Advanced Interpretation of XPS Spectra

Paul S. Bagus, Eugene S. Ilton, Connie J. Nelin

Department of Chemistry, University of North Texas, Denton, TX 76203-5017, USA

Paciﬁc Northwest National Laboratory, Richland, WA 99352, USA

C. J. Nelin Consulting, Austin, TX, USA

Abstract

We review basic and advanced concepts needed for the correct analysis of XPS features. We place these concepts on rigorous foundations and explore their physical and chemical meanings without stressing the derivation of the mathematical formulations, which can be found in the cited literature. The signiﬁcance and value of combining theory and experiment is demonstrated by discussions of the physical and chemical origins of the main and satellite XPS features for a variety of molecular and condensed phase materials.

Contents

Introduction
Theoretical models and methods

Initial and ﬁnal state effects and Koopmans’ theorem
One-electron features
1. Core-hole screening and relaxation effects
2. Initial state mechanisms leading to BE shifts
3. BE shifts in molecules, clusters, and solids.
XPS satellites: covalent interactions and screening
1. 1. XPS satellites of chemisorbed molecules
  2. XPS satellites for the NO dimer
Metal oxides and covalency
XPS satellite features and many-body effects
1. 1. Multiplet splitting
  2. Near-degeneracy effects: angular momentum recoupling and FACs
  3. Satellites in closed shell oxides
Conclusions

Acknowledgments

Appendix A. List of acronyms and abbreviations

References

1. INTRODUCTION

The fundamental relationship in Photoemission Spectroscopy, PES, follows from the photoelectric effect established by Einstein. In its simplest form, this relationship is

BE = hν–KE (1)

where hν is the energy of the incident photon, KE is the kinetic energy of the photoemitted electron, and BE is the binding energy, or ionization potential, IP, of the photoemitted electron. It is important to stress that the BE is not a one electron quantity; rather it is the difference in the energies of the N electron initial, un-ionized, state and the N−1 electron ﬁnal, ionized, state:

BE = E^N−1(final)−E^N (initial) (2)

where, by convention, BE >0 for a bound state. In order to obtain absolute BEs, corrections to Eq. (1) may include a work function and an energy reference for the BE. For an explicit discussion of these corrections, the reader is referred, for example, to Refs. [1,2]. In the present paper, we are mainly interested in relatively low intensity, which are not shown [19]. Much more complex XPS spectra are found for the core-level spectra of CeO2 [20–22]. In Fig. 2, the Ce 4s XPS of CeO2 is shown [21]. Since CeO2 is a closed shell oxide, one would have expected a single peak; instead, a distinct triplet is observed. There is agreement that the complex CeO2 XPS spectra cannot be explained by a one-electron model, as was the case for the Ne XPS (Fig. 1). In order to explain the CeO2 XPS, it is necessary to use a theoretical framework where several electronic conﬁgurations contribute to the XPS spectra. Because they rely on several conﬁgurations, these theoretical methods are described as many-body to distinguish them from one-body methods where a single electronic conﬁguration for the photoionized system is associated with each XPS peak. However, as we discuss in Section 5, there is disagreement about the character of the many-body effects, especially interested in the relative BEs, BE rel, of different PES features depending on whether rigorous or semi-empirical theory is used to analyze the XPS spectra. rather than with the absolute BEs.

We restrict the discussion to the analysis and interpretation of core-level spectra. We use the acronym XPS, X-ray photoelectron spectroscopy, in place of PES when we emphasize that X-rays are used, whether they are obtained with conventional laboratory or synchrotron sources; we also use the phrase “XPS Spectra” to refer to the core-level spectra, themselves. Our principle objective is to relate features of XPS BEs both to physical processes, such as multiplet splittings [3,4], and to chemical interactions. Such correlations go beyond the use of XPS to identify the elemental composition of the sample studied [1]. In fact, relating XPS to chemical bonding has been an objective essentially since the inception of modern XPS studies [5,6]. However, in order to correctly connect XPS with materials properties, one needs deﬁnitive understandings of a number of concepts which, unfortunately, have sometimes been elusive. Three important concepts to be considered in detail are (1) screening of a core-hole and related matters of relaxation and Koopmans’ theorem [7–14]; (2) the extent to which the shift of a core-level BE of an atom in different chemical and physical environments can be attributed to charge transfer, CT, between the atom and its environment [11,15,16]; and (3) for open shell systems, including high spin oxides, multiplet splitting arising from the angular momentum coupling of the ionized core-level with the valence open shell [3,17,18].

An idealized XPS spectra for the Ne atom is illustrated schematically in Figure. 1. The ﬁgure shows peaks corresponding to the 2p, 2s, and 1s shells at BE≈10, 20, and 750 eV, respectively. The Ne XPS also contains satellite features with relatively low intensity, which are not shown [19]. Much more complex XPS spectra are found for the core-level spectra of CeO₂ [20–22]. In Fig. 2, the Ce 4s XPS of CeO₂ is shown [21]. Since CeO₂ is a closed shell oxide, one would have expected a single peak; instead, a distinct triplet is observed. There is agreement that the complex CeO₂ XPS spectra cannot be explained by a one-electron model, as was the case for the Ne XPS (Fig. 1). In order to explain the CeO₂ XPS, it is necessary to use a theoretical framework where several electronic conﬁgurations contribute to the XPS spectra. Because they rely on several conﬁgurations, these theoretical methods are described as many-body to distinguish them from one-body methods where a single electronic conﬁguration for the photoionized system is associated with each XPS peak. However, as we discuss in Section 5, there is disagreement about the character of the many-body effects, especially depending on whether rigorous or semi-empirical theory is used to analyze the XPS spectra.

Fig. 1. Schematic XPS of the Ne atom showing three peaks corresponding to the Ne 2p, 2s, and 1s shells. Fig. 2. Ce 4s XPS of CeO2 measured with 875 eV photons; see Ref. [21].

The different spectra in Figs. 1 and 2 illustrate two important features of XPS spectra that a proper analysis must explain and, ideally, relate to materials properties. The ﬁrst is the number of features, attributed to ionization of a given shell, that receive signiﬁcant intensity; in Fig. 1, there is only one peak while in Fig. 2, there are three. A second, closely related feature is the relative intensity, I rel, of peaks when the XPS spectra of a particular shell has multiple peaks. Why only one peak in one case and multiple peaks in another and why the differences in intensity among multiple peaks are questions that rigorous theoretical analysis can be called upon to answer and to relate to materials properties. Another feature of XPS spectra that theory can help to understand concerns the shifts of XPS BEs, denoted ΔBE, which have been extensively studied to determine whether and how the ΔBEs reﬂect the chemistry of the materials [5,6,23]. The ΔBE may be between inequivalent atoms in a single sample, as for example surface or bulk atoms in a solid [11,24,25], or of atoms in different compounds [26]. As we shall show, theoretical analyses of these features permit an understanding of the interplay of several effects including: physical, dominantly intra-atomic; chemical bonding; environmental; and initial as opposed to ﬁnal state, or screening, effects. In other words, an important goal is to relate the number, intensity, and energy of these features to the materials chemistry and we will indicate the directions that need to be taken to achieve this goal.

Figure 2. Ce 4s XPS of CeO2 measured with 875 eV photons; see Ref. [21].

Another feature of interest is peak broadening. Theoretical studies yield individual lines at particular energies with particular intensities while the experimental spectral features, or peaks, are broadened by both extrinsic and intrinsic mechanisms. Extrinsic broadenings arise from the resolution of the electron energy analyzer and from the energy distribution of the photon ﬂux. For a laboratory apparatus that uses unmonochromated Mg or Al Kα X-rays, the photon ﬂux will have a broadening of ∼1 eV full width at half maximum (FWHM) [27], although this width can be reduced considerably if a monochromator is used. With high resolution XPS at synchrotrons, the broadening due to the photon ﬂux and the energy analyzer can be reduced to ∼0.25 eV or even less; see, for example, Refs. [2,28]. Another extrinsic source of broadening for insulating samples arises from charging of the material. There are several origins for intrinsic broadening. One is the lifetime of the core-hole state created in the XPS photo-ionization [29]; another is the broadening due to inhomogeneities in the material studied. There is also the possibility of broadening from unresolved ﬁnal states for different multiplets in, for example, open shell transition metal compounds, as will be discussed in detail later [10,30]. Another mechanism is vibrational broadening, often described as the Franck–Condon (FC) broadening [31], which has been known since the mid-1970s for the C(1s) XPS spectra of gas phase CH4 [32] and for broad XPS peaks of bulk oxides [33,34]. Recently, it has been shown that chemical interactions with the substrate are responsible for variations in the FC vibrational broadening of thin oxide ﬁlms [35]. These broadenings may have the Gaussian or Lorentzian character [29] and are best described with a Voigt convolution [36]. The details of the broadening functional form are beyond the scope of this paper and when we discuss XPS broadening we will use a Gaussian broadening. As discussed above, several physical and chemical mechan- isms contribute to the complexity of XPS spectra. Since the BEs in Eqs. (1) and (2) are the difference of the energies of N and N−1 electron wavefunctions, all peaks are rigorously many-electron rather than one-electron quantities. However, it can be useful to distinguish the mechanisms according to one- body or many-body physics and chemistry. When the wave- function, Ψ, for a state is well described by a single conﬁg- uration, perhaps even a single anti-symmetric orbital product or determinant, we describe this as a state where one-body effects dominate. This case is distinguished from states where a single conﬁguration is not sufﬁcient and Ψ must be described as a linear combination of several conﬁgurations. Such a distinction is reﬂected in observed XPS spectra. Spectra where one-body effects dominate have, associated with each shell, a single “main” peak and weak satellites, as in Fig. 1, while spectra, as in Fig. 2, which have several intense peaks for a given shell are cases where many-body effects dominate. In cases where an effect could be described either as one-body or as many-body, we explain our choice. For example, we chose to include the multiplets arising from angular momentum coupling as a many-body effect because angular momentum coupling may also involve recoupling of the valence open shell [30,37], which is inherently a many-body effect; see the discussion in Section 6.1.

The overall organization of this review is as follows: In Section 2, we describe the models that are used. The descriptions are brief, but necessary for the reader to under- stand the physical and chemical signiﬁcance of the work; technical details for the models and methods are given in the cited references. This section includes a discussion of the sudden approximation (SA) [38,39], method that we use to determine the relative intensities of the XPS peaks. We also compare our methods and models to those of others. In particular, we contrast the analyses that we present, based on wavefunction theory, to analyses based on density functional theory (DFT). In Section 3, we discuss the basic concepts necessary to understand the origins of BE shifts in molecular and condensed phase XPS spectra and we review several examples of these BE shifts. In particular, we present rigorous deﬁnitions of initial and ﬁnal state effects and show how Koopmans’ theorem is a way of separating these effects [40]. This separation is necessary to make it possible to distinguish initial and ﬁnal state contributions to BE shifts. The driving force for ﬁnal state effects is the response of the “passive”, un- ionized, electrons to the hole created by core-level ionization, which is described as relaxation and as screening of the core- hole. We relate the relaxation and core-hole screening to the concept of an equivalent core model introduced by Jolly [41,42]. It is common to describe screening, especially in high spin oxides [7,8,12–14,20,43], in terms of charge transfer from ligands to the unoccupied levels of the core-ionized atom. This usage of screening by charge transfer invokes a change of conﬁguration and it neglects orbital hybridization and the closed shell screening which arises because of this hybridiza- tion. The concept of closed shell screening is introduced in this section, although major applications and use of this concept are deferred to later sections after the discussion of covalency in oxides, which is presented in Section 5. In order to relate BE shifts to chemistry and chemical bonding, we introduce and use methods to decompose the properties of the wavefunctions into the contributions from different chemical and physical mechanisms. This is accomplished using constrained variations [44–46] where the orbitals to be varied and the orbital space of the variation are constrained so that certain types of chemical interactions are allowed and others are explicitly excluded. In Section 4, the interpretation of satellites is introduced for examples where these satellites can be explained in terms of screening and covalency as discussed in Section 2. The analysis of the complex satellite structure in oxides is presented in a later section. In the following section, Section 5, rigorous criteria for determining the extent of the covalent character in nominally ionic systems, e.g., metal oxides and halides are discussed; this analysis is necessary to understand the satellite XPS spectra of these materials. We also describe how uncertainties may be estimated for the assignments of effective charges on the various atoms. We examine the covalency in several oxides both with and without core- holes. This is necessary to lay the foundation for the discussion of the many-electron contributions to the XPS of oxides. In Section 6, several topics are considered. We introduce, the contributions of angular momentum coupling and multiplets to XPS. This starts with a discussion of the N(1s) and O(1s) XPS of the NO molecule and shows that the singlet and triplet multiplets may or may not be resolved. We also discuss angular momentum coupling and recoupling as well as near- degeneracy effects for transition metal [3,30,37,47] and actinide cations [48–50] and, surprisingly, even for rare-gas atoms [51]. We describe selection rules for XPS allowed and XPS forbidden conﬁgurations and show how “forbidden” states can steal intensity from XPS “allowed” states through conﬁguration mixing of the allowed and forbidden conﬁgura- tions. In addition, effects that lead to satellites in closed shell oxides where angular momentum coupling and recoupling will not play dominant roles are considered. For these systems, a particular concern is to contrast interpretations in terms of CT satellites or shake satellites [52,53]. Finally, in Section 7, we summarize the various contributions that we have identiﬁed for the interpretation of the complex XPS spectra of metal oxides. We point out that, in order to correctly interpret the physical and chemical origins of XPS spectra it is necessary to take the interplay of several mechanisms into account. Our major objective with this review is to show that valuable information can be gained by a proper, non-empirical analysis of this interplay. In Appendix A, we give an alphabetical list of abbreviations and acronyms used in the paper.

Theoretical models and methods

In this section, we review the underlying quantum mechan- ical foundations of the theoretical approaches used to model the XPS spectra as described in the following sections. In this discussion, we stress the physical and chemical content of the approximations that are used rather than the details of the mathematical formulations, which, in any case, can be found in the referenced literature. As a result, this section contains only three equations; this is the minimum number needed to understand the signiﬁcance of the theoretical considerations presented. Three topics are discussed. First, we discuss the materials models used to describe the physical systems that will be described in the rest of the paper. As we point out, it is necessary to formulate substantive technical reasons that these models contain the necessary physical and chemical features to properly describe the XPS properties of interest. Second, we discuss the general approaches taken for the determination of the electronic structure underlying the XPS processes; this includes placing the methods that are used in the later sections of this paper in the context of other methods that have been used for the theoretical analysis of XPS. Finally, there is a brief discussion of the determination of the intensity of the transi- tions to the XPS photoionized states. It is important to stress that a theoretical analysis is incomplete without theoretical predictions of the XPS intensity.

The key considerations underlying the choices of materials models, which are used for the analysis of XPS presented in the rest of this review are the following. For isolated molecules, we use the experimentally determined geometries since our concern is not to establish the geometries of these molecules but to relate properties of the XPS to the chemical bonds in the molecules. For bulk, surface, and chemisorption studies, we use clusters that contain the critical chemistry that determines the XPS features of interest. For the interactions of atoms and molecules with surfaces, we have examined how the properties of the interaction are affected by the size of the cluster used to simulate the substrate [54–56]. We ﬁnd that the chemical character of the interaction converges very rapidly with the number of atoms used in the cluster model of the solid substrate. The rapid convergence of the character of the bond formed has allowed us to tailor the clusters to correctly model the XPS features that we are studying and to relate these features to the chemical interactions and bonding. Sometimes dominant aspects of the XPS of cation core levels in high spin oxides are adequately represented by isolated cations [30,48,50,57], due to the importance of the multiplet splittings caused by angular momentum coupling of the core-hole with the unpaired valence electrons. In Fig. 3, we show three representative clusters. Fig. 3(a) shows an 18 atom cluster to model the XPS of bulk and surface atoms of Cu(100); the bond distances and the geometry of the atoms are taken from those of a bulk Cu crystal. The surface and bulk atoms whose core levels are studied, labeled S and B, have coordinations of 8 and 12, respectively. This cluster was used to analyze the core- level BE shifts of surface atoms relative to the bulk [24,25]. The clusters in Fig. 3(a) and (b) as well as a smaller, 13 atom, cluster have also been used to study BE shifts due to particle size and lattice strain in small particles [58,59]. In order to study lattice strain, the bond distances were allowed to breathe while keeping the fcc crystal geometry. Fig. 3(c) shows a cluster used to model the XPS of CeO2 [21,22], which contains a central, nominally 4+, Ce cation, although the actual effective charge on the cation is determined from variational calculations of the cluster wavefunctions [10]. The metal cation is surrounded by the eight nearest neighbor oxygen anions and by a large set of point charges to provide the Madelung potential of the extended crystal; all atoms and point charges are placed at lattice positions of the ﬂuorite crystal. Other metal, M, dioxides, MO2, with a ﬂuorite structure can be modeled with analogous embedded MO8 clusters; for example, UO2 [50]. In a similar fashion, embedded MO6 clusters, not shown in Fig. 3, can be used to model octahedral metal oxides. These clusters contain a central metal cation, six nearest neighbor O anions, and embedding point charges. These clusters have been used to model MnO [10] and UO3 [60]; these oxides will be discussed later in this paper. Embedded clusters with even larger numbers of atoms explicitly included were used to model
[35], also discussed later.

