵² which is close to the (maximal) 0.3 eV difference we observe here. However, similar to our results, the deviation from experiments is larger for C1/C2 than for C3/C4. Since adding the core functions seems to be more relevant for the descreened environments, the chemical shifts improve too, see Table 3. Some of the difference must be also attributed to the inclusion of the anti-gauche conformer in ref (45), which has slightly higher C 1s BEs.¹⁰²

We note here that ref (45) contains also ETFA results with G₀W₀@PBEh(α = 0.45) employing core-rich basis sets. However, the α values were tuned with respect to experiment using an extrapolation scheme with the cc-pVnZ basis sets.⁴⁹ An insufficiency in the basis set description (i.e., here a systematic underestimation) would be partly absorbed in the α value, which is the reason why our G₀W₀@PBEh results will agree better with experiment.

It has been recently shown that the SCAN functional yields excellent absolute and relative CLBEs for molecules¹⁷ and solids.¹⁹ We find that this is also true for the ETFA molecule. ΔSCAN yields the best MAE for absolute CLBEs, which is, however, very close to the evGW₀@PBE results⁴⁵ with the core-function-rich basis set. For the relative CLBEs, the ΔSCAN is outperformed by partial self-consistent and RS GW approaches as well as coupled cluster.

The absolute ΔCCSD(T) core excitations⁸ are underestimated by 0.23–0.35 eV. However, these results were obtained at the cc-pVTZ level and are probably not fully converged.²⁸ It is thus difficult to judge the performance of the method for absolute CLBEs. The chemical shifts on the contrary are often less affected by the basis set choice and ΔCCSD(T) yields together with G_RSW_RS(@PBE0 or @B3LYP) MAEs <0.1 eV.

4.4. Comparison of the Computational Cost

Comparing the computational cost of the five GW flavors employed in this work (G₀W₀, G_ΔHW₀, G_RSW_RS, evGW₀, and evGW), the G₀W₀ scheme is the computationally least expensive one. For core-level calculations, highly accurate frequency techniques are required,²² such as the fully analytic evaluation of the self-energy via eq 6 or the contour deformation. This increases the computational cost compared to valence excitations, where schemes like the analytic continuation³¹ (AC) can be used. In conventional implementations, the AC scheme scales with respect to system size N but fails to reproduce the self-energy structure for deep core states.²² The scaling of the fully analytic approach (eq 6) is with respect to system size N. The complexity of the contour deformation technique increases for core levels from (valence states) to A detailed analysis of the scaling behavior of the contour deformation approach can be found in ref (22). For both techniques, fully analytic approach and contour deformation, the polarizability, which is the computational most expensive step in the GW calculation, is explicitly constructed in each iteration step of eq 1 (the QP equation). Typically, the QP equation converges within 10 steps; that is, the self-energy must be computed 10 times.

The computational cost of G_ΔHW₀@PBE is only slightly larger than for G₀W₀. The Hedin shift ΔH (see eq 12) is computed from once before the iteration of eq 13. Given that the latter converges also in 10 steps, the self-energy needs to be calculated 11 instead of 10 times.

The G_RSW_RS requires the subspace diagonalization of the RS Hamiltonian (eqs 21 and 22), which scales only , which is at least two orders of magnitudes lower than the frequency integration of the self-energy. In practice, the computational cost of a G_RSW_RS calculation is only marginally larger than for a G₀W₀ calculation.

The computationally most expensive approaches discussed here are evGW₀ and evGW. In evGW₀, the eigenvalues are iterated in G (outer loop), and in each step of the outer loop, we iterate the QP equation, that is, eq 1, not only for the core state of interest but for all states. Assuming again that the iteration converges within 10 steps, this implies that even for small molecules we evaluate the self-energy in the evGW₀ procedure several hundred times. evGW is even more expensive because W is rebuilt in each step.

For core-level calculations, the eigenvalue self-consistent schemes are restricted to molecules with less than 20–25 atoms, while G₀W₀@PBEh, G_ΔHW₀, and G_RSW_RS are equally applicable to larger systems. In our previous work,²⁴ we computed CLBEs of structures of up to 110–120 atoms at the G₀W₀@PBEh level.

