Chemical significance of x-ray photoelectron spectroscopy binding energy shifts: A Perspective

The primary use of x-ray photoelectron spectroscopy (XPS) is to identify the presence of different elements in a sample but it can also be used to determine the composition, or the stoichiometry, of the elements within a compound. However, without a good understanding of the spectra, including background removal, accuracy may be seriously compromised.^1–3 Beyond determining elemental composition, which is not the subject of this Perspective, XPS, using BE shifts can also be used to obtain information about the electronic structure and the chemical bonding of the atoms in the sample studied. This was recognized in the pioneering research of the Uppsala group who coined the name, electron spectroscopy for chemical analysis or ESCA, as an alternative name for XPS, where the name ESCA was intended to describe both elemental composition and chemical bonding. These two uses are clearly described in the two books on ESCA from the Uppsala group,^4,5 where theory was used to relate the XPS spectra to the chemical bonding, especially for free molecules.⁵ Given the limited computational capabilities in those early days, most of the theory in the ESCA volumes was semiempirical. With the present computational capabilities, rigorous methods of molecular orbital (MO) theory can now be used to describe the chemical bonding and relate XPS features to this bonding (see, for example, Refs. 6 and 7, and references therein). A description of how XPS binding energy shifts can be interpreted to obtain information on electronic structure and chemical bonding is the primary objective of this Perspective.

We begin with a brief review of the key considerations that are used to describe XPS spectra, introducing the terms and concepts necessary to describe the XPS spectra, and the interpretation of the spectra, in relation to the chemical bonding and interactions; more details and precise definitions will be given later in the paper. A simplified schematic spectrum of the features, or peaks, of the XPS of an isolated Ne atom is shown in Fig. 1. The key points for this figure are as follows:

1. The number of peaks in the schematic, three, corresponds to the three occupied shells in the Ne atom, 1s, 2s, and 2p.

2. The peaks are not delta functions but have widths.

3. The relative intensities have been drawn to be proportional to the number of electrons in each shell, 2:2:6.

(1) The theoretical calculations to model the number and BEs would be calculations of the wavefunctions (WFs) and energies of the ground state of the Ne atom and the three ions (see Ref. 8 for the first applications of the HF method to XPS). In many cases, the Hartree–Fock (HF) method is adequate to describe these WFs. This method assumes that a single configuration, which is a single distribution of electrons over orbitals, can be used to describe the WFs for the initial (before ionization) and the final state (after ionization). Thus, for the Ne example in Fig. 1, the initial state has the configuration 1s²2s²2p⁶, and the final state ions have the configurations 1s²2s²2p⁵, 1s²2s¹2p⁶, and 1s¹2s²2p⁶. For each configuration, the HF equations are solved to give the orbitals that best describe that configuration. See the papers by Roothaan that describe the modern methods to determine HF WFs for closed shell systems⁹ and for open shell systems¹⁰ or see a modern text on quantum chemistry (for example, Ref. 11 for descriptions of the HF method). Although it may be necessary to go beyond a single configuration description of the initial and final states in order to include what are termed many-body effects (see later), the HF method can be used to study the physical and chemical significance of XPS BE shifts for a large number of cases and this is the focus of the present paper. For heavy atoms or for ions where the core shell ionized is significantly spin–orbit split, e.g., the np_1/2 and np_3/2 or nd_3/2 and nd_5/3, it is necessary to use Dirac Hartree–Fock (DHF), methods to properly include the spin–orbit effects.¹² 

The theoretical value for the BEs or ionization potentials is the difference in the energies of the initial state neutral atom and the appropriate ion. These would be single values, i.e., delta functions, rather than the broadened peaks shown in Fig. 1.

The theory also allows the separation of what are described as initial state effects from final state effects, and this distinction may be important for the analysis of the XPS as described in detail in this Perspective. The physical significance of initial state BEs is that they do not take into account the screening of the XPS core hole by the valence electrons, while the final state BEs do take this screening and the closely related relaxation energy into account. Rigorous definitions of initial and final state effects and WFs as well as their significance in terms of the interpretations of the XPS are given in Secs. III and IV.

In this context, it is appropriate to point out that density functional theory (DFT) and WF methods, in particular HF methods, are used to provide theoretical results for XPS BEs. The representative references and references contained in them provide an indication of the use of DFT methods for the XPS of molecules in Ref. 13 and for the XPS of condensed phase systems in Ref. 14. Indeed, there is evidence that HF and DFT yield results for BE shifts of comparable accuracy (see, for example, Refs. 13 and 15). However, some care needs to be used with DFT for the XPS BEs of open shell systems since DFT does not naturally treat multiplets. The general treatment of multiplets within a framework of DFT orbitals has been addressed, with different approaches, by Daul¹⁶ and Ikeno et al.¹⁷ Other electronic structure methods, for example, the GW approach based on Green’s function methods has been applied to the calculation of XPS BEs¹⁸ However, the focus of the present paper is to show that chemical information can be extracted from a proper analysis of the XPS; comparison of the computational advantages and disadvantages of different theoretical methods is outside the scope of this paper. These methods have been briefly described only to show the range of theoretical studies of core-level processes.

(2) The broadening of XPS peaks is best quantified by the full width at half maximum (FWHM) of an XPS peak and of the individual contributions to the broadening. This broadening arises from the experimental resolution and from the finite lifetimes of the core-level ions (see, for example, Refs. 6 and 19, and references therein). Furthermore, there is another origin of the broadening of XPS peaks, which arises from what is best described as a Franck–Condon (FC) vibrational broadening (see Refs. 20–25). The physical reason for the FC broadening is that the neighbors of the core-ionized atom see a stronger Coulomb attraction and are drawn toward the ionized atom leading to a different potential energy surface for the ionized than for the neutral configuration. As shown graphically in Ref. 22, the shift of the ionized potential surface leads to excitation to highly vibrationally excited levels of the system with the core ion and these excitations may have a modest range of energies. It is possible to view the FC broadening as another contribution to the Gaussian broadening.²² Values of the FWHM for the Gaussian broadening due to experimental resolution are typically fractions of an eV (Ref. 26). Values for the FWHM of FC broadening in a range of metal oxides²⁴ were found to be a significant fraction of an eV. When considering magnitude for these two different Gaussian broadenings, the sum of the FWHM due to Gaussian broadenings is the square root of the sum of the squares of the individual broadenings. The Lorentzian broadening due to lifetimes depends strongly on which core level is ionized and on the atom concerned; it ranges from a fraction of an eV to several eV.²⁷ The Gaussian and Lorentzian broadenings can be combined with a Voigt convolution of the theory to compare to experimental peaks in a spectrum.^28,29 The main point is that the peaks or features in XPS spectra are always broadened rather than being a single ionization energy that would arise from a theoretical calculation. In addition, there may be asymmetric broadening due to unresolved overlapping peak structure, or, if the experimental resolution is good enough, vibrational broadening.

(3) The relative intensities in XPS of the different core levels depend on several factors; the number of electrons in the shell concerned, the photoionization cross section, α, for an electron in that shell, and the instrumental transmission function, which is usually a function of KE. In this Perspective, since we are not primarily concerned with elemental quantification, we will not address quantification procedures using relative intensities from different atoms in a material sample. We note that in the schematic of Fig. 1, the relative intensities of the 1s:2s:2p ionizations are taken simply as the occupations of the shells, (1s)2:(2s)2:2p (6). This does not take into account the differing photoionization cross section of the different orbitals, which leads to large changes in the relative intensities. In the case of Ne, if using the usual Al K_α lab source at 1486.6 eV, the (1s) intensity is increased ∼60-fold, compared to (2s).

Returning to item 1 above, the situation changes and can become more complex when the system has an open valence shell. These complications arise even for the XPS of the NO molecule, which has the valence occupation in the π space of 1π⁴2π¹ configuration and the total symmetry of ²Π.³⁰ When either the O(1s) shell, the 1σ orbital, or the N(1s) shell, the 2σ orbital, is ionized, the total symmetries for the ion will involve the spin coupling of the open 1s core shell with the open 2π valence shell. This leads to two final states with slightly different energies, which, for the XPS of NO, are described as ¹Π and ³Π.³¹ Each is termed a “multiplet” with the superscript identifying the spin multiplicity, i.e., singly degenerate and triply degenerate. For a description of multiplets in the simple case of atoms see Ref. 32. Such multiplets, and their energy splittings, are more complicated when the valence open shell has a higher multiplicity, such as for 3d or 5f metals. An example is the Mn²⁺ cation (five unpaired 3d electrons), which can be used to model MnO (Ref. 33) and, in particular, the Mn 2p XPS of MnO.³⁴ As will be discussed in Sec. II, the Mn XPS can also be modeled by an embedded MnO₆ cluster.^33,34 In Fig. 2, the experimental Mn 2p_3/2 XPS of MnO (Ref. 34) is compared to the Voigt broadened calculation of 2p_3/2 obtained with an embedded MnO₆ cluster, where only a limited region of the XPS, going to 10 eV above threshold, is shown. The spectrum in this region is dominated by Mn 2p_3/2 ionization.³⁴ The first, lowest, band in this XPS spectrum is rather broad because it contains unresolved BEs to several multiplets. The theoretical XPS related to the individual multiplets is shown in Fig. 2 as light curves below the bold solid black curve, that is, the sum of the intensity from all multiplets. The different multiplets arise from the different couplings of the Mn 3d⁵ open shell with the ionized 2p_3/2 core shell, as will be discussed in more detail in Sec. III. We also note the feature at the higher BE of ∼7 eV relative to the first peak, which is placed at a relative energy of 0 eV. This feature has significant contributions from satellites, described as shake satellites, which are one type of a many-body effect, involving excitation of a valence electron along with ionization of a 2p electron. These satellites are not considered in this Perspective but are discussed in detail elsewhere for the cation 2p XPS of NiO (Ref. 35) and of Fe₂O₃ (Ref. 36). We do not consider these shake features further here for two reasons: First, our principal concern is for the BE shifts of main peaks. In this Perspective, we mean the specific change, ΔBE, between a given element present in the two or more different electronic, chemical, or physical forms being compared. As we show, these BE shifts, properly interpreted, provide insights into the electronic structure of the material studied. Second, the theoretical framework needed to describe the shake satellites involves complex theoretical considerations that go well beyond what can be described within the usual HF and DFT theoretical frameworks (see, for example, Refs. 35 and 36). Indeed, establishing the relationship between the intensities and energies of shake features is an area of ongoing research.

