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. 2024 Jul 6;20(14):6134–6143. doi: 10.1021/acs.jctc.4c00535

Real-Space Pseudopotential Method for the Calculation of Third-Row Elements X-ray Photoelectron Spectroscopic Signatures

Liping Liu †, Qiang Xu ‡, Leonardo dos Anjos Cunha ¶,‡, Hongliang Xin †, Martin Head-Gordon ¶,‡, Jin Qian ‡,*
PMCID: PMC11270745  PMID: 38970155

Abstract

graphic file with name ct4c00535_0005.jpg

X-ray photoelectron spectroscopy (XPS) is a powerful characterization technique that unveils subtle chemical environment differences via core–electron binding energy (CEBE) analysis. We extend the development of real-space pseudopotential methods to calculating 1s, 2s, and 2p3/2 CEBEs of third-row elements (S, P, and Si) within the framework of Kohn–Sham density-functional theory (KS-DFT). The new approach systematically prevents variational collapse and simplifies core-excited orbital selection within dense energy level distributions. However, careful error cancellation analysis is required to achieve accuracy comparable to all-electron methods and experiments. Combined with real-space KS-DFT implementation, this development enables large-scale simulations with both Dirichlet boundary conditions and periodic boundary conditions.

1. Introduction

X-ray photoelectron spectroscopy (XPS) is a widely used characterization technique that provides core–electron binding energies (CEBEs) in both molecules1 and condensed matter.2−4 In addition to element specificity, the same element in different local chemical environments can possess distinct CEBEs, differences of which can be referred to as binding energy shifts or chemical shifts1 and can provide guidance in probing the local configurations of the elements.5,6 This technique is traditionally applied in an ultrahigh vacuum setting. In recent years, ambient pressure XPS (apXPS) has been developed to accommodate in situ and operando experiments.7,8 These exciting developments allow for simultaneously probing the electric field, the atomic concentration, and the interfacial structure.9 In addition, time-resolved XPS (trXPS) is emerging as an effective probe for understanding ultrafast charge dynamics by tracking electronically and chemically sensitive, time-dependent CEBE shifts.10 As more nuanced experiments lack well-established benchmarks and often require ab initio guidance, the need for theoretical prediction is surging.11,12

A variety of theoretical approaches are available for the estimation of CEBEs. The simplest approach is Koopman’s theorem5,13,14 using the negative of eigenvalues of the ground states as CEBEs. Slater’s transition-state theory15,16 or generalized Slater’s transition-state theory17,18 can estimate CEBEs by employing modified self-consistent field calculations for states with partially occupied core orbitals. A well-established approach is the Δ self-consistent field (ΔSCF) method19−28 where the CEBEs are determined by the total energy differences between ground and core-excited states. Post-HF methods represents another popular class of approaches for estimating CEBEs, which includes but not limited to, configuration interaction,29,30 coupled cluster singles and doubles (CCSD)31−37 or equation of motion coupled cluster (EOM-CCSD).38−40 Lastly, the quasiparticle GW methods41−45 based on the Green’s function can provide an accurate and dynamic response of a system to a core-hole but suffer from high computational cost.

Here we focus on one of the most widely used KS-DFT based ΔSCF approach as it strikes a good balance between the accuracy and efficacy in CEBEs calculations. The ΔSCF approach within an all-electron (AE) framework has been extensively benchmarked and studied for both molecules23,25,46 and condensed matter.28,47,48 Recent studies17,47 showed that the SCAN exchange-correlation functional49 can achieve high numerical accuracy, with a mean absolute error (MAE) of ∼0.2 eV, for calculating absolute CEBEs (Eb) of elements in both molecules and solids. However, AE calculations suffer from variational collapse toward the lower energy states or ground states50 unless appropriate SCF solvers are not used, such as maximum overlap method (MOM),51 σ self-consistent field method (σ-SCF),52,53 and so on. Furthermore, AE calculations are computationally intractable for large-scale systems, thereby hindering its practical application. Parallel to the AE approaches, ΔSCF calculations based on pseudopotentials (PP)24,54−63 have also been performed, showing general agreement with experimental measurements in terms of CEBE shifts (ΔEb = Eb – Eref, Eref is an arbitrary reference system).

