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Core level binding energies, directly measured by X-ray photoelectron spectroscopy (XPS), provide unique information regarding the chemical environment of atoms in a given system. However, interpretation of XPS in extended systems may not be straightforward and requires assistance from theory. The different state-of-the-art theoretical methods commonly used to approach core level binding energies and their shifts with respect to a given reference are reviewed and critically assessed with special emphasis on recently developed theoretical methods and with a focus on future applications in materials and surface sciences.
Scheme 1 Processes of 1s core level photoemission upon X-ray hν radiation on H2O molecules (upper part) and on a materials crystal surface (lower part). In the case of the surface, the excited electron places first in the Fermi level, EF, upon which it is emitted to vacuum surpassing the energy cost of the surface work function, ϕ.
In the case of surfaces or solids described through periodic models, a common choice in computational materials science, the situation is less clear since the unit cell corresponding to the core-ionized system is charged and some action is needed to avoid artefacts arising from the coulombic repulsion between the periodically repeated infinite unit cells; these will be briefly described at the end of Section 4. Also, it is important to point out that in the case of extended systems the experimental measurements are often relative to a given arbitrary reference. For instance, it is customary to use the C(1s) core level as reference and to assign to be 285 eV.12 In these situations the quantity of interest is ΔBE rather than BE and this is the reason why most attention has been devoted to the calculation of the former values.5,13,14 Nevertheless, it is important to stress that interpretation of XPS is not always easy. The case of atomic O on Al(111) is a paradigmatic example where naïve interpretations of two well-defined features are in contradiction with theoretical results leading in turn to a fully consistent alternative interpretation.15 In Section 2 of this paper the available theoretical methods are described and critically assessed. Section 3 provides an example which can be used as a practicum, whereas Section 4 reports an overview of relevant results in the literature.
with BE being positive for a bound state. In principle, the exact solution of the energy for the many-electron initial and final states could be obtained by full Configuration Interaction (CI) calculations within a relativistic Hamiltonian and an infinitely complete basis set. Obviously, this is not practicable and approximated methods are needed to estimate the total energy of a given system. Even within limited basis sets and/or truncated excitations, CI calculations are extremely complex since the proper solution for core level BEs corresponds to a highly excited root and convergence problems will certainly appear.
At the HF level of theory, core level as well as valence BEs can be approached in two ways. From Koopmans’ theorem (KT) it follows that the negative value of the orbital energy (−εi) from which the electron is removed provides an estimate of the BE. It can be mathematically proven that the KT BE corresponds to the difference between the HF energy of the neutral molecule and the energy of the ionized system computed using the orbitals of the neutral molecule; the latter is often referred to as a Frozen orbital (FO) approach. Clearly, KT predicted BEs are significantly larger than the experimental values since (final state) electron density relaxation effects in response to the presence of the electron hole are neglected. As a matter of fact, BEs computed based on KT give an estimation of the so-called initial state effects. These are mainly related to the chemical bonding and arise from the core and valence electron density of the atom before ionization. More accurate absolute values of BEs can be computed by taking the energy difference between the final ionized system and the initial neutral ground state as stated in eqn (3), both obtained at the same HF level. The resulting approach is usually referred to as ΔSCF. This approach accounts for relaxation effects caused by the response of the passive electrons to the presence of the core hole and thus includes the so-called final state effects. Within the HF formalism, the relaxation energy, ER, is unambiguously defined as the difference between the KT and ΔSCF BEs and, consequently, is always a positive value. For gas phase molecules containing light atoms, the errors of HF ΔSCF computed BEs are typically of the order of 0.5–1.0 eV, the difference with respect to experiment due to the neglect of electron correlation and relativistic effects.16,17 As mentioned in the introduction, BE shifts (ΔBE) for a given atom in different chemical environments are often considered since absolute values of BEs are available for gas phase molecules only. A final comment is needed concerning spin symmetry since the neutral molecule exhibits most often a closed shell singlet ground state and, consequently, the core ionized molecule will exhibit a doublet ground state. Unless specified, all calculations described in the present work are spin restricted. Spin polarization effects can be taken into account by means of spin unrestricted calculations; these are most often negligible and indeed introduce uncontrolled spin contamination effects. Hence, the common practice is to rely on spin restricted calculations.13 Note, however, that in most solid state based codes, spin polarization is introduced with respect to an artificial closed shell reference state with half occupation in alpha and beta orbitals. In that case the effect can be as large as 10 eV 18 but we would argue that it is not appropriate to define it as spin polarization.