Other general considerations concern the consequences of spin–orbit splittings on the energies of the multiplets and the splittings, which arise from including, as well, ligand field splitting. These consequences are illustrated in the schematics in Figs. 3–5. The spin–orbit splitting can be described most easily by considering a model system of six-electron atomic ions with the configuration 1s²2s²2p² from C⁰ through to U⁺⁸⁶. The coupling of the 2p² open shell leads, neglecting spin–orbit splitting, to three Russell–Saunders (RS) multiplets, ³P, ¹D, and ¹S (left side of Fig. 3).³² This is the case at low Z, such as C, where the spin–orbit splitting is very small compared to the RS splitting. If one wants to add spin–orbit splitting to the RS multiplet splitting, it can be taken into account by coupling the total spin, S, with the total orbital angular momentum, L, to give a total angular momentum J, leading to multiplets ³P₀, ³P₁, ³P₂, ¹D₂, and ¹S_0, However for the very light C⁰, Z = 6, the splitting of the three ³P multiplets is very small, ∼1–2 meV, but as Z increases the splitting grows very rapidly so that when P⁺⁹ (still six electrons) is reached, the splitting of these multiplets has increased from meVs to ∼0.5 eV.³⁸ In fact, the only good quantum number now is J, with L and S no longer good quantum numbers. For larger Z, at the right of Fig. 3, j-j coupling, to give J, becomes dominant, and the configurations are best described as (2p_1/2)², (2p_1/2)¹(2p_3/2)¹, and (2p_3/2)², where the J values for these configurations are shown in Fig. 3. This is because the spin–orbit splitting of 2p_1/2 and 2p_3/2 has become much larger. In fact, by the time one reaches U⁺⁸⁶, the splitting of the 2p_1/2 and 2p_3/2 BEs is over 3500 eV.³⁹ 

There is also the interplay of spin–orbit splitting with ligand field splitting that needs to be taken into account. This interplay is shown in the schematics of the orbital energies of the spin–orbit and ligand field split 3d orbitals (Fig. 4) and the equivalent for 5f (Fig. 5). These figures do not give values of the splittings for any particular system but are meant to show trends. The values in the plots have magnitudes that are representative for those in our calculations on 3d transition metal oxides and actinide oxides (see below). On the left and right of the figure are extremes. On the left, the spin–orbit splittings of the isolated atom but without crystal field splittings (see Ref. 38) are shown, and on the right, the crystal field splittings for cubic oxides but without spin–orbit splittings^33,40,41 are shown. The 3d crystal field splittings are those appropriate for MnO or NiO or other cubic 3d oxides. The 5f crystal field splittings are those appropriate for actinide dioxides (see, for example, Ref. 42). The center gives the orbital energy splittings when both spin–orbit and ligand field splitting are taken into account.⁴³ The orbital energies are for Dirac Hartree–Fock calculations on the isolated ions of Mn²⁺ for the 3d and Pu⁴⁺ for 5f. The embedded MnO₆ and PuO₈ cluster models represent MnO^7,34 and PuO₂,⁴⁴ respectively. The relative energies of the orbitals for the three different models are preserved in the figures, but the absolute energies have been shifted for convenience to display the splittings between the orbital energies for the different models. These schematic figures follow the usage in Abragam and Bleany⁴¹ to show how the environment of the crystal affects the energies (termed ligand field) of the valence open shells in ionic compounds. The symmetry designations for the orbitals on the left are the usual for spin–orbit split atomic shells; the designations on the right are the usual crystal field notation for orbitals in a cubic environment.⁴⁰ In the usual notation for cubic groups, orbitals of a symmetry are twofold degenerate, the e symmetry orbitals are fourfold degenerate, and the t orbitals are sixfold degenerate, where this degeneracy includes spin-up and spin-down orbitals. The orbital symmetries in the center are for spin–orbit and ligand field split orbitals in a cubic environment.⁴³ The symmetry designations for the combination of these splittings, which are generally less familiar than for the extreme cases in Figs. 4 and 5, are reviewed by Boca.³⁷ It is noted that the orbitals in the figures labeled γ₆ and γ₇ are twofold degenerate while those labeled γ₈ are fourfold degenerate. The connections in dotted lines show the main relationships of the orbital character for the different cases. These relationships, with the combination of spin and ligand field splittings, are also shown on the labels of the free ion and ligand field split symmetries, The most important conclusion from Figs. 4 and 5 is that the combination of spin–orbit and ligand field splitting needs to be taken into account in order to have a full understanding of the splittings of orbitals and states, even for the relatively light 3d transition metal oxides.

The spin–orbit splitting of the 3d orbitals of transition metal cations, on the left in Fig. 4, is small, ∼0.05 eV, but the ligand field splittings, on the right in Fig. 4, are larger, ∼1 eV.^40,45 When both spin–orbit and ligand field splitting are taken into account, the ligand field splitting is modestly reduced and there is a small spin–orbit splitting of the dominantly t_2g orbitals. The splittings are somewhat different for the actinide 5f orbitals as shown in Fig. 5. The spin–orbit splitting of the 5f_7/2 and 5f_5/2 orbital energies in isolated actinide cations is almost twice the spin–orbit splitting of 3d_5/2 and 3d_3/2 orbital energies in transition metal cations. However, the ligand field splitting is somewhat smaller than that for the 3d case and only the a_2u component of the 5f orbitals has a significant ligand field splitting. The smaller ligand field splitting for the 5f orbitals can be understood from considerations of the fluorite geometry of the actinide dioxides, which was used for the plots of Fig. 5 as compared to the octahedral geometry used for the plots of Fig. 4. The ligand field splittings of the 5f orbitals in the fluorite geometry of PuO₂ have been considered in detail in Ref. 44. When spin–orbit and ligand field splitting are combined, center panel of Fig. 5, the five different energy levels for the nominally 5f open shell orbitals span a range of over an eV.

The important conclusion from both Figs. 4 and 5 is that there may be several populated final states, with energy differences that are roughly of order an eV, when a core level is ionized. This is because there can be multiplets, where there are different occupations and couplings of the underlying spin–orbit and ligand field split orbitals. This can, in fact, already be seen for the 2p_3/2 XPS of MnO shown in Fig. 2, where the theoretical individual multiplets for an embedded MnO₆ cluster model of MnO are drawn as light curves. These light curves include spin–orbit and ligand field split multiplets that have very small splittings. The light curves are summed to a dark total theoretical curve and compared to the experimental data as dots.³⁴ The initial, unionized multiplet is a high spin coupled ⁶A_1g with a tiny, less than 1 μeV, splitting into twofold and fourfold degenerate multiplets. The angular momentum coupling of the 2p_3/2 hole with the five d electrons coupled to their lowest ⁶A_1g multiplet leads, neglecting ligand field splitting, to four multiplets. However, the ligand field splitting coupled to the spin–orbit splitting leads to a larger number of multiplets that have very small splittings, with only one of these small splittings resolved in Fig. 2. An important consequence of these overlapping and largely unresolved features, as clearly seen in Fig. 2, is the large observed FWHM of the observed XPS “peak.”

While orbital energy differences do not map directly onto multiplet energies, they do provide a guide to the energy differences that may exist for different occupations of these orbitals. As is discussed in Secs. III and IV, the actual energies of multiplets are given only when one considers the full angular momentum coupling of the open shell electrons. In this context, we use the descriptions of one-body and many-body effects to distinguish between the properties of orbitals, which are functions of only a single electron, from the properties of WFs that are functions of all N electrons in the system. We recall that many electron WFs are commonly constructed from determinants, more rigorously described as Slater determinants, since this assures the antisymmetric character of the WFs required by the Pauli principle.¹¹ In particular, it is also useful to distinguish the N electron WFs that can be described by a single determinant composed of orbitals³² from the WFs that are appropriate sums over different determinants. For closed shell systems, the HF WFs are single determinants and may give useful descriptions of many fundamental properties related to the electron distribution. On the other hand, WFs of open shell systems that have the angular momentum couplings needed to have the correct symmetry may need to be sums over several determinants (see, for example, Ref. 32). In this sense, we describe the open shell WFs that include the correct spin and orbital angular momentum coupling as including a key “many-body” effect because they will be, in general, sums over several determinants since this coupling usually cannot be described by a single determinant. These many-body effects can be seen, for example, within ligand field theory, in the Tanabe–Sugano treatments of the multiplets of transition metal compounds where the energies are associated with quantum numbers that describe the total angular momentum of a compound.^40,45 There are, of course, additional many-body effects that involve effectively mixing determinants that are formed by distributing the electrons in different ways over shells of electrons.¹¹ Such treatments including many-body effects can lead to quite accurate energies and energy differences (see, for example, Ref. 11). A detailed treatment of the many-body effects needed to obtain very accurate energies is outside the scope of the present paper. It is important to note that we explicitly use for the HF wavefunctions, orbitals separately optimized for the different initial and final, ionized, configurations. Indeed, the different orbital sets for these two configurations play an important role in determining the XPS intensities,^46,47 as will be discussed in Sec. III. While this can be viewed as a many-body effect since different orbital sets are used for the initial and final configurations, we prefer to view this as still an orbital picture, which takes account of the different potentials for the two configurations. The only extension beyond a simple orbital picture that we will normally consider is the angular momentum coupling of core and valence open shells, which is viewed as a many-body effect. The XPS consequences of this angular momentum coupling were described briefly above for the Mn 2p XPS of MnO and will be considered in more detail when we discuss examples.

