- Email: [email protected]
X-Ray Spectroscopy
9.16
X-Ray Spectroscopy
S DeBeer, Max Planck Institute for Chemical Energy Conversion, Mu¨lheim an der Ruhr, Germany; Cornell University, Ithaca, NY, USA F Neese, Max Planck Institute for Chemical Energy Conversion, Mu¨lheim an der Ruhr, Germany ã 2013 Elsevier Ltd. All rights reserved.
9.16.1 9.16.2 9.16.2.1 9.16.2.2 9.16.2.3 9.16.2.4 9.16.2.5 9.16.3 9.16.4 9.16.4.1 9.16.4.1.1 9.16.4.1.2 9.16.4.1.3 9.16.4.1.4 9.16.4.1.5 9.16.4.2 9.16.4.3 9.16.4.4 9.16.4.5 9.16.4.6 9.16.5 References
Introduction X-Ray Spectra and Data Interpretation Ligand K-Edge Spectra Metal K-Edge Spectra Metal L-Edge Spectra X-Ray Emission EXAFS Experimental Methods Theoretical Calculation of X-Ray Spectra General Considerations Transition energies Transition intensities Spin–orbit coupling Continuum states The rising edge Time-Dependent DFT Ligand K-Edge Spectra Metal K-Edge Spectra Metal L-Edge Spectra X-Ray Emission Conclusion
Abbreviations DFT ECP EXAFS FT SOC SOMO
9.16.1
Density functional theory Effective core potentials Extended X-ray absorption fine structure Fourier transform Spin–orbit coupling Singly occupied molecular orbital
Introduction
X-Ray core-level spectroscopy provides a fundamental experimental means to obtain insight into the geometric and electronic structure of inorganic systems. These methods can provide information on metal oxidation state, spin state, and covalency, as well as identify the nature and distance of coordinated ligands. In addition, X-ray spectroscopy is inherently element specific, providing far greater selectivity than many other spectroscopic methods. This chapter focuses primarily on X-ray absorption (XAS) and X-ray emission spectroscopies (XESs), with a focus on the fundamental physical phenomena, the experimental methods, and the more detailed information that can be obtained through the application of quantum chemical calculations.
Comprehensive Inorganic Chemistry II
427 428 428 428 429 429 430 431 432 432 433 433 433 433 433 434 434 435 435 437 439 439
TDA TD-DFT XAS XES ZORA
Tamm–Dancoff approximation Time-dependent density functional theory X-Ray absorption spectroscopy X-Ray emission spectroscopy Zeroth-order regular approximation
XAS involves the excitation of core electrons to bound states localized on the photoabsorber and the eventual excitation of the photoelectron to the continuum.1 The resulting spectra are typically divided into two regions – the edge region, which provides electronic structure information and the extended X-ray absorption fine structure (EXAFS) region , which provides information about the distance, number, and type of ligands. By selectively tuning the incident X-ray energy to the binding energy of an absorber, one can, in principle, obtain XAS data on any element of interest. Thus for transition-metal complexes, measurements can be made both from the perspective of the ligand and from the metal. Here we discuss the information that can be obtained utilizing ligand K- and/or metal K-/L-edge XAS. XES involves monitoring the processes which occur after ionization of a 1s core electron.2 The most likely event is the
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X-Ray Spectroscopy
observation of a Ka fluorescence line, which occurs when a 2p electron refills a 1s core hole. Kb (3p–1s) and valence-to-core transitions are also possible, with each region of the XES spectrum providing different but complementary information. For much of X-ray core spectroscopy, the analysis of experimental data has relied largely on empirical observations and semi-quantitative correlations. Herein, we briefly review the more qualitative developments. We then discuss the more recent developments in quantitative analysis. It is the close correlation of the experimental data to quantum chemistry calculations, which greatly enhances the information that can be obtained, providing a detailed picture of the electronic structure. The complementary information that can be obtained from the EXAFS region is also briefly discussed. Recent theoretical and experimental developments are highlighted.
presence of an unoccupied molecular orbital, which contains both Cl 3p and Cu 3d characters. The intensity of the pre-edge feature can be correlated to the oscillator strength, which can be factored into two components: (1) the covalency and (2) the radial transition moment dipole integral.4 If one assumes a constant transition moment dipole integral, then the intensity reflects the covalency of the metal–ligand bond. However, as the total charge on the ligand changes, the assumption of a constant transition moment dipole integral fails and a critical point in the quantitative analysis of pre-edge intensities as a reflection of covalency is how this factorization is handled. This has been shown to have a particularly pronounced effect in the analysis of S K-edge spectra of highly covalent transitionmetal dithiolene complexes.5,6 The details of this analysis are discussed further in this chapter.
9.16.2
9.16.2.2
An XAS edge results when a core electron absorbs energy equal to or greater than its binding energy. XAS edges are labeled according to the shell the electron originates from. A K-edge corresponds to a 1s core level; L-edges correspond to 2s and 2p levels; and M-edges to 3s, 3p, and 3d levels, as shown in Figure 1. For reference, the Fe K-edge occurs at 7.1 keV, the L-edges at 700 eV, and the M-edges between 50 and 100 eV. The large range of energies spanned by K-, L-, and M-edge core excitations require different experimental setups and also provide access to a much wider range of electronic structure information. Here, the content of ligand K-, metal K-, and metal L-edges are described, with a focus on the complementarity of the information that can be obtained.
9.16.2.1
Ligand K-Edge Spectra
Ligand K-edge XAS involves monitoring the processes occurring after the excitation of a ligand 1s electron. By monitoring the absorption coefficient, m, as a function of energy, one can obtain an insight into the effective charge or oxidation state on an absorbing atom. In addition, lower-energy pre-edge features provide information about the nature of bound open-shell metal ions. This was clearly illustrated by the early work of Hodgson, Hedman, and Solomon, which examined the Cl K-edge spectra of D2d-[ZnCl4]2 and [CuCl4]2, as shown in Figure 2.3 The primary edge jump at 2827 eV corresponds to a dipole-allowed Cl 1s to 4p transition. In the case of a d10 metal system as D2d-[ZnCl4]2, no lower-energy transition is observed. However, in the case of d9 [CuCl4]2– a pre-edge feature occurs at 2820 eV. This feature arises from the
Metal K-Edge Spectra
In the case of a metal K-edge for a first-row transition metal, the intense edge jump results from an electric-dipole-allowed 1s to 4p transition. Superimposed on the rising edge are weak preedge transitions, which correspond to 1s to 3d transitions. These transitions are formally electric-dipole forbidden, but quadrupole allowed and can gain intensity through 3d–4p mixing in suitable symmetry.7,8 Hence pre-edges of centrosymmetric complexes (e.g., Oh) tend to be weak, while the pre-edge of noncentrosymmetric complexes (e.g., Td) typically have significantly higher intensities (Figure 3). Metal K-edges are most frequently used to identify the oxidation state of an absorbing atom. In a simple picture, as a metal becomes more oxidized, it takes more energy to ionize a core electron and the edge shifts up in energy. Figure 4(a) compares the Fe K-edge of ferrous and ferric FeS4 model complexes. For a change of one unit of oxidation state, the pre-edge shifts by 1 eV.8,9 However, even when the oxidation state is constant, changes in the local geometry can induce a pronounced change
2
D2d[ZnCl4]2-
Normalized absorption
X-Ray Spectra and Data Interpretation
1.5
D2d[CuCl4]2-
1
0.5
L-edges 0 2815
K-edge
2820
2825
2830
Energy (eV) 1s
2s
2p
3s
3p 3d Continuum
Energy Figure 1 Atomic energy level diagram for an absorbing atom and the corresponding X-ray absorption edges.
Figure 2 Comparison of the Cl K-edge spectra for D2d-[ZnCl4]2 and [CuCl4]2. Reprinted with permission from Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 1990, 112, 1643–1645. Copyright 1990 American Chemical Society.
X-Ray Spectroscopy
in the edge. This is clearly illustrated when comparing the data for ferro- and ferricyanide (Figure 4(b)) to the data for ferrous and ferric FeS4 model complexes (Figure 4(a)). A detailed analysis of the intensity and energy distributions of 40 ferrous and ferric model complexes, by Westre et al., has shown that the Fe K-pre-edge features may be correlated directly with the coordination number.8 In general, centrosymmetric six-coordinate complexes will have much weaker pre-edge intensities than noncentrosymmetric five-coordinate complexes. These model studies have thus been extended to numerous studies of nonheme iron enzyme active sites to determine the coordination number of active sites at various stages of an enzymatic reaction. The shifts in XAS pre-edge energies have also provided essential information for determining the oxidation states of high-valent metal species. In particular, Fe(IV), Fe(V), and Fe (VI) species have been characterized using XAS.10–13 For a series of complexes with similar coordination environments, an increase in one unit of oxidation state results in an 1 eV increase in the pre-edge energy.8,14 However, such simple rules
1.2
Normalized absorption
1.0 0.8 0.6
Edge 1s to “4p”
0.4
Pre-edge 1s to 3d
0.2 0.0 7105
7110
7115
of thumb are not applicable when the coordination environment changes. For example, Fe(V) species have been identified in which the pre-edge energy falls in the same range as typical Fe(IV) species.11,12 Thus care must be taken when using such generalized fingerprints for electronic structure. These discrepancies have motivated our interest in more rigorous interpretation of the pre-edge region, in order to obtain more detailed insights, and also to enable these spectra to be used in a predictive fashion.15–17 Recent density functional theory (DFT)-based approaches to understanding the edge region are described below.