where EDFT denotes the energy computed with a particular density functional for both the ground state density and the density corresponding to the system with the core hole but with the density of the neutral molecule. This approach to quantify initial state effects in DFT computed BEs is equivalent to the use of KT in HF calculations. Accordingly, the relaxation energy, ER, is defined as the difference between the FO and DFT calculated BEs both obtained by applying eqn (3).
A potential approach beyond DFT based calculations to estimate ionization energies is the so-called GW quasiparticle approach introduced long ago by Hedin26 and reviewed later in a rigorous and pedagogical way by Aryasetiawan and Gunnarsson.27 This method is based on a generalization of the HF equations in terms of Green's functions (G), where the self-energy term is non-local and energy dependent and the electrostatic potential is dynamically screened (W) thus including explicitly many body effects. This methodology has been implemented at different levels of approximation, depending on whether the method is applied non-self-consistently or self-consistently28 and, in any case, involves a power expansion of W usually truncated at the second order. A major drawback encountered by the approximated implementations of this approach is the possible dependence of the results on the starting electron density. This is the case when the method is not applied self-consistently although reliable results for the lowest ionization potential of 100 test molecules (the GW100 database)29 have been reported at the simplest G0W0 level, where both G and W are computed from the electron density obtained from a standard DFT calculation within the generalized gradient approximation (GGA) type functionals. On the other hand, the computational resources needed increase drastically for quasiparticle self-consistent-GW calculations, although the results obtained applying this approach exhibit better accuracy. The GW method has been successfully applied in the field of materials and surface sciences in order to compute band gaps and electronic excitations. Concerning the calculation of BEs, the advantage of this method lies in the fact that quasiparticle energies of occupied levels effectively represent ionization potentials while for unoccupied states they represent electron affinities. Nevertheless, applications of this methodology to the prediction of core level BEs is still in its infancy with just one study recently published.30 For a representative series of molecules, this study shows that self-consistent GW quasiparticles provide a reliable estimate of core level BEs although the accuracy is still inferior to ΔSCF with either HF or the various available, broadly used, density functionals, as exemplified in a forthcoming section.
As an illustrative example of the accuracy of the different computational approaches to compute BEs described above, we have studied the O(1s) core hole in the water molecule. Calculations have been performed at the optimized geometry as obtained using the PBE33 functional and a tight Tier 2 numerical atom-centred orbitals basis set34–36 (an optimized H–O–H angle of 104.12° and an O–H distance of 0.969 Å, in good agreement with the experimental values, an H–O–H angle of 104.48° and an O–H distance of 0.958 Å). The basis set used in the calculations is an aug-cc-pcVQZ Gaussian basis set corresponding to a primitive (16s, 10p, 6d, 4f, 2g) set contracted to [9s, 8p, 6d, 4f, 2g].37 In Table 1, the computed non-relativistic BEs are reported at the HF level of calculation and by applying three different DFT methods. Those include a GGA derivation of the exchange–correlation functional, the PBE functional,33 and two hybrid implementations, the 3-parameter Becke–Lee–Yang–Parr (B3LYP)38,39 and the hybrid PBE (PBE0),40 with inclusions of 20% and 25% of HF exchange, respectively. From Table 1 it can be seen that BEs derived from the application of Koopman's theorem for εi-KS are meaningless and do not represent initial state effects. Instead, these effects can be accounted for by computing the BE by means of a FO wavefunction, where the core hole is frozen as stated in eqn (5). In such a case, initial state effects computed by DFT are similar to the value obtained by HF calculations, where BEs obtained by KT are well-defined and have a physical meaning. BEs computed by the ΔSCF approach include final state effects and are closer to the experimental value, 539.70 eV.41 As a matter of fact, final state effects can be evaluated by calculating the relaxation energy as the difference between the ΔSCF BE and the initial state BE computed as BE(FO). Relaxation energy at the HF level is 20.48 eV while its magnitude is 20.70 eV for PBE calculations. Hybrid functionals give slightly lower values, 20.61 and 20.64 for B3LYP and PBE0, respectively. As shown in Table 1, final state ΔSCF BEs computed by HF and B3LYP show rather better agreement with experiment that provides PBE and PBE0 results. However, it has to be noticed that relativistic effects are not accounted for. As commented previously, relativistic corrections to the BE can be accurately estimated by fully relativistic atomic calculations. For the O(1s) core hole of the isolated O atom, relativistic effects increase the BE by 0.45 eV. Hence, inclusion of the atomic relativistic corrections in the O(1s) BE of the water molecule brings the HF value to 539.63 eV and the B3LYP result to 539.84 eV, both very close to the 539.70 eV measured value.