In this Perspective, the focus is on the interpretation of the number and width of peaks in an XPS spectrum, and especially on the BE shifts, ΔBE, between the BEs of an atom in different environments and/or in different compounds. There will be an emphasis on the identification of the origin of the BE shifts in terms of the chemical bonding in the system studied. However, we will not consider shake satellite features, which may also reflect the chemical interaction (see, for example, Ref. 35). We choose to focus in this Perspective on the main rather than on satellite features in an XPS core-level spectrum, especially since the relationship of the satellite intensity to the chemical bonding is still the subject of ongoing research.^35,36

It is common to assume that there is a direct relationship between the effective charge or ionicity of the core-ionized atom, denoted as Q, and the XPS BE. While the charge, Q, certainly does contribute to shifts of the BEs between different compounds and different environments within a sample, it is not the only reason for BE shifts and it can be misleading to assume that the BE shift, ΔBE, is proportional to ΔQ of the ionized atom. In order to establish the correct origins of the XPS ΔBE, it is necessary to rigorously define and separate what is described as initial state and final state mechanisms and this will be done in Sec. IV.^6,7 Often, the most useful and detailed chemical information is obtained by combining measurement and full ab initio theory (see, for example, Refs. 1, 2, 35, 48, and 49). However, even without computational results for the electronic structure of a system, it may be possible to use the mechanisms described in Sec. III to draw some conclusions about the electronic structure and to suggest possible theoretical calculations to support these conclusions. In this Perspective, we will identify these mechanisms and explain how they modify XPS features. Though XPS studies are often used to attempt to obtain information about the chemical interactions and bonding in different situations through the use of BE shifts, the BE shifts can only be correctly related to the chemical properties when there is a clear understanding of the mechanisms that are responsible for the observed BE shifts. Explaining these mechanisms and illustrating their importance for different systems is the objective of this paper. Once these mechanisms are understood, it should then be possible to avoid incorrect interpretations about the chemical bonding from chemical shifts

The theory that will be considered here is based on molecular orbital WFs for atoms, molecules, and cluster models of extended systems.^6,7 For condensed systems, band structure (see, for example, Refs. 14, 50, and 51) and other ab initio theoretical treatments^52,53 have also been used to describe XPS. In this Perspective, we choose to focus entirely on MO theory since, as we shall demonstrate, with the framework of MO theory it is possible to establish clear and direct connections between the observed XPS BE shifts and changes in the electronic structure. We stress that our focus will not be on the mathematical formulations of the analysis but, rather, on the chemical and physical content of the various quantities that are defined and used to analyze and interpret the XPS BE shifts.

The organization of the paper is as follows: In Sec. III, the XPS process and selection rules are reviewed, and the sudden approximation (SA)^46,47 for the XPS intensities is described. This is done to establish the MO basis for the analysis of both the XPS energies and intensities provided that the many-body effects of the angular momentum coupling and ligand field splitting are included. This analysis is in strong contrast to claims that accurate many-body analyses that include theoretical effects that go beyond a simple distribution of electrons over the core and valence shells must be made, and, thus, the use of orbitals to describe the XPS is strongly limited.⁵⁴ It will also serve to alert the reader that it may be difficult to define BE shifts for systems with valence open shells, for example, transition metal oxides like FeO_x (Ref. 49) and NiO,³⁵ where the ionization of a core shell may lead to several distinct, resolved, and, possibly, unresolved XPS peaks.⁷ In Sec. IV, Hartree–Fock WFs for the initial and final states or configurations will be described. It will be stressed that the energies of these HF WFs are degenerate for the individual states within a multiplet. Furthermore, as we discuss in one of the examples studied, the XPS intensity associated with a multiplet is nearly proportional to the degeneracy of the multiplet and this can be used to identify the origin of XPS features.³¹ Furthermore, the concepts of initial and final state BEs will be given rigorous definitions and the Koopmans’ theorem (KT), BE, which is normally extracted from ab initio calculations, will be defined in the context of initial and final state effects. In Sec. V, the different kinds of initial state effects that modify BEs and, hence, lead to BE shifts, ΔBE, will be described for the model system of an isolated Cu atom. For this system, the ΔBE obtained considering only initial state effects will be shown to be very similar to the ΔBE obtained when the final state, as well as the initial state effects, are included. In particular, a criterion for when final state effects will be important for the ΔBE is established. In Sec. VI, the mechanisms and concepts developed for the Cu atom model system will be applied to explain and interpret the physical significance of the XPS ΔBE found for five real systems. The importance for the ΔBE of the different kinds of initial state effects, or mechanisms, as described in Sec. V for the ΔBE of the model Cu atom system, will be described for each of these five real systems. Finally, our conclusions will be summarized in Sec. VII, where we will stress again that the primary objective is gaining an understanding of the underlying physical and chemical reasons, including the chemical bonding, that are responsible for the observed XPS features. Since we have introduced several terms to describe the XPS features and the theoretical interpretation of the BE shifts in terms of the chemistry of the system, we have added a glossary after Sec. VII as a convenience to recall the definitions and the significance of many of the terms that have been introduced.

An important additional effect on the atomic multiplet splittings is the covalent character of the open shell 3d orbitals, which, for MnO, are antibonding combinations of Mn(3d) and the ligand, O(2p).³⁴ This is because the exchange interaction, which drives the multiplet splittings of the Mn 2p core ions, is largely an atomic effect and it is reduced as the covalency increases.^7,34 Thus, we would predict that the Mn 2p_3/2 XPS in compounds where there are different extents of covalency would have different final, core ionized, state multiplet splittings. Indeed, this has been observed for islands of thin MnO films where contributions from bulk, surface, and edge atoms have been observed in the Mn 2p_3/2 XPS.⁵⁸ However, for both the atomic, Mn²⁺ and MnO₆ cluster models, a general rule for the XPS allowed symmetries for the ionized system is that they are given by the angular momentum coupling of the ionized core shell with the valence open shell. It is also shown in Fig. 2 that the energy splittings of the allowed symmetry couplings are up to ∼1 eV for Mn²⁺ 2p_3/2 XPS. Because they may not be fully resolved, ionization to the different couplings between 2p_3/2 and the open 3d shell, which for Mn²⁺ are the different J values, may lead only to an observed broadening of features in the XPS spectra. The broadening and the position of the maximum of a feature, which contains unresolved XPS multiplets may affect the definition of the BE shifts derived from the experimental data. In this connection, it is worth examining the FWHM of XPS peaks, particularly asymmetric ones, rather than just the apparent BE shifts of what might be unresolved features.⁷ 

In order to calculate BEs, we pointed out earlier that it is necessary to have WFs and associated energies for the initial and the final ionized configurations. In Sec. III, we described WFs for individual states and groups of states, but we did not describe how these WFs might be obtained and we did not describe the physical model systems that the WFs were obtained for. These aspects are discussed here with a special view to rigorously separating initial and final state effects for XPS BEs within the context of MO theory.

Obviously, when the system of interest is an atom or a molecule, no physical modeling is needed. For condensed systems, we use, as models, clusters of atoms to describe the element or compound being studied. In Fig. 6(a), we show the simplest cluster to model bulk Cu as a single atom and in Fig. 6(b), an embedded cluster of FeO₆ to model Fe₂O₃ as used in Ref. 48. Of course a more realistic cluster for the XPS of bulk fcc Cu would contain 13 atoms, the central Cu atom that will be ionized and its 12 nearest neighbors to model the environment in bulk Cu; this cluster has been used in Ref. 60. Larger clusters for fcc Cu have also been studied to examine size convergence effects (see Ref. 61). The cluster model for Fe₂O₃ contains a central Fe cation surrounded by its six nearest O anion neighbors and embedded in point charges to represent the electrostatic environment in the oxide. To model oxides, larger clusters have also been used to provide a more realistic representation of the environment, but detailed studies^15,62–64 have shown that the size cluster shown in Fig. 6(b) is actually sufficient to represent a very large part of the effects involved in determining the observed XPS of oxides. In general, the known bond distances are used for these materials models, although in certain circumstances distances may be varied to test the effects on the calculated BE shifts (see, for example, Ref. 61 and Sec. VI).

For these materials models, ab initio WFs for the initial state of the system, with N electrons before photoionization, and for the final state, with N − 1 electrons, after photoionization, are calculated. The energies of these WFs are then used to calculate the XPS BEs and ΔBEs. The properties of the orbitals in these WFs and the possible many-configuration character of the total WFs are analyzed to determine the character of the states probed with XPS. An important property of the valence orbitals is their degree of localization and their covalent character (see, for example, Refs. 49 and 65). We consider HF WFs and then move on to more complex WFs that would give more accurate BEs. However, for many purposes, these HF WFs are satisfactory to give qualitative, and near quantitative, values for the ΔBE, and to permit the physical and chemical origins of the shifts to be determined.

