Chapter 2.7. Pre-edge structure, selection rules and quadrupole contributions

The spectrum obtained from an X-ray absorption near-edge structure (XANES) experiment is generally heavily influenced by the coordination environment (within several ångströms) of the target element. Characteristic peaks that appear near the energy regions just below an absorption edge, which are called pre-edge peaks (prepeaks), have received much attention because they provide highly sensitive information about the configuration of the atom, especially relating to the symmetry of a central atom (Fig. 1). XANES spectroscopy has been widely utilized for the analysis of various target elements in order to investigate electronic and geometrical structures and oxidation states, to identify chemical species and to perform quantitative analyses (Iwasawa, 1996; Cabaret et al., 2010; van Bokhoven & Lamberti, 2015), among other things. In this chapter, we describe the characteristics of pre-edge peaks with regard to their intensity, assignment and general tendencies on the basis of the coordination symmetry and oxidation number of the target atom.

Figure 1 XANES pre-edge peaks of different coordination environments and oxidation states.

Fig. 2 shows the K-edge XANES spectra of sixfold- and fourfold-coordinated 3d transition-metal compounds. As is well known, the pre-edge peak intensities of four-coordinated species are larger than those of six-coordinated species for all elements. The pre-edge peaks of the early 3d transition elements (vanadium, chromium and manganese) are much larger than those of the later elements (iron and copper). Pre-edge peaks have been confirmed, even for sixfold coordination, with maximum intensities as low as <0.1. The existence of multiple peaks has been clearly confirmed in the sixfold-coordination complexes of some elements, such as chromium. However, pre-edge peaks have not been confirmed in the Zn K-edge XANES spectra of any fourfold- or sixfold-coordinated compounds.

Figure 2 Pre-edge features in the K-edge XANES spectra of selected 3d transition-metal compounds with fourfold- or sixfold-coordinated metal–oxygen polyhedra. CuAl2O4 is 66.5% Td and 33.5% Oh.

Detailed systematic experimental and theoretical studies of pre-edge peak features and coordination number have been carried out for Ti⁴⁺ species using Ti K-edge XANES spectra (Farges et al., 1996). As shown in Fig. 3(a), the peak intensity decreases with coordination number (from six to four), with an accompanying shift to higher energy. Similar tendencies have been confirmed for vanadium (Yoshida et al., 1992), iron (Jackson et al., 2005) and nickel (Farges et al., 2001) K-edge XANES. These characteristic features, especially in the peak intensity, have been utilized as a powerful fingerprint for investigating the coordination number of a central atom in unknown samples. For example, the vanadium species in heterogeneous catalyst materials have been identified using XANES pre-edge features (Fig. 3b; Yoshida et al., 1992).

Figure 3 Dependence of pre-edge peak features on the coordination number. (a) Ti K-edge XANES for Ti4+ compounds. Reprinted from Farges et al. (1996) with permission from Elsevier. (b) Application of pre-edge intensity and position in V5+ reference samples to estimate the coordination number. Reprinted from Yoshida et al. (1992). Copyright 1992 Springer Nature.

As shown in Fig. 2, the pre-edge peak intensity of FePO₄ in a fourfold coordination environment is much smaller than those of early 3d transition-metal compounds. This intensity level (∼0.2, normalized to X-ray absorption) is valid because various kinds of tetrahedral Fe³⁺ compounds yield the same pre-edge peak intensities (Roe et al., 1984; Westre et al., 1997; Jackson et al., 2005). The weak pre-edge peak intensity has been reported to be related to the oxidation state of the element (Fig. 4; Yamamoto, 2008).

Figure 4 Effects of the oxidation number on the pre-edge intensity at the K edge (Yamamoto, 2008) for four-coordinated tetrahedral compounds: Fe2+ in Fe1.5Mg0.5Al9Si3.9Al0.1O22(OH)2 (Jackson et al., 2005), Fe3+ in γ-Fe:LiAlO2 (Jackson et al., 2005), Fe6+ in SrFeO4 (Tanaka, 1994), Mn2+ in MnFe2O4 (Farges, 2005) and Mn7+ in KMnO4.