9.16.2.3
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Metal L-Edge Spectra
Metal L-edge XAS corresponds to monitoring dipole-allowed 2p–3d transitions of a transition-metal absorber. In contrast to the weak dipole-forbidden 1s–3d pre-edge transitions at a metal K-edge, the metal L-edge provides an intense dipoleallowed probe of the 3d-manifold. Due to spin–orbit coupling (SOC) of the 2p hole, 2p3/2 and 2p1/2 final states result, giving rise to the L3 and L2 edges.18 In a manner similar to the ligand K-edge, it has been shown that the intensity of these features should reflect the total metal 3d character in the unoccupied molecular orbitals. Therefore, the more covalent a transitionmetal complex is, the lower the L-edge intensity will be.19,20 This is, of course, a very qualitative picture, and again one must take care in accounting for the changes in the radial transitionmoment integral which will also contribute to the observed intensities. In the case of d1–d8 systems, additional splitting of the L3 and L2 edges due to the multiplet effect is observed.
9.16.2.4
7120
X-Ray Emission
XES involves following the emission of photons by the decay of electrons after ionization of the metal 1s electron. The most likely event is a Ka emission line, which corresponds to a dipole-allowed 2p–1s transition. Experimentally, the Ka line will be split into two features due to the SOC of the 2p hole, resulting in Ka1,2 and Ka2,3 lines. Approximately one order of
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Energy (eV) Figure 3 The edge and pre-edge region for Td (solid line) and Oh (dashed line) ferric complexes. 1.2
2.0
Normalized absorption
Normalized absorption
1.0
0.8
0.6
0.4
1.5
1.0
0.5
0.2
0.0 7105 (a)
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7115
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Energy (eV)
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0.0 7105
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Energy (eV)
Figure 4 Comparison of the normalized Fe K-edges for oxidized (solid line) and reduced (dashed line) FeS4 model complexes (a) and ferri- (solid line) and ferrocyanide (dashed line) (b).
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X-Ray Spectroscopy
Kb /Kb”
and
2,5
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kb
Fe 3d
Normalized intensity
1,3
k¼
Lp Ls
Fe 1s
´ 100 7040
7060
7080
7100
7120
7140
Energy (eV) Figure 5 Fe Kb XES spectrum for a high-spin ferric model complex. The box indicates the valence-to-core region of the spectrum, which has been expanded by 100-fold above. A simplified energy diagram showing the origin of the Kb1,3 and valence-to-core transitions is given on the right. Adapted from Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727.
magnitude lower in intensity than the Ka lines are the K b lines, corresponding to dipole-allowed 3p–1s transitions (Figure 5). The Kb1,3 emission line corresponds to an electric-dipoleallowed 3p–1s transition, which will have significant contributions from 3p–3d electron exchange interactions as well as 3p SOC contributions. To higher energy are transitions that correspond to valence electron transitions into the metal 1s core hole, the so-called Kb2,5/Kb" (or valence-to-core) region. These transitions have been previously assigned as ligand np to metal 1s (Kb2,5) and ligand ns to metal 1s (Kb’ or ‘satellite’) transitions.2 Recently, systematic studies on a series of low-molecularweight ferrous and ferric complexes were carried out.21 These results demonstrate that the Kb main-line spectra are dominated by spin-state contributions (Figures 6(a) and 6(c)), while the valence-to-core spectra (Figures 6(b) and 6(d)) show a greater sensitivity to the chemical environment. Significant efforts have been placed on understanding the valence-to-core region. The experimental data demonstrate that the energy and intensity distributions are affected by spin state, oxidation state, and coordination environment and thus provide a detailed ‘fingerprint’ of the local geometric and electronic structure.
9.16.2.5
EXAFS
The EXAFS region primarily provides metrical information for the number and type of ligands surrounding a photoabsorber, as also discussed in Chapter 9.05.1 When the photoelectron has been excited to the continuum, it can be modeled as a wave, which propagates out from the photoabsorber and is backscattered by the electron density surrounding the neighboring atoms. The EXAFS region is defined in terms of w(k), where1 wðkÞ ¼
mðkÞ m0 ðkÞ , m 0 ðk Þ
wðkÞ ¼
X Nf ðkÞ 2 2 e2k s sin ½2kR þ dðkÞ 2 kR s
[2]
In eqn [1], m(k) is the observed absorption coefficient and m0(k) is the atomic absorption (or the absorption coefficient that corresponds only to the absorbing atom, without any surrounding ligands). For convenience, EXAFSs are expressed in terms of photoelectron wave vector space (k) rather than energy space. The relationship between energy (in eV) and k (in A˚1) is given in eqn [2]. Here E0 defines the point at which the EXAFSs start (i.e., k ¼ 0) and represents the ionization threshold energy for the electron. In its simplest form, EXAFSs can be approximated as photoelectron scattering, where
Fe 3p
7020
2me ðE E0 Þ ħ2
[1]
wðkÞ ¼
i i2kR X ei2kR h idðkÞ e kf ð k Þe s kR kR
[3]
This equation can be expressed as a sine function, where wðkÞ ¼
X Nf ðkÞ sin ½2kR þ dðkÞ s kR2
[4]
Here N refers to the number of scatterers of a given type, f(k) is the amplitude function for the backscattering atom, d(k) is the phase shift for the absorber–backscatterer pair, and R is the absorber–scatterer (i.e., metal–ligand) distance. Averaging over all atoms and introducing the disorder parameter s2 then gives the EXAFS equation for a single absorber–scatterer pair, as wðkÞ ¼
X Nf ðkÞ 2 2 e2k s sin ½2kR þ dðkÞ s kR2
[5]
The contributions of each parameter to the EXAFS data are best illustrated through examples. Figure 7 shows the dependence of the EXAFS data on the metal–ligand distance (R) together with the Fourier transform (FT) of the data into R-space (i.e., A˚). The FT allows one to visualize the radial distribution of electron density with respect to the central absorbing atom. As the Fe–O bond is elongated from 2 to 3 to 4 A˚, the amplitude of the EXAFS decreases and the frequency increases (Figure 7, inset). This is more readily visualized by examining the FT of the data, which shows the intensity of a scattering contribution at a given distance from the photoabsorber. It should be noted that the FTs are not phase-shift corrected, and thus the distances cannot be read directly from the FT. In any case, this plot clearly indicates that the decrease in intensity as the Fe–O distance is increased. As illustrated by eqn [5], the data have a 1/R2 dependence demonstrating that this is a local probe of metrical structure. As also indicated by eqn [5], the EXAFS will have a linear dependence on the coordination number. As the coordination number (N) of a given backscatterer increases, the amplitude of the EXAFS signal and the intensity of the FT increase linearly. A similar effect is observed for a change in the disorder parameter (s2), where an increase in disorder decreases the amplitude of both the EXAFS and the FT, with an exponential dependence. The identity of the backscatterer will affect both the phase and amplitude of the EXAFS signal. Figure 8 shows the calculated FTs and EXAFS for Fe–O, Fe–S, and Fe–Fe vectors. For
X-Ray Spectroscopy
1.5
0.12
[Fe(lll)Cl4]
1-
Normalized intensity
[Fe(lll)(acac)3] 0.08
Normalized intensity (X 1000)
0.1
431
0 0
[Fe(lll)(tpfp)Cl]
0.06
0.04
1
0.5
0.02
0 7020
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Energy (eV)
(a)
0 7085
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7115
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7110
7115
7120
Energy (eV)
(b) 4
0.12
3.5
[Fe(lll)(CN)6]
3-
Normalized intensity
[Fe(lll)(tacn)2] 0.08
Normalized intensity (X 1000)
0.1
3+ 1+
[Fe(lll)(tpp)(ImH)2]
0.06
0.04
3
2.5
2
1.5
1 0.02 0.5
0 7020
(c)
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7040
7050
7060
7070
7080
0 7085
7090
Energy (eV)
(d)
7090
7095
7100
7105
Energy (eV)
Figure 6 Fe Kb main line XES spectrum for high spin ferric complexes (a). Valence-to-core region for the complexes shown in (a) (b). Fe Kb main line XES spectrum for low-spin ferric complexes (c). Valence-to-core region for the complexes shown in (c) (d). Adapted from Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727.
higher atomic number backscatters, the FT intensity increases and the amplitude envelope of the EXAFS peaks at a higher k-value. This allows EXAFS to be used to determine the identity of the atoms ligated to an absorber. However, it should be noted that similar backscatterers will have similar phase and amplitude profiles; thus while EXAFS can distinguish oxygen from sulfur, it cannot distinguish oxygen from nitrogen. In general for first-row atoms, atomic numbers of Z 1 cannot be distinguished. Moving down the periodic table, the required change in Z becomes larger, as the change in one electron becomes a smaller and smaller perturbation on the change in overall electron density surrounding the scatterer. In fitting EXAFS data, the goal is to determine the sum of all the absorber–scatterer interactions that make up the total EXAFS signal. Using the EXAFS equation, this may be described
as a sum of damped sine waves in k-space. The radial distribution of backscatterers with respect to the absorber is most readily visualized by examination of the FT.