HF WFs are normally defined for a single configuration of shell occupations and the orbitals are determined as self-consistent field (SCF) solutions of the variational HF equations.¹⁰ These solutions may be nonrelativistic,¹⁰ or they may be relativistic solutions of the Dirac–Fock equations.¹² There are scalar and spin–orbit relativistic effects. The scalar effects are important for all atomic symmetries and, as discussed below, can lead to large changes in the absolute BEs, especially for 1s BEs of heavier atoms. The spin–orbit splittings are important for the core levels where the orbital angular momentum, ℓ, is ℓ ≠ 0, and for even modestly heavy atoms can lead to very large splittings for the ℓ + 1/2 and ℓ−1/2 BEs (see Ref. 39 and the discussion below). Our discussion of relativistic effects is limited since most of the fundamental physics and chemistry, which affect the XPS BE shifts considered can be explained without the need to include these relativistic effects. The HF equations that we use yield WFs that have the full symmetry of the system. Therefore, the HF orbitals and WFs are a basis for the irreducible representations of the point group of the system,⁴⁵ which ensures that the WFs have the appropriate spin and orbital angular momentum symmetry. We do not consider unrestricted variants of HF theory where the orbitals and WFs do not have appropriate symmetry.⁶⁶ Thus, the energies of the HF orbitals and the total WFs are in sets, or groups, which have the degeneracies of the irreducible representations, or symmetries, for the point group of the system. The groups of total WFs, which are degenerate by symmetry are termed multiplets.⁴⁵ For nonrelativistic solutions, spin–orbit splitting is neglected, and the multiplets are called Russell–Saunders multiplets, which have the total spin, S, and total orbital angular momentum, Λ, as good quantum numbers. The notation for the Russell–Saunders multiplets is ^2S + 1Λ. The spin angular momentum of an orbital is ½ and the spin angular momentum of the total WF, S, depends on the coupling of unpaired electrons within and between open shells and follows the quantum mechanical rules of angular momentum addition constrained by the Pauli exclusion principle (see, for example, Refs. 45, 67, and 68). The spatial symmetry of the total WFs is denoted Λ, and the symmetry of the orbitals is denoted with lower case λ. Both Λ and λ describe irreducible representations of the same symmetry group and, hence, have the same sets of degeneracies.⁴⁵ The transition from the case where Russell–Saunders coupling dominates, to one where spin–orbit splitting is sufficiently large that j-j coupling dominates, is shown in the schematic of Fig. 3. In Fig. 3, the character and splitting of the multiplets for an atomic 2p² configuration from Z = 6 where spin–orbit splitting is small and RS coupling dominates to large Z where for sufficiently large Z, the spin–orbit coupling will dominate. For large spin–orbit coupling, i.e., large Z, only the total angular momentum, J, where J = L + S, is a good quantum number and the 2p shell is split into 2p_1/2 and 2p_3/2 shells that have huge energy splittings for large Z. The spin–orbit splitting of atomic orbitals as a function of Z is given in Desclaux’s Dirac–Fock study of the atoms up to Z = 120.⁶⁹ For Z = 6, the difference of the 2p_1/2 and 2p_3/2 orbital energies is 8 meV (Ref. 69) and RS coupling is clearly appropriate. The small spin–orbit splittings at low Z are taken into account in RS multiplets by coupling the total L and the total S to yield a total J, which is denoted as a subscript to the RS notation. For 2p², the ³P multiplet becomes ³P₀, ³P₁, and ³P₂ with miniscule splittings. For Z = 94, the Pu atom, the 2p_1/2 and 2p_3/2 orbital energy splitting is 4266 eV,⁶⁹ an increase of more than 5 orders of magnitude from C. For orbitals at large Z, RS coupling cannot be used since only the total angular momentum is conserved. Here j-j coupling becomes appropriate and the electrons are distributed over the 2p_1/2 and 2p_3/2 sub-shells. The composition of the J levels for the open shell 2p² in the limit of large Z is shown on the right of Fig. 3. In the intermediate Z region, neither RS or j-j coupling is appropriate, and we must allow mixings between multiplets in these limits with the same total angular momentum which for atoms is denoted J. Thus, for J = 0 in the example in Fig. 3, the RS multiplets ³P₀ and ¹S₀ can and do mix or, equivalently, the j-j configurations (2p_1/2)² and (2p_3/2)² can and do mix.

While much of the discussion above has been for the orbital and WF spatial symmetries for atoms, s, p, d, …, these spatial symmetries and their notation are different for molecules and compounds.^33,43,45 The nomenclature and the degeneracies are briefly reviewed for the O_h cubic group, which represents the local symmetry of many metal oxides, for example, NiO and UO₂.⁷⁰ For this cubic group, there are only nondegenerate, twofold degenerate, and threefold degenerate spatial symmetries.⁶⁸ Atomic d orbitals are split into a twofold degenerate e_g and a threefold degenerate t_2g symmetry and atomic f orbitals are split into a nondegenerate a_2u, and two threefold degenerate, t_2u and t_1u, symmetries (see the rightmost columns in Figs. 4 and 5). The degeneracy of the sets of total WFs with the same symmetry, as given by the Russell–Saunders multiplet, is the product of the orbital degeneracy and spin degeneracy, 2S + 1. When spin–orbit splitting is taken into account, then there is further splitting into symmetries described, for the O_h cubic group, as γ_i for the orbitals and Γ_i for the WFs; the splittings of orbital energies are shown in the central panels of Figs. 4 and 5.

For a given core-level ion, the difference between the BE(FO) and the BE(ΔSCF) is described as the relaxation energy, E(R), where

There is a significant amount of information contained in E(R), BE(ΔSCF), and BE(FO). We describe the nature of this information here and then consider applications to specific cases of XPS in Secs. V and VI.

In general, E(R) is well defined and Eq. (7) is best applied to the electronic states that describe the main, most intense, peaks in an XPS spectrum (see, for example, Ref. 35). Thus, we consider first the case of closed shells where there is a single main peak for each shell (see Fig. 1). Clearly, E(R) must be greater than zero because the relaxation, the response to the presence of the core hole, must act to lower the energy.⁶ The magnitude of E(R) is related to XPS “losses” of intensity from main peaks to the satellite peaks, as has been discussed elsewhere.⁷⁵ However, since the BE(FO)s reflect only the valence charge density in the initial state before ionization, differences in the BE(FO)s for a given atom in different environments or different compounds, ΔBE(FO), directly reflects the differences in the chemical environment of that atom in its initial state. The absolute values of the observed XPS BEs, on the other hand, are determined not only by the initial state chemistry but also by the response, E(R), to the presence of the core hole, and are theoretically included only in the BE(ΔSCF). There are situations where the E(R) will be very similar for different systems and situations where E(R) will be very different for different systems [see Ref. 13 for a study of the N(1s) BE(FO) and BE(ΔSCF) for a large number of compounds]. There is a criterion, described below, to estimate when changes in the BE(FO) will dominate the shifts of the BE(ΔSCF) of an atom in different environments and compounds; i.e., when changes in E(R), ΔE(R), are small and the BE(FO) shifts are similar to the XPS observed BE shifts. In such cases, the ΔBE(FO) can be used to identify the origins of the observed ΔBE. The important consideration for the BE(FO) shifts to dominate is that the systems whose BE shifts are studied need to be of comparable size and contain a similar number of atoms. This follows because E_R generally increases as the number of atoms that can relax in response to a core-hole increase.^1,2 The dependence of E_R on both the size of a system and on the character of the chemical bonding will be examined in Sec. VI where BE shifts between N atoms in two related compounds are compared and where the shifts between surface and bulk atoms of a metal are compared.

For open shell systems, the situation with respect to the definition of E(R) is, in general, more complicated. Here, the BE splittings to different multiplets of the core ion (see Fig. 2), where this is shown for the 2p_3/2 XPS of a cluster model of MnO, are likely to be quite important and make it more difficult to identify a single “main” peak. Thus, for these systems, there is less focus on interpreting the ΔE(R)s.

Several things are immediately clear from Table I. There is a large ΔBE for the 1s BE between Cu⁰ and Cu⁺, which, for the AVG coupling, is 8.6 eV for ΔBE(FO) and 9.9 eV for ΔBE(ΔSCF). Clearly, this ΔBE is dominated by an initial state effect. Furthermore, very similar behavior also holds for the 2s, 2p, and 3s ionizations; ΔBE(FO) for the ionizations range between 8.4 and 8.6 eV while the ΔBE(ΔSCF) range between 9.7 and 9.9 eV. The difference between the Cu⁰ and Cu⁺ relaxation energies [Eq. (7)] is also very similar for the AVG BEs for all four of these shells; the difference is 1.2 eV for 3s ionization and 1.3 eV for the 1s, 2s, and 2p ionizations. However, the magnitude of E(R) is quite different for the different shells, largest for the deepest core, 1s ionization with E(R) ∼ 50 eV and smallest for 3s ionization with E(R) ∼ 7 eV. The shifts, ΔBE, of the Cu⁰ BEs from the Cu⁺ BEs, when the multiplet splitting for the core ionization of Cu⁰ to singlet and triplet coupled core-level ions is included, are not significantly different from the ΔBE obtained using the average coupling. This is examined in Table I for the 3s ΔBE. For Cu⁰, the difference of the ³S and ¹S coupled 3s BEs with respect to the BEs for the average, AVG, coupling taken as 0.00, is given for both the ΔBE(FO) and ΔBE(ΔSCF). For ΔBE(FO), the coupling of the ion to ³S reduces the BE by 0.11 eV and the coupling to ¹S increases the BE by 0.34 eV. Both the small magnitude of the BE differences and the difference of a factor of three for the ΔBE(FO) changes for ³S and ¹S are explained because the BE changes are due to a small exchange integral that adds differently to the energies of the two couplings.³² For ΔBE(ΔSCF), the changes due to coupling to ³S and ¹S rather than average are still small but somewhat larger and different by more than a factor of 3. This is because the variational process acts to increase the lowering of the energy of the triplet coupled multiplet and to reduce the raising of the energy of the singlet coupled state.^11,32 Overall, the changes in the ΔBE(FO) due to using specific coupling rather than average coupling are less than 4% and the changes in ΔBE(ΔSCF) are less than 8%. Thus, we can use the AVG couplings to analyze and understand the BE shifts for this model system, although there are circumstances where the multiplet splittings do need to be taken into account as discussed in Sec. VI E. Clearly, the BE shifts between ionization of Cu⁰ and Cu⁺ are almost purely due to initial state effects. We turn now to an explanation of the origin of the FO ΔBE, the initial state effect, and then explain why ΔBE(ΔSCF) has an almost constant increase of ∼1.3 eV for ionizations of all the shells.

The key difference between an isolated Cu⁰ atom and an isolated Cu⁺ cation is that the diffuse 4s orbital is occupied in Cu⁰ but not in Cu⁺. The potential of this 4s orbital at the Cu nucleus is the main contribution that it makes to the BE shifts from Cu⁺ to Cu⁰. This 4s orbital has a mean value of $⟨$r $⟩$ = 1.76 Å. While the potential of an electron at the nucleus, $⟨$1/r $⟩$ determines BE shifts, this can be roughly approximated by 1/ $⟨$r $⟩$⁠, which is useful since the size of an orbital is a more familiar quantity that one can easily estimate. For the specific case of a Cu⁰ 4s orbital, 1/ $⟨ r⟩$ = 0.72 Å⁻¹ is a rough guide to the potential of the 4s orbital at the Cu nucleus which is $⟨$1/r $⟩$_4s= 0.57 Å⁻¹_. To determine whether we can focus on the contribution of the 4s orbital to the ΔBE shifts between Cu⁰ and Cu⁺, we need to know the difference of the 1s to 3d orbitals of Cu⁺ and Cu⁰. We expect that they will be very similar since their relaxation from the pure Cu⁰ orbitals when the diffuse Cu 4s is removed is small. To confirm this, we calculated the diagonal overlaps of the 1s to 3d orbitals between those optimized for Cu⁰ and those optimized for Cu⁺. These overlap integrals, which are given in the supplemental material, are essentially 1.0 for all orbitals;¹¹⁰ even for the 3d orbitals, which differ most, the overlap is still 0.9999. In other words, the inner orbitals of Cu⁰ and Cu⁺ are essentially identical and the XPS ΔBEs must arise almost entirely from the contribution of the 4s orbital to the Cu⁰ BEs. The main effect of having the 4s orbital occupied (Cu⁰ cf. Cu⁺) is to reduce the core orbital BEs (1s through 3p) by its potential at the nucleus, with that reduction being proportional to $⟨$1/r $⟩$⁠. Converting the $⟨$1/r $⟩$ = 0.72 Å⁻¹ for the HF Cu 4s orbital of Cu⁰ to an energy in eV^11,66 leads to a shift to lower BE of 10.3 eV. This simple model for ΔBE is within 20% of the accurate HF calculated ΔBE(FO) of ∼8.5 eV in Table I. This shows not only that the dominant origin of the ΔBE is the occupation and the size of the 4s orbital but that its value can be reliably estimated using a simple model.