For the molecules or clusters, wavefunctions are determined using either the non-relativistic program system CLIPS [61], or the relativistic program system DIRAC [62]. The choice of using relativistic or non-relativistic wavefunctions depends on whether the atoms in system are heavy or light and on whether we are modeling core holes in s shells or in shells with non-zero orbital angular momentum where it is normally necessary to take spin–orbit splitting into account.

Fig. 3. Representative clusters used to model XPS BEs for different condensed systems: (a) A Cu18 model of the Cu(100) surface where the representative bulk and surface atoms are marked with B and S, respectively. (b) A Cu115 model of a Cu nanoparticle. (c) An embedded cluster model of CeO2 with a central Ce cation, 8 nearest nearest O anions, and point charge embedding.

Thus, for example, the study that we describe below of the XPS of the NO molecule is based on non-relativistic wavefunctions while the study of CeO2 used relativistic wavefunctions. One of the special features of theoretical calculations of wavefunctions is that it is possible to decompose a property into contributions from different chemical and physical mechanisms. This is done by imposing constraints on the orbitals that are varied and on the space in which these orbitals can be varied. The name that we have given for these constraints is CSOV for constrained space orbital variations [44–46], although similar methods of decomposition have been developed by others [63].

The orbitals or spinors are expanded in terms of sufﬁciently large sets of Gaussian basis functions [64], so that the results are reliable. Pseudo-potentials, described as effective core potentials (ECPs) [65,66] represent the core electrons of the “environmental” atoms [67] but are not used for atoms where a core electron may be ionized. In every case, variationally optimized orbitals or spinors are determined through either Hartree–Fock (HF) [64] or Dirac–Hartree–Fock (DHF) [68] calculations. In the case of open shell systems, the non- relativistic orbitals are normally determined taking into account the angular momentum coupling and solving the HF equations for a speciﬁc multiplet [69] while for the relativistic spinors, the DHF equations are solved for the average of conﬁgurations [68] and the angular momentum coupling is introduced at a later step. HF and, by direct extension, DHF wavefunctions provide reasonably accurate absolute core-level BEs with uncertainties of ∼1 eV [70–72] but, even more important for our purposes, accurate relative BEs for BE shifts [25] and for multiplet splittings [71]. Of course, when the angular momentum coupling becomes more complex [30,37] and when other many-body effects need to be treated [12–14,47,52,53], it is necessary to go beyond one conﬁgura- tion HF wavefunctions. This is done through the method of conﬁguration mixing or conﬁguration interaction (CI) where the wavefunction is expanded as a sum over determinants or over angular momentum coupled conﬁgurations called conﬁg- uration state functions (CSFs) [73].

The general form of a CI wavefunction, Ψ, is where the index i denotes the ith state and the coefﬁcients Ci,k are determined by solving the exact Hamiltonian in the space of the Φk determinants or CSFs [64]. In principle, for a sufﬁciently large expansion, the Ψi can be exact solutions. However, the accuracy and reliability of the CI wavefunctions of Eq. (3) depend on the selection of orbitals to construct the determinants Φk and of the Φk to be used in the sum. Often, it is possible to base these choices on chemical and physical considerations of how particular many-body effects contribute to the XPS spectra. These considerations will be discussed in the appropriate sections below.

The standard approximation for the calculation of XPS Irel is the sudden approximation (SA) [38], which is exact in the limit of inﬁnite photon energy. However, the SA provides accurate Irel for photon energies that are ∼100–200 eV above the ionization threshold for a core or valence level [74]. The SA starts with the assumption that the ﬁnal state N electron wavefunction after photo-ionization can be written as an anti-symmetrized product of an N−1 electron bound state, ΨN−1, and a one electron continuum orbital, ε. Further, it is assumed that at time t 0 the N−1 electron bound state is given by suddenly removing an electron from the kth shell and leaving the other electrons unchanged. This is written formally by using the annihilation operator, ak, such that

This wavefunction is often called a frozen orbital (FO) wavefunction [40] since the “passive” orbitals are frozen, or ﬁxed, as they are in the initial state where the core-shell is ﬁlled. However, ΨN−1 is not an eigenfunction of the N−1 electron Hamiltonian. Since energies are measured in XPS, it is necessary to expand the FO wavefunction, ΨN−1, in terms of a complete set of N−1 electron wavefunctions that are solutions of the N−1 electron Hamiltonian:

The relative intensity of an XPS peak corresponding to the N−1 electron ion being in a ﬁnal state ΨN−1 is simply the probability of ﬁnding the system in this state. From Eq. (5), Irel(k,α) |〈ΨN−1|ΨN−1〉|2; i.e., the square of the overlap of the N−1 electron FO wavefunction with N−1 electron eigenstate ΨN−1. For the discussion of intensity losses from main XPS peaks, it is convenient to number the eigenstates in Eq. (5) so that α 0 is the lowest energy state with a hole in the kth shell and, hence, normally represents the “main” XPS peak. An important quantity is the deviation of Irel(k,0) from 1, which indicates the loss of intensity from the main peak to satellites; it is common to describe this loss as a percent of the total intensity, into all ﬁnal ionic states, for ionization of the kth shell or [1-Irel(k,α)] 100. For PES calculations based on Green’s function methods to treat satellites [75,76], the relative intensities, Irel(k,α), are described as pole strengths. It is important to point out that we use two different sets of orbitals, one variationally optimized for the initial state, where the core shells are ﬁlled, and the other for the ﬁnal, core-hole, state [10,40]. The use of two sets of orbitals gives a compact and, as we show in the following sections, a chemically meaningful description of “closed shell” screening. Because the two different sets of orbitals are not orthogonal to each other, the evaluation of the overlap integral between ΨN−1 and ΨN−1 is

The overlap integral for Irel(k,α) provides selection rules; when the integral is zero by symmetry for the ﬁnal state ΨN−1, then Irel will be zero. The selection rules are stronger if we are able to neglect closed shell screening and assume that the same orbital set is used for both initial and ﬁnal states. This assumption is most useful for studying the XPS multiplet features that arise from the angular momentum coupling between the core-hole and open valence shells. Here, as discussed in Section 6, these stronger selection rules can be used to qualitatively understand the XPS multiplet features. However, we stress that closed shell screening is neglected only to allow qualitative analyses to made; the exact SA Irel that are used to predict the XPS spectra take full account of the closed-shell screening.

Implicit in the discussion above is that Hartree–Fock or Dirac–Hartree–Fock methods will be used to determine the orbitals for conﬁgurations and, in some cases, for multiplets. Then, CI wavefunctions will be used to determine additional many-body effects and, if it has not already been done with the Hartree–Fock calculation, to determine the multiplets. As well as wavefunction theory, based on HF and CI wavefunctions, DFT has also been used to study core-level spectroscopies and has been very successful determining the main XPS peaks of a very large set of organic molecules [79–81]. The comparison of HF and CI with DFT methods is not central to the main object of this review, which concerns the relationships between features of an XPS spectra with the electronic structure and properties of the material. However, since DFT is a widely used approach to determine the electronic structure of a wide range of materials, it is appropriate to contrast and compare HF and CI methods, for the states involved in XPS, with DFT methods.

DFT is inherently a one conﬁguration theory [82,83], and multiplets and multiconﬁguration effects do not arise naturally unless one uses time-dependent DFT (TDDFT) [84]. Further- more, it has been argued [85] that not all excited states are included in a TDDFT treatment and it is not clear how this limitation will affect states relevant for XPS. In contrast, all excitations can, in principle, be included in a CI wavefunction. It is possible to use DFT energies for different determinants to determine multiplet energies for open shell systems; see, for example, Refs. [86,87]. The equations for the multiplet energies obtained in this way are obtained from the energy expressions for the angular momentum coupled wavefunctions for a single conﬁguration [88]. Thus the multiplet splittings given by these equations may be expected to have limitations in their accuracy similar to the splittings determined with HF wavefunctions. While HF energies for the different multiplets arising from an open shell conﬁguration often give the correct order and magnitude of the multiplet splittings, additional many-body effects must be taken into account to obtain very accurate values for these splittings. For many-body corrections to the HF multiplet splittings for pn open shell conﬁgurations see Refs. [89–93]; for corrections for dn open shell conﬁgurations see Ref. [94]. It is also possible to use the orbitals obtained as solutions of the DFT Kohn–Sham equations to carry out CI calculations for the energies and wavefunctions for core-excited states. These CI wavefunctions can include full angular momentum coupling of the multiplets arising from the open valence and core open shells as well as other many- body effects. Such CI calculations using Kohn–Sham DFT orbitals have been carried out for core excited states arising in X-ray adsorption spectroscopy (XAS) by Ikeno et al. [95,96]. Unfortunately, since it is necessary to compute CI wavefunc- tions with the Kohn–Sham orbitals to treat many-body effects for the core excited states, some of the computational advantages of a DFT treatment are lost. An advantage of HF wavefunctions is that the HF orbital energies give initial state BEs while this is not the case for Kohn–Sham orbital energies; see the comparison of the meaning of HF and DFT orbital energies discussed in Section 3.1.

A theoretical framework based on CI wavefunctions to represent many-body effects is also the basis of the semi- empirical Anderson Model methods used to describe the XPS of metal oxides and halides [12–14,97]. However in the semi- empirical formulation, it is assumed that orbitals used are either pure metal or pure ligand and there are fundamental issues concerning the validity of this assumption. The metal and ligand orbitals have a non-zero overlap and, hence, cannot, as is done in the Anderson Model Hamiltonian calculations, be treated as though they were orthogonal. Especially in cases where the overlap is large, see the discussion in Section 5, errors may be introduced by neglecting the overlap. In principle, it could be possible to add parameters to the model Hamiltonian to take account of this orbital overlap; possibly in the context of a non-orthogonal CI [98,99]. However, this has not been done. A second, but closely related, concern is that the assumption of pure metal or pure ligand orbitals neglects the covalent mixing of these orbitals to form ﬁlled bonding orbitals and open-shell or empty anti-bonding orbitals [10]; also see Section 5. This covalent mixing is fundamental to our understanding of chemical bonding. The consequences of neglecting covalent mixing of orbitals could be avoided by allowing all possible distributions of the electrons over the nominally pure metal and ligand orbitals that are involved in the many-body treatment. This is because mixing of a complete set of conﬁgurations is invariant to the rotation of the orbitals among each other [77]. Unfortunately, such complete mixings are not practical from a computational point of view since the number of the conﬁgurations to be mixed grows exponentially. In addition, parameters in the Anderson model Hamiltonians are normally adjusted to ﬁt experimental data [12–14] and one can get good ﬁts to XPS spectra for the wrong chemical and physical reasons [47]. In principle, one could adjust the model Hamiltonian parameters to ﬁt the results of rigorous, non-empirical calculations [97], so that the meaning of the parameters could be placed on a sound footing [97]. However, this is rarely done.

Green’s function methods have also been used to simulate the PES [75,76] of isolated and chemisorbed molecules but not, to our knowledge, to the high spin oxides of transition, lanthanide, and actinide metals. Green’s function methods are rigorous and, since they are based on perturbation theory, have a path to exact solutions by increasing the effective order of the perturbation, normally described as the level of (n+1)hole–(n) particle excitations. A difﬁculty for application to the XPS of core-holes in transition and heavy metals is that high order hole-particle excitations will be required to give accurate BEs and Irel. These high order excitations are needed because Green’s function methods use a single set of orbitals where closed-shell orbital screening is not taken into account.

One-electron features

In this section, we deﬁne important concepts needed for the analysis of XPS spectra and then, in Section 3.4, we consider applications of these concepts to interpret two features of XPS spectra, BE shifts and vibrational, or Franck–Condon, broad- ening of the XPS features. The concerns in this section are for one-electron features of the XPS; i.e., features that can be understood on the basis of a wavefunction that is a single determinant. In contrast, many-electron, or many-body, fea- tures require the use of wavefunctions described by several determinants; many-body effects are described in later sec- tions. In Section 3.1, deﬁnitions are given to rigorously distinguish initial and ﬁnal state effects; ﬁnal state relaxation, which screens a core hole is also deﬁned and discussed. The use of Koopmans’ theorem as a means to separate initial and ﬁnal state contribution to BEs is also discussed in this sub-section. In Section 3.2, the different ways that a core-hole may be screened are described and different theoretical models for this screening are compared and contrasted. Furthermore, the description of screening that can be obtained by replacing a core-ionized atom with the next atom in the periodic table is discussed and limitations of this model are considered. In Section 3.3, the initial state electronic mechanisms that lead to chemical shifts of BEs are presented; in particular, it is shown that hybridization can make substantial contributions to BE shifts.

3.1 Initial and ﬁnal state effects and Koopmans’ theorem

It is very important to distinguish between initial and ﬁnal state contributions to the BEs, especially for the interpretation of the physical and chemical meaning of BE shifts, ΔBE [24,25,58,59,100–103]. As we show later in this section, the separation may also help to understand a class of XPS satellites. The initial state effects are obtained from the BE for the FO wavefunction of Eq. (4) as

here the subscript k indicates a hole is created in the k shell. If there are open valence shells, the index k may also represent the angular momentum coupling of the core-hole with the valence electrons. Since, in the FO wavefunctions, the potentials seen by the core electrons are ﬁxed using the initial state orbitals, it is proper to describe the BEs as initial state BEs. It can be shown with Koopmans’ theorem, and its extensions for open shell systems [104], that BEk(FO) ¼−εk,,where εk is the orbital energy of the kth shell in the initial state Hartree–Fock wavefunction; we shall return shortly to discuss Koopmans’ theorem further. First, we contrast the initial state BE(FO) with a “ﬁnal” state BE. When we solve the variational equations for the core-hole state and allow the “passive” orbitals to relax in the presence of the core-hole, we obtain
the ΨN−1 of Eq. (5) where the lowest energy core-hole state with α 0 is normally the state of interest. This wavefunction, since it has allowed the “passive” electrons to relax or respond to the core hole, includes both initial and ﬁnal state effects. The BE obtained with the relaxed N−1 electron wavefunction is called a ΔSCF BEα and is given by

The difference between BE(Initial) and BE(ΔSCF) is the relaxation energy ER,

where ER is normally deﬁned for the lowest energy core-hole state with α 0. The FO BE(Initial) is an important quantity since it reﬂects the potential arising from the physical and chemical environment of the XPS core-ionized atom, and hence, it contains direct information about this environment. On the other hand, the variationally optimized wavefunctions for the hole state conﬁgurations, ΨN−1, are, as discussed below, considerably changed from the those for the initial state. With XPS, only the BE including both initial and ﬁnal state contributions, as given by ΒΕα(ΔSCF) in Eq. (7), can be measured; BE(FO) or ER must be obtained from the theory. However, if one combines experimental XPS and Auger data, it is possible to get information on the initial and ﬁnal state contributions to the shifts of BEs [100,102,103]; this is discussed at the end of this sub-section. Since orbital variation within the k hole subspace leads to decrease in the energy, the relaxation energy ER, Eq. (8), must be a positive quantity [70,105]. The ER may be very large and it depends on the size of the system [58] and on how deep the core-hole is. To illustrate the magnitude of ER, we give values of ER obtained for different core-holes on an isolated Cu+ cation. The closed shell 1s22s2…3p63d10 conﬁguration was chosen to avoid the multiplets that would arise for the core ions of open shell neutral Cu. The BE(ΔSCF) and the ER in Table 1 are obtained from non-relativistic HF calculations for the ground state and the core-hole states of Cu+. The ER are reasonably similar for a given principle quantum number and increase as one goes from M to L to K shells. Even for light atoms, the initial state BE(FO) for the core levels are

Table 1

signiﬁcantly larger than the observed XPS BEs. It has been pointed out, correctly, that the BE(FO) have signiﬁcant errors when compared to experiment; see, for example, Refs. [79,80]. In this sense, one may ask if the BE(FO) have any value or scientiﬁc use. The answer is yes because, in many cases, the shifts of BEs are dominated or have, at least, major contributions from initial state effects; several examples will be given later in this section, Furthermore, the initial state effects can be directly related to the chemistry and the bonding in the material being studied. A common assignment is that a shift to larger BE, ΔBE 40, indicates that the atom with the larger BE has a larger positive or a less negative effective charge while a shift to lower BE, ΔBEo 0, indicates that the ionized atom has more electronic charge and its effective charge is either less positive or is negative. This relationship between the charge state of the ionized atom and the shift of the BE is intuitive and it is supported by a large body of data for atomic ionization potentials [106]. However, it must be used with care because there are other reasons for shifts in BE. Indeed, using only the relationship that ΔBE reﬂects the charge of the core- ionized atom can lead to contradictory conclusions. Thus, Wertheim [107], on the basis of BE shifts in the cation levels, concluded that BaO had considerable covalent character, while Barr and Brundle [108], on the basis of BE shifts of the anion levels, concluded that BaO was “very ionic”. This apparent contradiction was resolved by the work of Pacchioni and coworkers [16,109], where other contributions to BE shifts, in addition to effective charges were taken into account.