The comparison of the computational cost between G₀W₀@PBEh and ΔSCF depends on (i) the system size due to the different scalings of both methods and (ii) the choice of the functional for the ΔSCF calculations. A comprehensive assessment of the computational cost was given in our previous work,²⁴ where we compared G₀W₀@PBEh to Δ Kohn–Sham¹⁰⁷ (ΔKS) calculations with the PBE functional. (ΔKS is the projector augmented wave variant of the all-electron ΔSCF method.) We found that G₀W₀@PBEh is already ≈50 times more expensive for smaller molecules of around 20 atoms, while for isolated structures of ≈100 atoms the factor increases to 20,000. When using hybrid functionals for the ΔSCF calculation, the computational cost of G₀W₀@PBEh is similar to ΔSCF for small molecules since the evaluation of the exchange is the computational bottleneck in both cases. With increasing system size, the higher-scaling steps in G₀W₀ start to dominate. For structures with ≈100 atoms, we estimate that G₀W₀@PBEh is 1–2 orders of magnitude more expensive than hybrid-based ΔSCF calculations.

5. Conclusions

We have presented a benchmark study of different GW approaches for the prediction of absolute and relative CLBEs. In addition to the evGW₀@PBE and G₀W₀@PBEh(α = 0.45) methods, which were already investigated in ref (49), we have included evGW@PBE and two new methods, namely, G_ΔHW₀ and G_RSW_RS, in our study. G_ΔHW₀ is an adaption of the “Hedin shift”^74,108 to core levels and can be considered as computationally less expensive approximation to evGW₀. In the G_RSW_RS approach, the RS Green’s function is used as a new starting point and, in contrast to our previous work,⁶⁷ also used to compute the screened Coulomb interaction. The purpose of introducing the RS scheme is to reduce the dependence on the starting point, and the method has thus been tested with four different DFAs (PBE, PBE0, B3LYP, and PBEh (α = 0.45)).

By investigating the self-energy matrix elements and spectral functions, we have confirmed that evGW₀@PBE, G₀W₀@PBEh, G_ΔHW₀@PBE, G_RSW_RS@PBE, G_RSW_RS@PBE0, G_RSW_RS@B3LYP, and G_RSW_RS@PBEh yield a unique solution. G₀W₀ schemes starting from a GGA or hybrid DFT calculation with a low amount of exact exchange do not yield a distinct QP solution. A meaningful physical solution can thus not be obtained with standard approaches such as G₀W₀@PBE, G₀W₀@PBE0, and G₀W₀@B3LYP for CLBEs.

We have studied the CORE65 benchmark set and the C 1s excitations of the ETFA molecule with all 8 approaches, for which a physically reasonable solution behavior was confirmed. For the CORE65 set, evGW₀@PBE, G₀W₀@PBEh, and G_ΔHW₀@PBE yield with MAEs of 0.30, 0.33, and 0.25 eV, respectively, the best results. evGW and G_RSW_RS overestimate the absolute CLBEs by several electronvolts and are thus not suitable for the prediction of the absolute BEs. Nevertheless, the RS approaches significantly reduce the starting point dependence as intended. The relative CLBEs are reasonably reproduced with all methods, but in particular with the eigenvalue self-consistent schemes and G_ΔHW₀@PBE (MAEs < 0.2 eV). The methods exhibit a similar performance for the ETFA molecule, except that the RS approaches with standard hybrid functionals yield here the best chemical shifts.

The G₀W₀@PBEh(α) approach was introduced in our previous work⁴⁹ as computationally affordable alternative to evGW₀ that can mimic to some extent the effect of eigenvalue self-consistency in G. However, the α-tuning is methodologically unsatisfying since the optimal α increases with the atomic number and an individual tuning for each element is in principle required and in fact mandatory for heavier elements. We therefore recommend to use the G_ΔHW₀@PBE approach instead, which is in terms of computational cost comparable to G₀W₀.

Finally, we found that evGW₀@PBE and G_ΔHW₀@PBE systematically underestimate the experiment. Our comparison to the ETFA literature results and very recent work⁵² suggest that this slight but systematic underestimation can be cured by very large, core-rich basis sets, which might improve the agreement with experiment even further. Future work will consider this and focus on the development of compact and computationally efficient NAO basis sets for core-level GW calculations.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.2c00617.

• CORE65 benchmark results, solution behavior of G_RSW₀, errors of KS-DFT for predicting absolute CLBEs and relative CLBEs, starting point dependence on the tuning parameter in PBEh, and geometry of ETFA (PDF)

The authors declare no competing financial interest.

Notes

The data are available in the Supporting Information, and the input and output files of the FHI-aims calculations are available from the NOMAD database.^109,110