In order for the final state effects on the BE shift to be both small and similar for all core levels (as they are), the difference of relaxation energies for ionization of a given shell in Cu⁺ and in Cu⁰ must be small and similar for ionization of all shells. The E(R) [see Eq. (7)] for the average couplings of ionization for the different shells in Cu⁰ and Cu⁺ are given, in eV, in Table II. From the data in the table, it is clear that there is a huge difference in the E(R) from ∼48 eV for 1s ionization to ∼7 eV for 3s ionization. These large differences in E(R) are certainly to be expected based on the equivalent core model^76,77 since fewer shells of electrons see a larger effective charge when the outer shells are ionized and more electrons see this effective charge when inner shells are ionized. However, the main contribution to the difference in the relaxation of Cu⁰ and Cu⁺ is from the 4s orbital, which is present only in Cu⁰. Furthermore, the relaxation of the 4s orbitals should be similar for the ionization of all core shells because the 4s orbital is mostly outside of all the core shells, as shown by the $⟨ r ⟩$ in Table III; thus, from the equivalent core model,^76,77 the 4s orbital sees a larger effective nuclear charge for all core ionizations. This is why the difference of the Cu⁰ and Cu⁺ E(R) for the ionization of all shells is so similar, between 1.2 and 1.3 eV (see Table II). In fact, the ΔE(R) = 1.2 eV for the 3s ionization is slightly smaller than the ΔE(R) for the other orbitals. This is because the 3s orbital is the most diffuse of the core orbitals (see Table III). Thus, it is a poorer approximation to shrink the 3s orbital into the nucleus; hence, the equivalent core model is a less good approximation for diffuse core orbitals than for the very contracted deeper core orbitals.

The main conclusion of this analysis has been to explain that the BE shift due to changing the charge state of an atom is significant but similar for ionization of all core shells and, most important, is largely an initial state effect. It has, however, also provided a direct physical understanding of the origin of the increase in BE with the effective charge of the ionized atom and it has shown that the BE shift is very similar for all core shells. This analysis is fully consistent with the common understanding that core BEs increase as the charge of an ionized atom is increased. Of course, there are exceptions to this simple correlation. For example, for solid Cu⁰ metal compared to Cu⁺ in Cu₂O, there are the additional effects of having a Madelung potential in Cu₂O, but not in metallic Cu⁰, and a difference in relaxation between the metal and oxide. This is why, for instance, ΔBE for Cu2p between Cu metal and Cu₂O is much smaller than for ΔBE(2p) between the isolated atoms of Cu⁰ and Cu⁺ (see Table I). However, even for an isolated atom, we show in the following that the total charge of an atom is often not the only initial state effect leading to BE shifts.

We demonstrate that there can be other significant initial state effects by considering the core BEs for different low-lying configurations of the neutral Cu atom. We consider: (1) the ground state configuration, …3d¹⁰4s¹, already discussed; (2) the first excited configuration, …3d⁹4s² where a 3d → 4s promotion from the ground state has been made; and (3) the second excited configuration …3d¹⁰4p¹, where a 4s → 4p promotion from the ground state has been made. For the two excited initial state configurations, Koopmans’ theorem BE shifts, ΔBE(KT), from the ground state, 3d¹⁰4s¹, BEs are calculated for the 1s, 2s, 2p, and 3s ionizations. For all ionizations, the average coupling of the open shells is used. ΔBE(ΔSCF), not shown, follows the same trends as the ΔBE(KT).

If the only reason for BE shifts was a change in the effective charge of the atom, which is core ionized, then it follows from the relationship of Eq. (8) that ΔΒE(KT) for the excited configurations in Table IV should all be 0. This is because, for both the excited initial state configurations, the Cu atom has the same charge; it is Cu⁰. However, they are large; ∼7 eV for the 3d⁹4s² configuration and ∼2.5 eV for the 3d¹⁰4p¹. Both the large values of ΔBE for the excited Cu⁰ configurations and the difference of the ΔBE for the 3d⁹4s¹ and 3d¹⁰4p¹ initial configuration have a common origin, which is, again, the size of the orbitals which have different occupations in the configurations. This origin is very closely related to the origin of the ΔBE between Cu⁰ and Cu⁺, discussed above, where we considered the size of the 4s orbital using $⟨ r ⟩ 4 s$ as a measure of the change in the potential at the nucleus when the 4s electron is removed from Cu⁰ to form Cu⁺. In the present case, for …3d⁹4s² configuration, a compact 3d orbital with $⟨ r ⟩$ = 0.52 Å has been promoted into a diffuse 4s orbital with $⟨ r ⟩$ = 1.76 Å (see Table III). While the $⟨ r ⟩$ in Table III are for the ground state and the $⟨ r ⟩$ for the excited …3d⁹4s² configuration, are ∼10% smaller, the important fact is that the $⟨ r ⟩ 4 s$ is a factor of three larger than $⟨ r ⟩ 3 d$ for both configurations. Although the change in potential at the nucleus from the 3d and 4s shell occupations for the 3d → 4s excitation is not as great as for promoting a 4s electron to infinity, as in the case when comparing the BEs of Cu⁰ and Cu⁺, it is still large. Thus, excitation from the compact 3d to the diffuse 4s leads to a change in the BEs of ∼75%– 85% of that in removing the 4s electron entirely. The size of the 4p orbital, with $⟨ r ⟩ 4 p$= 2.7 Å from the HF calculation for the …3d¹⁰4p¹ configuration, is 75% larger than the $⟨ r ⟩ 4 s$ for the …3d¹⁰4s¹ configuration. Using 1/ $⟨ r ⟩$ as rough guide to the orbitals’ contribution to the potential at the nucleus would lead to an increase for the core-level BE of ∼2.9 eV for the …3d¹⁰4p¹ configuration, consistent with the ΔBE(KT) in Table IV.

Summarizing this section, the changes in the extent, or diffuseness, of the valence orbitals lead to significant ΔBE for the same physical reason that changing the charge of the ionized atom leads to a ΔBE. Moreover, the changes, ΔBE, from excitation of a valence electron are of a comparable magnitude to the ΔBE for the removal of a valence electron (75% and 30% for the 2 excited states, respectively).

So far, we have discussed the effects of ionization and electron configuration for the “toy” problem of an isolated Cu atom, which has allowed us to directly understand the origin of the ΔBE. Below, we extend these insights to the XPS of real molecules and solids.

In this section, we briefly review five cases where BE shifts have been observed, identify the mechanisms that lead to the ΔBE, and give references to more complete analyses. In most cases, the origin of the BE shifts is directly related to the mechanisms described above for the “toy” model of the Cu atom. In one case, for the BE shifts for I/Pt, a new mechanism for shifts, which is the presence of internal electric fields, will be identified and this will be given a somewhat longer description. In another case, it will be shown that multiplet splittings of the XPS peaks in an open shell heterogeneous molecule can be different for the different elements. Furthermore, with an extension of the logic in Sec. V, it will be shown how these multiplet splittings reflect the electron distribution in the valence shells because the splittings reflect the charge distribution in the valence open shell orbitals. In all cases, it will be shown that initial state effects make major contributions to the ΔBE; in most cases, initial state effects make the dominant contributions to the observed ΔBE.

The relationship between the different bonding in two compounds, pyrrole, C₄H₅N, and pyridine, C₅H₅N, and the 1.2 eV shift of the N(1s) BE, being greater in pyrrole has been studied by Bagus et al.^78,79 The schematic of the two molecules in Fig. 7 clearly shows that the character of the N(2s) to N(2p) hybridization will be different between the two compounds; the traditional descriptions of the N hybridization in these molecules are discussed in detail in Ref. 79. It has been known from the early days of the use of XPS that hybridization from ns to np could lead to BE shifts possibly as large as 1 eV.⁴ Our object in this brief review is to present a novel theoretical approach, which allows us to quantify the N(1s) ΔBE due to the different hybridization in the two molecules and to exclude that different effective charge transfer contributes to the ΔBE. In addition, we show that the reason for the ΔBE due to hybridization follows the same logic described in Sec. V for the ΔBE for different configurations of the Cu atom. Furthermore, this type of theoretical analysis can be applied to other cases to quantify the BE shifts due to hybridization.

The experimental N(1s) for these two molecules is ΔBE = +1.2 eV, the theoretical ΔBE(ΔSCF) = +1.2 eV, and the ΔBE(KT) = 0.9 eV, all with pyrrole being larger. The dominantly initial state, KT, origin of the ΔBE is consistent with the fact that the two molecules are of comparable size and shows that the difference in the chemical bonding is the origin of the ΔBE. Using a special method of constrained orbital variations,^80,81 it was possible to separate the contributions of changes in the nominal charge transfer from N to the C atoms and changes in the N 2s → 2p hybridization to the KT BEs. The charge transfer alone would actually lead to a slightly larger BE, 0.15 eV, for pyridine than for pyrrole, opposite to the observed 1.2 eV larger BE for pyrrole. However, the constrained variations also explicitly show the magnitude that the N 2s → 2p hybridization is more important for the bond in pyrrole than pyridine. The pyrrole hybridization lowers the total molecular energy by 3.9 eV compared to the hybridization in pyridine where the total molecular energy is lowered by only 2.3 eV. Following the analysis for the ΔBE for Cu⁰ between the …3d¹⁰4s¹ and …3d¹⁰4p¹ configurations, a larger promotion of N 2s to the more diffuse N 2p orbital should lead to a larger BE for pyrrole, where the 2s → 2p promotion (hybridization) is greater than in pyridine. In fact, the change in ΔBE due to this hybridization is 1.05 eV larger for pyrrole than for pyridine. Combining the ΔBE due to hybridization of +1.05 eV with the ΔBE due to charge transfer of −0.15 eV leads to a prediction for the ΔBE = +0.9 eV larger for pyrrole, consistent with the observed N(1s) XPS BE shift. Thus, the information from the “toy” exercise for the Cu atom has been a guide to understanding the chemistry in real compounds and is a caution against relating ΔBE purely to changes in the charge on the ionized atom.