9.16.3
Experimental Methods
A brief overview of the experimental setup and sample requirements for XAS and XES measurements is described. Typically both XAS and XES measurements utilize intense tunable X-ray sources provided by synchrotron facilities. The experimental setup varies depending on the energy region of interest, with <2 keV measurements requiring ultra-high vacuum condition, 2–5 keV measurements requiring a helium environment, and > 5 keV measurements being performed in air.
432
X-Ray Spectroscopy
0.4
2Å 3Å 4Å EXAFS * k3
FT magnitude
0.3
0.2
2
4
6
8
10
12
14
-1
k (Å )
0.1
0 0
1
2
3
4
5
6
R (Å) Figure 7 Calculated FTs and EXAFS (inset) for an Fe–O interaction as it is elongated from 2 to 3 to 4 A˚.
1.6
O S EXAFS * k3
FT Magnitude
Fe
1.2
higher resolution, it is worth commenting on the process of fluorescence-detected XAS in more detail. After the excitation of a core electron, the core hole will be refilled by an electron from a higher-energy shell and a fluorescent photon will be produced. This assumes that the number of emitted Ka-photons is proportional to the number of photons absorbed, which is a reasonable assumption in the dilute limit. As a standard solid-state detector will have an E/DE of 50, no fine structure is observed on the emission lines. XES measurements also use fluorescence detection, but rely on high-resolution Bragg optics spectrometers (E/DE of 5000 or greater) to spectrally analyze the fluorescence, which is then detected by a standard solid-state detector. Unlike XAS, XES typically utilizes a fixed-energy incident beam with a photon energy well above the absorption of the absorber. The spherical crystal spectrometer is then scanned in energy in a y–2y relationship defined by a Rowland circle. Using a Von Hamos geometry with a cylindrical analyzer together with a positionsensitive detector, dispersive XES measurements are possible. Variable energy XES which scans through the incident energy is also possible, and is more commonly referred to as resonant XES or resonant-inelastic x-ray scattering. Sample requirements for XAS and XES measurements in the hard X-ray range are similar. Samples can be run in almost any form, but are most typically run as dilute solids or solutions. Typically concentrations of 1 mM or greater in the absorber of interest are needed. Required volumes will depend on the individual beam line and the size of beam spot, though sample volumes of 50–150 microliters are common.
9.16.4
0.8 5
10 k (Å-1)
0.4
0 0
1
2
3
4
Theoretical Calculation of X-Ray Spectra
15
5
6
R (Å) Figure 8 Calculated FTs and EXAFS (inset) for Fe–O, Fe–S, and Fe–Fe interactions all at a distance of 2.5 A˚.
Currently, most XAS measurements are performed by scanning the energy of the incident X-ray beam from just below the ionization threshold of the absorbing atom to well above the edge. This is achieved using either a crystal monochromator or a grating (at lower energies) for energy selection through Bragg reflection. A standard XAS scan spans an energy range of 100 eV (to over 1000 eV for EXAFS). At high energies (> 2 keV), the incident-beam intensity is measured using a gas-filled ionization chamber, whereas at lower energies it is often more convenient to measure total electron yield from a standard (such as a metal grid). Absorption by the sample is then measured either directly by transmission or indirectly by fluorescence or electron yield. For fluorescence measurements, most often a solid-state Ge or Si detector is used. As XES measurements are enabled by the ability to detect fluorescence at
In recent years it has become increasingly commonplace to supplement XAS and XES measurements with quantum chemical calculations. The mostly used method is DFT.22 Hence the focus of the description will be on DFT-based methods. In this section, the scope and limitations of the available techniques are briefly described. Since the chapter cannot be comprehensive, we focus on molecular systems. Molecular techniques can also beneficially be used to study defects in solids. However, for calculations on extended solid-state systems and infinitelattice compounds, special techniques are in use but are outside the scope of this chapter.
9.16.4.1
General Considerations
X-Ray spectroscopic techniques involve the excitation of coreelectrons from deep-lying core-orbitals into semi-occupied or unoccupied valence orbitals. Both bound final states and continuum final states are accessed. In addition, strong core-level SOC effects can prominently influence the spectra shapes. The requirements on the theoretical methods that are able to deal with these challenges are very different. Hence, it is customary to divide the discussion into three separate areas dealing with the calculation of transition energies and intensities, SOC effects, and continuum states. We start by considering the excitation of a core s-electron into a semi-occupied or empty valence orbital. In this case, there are no important effects of SOC and the final state is
X-Ray Spectroscopy
bound. The overall goal of the theoretical calculations is the prediction of: (a) transition energies, (b) transition intensities, and (c) the correct number of spectral features. While (c) appears to be a trivial requirement, it is among the most difficult objectives to meet.
9.16.4.1.1 Transition energies In calculating transition energies, it is important to determine beforehand whether one is aiming at accurate, absolute, or relative transition energies. ‘Relative’ is, in this context, understood as the transitions occurring in one compound or analogous transitions occurring in a series of compounds. The accurate calculation of absolute transition energies, on the other hand, is a formidable challenge. To succeed in this, undertaking requires to accurately capture all differential energy contributions between the ground state and the core-hole excited state. Within DFT, this first of all involves the proper description of core orbitals in an all-electron treatment. Many contemporary calculations are based on effective core potentials (ECPs), which require additional approximations and assumptions in the treatment of core-level excitation events. Unfortunately, contemporary DFT potentials fall short of being accurate in the core region (as well as in the outer region of an atom or molecule) and hence core-orbital energies deviate from their exact Kohn–Sham values by dozens if not hundreds of eV. Secondly, the proper incorporation of kinematic (scalar) relativistic effects into the treatment is required. These spin-free relativistic effects are known to preferentially stabilize s- and p-orbitals and destabilize d- and f-electrons.23 The lower the orbital energies, the larger will be the associated effects. Hence, scalar relativistic effects will make important contributions to the energies of, say, s!d excitations. Unfortunately, contemporary methods to treat relativistic effects, such as the zerothorder regular approximation (ZORA)24 for relativistic effects, do not treat the core orbitals properly and overstabilize them.24 To some extent, the introduced errors tend to cancel with the inherent errors in the DFT potentials. However, such a cancelation is not systematic or reliable. Third, the treatment should properly capture the differential correlation energies of the states, which is not a simple task as the correlation energies of, say, (s2dn) configurations and (s1dnþ1) configurations in transition-metal complexes may be very different. Fourth, the relaxation of the electronic structure due to the creation of a core hole must be treated accurately and this is, again, anything but a simple undertaking. It is probably fair to state that the systematically accurate calculation of absolute transition energies is not feasible, except when using the most advanced theoretical methods (based on correlated wavefunction theory) on the smallest systems (e.g., atoms, ions, and very small molecules). It will be argued below that the calculation of transition energies that carry large, but highly systematic errors is much more feasible and leads to very useful predictions, provided that consistent computational protocols are employed.
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compared to molecular dimensions (a few Angstro¨ms). Thus, one has to recall that the interaction between light and matter is based on a multipole expansion that is usually terminated after the first term, which is represented by the electric-dipole operator. However, a precise treatment shows that the form of the electric-dipole operator that arises naturally from the treatment is the dipole-velocity form, while in practice the dipolelength operator is commonly employed.25 In the limit of exact molecular Eigen states, the two forms are equivalent, but in an approximate treatment they are not. Fortunately, this seems to be of limited relevance for the calculation of X-ray spectra because the dipole length and velocity forms yield almost indistinguishable results. The next terms in the multipole expansion are represented by the electric quadrupole and magnetic-dipole intensity mechanisms. While both are expected to be about four orders of magnitude smaller than the electric-dipole mechanism, it is important that they follow different selection rules. Thus, the requirement for an allowed electric-quadrupole transition is that the direct product of the initial and final states with the six components of the quadrupole operator (transforming as xx, yy, zz, xy, xz, or yz) contain the totally symmetric representation.26 Magnetic dipole transitions are allowed when the direct product with the molecular-rotation operators (Rx, Ry, and Rz) contain the totally symmetric representation. In practice, electricquadrupole transitions have clearly been observed in XAS spectra and can be experimentally distinguished from electricdipole transitions by the angular dependence of the transition moment in measurement on oriented crystals.27 Evidence for dominant magnetic-dipole mechanisms has not been obtained to the best of our knowledge.