It is common to use −ε from Hartree–Fock and Kohn–Sham orbitals as an approximation to BEs and to describe this as a Koopmans’ theorem, KT, BE [9,70,104,110]. It is important, however, to give a rigorous deﬁnition of the KT BEs. For HF wavefunctions of closed shell systems, it is straightforward to show the initial state or FO BE is identical to the KT BE; i.e.,

BEkðinitialÞ ¼ BEkðFOÞ ¼ BEkðKTÞ ¼ −εk (9)

For open shell systems, where there are several ﬁnal state multiplets from the angular momentum coupling of the core- hole with the valence open shell electrons, an additional qualiﬁcation is required for Eq. (9). The KT BE is the weighted average of the BEs of the different ﬁnal conﬁguration multiplets [104]. If the multiplet splitting is not of direct interest, then KT can be used for open as well as closed shell systems; if the multiplet splitting is of interest, the average KT BE must be corrected with suitable exchange integrals [88]. The important conclusion from Eq. (9) is that the KT BEs for different systems, obtained from HF wavefunctions, directly indicate the initial state contributions to the changes or shifts of the BEs between these systems. This is different from the interpretation of the DFT Kohn–Sham orbital energies [9,80]. The DFT orbital energies are an approximation to the exact BEs, which include ﬁnal state relaxation as well as initial state effects; see Ref. [80] and references therein. In particular, the exact Kohn–Sham orbital energy for the highest occupied molecular orbital (HOMO) obtained when the exact density functional is used, is the exact ﬁrst ionization potential of a system. Unfortunately, except for a few special cases, the general form of the exact density functional is not known [80]. The analysis that we have described above for BEs also applies to electron afﬁnities. In particular for DFT, the Kohn–Sham orbital energy for the lowest unoccupied molecular orbital (LUMO) is a good approximation to the electron afﬁnity [9]. However, if one uses DFT orbital energies as representing only initial state effects in the sense of Koopmans’ theorem for HF wavefunctions, one may get misleading results.

We illustrate both the accuracy of DFT BEs [79–81] obtained with ΔSCF calculations and the limitations of Kohn–Sham orbital energies as reﬂecting initial state effects by considering the case of the closed shell CO molecule where multiplet splittings are avoided. In Table 2, the O(1s) BEs for the CO molecule from HF and DFT ΔSCF calculations are compared with the values of the HF and DFT orbital energies with sign changed, −ε, and with experiment. The differences of the ΔSCF BEs and the values of −ε are discussed in terms of the relaxation energy, ER, as deﬁned in Eq. (8). The DFT results are taken from Refs. [79,111] and the HF results have been calculated using good quality basis sets; all calculations are for the experimental C–O bond length. While both DFT and HF BE(ΔSCF) give reliable values for the O(1s) BE, the DFT value is much more accurate; this is not surprising since the DFT result includes correlation effects. However, if we treat the DFT O(1s) −ε as a KT initial state BE and use Eq. (8) to calculate ER then ERo 0, which is not physical. The incorrect sign of ER is compelling evidence that Kohn–Sham orbital energies do not represent initial state effects and, thus, cannot be used to separate initial and ﬁnal state effects.

In our discussion thus far, we have treated the separation of initial and ﬁnal state effects through calculations of the quantities BE(FO) and BE(ΔSCF). It is, however, possible to deduce initial and ﬁnal state contributions to BE shifts by combining experimental shifts of XPS and Auger lines [100,102,103]. The experimental data is analyzed using an Auger parameter and the separation between initial and ﬁnal state effects follows from arguments about the scaling of the relaxation energy for one hole, XPS, and two hole, Auger, ﬁnal states. These arguments are based on an analysis of the difference of the extra-atomic relaxation energy for the singly charged XPS hole state and the doubly charged Auger hole state [100–103]. However, there is strong evidence that the assumptions about the relaxation energy for the Auger state do not hold if the Auger decay involves an electron from a high lying level in a valence or conduction band [59,100]. Thus, care must be taken in the formulation of the Auger parameter and in the choice of Auger lines used [58,59,100]. Depending on these choices one may obtain either a useful or a misleading decomposition

Table 2

Core-hole screening and relaxation effects

We have shown that the ER may be quite large and that this implies large screening effects of the core-hole. One way that screening has been described, especially in ionic materials, is in terms of charge transfer (CT) from ligands or neighbors to the core-ionized atom; see, for example, Refs. [7,8,12– 14,20,43,112]. Indeed, there is a nomenclature to describe the screening in terms of the CT from ligand to metal that we illustrate with an example for the high-spin material MnO. The ground state is dominated by a conﬁguration that is Mn2+ and O2− and is denoted as

…½cores]3d5 (10)

where the occupation of the O anions is assumed to be 1s22s22p6. When a core electron is removed from the Mn several ﬁnal state conﬁgurations are considered:

In Eq. (11), [cores−1] indicates a core-hole. In the conﬁg- uration of Eq. (11a), an electron has been removed from Mn, which is now an Mn3+ cation and the O ligands are still O2− as in the initial state. In Eq. (11b), an electron has been removed from a core level but, in addition an electron has been transferred from an O anion to the 3d shell, the Mn occupation is now 3d6 and the CT of an electron from a ligand is denoted L. The CT of two electrons from the ligands to the Mn 3d, Eq. (11c), is denoted L2. The charges on the Mn cation, including the core-hole, in Eqs. (11b) and (11c) are Mn2+ and Mn1+, respectively. In fact, it is possible to have higher orders of CT up to conﬁgurations where 5 electrons have been transferred from the ligands to Mn, … cores−1 3d10L5; although, such high order excitations are not normally used. The extension to other 3d occupations, to Lanthanides with an open 4f shell, and to actinides with an open 5f shell is straightforward. However, the theory that uses the CT model conﬁgurations of Eq. (11) is semi-empirical; see references cited above. Indeed, the analysis of screening as CT assumes that the orbitals are either pure metal or pure ligand in character whereas the chemical bonds in most oxides have some degree of covalent character, see Section 5. Consequently, rigorous methods cannot use the CT model, as described above. Since the conﬁgurations of Eq. (11) are allowed to mix, the use of CT as the major form of screening belongs with many-electron effects and it core-holes and so that we can contrast it with an alternative understanding of screening that explicitly recognizes and makes use of covalent chemical bonds.

The physical and chemical basis of screening through change in the covalent character between the initial and ﬁnal, core-hole, states is best understood using the ideas that form the basis of the equivalent core approximation introduced by Jolly et al. [41,42]. The fundamental physical concept of this valence electrons is increased by 1 when a hole is created in approximation is that the effective nuclear charge seen by the one of the core levels of an atom. Hence, we can replace the nucleus by that of one atom higher in the periodic table but with the core shells all ﬁlled. Thus C, which has a nuclear charge of Z=6, with a 1s core-hole and conﬁguration 1s¹2s²2p² will be equivalent to N+, which has Z=7, and conﬁguration 1s22s22p2. The equivalent core approximation is sometimes referred to as the Z+1 model. Thus, CO with a C(1s) core-hole is approximated by NO+ and with an O(1s) hole by CF+. It is entirely expected that the character of the orbitals in NO+ and CF+ is different from that in the parent molecule CO. We caution that there is penetration of the orbitals into the region of the nucleus; in particular, orbitals that have atomic s character have non-zero values at the nucleus. Hence, the Z+1 model is indeed an approximation but it does provide a way of understanding how changes in the covalent character of the “passive” orbitals screen a core-hole without the need for a change of conﬁguration as is explicitly required in the CT model described above.

It is possible to examine the changes in the covalent character of the “passive” orbitals by examining how the 〈z〉 for the orbitals change for the CO molecule where the z-axis is taken as the inter-nuclear axis and the C atom is at the origin and the O atom at positive z. The 〈z〉 indicates the center of charge for an orbital or for a sum of orbitals. If the change is toward a larger value, this indicates a motion of charge from C toward O. (With the use of 〈z〉 to indicate motion of charge, it is possible to avoid uncertainties associated with population analyses [113].) We show in Table 3, the 〈z〉 for the valence, 3s, 4s, 5s, and 1π, orbitals for the ground state and for the C (1s) and O(1s) core-hole states; these expectation values are also given for the equivalent core molecules, NO+ and CF+. All calculations are for the CO experimental distance of 1.13 Å. “Ideal” covalent orbitals, as for N2, would have the center of charge at the center of the bond with 〈z〉 0.57 Å. We also give in the table the sum of 〈z〉 for these 10 valence electrons where the individuals 〈z〉 are weighted by the orbital occupations. We consider ﬁrst the properties for the initial, or

Table 3

ground, state. The deepest lying 3s orbital is a bonding orbital polarized toward O. The 4s is a lone pair on O with 〈z〉 slightly outside the position of the O nucleus while the 5s is a lone pair on C with 〈z〉 well outside of the position of the C nucleus. The bonding 1π orbital is strongly polarized toward O. The 〈z〉 are fully consistent with the usual view of bonding in the CO molecule [64]. When a C(1s) electron is ionized, all the 〈z〉 become closer to zero indicating a motion of charge toward the C center. This covalent screening of the core-hole has been described as “closed-shell” screening [10] to distinguish it from the CT screening described above which involves moving an electron into an open shell. The direction of the changes of 〈z〉 for the equivalent core molecule NO+ are the same as for the C(1s) core-hole ion of CO. While the 〈z〉 for the high lying 5s and 1π orbitals are very similar for the C(1s) core-hole state and for NO+, the changes in 〈z〉 for the deeper lying 3s and 4s as well as for the sum of the 〈z〉 are more different. Analogous comments can be made for the 〈z〉 of the O(1s) hole state and the Z+1 model of CF+. The center of charge moves toward O to provide closed shell screening of the O(1s) core-hole and the Z+1 model qualitatively shows the same trends.

The Z+1 model has been used to explain core-level BE shifts between surface and bulk atoms using a Born–Haber cycle to change the position of the Z+1 equivalent core atom between surface and bulk [23,114,115]. The Z+1 model has also been used to interpret resonances in X-ray adsorption spectra [116] where the electron that is excited by the incident light moves in the potential of the Z+1 model of the system with a core-hole. The results in Table 3 suggest that while the Z+1 model qualitatively describes the screening of the core- hole by the valence or conduction band electrons, there are quantitative limitations to the description of the screening. Furthermore, since the core shell is ﬁlled in the equivalent core atom, it is not possible to describe the multiplets arising from the angular momentum coupling of the open core level with the open valence levels. As we show, especially in Section 6, these multiplets must be taken into account for a correct description of XPS spectra.

Initial state mechanisms leading to BE shifts

In this sub-section, our concern is to understand the origins of shifts in core-level BE for an atom in different environ- ments. Such shifts are especially relevant for cases where the XPS spectra are dominated by a main peak and where satellites have low intensity. It is common to discuss these shifts in terms of the charges of the atom where the atoms that have a CF+. The sum of the 〈z〉 weighted by the orbital occupations is also given; if the center of charge were at the center of the molecule, this sum would be 5.65 Å.

CO–GS CO+–C(1s) NO+ CO+–O(1s) CF+

more negative charge or a less positive charge have smaller core level BEs. While core-level BEs do reﬂect the effective charge of the ionized atom, there are other initial state mechanisms that can induce BE shifts. The electric ﬁelds generated by charged species in a molecule or in a compound have been recognized, since the early efforts in interpreting XPS, as leading to ΔBE and estimates of the electrostatic effects of charged species were developed to explain ΔBE in molecules [5,6]. For ionic crystals, the Madelung potential must be taken into account to properly describe BE shifts in these crystals [16,109]. However, the hybridization of elec- trons in deeper valence shells or, more generally, the involve- ment of these electrons in the covalent bonding, has not been recognized until rather recently [24,25,58,59]. In the following paragraphs, we will show why this hybridization can lead to BE shifts and also show that it is an initial state effect. We will also indicate how changes in the chemical environment can induce changes in the hybridization.

Core electrons are so close to the nucleus that, to a good approximation, contributions to core-level BEs due to valence and conduction band electrons are simply the potential of these electrons at the nucleus. Thus the N electrons in a valence shell orbital, φ, have a potential at the nucleus of atom A, VA(φ),

where the minus sign signiﬁes that the electrons in φ reduce the BE and the approximate equality on the right allows us to relate the potential to our intuitive view of the size of the orbital. Thus, the larger a valence orbital is then the less it reduces the BEs of core electrons and therefore the BE becomes larger. Of course, if an electron is removed entirely to form a cation, then the potential of Eq. (12) is no longer present and the BE is raised. Similarly, if an electron is added to form an anion, a new term of the form of Eq. (12) is added and the BE is reduced. The magnitude of the effect of hybridization is indicated in Table 4 where the shifts of the deep core 1s BE of the isolated Cu atom are given for three low-lying states. These states are the ground state with conﬁguration …3d104 s1, the excited state where the 4s electron is promoted to the somewhat more diffuse 4p orbital, …3d104p1, and the excited state where a 3d electron is promoted into the 4s shell, …3d94s2. These are the three lowest states of the Cu atom. The ΔBEs, both the initial state, ΔBE(KT), and the total shift including both initial and ﬁnal state effects, ΔBE(ΔSCF), are given for ionization to the weighted average of the singlet and triplet 1s-hole multiplets. The excitation energies from the ground state to the excited states, ΔE(Total), are also given. All results are from Hartree– Fock calculations on the initial and ﬁnal states. The Cu 1s KT BE shifts, ΔBE(KT) in Table 4 are 2.7 eV when the 4s electron is promoted to the 4p shell, 4s–4p, and 6.8 eV when a 3d electron is promoted to the 4s shell, 3d–4s; the ΔBE for the 3d–4s excitation is more than twice as large as for the 4s–4p excitation. While the relaxation energies are somewhat different for a 1s-hole in each of these states, the ΔBE are dominated by initial state effects as can be seen from the fact that the ΔBE(ΔSCF) are similar to the ΔBE(KT); see Table 4.

Table 4

Table 5

From the logic of Eq. (12), it would be expected that 〈r〉4p is larger than 〈r〉4s which, in turn, is larger than 〈r〉3d. The 〈r〉, 1/〈r〉, and 〈1/r〉 for these orbitals are given in Table 5; the values for the 3d and 4s orbitals are taken from the conﬁguration … 3d104s1 while the 4p values are from …3d104p1. From these sizes, it is clear that the BE order will be BE(…3d104s1) o BE (…3d104p1)⪡ΔBE(…3d94s2).

The practical question is how we may a priori relate the degree of hybridization to the chemical and physical environ- ment of atoms in the system. Hybridization involves promotion of electrons in more strongly bound atomic shells, i.e. deeper valence levels, into shells that are less strongly bound, i.e. shallow valence levels, in order to facilitate chemical bonding with neighboring atoms. Often, one thinks of the promotion of an entire electron, as in the case of sp3 hybridization of the C atom in the CH4 molecule. However, hybridization can also involve promotion of only a fractional number of electrons from, for example, the 3d shell into the 4s and 4p shells of transition metal atoms [24]. The essential fact is that hybridi- zation enhances the chemical bonding between atoms. Thus if an atom has fewer neighbors, we would expect a smaller hybridization of the deeper valence to the shallow valence levels directly involved in the bonding. Since surface atoms have fewer neighbors than bulk atoms, hybridization should act to make the core-level BEs of surface atoms smaller than those of bulk atoms. The degree of hybridization should also increase as the bonds become shorter since stronger bonds are normally formed at shorter distances. For this reason, lattice- strain in nanoparticles contributes to BE shifts as a function of particle size [58].

In contrast, the environmental charge density surrounding the core-ionized atom can offset BE shifts due to hybridization [24,117,118]. From the same reasoning that leads to Eq. (12), the environmental charge density surrounding an atom, neglecting any changes in the atom’s electronic structure that might facilitate bonding, will create a potential that reduces the core-level BEs on the atom. The environmental charge density is affected by the both the number of neighboring atoms and by the distance of these atoms from the atom to be ionized. However, the effect of the environmental charge density on the BE shifts cancels the effect of the hybridization. Thus, it is possible that this could lead to a net value of ΔBE which is small. We shall examine how this cancellation occurs in detail in the following sub-section.