In a previous study,50 we proposed a PP approach within the ΔSCF scheme of real-space KS-DFT for the calculations of 1s CEBEs of second-row elements. Using Dirichlet boundary condition (DBC), the PP approach showed comparable MAE of ΔEb with those obtained from the AE approach. However, the applicability of this method for heavier elements, as well as its implications under periodic boundary conditions (PBC) are still unclear. Herein, we extend this PP strategy to the calculations of 1s, 2s, and 2p3/2 CEBEs of third-row elements (S and P), benchmarked with state-of-art AE approaches64−66 and experiments. Compared to AE calculations, this PP approach exhibits excellent numerical stability and simplifies selections for core-excited orbitals, especially for condensed orbital spaces where multiple orbitals display similar energy levels, thereby significantly lowering the adoption barrier for users. Furthermore, it shows comparable accuracy to the state-of-art AE calculations in terms of ΔEb. Lastly, we devise strategies to extend this PP approach to PBC by carefully discussing and leveraging error cancellation. By either correcting Coulomb interactions between periodic replicas or employing large, fixed supercells, we can arrive at a satisfactory ΔEb for periodic systems. Hence, embedded in real-space KS-DFT, this method enables the calculations of CEBEs of large-scale systems using both DBC or PBC, which will be useful for applications in areas such as surface science, catalysis, and energy storage.

2. Results and Discussion

2.1. Computational Details

Pseudopotential Approach

Within the ΔSCF scheme,23 the Eb can be obtained by eq 1,

graphic file with name ct4c00535_m001.jpg 1

where EN[nI] and EN–1[nF] are the total energies of the initial states (ground states) and corresponding final states (the core-excited states with core-holes), respectively, and N is the total number of electrons of the initial state.

Our recent work50 derived the AE total energies for the final and initial states using the definition of cohesive energy:

graphic file with name ct4c00535_m002.jpg 2

where Na is the number of atoms in the simulation systems, n and ρ denote the AE and PP electron densities, respectively, and E[na] and E[ρa] are the AE and PP energies of the a-th isolated atom, respectively, which can be conveniently obtained during the PP generation. Combining eqs 1 and 2, the Eb are obtained from PP calculations for final and initial states. A detailed derivation can be found in our previous work.50 The success of this approach is attributed to the fact that the PP calculations can provide a good approximation of cohesive energies as those obtained from AE calculations. The PPs derived from the AE calculations of isolated atoms can accurately replicate the AE potentials outside of pseudocore regions.67 Given the inertness of core electrons in chemical reactions, this alignment facilitates the accurate evaluation of the interaction energies between valence electrons and pseudocores, ensuring the transferability of these PPs across different chemical environments. Note that if core–valence mixing becomes considerable, a harder PPs should be employed, where more core electrons are excluded from the frozen pseudocores during the PP generation.

All calculations via the PP approach are performed in the real-space ARES package.68 As structures from the B3LYP functional only marginally improve the accuracy of the Eb predictions (Table S1), geometry optimization is performed in Q-Chem69 with the PBE functional70 and the cc-pVTZ basis set.71 Troullier-Martins (TM) PPs72 with 1s, 2s, and 2p core-holes are generated using the FHI98PP code73 to represent the atoms with core-holes, respectively. General TM PPs are used to describe the atoms that are not core-excited. This assignment intrinsically localizes a core-hole to a specific core-excited atom, which is analogous to the mixed basis strategy used in the AE calculations.65 Especially for species with equivalent atoms, this approach effectively breaks the equivalence, thereby avoiding hole delocalization among equivalent atoms. With the fully screened core-hole assumption,5 these PPs can be implemented into the ARES package for calculations of initial and final states via the traditional SCF iterations. Note that the PPs with 2p core-holes are spherically symmetric and can thus be interpreted as the average of those with 2p1/2 and 2p3/2 core holes. Consequently, 2p3/2 CEBEs can be effectively extrapolated from the calculations with the TM PPs. This method resembles the multiplet average scheme66 used in the AE approaches. To improve the accuracy of the calculations of ΔEb at a low computational cost, the one-shot B3LYP-refining step proposed in ref50 is adopted by using the PBE-optimized KS orbitals and electron density as input, and B3LYP energy as output. The PP methods using PBE functional and B3LYP-refining step are denoted as PP–PBE and PP–PBE(B3LYP), respectively, adopting a consistent nomenclature with our previous work.50,74

All-Electron Approach

AE calculations, together with experimental data,1 serve as valuable benchmarks for the developed PP approach. Within the AE framework, the total energy can be obtained by explicitly relaxing all the core and valence electrons. As the core-holes represent non-Aufbau solutions to the SCF equations, specialized solvers need to be used to avoid variational collapses and refilling of the core-holes in AE calculations.