9.16.4.1.3 Spin–orbit coupling Large SOC effects arise if the excitation does not arise from a core-level s-orbital but from a core-level p-orbital. The in-state SOC of the p5 configuration leads to very large splittings and a pronounced multiplet structure in the spectra. This is difficult to model with good accuracy theoretically. In particular, multiplet treatments in DFT are scarcely developed and the most common approach is based on heavily parameterized quasi-atomic multiplet calculations28,29 as will be further elaborated below.
9.16.4.1.4 Continuum states Excitations into the continuum of unbound states cannot be described with traditional techniques of quantum chemistry. Here the language of scattering theory must employed, that is largely a domain of physics. The necessary techniques are so different from the apparatus used to calculate the pre-edge region of XAS spectra that even their cursory description is outside of the range of this chapter. Most research groups involved in EXAFS employ the FEFF code.30 The underlying theory is described in the literature.30–35
9.16.4.1.5 The rising edge 9.16.4.1.2 Transition intensities The calculation of transition intensities in core-level spectra is somewhat more involved than for valence-to-valence transitions in molecular systems. The underlying reason is that the wavelength of the employed radiation is no longer large
The rising edge region of XAS spectra represents weakly-bound states close to the continuum. These states are difficult to calculate with either molecular, solid state, or scattering techniques. Since the focus of this chapter is X-ray spectroscopy of molecular systems, we focus below on the calculation of the
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X-Ray Spectroscopy
pre-edge region of XAS spectra and the valence-to-core region of XES spectra.
with ð IÞ
Akl ¼
9.16.4.2
1 5
Ck ¼ oI
Time-Dependent DFT
The protocol outlined above is applied below to ligand and metal K-edge spectra and is therefore described briefly. In DFT linear-response theory, the transition frequencies o are calculated from the nonstandard eigenvalue-type equation:22 A B X 1 0 X ¼o [6] B A Y 0 1 Y
[7]
f XC jb cHF jckb Bia, jb ¼ ia~
[8]
And the effective two-electron exchange-correlation operator 1 f~XC ðr1 ; r2 Þ ¼ r12 þ
d2 EXC drðr1 Þdrðr2 Þ
[9]
For some two-electron operator g(r1,r2), the integrals are defined as ðð pqgrs ¼ cp ðr1 Þcq ðr1 Þgðr1 ; r2 Þcr ðr2 Þcs ðr2 Þdr1 dr2 [10] Labels i,j refer to occupied spin–orbitals of the Kohn–Sham solution, a,b to unoccupied orbitals (p, q, r, s to general spin orbitals), ep are the orbital energies, and cHF is the fraction of the Hartree–Fock exchange in the density functional. EXC is the exchange correlation functional, r(r1) the ground-state electron density at position r1, and r12 is the interelectronic distance. The solution of eqn [6] yields the transition frequencies o and transition amplitudes Xai, Yai from which one can calculate dipole- and quadrupole-transition moments as X ^ a ðXia þ Yia Þ iD [11] D¼ ia
Q¼
X
^ a ðXia þ Yia Þ iQ
[12]
ia
^ k ¼ rk D ^ kl ¼ rk rl 1r 2 dkl Q 3
[13] [14]
where the sum r is the position operator of the electron with components (k ¼ x, y, z, r ¼ (rx2 þ ry2 þ rz2)1/2). The calculation of transition-dipole moments is straightforward, but the calculation of quadrupole-transition moments presents some difficulty, since the results depend on the choice of origin. Since the quadrupole transition-moment mechanism arises from a series expansion, it seems natural to choose the origin such that the expansion yields optimum convergence. As discussed in detail by DeBeer George et al.16, it is possible to readjust the origin for each transition (I) from the solution of a simple linear equation system as: AðIÞ R ¼ CðIÞ
[15]
X
ðIÞ
ðIÞ
Qkl Dl
l
2 3
4 ðIÞ ðIÞ o I Dk Dl 15
X
ðIÞ
eklm Ml DðmIÞ
[16] [17]
lm
Here eklm is the Levi–Civitta tensor and Ml is the magnetic transition-dipole moment. This procedure was followed throughout the present work. Neglecting the B-matrix amounts to the so-called Tamm–Dancoff (TDA) approximation. One then simply solves AX ¼ oX
With the supermatrices f XC jb cHF ijab Aia, jb ¼ ðea ei Þdai, jb þ ia~
ð IÞ 8 oI D 2 dkl 15
[18]
which resembles the configuration interaction with a single excitation approach. The key point in solving eqn [6] is to project the solution onto the manifold of core-excited single excitations. This obviously neglects the coupling of the coreexcited K-edge states to L- and M-edge, as well as valenceexcited states. The plausibility of this approximation has been discussed.17 It is most important to emphasize that due to the factors discussed above, the calculations do not yield accurate absolute transition energies. The errors are large but very systematic such that they can be compensated for by calibration. Such a calibration involves the comparison of theory and experiment for a set of ‘test molecules’. Obviously, the resulting calibration curve must be done for each element separately and depends on the basis set and the specific-density functional used in the calculation and on whether relativistic effects have been included in the treatment or not. It is therefore necessary to employ a consistent protocol. For example, a recent proposal involves the BP functional, the def2-TZVP basis set, and the ZORA treatment for relativistic effects.15
9.16.4.3
Ligand K-Edge Spectra
Ligand K-edge absorption spectra are obtained by exciting a ligand core-electron into semi-occupied or empty valence orbitals. In practice, this is best done at the chlorine, sulfur, and phosphorous edges. The most salient points are readily appreciated by taking a system with a single hole as an example.4 The starting point is the frozen orbital approximation to the initial and final states. The initial state is a single Slater determinant: jIi ¼ jc1 c1 . . . ck ck . . . cn cn cSOMO j
[19]
and the final state is the one where an electron is promoted from a ligand orbital with 1s character, ck, to the singly occupied molecular orbital (SOMO) jFk i ¼ jc1 c1 . . . ck cSOMO . . . cn cn cSOMO j
[20]
In the dipole-length approximation to the transition-dipole operator and using the Fermi golden rule, an electric-dipole transition has intensity proportional to: 2
! [21] Dk / cSOMO r ck
X-Ray Spectroscopy
where A sums over nuclei and i over electrons and ZA is the ! ! charge of atom A. R A is the position of the A0 th nucleus and r denotes an electron position. The ground-state singly occupied orbital, cSOMO, is written pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cSOMO ¼ 1 a2 jMi ajLi [22] where the overlap between the metal-centered part of the SOMO, |Mi, and the ligand-centered part, |Li, is neglected. Therefore 100 a2 is the percentage ligand character in the SOMO which is distributed over the individual ligands that contribute to the SOMO and 100 (1 a2) is the percentage metal character in the SOMO. The ligand part will in general be a linear combination of valence ns and np orbitals jLi ¼ kjnsi þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 jnpi
[23]
with hybridization coefficient k. The donor MO, ck, is assumed to be a pure ligand 1s orbital (see below). The square of the transition-dipole moment becomes D E2 1 2 ! 2 2 c [24] SOMO jr jck ¼ 3 a 1 k hr inp, 1s
435
instrumental for establishing whether the ligand behaves in a noninnocent way or not.5
9.16.4.4
Metal K-Edge Spectra
The main difference between metal- and ligand K-edge spectra in terms of the computational protocol is the intensity mechanism that is operative. At the metal K-edge, the dominating transitions in the pre-edge region are of metal-1s!metal-nd character. These transitions are electric quadrupole allowed and electric-dipole forbidden. Dipole character is induced in these transitions due to mixing of d- and p-orbitals at the metal, which, in turn, is induced by deviations from centrosymmetry. These are fairly subtle effects since the p-mixings are small and effectively compete with the quadrupole-intensity mechanism. These factors were all systematically addressed in a study dealing with iron K-edge spectra.17 It was found that the agreement between theory and experiment is also fairly good (Figure 10). Transition energies are typically computed to an accuracy of better than 0.5 eV and transition intensities correlate with experiment with a correlation coefficient R ¼ 0.98–0.99.17
with
hr inp, 1s ¼ Radnp jrjRad1s
[25]