Calculations based on the GW quasiparticle approach have also been carried out using the same geometry as in the HF and DFT calculations but results have been extrapolated to the complete basis set limit. Different GW approaches have been considered: G0W0 with a linearized quasiparticle equation, starting either with PBE or PBE0 densities, and quasiparticle self-consistent GW. The O(1s) core hole BE corresponding to the linearized G0W0 method starting with PBE and PBE0 densities are respectively, 529.45 and 534.90 eV, whereas the self-consistent implementation improves the value to 542.13 eV providing a rather accurate result yet 2.4 eV larger than the experimental one.
In this section we critically revise the so far accumulated literature on core level BE prediction by the above-commented methodologies, with a final overview on future work and applicability in materials and surface sciences. As aforementioned, the HF method has been and is the working horse in obtaining core level BEs, either within the FO approximation to evaluate initial state effects, or by ΔSCF methodology to acquire final state BEs. The Hartree–Fock orbital energies (KT-εi) and ΔSCF values have been used to understand the XPS of a broad range of systems, although here we will focus on several systematic studies evaluating the peculiarities and suitability of HF to predict core level BEs. One of these earliest systematic studies is that of Besley et al.,42 who studied the performance of HF for 1s core level BEs of a series of 14 simple organic molecules containing C→F elements using a 6-311G** basis set. Results show a mean absolute deviation (MAD) of 0.60 eV with respect to experiments, a value which is only slightly changed (0.64 eV) when using a larger, more complete, cc-pCVQZ basis set. This indicates that basis set convergence seems to be achieved. However, better values are obtained when using a fully uncontracted basis set with the same primitive set (named as u6-311G**) which yielded a somewhat better performance with a MAD of 0.44 eV. This is not surprising since an uncontracted basis allows for a better relaxation of the core electron density upon ionization.
A posterior study, for a series of 19 N(1s) core levels of 17 N-containing organic molecules,16,17 evaluated the performance of the HF ΔSCF methodology using a fully uncontracted basis set of polarized valence triple zeta (pVTZ) quality for the N atom. Results consistently yielded a MAD value of 0.8 eV, highlighting how the choice of the explored data set, even the particular core level, can lead to different accuracies. Indeed, further work on the same spirit expanded the study to C→F elements —20 molecules and 39 core levels—, with a slightly improved MAD of 0.67 eV.43 In that work it was highlighted that the agreement between theory and experiment is high for C(1s), although it gets slightly worse for the other elements. In the most exhaustive study published to date in this subject, a total of 68 molecules containing elements from B to F were explored with a total of 189 1s core level BEs being analyzed.21 At the HF level of theory, results yielded MAD values of 0.22, 0.30, 0.49, 0.79, and 1.09 eV for B→F, respectively, with an overall performance with a MAD value of 0.44 eV.
where K = 2.198 × 10−7, N = 2.178, and Inr is the non-relativistic BE with the final value of Crel in eV. By this approximation, the relativistic corrections are about 0.02, 0.05, 0.10, 0.19, and 0.34 eV for 1s cores in atoms B→F. These are clearly underestimated by more than half with respect to the fully relativistic Dirac–Fock calculations, a somehow expected issue when using a phenomenological equation.44 To a large extent, relativistic and electron correlation effects not included in the HF wavefunction cancel out when considering ΔBEs. Along this line, the performance on HF ΔBEs on a reduced set of data yield MAD values of 0.38 eV, in comparison to the value of 0.67 eV for absolute core level BEs.43 By considering a more complete set of data, the ΔBEs MAD drop to values of 0.27 eV, and, as expected, no significant discrepancies are found for the different considered elements.