As for the general rule regulating the pre-edge peak intensity in four-coordinated tetrahedral symmetry, the intensity has been proposed to be closely related to the number of 3d electrons and is almost independent of the element itself or the oxidation number (Fig. 5; Yamamoto, 2008). The d⁰ compounds give the highest intensity peaks and the intensity decreases as the number of 3d electrons increases, reaching an intensity of zero at d¹⁰.

Figure 5 Dependence of the pre-edge peak height of fourfold-coordinated metal–oxygen tetrahedron complexes on the number of 3d electrons in the K-edge XANES (Yamamoto, 2008). Copyright 2008 John Wiley & Sons, Ltd.

The bond-length dependencies for Mn and V K-edge XANES are shown in Fig. 6 (Yamamoto, 2018). Wong and coworkers have systematically investigated the pre-edge peak features of vanadium compounds with different oxidation numbers and symmetries around the central atom. They found clear empirical `molecular-cage' effects; that is, the shorter the averaged bond length, the higher the pre-edge peak intensity among compounds with the same symmetry (Wong et al., 1984). The interatomic distance of a four-coordinated tetrahedral unit is generally shorter than that of a six-coordinated octahedral unit with the same oxidation number, which is due to steric hindrance. It could also be regarded that four-coordinated species with shorter bond lengths would give rise to more intense pre-edge peaks than six-coordinated species with longer bond lengths.

Figure 6 Relation between the pre-edge peak intensity and the bond length of a metal–oxygen polyhedron (reproduced from Yamamoto, 2018).

The interatomic distance of polyhedral units would be shorter for compounds with a higher oxidation number because the ionic radii of atoms generally decrease with an increase in the oxidation number. Recent theoretical calculations for K-edge XANES spectra of MnO₄ units gave identical pre-edge peak intensity tendencies for inter­atomic distances in the range 2.01–1.63 Å (the average oxidation number increased from ∼3 to 7, in accordance with decreasing distance; Bunker, 2010). This shows that the dependence of pre-edge peak intensity on interatomic distance might also be considered to be intrinsically related to a function of the number of d electrons.

The transition moment related to X-ray absorption consists of an electric dipole, an electric quadrupole and magnetic dipole transitions (where the contribution of the magnetic dipole transition is negligible; Brouder, 1990). The electric dipole transition is the main component and its value is strictly regulated by electric dipole selection rules (Δj = 0, ±1, Δl = ±1). An intense pre-edge peak appears in cases where inversion symmetry is broken. The transition probability between s and d orbitals would be much lower than that of s–p transitions, for example by three orders of magnitude for 3d transition elements (Cotton, 1971; Bair & Goddard, 1980), as shown in Table 1 (Kawai, 2000). This tiny, but nonzero, probability is due to an electric quadrupole transition.

If a d orbital hybridizes with a p orbital, an electric dipole transition between the s orbital and the p component in a d–p orbital would be possible. The availability can be qualitatively evaluated using character tables from group theory. An irreducible representation of the character table (Cotton, 1971) for T_d, O_h, C_4v and D_4h point groups is shown in Table 2. For T_d, p_x, p_y, p_x and d_xy, d_xz, d_yz orbitals belong to T₂. It shows that p–d hybridization could occur among orbitals lying in a similar energy range (for example 3d–4p, 5d–6p). In contrast, there are no p and d orbitals with the same symmetry for O_h and D_4h, indicating that d–p hybridization would not occur. An intense pre-edge peak observed in the K-edge XANES spectra of 3d transition-metal compounds with four-coordinated tetrahedral units can then be assigned to the electric dipole transition of a 1s electron to the p component in the 3d–4p orbital. Only tiny peaks due to the quadrupole transition to a vacant 3d orbital would be visible for compounds with regular O_h and D_4h symmetry.

The five d orbitals are split into t_2g and e_g orbitals (3:2) in the presence of an octahedral crystal field. Fig. 7(a) shows the K-edge XANES spectra of rock-salt-type metal monoxides and an inverse spinel-type binary oxide, the central atom of which has O_h symmetry (Yoshida & Tanaka, 1996). As expected, weak peaks due to electric quadrupole transitions to vacant split 3d orbitals were confirmed. The peak near 1 eV is now believed to contain some electric dipole component (Modrow et al., 2003; Wu et al., 2004; de Groot et al., 2009) caused by the breakage of perfect inversion symmetry owing to neighbouring atoms, the effect of the core-hole potential and so forth. Recent progress in high energy resolution XANES spectroscopy has allowed the observation and characterization of precise pre-edge structures (van Bokhoven & Lamberti, 2015). For example, the quadrupole transitions in octahedrally coordinated 3d transition-metal compounds, such as CoO, were well reproduced by theoretical calculations with a charge-transfer multiplet program, as shown in Fig. 8(b) (de Groot et al., 2009).