where ‘Rad’ denotes the radial part of the orbitals. The expression has a straightforward interpretation – the pre-edge intensity increases as the ligand character the SOMO increases (a2), as the ligand in question makes a larger contribution to the total 2 ) and as the p-character ligand character in the SOMO (cSOMO 2 of this contribution increases (1 kSOMO ). In absolute terms, for a particular ligand atom, the factor 1 =3 hr i2np, 1s is the intrinsic intensity of an atomic 1s!npx,y,z transition. Its value does, however, not reflect a real atomic transition because the radial wavefunctions distort in the molecule according to the atoms’ environment. Since 1 =3 hr i2np, 1s depends on the molecular environment, Solomon and coworkers have found it necessary to determine its value for different types of bonding situations (e.g., for sulfur ligands: R–S, R–S–S–R, S2 etc.) separately before values of a2 can be determined from the areas under the experimental XAS absorption curves. However, neither 2 2 1 =3 hr inp, 1s itself nor the value of a is a physical observable. Only their product is directly related to the oscillator strength which is an observable. We have therefore chosen a different approach and directly calculated the physically observable quantity, for example, the oscillator strengths of the observed transitions using a simple protocol outlined above. The TD–DFT protocol outlined above was successfully applied to a substantial number of sulfur and chlorine containing transition-metal complexes and was found to be surprisingly successful. Transition energies are typically predicted within 0.2–0.3 eV and the correlation between calculated oscillator strength and properly normalized experimental intensities is on the order of 10%. An example for the comparison between theory and experiment is given in Figure 9. Following the detailed study of Ray et al.,5 DeBeer, Wieghardt, and coworkers have studied a number of transition-metal dithiolene complexes where the comparison between calculated and experimental sulfur K-edge spectra was
9.16.4.5
Metal L-Edge Spectra
The same TD–DFT protocol can obviously not be applied to the calculation of metal L-edge spectra as it (a) does not contain any SOC effects and (b) does not properly treat multiplet splittings. The first shortcoming could be fixed by resorting to twocomponent TD–DFT methods. However, this approach can still not be successful because the multiplet effects are not properly accounted for. The term ‘multiplet effects’ describe the fact that a number of final states of different energy arise from a given electronic configuration. If the number of unpaired electrons in a given electronic configuration exceeds the value of 2S, where S is the total spin of the target state, then several spin couplings arise, all of which lead to a given total spin S. In a similar way, the orbital angular momenta couple to different total values of L. In atoms and ions this behavior is well understood and summarized in the Russell–Saunders (LS) coupling scheme. The problem of transferring this scheme to molecules is that the different Russell– Saunders-like multiplet states are not single determinants, even if the ground state is well described by a single Slater determinant. The necessary determinants involved in properly building multiplet states may well be double- or triply excited relative to the electronic ground state. Hence, they are not accessible by TD–DFT that only incorporates single excitations. In fact, a DFT based theory that would properly include these effects has apparently not been developed. In principle, one can of course resort to wavefunction-based configuration interaction methods in which multiplet effects, spin–couplings, and higher excitations can be incorporated explicitly. These methods suffer from two defects: (a) the highcomputational cost of the calculations and (b) the poor quality of the Hartree–Fock–type reference wavefunction from which these calculations must be started. Thus, enormous levels of configuration interaction would be necessary to capture the physics of the problem correctly. These extensive calculations are not feasible and would also be of questionable quality because of the size-consistency problems that are inherent in configuration
436
X-Ray Spectroscopy
2.0
Normalized absorption
D2d-[CuCl4]2D4h-[CuCl4]21.5
[NiCl4]2[CoCl4]2-
1.0
[Fe(II)Cl4]2[Fe(III)Cl4]1[TiCl4]
0
0.5
0.0 2818
2819
2820
2821
2822
2823
2824
Energy (eV)
Calculated pre-edge intensity
6 5 4 3 2 1 0 2733
2734
2735
2736
2737
2738
2739
Energy (eV) Figure 9 Comparison of experimental (top) Cl K-pre-edge XAS data to the calculated spectra (bottom). Adapted from DeBeer George, S.; Petrenko, T.; Neese, F. Inorg. Chim. Acta 2008, 361, 965–972.
0.20
1+,0,1-,23-,4Calculated quadrupole + dipole contributions
Calculated pre-edge enery (eV)
7115
7114
7113
7112
7111
7110 7110
[FeCl4]1-
0.15
0.05
0.00 7111
7112
7113
7114
Experimental pre-edge energy (eV)
7115
[FeCl4]2-
0.10
[Fe(salen)Cl]
[Fe(acac)3] [FeCl6]3[Fe(CN)6]3[FeCl6]4[Fe(prpep)2]+ [Fe(CN)6]4- [Fe(prpep)2]
0
5
10
15
20
25
Experimental intensity
Figure 10 Correlation of experimental and theoretical transition energies (left) and intensities (right). Adapted from DeBeer George, S.; Petrenko, T.; Neese, F. J. Phys. Chem. A 2008, 112, 12936–12943.
X-Ray Spectroscopy
interaction treatments. While these problems are all severe, they may be surmountable in the foreseeable future. An alternative approach that has seen much use is the parametric multiplet approach as prominently exercised by Sawatzky, de Groot, and coworkers.28,29,36–39 In these approaches, one essentially performs an atomic configuration interaction calculation that leads to the correct LS multiplets but perturbs these multiplets with ligand-field splittings and charge-transfer interactions. The necessary parameters involve metal–ligand covalencies, which lead to anisotropy in the ligand field and the electron–electron interaction parameters. Together with the ligand-field and charge-transfer parameters that enter the model, there are a substantial number of empirical parameters that enter the calculations. These parameters are not straightforward to predict from DFT or ab initio calculations, which means that the main domain of application of the multiplet model is the fitting of known spectra. Good agreement with experiment can be achieved in this way, as shown in Figure11, which shows the calculated and experimental L-edge spectra of a low-spin Fe(III) model complex.40
9.16.4.6
X-Ray Emission
A physically rigorous modeling of the XES experiment would involve a calculation of the emission process following the core-level ionization. This is a fairly involved task as it will require the treatment of the electronic relaxation following the ionization event together with possible spin recouplings to give spin- and space-symmetry adapted final states. In practice, it has become evident that a much simpler approach is successful that is described below.21 The total number of photons emitted per unit time IIF that correspond to the transition of an initial core-ionized or coreexcited state |Ii to the final state |Fi in the limit of low concentration of the absorbing species and small path length of the incident beam is given by IIF ¼ WGI DN
AIF DI
2
720
725
The emission rate corresponding to the transition of state |Ii to state |Fi is given by the Einstein coefficient AIF which is proportional to the emission oscillator strength fIF AIF ¼ 3fIF gel
[28]
where gel is the classical radiative decay rate of the single-electron oscillator at frequency oIF which is equal to the transition frequency between the initial and final states (CGS units) gel ¼
2e2 o2IF 3me c3
[29]
Here, me is the mass of the electron, e is the electron charge, D is the speed of light, DI is the total decay rate of the coreexcited state |Ii which is the sum of the radiative (DIR) and nonradiative (DIN) decay rates R N DI ¼ D I þ DI X DRI ¼ AIF0
[30]
F0
According to eqns [26]–[30], the relative intensities of different transitions in the XES spectrum can be written as o2 fIF IRIF ¼ X IF 2 oIF fIF F
[31]
0
730
4
715
[27]
where L(ocm,oIF,G,s) is a lineshape function which, in general, depends on homogeneous (G) and inhomogeneous (s) broadening parameters. From eqn [7], it follows that Ðthe calculated XES spectrum has an integrated area of unity ( IR(oem) ¼ 1), and thus can be directly compared with the normalized experimental XES spectrum. The emission oscillator strength fIF is related to the absorption oscillator fFI strength as fIF ¼ fFI. The oscillator strength can be expanded as a power series in terms of the dimensionless fine-structure constant a (¼1/137.03599). The first term in this expansion represents the electric-dipole oscillator strength (fIFed). For orientationally averaged transition rates, the term linear in a is zero, and the next leading term a2 can be written as the sum of the electric quadrupole (fIFeq) and magnetic dipole (fIFmd) oscillator strengths:21,41
6
710
WGI ¼ sGI ðo0 ÞI0
[26]
8
705
where DN is the total number of absorbing species in the irradiated volume, and WGI is the transition rate between the ground state |Gi and state |Ii (capital letters are used for many electron states, lowercase letters for orbitals). WGI is related to the photoionization cross section sGI(o0) at the incident photon frequency o0 as
Then the normalized XES spectrum IIR(oem) corresponding to emission from state |Ii is X IRI ðoem Þ ¼ IRIF Lðocm ; oIF G; sÞ [32]
10
0 700
437
Energy (eV) Figure 11 Comparison of calculated (dashed line) and experimental (full line) L-edge X-ray absorption spectra of [Fe(tacn)2]3þ. Reprinted with permission from Wasinger, E. C.; de Groot, F. M. F.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 2003, 125, 12894–12906. Copyright 2003 American Chemical Society.