Surface core-level shifts (SCLSs) have been observed both for metals^82–85 and for oxides.^15,63 For metals, there is a small shift, ∼0.5 eV with the surface atom normally having a smaller BE than a bulk atom. It has been shown that this small ΔBE is due to a cancellation of two much larger shifts.^83,86 A surface atom has a smaller environmental charge density than the bulk atom, since it is surrounded by fewer atoms. In the cubic crystal structure, a bulk atom has 12 nearest neighbors while a (100) surface atom has only 8 nearest neighbors.⁷⁰ This different environment can be seen from the Cu18 cluster used to model the surface core-level shifts shown in Fig. 8. Since we want to identify the role of d-electron hybridization or promotion from Cu 3d to Cu 4sp, we designed calculations of the Cu₁₈ wavefunctions where this hybridization was forbidden and then where it was allowed. The d hybridization of the representative surface and bulk atoms in the cluster (see Fig. 8) was prevented by fixing the 28 1s to 3d core electrons of these atoms to have exactly the same orbitals as they have in an isolated Cu atom.⁸³ For these two surface and bulk atoms, only their 4s electrons are, thus, included in the variations to optimize the HF wavefunctions and the hybridization of their d electrons is prevented. The only contribution to different core-level BEs can be regarded as the different environmental charge density of the conduction band electrons, since the 3d electrons are forced to maintain their atomic character and their atomic occupation. In a second calculation of the wavefunction for the Cu₁₈ cluster, only the 18 Ar core electrons of the representative bulk and surface Cu atoms were frozen as for the isolated Cu⁰ atom, but their 10 d and one 4s electron were included in the variation to determine the HF wavefunction and orbitals. For this wavefunction, 3d to 4sp hybridization is possible. In a final calculation, none of the orbitals are frozen and all the 29 electrons of the representative bulk and surface atoms were allowed to be varied. The results of this last calculation for the BEs were essentially identical to those where the Ar core electrons were frozen. This is fully consistent with the fact that the Ar core electrons of Cu do not participate in the interaction and bonding of Cu atoms. We should stress that the BE separation for bulk and surface atoms from these calculations is only at the KT level but we have verified that the KT and ΔSCF BE shifts between ionization of the bulk and surface atoms are similar for the calculation where no electrons are frozen. Thus, the KT separation with these calculations where core electrons are fixed or frozen is a reliable guide to the involvement of the 3d and Ar core electrons in the XPS BE shifts. For the wavefunction where the 28 Ar + 3d core electrons were frozen the different environment of the surface and bulk atoms leads to a larger BE(KT) for a surface atom by ∼0.55 eV. This is because the bulk atom is surrounded by the conduction band electrons coming from 12 nearest neighbor atoms, while the surface atom is surrounded by conduction band electrons coming from only 8 nearest neighbors (see Fig. 8). However, the 3d → conduction band hybridization is greater for a bulk atom than for a surface atom simply because there are more atoms to form bonds with for the bulk atom. The hybridization increases the BE for exactly the same reasons as we showed above for the “toy” cases of different Cu atom configurations. The 3d hybridization for the bulk atom increases the BE(KT) by almost 4 eV while the smaller 3d hybridization for a surface atom increases BE(KT) by only ∼2.5 eV, a difference of 1.5 eV in the opposite direction to the purely environmental shift. It is the cancellation of these two mechanisms that leads to the small negative observed SCLS BE shifts for metals.⁵⁴ 

The situation is quite different for ionic compounds where the bulk and surface BEs have been studied for MgO⁶³ and CaO.¹⁵ If one simply considers the different Madelung potentials at bulk and surface atoms, the SCLS of the cation should be a shift to higher BE for the surface atom because there are fewer anions surrounding the surface ion (5) than a bulk ion (6). Similarly, the SCLS of the anion, O²⁻, should be a shift of the same magnitude but to lower surface atom BE because the surface anion is surrounded by fewer cations. In fact, the SCLS of the surface cation is, as expected, to higher BE by roughly the expected magnitude. However, the shift of the surface anion is not as large as expected but is almost zero. The reason for this difference is due entirely to the different charge distribution of surface and bulk anions and is entirely an initial state effect. The surface cation retains the spherical behavior of the bulk cation and the SCLS BE shift is nearly as expected from the change in the Madelung potential at bulk and surface atoms. This is because the charge density of the cations is largely that of the isolated cation and is not especially influenced by its environment. The situation is quite different for the charge distribution of the electrons of the very loosely bound O²⁻anion. In this context, it is important to remember that the O²⁻ anion is not bound as an isolated atomic anion. It exists only because of the environment of the crystal; in particular, the six nearest neighbor cations, which forces the weakly bound O 2p electrons to remain localized around the O nucleus. Obviously, the potential around a surface O²⁻ is quite different since the cation that would be immediately above the anion are missing at the (100) surface of a cubic oxide. Thus, the O²⁻ anion charge can, and does, expand outward from the surface and this distortion makes the charge distribution more diffuse. Specifically, this expansion leads to an ellipsoidal, cigar like, charge distribution of a surface O²⁻ anion, which extends above the surface, as opposed to the nearly spherical distribution of a bulk O²⁻ anion.⁶³ With this distorted charge distribution of a surface anion, the average distance of the electrons from the O nucleus is larger for a surface than for a bulk anion and the potential due to the O valence electrons on the core electrons, which, as we showed in Sec. V, lowers the core-level BEs is reduced in magnitude. This change in potential raises the KT BE of a surface O²⁻ anion and cancels, in large part, the SCLS BE increase due to the reduced number of cations around a surface anion.⁶³ This logic is simply an extension of the BE shifts for different electron configurations that we showed for the “toy” case of Cu atoms in Sec. V.

Another example of the use of XPS BE shifts to identify bonding character concerns the XPS BE shifts of the atoms in small metal particles supported on “inert” metal surfaces as the size of the clusters grows from small to continuous metal overlayers (see the representative Refs. 61 and 87–91, and references therein). These BE shifts with cluster size are a decrease in the BE of ∼1 eV from the smallest supported clusters to large clusters and monolayers of the metal adsorbate. If the origins of this decrease in BE with increasing cluster size are understood, it may provide a guide to understanding how the electronic structure and morphology change with the size of the supported metal particles. This BE shift had originally been explained by treating the metal cluster as an ideal classical metal conductor where the positive charge of the ionized cluster must remain on the surface.^88,92,93 The role of such a surface charge can most easily be understood by imagining that the cluster is a sphere where the classical electrostatic potential of a spherical unit charge is constant within the sphere with a magnitude proportional to 1/R where R is the radius of the sphere. Since this is the potential of a positive charge, it will raise the core-level BEs of the atoms in the cluster by an amount proportional to 1/R, the radius of the cluster. Clearly, this increase will be smaller for larger clusters, and depending on assumptions about the cluster size, it can be consistent with a decrease in the BEs of the atoms in a large cluster of ∼1 eV as 1/R → 0 for large clusters. Indeed, the shift in the BE could be, and was, taken to reflect the sizes of the supported clusters.^87,92 This classical interpretation can be viewed as assigning the BE shift with cluster size as a final state effect since the response that moves the electrons toward the core hole to leave a net positive charge at the surface of the cluster is a final state relaxation. For a cluster to be significantly large for this classical argument to hold, it would have to be so large that the BEs have reached the bulk value.⁹⁰ Moreover, this classical explanation does not take into account the quantum effects, which dominate for the relevant small clusters of tens to hundreds of atoms; it neglects the atomistic character of the metal atoms in small metal clusters involved in this shift. The influence of cluster size can be taken into account with the BEs obtained from electronic structure calculations of the wavefunctions and energies of metal clusters of increasing size. Clearly, final state effects will be important for particles of different sizes since the relaxation energy, which lowers the BE(ΔSCF) from the BE(KT) values, is larger for clusters that contain more atoms. Indeed, the shifts of KT and ΔSCF BEs determined for these clusters can separate the contributions of initial and final state effects to the BE shifts with cluster size.^60,61,90 We will not discuss this separation of initial and final state effects here and will consider only the ΔSCF BEs for the atoms of metal clusters of increasing size. This is sufficient to show that lattice strain, the change of interatomic distances with cluster size, is a key reason responsible for the BE shifts with size of the supported clusters.^60,61 As a first step, we assumed that the interatomic distances did not change with cluster size and for the different size clusters we used interatomic distances in the clusters that were the same as in the bulk. However, the decrease of the BE with respect to cluster size assuming that the interatomic distance is constant does not lead to the observed ∼1 eV shift to smaller BE from very small clusters to the bulk BE but only gives about two-thirds of the shift. Clearly, there must be another contribution to the BE shift. It is shown⁶¹ that this contribution for the remaining one-third of the BE shift comes from the fact that bond distances do change from small metal clusters to bulk metal. This is logical since the bond in smaller clusters is spread over a fewer number of nearest neighbors and hence the individual atom–atom bonds will be stronger because there are, on average, fewer than the 12 neighbors in the bulk. It was also determined from low energy electron diffraction (LEED) and related surface science measurements of supported clusters,^94,95 that the interatomic distances are shorter for small clusters than for larger clusters and for the bulk metal. When the lattice strain, the change of interatomic distance, is taken into account for the cluster models, the BE shift between small and large clusters is more nearly consistent with the measured shifts.⁶¹ Furthermore, the change in BE due to changing the interatomic distance is largely an initial state effect. Thus, the BE shifts with particle size are a way of understanding the morphology of particle growth.⁶¹ 

The changes in the BEs of an adsorbate with increasing surface coverage have been measured and related to the character of the chemisorption bond at different adsorption sites. Specifically, the system considered here is I/Pt(111). LEED measurements showed two patterns for the I adsorbate;^96,97 one at lower coverages of up to 0.33 I adsorbates per surface Pt atom, and the second for additional coverages up to 0.43 I adsorbates per surface Pt atom. This has been interpreted as I at threefold sites until the coverage of 0.33 and then the additional I atoms at higher coverages are adsorbed at on-top sites (see Ref. 98). The different sites are shown in Fig. 9 for cluster models that we used to study the interaction of I/Pt(111) at both threefold and on-top sites.⁹⁹ The cluster model study was to analyze and interpret the properties reported by Jo and White,⁹⁸ who had performed a detailed study of the coverage dependence of I/Pt including XPS and work function measurements. They were able to determine that the I(3d_5/2) BE was different for I at a threefold site, 619.3 eV, and at an on-top site, 618.2 eV. The relatively low BEs suggest that the I is negatively charged although the extent of the negative charge is not obvious. Indeed, one interpretation of the 1.1 eV smaller I(3d_5/2) BE at an on-top site is that the I at this site has a larger negative charge, consistent with the idea that a smaller BE reflects a larger negative charge on Iodine. The focus of this review of our earlier theoretical study⁹⁹ is the interpretation of the origin of the I(3d) BE shift and whether the large BE shift does indeed indicate a significant difference of anionicity of the I at the different sites. The 31 Pt atom cluster shown in Fig. 9 contains atoms in three layers of an unreconstructed Pt(111) surface and was designed to provide equivalently accurate representations for I adsorbed at both sites.