BE shifts in molecules, clusters, and solids

In this sub-section, we use the concepts developed above to analyze and interpret the chemical information that can be obtained from observed ΔBE. The ﬁrst case that we consider is the shift of the N(1s) BE in two closely related molecules, pyridine, C5H5N, and pyrolle, C4H5N [26,119], where the N (1s) BE in pyrolle is ∼1 eV larger than the N(1s) BE in pyridine. The original assignment for the origin of this ΔBE was a charge transfer from the N in pyrolle to the C atoms in the ring. From a theoretical analysis, we are able to show that this assignment is incorrect and to identify the correct origin of the ΔBE. In Table 6, we compare the XPS measured N(1s) BEs with the BE(KT) and BE(ΔSCF) from HF calculations on the two molecules. The absolute and relative values of the BE (ΔSCF) are close to the measured XPS indicating that the theoretical treatment is reasonably accurate. However, the ΔBE(KT) is also close to the observed BE shift. Despite the fact that the ER is large, it is similar for the two molecules. The agreement of the ΔBE(KT) with ΔBE(ΔSCF) and with experiment establishes that initial state mechanisms are the dominant origin of the ΔBE; it does not, however, establish the mechanism. Possible reasons for the initial state N(1s) BE shift between pyridine and pyrolle are as follows: (1) there is charge transfer from the pyrolle N to the ring C atoms that leads to a larger BE than for the N atom in pyridine; (2) there is a chemical bonding between N and H in the N–H unit in pyrolle while this unit is not present in pyridine; and (3) there is some other difference in the chemical bonding of the N atom in these two molecules. We describe a theoretical analysis that decom- poses the different contributions to the chemical bonding [44–46] and makes it possible to quantify how these different contributions affect the N(1s) BEs in the two molecules [26,119]. This analysis makes it possible to rule out charge transfer as the origin of the N(1s) BE shift; furthermore, the analysis deﬁnitively identiﬁes hybridization as the chemical bonding mechanism that leads to the N(1s) shift to higher BE in pyrolle. In order to decompose the various individual contributions to the BE shift, we constrain the variation allowed in the calculation of the wavefunctions to exclude N 2s–2p hybridization by ﬁxing the N(1s) and N(2s) orbitals as they are for the isolated N atom in its 4S ground state. This wavefunction and related properties are described as Frz N(s). Although N 2s–2p hybridization is excluded from Ψ[Frz N (s)], charge transfer from and covalent bonding between the N (2p) and the C atoms is allowed. (The ﬁxing of the N(1s) orbitals is simply a convenience in the application of the constraints within the CSOV formalism [44–46]. There is no signiﬁcant effect on the KT N(1s) BE since the N(1s) is not involved in the chemistry of either molecule.) The properties of Ψ[Frz N(s)] are contrasted with those of the wavefunction

Table 6

where no constraints are imposed denoted Ψ[Full SCF]. We use the KT BEs since we have shown in Table 6 that the ΔBE is dominated by initial state effects described with KT. The ﬁrst indication that 2s–2p hybridization is more important in pyrrole than in pyridine comes from the energy improvement when hybridization is allowed; namely E[Frz N(s)]−E[Full SCF]. This difference is 2.3 eV for pyridine but almost twice as large, 3.9 eV, for pyrrole. The consequences of this greater N 2s–2p hybridization for pyrrole for the BE shift are shown in Table 7. The Full SCF BE for both pyridine and pyrrole are larger than the Frz N(s) BEs. This is entirely consistent with the prediction of Eq. (12) for the contribution to the potential at the N center for a larger hybridized orbital than for an orbital ﬁxed to be a N(2s) orbital. However, the increase is much larger for pyrrole than for pyridine simply because there is a larger 2s–2p hybridization in pyrrole. We emphasize that even though CT from the N atom to the C atoms is allowed with the Frz N(s) wavefunction, the N(1s) BEs are almost the same. It is only when 2s–2p hybridization is allowed that the Pyrolle BE becomes larger than the Pyridine BE. We now consider two further examples to demonstrate the importance of hybridization in determining BE shifts.

The shift of core level BEs between surface atoms at the (100) face of Cu from bulk atoms or surface core level shift (SCLS) is such that the bulk atoms have a larger BE than the surface atoms; ΔBE(B−S) 40. For Cu 2p3/2, the difference is 0.24 eV [24]. Usually [23], the ΔBE(B−S), for metal surfaces, are small, well under an eV; however, they may be either positive or negative depending on the metal being studied [23]. The fact that the SCLS has different signs is consistent with a cancellation of positive and negative contributions. Indeed, the theoretical analysis that we discuss [24,25] demonstrates that there is just such a cancellation and it identiﬁes the chemical mechanisms that are responsible for the cancellation. Our model for the SCLS of Cu(100) is shown in Fig. 3(a). We consider the deepest 1s Cu core level, although the SCLS of other core levels are similar. The KT SCLS is ΔBE(B−S; KT) +0.63 eV, while the ΔSCF SCLS is ΔBE(B−S; ΔSCF) +0.36 eV. Again, this is a strong indication that initial state effects dominate the SCLS. This is not surprising; the relaxation, or screening, of a core-hole on a surface or a bulk atom should be similar since the conduction band electrons can just as easily move to the location of the core hole at the surface or in the bulk. The difference between the KT and ΔSCF SCLS, see the values given above, shows that ER for a bulk atom is only 0.27 eV ( 0.63−0.36 eV) larger than ER for a surface atom. Clearly, this difference in ER is a very small fraction of the total ER≈50 eV for the Cu(1s) BE of either a bulk or a surface atom. Thus, we are able to use the KT

Table 7

BE to analyze individual chemical contributions to the SCLS. In order to avoid large numbers for the absolute values of the Cu(1s) BEs, we take the zero of BE as the KT BE of a free Cu atom. All BEs are for multiplet weighted averages of the spin couplings of the core-hole with the open shell valence level electrons. As well as the Full SCF SCLS, we also consider two constrained variations. These constrained variations follow the logic used in our analysis of the BE shift between pyridine and pyrrole; however, here the constraints are to either exclude or include 3d hybridization to the 4sp “conduction band” of Cu. The ﬁrst constraint, denoted Frz Core, ﬁxes the orbitals for the 28 electron core of the surface and bulk Cu atoms as they are for the isolated Cu atom. Only 1 electron for each of these atoms is allowed to vary and interact with the neighboring Cu atoms in the Cu18 cluster model of Cu. For this Frz Core constrained variation, 3d–4sp hybridization is excluded. The second constraint, denoted Frz Ar Core, includes the 3d orbitals and electrons of the surface and bulk Cu atoms in the variation and ﬁxes only the orbitals of the 18 Ar core, 1s2…3p6, electrons to be as they are for the isolated Cu atom. This variation allows the 3d orbitals of the representative surface and bulk atoms to hybridize and to participate in the chemical bonding with the other Cu atoms. The BEs obtained with the Frz Ar Core constraint are reasonably similar to the values for the full, unconstrained SCF calculation on the cluster [24]. In Fig. 4, we compare the surface and bulk BEs for these two constrained variations. When the 28 electron cores are frozen and 3d hybridization and participation in the bonding is excluded, both bulk and surface atoms have BEs signiﬁcantly smaller than the isolated Cu atom and the surface atom BE is larger than the bulk atom BE, ΔBE(B−S)−0.6 eV. These are exactly the shifts that are expected from the potential due to the environmental charge density that surrounds the bulk and surface atoms. This potential lowers the BEs because the potential due to electrons lowers BEs, see Eq. (12), and it has a larger magnitude for a bulk atom, which has 12 nearest neighbors, than for a surface atom which has only 8 nearest neighbors. However, the SCLS for this constraint has the opposite sign than measured or than predicted by theory when no constraints are imposed.

Fig. 4. The KT 1s BEs for Bulk Cu and Surface Cu atoms for the (100) crystal face; the BEs are for the Frz Core and Frz Ar Core wavefunctions where 3d– 4s hybridization is excluded or allowed, respectively.

The situation is completely different with the Frz Ar Core constraint when the Cu 3d electrons are allowed to hybridize and to participate in the chemical bonding. Now, the BEs of both bulk and surface atoms are larger than the BE of an isolated atom and the SCLS changes sign and becomes ΔBE(B−S) 40 consistent with XPS measurements. Exactly as predicted by Eq. (12), the hybridization reduces 〈1/r〉 for the orbitals that are dominantly 3d in character and hence allows the BE to become larger. However, the 3d hybridization shift is differential; the BE of the bulk atom is raised by 3.9 eV but the BE of the surface atom, with fewer nearest neighbors and hence a reduced 3d–4sp hybridization, is only raised by 2.6 eV. The small SCLS arises from a cancellation of the environmental SCLS of −0.6 eV and the hybridization SCLS of +1.3 eV. This cancellation explains both the small SCLS and the fact that the sign of the SCLS may change for different elements [23].

Changes in hybridization with the size of nanoparticles also contribute to shifts of core-level BEs [58,59,120]. However, the reason for the change in the hybridization with particle size is different from that discussed above for the SCLS. For nanoparticles, theoretical analysis shows that changes in hybridization are due to bond distance variations as the particle size changes rather than to the difference of coordination as was the case for SCLSs. The core-level BEs shift by ∼1 eV to lower BE as the metal particle size increases from very small to large with several atomic layers. This BE shift was originally ascribed to a ﬁnal state effect arising from the charging of the nanoparticle during the XPS photo-ionization. Regarding the metal particle as a perfect macroscopic con- ductor, the charge must reside on the surface of the particle. Furthermore, assuming that the particle is spherical, this positive charge at the surface would lead to an increase in the BE∝1/R, where R is the radius of the particle [121]. However, the assumption that the particle can be treated as a macroscopic conductor is not valid for very small particles since it neglects the atomistic structure of matter [117,122]. Thus, it is not possible to use the BE shifts of small nanoparticles as a measure of the particle size. In a recent joint experimental and theoretical effort [58], the measured BE shifts with particle size were compared with theoretical predictions. The theoretical models were for Cu particles, up to the large cluster shown in Fig. 3(b), where the ﬁnal state screening or relaxation energy has converged to within ∼0.1 eV of the bulk value. It was found that the shift to smaller BE with increasing cluster size was only 2/3 of the expected shift of ∼1 eV. However, an assumption made for this theoretical study of cluster size was that the “lattice constant” for the particles was the same as in the bulk. This choice allowed the separation of the consequences of simply changing cluster size from other effects. The ΔBE for the ﬁxed lattice constant show clearly that the ﬁnal state charging argument proposed earlier [121] is incomplete. From transmis- sion electron microscopy studies, it is known that there is a 5–10% contraction of the bond distances between atoms in small nanoparticles relative to those in the bulk metal [123,124] .The change in bond distance for nanoparticles is described as a “lattice strain” since the lattice constant changes from the bulk value. It is reasonable that bond lengths are longer for higher coordinations where the bond strength is distributed over a larger number of bonds than for lower coordinations where the bond strength is concentrated in fewer bonds. For small particles, where the average coordination is less than for the bulk, it is logical that the bond distance will be contracted relative to bulk.

Lattice strain in nanoparticles was modeled by a breathing motion for the Cu18, Fig. 3(a) and the smaller Cu13 cluster, which contains only one fully coordinated bulk Cu atom [58,120]. The objective of this work was to identify the magnitudes and the chemical reasons for BE shifts arising from changes in the bond distances. The BE(KT) and BE (ΔSCF) were examined for changes in the bond distance ranging from an increase of 2% to a decrease of 6% with respect to the bulk values, As for the analysis of the SCLS, we also consider the KT BE shifts for the Frz Core and Frz Ar Core constrained variations. For the bulk atom in Cu13, the Cu 2s ΔBE are summarized in Table 8, where the BE for the bulk lattice constant is taken as reference for the shifts and ΔBE 0 for this distance. The small differences between the ΔBE(ΔSCF) and ΔBE(KT) show that the changes in the BE with lattice strain are dominated by initial state affects. The ΔBE are almost linear with the change in bond distance, with the BE decreasing for larger bond distances and increasing as the bond distances are increased. The ΔBE for the Frz Ar Core, not given in Table 8, are identical to the Full SCF ΔBE (KT). This shows that as soon as the d hybridization and participation in the covalent bonding is included, there are no further signiﬁcant changes in the ΔBE. As for the SCLS, the constrained variation that allows only the BE shifts due to the environmental charge, ΔBE(Frz Core), leads to a reduction in the BE as the atomic distances are decreased. This is to be expected since the charge density surrounding the ionized atom increases as the inter-atomic distance is reduced. How- ever, the increases in the BE due to the increase in the d hybridization as the inter-atomic distance is reduced are almost twice as large as the decrease due to the environmental charge. Thus, the hybridization contribution to ΔBE due to lattice strain in small particles dominates and the shift is to larger BE. The effects described above are general for noble metals and for systems where the d shell is nearly ﬁlled [58,120,125]. The SCLS and the ΔBE due to lattice strain show that hybridiza- tion of deeper valence levels into higher lying, more diffuse valence levels makes an important contribution to BE shifts.

Table 8

The ﬁnal topic in this section concerns another consequence of changes in bond distances, in this case for vibrational broadening of the MgO XPS. Since MgO is nearly an ideal ionic oxide [35,126], when a hole is created on either the metal cation or the oxygen anion, there will be a signiﬁcant change in the electrostatic attraction of the counter-ions and hence a change in the Mg–O bond distance. Qualitative studies [33,34] have suggested that bond distance changes in ionic compounds lead to a substantial Franck–Condon, FC, broadening of the XPS lines because the ﬁnal, ionic states will be in highly excited vibrational levels. This may also be described as the ionic state having large phonon excitations accompanying the core level ionization. A recent combined theoretical and experimental study of the Mg 2p XPS for MgO thin ﬁlms shows that the line width depends on the ﬁlm thickness [35]. From the theoretical analysis, it was possible to identify the reason for the reduced broadening of the Mg 2p XPS lines in very thin ﬁlms in terms of the character of the chemical bonding of the thin ﬁlm to the substrate. This relationship between the broadening of the XPS peaks and chemical bonding of the ﬁlm adds a new dimension to the usefulness of FC broadening. For the Mg 2p ionization, bulk MgO is described with an MgO6Mg18 cluster, embedded in a point charge ﬁeld; where each O is surrounded by six Mg cations. This is distinct from clusters used in other studies of oxides where only one of the neighbors of the O anions is a real cation and the others are point charges. The additional extended cations are needed in the present study because a new feature of the metal–oxygen interaction, not treated in the studies that used clusters like that shown in Fig. 3(c), had to be investigated in this study of FC broadening. The new feature is the dependence of the metal–oxygen distance on whether a core-hole is present on the central metal cation. If changes in the position of the O are to be studied, it is not adequate to move the O toward or away from point charges since point charges do not provide the steric repulsion that arises from the extended charge distribution of the electrons associated with the cation. As is shown below, once the O’s are fully surrounded by real cations, this steric repulsion is properly represented and reliable metal–oxygen distances are obtained. A potential curve was determined for the breathing motion of the 6 O nearest neighbors of the central Mg atom. For the ground state of MgO, the equilibrium re(Mg–O) is 2.09 Å, which is quite close to the distance in the crystal, r(Mg–O) 2.11 Å. When a Mg 2p electron localized on the central Mg cation in the MgO6Mg18 cluster is ionized, the Mg cation becomes effectively Mg3+ The minimum on the breathing potential curve is now re(Mg–O) 1.97Å a decrease of 0.12 Å. There are almost identical reductions in re when an Mg 2s or an Mg 1s electron is ionized providing conﬁrmation that the changes in the Mg–O bond distance are electrostatic in origin. Conversely, ionizing an O2− core electron using an O centered cluster reduces its electrostatic attraction with Mg2+ which increases the Mg–O distance by 0.1 Å. Fig. 5 is a schematic view of the potential curves for the ground state and for two ionic states. In case I, the Mg–O distance for the ion is shorter than in the ground state, as in the Mg 2p XPS, and in case II, the Mg–O distance for the ion is longer than in the ground state, as in the O 1s XPS. The horizontal lines on the ground state curve mark the energy of the v 0 vibrational level and the classical turning points for this level are the intersections with the potential curve. Since the electronic transitions are much faster than the nuclear motion [31], the XPS transitions are vertical. However, the system may be at any point allowed on the v 0 ground state vibrational level. Excitations from the classical turning points are a good approximation to the extremes of the vertical transitions. Using information for the harmonic vibrational frequencies [35], FC broadening of the Mg(2p) XPS and O(1s) XPS are predicted to be 0.8 eV FWHM.

Fig. 5. Schematic potential curves for the ground and hole-state potential energy curves showing the FC vibrational broadening that arises when the Mg–O distance changes in the ionic state.

In order to describe a monolayer of MgO/Ag(100), a point charge embedded MgO4Mg8 cluster modeling an MgO mono- layer was placed above an Ag25 cluster representing the Ag(100) surface. The calculated FC broadening of the Mg 2p XPS is 0.8 and 0.4 eV for the bulk and thin ﬁlm MgO, respectively. Furthermore, the theory indicates that it is the chemical interac- tion between the MgO ﬁlm and the Ag(100) surface that affects the FC broadening. It was possible to exclude other mechanisms such as a change in the ﬁnal state screening of the Mg core-hole by the metal support or a difference between the bulk and isolated monolayer MgO. The measured increase in FWHM going from 1 monolayer MgO to thick MgO ﬁlms with bulk properties is 0.15 eV measured with an unmonochromated laboratory XPS system [35]. If one takes into account an additional broadening of 1 eV for instrumental resolution and convolutes this with the calculated FC vibrational broadening, the theory would predict a reduction in the broadening going from bulk to one monolayer MgO of 0.18 eV. This is close to the experimentally measured difference and indicates that the assignment of the chemical origin of the change is correct. Although the changes in FWHM due to the chemical interaction of the ﬁlm with its support are small, they are easily measurable. This work has demonstrated a new use of XPS to characterize the chemistry of materials.