AE calculations for CEBEs of third-row elements are performed with the Q-Chem package.69 Molecular structures are optimized with the PBE functional70 and the cc-pVTZ basis set71 as the B3LYP functional75 only slightly improves the accuracy of AE calculations (Table S2), consistent with those in the PP calculations. It is worth pointing out that the AE methods adopted in our previous report for second-row elements50 are not sufficiently accurate for the third-row elements and shows a high MAE (∼0.3 eV) for calculating ΔEb (Table S2). The low accuracy of these AE methods is largely due to incompleteness of the cc-pVTZ basis set and the omission of nontrivial relativistic effects, which are important for heavier elements. Moreover, the selection of basis set has been shown to impact both the accuracy and efficiency of AE calculations.44 Thus, in this manuscript, the aug-pcX-2 basis set76 is used, and scalar relativistic effects are incorporated through the exact two-component (X2C) Hamiltonian.66 The semilocal meta-GGA SCAN functional49 together with the PBE70 and B3LYP75 functionals are employed for benchmarking. Local exchange-correlation integrals for all density functional approximations are calculated over a radial grid with 99 points and an angular Lebedev grid with 590 points on each radial sphere. ΔSCF calculations with the maximum overlap method51 (MOM) are carried out to obtain core-ionized states. For more complex cases, such as S8 and P4, the square gradient minimization (SGM)65,77 protocol is employed to prevent variational collapse. For species with more than a single equivalent atom (e.g., S in CS2), the Boys localization78 procedure is used to localize the core-hole onto a single site. Finally, spin–orbit coupling effects are accounted for through the multiplet average scheme introduced in ref66 to obtain the 2p3/2 CEBEs. The convergence threshold for all core-ion calculations is set to 10–5 a.u.

The ΔEb for 1s core-hole of S and P elements performed by AE calculations with different input parameters are summarized in Table S2. Interestingly, the SCAN functional with the aug-pcX-2 basis set and inclusion of scalar relativistic effects (denoted as AE-SCAN) is the best AE approach for the calculations of both Eb and ΔEb of 1s core electrons of the third-row elements. AE methods using PBE and B3LYP functionals are labeled as AE-PBE and AE-B3LYP, respectively.

2.2. Numerical Results within DBC

The real-space KS-DFT code ARES68 allows for calculations with either PBC or DBC. However, it is challenging to model the core-excited states of systems within PBC because of the long-range electrostatic interactions between the periodic replicas. Even if uniform compensation charges are included in the supercell, the nonconvergence of the periodically charged final states is demonstrated in this work (Figure 3b). Adopting one of the many mathematical schemes28,43,79−81 that numerically correct or counteract the Coulomb interactions is usually imperative but daunting. Thus, we first benchmark our PP development within DBC for calculating ΔEb of third-row elements in isolated molecules. Thereafter, we compare our results with the AE and experimental results. In the following section, we will discuss the implications for extending this approach to PBC.

Figure 3.

Figure 3

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Numerical results within PBC. (a) Illustration of charged molecules within the PBC. (b) Total energy convergence with respect to the cell sizes of initial and final states with 1s core-holes for H2S and SF6 molecules. The total energy differences are computed by the energy differences between the final/initial states within PBC and the corresponding states within DBC. (c) the convergence of 1s CEBEs of SF6 with respect to the cell size. (d) 2p3/2ΔEb of S-containing molecules within PBC with the cell sizes of 15 and 25 Å, respectively, in comparison to experiments and that within DBC. Eb and ΔEb are available in Table S5. MAE of PBC (15 Å) and PBC (25 Å) are 0.15 and 0.14 eV, respectively, which shows excellent consistency with that of DBC. The results are generated using the PP–PBE(B3LYP) method.