F0
eq
fIF ¼ fIFed þ fIFmd þ fIF
[33]
The mathematical form of the electric-dipole, magneticdipole, and electric-quadrupole contributions to this oscillator strength are the same as described above for the absorption process.
438
X-Ray Spectroscopy
The one-electron approximation consists of employing a set of frozen ground-state orbitals for the calculations. These are conveniently taken to be Kohn–Sham DFT orbitals. Associated with the Kohn–Sham procedure is the ground-state Kohn– Sham determinant jGi ¼ ji . . . j . . . nj
[34]
In the one-electron approach, the initial core-excited and final states are constructed as jIi ¼ ja . . . j . . . nj
[35]
jFi ¼ ja . . . i . . . nj
[36]
For the specific application at hand, orbital i is a coreorbital, for example, the metal 1s-derived orbital of a given absorber atom, a is an arbitrary unoccupied orbital (or a continuum unbound one-electron level), and j is another semicore or valence orbital of the same atom. Rather than trying to obtain accurate total energies for all of these states, the simple approximation is adopted for the emission energy
EsIF ¼ ħosIF ¼ esj esi
This is of course fairly simplistic. In terms of time-dependent DFT, it can be shown that the orbital energy difference is a welldefined approximation to the state energy difference if no Hartree–Fock exchange is present in the DFT potential.22 Calculations indicate that DFT in conjunction with the oneelectron approach does not provide an accurate modeling of the Kb main line. This is presumably due to: (a) a general underestimation of the core-level spin-polarization and (b) the lack of a proper treatment of the multiplet structure giving rise to the actual Kb main line peaks. On the other hand, the one-electron approach has been found to be very successful in the modeling of the Kb2,5 region, representing valence-to-core transitions. A comparison between calculated and measured spectra for a series of Fe(II) and Fe(III) complexes is shown in Figure 12 and demonstrates that the main features of the spectrum are all faithfully reproduced by the calculations. Thus it appears that this region mainly reflects the ground-state electronic structure. The same comments concerning the actual calibration procedure apply
80
1.5
[Fe(III)Cl4]
10
70
[Fe(III)(acac)3]
0
[Fe(III)(tpfp)Cl]
60
Calculated intensity
Normalized intensity (X 1000)
[37]
1
0.5
50 40 30 20 10
0 7085
7090
7095
7100
7105
7110
7115
Energy (eV)
(a)
0 7085
7120
7090
7095
7100
7105
7110
7115
7120
7110
7115
7120
Energy (eV)
(b) 150
4 3+
[Fe(III)(tacn)2]
3
3-
[Fe(III)(CN)6]
[Fe(III)(tpp)(ImH)2]
1+
Calculated intensity
Normalized intensity (X 1000)
3.5
2.5 2 1.5
100
50
1 0.5 0 7085
(c)
7090
7095
7100
7105
Energy (eV)
7110
7115
0 7085
7120
(d)
7090
7095
7100
7105
Energy (eV)
Figure 12 Experimental valence-to-core spectra for high-spin ferric complexes (a) and the corresponding calculated spectra (b). Experimental valence-to-core spectra for low-spin ferric complexes (c) and the corresponding calculated spectra (d). A 2.5 eV broadening and a constant shift of 182.5 eV have been to all calculated spectra. Reprinted with permission from Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727. Copyright 2010 American Chemical Society.
X-Ray Spectroscopy
as in the case of the TD–DFT-based XAS calculations. In the calibration study by Lee et al., 21 it was found, however, that the errors in the calculated transition energies are somewhat larger than in the case of XAS and amount to about 1 eV. The reasons for this increase in the error are not fully understood and may involve the limitations of the one-electron approach but also uncertainties in the experimental calibration procedure.
9.16.5
Conclusion
We hope that this chapter has demonstrated the complementarity of XAS, XES, and EXAFS. When the results of these techniques are viewed together, one can obtain a fairly detailed picture of the geometric and electronic structure of the investigated systems. Most importantly, the techniques are highly element specific and such highly selective measurements can even be performed on complex systems. In order to employ synchrotron-based techniques, it is not necessary to use isotopically enriched samples and it is also not necessary for the species to be paramagnetic. All of these advantages have added to the growing popularity of synchrotron-based X-ray spectroscopic techniques over the past decades in several research fields ranging from material sciences over biological chemistry to heterogeneous and homogenous catalyses. In this chapter, we have provided a cursory overview of the experimental and theoretical techniques that are presently in large-scale use. We hope that it became evident or at least plausible that one can take the analysis of the information content of the experimental data to the next higher level through the combination with quantum chemical calculations. These calculations can now efficiently be performed with user-friendly programs and are no longer confined to ‘expert territory.’ Together with the increasing availability of bright beam lines and ever improving X-ray detectors at the various synchrotrons worldwide, there appears to be every reason to be optimistic about the future of synchrotron-based spectroscopic techniques.
References 1. Koningsberger, D. C.; Prins, R. X-ray Absorption, Principles, Applications Techniques of EXAFS, SEXAFS, and XANES. Wiley: New York, 1988. 2. Glatzel, P.; Bergmann, U. Coord. Chem. Rev. 2005, 249, 65–95. 3. Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 1990, 112, 1643–1645. 4. Neese, F.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. Inorg. Chem. 1999, 38, 4854–4860. 5. Ray, K.; DeBeer George, S.; Solomon, E. I.; Wieghardt, K.; Neese, F. Chem. Eur. J. 2007, 13, 2783–2797. 6. Ray, K.; Petrenko, T.; Wieghardt, K.; Neese, F. Dalton Trans. 2007, 1552–1566. 7. Roe, A. L.; Schneider, D. J.; Mayer, R. J.; Pyrz, J. W.; Widom, J.; Que, L. J. Am. Chem. Soc. 1984, 106, 1676–1681. 8. Westre, T. E.; Kennepohl, P.; DeWitt, J. G.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 1997, 119, 6297–6314. 9. DuBois, J. L.; Mukherjee, P.; Collier, A. M.; Mayer, J. M.; Solomon, E. I.; Hedman, B.; Stack, T. D. P.; Hodgson, K. O. J. Am. Chem. Soc. 1997, 119, 8578–8579.