XPS satellites: covalent interactions and screening

In this section, we consider two cases from early work: (1) CO and N2 chemisorbed on metal surfaces [127–131] and (2) the NO dimer [129], (NO)2. In the ﬁrst case, we focus on the interpretation of the satellites in terms of the covalent character of the substrate–adsorbate bond in both the initial and the ﬁnal states. In the second case, we focus on the character of the screening as indicated by the positions of the center of charge in the NO dimer. We have two objectives for this study. The ﬁrst is to establish the essential multiconﬁgurational character of the dimer wavefunctions and the second is to distinguish between two types of screening, namely inter-unit and intra- unit screening. At the time the original work was performed, it was necessary to use much smaller and simpler molecular models for the wavefunctions. This could, in fact, have been an advantage since it was necessary to carefully select the models that were used and to explicitly include the critical physical and chemical mechanisms. The original papers used the common jargon of the time. In particular, the distinction was made between “screened” and “unscreened” ﬁnal states. We now know that there are different types of responses to core- holes, where inter-molecular or intra-atomic screening constitute a more appropriate terminology.

XPS satellites of chemisorbed molecules

The C(1s), N(1s), and O(1s) XPS of CO or N2 chemisorbed on metal surfaces have a satellite at ∼5 eV higher BE than the main peak where the intensity of the satellite increases as the adsorption bond becomes weaker [127,132]. A general dis- cussion of the XPS of CO in compounds and as an adsorbate on metal surfaces has been given by Freund and Plummer [133] as part of an effort to relate the features of the XPS to the character of the metal carbonyl interaction. The study of CO on metal surfaces is of particular importance because it can be extended to the complex surfaces that are involved in catalytic processes [134]. The original theoretical study of the satellite structure of weakly bound CO adsorbates was based on the model Hamiltonian calculations [135] where the relationship between the XPS and the character of the chemical interaction was indirect. Here, we review the work of Bagus and Hermann [131] on the XPS of CO/Ni where the connection between the satellite features and the character of the metal–CO interaction could be established in a clear and direct fashion. Based on extensions of this early work [127,128,130] and on the understanding of the CO XPS features [133], we expect that the analysis that is described will have a general validity. The model, used in this work, was a simple linear cluster with one Ni atom and one CO molecule, NiCO, which allowed an easy interpretation of the XPS in terms of covalent bonding between the metal substrate and CO. The natural state to use as the initial state, before core ionization, is one of the low-lying states that can be derived from the ground state 3d94s1 conﬁguration of the Ni atom plus the closed shell CO molecule. However, with this conﬁguration back-donation from the Ni conduction band to the CO(2πn) cannot occur since the one Ni atom does not have any sp electrons with π symmetry. Yet, as conﬁrmed later [54,130], back-donation is quite important, especially for hole-states of chemisorbed CO [130]. Consequently, for this cluster, an excited state, denoted GS(π), derived from the 3d94p1 conﬁguration of Ni was used as the initial state [131]. This state allowed back-donation or covalent bonding between the metal sp conduction band and CO(2πn) and provided a direct explanation of the origin of the XPS satellite structure of chemisorbed CO.

The GS(π) initial state has a conﬁguration:

……ðvπÞ1…ð1sÞ2ð2sÞ2…ð1πÞ4 (13)

where only the Ni and CO orbitals of interest are shown. The orbital labeled vπ is bonding between Ni and CO and, for Ni–C distances near that of CO from the Ni surface, vπ has large amounts of both Ni 4pπ and CO 2πn character. Thus, vπ represents the back-donation from the Ni sp conduction band to CO(2πn). The orbitals labeled 1s and 2s are the O(1s) and C(1s) orbitals of CO, respectively and 1π is the occupied CO π orbital, albeit modiﬁed by the presence of the Ni. The conﬁgurations for the XPS core-hole states are as follows:

……ðnπ′Þ1…ð1sÞ1ð2sÞ2…ð1πÞ4 and (14a)

……ðnπ′Þ1…ð1sÞ2ð2sÞ1…ð1πÞ4 (14b)

for each hole, the Hartree–Fock equations are solved for two different states. For the lowest state the valence π orbital, denoted 1π′, is occupied and for the excited state, the 2π′ orbital is occupied. We use the notation nπ′ to avoid confusion with the 1π orbital associated with CO. The main difference between the two states is for the 1π′ and 2π′ orbitals. The character of the 1π′ and 2π′ orbitals are best understood by ﬁrst considering CO at a large distance from Ni so that there is little interaction. The ionization potential of a Ni atom, as well as the work function of a Ni surface, are ∼5 ev and the electron afﬁnity of CO with an O(1s) or a C(1s) core-hole, based on the energies of the equivalent core approximation molecules [41,42], is ∼10 eV; see Ref. [131]. Thus, at large separation

Fig. 6. BEs and relative intensities, I, for the two C(1s) hole states as a function of the Ni–CO distance, dNi–C; the states are indicated with subscripts as BE1, BE2, I1, and I2. (a) Energies where the left scale is for the BE and the right for ΔBE¼BE2−BE1. (b) Relative intensities where the left scale is for the 1π′ is simply the CO(2πn) orbital in the presence of the CO core hole and 2π′ is the Ni 4pπ orbital.

From the energy balance above, the difference in energy of the two states is ∼5 eV. As the distance between Ni and CO is shortens, the 1π′ orbital acquires Ni 4pπ character and is bonding between Ni (4pπ) and CO(2πn). On the other hand, the 2π′ orbital acquires CO(2πn) character and is anti-bonding between Ni and CO. In Fig. 6(a), we give the C(1s) BEs for the two ﬁnal states, BE1 and BE2, as a function of the distance between Ni and C, dNi–C; ΔBE BE2−BE1 is also given. For convenience, dNi−C 0 is chosen as a distance near the distance of CO above Ni(100); thus dNi−Co 0 and dNi−C40 correspond to distances shorter and longer than equilibrium. The BEs and the ΔBE depend only weakly on dNi−C. Similar weak distance dependence is found for the O(1s) BEs [131]. The situation is quite different for the XPS relative intensities [38]. Neglecting the relaxation of the other passive orbitals, Irel(n)≈|〈vπ|nπ′〉|2 where vπ is the GS(π) ground state valence π orbital deﬁned in Eq. (13) and the nπ′ orbitals are the ﬁnal, core-hole state orbitals deﬁned in Eq. (14). At large dNi−C, the limiting values are Irel(1) 0 and Irel(2) 1. For small separations, Irel(1) becomes close to 1 since both vπ and 1π′ are bonding between Ni 4pπ and (2πn) and Irel(2) becomes close to 0 since 2π′ is anti-bonding. At an intermediate distance, the intensities of the individual I1 and I2 and the right scale is for the ratio I1/I2. two states will be Irel(1) Irel(2) 0.5. The Irel are shown as a function of CO dNi–C in Fig. 6(b). The main conclusion is that one must compare the covalent character of the ground state and the core-hole state orbitals to understand the intensities of main and satellite peaks. It is not sufﬁcient to look only at the covalency in the initial state. We will return to the importance of considering both initial and ﬁnal state covalency in a later section concerning the satellite intensity in the XPS of oxides.

XPS satellites for the NO dimer

Thin ﬁlms of NO adsorbed on Ag surfaces and nanoparticles are known to form dimers; see, for example, Refs. [129, 136–139] and references therein. There is a signiﬁcant and ongoing interest in the photochemistry of NO on Ag; in particular, since these studies have implications for catalytic activity [136–139]. Since the NO dimer is weakly bound [140], it is expected, as discussed above, that the XPS of the dimer will be complex with intense satellites. Furthermore, there is a potential that the XPS, properly interpreted, can help to identify presence of dimers. To demonstrate this potential, we describe an early combined theoretical and experimental XPS study [129] that explained the origin of the complex O (1s) and N(1s) spectra of (NO)2 in terms of the screening of the core-hole. The XPS measurements discussed are for condensed multilayers of NO on Ag(111) where dimers are formed. The theory considers an isolated (NO)2 dimer at the experimental geometry where the core-hole is localized on the N or O atoms on one of the two NO units [72]. In contrast to the NiCO model of CO/Ni, where we could examine changes in the covalent character of a single valence electron, the interaction and the covalent bond between NO units in (NO)2 involves two electrons arising from the 2π orbitals on each of the NO units. The wavefunctions that describe the weak NO–NO interaction for both the initial and ﬁnal core-hole conﬁgura- tions are inherently multiconﬁgurational. They involve mix- tures of conﬁgurations with different distributions of the two valence electrons over the two 2π orbitals. This mixing is large because the conﬁgurations are nearly degenerate. Recall that near degeneracy is not measured by the difference in diagonal energies but by the ratio of the off-diagonal matrix element, Hij, to the difference of the diagonal matrix elements, Hii−Hjj. In Section 6, we describe other cases where conﬁguration mixing in multiconﬁgurational CI wavefunctions, Eq. (5), is essential to properly describe XPS.

Fig. 7. XPS for condensed NO/on Ag(111) compared to multiconﬁgurational

We found that there are two states, separated by ∼4 eV, that carry signiﬁcant intensity for both N(1s) and O(1s). There is a third ﬁnal state for the N(1s) XPS that carries modest intensity but is very near the main satellite. In Fig. 7, we compare the XPS of condensed NO on Ag(111) with our theoretical predictions. The theory gives intense satellites and correctly predicts greater satellite intensity for N(1s) compared to O(1s) The main limitation of the theory is that the satellite BEs are ∼1 eV too high. It is possible that this error arises because the calculations do not include the environmental effects present in the condensed NO dimers on the Ag surface. The close correspondence between theory and experiment is a clear indication that NO dimers are formed in the overlayers.

Furthermore, and perhaps even more important, it is possible to examine the character of the wavefunctions for the different states in order to gain an understanding of the character of the screening in each of the core-hole states. It was relatively straightforward to do this for the previous example of CO/Ni because it was possible to focus on a single orbital and to show how the metal and CO character of this orbital changed as a function of the distance between the CO and the Ni surface. For (NO)2, there are two electrons, one associated with each one of the in-plane 2πn orbitals on each NO unit and it is not possible to follow a single orbital. For this reason, the positions of the centers of charge are used to distinguish the characters of the different ﬁnal states. These positions indicate the character of the response or screening of the “passive” electrons to the core hole. The signiﬁcance of the centers of charge can be understood by considering the FO ﬁnal state of Eq. (4). Since the core hole is localized on one of the N or O atoms, the center of charge will be essentially on the nucleus of the core-ionized atom. Of course, once the response of the passive orbitals is taken into account, the position of the center of charge will change signiﬁcantly, As we discuss below, this displacement of the center of charge for the various states indicates the character of the screening of the core-hole in the different states. When we discuss the covalent bonding and the screening in metal oxides in the next section, it will be necessary to use still another measure to distinguish the character of the states.

In Fig. 8, the centers of charge for the O(1s) and N(1s) hole states are indicated by a ■ and a Δ, respectively, and the core hole is on the right NO denoted β in the ﬁgure. For the O(1s) holes, the center of charge for the ﬁrst excited state is shown even though its XPS Irel is small and it is not plotted in Fig. 7. The positions of the centers of charge for the O(1s) holes are at the level of the N atoms, even though the hole is created on the O atom in the lower right of the ﬁgure. This shows that the charge moves from the N atoms to fully screen the core hole on O. The equivalent core molecule of NO with an O(1s) core- hole is NF+, and the position of the center of charge being at the level of the N atoms, indicates that the NF+ molecules is best described as being N+ and F0 which is not surprising since it is easier to remove an electron from N than from F. The center of charge for the 2nd excited O(1s) hole state, corresponding to the ∼3 eV XPS satellite is almost at the N atom of NO β the core-ionized NO, and shows, for this state, that the electrons of NO α, which is not core-ionized, are not signiﬁcantly involved in the screening of the O(1s) core-hole theory for isolated (NO)2; see Ref. [129].

Figure 8. Centers of charge for the various hole-states of (NO)2. See Ref. [129]

The screening in this case is purely intra-unit and the electrons on the other NO are essentially passive. The screen- ing of the low intensity 1st excited O(1s) hole state is also intra-unit. On the other hand, the center of charge of the lowest O(1s) hole state is almost halfway between the two NO units. This shows there is signiﬁcant involvement of the electrons on the un-ionized NO α to screen the core-hole on NO β; i.e., there is signiﬁcant intra-unit screening. For the N(1s) core- hole, the y coordinates, see Fig. 8, of the centers of charge for all three states are roughly mid-way between the N and O atoms. Since the equivalent core molecule for this case is O+, a homopolar molecule, it is expected that the center of charge would be near the center between the N and O atoms. For the N(1s) hole states, the centers of charge are either near the center between the two NO units or, in the case of the lowest state, even closer to the NO α unit than to the core-ionized NO β unit indicating considerable inter-unit screening. The lowest N(1s) state, corresponding to the main XPS peak is very close to NO α indicating that NO α has almost lost all of its 2π electron and is close to NO+, while the 2π occupation on core-

strong dependence on the parameters of the basis sets used to expand the orbitals in the wavefunctions for oxides and other systems [113]. Another approach used to characterize the distribution of charge in a molecular system is based on the analysis developed by Bader [147,148]. The Bader analysis assigns the charge in certain regions of space to be assigned to a particular atom in the system based on the topological properties of the charge distribution. However, charge in a molecule or in a condensed system does not belong to individual atoms but is shared between atoms; this is the essence of a chemical bond. Thus, as well as assigning an effective charge to an atom, it would be desirable to be able to estimate the uncertainty in the assignment of the atomic charge. Here, we focus on the use of orbital projections [10,149] to estimate the number of electrons or, equivalently, the effective charge to be associated with an atom or a fragment of the complete system. This approach is taken because, as we show below, projection often allows us to estimate the uncertainty of the assignments of charge [60]. The projection involves constructing an projection operator, φ φ†, ionized NO β is almost 2. The centers of charge provide a better understanding of the inter-unit screening of a core-hole than the simpliﬁed picture of charge transfer, CT, where a binary view that there is CT or there is not CT, see Eq. (11), is often used and the covalent character of the orbitals is neglected.

Metal oxides and covalency

It is common to describe oxides, as well as other ionic compounds, in terms of their oxidation states; thus, for MnO, one describes the Mn oxidation state as +2 with a 3d5 occupation. For high spin oxides, the open shell electrons are commonly assumed to have a maximum spin alignment although, for heavy metal oxides, this assumption, which neglects the spin–orbit coupling in the open shell, has been questioned [141]. Our principle concern in this section is to demonstrate that the purely ionic view is misleading and that, very often, there is a signiﬁcant covalent character to the cation-oxygen bonds in metal oxides. Moreover, we will show that the covalent character may be strongly different for initial states and for core-hole states [10,22,60]. The discussion of the covalent character of oxides is key to understanding the signiﬁcance of the XPS of these materials, especially their satellite structure [52,53]. A knowledge of the covalent character of the interactions in oxides is also needed in properly characterize the electronic structure of these ionic systems.

First, it is necessary to establish theoretical methods and criteria that are suitable to characterize and to quantify the covalent character of an interaction. The covalent character of bonds have commonly been quantiﬁed using population analyses [142–145] to determine cation shell occupations and charges in oxides; however, these analyses can give misleading information; for a particularly serious error in the charges from a Mulliken population analysis, see Ref. [146]. Clearly, the atomic charges obtained from population analyses have a where φi is an orbital of an atom or a fragment whose occupation is to be determined. The expectation values of this operator are then taken for the wavefunctions of the total system to determine the desired occupation; the methodology is described in more detail below in connection with applications to selected oxides. It is appropriate to address why a reliable quantiﬁcation of the departures from nominal oxidation states is a key piece of information for understanding the properties of this class of materials, in general, and the analysis of their XPS, in particular. The nominal charges obtained from the stoichiometry of an oxide, or other ionic compound, carry no information about the covalent mixing of metal and ligand orbitals. Indeed, these nominal charges neglect the overlap of metal and cation orbitals, which is critical in the formation of chemical bonds. However, a knowledge of the covalent mixings has the potential to give direct insight into the reactivity and catalytic activity of the material. For the interpretation of the XPS features, we show, in this and the following sections, how the covalent character is related to the distribution of intensity between main peaks and satellites. This opens the way to use the XPS spectra directly to draw inferences about the electronic structure and the chemical and physical properties of the material.

We ﬁrst use graphical views of the charge densities and orbitals to contrast the properties of MgO, a nearly ideal ionic oxide [35], and MnO, where there is some covalent character [10]. Then, we consider quantitative estimates of the cation charge states, which are obtained by the projection of orbital character on the oxide wavefunctions. As well as these simpler oxides, we also consider lanthanide and actinide oxides to show that their covalent character is unexpectedly large; in particular for core-hole conﬁgurations. In Section 3, we introduced the term closed-shell screening to describe the changes in covalent character for core-hole conﬁgurations. Here, we discuss how the large closed shell screening in CeO2 affects the XPS Franck–Condon vibrational broadening (ﬁrst introduced in Section 3 for MgO). In Section 6, we consider other, more general, effects of covalency on the XPS of metal oxides.