Prior to the CEBE calculations for third-row elements, we tested our AE and PP methods on ethyl trifluoroacetate (Table S3), also known as the ESCA molecule, as its C 1s chemical shifts serve as a critical reference and benchmark for XPS calculation methods.45 Compared to experiments,82 our developed PP–PBE(B3LYP) method achieves a high accuracy with MAE of 0.14 eV, which is comparable to that (0.13 eV) of ΔPBEh in ref.45 This result reaffirms the efficacy of our PP approach for second-row elements.50 To further validate the transferability and generalizability of this PP approach for heavier elements, a broad range of S- and P-containing molecules with distinct chemical environments were chosen in this study. Consistent with our previous report,50 the Eb values obtained via the PP approach deviate considerably from experimental results, partly due to the neglect of spin polarization and relativistic effects when constructing the pseudopotentials (Table S1). Fortunately, these errors tend to cancel out when computing ΔEb.50 The 1s, 2s, and 2p3/2ΔEb of S- and P-containing molecules are calculated with respect to the molecules with the lowest Eb in each category: H2S and P(CH3)3, S8 and P4, S(CH3)2 and P(CH3)3, respectively. All values of Eb and ΔEb are summarized in Table S1–S5. The calculated ΔEb of the corresponding core-holes from the PP approach in ARES, AE calculations in Q-Chem, and the experiments are illustrated in Figure 1. There is a systematic overlap between the results from the PP–PBE and those from AE-PBE, indicating that the PP approach can achieve the accuracy level of the AE approach, if the same functional is used. All PP results can reproduce the trends of ΔEb in experiments (Figure 1).

Figure 1.

Figure 1

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Binding energy shifts of 1s (a, b), 2s (c, d), and 2p3/2 (e, f) core excitations of molecules for S and P elements, respectively. More details can be found in Tables S1, S2, S3, and S4 in the Supporting Information.

It is worthwhile to point out that the PP approach with frozen core-holes intrinsically exhibits high numerical stability for all calculations. On the other hand, the AE calculations for core excited states, especially for molecules with multiple equivalent atoms, require careful use of specialized algorithms to avoid variational collapse as well as to localize the core-holes, as discussed in the Computational Details subsection. Furthermore, the PP approach simplifies the selection of the desired core excitation by assigning specific core-hole pseudopotentials to the targeted atom. By contrast, at present, the AE approach requires manual identification of the core excitation orbital, which is nontrivial when the orbital space is dense, such as the 2s excitation of P in PSCl3.

The MAEs of ΔEb from the PP approach are summarized in Table 1, for quantitative comparison with the AE approach. The PP–PBE has comparable accuracy as AE-PBE. Consistent with our previous report, the AE-PBE(B3LYP), using a nonself-consistent B3LYP refining step, demonstrates systematically improved accuracy for ΔEb of 1s, 2s, and 2p3/2 core-holes, as compared to the AE-PBE, proving its universality and robustness in calculating ΔEb. The only exceptions appear to be the 2s excitations of P, but we observe that those systems also exhibit apparently poor results using the AE methods (compare AE-SCAN and AE-PBE in Figure 1d and Table 1 with experiments). This consistent exception may be attributed to the poor quality of the experimental data set for 2s core-hole of P. Compared to the AE-SCAN method, the AE-PBE(B3LYP) shows slightly worse MAEs of ΔEb. Evidently, the AE approach with appropriate XC functionals, state-of-art basis sets and explicit relativistic effect can accurately describe the physics of final core-excited states, and hence exhibit much better Eb and slightly better ΔEb than the PP approach. In summary, on par with the AE approach, the PP approach with frozen core-holes is a robust methodology for calculating XPS binding energies with high numerical stability and ease of usage, especially for large-scale systems. However, this comes at the cost of relying on error cancellation and the sacrifice of part of physical and numerical accuracy.

Table 1. Mean Absolute Errors of 1s, 2s, and 2p3/2 Binding Energy Shifts of S and P in Molecules with Respect to the Experimentsa.

