On the prediction of core level binding energies in molecules, surfaces and solids

Core level binding energies, directly measured with 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 recent developed theoretical methods and with a focus on future applications in materials and surface sciences.


Introduction
The Binding Energy (BE) of core electrons is characteristic of each chemical element and its accurate measurement was made possible thanks to the pioneering work of Siegbahn et al. back in 1957, 1  Because of the accurate measurement of the surface work function is not simple, the application of this technique to materials science usually focus on the difference of BE for a given atom in different chemical environments. The quantity of interest is now ΔBE, usually taken with respect to a given reference; for instance, the BE of the element in the bulk most stable form. In the context of materials science, this technique is usually referred to as X-Ray Photoelectron Spectroscopy (XPS) and it is widely used since ΔBE values provide information about the local environment of the element from which the photoelectron is removed, this involves for instance oxidation states 4 and other bonding features which can 3 only be disclosed by theoretical approaches. 5,6 Moreover, the fact that the technique is surface sensitive 2, 3 makes it especially suitable to investigate surface phenomena. Recently, XPS has been used to observe in situ the evolution of heterogeneously catalyzed reactions providing invaluable atomic level information about reaction mechanisms. [7][8][9] The broad use of XPS in materials and surface sciences together to the applications in heterogeneous catalysis has triggered also a number of theoretical approaches aimed at predicting BEs or, most often, ΔBEs. The prediction of absolute BEs is straightforward for finite systems such as atoms, molecules, clusters or nanoparticles, provided a reliable method exists to approach the total energy of the neutral and core-ionized systems as described in detail in the next section. Starting from the seminal work of Bagus 10 on the calculation of BEs at the Hartree-Fock level of theory, recent advances in the first-principles based methods of electronic structure will be emphasized. In these systems, comparison between theory and available experimental data (see for instance Ref. 11) can be carried out in a clear-cut way. A similar strategy is also possible for surfaces and solids as long as they can be described through an appropriate finite representation as in the cluster model approach. 5 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 it a BE of 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 is in contradiction with theoretical results leading in turn to an fully consistent alternative interpretation. 15 In Section 2 of this perspective 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.

Theoretical approaches to compute core level binding energies
The BE of a given (core or valence) electronic level corresponds to the energy difference between the initial N-electron non-ionized state and the N−1 electron final ionized state: 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.
Two different situations can be encountered when computing BEs depending on whether the electronic states of interest are well described by a single configuration wave function or require a linear combination involving several configurations. 13 In the first case one-body effects dominate and the measured spectra show a single main peak for each ionized electronic shell with weak satellites only. On the contrary, in the second case, many-body effects are important and the spectra usually show several intense peaks for a given shell.
Here we focus mainly on the first kind of systems, which is the most common situation in 5 molecules containing main group elements. To describe the central features of the XPS spectra for these systems, monoconfigurational quantum chemistry methods can be used. In fact, self-consistent field (SCF) Hartree-Fock (HF) calculations have been extensively applied in the computation of core level BEs as a tool to understand and assign the peaks in XPS spectra. 5 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.
Core level BEs can also be approached by Density Functional Theory (DFT) based methods. For molecular systems, choosing the appropriate density functional to compute the total energies and determining BEs by applying Eq. 3 leads to calculated BEs comparable, or slightly better, to those obtained from the HF method. [19][20][21] Note that DFT methods implicitly include exchange and correlation effects, even if to an unknown extent. However, important differences exist in the definition of initial and final state effects at the DFT level. It has been shown that Kohn-Sham orbital energies (KS-ε i ) cannot be interpreted as a measure of absolute initial state contributions to the BEs, although they display correct ΔBE shifts with respect to a given reference. 16 In fact, there is an alternative, more physical, view to interpret KS orbitals as an approximation to Dyson orbitals. 22,23 This point of view has been described in detail in the review papers by Ortiz 24 and by Ortiz and Öhrn. 25 Initial state effects in DFT calculations can be obtained by invoking the same physics, which means making use of a Frozen Orbital 7 (FO) approach, where no response to the core hole is allowed. 17 Hence, the FO density, ρ(FO), is defined as: which corresponds to the ground state density, ρ(gs), but with a core-hole in the proper core orbital ϕ i . In this way, initial effects BE values can be obtained from: where E DFT 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, E R , is defined as the difference between the FO and DFT calculated BEs both obtained by applying Eq. 3.
A potential approach beyond DFT based calculations to estimate ionization energies is the so-called GW quasiparticle approach introduced long ago by Hedin 26 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-selfconsistently or self-consistently 28 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 G 0 W 0 level where both G and W are computed 8 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 about to be 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.
Finally, one important issue that has to be considered in the computation of BEs is the influence of the relativistic effects as discussed in several reviews 13 and highlighted since the days of the early ΔSCF calculations. 31 The importance of relativistic effects depends on the particular core under study, whether the ionized atom is heavy or light and the core hole involves s shells or shells with non-zero orbital angular momentum. These effects can be quantified by explicit calculations of the relativistic effects, both scalar and spin-orbit contributions, on the molecular system, or by considering the relativistic correction on the BE determined in fully relativistic calculations of the isolated atoms. The latter approach has been shown to be reliable for molecular systems, like the CO molecule. 32