The first important conclusion of our cluster model theoretical study^99,100 is that the adsorbed I at either threefold or on-top sites is essentially a full I⁻ anion and that there is essentially no difference between the ionicity at either site. This complete ionicity of the I adsorbate is quite consistent with the electron affinity of halides. Thus, a difference in ionicity cannot be the origin of the 1.1 eV shift of the I(3d_5/2) BE to lower BE at the on-top site. However, it was also found that the equilibrium distance of the adsorbed I above the surface, z_e, was greater by 0.3 Å at the on-top site than at the threefold site. This is not at all surprising because the distance between the adsorbed I and the nearest surface layer Pt atoms would be less at the on-top site than at the threefold site for I at the same distance, z, above the surface. Thus, the steric repulsion between the electronic charge of the I⁻ anion and the electronic charge of the Pt surface, for the same distance of I above the surface layer of Pt atoms, is greater at the on-top site and this favors a larger z_e at this site. The difference of z_e at the two sites above a metal surface is very important for the difference in the I(3d_5/2) BEs at these sites. Indeed, this change in distance, not considered by Jo and White, is, in fact, the reason for the 1.1 eV BE shift to lower BE at the on-top site. Since I/Pt(111) is an ionic adsorbate on a conducting metal surface, the origin of the change in BE can best be understood in terms of image charge theory (see, for example, Ref. 101). Within image charge theory, the metal substrate electronic charge is viewed as a continuous background of negative charge, and the atomistic structure of the surface is neglected; the ion is viewed as a point charge. With this model, we can use classical electrostatics to understand the electric field above the surface, which is the electric field that the I⁻ adsorbate sees, as shown in Fig. 10. Since the substrate is assumed to be a perfect conductor, the electric field arising from the charged adsorbate must terminate at the surface [Fig. 10(a)] and the electronic charge in the conductor must modify to insure this termination. Classical electrostatics shows that the electric field above the surface, seen by the adsorbed I⁻, is equivalent to placing a positive point charge a distance z below the surface [Fig. 10(b)]. However, since I⁻ is in the presence of an electric field, its core-level BEs will be modified as discussed in Sec. IV. The image charge field at the I nucleus is proportional to 1/(2z_e), where 2z_e is the distance between the adsorbate I⁻ and its image charge. The significance is that the potential of the “image charge,” is greater at the threefold site, where z_e= 2.5 Å, than at the on-top site, where z_e= 2.8 Å. Just as the potential of the valence electrons (negative charge) lowers the BE of a core level, the potential of the positive image charge raises the I(3d) BE. Indeed, if one computes the electric potential differences, using standard electrostatics, at the two sites to estimate the effect of the change in z_e, one calculates a BE shift of 0.7 eV between the two sites, which is similar to the observed BE shift of 1.1 eV. Since image theory neglects the atomic structure of the surface, we should not expect that it will give quantitative agreement with the observed BE shifts due to I⁻ adsorbed at different sites. For such an agreement, we would need to directly compute the BE shifts for the cluster models shown in Fig. 9.

In Ref. 99, the KT I(3d) BEs were calculated for the two I⁻ surface sites as part of the study confirming that the reason for the BE change is due essentially entirely to the z_e differences between the threefold on-top sites. Since the two models for these sites have exactly the same number of atoms, it is reasonable to expect that the relaxation energies will be essentially the same for the two sites and that ΔBE(KT) will be almost the same as ΔBE(ΔSCF). See Sec. VA, where the equivalence of ΔBE(KT) and ΔBEΔSCF) was explicitly shown for the N(1s) BEs of pyridine and pyrrole. In fact, the shift of the BE(KT) between the two sites is 1.1 eV fully consistent with the measured ΔBE and close to the internal electric field model. This clearly demonstrates that the internal electric fields, which arise with charged adsorbates on metal surfaces, as is the case with alkali metal or halogen adsorbates,⁸² can lead to significant shifts of adsorbate BEs. In particular, the I(3d) ΔBE for I/Pt shows us that these shifts reflect different adsorbate distances above the surface.

As a final example, we consider the multiplet splitting of the N(1s) and O(1s) BEs for the open shell molecule NO molecule, with configuration 1σ²(O1s)2σ²(N1s)…2π¹, where only the O(1s) and N(1s) core orbitals and the singly occupied 2π orbital are given. The purpose of this analysis is to show the origin of the multiplet splitting, and the differences in this splitting between N(1s) and O1s), in terms of the chemical bond in the NO molecule, and to show that, properly interpreted, they give direct evidence for the character of the chemical bond. The coupling of the ionized 1σ or 2σ orbital with the singly occupied 2π orbital leads to ³Π and ¹Π multiplets. Figure 11(a) shows the experimental N1s spectra. The energy splitting of these multiplets has been studied in detail.^31,72,102 Theoretical plots of the different multiplet splitting of the N(1s) and O(1s) BEs are shown in Figs. 11(b) and 11(c), where the intensity of the broadened peaks have intensity ratios of very close to 3:1 consistent with the intensity expected for the different multiplicities.³¹ As can clearly be seen in Figs. 11(b) and 11(c), the splitting of the ³Π and ¹Π BEs for the N(1s) ionization is 1.4 eV while the splitting for the O(1s) ionization is only 0.5 eV and the O(1s) multiplet peaks are unresolved experimentally. The difference in the N(1s) and O(1s) multiplet splittings is largely an initial state effect since the BE(KT) splittings¹⁰² are 0.7 and 1.2 eV for the O(1s) and N(1s) ionizations, respectively, compared to BE(ΔSCF) splittings of 0.5 and 1.4 eV. This difference in the O(1s) and N(1s) multiplet splittings can be related to the character of the 1π and 2π orbitals, where the 1π orbital is bonding between N(2p) and O(2p) and the 2π orbital is antibonding between N(2p) and O(2p). This is exactly as expected from the quantum mechanical rules for molecular orbitals within the linear combination of atomic orbitals proposed by Mulliken,^11,104,105 where only when the bonding orbital is filled are electrons placed in the antibonding orbital. However, only for homopolar diatomic molecules, such as O₂, do the bonding and antibonding orbitals have equal weight on both atoms. In the HF wavefunction for the ground state of NO, the 1π orbital has a population, that is, ∼30% on N and ∼70% on O, based on Mulliken orbital populations.^106,107 This means that the open shell 2π orbital, to be orthogonal to the 1π orbital, must be ∼70% on N and 30% on O. This is shown graphically in a contour plot of the 2π orbital in Fig. 12. The plot is for the orbital values in a plane that includes the internuclear axis and the lines are contours of equal value of the orbital; it was made with the plotting package developed by Hermann.¹⁰⁸ The contours change in magnitude by a constant value Δ between the minimum value of the absolute magnitude of the orbital, | φ |, to the maximum value of | φ | used in the plot where the minimum value of | φ | is at the outermost contour and the maximum is at the contour nearest the nuclei. Contours of positive values are shown by full lines and contours of negative values are shown by long dashed lines. The contours for φ = 0 are shown by short, dashed lines. There are two contours of φ = 0; one is a horizontal line along the internuclear axis as must be the case for an orbital of π symmetry¹¹ and the other is a curved, but nearly vertical line between the positions of the N and O nuclei showing that the orbital is antibonding. The critical point in Fig. 12 is that the electron density around N is larger than the electron density around O, which is fully consistent with the orbital populations.

In this Perspective, we have reviewed the understanding and significance of XPS BE shifts and the selection rules for XPS as well as the sudden approximation (SA) for determining intensities. The main purpose of these discussions has been to demonstrate that there is an essential foundation for using orbitals to understand the physical and chemical meaning of XPS features (see Bagus and Illas¹⁰⁹). This role has been questioned by Truhlar et al.,⁵⁴ who quite correctly pointed out that a high level of electron correlation is required to reproduce exact XPS measurements of BEs. However, our Perspective has been devoted to establishing that orbitals are the key to relating XPS measurements to the physical and chemical properties of the systems measured. We have provided a generalized definition of Koopmans’ theorem for BE(KT) so that it may be applied not only to closed shell systems, for which Koopmans’ theorem was originally proposed but to open shell systems as well. In particular, we have established a basis for comparing and contrasting initial state BE shifts using KT, ΔBE(KT), to those which include final state relaxation, E_R, termed ΔBE(ΔSCF).

We examined the physical and chemical properties that lead to the ΔBE(KT), using a simple one atom model, Cu in the zero valent state, Cu⁰, and in the monovalent state, Cu⁺. This is first established for several toy problems of the ΔBEs for different states and ionicity of a Cu atom, Cu⁰ and Cu⁺. This illustrated why, as expected, core-level BEs depend on the charge of an atom and also showed why the total charge is not the only reason for BE shifts. The differences in the character and spatial extent of the outermost valence orbital can also lead to large ΔBE, even when the total charge on the atom is not changed. For specific real solid material examples, we then established that initial state shifts dominated, provided the number of atoms involved in the compared situations was similar. However, in one of our examples, we also show that even when the size of a system changes (nanoparticles of different sizes), the change in final state relaxation is not the only contribution to changes in the BE but initial state effects are still an important contribution.