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10. Aliaga-Alcalde, N.; Mienert, B.; Bill, E.; Wieghardt, K.; George, S. D.; Neese, F. Angew. Chem. Int. Ed. 2005, 44, 2908–2912. 11. Berry, J. F.; Bill, E.; Bothe, E.; DeBeer George, S.; Mienert, B.; Neese, F.; Wieghardt, K. Science 2006, 312, 1937–1941. 12. de Oliveira, F. T.; Chanda, A.; Banerjee, D.; Shan, X.; Mondal, S.; Que, L., Jr.; Bominaar, E. L.; Mu¨nck, E.; Collins, T. J. Science 2007, 315, 835–838. 13. Rohde, J.-U.; In, J.-H.; Lim, M. H.; Brennessel, W. W.; Bukowski, M. R.; Stubna, A.; Mu¨nck, E.; Nam, W.; Que, L., Jr. Science 2003, 299, 1037–1039. 14. DuBois, J. L.; Mukherjee, P.; Stack, T. D. P.; Hedman, B.; Solomon, E. I.; Hodgson, K. O. J. Am. Chem. Soc. 2000, 122, 5775–5787. 15. DeBeer George, S.; Neese, F. Inorg. Chem. 2009, 49, 1849–1853. 16. DeBeer George, S.; Petrenko, T.; Neese, F. Inorg. Chim. Acta 2008, 361, 965–972. 17. DeBeer George, S.; Petrenko, T.; Neese, F. J. Phys. Chem. A 2008, 112, 12936–12943. 18. de Groot, F. M. F. Coord. Chem. Rev. 2005, 249, 31–63. 19. DeBeer George, S.; Metz, M.; Szilagyi, R. K.; Wang, H. X.; Cramer, S. P.; Lu, Y.; Tolman, W. B.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 2001, 123, 5757–5767. 20. George, S. J.; Lowery, M. D.; Solomon, E. I.; Cramer, S. P. J. Am. Chem. Soc. 1993, 115, 2968–2969. 21. Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727. 22. Neese, F. Coord. Chem. Rev. 2009, 253, 526–563. 23. Hess, B. A.; Marian, C. M. In Computational Molecular Spectroscopy; Jensen, P., Bunker, P. R., Eds.; Wiley: New York, 2000; p 169ff. 24. van Lenthe, E.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys. 1996, 105, 6505–6516. 25. Steinfeld, J. I. Molecules and Radiation. Harper and Row: New York, 1979. 26. Cotton, F. A. Chemical Applications of Group Theory. Wiley Interscience: New York, 1990. 27. Solomon, E. I. Comments Inorg. Chem. 1984, 3, 227. 28. Glatzel, P.; Bergmann, U.; de Groot, F.; Cramer, S. P. Abst. Pap. Am. Chem. Soc. 2002, 224, 025-INOR. 29. Degroot, F. M. F. J Electron Spectros Relat Phenom 1993, 62, 111–130. 30. Ankudinov, A. L.; Ravel, B.; Rehr, J. J.; Conradson, S. D. Phys. Rev. B 1998, 58, 7565–7576. 31. Ankudinov, A. L.; Conradson, S. D.; de Leon, J. M.; Rehr, J. J. Physical Review B 1998, 57, 7518–7525. 32. Rehr, J. J.; Ankudinov, A.; Zabinsky, S. I. Catal. Today 1998, 39, 263–269. 33. Ankudinov, A. L.; Rehr, J. J.; Low, J. J.; Bare, S. R. Top. Catal 2002, 18, 3–7. 34. Ankudinov, A. L.; Bouldin, C. E.; Rehr, J. J.; Sims, J.; Hung, H. Physical Review B 2002, 65, 104107. 35. Ankudinov, A. L.; Rehr, J. J.; Low, J. J.; Bare, S. R. J. Chem. Phys. 2002, 116, 1911–1919. 36. Esteva, J. M.; Karnatak, R. C.; Fuggle, J. C.; Sawatzky, G. A. Phys. Rev. Lett. 1983, 50, 910–913. 37. Kotani, A.; Ogasawara, H.; Okada, K.; Thole, B. T.; Sawatzky, G. A. Physical Review B 1989, 40, 65–73. 38. Vanderlaan, G.; Thole, B. T.; Sawatzky, G. A.; Verdaguer, M. Physical Review B 1988, 37, 6587–6589. 39. Hocking, R. K.; Wasinger, E. C.; Yan, Y. L.; de Groot, F. M. F.; Walker, F. A.; Hodgson, K. O.; Hedman, B.; Solomon, E. I. J. Am. Chem. Soc. 2007, 129, 113–125. 40. Wasinger, E. C.; de Groot, F. M. F.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 2003, 125, 12894–12906. 41. Berry, J. F.; DeBeer George, S.; Neese, F. Phys. Chem. Chem. Phys. 2008, 10, 4361–4374. http://www.cec.mpg.de – Max-Plank-Institut fu¨r Chemische Energiekonversion.
Relevant Website http://www.cec.mpg.de – Max-Plank-Institut fu¨r Chemische Energiekonversion.
X-Ray Spectroscopy
S DeBeer, Max Planck Institute for Chemical Energy Conversion, Mu¨lheim an der Ruhr, Germany; Cornell University, Ithaca, NY, USA F Neese, Max Planck Institute for Chemical Energy Conversion, Mu¨lheim an der Ruhr, Germany ã 2013 Elsevier Ltd. All rights reserved.
9.16.1 9.16.2 9.16.2.1 9.16.2.2 9.16.2.3 9.16.2.4 9.16.2.5 9.16.3 9.16.4 9.16.4.1 9.16.4.1.1 9.16.4.1.2 9.16.4.1.3 9.16.4.1.4 9.16.4.1.5 9.16.4.2 9.16.4.3 9.16.4.4 9.16.4.5 9.16.4.6 9.16.5 References
Introduction X-Ray Spectra and Data Interpretation Ligand K-Edge Spectra Metal K-Edge Spectra Metal L-Edge Spectra X-Ray Emission EXAFS Experimental Methods Theoretical Calculation of X-Ray Spectra General Considerations Transition energies Transition intensities Spin–orbit coupling Continuum states The rising edge Time-Dependent DFT Ligand K-Edge Spectra Metal K-Edge Spectra Metal L-Edge Spectra X-Ray Emission Conclusion
Abbreviations DFT ECP EXAFS FT SOC SOMO
9.16.1
Density functional theory Effective core potentials Extended X-ray absorption fine structure Fourier transform Spin–orbit coupling Singly occupied molecular orbital
Introduction
X-Ray core-level spectroscopy provides a fundamental experimental means to obtain insight into the geometric and electronic structure of inorganic systems. These methods can provide information on metal oxidation state, spin state, and covalency, as well as identify the nature and distance of coordinated ligands. In addition, X-ray spectroscopy is inherently element specific, providing far greater selectivity than many other spectroscopic methods. This chapter focuses primarily on X-ray absorption (XAS) and X-ray emission spectroscopies (XESs), with a focus on the fundamental physical phenomena, the experimental methods, and the more detailed information that can be obtained through the application of quantum chemical calculations.
Comprehensive Inorganic Chemistry II
427 428 428 428 429 429 430 431 432 432 433 433 433 433 433 434 434 435 435 437 439 439
TDA TD-DFT XAS XES ZORA
Tamm–Dancoff approximation Time-dependent density functional theory X-Ray absorption spectroscopy X-Ray emission spectroscopy Zeroth-order regular approximation
XAS involves the excitation of core electrons to bound states localized on the photoabsorber and the eventual excitation of the photoelectron to the continuum.1 The resulting spectra are typically divided into two regions – the edge region, which provides electronic structure information and the extended X-ray absorption fine structure (EXAFS) region , which provides information about the distance, number, and type of ligands. By selectively tuning the incident X-ray energy to the binding energy of an absorber, one can, in principle, obtain XAS data on any element of interest. Thus for transition-metal complexes, measurements can be made both from the perspective of the ligand and from the metal. Here we discuss the information that can be obtained utilizing ligand K- and/or metal K-/L-edge XAS. XES involves monitoring the processes which occur after ionization of a 1s core electron.2 The most likely event is the
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X-Ray Spectroscopy
observation of a Ka fluorescence line, which occurs when a 2p electron refills a 1s core hole. Kb (3p–1s) and valence-to-core transitions are also possible, with each region of the XES spectrum providing different but complementary information. For much of X-ray core spectroscopy, the analysis of experimental data has relied largely on empirical observations and semi-quantitative correlations. Herein, we briefly review the more qualitative developments. We then discuss the more recent developments in quantitative analysis. It is the close correlation of the experimental data to quantum chemistry calculations, which greatly enhances the information that can be obtained, providing a detailed picture of the electronic structure. The complementary information that can be obtained from the EXAFS region is also briefly discussed. Recent theoretical and experimental developments are highlighted.
presence of an unoccupied molecular orbital, which contains both Cl 3p and Cu 3d characters. The intensity of the pre-edge feature can be correlated to the oscillator strength, which can be factored into two components: (1) the covalency and (2) the radial transition moment dipole integral.4 If one assumes a constant transition moment dipole integral, then the intensity reflects the covalency of the metal–ligand bond. However, as the total charge on the ligand changes, the assumption of a constant transition moment dipole integral fails and a critical point in the quantitative analysis of pre-edge intensities as a reflection of covalency is how this factorization is handled. This has been shown to have a particularly pronounced effect in the analysis of S K-edge spectra of highly covalent transitionmetal dithiolene complexes.5,6 The details of this analysis are discussed further in this chapter.
9.16.2
9.16.2.2
An XAS edge results when a core electron absorbs energy equal to or greater than its binding energy. XAS edges are labeled according to the shell the electron originates from. A K-edge corresponds to a 1s core level; L-edges correspond to 2s and 2p levels; and M-edges to 3s, 3p, and 3d levels, as shown in Figure 1. For reference, the Fe K-edge occurs at 7.1 keV, the L-edges at 700 eV, and the M-edges between 50 and 100 eV. The large range of energies spanned by K-, L-, and M-edge core excitations require different experimental setups and also provide access to a much wider range of electronic structure information. Here, the content of ligand K-, metal K-, and metal L-edges are described, with a focus on the complementarity of the information that can be obtained.