CH Element AE-PBE AE-SCAN PP-PBE PP-PBE (B3LYP)
1s S 0.65 0.22 0.70 0.20
  P 0.30 0.09 0.32 0.19
2s S 0.38 0.26 0.40 0.23
  P 0.24 0.49 0.22 0.65
2p3/2 S 0.32 0.16 0.33 0.14
  P 0.25 0.13 0.31 0.15
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a

The numbers in this table are indicated in eV.

To further rationalize the generalizability of the present methodology for estimating CEBEs of elements in a wide range of chemical environments, we can employ a simple charge model due to Siegbahn.83 This model pointed out the positive correlation between atomic charges and XPS binding energies. The calculated ΔEb versus Mulliken charges are illustrated in Figure 2. We show that the ΔEb in a broad range generally exhibits a positive correlation with the atomic charges. This phenomenon arises because a positively charged atom experiences reduced electrostatic repulsion, allowing it to tightly hold its core electrons, ultimately resulting in larger values for its CEBEs.

Figure 2.

Figure 2

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Binding energy shifts of 1s (a, b), 2s (c, d), and 2p3/2 (e, f) core excitations versus the Mulliken charges of molecules for S and P elements, respectively.

2.3. Implication for Periodic Systems

XPS has also been extensively utilized in a range of condensed matter systems, including bulk solids, surfaces, and interfaces. As these systems are inherently periodic and usually lack well-established benchmarks, accurate prediction of CEBEs of these systems is pivotal for interpreting XPS data and discerning the local chemical environments of atoms. This holds significant applications in diverse scientific topics, such as surface science,84 catalysis,85 and energy storage.86 In the final core-hole states within PBC, the localization of the core-holes within PPs’ pseudocores breaks the translational symmetry in the systems,87 thus resulting in the nonconvergence of their total energies (Figure 3a, b). By treating these localized core-holes as point charges, the electrostatic interactions among periodic core-holes (Ecorr)28 can be accurately assessed by using the Makov-Payne equation:79

2.3. 3

where q is the charge of the core-hole and equal to 1 e, α is the Madelung constant, ϵ is the dielectric constant of the material, and L is the cell size. The Madelung constant is dependent on the shape of the supercell. It is imperative to acknowledge that the Makov-Payne equation is not applicable for periodic systems where their electron holes are delocalized.87

Based on eq 3, two strategies can be introduced to ensure the accurate prediction of CEBEs of atoms within large-scale systems using PBC, including: 1) applying the Ecorr to the final core-hole states, 2) calculating the ΔEb within an identical supercell by canceling out the Coulomb interactions. To examine the performance of these strategies, we apply PBC with different cell sizes to the calculations of 1s CEBEs of S-containing molecules. The value of Inline graphic of eq 3 in our system is determined by the Ecorr at the cell size of 50 Å, where Ecorr is equal to Eb (PBC)–Eb (DBC) . Notably, while the Eb of SF6 under PBC fails to coverage, both Eb – Ecorr and the ΔEb demonstrate convergence at the cell size of 25 Å (Figure 3c). This result suggests that for a large supercell, in this case >25 Å, the final state core-hole can be sufficiently represented by point charges, and therefore, the interaction errors can be safely canceled out when one employs a sufficiently large supercell.

From a practical point of view, we now analyze and compare the convenience of usage for these two strategies. We note that significant attention is needed for strategy 1). The calculation of Ecorr is nontrivial because of the difficulty associated with obtaining both the dielectric constant and the Madelung constant. To obtain the Inline graphic of eq 3, linear fitting Eb with Inline graphic is required.28 In contrast, the strategy 2) ensures the accurate prediction of ΔEb by employing a large, identical supercell and thus canceling out the Ecorr. To further validate the applicability and generalizability of the strategy 2), the 2p3/2ΔEb of S element across various chemical environments are calculated by ARES within PBC with different cell sizes (Figure 3d). As expected, when setting the supercell size to 25 Å, the 2p3/2ΔEb of S within PBC is consistent with the results obtained within DBC. Interestingly, we also observe that a large part of the Coulomb interaction error cancels out for systems with an even smaller supercell, such as 15 Å. In this context, the large difference of numerical values obtained in DBC (or PBC at 25 Å) and PBC at 15 Å, for example in H2S, SF6, and N2S2, results from large bond dipoles that interact strongly with nearby core-holes.