Charge density difference plots are shown in Fig. 9 for embedded cluster models of the cubic oxides MgO and MnO [60], where experimental geometries and bond distances are used [150]. The plots are through a plane (denoted xy) that contains a central metal cation and 4 nearest neighbor O anions near the corners of the square. The density difference, Δρ, is the difference between ρ for the HF Ψ and an anti-symmetric Ψ formed from superposed spherical atoms, which are HF solutions for the isolated ions: Δρ ρ(HF)−ρ(Spherical). The wavefunctions for the superposed atoms take the Pauli exclusion into account [151,152] and, hence, the Δρ shown in Fig. 9 represent chemical changes in ρ(HF). The solid lines in Fig. 9 are for contours of constant Δρ 40, the dashed lines for contours of constant Δρ o 0, and the dotted lines are for Δρ 0. The contours are for uniform steps of Δρ and there are cutoffs for large magnitudes of Δρ. The simpler case of MgO, in Fig. 9(a), is discussed ﬁrst. There is a polarization of the spherical O towards the Mg2+ cation at the center. There is also a departure of the O charge density from spherical away from the central Mg and toward the neighboring O anions indicating the formation of the O(2p) band. The small changes that occur in the immediate region of the central Mg arise as a result of the incomplete basis sets [153,154] used and do not represent actual charge ﬂow. There is, however, no buildup of charge between O and Mg as would occur if a covalent bond were formed. Another indication of the ideal ionicity of MgO comes from the difference in the energies of the HF and superposed spherical atoms wavefunctions; the energy of the HF wavefunction is lower by only 2.6 eV. Further, a CSOV decomposition [44–46] shows that 90% of this energy change comes from the reorganization of the O charge due to the presence of the Mg2+ cation and not from a departure from the perfect ionicity assumed in the superposition of spherical charges. The situation is very different for MnO, Fig. 9(b), where there is a buildup of O charge between O and Mn. This buildup appears to have a symmetry that would arise from orbitals of e symmetry in the Oh point group of MnO. Furthermore, the energy change for the wavefunctions going from the superposed spherical atoms to the HF Ψ is over 10 eV, almost 4 times the change for the MgO wavefunctions. An important cause for this large energetic improvement for MnO is the covalent character of the MnO cluster orbitals. We give in Fig. 10, contour plots of two eg orbitals taken from HF wavefunctions for the embedded MnO6 cluster [10]; the orbitals for the initial state conﬁguration are plotted in Fig. 10(a) and (b) and the orbitals for the 2p-hole conﬁguration are plotted in Fig. 10(c) and (d). The same conventions are used for Fig. 10 as for the plots in Fig. 9 except that the signs shown by the solid, dashed, and dotted lines now refer to the orbital, φ, rather than the density difference, Δρ. The two orbitals plotted are the closed shell, e4, orbitals, Fig. 10(a) and (c), that are dominantly of O(2p) character and the open shell, e2, orbitals, Fig. 10(b) and (d), that are dominantly of Mn(3d) character. We have plotted the xy component of the degenerate pair of e orbitals since these components can form covalent combinations with the O ligands in the plane shown. In the semi-empirical Anderson Model formulation [12–14], these orbitals are treated as though they were pure metal or ligand orbitals. Clearly this is not the case. Although, the closed shell orbitals, shown in Fig. 10(a) and (c), are dominantly p orbitals on O, there are contours that extend to the Mn atom. Further, the combination is a bonding combination of O(2p) and Mn (3d) in this closed shell orbital. The open shell orbitals, shown in Fig. 10(b) and (d), are dominantly Mn(3d) and the d(xy) character of these orbitals are obvious. However, there is some O(2p) character and the node, or zero, between O and Mn demonstrates that the orbitals are anti-bonding. Before turning to quantitative estimates of the covalent mixing, we use orbital contour plots, to show that the covalent character between O (2p) and Mn(3d) is different for the initial and the ﬁnal, core- hole conﬁgurations. Comparing Fig. 10(c) and (d) with Fig. 10 (a) and (b), it is clear that the covalent character, for both the ﬁlled bonding orbital and the half ﬁlled anti-bonding orbital, is larger for the 2p-hole conﬁguration than for the initial state conﬁguration. The orbital plots in Fig. 10 are for non- relativistic HF wavefunctions [10]. Although, these plots are only for eg orbitals, there is also closed shell bonding and open shell anti-bonding covalent character for the t2g orbitals [10].

Fig. 9. Charge density difference contours for (a) MgO and (b) MnO; see text.

Fig. 10. Orbital contour plots of the dominantly O(2p) closed shell, denoted e4, and the dominantly Mn(3d) open shell, denoted e2, orbitals for the embedded MnO6 cluster model of MnO: (a)e4 orbital for the initial state MnO6 conﬁguration; (b) e2 orbital for the initial state; (c) and (d) e4 and e2 orbitals for the ﬁnal state 2p-hole conﬁguration.

In order to quantify the covalent character of the metal– oxygen interaction, we turn to a discussion of the projection of the orbitals of the isolated cation on the cluster model orbitals to determine the occupation of the cation orbitals in the wavefunctions for the condensed phase [149].

The speciﬁc summation to obtain the 3d occupation in MnO from projection, NP(3d) is in Eq. (15) can be limited to a subset of the orbitals or even to a single orbital; as we show below, the properties of subsets of the orbitals may help explain the chemistry. Second, the occupation NP depends weakly on the choice of conﬁguration for the isolated atom [149,155] and all reasonable choices will give the same physical picture. Third, as with all assignments of atomic charges, the value of NP is inﬂuenced by the overlap of orbitals on different atoms [142–145,149]. In other words, because chemical bonding involves the sharing of electrons where the projection operator, φ(3d)φ†(3d), is for a 3d orbital obtained for an isolated atom and the matrix element is simply the square of the overlap integral 〈φi|φ(3d)〉. The sum is weighted by the occupation of the orbitals in the MnO conﬁguration; the sum over the different atomic 3d orbitals is not given explicitly in Eq. (15) but, in practice, is always taken. As written, Eq. (15) is strictly valid for a single conﬁguration or for a single CSF. The projections could also be made for the more general CI wavefunctions of Eq. (3) but we restrict ourselves to the simpler formalism for a single conﬁguration since our primary interest is to identify the covalent character of the bonding. The generalization of Eq. (15) to other projections and other systems is straightforward [149]. Three comments are in order. First, the summation between atoms, it is not possible to uniquely assign electrons to atoms when covalent bonds are present. We describe this an uncertainty in the assignment of charges to the atoms in a compound and we describe below ways to use internal consistencies in order to obtain estimates of the uncertainties. The NP(3d) for MnO obtained from projections on relati- vistic and non-relativistic wavefunctions for the ground and 2p-hole conﬁgurations of an embedded MnO6 cluster are listed in Table 9 [10]. The projections are grouped into the closed shells of the cluster where the nominal number of d electrons is zero and the open shells where the nominal number of ﬁve d electrons is distributed, in the ground state, into 3t2g and 2eg. For both the relativistic and non-relativistic wavefunctions, the projections are for the average of conﬁgurations with orbitals

Table 9

optimized for this average [68]. For the relativistic case, the division into t2g and eg is approximate since, in the Oh double group [156], the appropriate representations are the doubly degenerate γ7 and the four fold degenerate γ8, where t2g and eg can mix in the γ8 spinors; however, the mixing of eg and t2g is small for MnO.

Not surprisingly, the relativistic and non-relativistic projec- tions are quite similar and our discussion focuses on the relativistic projections. For the initial state conﬁguration, the covalent bonding in the t2g and eg closed shells leads to an Mn 3d occupation of almost half an electron in these shells. Because of the anti-bonding character of the t2g and eg open shells, the open shells have an NP(3d) 4.75, reduced from the nominal value of 5. As shown in Table 9 for the open shell orbitals, the covalent character of the t2g orbitals is only about 25% of that for the eg orbitals. The smaller covalent mixing of the t2g orbitals follows because the t2g and eg orbitals are directed between and towards the O atoms, respectively [10,157]. Summing the t2g and eg covalent mixings, the open shells have lost ∼0.25 d electrons because they now have anti- bonding mixing with the O electrons. However, the total number of Mn 3d electrons has increased by 0.2 electrons over the nominal value of 5, see Table 9, because of the covalent bonding character of the fully occupied closed shell orbitals. This leads to a reduction of the net charge on the Mn cation below the nominal value of +2. Similar comments apply to the 2p-hole state orbitals except the covalent mixing of these orbitals is substantially larger and there are now almost 5.5 3d electrons associated with Mn. The 0.26 increase of NP(3d) for the 2p core-hole conﬁguration is a “closed shell” screening of the 2p core-hole. Because of this closed shell screening, there may not be a need to invoke a change of conﬁguration with a transfer of one or more electrons from the ligands to the metal 3d; see Eq. (10), as is done in the semi-empirical Anderson model approach.

When the metal 3d and ligand orbitals overlap, the bonding and anti-bonding character of orbitals affects the apparent occupations, NP, determined from projection [149]. Speciﬁ- cally, the overlap of the ligand and metal orbitals will act to increase the NP(3d) for the dominantly ligand bonding orbitals and decrease NP(3d) for the dominantly metal open shell orbitals. The net effect is that the overlap of the metal 3d and ligand 2p orbitals will exaggerate the covalent character; although, we would expect to see similar exaggerations for both the initial and the ﬁnal state conﬁgurations. An estimate of the effect of the overlap on the value of NP(3d) can be made by projecting the Mn 3d orbitals on the wavefunction for O6 [10,158]. This projection is 0.17 and it is very likely to be an upper bound for the “error” of the value of NP(3d) [10,158]. While there are formulations which use projections based on the overlaps to form a population analysis [159,160], we prefer to use the overlap projection as an estimate of the “uncer- tainty” of assigning 3d electrons to the cation because the electrons are shared between the cation and ligand. Although, this uncertainty reduces the covalent character of the interac- tion, a modest amount remains. There is clearly a signiﬁcant covalent character to the interaction for both the initial and ﬁnal, core-hole, conﬁgurations. Furthermore, the increase of the total NP(3d) for the core-hole conﬁguration, shows that there is closed shell screening of the Mn 2p-hole by the O electrons. The overall view is that MnO is dominantly an ionic system with modest covalent character. In the following paragraphs, it will be shown that closed shell screening is even larger for heavy metal oxides and contributes to the complexity of their XPS.

We next consider closed shell CeO2 and UO3 modeled with embedded CeO8 (Fig. 3c) and UO6 (with the δ-UO3 coordina- tion according to Ref. [161]) clusters and discuss covalency only for relativistic wavefunctions [60]. The nominal oxidation states and occupations are Ce(IV)–(4f)0 (5d)0 and U(VI)–(5f)0 (6d)0, respectively. Since these are closed shell systems, the occupations given for the Ce 4f and 5d and for the U 5f and 6d can only arise from closed shell orbitals that have bonding character between the metal and ligand. A novel approach is used to estimate the uncertainty of the occupations, NP, obtained with the projection analysis [60]. As well as project- ing the cation orbitals on the embedded clusters, we also project the orbitals of an O6 cluster (for UO3) or an O8 cluster (for CeO2). Bare On clusters were also used in earlier work to determine the energetic importance of the O polarizations and bonding in oxides with lighter metals [109,126,162]. The present interest is to use the projections of the O(2p) character as a guide to counting electrons involved in the covalent interactions [60]. Since the On clusters also have cubic symmetry, the orbitals, or spinors, belong either to g or u point group representations. By symmetry, only the O(2pg) electrons can mix with the central metal atom d orbitals and only the O(2pu) electrons can mix with the central metal atom f orbitals. For the O(2p) projections, the quantity ΔNP(O2p), the number of 2p electrons lost from the full 2p occupation in the bare On cluster, is used rather than the total number of 2p electrons, NP(O2p). If there were no uncertainties in the projections, then NP(metal cation orbital) −NP(O2p), where the metal cation orbital is one of the d or f orbitals listed above and the O2p projection is restricted to either g or u orbitals depending on whether the metal orbital is d or f. In other word, the number of electrons gained by the cation orbital is equal to the number lost from the O 2p ligand orbitals. Because there are uncertainties in the projection due to the overlap of the metal and ligand orbitals, this equality will not be satisﬁed exactly. The extent of the uncertainty in the assignment of occupation based on the projections can be estimated from the extent to which the magnitudes of the two projections are not equal. We consider ﬁrst the projections, in Table 10, related to the occupation and covalent bonding with the outermost metal nf orbital for both the initial and the ﬁnal core-hole conﬁg- urations. The difference of the NP between the initial and the ﬁnal conﬁgurations is given as Δ(Initial) and it provides a direct measure of the closed shell screening.

For the initial state of UO3, the occupation of the “nominally empty” 5f shell is 1.4 electrons and the loss of the O(2pu) electrons is −0.8. Although these two estimates of the covalent interaction of the U(5f) are different, they both indicate that the covalent mixing in bonding orbitals leads to a large 5f occupation of nearly 1 electron. It is critical to stress that this 5f electron character is not in an open shell; the UO6 cluster does not have open shells. Rather, the 5f character arises from the covalent bonding in the dominantly O(2p) closed shells. For the 4f-hole state, there is a large increase of NP(5f) by 1.2 electrons and a corresponding reduction of ΔNP(O2pu) by 0.9 electrons. The magnitudes of the changes in NP(5f) and ΔNP(O2pu) occupations between the initial and 4f-hole con- ﬁgurations, Δ(Initial), are more similar than the agreement of the absolute values of NP and ΔNP for either the initial or 4f core-hole conﬁgurations. As we noted in the discussion above for the MnO 3d occupation, the errors or uncertainties in the NP will be similar for the initial and ﬁnal, core-hole conﬁg- urations. We can view this as a cancellation of errors in the NP which give more accurate values for the closed shell screening; there is a similar cancellation for CeO2, see below. The |Δ(Initial)| differ by ∼0.3, which is less than half of the differences between |NP(5f)| and |ΔNP(O2pu)|; see Table 10.

The Δ(Initial) values give strong support to a closed shell screening of the 4f-hole by∼1 electron. The large initial state participation of the U(5f) in covalent mixing with the O(2pu) orbitals is, at least in part, a consequence of the large nominal positive charge of +6 on the U cation. For CeO2, where the nominal Ce charge is +4, NP(4f) 0.32 is smaller than the cation occupation for UO3 but 50% larger than the cation occupation in MnO where the nominal ionicity is only +2; see Table 9. When a 3d or 4d core-hole is made on Ce, the closed shell screening is large as judged from both NP(4f) and ΔNP(O2pu).

Furthermore, the closed shell screening is larger for the deeper 3d core-hole than for the 4d core-hole; see Table 10. The different screening occurs naturally with our wavefunction based theory [21,22,60] while for semi-empirical Anderson model theories, shallower or deeper core-holes are distinguished by empirically adjusting parameters to ﬁt observed XPS spectra [112,163]. Furthermore, the core-hole is screened through a change in the chemistry of the core- ionized atom with its surroundings rather than a change in conﬁguration, see Eq. (10), as required in semi-empirical theories [8,12–14,112]. Finally, we point out that the uncer- tainty in the assignment of charges to atoms is a natural consequence of chemical bonding and interactions; when bonds form, electrons are shared between atoms and cannot be uniquely assigned at one atom or another.

The uncertainty of the assignments of metal character due to covalent mixing with ligands is seen very clearly for the analogous projections for the outermost d shells of U and Ce and the O(2pg) shells shown in Table 11 [60]. The projection of the nominally unoccupied cation 5d or 6d metal orbitals on the initial state wavefunctions for the embedded cluster models of UO3 and CeO2, give large NP(nd)≈3. However, the magnitude of ΔNP(O2pg)≈−0.6 is much smaller and indicates that there is substantial uncertainty in the actual occupation of this d shell. This uncertainty arises because of the larger spatial extent of the d orbitals than of the f orbitals and, thus, a much larger overlap of the d and the O(2p) orbitals. While, the d orbitals do have some participation in the covalent interaction with oxygen, it is not possible to give a precise value for the d occupation. The important fact is that the NP(nd) and ΔNP(O2pg) for the core-hole states are very similar to the values for the initial states; see Table 11. The changes between initial and ﬁnal states are not especially large and are considerably smaller than for the projections of the nf and O (2pu) shells shown in Table 10. The participation of the d shells in the closed shell screening of core holes is much smaller than that for the f shells. There is a simple chemical explanation for this difference.

Table 10

Table 11

The outer f shells are more contracted than the outer d shells [164]. Since, there is an increase in the number of f electrons by ∼1, the effective charge seen by the nd electrons is, to a large extent, unchanged. Thus, the closed shell screening by covalent mixing to form bonding orbitals between the ligand and nf orbitals cancels the increase in charge due to the core-hole and the electrostatic driving force for closed-shell screening into the nd orbitals is largely absent.

The large closed shell screening in the heavy metal oxides has important consequences for the Franck–Condon vibra- tional broadening of the XPS peaks. In a previous section, it was shown that a core-hole on the Mg cation in MgO leads to a 0.12 Å decrease in the Mg–O distance and this leads to a vibrational broadening of the Mg 2p FWHM by ∼0.8 eV [35]. Because, as discussed above, the closed shell screening of a core-hole in MnO is only modest in size, ∼0.2 electrons, the Mn–O distance for a 2p core-hole on Mn is reduced by 0.13 Å and this leads to a broadening of the Mn 2p FWHM by 0.9 eV [165]. On the other hand, calculations of the Ce–O distance for a core-hole on Ce lead to only a small change in the Ce–O distance. In fact, for a 4s core-hole on Ce, the Ce–O distance for the core-hole state is actually larger, but by only 0.01Å, than for the initial state [165]. This is because, as suggested by the projections, the core-hole is actually over-screened by the ﬁnal state covalent interaction between the O(2p) and the Ce (4f). With this small change in the Ce–O distance, the vibrational broadening must be small and there must be a different origin for the large broadening observed in the CeO2 XPS [20–22].