The H 2 O molecule O(1s) core level as a textbook example
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.

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Calculations have been performed at the optimized geometry as obtained using the PBE 33 functional and a tight Tier 2 numerical atom-centred orbitals basis set [34][35][36]

A survey of examples and perspectives
In this section we critically revise the so far accumulated literature on core level BEs prediction by the above-commented methodologies, with a final overview on future work and applicability on materials and surface sciences. As aforementioned, 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-ε i ) and ΔSCF values have been used to understand 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 works 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. Notice that such discrepancies with experimentally measured BEs are explained, in part, due to the absence of relativistic effects in the calculations, which can actually be significant for such low-energy core levels and become more important as Z increases.
Inclusion of relativistic effects bridges the disagreement gap by 0.06, 0.13, 0.25, 0.45, and 0.75 eV for B→F, respectively, and so, when accounted for reduces the MAD to 0.3 eV only.
Notice that these explicit relativistic corrections differ from those obtained from previously raw approximations where the correction, C rel , 44 is evaluated as Indeed, further work on this field targeted the description of other type of core levels.
In particular, we mention the extensive work of Segala et al. 50  At this point, the performance of the ΔSCF approach, either using HF or DFT based total energy calculations, in predicting BEs and ΔBEs has been clearly established indicating that quite accurate predictions for small to medium size organic molecules is to date at hand.
These theoretical approaches are equally applicable to more complex systems involving transition metals, lanthanides or actinides although the calculation of core level binding energies is likely to be complicated by the presence of multiplet splitting and/or intrinsic satellites; the interested reader is addressed to the review papers by Bagus et al. 13,15 Further steps are necessary where 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 in 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  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 approach to estimate ΔBEs, applications to surface science are scarce with pioneering studies on CO adsorbed on Ni(001) 60 or Rh (111) surfaces. 61  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.

Conclusions
Core level Binding Energies (BE) are available from X-ray Photoemission Spectroscopy (XPS) and provide important information regarding the chemical environment of the atoms is particular systems. Yet, direct interpretation of XPS is not always possible, especially when different and even contradictory hypothesis can be formulated. In these cases, theory is like 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 element 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 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 on core level binding energy shifts being generally smaller due to error cancellation effects, most specifically of relativistic effects, which are quite specific of 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 estimate 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 to obtain 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.

Scheme 1.
Processes of 1s core level photoemission upon X-ray hν radiation on H 2 O molecule (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, E F , upon which is emitted to vacuum surpassing the energy cost of the surface work function, φ.