2.4. Demonstration of a Large System

Lastly, to illustrate the advantage of the present real-space KS-DFT PP development in simulating large-scale, periodic systems, we turn to ZSM-5 zeolite series which are widely used as catalysts or supports of catalysts for various industrial processes.88−90 The Al/Si ratios and the densities of Brønsted acid sites (the protons on the O linking to the framework Al3+) are key features of ZSM-5 zeolites, determining their efficacy in catalysis and their capabilities in anchoring catalysts. However, it is often challenging to quantify these features in experiments. XPS can serve as a powerful tool for discerning the chemical environments of Si cations adjacent to the framework Al3+ or Brønsted acid sites, facilitating the identification of these important sites.

To ensure the applicability of the aforementioned strategy 2), we adopt a p(1 × 1 × 2) supercell with ∼580 atoms and a large cell size of 20.383 × 19.534 × 26.994 Å. Computational details for the structure optimizations of the ZSM-5 zeolite series can be found in ref.88 We then calculate the Si-2p ΔEb in Al-substituted ZSM-5 with Bronsted acid sites (HZSM5) and deprotonated counterparts (HZSM5-deH) with respect to the pristine ZSM-5 (ZSM5) (Figure 4). The results show that Si-2p Eb of HZSM5 is elevated relative to ZSM5, while HZSM5-deH exhibits the lowest value. To elucidate the observed shifts in the Si-2p Eb, we employ the orbital centers of Si valence electrons91 as a descriptor, which can be computed from the total DOS of the Si cations (Figure 4b). The position of valence orbital center is a representation of the averaged energy level of Si valence electrons. The Si centers with the lower valence electron levels is expected to also have lower core electron levels, thereby resulting in larger (more positive) 2p CEBE. In accordance with the Koopman’s theorem,5 we rationalize that the valence orbital centers are inversely correlated with 2p CEBEs, as shown in Figure 4c. Our calculations suggest that theoretical XPS predictions have the exciting potential for differentiating important catalytic active centers, at a resolution that was traditionally unachievable via experiments alone.

Figure 4.

Figure 4

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Si-2p CEBEs in the ZSM-5 zeolite series. (a) Structures of the ZSM5, HZSM5, and HZSM5-deH. (b) Total DOS of the Si cations. (c) Si-2p ΔEb with respect to the valence orbital centers. The Si cations with core-holes are highlighted with enlarged representation. Color code: Si, blue; O, red; Al, green; H, yellow.

3. Conclusion

In sum, a real-space PP method for the calculation of 1s, 2s, and 2p3/2 CEBEs of the third-row elements has been developed. The performance of this PP development within DBC and PBC are benchmarked with high quality AE methods as well as experiments. Roughly 0.2 eV accuracy (MAE) has been achieved with the one-shot B3LYP strategy and careful error-cancellation analysis has been presented. The developed PP approach exhibits superior numerical stability and simplifies selections for core-excited orbitals. Combined with the efficient real-space KS-DFT implementations, this method provides advantages for calculating accurate core–electron binding energies of large-scale systems.

Acknowledgments

This work was primarily supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, under Contract No. DE-AC02-05CH11231. J.Q was supported by the DOE Early Career Research Program, and M.H.G. and L.A.C. were supported by the Atomic Molecular and Optical Sciences Program for development of the X2C capability. M.H.G. and L.A.C. received additional support from DOE BES under Award DE-SC0021266 (Liquid Sunlight Alliance), for the assessment of different density functionals. L.L. and H.X. acknowledge support from the NSF Chemical Catalysis program (CHE-2102363). L.L. thanks the NSF Non-Academic Research Internships for Graduate Students (INTERN) program for supporting his work in the Lawrence Berkeley National Laboratory under the guidance of J.Q.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jctc.4c00535.

  • Summary of calculated Mulliken charges and all CEBE data (PDF)

  • Optimized structures, PPs for ground states and core-excited states, AE calculation inputs, and PPs calculation inputs (ZIP)

The authors declare no competing financial interest.

Supplementary Material

ct4c00535_si_001.pdf (114KB, pdf)
ct4c00535_si_002.zip (72.1KB, zip)

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