XPS satellite features and many-body effects

In prior sections, we primarily considered PES main peaks except for Section 4 where satellites were considered speciﬁ- cally in relation to covalent bonding and anti-bonding orbitals, especially for cases of weak chemical interactions. In the present section we expand this earlier treatment to consider a wide variety of many-body effects that can lead to satellites, often with very large intensities. A suitable deﬁnition for many-body effects is that they require the use of wavefunc- tions that cannot be represented by a single conﬁguration or a single CSF. In particular, for closed shell systems, this means that we must go beyond a single determinant description of the wavefunction. Final states that correspond to multiplets are also considered a many-body effect, although the reasoning requires explanation. A multiplet can be regarded as a degenerate set of states that arise from the angular momentum coupling of the open shells in a single conﬁguration [88] and, hence, not a many-body effect. On the other hand, wavefunc- tions that are eigenfunctions of the orbital and spin angular momentum operators normally are combinations of determinants [88] and, hence, multiplets can be grouped with many- body effects. Furthermore, there is a type of angular momentum coupling that is a pure many-body effect. This occurs when the valence open shell can couple to multiplets other than the ground state multiplet of the initial state conﬁguration. It is often possible to couple the core-hole with the re-coupled valence open shell to give a total multiplet the same as when the core-hole is coupled to the ground state multiplet of the valence open shell [30,166,167].While these angular momentum re-coupled multiplets are XPS forbidden, they are able to mix with the XPS allowed multiplets and thus lead to “satellites” with large intensity [30]. An example of the many-body mixing of XPS forbidden with XPS allowed multiplets in given in Section 6.2. For all these reasons, we chose to group the discussion of XPS multiplets with many- body effects. In Section 6.1, we consider the multiplet splitting of the N(1s) and O(1s) XPS of NO [17]. We also examine the Mn 3s XPS [4], where multiplet theory is not sufﬁcient. In Section 6.2, angular momentum recoupling is introduced as a major correction to the simple angular momentum coupling theory. This recoupling is important because the re-coupled CSFs are nearly degenerate with the XPS allowed multiplets. Other types of near degeneracy are also discussed that provide a more complete description of “atomic” processes than multiplet theory alone. In this section, the effects largely depend on orbitals with dominantly atomic character but which can be modiﬁed by ligand ﬁeld splittings and covalent bonding. In Section 6.3, we consider very recent results [21,22,52,53] for the XPS of the two closed shell oxides, CeO2 and UO3 whose covalent character was discussed in Section 5.

Multiplet splitting

NO is an open shell molecule with a ﬁlled bonding 1π orbital and a single electron in the anti-bonding 2π orbital. The conﬁgurations of the 2Π ground state and the N(1s) and O(1s) hole states are

where 1s is the O(1s) shell and 2s is the N(1s) shell. The spins of the single 1s electron and the single 2π electron can couple parallel to 3Π or anti-parallel to 1Π multiplets [17,71,88]. In the HF approximation, the energy difference, ΔE between 3Π and 1Π multiplets is 3K(ns,2π) [88], where K is the exchange integral between the core-hole and the open valence 2π shells. Since there are 6 states in the 3Π and but only 2 states in the 1Π multiplet, the ratio of the intensities is expected to be I(3Π)/I(1Π) 3 [17,168]. The 3Π−1Π multiplet splittings in NO were calculated with Hartree–Fock (HF) wavefunctions determined separately for each multiplet, Eq. (15) [71]. The measured (calculated) multiplet splittings are ΔE(O1s) 1.41 (1.35) eV and ΔE(N1s) 0.53(0.48) eV [71], where the HF multiplet splittings are just outside of experimental error. With the resolution available using a laboratory XPS system for the measurements on NO [17], the multiplet, or exchange, splitting of the N(1s) XPS peaks could not be resolved as two separate peaks and the splitting of 0.48 eV was determined by ﬁtting the measured spectra with two Gaussian peaks [17,71]. On the other hand, the 1.35 eV separation of the multiplet split O(1s) peaks could be resolved in the measured XPS spectra [17]. If the multiplet splittings are to be used as a guide to the chemical bonding in NO, it is essential that the very different multiplet splittings be understood in terms of the characters of the bonding NO 1π and the anti-bonding NO 2πn orbitals. Indeed, based on the character of the 2πn orbital, there is a simple reason that the O(1s) splitting is ∼3 times larger than the N(1s) splitting, The 2πn orbital is dominantly on N; for the initial state, 2πn is ∼70%, on N with only ∼30% on O. The magnitude of the exchange integral, K(ns,2πn), depends largely the local character of the 2πn orbital at the center of the ns 1s orbital and, thus, K(2s,2πn) 4 K(1s,2πn). Of course, the distribution of the 1π orbital does not contribute to the multiplet splitting since the 1π shell is ﬁlled and makes identical contributions to the energies of both the 1Π and 3Π ﬁnal XPS multiplets [88]. However, once we know that the 2πn orbital is dominantly on the N atom, then, from orthogon- ality, it follows that the 1π orbital must be dominantly on the O atom. The experimental and theoretical XPS intensity ratios, I (3Π)/I(1Π), have also been compared with each other and with the statistical ratio of 3. The calculated and observed intensity ratios are both somewhat larger than the expected statistical value of 3 but the deviations are less than 25%. This small deviation from the statistical is relevant for the Mn 3s XPS multiplet splitting, discussed next. Other examples of multiplet splitting in molecules can be found in Ref. [6].

The dominantly ionic character of MnO, Section 5, where the 5 open shell electrons are ∼90% Mn 3d, even for a deep 2p core-hole, suggests that an isolated Mn2+ cation with a 3d5 open shell coupled to the high spin, 6S, multiplet would be a suitable model for the Mn XPS. A direct comparison of the measured 3s XPS in Mn atoms and in MnF2 and MnO crystals shows that the spectra are very similar for all these systems, which justiﬁes the belief that Mn2+ is a good model for the XPS of MnO [169]. It should be noted that the atomic XPS data in Ref. [169] was for Mn atoms and not the Mn2+ cations appropriate for ionic crystals and that the energy separation of the ﬁrst two peaks is slightly larger for Mn0 than for MnF2 and MnO. This slightly larger separation is reproduced in the calculated separation of the ﬁrst two peaks between Mn0 [47] and Mn2+ [3]. One expects that the energy separations and the relative intensities of the 3s XPS peaks should be similar for Mn2+ and Mn0 since the 4s electrons, present in Mn0, are spectators for the core-level ionization, However, there will be some differences, which arise because a portion of the 4s charge density penetrates toward the core of the Mn atom. It is important that the differences between the theoretical results for Mn0 and for Mn2+ are mirrored in the differences between the XPS for the MnO or MnF2 crystals and the Mn atom. Following the logic for NO, discussed above, the 3s XPS should have two peaks. 7S and 5S split by 6 K(3s,3d) [88]. HF theory predicts a splitting of ∼14 eV and an intensity ratio of 1.4:1 while the XPS measurements for several ionic crystals give a splitting of ∼6 eV with an intensity ratio of ∼2:1 [4,169]. This ratio is larger, by over 40%, than the statistical ratio of the two multiplets and is outside the range of deviation suggested from the results for NO; see above. The ∼8 eV error in the multiplet splitting is also much greater than would be expected from the errors of BEs calculated from atomic HF calculations [51,70]. Clearly, many-body effects other than multiplets must make important contributions as discussed in the following sub-section.

Near-degeneracy effects: angular momentum recoupling and FACs

Before turning to the Frustrated Auger Conﬁgurations, FACs, that are needed to address the Mn 3s XPS multiplet splitting discussed above, we consider the angular momentum recoupling needed to treat core-holes in non-spherically sym- metric shells. Such open core shells lead to a much richer angular momentum coupling than the s or s levels discussed previously. Freeman et al., FBM [37], considered the 3p holes for Mn2+ where 3p5(2P) couples with 3d5(6S) to give 7P and 5P multiplets. However, it is possible to ﬁnd total 5P multiplets by recoupling the d5 shell to either 4D or 4F. These conﬁgurations can be viewed as a spin ﬂip within the d shell coupled to a simultaneous spin ﬂip within the 3p shell. While the spin-ﬂip conﬁgurations are XPS forbidden, they can and do mix with the one allowed conﬁguration where the d5 electrons remain coupled to 6S. This leads to XPS peaks beyond the two XPS allowed 7P and 5P multiplets. Hence, it is properly described as an angular momentum recoupling. Bagus et al. [30] also used the atomic model of Mn2+ to describe MnO but they extended the FBM treatment to include relativistic effects with a full 4 component treatment including spin–orbit coupling, which enabled study of both the 2p and 3p XPS. The treatment of angular momentum recoupling was made in the following way: All determinants were formed by distributing the 5 electrons in the ionized NP shell in all possible ways over the 6 np1/2 and np3/2 spinors and the 5 electrons in the 3d shell in all possible ways over the 10 3d3/2 and 3d5/2 spinors. A complete conﬁguration mixing was made over this set of determinants and all the wavefunctions were determined by diagonalizing the CI Hamiltonian matrix. The intensities were determined using the sudden approximation and broadened with a Voigt [36] convolution of a 1.0 eV FWHM Gaussian and a 0.7 eV FWHM Lorentzian. [30] The theory for Mn2+ and the XPS experiment for MnO are compared in Fig. 11 where individual peaks with large intensity as well as the sum of all contributions are shown. No adjustments of the relative energies, Erel, or the relative intensities, Irel, were made. Note that the doublet predicted for the leading edge is also found in the higher resolution experimental XPS of Ref. [28]. Considering that the theory takes no solid state effects into account, the agreement with experiment is remarkable. In particular, there is no need to invoke, as have others [14,170], conﬁgurations for charge transfer from O to Mn, see Eq. (11), in order to explain the main features of the spectra, which arise from a purely atomic model of MnO. The two most intense composite peaks are due almost entirely to ionization of 2p3/2 and 2p1/2. The intermediate satellite has roughly equal con- tributions from both 2p3/2 and 2p1/2. The origin of the broad

Fig. 11. Comparison of the theoretical Mn 2p XPS for Mn2+ with experiment for MnO. The dotted line is experiment and the solid lines are from the theory; the dark solid line is the sum of the intensities for all ﬁnal states while the light solid lines are for individual states with large intensity. The curves are replotted from the data in Ref. [30].

leading edge is the multiplet splitting of the coupling of 2p3/2 with the 3d5 shell high spin coupled to 6S with J=5/2. This coupling leads to the 4 multiplets shown with J=4, 3, 2, and 1 with intensities that almost follow the (2J+1) statistical ratio. This is an excellent example of how unresolved multiplets can lead to major broadening of XPS features. On the other hand, a parallel analysis of the main XPS Mn 2p1/2 peak at Erel∼15 eV, see Fig. 11, in terms of the multiplet splitting of the 2p1/2 and 3d open shells does not hold. Based on the coupling of j 1/2 for the hole in the 2p1/2 shell with j 5/2 for the 3d5shell, we would expect 2 peaks with J 3 and 2 with an intensity ratio of 7:5. It is clear from Fig. 11 that several individual ﬁnal states contribute to this peak, not just 2; within the peak, there are large contributions from 5 individual ﬁnal states. The difference of this 2p1/2 peak from the leading edge 2p3/2 peak shows that many-body effects make a much larger contribution for the 2p1/2 than for the 2p3/2 peak. These effects may well include recoupling of the 3d5 electrons away from the XPS allowed 6S5/2 coupling to other, XPS forbidden, multiplets.

The effect of covalency in MnO, see Section 4, on the 2p XPS was studied with an embedded MnO6 cluster [10]. The main effect is to reduce the splittings of the different peaks largely because the partial ligand-character of the Mn 3d orbitals decreases the exchange integrals. The reduction of the multiplet splitting improves agreement between the theoretical simulations and measurement [10]. With the MnO6 cluster modiﬁed to include the steric repulsion of the next shell of cations, Franck–Condon vibrational broadening, see Section 4, contributes a large fraction of the ad hoc broadening of each multiplet used to compare theory with experiment [165]. This is important since it provides an independent justiﬁcation for the choice of FWHM for the calculated multiplets.

In the previous sub-section, we showed that the angular momentum coupling of the Mn 3s-hole with the open 3d5 valence shell to form high and low spin multiplets did not correctly describe either the relative energy or the relative intensity of the leading peaks in the Mn 3s XPS. Since there is very good reason to expect that the isolated cation should be a good ﬁrst order model for Mn in an ionic crystal, it is appropriate to ask if there are atomic many-body effects that have been neglected in the multiplet analysis. Consider the conﬁguration, 3s23p43d6, formed by the “excitation” of two 2p electrons, one to ﬁll the 3s shell and one promoted into the half-ﬁlled 3d shell, denoted 3p2–3s3d. For this conﬁguration, several angular momentum couplings are possible to the low spin 5S multiplet but not to the high spin 7S multiplet [3,37,88,171]. Thus, mixing this excitation with the XPS allowed 3s13p63d5 conﬁguration can lower the energy of the 5S 3s-hole multiplet but it cannot change the energy of the high-spin 7S multiplet. This would change the Erel in the correct direction from the pure multiplet splitting results. In recent work [47], the acronym FAC, frustrated Auger conﬁg- uration, has been applied to this type of excitation because of its similarity to an Auger transition except an electron is promoted into a bound state not into the continuum. This is a useful acronym since it can direct us to identify conﬁgurations that may lead to low lying, nearly degenerate, excited states.

Indeed, the energies of the angular momentum coupled CSFs for the 3p2–3s3d excitation are ∼15 eV above the … 3s13p63d5 CSF, which is comparable to the non-zero off-diagonal Hamiltonian matrix elements [3,37]. Including this FAC in the CI wavefunctions for the 3s-hole states, originally reported in Ref. [3], leads to a major reduction in the separation of the ﬁrst two XPS peaks as well as two satellites that are observed [3]. The comparison of the theory including the 3p2–3s3d FAC with the experiment of Ref. [169] is shown in Fig. 12 taken from Ref. [47]. The dark vertical bars represent the 1973 theoretical results from Ref. [3]. The improved agreement over the multiplet splittings without the

Fig. 12. Experiment for the Mn 3s XPS for MnO, MnF2, and Mn gas, from Ref. [169], compared to theory, solid vertical bars from Ref. [3], and dotted vertical line from Ref. [47].

FACs is evident but the separation of the ﬁrst two peaks is below experiment by ∼2 eV. Okada and Kotani [13] were able to reproduce the experimental splitting by reducing a key atomic interaction integral to 75% of the value obtained from a rigorous atomic calculation but they did not introduce any new many-body effects beyond the 3p2–3s3d FAC of Ref. [3]. In 2004, Bagus et al. [47] recognized that there is another important many-body effect that involves a FAC where a 3p electron still ﬁlls the 3s shell but now a 3d electron is promoted to a 4f shell. The 3p3d–3s4f FAC has a differential affect and lowers the high spin state by 2 eV more than the excited low spin 3s-hole state bringing the low spin 3s-hole state to a correct relative energy without the need to empirically adjust directly computed integrals. The position of the lowest 5S 3s-hole state is show by a dotted vertical line in Fig. 12. This example shows that empirical adjustments to bring theory into agreement with experiment may get the “right” answer for the “wrong” reason.

Satellites in closed shell oxides

Shake excitations have been the traditional explanation for an important class of XPS satellites in atomic and molecular systems [5,6,19,38]; although this interpretation may also be applied to condensed systems, including oxides [21,22,39] and to CO chemisorbed on metal surfaces [133,172,173]. The logic of the name shake arises because the photoelectron, as it exits the system, has some probability to excite, or shake, another electron from an occupied orbital into a partially occupied or an unoccupied level; from this excitation, the photoelectron loses some of its kinetic energy. There is a probability that the electron leaves the system without making this excitation of a valence electron and there are probabilities for exciting the valence electrons into different excited levels. A more rigorous treatment than this simple view involves mixing the shake excited conﬁgurations with the conﬁgurations for the other many-body effects discussed in the previous sections. If the covalent mixing of metal cation and ligand orbitals is neglected, then shake is equivalent to including the conﬁgura- tions of Eqs. (11b) and (11c) into the hole-state wavefunctions. This is the reason that these excitations have been called charge transfer or CT [7,8,12–14,43]. However, the covalent mixing may be very large [60]. Regardless, it is more fundamental to describe this many-body process as shake rather than as CT because the electron is excited from a molecular bonding orbital and not strictly from a pure ligand orbital. Since it normally costs energy to make this excitation, the outgoing electron loses kinetic energy and the shake satellites are normally to higher BE than the main line, However, it is possible to have shake-down satellites on the low BE side of the main XPS line, especially when more than one electron is involved in the shake excitation [21]. In this regard, papers based on semi-empirical Anderson model Hamiltonian treatments of the CT or shake effects often claim that the lowest energy core-hole states are dominated by the shake or CT conﬁguration of Eq. (11b) rather than by the conﬁguration without shake of Eqs.(11a) [7,8,12,20,43].