Figure 7 Pre-edge features of 3d transition-metal oxides with six-coordinated octahedra. (a) Observation of crystal field splitting of 3d orbitals. Reprinted from Yoshida & Tanaka (1996). Copyright 1996 World Scientific. (b) Precise observation of the pre-edge structure using the high energy resolution fluorescence detection (HERFD) technique. Reprinted from de Groot et al. (2009) with permission from Elsevier.

Figure 8 Dependence of the pre-edge peak area on the total Fe 4p mixing into 3d molecular orbital iron complexes with different coordination numbers. Reprinted with permission from Roe et al. (1984). Copyright 1984 American Chemical Society.

There are few compounds with completely regular symmetry. Even if the symmetry is completely regular in the first coordination sphere of a target element, the unsymmetrical lattice vibration (vibronic coupling) and/or the outer coordination sphere would reduce the regularity (i.e. break the inversion centre). Thus, in most cases, some extent of d–p hybridization would be expected for every compound. The degree of d–p mixing would be influenced by the pre-edge peak intensity. This was first demonstrated by Roe and coworkers in Fe K-edge XANES spectra of iron complexes and theoretical calculations (Roe et al., 1984). Fig. 8 shows the pre-edge peak intensities of iron complexes plotted as a function of the estimated p fraction in the hybridized orbital. It is obvious that the inclusion of the Fe 4p character into Fe 3d orbitals enhanced the transition momentum for the electric dipole transition. This was supported by more detailed experimental and theoretical investigations (Westre et al., 1997).

The extent of 4p_z mixing into the Fe orbital with different central atom symmetries was also investigated theoretically for [FeCl₆]³⁻ octahedral species as a function of the axial distortion (Westre et al., 1997). As shown in Fig. 9, the 4p_z component was not included in cases where one of the six Fe–Cl bonds was set around 2.5 Å, but it was mixed at shorter or longer distances. This result clearly indicated that 3d–4p mixing occurred in C_4v symmetry, the extent of which increased with distortion from O_h symmetry (ionic radii: Fe³⁺, 0.645 Å; Cl⁻, 1.81 Å). The irreducible representation shown in Table 2 suggests that orbital mixing occurred readily because both the p and d orbitals belonged to the a₁ orbital oriented in the same direction. Theoretical calculations of a series of [FeCl₆]³⁻ units for molecular orbital analysis indicated that a 1s electron transits to an a₁ orbital via the electric dipole transition (solid arrow) and to the other d orbitals via the electric quadrupole process (dashed arrow), as shown in Fig. 9.

Figure 9 Theoretically evaluated pre-edge features of Fe K-edge XANES of a six-coordinated octahedral unit with different axial distortions. (a) Degree of mixed Fe 4pz character in the orbital for [FeCl6]3−. (b) Electric dipole and quadrupole contributions. Reprinted with permission from Westre et al. (1997). Copyright 1997 American Chemical Society.

The selection rule for the inner-shell transitions of a 2s electron is the same as that for a 1s electron. Thus, the pre-edge features for L₁-edge XANES are also expected to be equivalent to those for the K edge. In fact, the pre-edge peak intensity for W L₁-edge XANES was confirmed to be proportional to the mixing extent of W p orbitals into the 5d orbitals (Yamazoe et al., 2008), similar to the Fe K-edge XANES shown in Fig. 8. As shown in Fig. 1, the distortion of octahedral coordination for 4d⁰ (Mo⁶⁺) and 5d⁰ (W⁶⁺) compounds results in an enhancement of pre-edge peak intensity, whereas the distortions are smaller than those with T_d symmetry. Note that no pre-edge peaks were observed in the W K-edge (69.5 keV) XANES spectra of T_d compounds (Yamamoto, 2008), mainly owing to the intrinsically broad natural width (K, 39.9 eV; L₁, 5.61 eV; Krause & Oliver, 1979).