In one way, our goal for theory can be viewed as an extension of the use of theory in the original ESCA publications,^4,5 where approximate semiempirical theories were used to support their XPS analyses. By using ab initio HF theory, we are able to avoid the uncertainties arising from the choices of parameters in semiempirical theory. Our conclusions can be summarized as follows:

1. MO theory at the HF or Dirac HF level, while not providing accurate absolute BEs for extended systems, because they do not include many-body effects, is quite sufficient in most cases to give the BE shifts between the situations being compared.

2. In many cases, the BE shifts are dominated by the changes in the initial ground state of the systems being compared.

3. Owing to (2) above, the KT calculation of the shifts, which does not include any final state relaxation effects, is often, maybe mostly, sufficient to provide semiquantitative agreement for the observed shifts for closed shell systems.

4. For open valence shell systems, multiplet splitting, involving angular momentum coupling between the unpaired electron left in the core level ionized and the electrons in the open valence shell, leads to several different XPS BEs for the ionization of a given core shell, a minimum of 2 as in NO. When spin–orbit splitting is significant, plus the effects of crystal field splitting for solids, the spectra become complex. However, the KT calculation for the shift between weighted averages of the multiplets for compounds with different environments but with the same nominal occupations still provides insight into the origins of the BE shifts.

5. The KT calculated values of the magnitude of the splittings between multiplets, plus their degeneracies and the application of the sudden approximation, give good guides to their experimental XPS relative intensities and relative positions. This is particularly useful when an experimental spectrum feature is actually a broad asymmetric band of partially resolved features (e.g., the 2p XPS of Mn2⁺ in MnO).

6. Understanding the mechanisms behind the BE shifts will sometimes lead to the possibility of developing simple empirical models, without any MO calculations, to extract the major electronic, or even structural change leading to a shift, such as variations in the shape and diffuseness of outermost valence orbitals or the change in the distance of an atom above a metal surface.

7. The computational requirements for MO calculations at the HF level are comparable to those for DFT calculations. Thus, they can be used to check DFT and to avoid uncertainties related to the choice of a density functional.

8. A qualification. We have largely avoided the discussion of satellites (shake) associated with a “main” peak, since a many-body effect is involved (excitation of a valence electron along with the core ionization), which cannot be handled at the HF/Dirac level. It is a very specific many-body effect, however, and can still be calculated by MO theory by going beyond the HF/DHF approximations. When this is done, information on the origin of satellites can be obtained. This whole area requires a further review or Perspective. Experimentally, satellites become important in quantitative analysis, using peak ratios, because they represent “lost” intensity from the associated main peak. How important depends on how different the losses are for the peaks ratioed and the accuracy required.

Overall, we continue to see a very bright future for the coupling of MO theory with measurement for the interpretation and practical use of XPS but emphasize that the theory should be used to understand the origins of the XPS features, not to provide highly accurate BEs, which is best done by experiments.

The various terms, especially related to the theoretical treatment of XPS, are briefly defined:

• Configuration: The distribution of electrons over shells. The ground state configuration for the C atom is 1s²2s²2p². Sometimes, the coupling of the open shell electrons to a particular symmetry, or multiplet, is added to the electron occupation as, for C, …2p²(³P).

• Degenerate states: Depending on the symmetry of the system, the wavefunctions of one or more states will have the same energy and these are described as degenerate. For example, for the ground state configuration of the B atom, 1s²2s²2p¹, there are, if one neglects spin–orbit splitting, six states with the same energy for the ²P multiplet, i.e., there are six degenerate states.

• DFT: DFT is a commonly used electronic structure method where an exchange-correlation potential, V_XC, which is a functional of the electron density is added to the one-electron potential. The advantage of DFT is that, especially for closed shell systems, it can provide energies that have a greater accuracy than Hartree­–Fock. A disadvantage is that the exact form of V_XC is not known.

• Determinant: The wavefunction for electrons distributed over orbitals is usually described as a determinant where the matrix rows are the spin-orbitals that are occupied, and the columns are the electrons that these orbitals are applied to. Thus, for column 1, the orbitals are functions of electron 1, for column 2, they are functions of electron 2, etc. This is important because the determinant satisfied the Pauli exclusion principle for electrons and other fermions, which are spin ½ particles. In other words, the determinant ensures that the wavefunction built from determinants changes sign when the coordinates of two electrons are exchanged.

• Density functional theory: See DFT.

• Dirac–Fock or Dirac–Hartree–Fock (DHF): Formulation of the Hartree–Fock equations, see entry below; to include relativistic effects, see entry below.

• Double group: Groups that describe the symmetry of an atom or a compound and which include the spin–orbit coupling of electrons; see group theory below.

• Electron correlation effects: It is common to use an approximation for electronic wavefunctions, that is, an anti-symmetrized product of orbitals, a determinant. This is a good approximation but it is not an exact solution of the Schrödinger or Dirac equations. One must go beyond this approximation, which is a single configuration or a single orbital product if one wishes to have high accuracy for the energies and wavefunctions. This extension is described as including electron correlation effects.

• Expectation value: This is the way that average, or expectation, values for a given property are obtained from the quantum mechanical wavefunction of the system. They are integrals of the operator for the property acting on the wavefunction times the wavefunction. For an orbital, φ, the expectation value for an operator, Op is the integral over ∫φ*(1)Op(1)φ(1)dv₁, where the integral is over space and spin coordinates. The expectation value is denoted $⟨$φ|Op|φ $⟩$

• Group theory: Point groups provide the symmetry information about compounds, which is different from, lower, than for atoms. When spin–orbit coupling is taken into account, one needs to use the double group that distinguishes even and odd electron systems.

• HF: Approximate variational wavefunctions for a single configuration. As distinct from DFT results that normally represent a single determinant, it is often necessary to use a few determinants to describe many configurations; thus, while a configuration ns¹ms¹ coupled to ³S is a single determinant coupled to ¹S, it requires two determinants. Often a good approximation and based on an orbital model that we use to interpret measured properties.

• Madelung potential: The electrostatic potential in an ionic compound like NaCl or NiO. It represents an infinite sum over the potential due to the anions and cations in a compound. Its value is normally considered at the ionic sites since it is this potential that affects XPS BEs.

• Many-body effects: Hartree–Fock wavefunctions, which assign electrons to orbitals, do not give exact solutions of the Schrödinger or Dirac equations. Many-body effects, also described as electron correlation effects, need to be included to achieve these exact solutions and exact energies. The term many-body is used to indicate effects not possible with a single configuration or a single determinant model. See also electron correlation effects discussed above. These many-body, or electron correlation effects, must normally be included to accurately describe shake satellites.

• Multiplet: The collection of electronic states with the same energy, degenerate states. Thus, the ³P multiplet of the ground state of the C atom with configuration 1s²2s²2p² contains nine degenerate states, neglecting the small spin–orbit splitting to ³P₀, ³P₁, and ³P₂.

• Multiplet splitting: The energetic separation of the different multiplets that arise from ionizing a shell in an open shell system.

• Nonrelativistic: Nonrelativistic wavefunctions have both the total spin and the total orbital angular momentum as good quantum numbers. For the carbon atom, the ground state is ³P, meaning that the total spin is S = 1 and the total orbital angular momentum is L = 1. An important effect neglected with nonrelativistic wavefunctions is the spin–orbit splitting of orbitals; see relativistic effects discussed below.

• O_h group: This is one of the point groups (see group theory above). It describes the symmetry of cubic crystals, including NaCl and NiO and of fluorite crystals, such as CeO₂. The O_h symmetries, formally irreducible representations, for d electrons are t_2g and e_g^. When spin–orbit splitting is included, the double group symmetries (see group theory above) are γ₇ and γ₈ (see Fig. 4).

• Relativistic: Relativistic wavefunctions include the spin–orbit coupling which, for an atom, is written as J = L + S. In principle, J is the only good quantum number but often S and L are almost good quantum numbers. Thus, adding spin–orbit coupling divides the ground state of the C atom into ³P₀, ³P₁, and ³P₂, but with very small splittings of ³P₁ and ³P₂ from the lowest ³P₀ of 16 and 43 cm⁻¹, respectively. However, for the core shells of heavier atoms, the spin–orbit splitting will be tens to hundreds of eV. The Cu 2p_1/2 and 2p_3/2 splitting is 13.1 eV and for U 2p_1/2 and 2p_3/2 splitting is almost 3800 eV. A full set of spin–orbit splittings can be found in Ref. 39.

• SCF: A method used to solve the Hartree–Fock equations (see entry above). Often used as an alternate description of Hartree–Fock equations and wavefunctions.

• SA: An approximate treatment of XPS intensities that does not explicitly include the ionized, continuum electron. It is accurate for all XPS studies where the KE of the photoelectron is greater than ∼100–200 eV.

• Voigt convolution: The BEs in XPS are not precise single energies but are broadened dominantly by two different physical effects. One is the resolution of the XPS system, which has a Gaussian character and the other is the short lifetime of the core-hole states created in XPS, which has a Lorentzian character. The combination, mathematically the convolution, of the Gaussian and Lorentzian broadenings is a Voigt convolution. This convolution is an integral and does not have a functional form, which the individual Gaussian and Lorentzian broadenings have.

• WFs: The solution of the Schrödinger or Dirac equations. They contain all the information that can be known about a system.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences (CSGB) Division through its Geosciences program at Pacific Northwest National Laboratory (PNNL). PNNL is a multiprogram national laboratory operated for the DOE by Battelle Memorial Institute under Contract No. DE-AC05-76RL01830. We acknowledge useful discussions with B. Vincent Crist at The XPS Library, Salem, Oregon.

The authors have no conflicts of interest.

Paul S. Bagus: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Connie J. Nelin: Conceptualization (equal); Investigation (equal); Methodology (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). C. R. Brundle: Conceptualization (equal); Data curation (equal); Methodology (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that supports the findings of this study are available within the article and its supplementary material.