Indeed, further work in this field targeted the description of other types of core levels. In particular, we mention the extensive work of Segala et al.50 for describing 2p core levels for 145 cases of Si, P, S, Cl, and Ar containing molecules. These authors evaluated the core level BEs at the ΔSCF approach by considering 26 xc functionals and including relativistic effects through the Zeroth Order Regular Approximation (ZORA).51,52 At variance with 1s BEs, the best performance is found for a combination of the OPTX functional for exchange,53 with the Lee–Yang–Parr (LYP) correlation,39 with a MAD value of 0.26 eV, whereas for these types of orbitals the performance of PW86/PW91 was poorer with a MAD value of 0.45 eV. Minnesota functionals were evaluated by Pueyo Bellafont et al.43 finding that hybrid M1154 and meta-GGA M06-L55 excelled over other with MAD values of 0.33 eV, thus, better than HF or B3LYP in the same study. The same MAD value of 0.33 eV was found for the meta-GGA xc of Tao–Perdew–Staroverov–Scuderia (TPSS),56 0.21 eV when including relativistic effects.
From the discussion so far it is clear that HF and the different DFT based methods exhibit a similar performance, at least as far as this is measured from the calculated MADs ranging from 0.2 to 0.8 eV depending on the method and on the ionized core. In relative terms the error is very small since these BEs are of the order 300–600 eV but in absolute terms this may still be too large when aiming at interpreting XPS data where the limit of accuracy of High-Resolution Photoemission Electron Spectroscopy (HR-PES) is of ∼0.1 eV. Regarding ΔBE values, DFT based methods seem to surpass in efficiency, HF,43 with MAD values as small as 0.16 eV for hybrid functionals B3LYP and M06,57 whereas other meta-GGA functionals such as M06-L and M11-L58 yield MAD values of 0.18 eV, and TPSS displays a value of 0.25 eV on the 185 systems data set,21 compared to a value of 0.24 eV for the GGA of Perdew–Burke–Ernzerhof (PBE),33 and a value for HF of 0.27 eV.
Further steps are necessary aiming at predicting these values for such molecules interacting with solid surfaces. This is especially the case when crystalline solids are described using codes exploiting periodic boundary conditions. As already commented, the ionization of one core electron in a given atom leads to a charged periodic cell, with the concomitant problematic convergence issues along with possible artefacts in the calculated values. In fact, a common way out of the problem consists of creating a countercharge inside the cell which can lead to unrealistic core level BEs and ΔBEs as a result of the perturbation created by the artificially added electric field. An additional, not minor, problem is that most of the available and broadly used periodic codes use pseudopotentials to take into account the effect of the atomic core in the valence electron density. In computational materials science, the Projector Augmented Wave (PAW)59 method is one of the most commonly used.
Despite this limitation, some steps towards approximate ΔSCF calculations in periodic calculations have been reported in the literature. In particular we highlight the paramount work of Pehlke and Scheffler60 on the use of a pseudopotential approach to study core holes at the Si(001) and Ge(001) surfaces and the more recent work of Ljungberg et al.18 within the GPAW. These authors point the necessity of carrying out spin-polarized calculations when exciting one hole electron, otherwise variations of up to 10 eV can be found, as reported for the H2O molecule. This is quite a surprising result since spin restricted (open shell) and unrestricted calculations indicate that the effect of spin polarization is in this case negligible. In spite of this, their implementation yielded a BE estimation only 0.8 eV far from the experimental value. An alternative implementation has been proposed by Köhler and Kresse61 although it actually models X-ray Absorption Spectroscopy (XAS) rather than XPS, given that the excited electron is placed in the molecular Lowest Unoccupied Molecular Orbital (LUMO), or, in periodic systems, in the band right above the Fermi level. The only systematic study on this procedure considered the 185 1s data set and focused on initial and finals state effects as predicted from the approach of Köhler and Kresse,61 as well as the Janak–Slater (JS) approximation, in which only half an electron is excited. The authors also inspected the case of removing the electron—half the electron in JS—from the system, with the charge in the unit cell counteracted by a uniform background of opposite charge. Results revealed that complete removal of one electron leads to better agreement with experiment for ΔBEs although this approach does not allow obtaining absolute values for the core level BEs. Applying the JS approach while removing the excited half electron leads to the MAD values of 0.37 and 0.21 eV for the ΔBEs with PBE and TPSS xc functionals, respectively,62 indicating that, as long as ΔBEs are concerned, this is a sound approach.