Polarized X-ray absorption experiments, including theoretical approaches, are also useful for the precise assignment of pre-edge peaks because the electron transition moment could be nonzero in cases where the electric field vector of an incident X-ray and the direction of the target orbital are the same (Brouder, 1990; Kosugi et al., 1984). For example, the polarized Cu K-edge XANES spectra of a square-planar [CuCl₄]²⁻ complex and the dependence of the pre-edge peak intensity on the rotation angle are shown in Fig. 10 (Hahn et al., 1982). The peak intensity varies with a 90° period against the xy plane. As discussed above, no d–p hybridization occurs in regular D_4h symmetry. This dependence on rotation angle, the shape of the d orbital, clearly shows that the component of the pre-edge peak was assigned as an electric quadrupole component to the orbital. Note that the extremely `large' pre-edge peak, as for electric quadrupole transition, was explained by the inclusion of electric dipole transitions due to vibronic coupling.

Figure 10 Rotation-angle dependence of the pre-edge peak intensity in the polarized Cu K-edge XANES spectra of a square-planar complex, [CuCl4]−. Reprinted from Hahn et al. (1982) with permission from Elsevier.

The assignment of the three tiny pre-edge peaks of TiO₂ (rutile) in the Ti K-edge XANES spectrum has been studied using both polarization experiments and theoretical calculations (Joly et al., 1999). As shown in Fig. 11, weak electric quadrupole contributions were confirmed at A₁ in both ɛ∥z and ɛ⊥z spectra and at A₂ peaks in ɛ∥z. The band calculation revealed that the Ti 2p_z orbital occupied the same energy as the A₂ and A₃ peaks but was absent in the A₁ peak. Thus, the A₁ and A₃ peaks were assigned to pure electric quadrupolar t_2g and electric dipole transitions, respectively. The A₂ peak consisted of a mixture of the quadrupolar e_g and dipole transitions.

Figure 11 Experimental and theoretical calculations of the polarized XANES spectra of a TiO2 (rutile) single crystal for pre-edge analysis. Reprinted with permission from Joly et al. (1999). Copyright 1999 by the Americal Physical Society. (ɛ, z) = (, [110])(ɛ⊥z), (ɛ, z) = ([001], [110])(ɛ∥z).

Polarization experiments could clarify the origin of pre-edge peaks associated with a specific bonding structure in some cases. Fig. 12(a) shows the pre-edge region of a polarized V K-edge XANES of a V₂O₅ single crystal and the components obtained by theoretical calculations (Šipr et al., 1999). An intense peak was experimentally observed along the z axis, the intensity of which was much larger than those for the other directions (by two orders of magnitude). A theoretical calculation indicated that weak electric quadrupole transitions to the d orbitals occurred in all directions of the V₂O₅ crystal structure, whereas an intense electric dipole transition appeared for the p_z orbital along the z direction only. Theoretical spectra for the model structure indicated that an intense peak could be reproduced only in the case where short V—O bonds existed along the z axis. In this model, the VO_x units (x = 1–5) utilized were obtained by removing some of the O atoms from a model VO₆ unit and/or were modified by changing the location of the selected atoms. The main component of the intense pre-edge peak for V₂O₅ was assigned to an electric dipole transition corresponding to one V=O vanadyl group. Appling this property to characterize the catalyst material [vanadium oxide supported on the surface of a ZrO₂(100) single crystal] using the polarized total-reflection fluorescence XAFS technique, the V=O bonds were suggested to be tilted by about 45° from the surface normal (Shirai et al., 1994).

Figure 12 Polarization-dependent pre-edge peaks due to V=O groups in the V K-edge XANES. (a) Pre-edge analysis for a V2O5 single crystal and estimated electric dipole and quadrupole contributions. Reprinted with permission from Šipr et al. (1999). Copyright 1999 by the Americal Physical Society. (b) Suggested orientation of vanadyl groups on the surface of a single crystal according to polarized total-reflection fluorescence XAFS. Reprinted with permission from Shirai et al. (1994). Copyright 1994 Springer Nature.