Although there are exceptions, [174] in most cases where rigorous, non-empirical theory is used, the lowest state does not have a signiﬁcant shake contribution [10,21,22,52,53,175]. In the remainder of this section, we consider two cases of closed shell oxides. One is the 4s XPS of CeO2 [21,22], where both a FAC as well as shake excitations contribute to the satellites. Here, we examine the different many-body contribu- tions to the XPS. The second case is the 4f XPS of UO3 [53], where we could not identify FACs that might contribute to the XPS and only contributions from shake excitations to the satellite structure were considered. In this case, we examine the inﬂuence of bond distance on the satellite intensity. In both cases, we use the sudden approximation (SA) [38,39] to determine the Irel, where important quantities are the overlap integrals between the orbitals of the initial and ﬁnal, core-hole states. As discussed in the previous section, the extent of covalency is quite different for the initial and ﬁnal states. Hence the SA intensities should be strongly affected; this is analyzed in connection with the XPS satellites for UO3.

The comparison of theory and experiment for the Ce 4s XPS of CeO2 [21] is shown in Fig. 13 where the theory includes FAC and shake many-body effects. The main peak (A) is aligned so that it is at the same energy as the leading XPS peak and it is broadened by a Gaussian with a FWHM chosen to match experiment; the same FWHM is used to broaden all the theoretical ﬁnal states. No adjustments to the relative theore- tical energies or intensities have been made. It is important to recognize that, in general, several rather than a single ﬁnal state contribute to each of the resolved XPS peaks. An important reason for this distribution of intensity is that many-body effects combined with spin–orbit and ligand ﬁeld splittings distribute the allowed XPS intensity over several ﬁnal states [10,21,22,50]. In particular for CeO2, except for the main peak, A, in Fig. 13, several individual ﬁnal states contribute to each of the features B, C, and D. The peaks B and C are at an Erel that is larger than experiment by ∼3 eV. This error arises from the choice of a single set of orbitals optimized for the main peak, A [21,22]. However, the wavefunctions allow the composition of the ﬁnal states to be analyzed in terms of the many-body effects that contribute to these states. The different determinants in the wavefunctions are placed into three groups: Shake(0), Shake(1), and Shake(2)

Fig. 13. Experiment, upper curve, and theory, lower curves, for the Ce 4s XPS of CeO2 [21]. The darker theoretical curve is the envelope of all contributions while the contributions of individual states with large Irel are shown below the envelope. The letters A–D indicate the four main theoretical features.

which are distinguished by the occupations of bonding, dominantly O(2p) orbitals, and anti-bonding, dominantly Ce 4f and Ce 5d orbitals. When there are no shake excitations, Shake(0), the occupations are N(bonding)=48 and N(anti- bonding)=0. The signiﬁcance of 48 is that the bonding orbitals arise from the 6 2p electrons associated with each of the eight O anions in the CeO8 cluster model of CeO2, Fig. 3(c), before the covalent mixing that forms occupied bonding and unoccu- pied anti-bonding orbitals. The Shake(1) group of determinants have occupations N(bonding) 47 and N(anti-bonding) 1, and the Shake(2) group have N(bonding) 46 and N(anti- bonding) 1. We now determine the contributions of these determinants to the wavefunctions of the different ﬁnal states. This is done by summing, for a given wavefunction, the squares of the coefﬁcients of the determinants in each of the groups giving Weight(i) for i 0, 1, and 2; where the sum of the three weights is exactly 1.0 since the wavefunctions are normalized. (A different notation of charge transfer neglecting the covalent character of the orbitals was used for these weights in Ref. [22].) The weights for representative states in each of groups A–D are given in Table 12. The state in peak A is 95% composed of determinants that have no shake character, Shake(0). This is the reverse of the semi-empirical assignments [20,112] where the main peak at Erel=0 is claimed to have almost no Shake(0) weight; instead, Shake(0) weight is placed in a highly excited peak that corresponds to peak C in Fig. 13. It would be interesting to see what choices of adjustable parameters would lead the semi-empirical model Hamiltonian calculations [112] to give the same composition for the ﬁnal states as the rigorous many-body theory (Table 12).

For the U 4f XPS of UO3, [52,53] only shake excitations were considered since other redistributions of electrons gave rise to considerably higher energies which did not mix with the single 4f-hole conﬁgurations. Only Shake(1) conﬁgurations (or determinants) were mixed with the one shake(0) conﬁguration. The materials model for the theoretical XPS studies was an embedded UO6 cluster model of δ–UO3. [161] The theoretical results at the experimental distance of d(U–O) 2.09 Å are compared in Fig. 14 to the XPS for Schoepite where a Tougaard background [176,177] has been removed from the experimental data [52]. The theory reproduces the main XPS features including the satellites at ∼3–4 eV. A careful examination shows that under the leading edge 4f7/2 peak there are two closely spaced peaks with an intensity ratio of ∼3:1.

Table 12

Fig. 14. Experimental and theoretical U 4f XPS for UO3.

This is because the 5f7/2 level is split by the ligand ﬁeld into three closely spaced levels with two doubly and one fourfold degenerate level. The fourfold and one of the twofold degenerate levels are grouped into a single peak leading to the 3:1 intensity ratio. Similar arguments apply to the multiple peaks under the main 4f5/2 peak but they are more involved [52]. The composition of the main peaks shows that they are dominated by the determinants where there are no shake excitations, while the shake excitations dominate the states that compose the satellite peaks. This is consistent with the assignments for the CeO2 XPS states, see above and Refs. [21,22], and lends conﬁdence to the assignments made for the character of the ﬁnal states contributing intensity to the primary and satellite peaks.

Given the large extent of covalency between U and O in UO3, see Section 4, this is an excellent system to investigate the role of covalency on the relative intensity of satellites and main peaks [53]. In order to make this connection, it is necessary to develop a measure for the loss of intensity to satellites and to use an approach that allows the degree of covalency to be varied independently of other changes in the U–O interaction. For a simple measure of losses to satellites, we determine the SA Irel for the Dirac–Hartree–Fock wave- functions for a 4f-hole in the closed shell conﬁguration of the initial state of UO6 neglecting shake many-body effects [53]. There are 14 wavefunctions that can be formed with these optimized spinors with an ∼11 eV spin–orbit splitting into a group of 6 and 8 states with further small splittings due to the ligand ﬁeld in UO3; see discussion above. If the orbitals of these ﬁnal states were the same as those for the initial state, where there is no core-hole, the SA Irel of each of the ﬁnal states would be 1. This follows from the fact that the matrix of the overlaps of the initial and ﬁnal state orbitals is, by construction, a unit matrix [17,38]. In other words, in the absence of closed-shell screening of the core-hole, there is no intensity lost to shake satellites. When shake conﬁgurations are allowed to mix with the determinants that do not include shake, the intensity of the main peak will change. However, since the main peak has only relatively small contributions from determinants where there is a shake excitation, the deﬁnition of the loss to shake satellites given above is a useful signiﬁcance of XPS spectra.

Table 13

There have been several recurring themes. One has been to provide clear deﬁnitions for various terms used in the analysis and interpretation of XPS spectra. These terms include: initial and ﬁnal state effects; core-hole screening; and covalency. Another recurring theme has been to relate the quantities measured in XPS to the chemical and physical properties of materials. The review has drawn a distinction between one-body and many-body effects; in this context, an important objective has been to clarify the concepts of many-body interactions and to provide a unifying descrip- tion of different many-body mechanisms.

The different mechanisms that can lead to BE shifts were guide to the intensity lost and it is an easy quantity to calculate. In order to separate the change in covalency from other effects, the distance d(U–O) in the UO6 cluster is changed and the distance to the embedding point charges is also changed; in effect, the lattice constant of UO3 is artiﬁcially changed so that the covalency can be changed. The distance variations are kept small to insure that the interaction remains dominantly ionic since, if d(U–O) is increased by too large an amount, the Madelung potential will not be large enough to support a dominantly ionic interaction. We expect that the initial state covalency will decrease as d(U–O) is increased since the covalent interaction depends, to ﬁrst order, on the overlap of the metal cation and O anion orbitals [64]. Following from the analysis of XPS satellites of CO chemisorbed on metal surfaces discussed in Section 3, it would seem reasonable to expect that as the covalent interaction between U and O is decreased then the loss to satellites will also decrease. However, this analogy turns out to be misleading.

In Table 13, the projected numbers of U(5f) electrons, NP(5f), see Section 4, and the XPS intensity losses from the “main” 4f5/2 and 4f7/2 peaks are given for different values of d(U–O) about the experimental value of 2.09 Å. The NP are given for the initial and ﬁnal, 4f-hole, conﬁgurations. As expected, the initial state cova- lency, as measured by NP(5f), decreases as d(U–O) becomes larger; see Table 13. However, there is the surprising result that the intensity loss to XPS satellites actually increases, rather than decreases, when the initial state covalency decreases. Further, the covalent character of the 4f-hole conﬁguration remains almost constant. This is not surprising since the equivalent core atom [41,42] for the core-ionized U is Nd and the greater electron afﬁnity of Nd increases the covalent character of the interaction independent of the relatively small changes in d(U–O). The correct correlation to satellite losses is the difference of the covalency of the initial and ﬁnal state conﬁgurations since it is this difference that changes the overlap integrals between the initial and hole-state sets of orbitals. For the geometric structure of UO3, it is clear that this difference increases as d(U–O) becomes larger and, thus, the XPS losses to satellites become larger. It would be interesting to examine data for various UOx compounds to determine if there is this correlation between d(U–O) and satellite intensity.

Conclusions

This review has presented a wide range of material on the interpretation and analysis of the physical and chemical discussed. An important conclusion is that an attempt to relate ΔBE to the effective charge on an atom is an oversimpliﬁca- tion and that several other chemical properties, in addition to charge, play important roles in determining the signs and magnitudes of BE shifts. In terms of materials properties, one needs to take into account the fact that BE shifts depend on bond distances and on the coordination, or number of nearest neighbors, of the ionized atom. One of the reasons that these materials properties are important for the ΔBE is because they inﬂuence the electronic structure. In particular, they modify the extent of hybridization of inner valence orbitals, which, in turn, modiﬁes the electrostatic potential that a core electron sees. These different effects are often canceling and the net BE shifts may be small, as is often the case for the shifts of the core-level BEs of surface atoms compared to bulk atoms.

Many-body effects may lead to complex XPS spectra with intense satellites, often sufﬁciently intense that it is not possible to identify a single main line with satellites. We have considered satellites, especially for oxides as representative of ionic crystals. For these oxides, we showed that covalent mixing of the metal cation and the ligand orbitals may be important and we gave examples that related this covalent character to the XPS of the oxides. For the oxides, three types, or groups, of many-body effects have been distinguished: (1) multiplets and generalized multiplets; (2) other near degeneracy effects; and (3) shake excitations. Examples were presented to show that the multiplet splittings in the XPS spectra may be large. This is true for condensed systems as well as for atoms and molecules. Indeed, sometimes, an isolated atom will actually be an acceptable model for important aspects of the XPS of a condensed phase system. By generalized multiplets, we mean the angular momentum recoupling of the electrons in the open valence shells, which then couple to the core-hole. While these angular momentum couplings are XPS forbidden, they can steal intensity by mixing with the XPS allowed multiplets. We have also discussed how ligand ﬁeld splittings modify the multiplets of an isolated atom. Many-body effects in groups (2) and (3) above are closely related since they both lead to satellites at relatively low BEs, especially when they involve conﬁgurations that are nearly degenerate with the lowest core-hole conﬁguration. The distinction between them is how the rearrangements of electrons, or excitations, occur. The shake excitations involve excitations of electrons from levels whose character is bonding between the metal cation and the ligand orbitals to levels whose character is anti-bonding between metal and ligand. The effects described as near degeneracy normally involve levels localized about the cation that are also normally semi-core levels. In treatments where the covalent character of the orbitals is neglected, the shake excitations are referred to as charge transfer from ligand to metal. While the shake excitations do involve a net motion of charge from ligand to metal, their description as shake stresses the importance of their covalent character. It has been pointed out that there is an interplay between these different effects so that no single one can be considered to the exclusion of others.

It is the interplay of all these different effects that complicates the interpretation and the analysis of XPS. However, it is necessary to properly treat this complexity in order to draw the correct inferences about a material’s electronic structure from XPS spectra. The ability to draw these inferences greatly expands the value of XPS beyond the simple identiﬁcation of elemental composition.

Acknowledgments

We acknowledge support by the Geosciences Research Program, Ofﬁce of Basic Energy Sciences, U.S. DOE. We also wish to express our gratitude to Prof. Hajo Freund for his support and encouragement in the preparation of this paper, which was essential for the successful completion of the work.

Appendix A. List of acronyms and abbreviations

BE: binding energy. The binding energy of an electron associated with a peak in a photoelectron spectra.
CI: conﬁguration interaction. The common use is to describe many-body wavefunctions that are the mixing of several determinants for different conﬁgurations.
CSF: conﬁguration state function. A determinant or combination of determinants that is an eigenfunction of the angular momentum operators. Normally CSFs are formed for Russell–Saunders, L–S coupling but they can also be formed for j–j coupling.
CT: charge transfer. Usually refers to the transfer of an electron from a ligand orbital into an unoccupied or partially occupied, metal orbital.
CSOV: constrained space orbital variation. A theoretical procedure for decomposing the contributions to various properties by constraining the space of orbitals varied and the space of basis functions in which they are varied.
ΔSCF: delta self-consistent ﬁeld. Normally refers to the difference in the properties of two states, an initial and a ﬁnal state, each determined from separate self-consistent ﬁeld variations.
DFT: density functional theory.
DHF: Dirac–Hartree–Fock.
ER: relaxation energy.
Erel: relative energy. Normally of the BE of a peak relative to some reference BE taken as zero.
FC: Franck–Condon. Normally used in connection with the vibrational broadening of peaks in photoemission spectra.
FO: frozen orbital. Used to describe wavefunctions and other properties obtained when the orbitals of the“passive” electrons are ﬁxed or frozen as they are for the initial state before ionization.
FWHM: full width at half maximum.
HF: Hartree–Fock. This is used to describe the wavefunc- tions and other properties that result from Hartree–Fock calculations.
Irel: relative intensity. In this context, the intensities of the different XPS peaks arising from the ionization of the same primary electron.
KT: Koopmans’ theorem. Used to describe energies obtained by freezing orbitals of the “passive” electrons when an electron is ionized; closely related to FO properties.
PES: photoemission spectroscopy
SA: sudden approximation. An approximation for the calculation of the relative intensity of the XPS ionization into a particular ionic state; the approximation is exact in the limit of ionization by very high energy photons.
SCF: self-consistent ﬁeld.
SCLS: surface core level shift. This is used to describe the BE shift between photoelectrons arising from the surface and from the bulk of a crystal.
XPS: X-ray photoelectron spectroscopy.

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CLIPS, is a program system to compute ab initio SCF and correlated wavefunctions for polyatomic systems. It has been developed based on the publicly available programs in the ALCHEMY package from the IBM San Jose Research Laboratory by P. S. Bagus, B. Liu, A. D. McLean, and M. Yoshimine.
DIRAC, a relativistic ab initio electronic structure program, Release DIRAC08, 2008, written by L. Visscher, H. J. Aa. Jensen, and T. Saue, with new contributions from R. Bast, S. Dubillard, K. G. Dyall, U. Ekström, E. Eliav, T. Fleig, A. S. P. Gomes, T. U. Helgaker, J. Henriksson, M. Iliaš, Ch. R. Jacob, S. Knecht, P. Norman, J. Olsen,

M. Pernpointner, K. Ruud, P. Sałek, and J. Sikkema (see the URL at http://dirac.chem.sdu.dk..
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Advanced Interpretation of XPS

Advanced Interpretation of XPS Spectra

1. INTRODUCTION

Theoretical models and methods

One-electron features

Core-hole screening and relaxation effects

Initial state mechanisms leading to BE shifts

BE shifts in molecules, clusters, and solids

XPS satellites: covalent interactions and screening

Metal oxides and covalency

XPS satellite features and many-body effects

Multiplet splitting

Near-degeneracy effects: angular momentum recoupling and FACs

Satellites in closed shell oxides

Conclusions

References

List of All Pages in The XPS Library

Vertical menu

xml site-map

Advanced Interpretation of XPS

Advanced Interpretation of XPS Spectra

1. INTRODUCTION

Theoretical models and methods

One-electron features

Core-hole screening and relaxation effects

Initial state mechanisms leading to BE shifts

BE shifts in molecules, clusters, and solids

XPS satellites: covalent interactions and screening

Metal oxides and covalency

XPS satellite features and many-body effects

Multiplet splitting

Near-degeneracy effects: angular momentum recoupling and FACs

Satellites in closed shell oxides

Conclusions

References

List of All Pages in The XPS Library

Vertical menu

xml site-map

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