In spite of the potential of these PAW based approaches to estimate ΔBEs, applications to surface science are scarce with pioneering studies on CO adsorbed on Ni(001)60 or Rh(111) surfaces.61 Subsequent successful applications involve interpreting HR-XPS of SOx species on Pd surfaces,63,64 or the different attachment modes of graphene on Ni(111).65 However, the most spread approximation found in the literature to evaluate BEs and ΔBEs on solid systems or molecules interacting with crystalline surfaces involves ΔSCF calculations (either at the HF or DFT levels) using cluster models.13 Along this line an extended set of examples are found in the literature, from the evaluation of the interaction of halides on Pt(111), using effective core potentials in the surrounding metal atoms not in direct contact with the halide atom to reduce the computational burden,66 passing by the effect on O(1s) in different alkaline and alkaline-earth oxides,67 to the effect of strain in the core level BEs of Pt–Ru nanoparticles.68 Indeed, coupling sufficiently accurate estimates of BEs and ΔBEs to XPS, PES, and HR-XPS experiments is of great importance, given that such experiments are broadly in use in solid state physics, materials science, nanotechnology, and catalytically relevant systems.69 However, the theoretical methods described in the present perspective have to be further tested on an extended variety of systems, including proper modelling, to ascertain the degree of applicability and reliability, a point that definitely is a matter of future research.
Core level Binding Energies (BE) are available from X-ray Photoemission Spectroscopy (XPS) and provide important information regarding the chemical environment of the atoms in particular systems. Yet, direct interpretation of XPS is not always possible, especially when different and even contradictory hypotheses can be formulated. In these cases, theory is likely to provide the clue to decide among the different interpretations. However, theoretical values need to be accurate enough and this has triggered a considerable amount of work that has been critically assessed in the present perspective article.
The state-of-the-art methods of computational chemistry allow for accurate predictions of core level BEs of 1s cores of main group elements containing molecular systems. The mean absolute error for Delta Self-consistent Field (ΔSCF) calculations at the Hartree–Fock level is roughly of 0.5 eV. This may be larger or smaller for ΔSCF calculations using Density Functional Theory (DFT) based methods with an average error of the order of 0.8 eV for the Perdew–Burke–Ernzerhof Generalized Gradient Approximation (GGA) type functional or slightly larger than 0.2 eV for the Tao–Perdew–Staroverov–Scuseria meta-GGA one.17,21 Errors in core level binding energy shifts are generally smaller due to error cancellation effects, most specifically for relativistic effects, which are quite specific for the atomic core.
The interpretation of orbital energies as an approximation to initial state core level binding energy holds for Hartree–Fock calculations but it is not correct for DFT based ones although a proper definition of initial state effects is also possible.17 On the other hand, orbital energies, either at Hartree–Fock or DFT levels, provide a rather good estimation of core level binding energy shifts.16 This provides rather fast and reliable information when handling extended systems for which ΔSCF calculations may be cumbersome. Methods to take into account final state effects on the core level shift in periodic systems have also been proposed and briefly discussed showing, however, the difficulty in obtaining the absolute values of the core level binding energies. In this respect, the emerging GW techniques are likely to provide a reliable and physically meaningful result even if at a quite high computational cost.
This research was supported by the Spanish MINECO/FEDER CTQ2015-64618-R grant and, in part, by Generalitat de Catalunya (grants 2014SGR97 and XRQTC) and by the NOMAD Center of Excellence project, which received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No. 676580. F. V. thanks MINECO for a postdoctoral Ramón y Cajal (RyC) research contract (RYC-2012-10129) and F. I. acknowledges additional support from the 2015 ICREA Academia Award for Excellence in University Research.
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