9.16.2.1
Ligand K-Edge Spectra
Ligand K-edge XAS involves monitoring the processes occurring after the excitation of a ligand 1s electron. By monitoring the absorption coefficient, m, as a function of energy, one can obtain an insight into the effective charge or oxidation state on an absorbing atom. In addition, lower-energy pre-edge features provide information about the nature of bound open-shell metal ions. This was clearly illustrated by the early work of Hodgson, Hedman, and Solomon, which examined the Cl K-edge spectra of D2d-[ZnCl4]2 and [CuCl4]2, as shown in Figure 2.3 The primary edge jump at 2827 eV corresponds to a dipole-allowed Cl 1s to 4p transition. In the case of a d10 metal system as D2d-[ZnCl4]2, no lower-energy transition is observed. However, in the case of d9 [CuCl4]2– a pre-edge feature occurs at 2820 eV. This feature arises from the
Metal K-Edge Spectra
In the case of a metal K-edge for a first-row transition metal, the intense edge jump results from an electric-dipole-allowed 1s to 4p transition. Superimposed on the rising edge are weak preedge transitions, which correspond to 1s to 3d transitions. These transitions are formally electric-dipole forbidden, but quadrupole allowed and can gain intensity through 3d–4p mixing in suitable symmetry.7,8 Hence pre-edges of centrosymmetric complexes (e.g., Oh) tend to be weak, while the pre-edge of noncentrosymmetric complexes (e.g., Td) typically have significantly higher intensities (Figure 3). Metal K-edges are most frequently used to identify the oxidation state of an absorbing atom. In a simple picture, as a metal becomes more oxidized, it takes more energy to ionize a core electron and the edge shifts up in energy. Figure 4(a) compares the Fe K-edge of ferrous and ferric FeS4 model complexes. For a change of one unit of oxidation state, the pre-edge shifts by 1 eV.8,9 However, even when the oxidation state is constant, changes in the local geometry can induce a pronounced change
2
D2d[ZnCl4]2-
Normalized absorption
X-Ray Spectra and Data Interpretation
1.5
D2d[CuCl4]2-
1
0.5
L-edges 0 2815
K-edge
2820
2825
2830
Energy (eV) 1s
2s
2p
3s
3p 3d Continuum
Energy Figure 1 Atomic energy level diagram for an absorbing atom and the corresponding X-ray absorption edges.
Figure 2 Comparison of the Cl K-edge spectra for D2d-[ZnCl4]2 and [CuCl4]2. Reprinted with permission from Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 1990, 112, 1643–1645. Copyright 1990 American Chemical Society.
X-Ray Spectroscopy
in the edge. This is clearly illustrated when comparing the data for ferro- and ferricyanide (Figure 4(b)) to the data for ferrous and ferric FeS4 model complexes (Figure 4(a)). A detailed analysis of the intensity and energy distributions of 40 ferrous and ferric model complexes, by Westre et al., has shown that the Fe K-pre-edge features may be correlated directly with the coordination number.8 In general, centrosymmetric six-coordinate complexes will have much weaker pre-edge intensities than noncentrosymmetric five-coordinate complexes. These model studies have thus been extended to numerous studies of nonheme iron enzyme active sites to determine the coordination number of active sites at various stages of an enzymatic reaction. The shifts in XAS pre-edge energies have also provided essential information for determining the oxidation states of high-valent metal species. In particular, Fe(IV), Fe(V), and Fe (VI) species have been characterized using XAS.10–13 For a series of complexes with similar coordination environments, an increase in one unit of oxidation state results in an 1 eV increase in the pre-edge energy.8,14 However, such simple rules
1.2
Normalized absorption
1.0 0.8 0.6
Edge 1s to “4p”
0.4
Pre-edge 1s to 3d
0.2 0.0 7105
7110
7115
of thumb are not applicable when the coordination environment changes. For example, Fe(V) species have been identified in which the pre-edge energy falls in the same range as typical Fe(IV) species.11,12 Thus care must be taken when using such generalized fingerprints for electronic structure. These discrepancies have motivated our interest in more rigorous interpretation of the pre-edge region, in order to obtain more detailed insights, and also to enable these spectra to be used in a predictive fashion.15–17 Recent density functional theory (DFT)-based approaches to understanding the edge region are described below.
9.16.2.3
7125
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Metal L-Edge Spectra
Metal L-edge XAS corresponds to monitoring dipole-allowed 2p–3d transitions of a transition-metal absorber. In contrast to the weak dipole-forbidden 1s–3d pre-edge transitions at a metal K-edge, the metal L-edge provides an intense dipoleallowed probe of the 3d-manifold. Due to spin–orbit coupling (SOC) of the 2p hole, 2p3/2 and 2p1/2 final states result, giving rise to the L3 and L2 edges.18 In a manner similar to the ligand K-edge, it has been shown that the intensity of these features should reflect the total metal 3d character in the unoccupied molecular orbitals. Therefore, the more covalent a transitionmetal complex is, the lower the L-edge intensity will be.19,20 This is, of course, a very qualitative picture, and again one must take care in accounting for the changes in the radial transitionmoment integral which will also contribute to the observed intensities. In the case of d1–d8 systems, additional splitting of the L3 and L2 edges due to the multiplet effect is observed.
9.16.2.4
7120
X-Ray Emission
XES involves following the emission of photons by the decay of electrons after ionization of the metal 1s electron. The most likely event is a Ka emission line, which corresponds to a dipole-allowed 2p–1s transition. Experimentally, the Ka line will be split into two features due to the SOC of the 2p hole, resulting in Ka1,2 and Ka2,3 lines. Approximately one order of
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Energy (eV) Figure 3 The edge and pre-edge region for Td (solid line) and Oh (dashed line) ferric complexes. 1.2
2.0
Normalized absorption
Normalized absorption
1.0
0.8
0.6
0.4
1.5
1.0
0.5
0.2
0.0 7105 (a)
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7115
7120
Energy (eV)
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429
0.0 7105
7135 (b)
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7115
7120
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Energy (eV)
Figure 4 Comparison of the normalized Fe K-edges for oxidized (solid line) and reduced (dashed line) FeS4 model complexes (a) and ferri- (solid line) and ferrocyanide (dashed line) (b).
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X-Ray Spectroscopy
Kb /Kb”
and
2,5
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Kb
Fe 3d
Normalized intensity
1,3
k¼
Lp Ls
Fe 1s
´ 100 7040
7060
7080
7100
7120
7140
Energy (eV) Figure 5 Fe Kb XES spectrum for a high-spin ferric model complex. The box indicates the valence-to-core region of the spectrum, which has been expanded by 100-fold above. A simplified energy diagram showing the origin of the Kb1,3 and valence-to-core transitions is given on the right. Adapted from Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727.
magnitude lower in intensity than the Ka lines are the K b lines, corresponding to dipole-allowed 3p–1s transitions (Figure 5). The Kb1,3 emission line corresponds to an electric-dipoleallowed 3p–1s transition, which will have significant contributions from 3p–3d electron exchange interactions as well as 3p SOC contributions. To higher energy are transitions that correspond to valence electron transitions into the metal 1s core hole, the so-called Kb2,5/Kb" (or valence-to-core) region. These transitions have been previously assigned as ligand np to metal 1s (Kb2,5) and ligand ns to metal 1s (Kb’ or ‘satellite’) transitions.2 Recently, systematic studies on a series of low-molecularweight ferrous and ferric complexes were carried out.21 These results demonstrate that the Kb main-line spectra are dominated by spin-state contributions (Figures 6(a) and 6(c)), while the valence-to-core spectra (Figures 6(b) and 6(d)) show a greater sensitivity to the chemical environment. Significant efforts have been placed on understanding the valence-to-core region. The experimental data demonstrate that the energy and intensity distributions are affected by spin state, oxidation state, and coordination environment and thus provide a detailed ‘fingerprint’ of the local geometric and electronic structure.
9.16.2.5
EXAFS
The EXAFS region primarily provides metrical information for the number and type of ligands surrounding a photoabsorber, as also discussed in Chapter 9.05.1 When the photoelectron has been excited to the continuum, it can be modeled as a wave, which propagates out from the photoabsorber and is backscattered by the electron density surrounding the neighboring atoms. The EXAFS region is defined in terms of w(k), where1 wðkÞ ¼
mðkÞ m0 ðkÞ , m 0 ðk Þ
wðkÞ ¼
X Nf ðkÞ 2 2 e2k s sin ½2kR þ dðkÞ 2 kR s
[2]
In eqn [1], m(k) is the observed absorption coefficient and m0(k) is the atomic absorption (or the absorption coefficient that corresponds only to the absorbing atom, without any surrounding ligands). For convenience, EXAFSs are expressed in terms of photoelectron wave vector space (k) rather than energy space. The relationship between energy (in eV) and k (in A˚1) is given in eqn [2]. Here E0 defines the point at which the EXAFSs start (i.e., k ¼ 0) and represents the ionization threshold energy for the electron. In its simplest form, EXAFSs can be approximated as photoelectron scattering, where
Fe 3p
7020
2me ðE E0 Þ ħ2
[1]
wðkÞ ¼
i i2kR X ei2kR h idðkÞ e kf ð k Þe s kR kR
[3]
This equation can be expressed as a sine function, where wðkÞ ¼
X Nf ðkÞ sin ½2kR þ dðkÞ s kR2
[4]
Here N refers to the number of scatterers of a given type, f(k) is the amplitude function for the backscattering atom, d(k) is the phase shift for the absorber–backscatterer pair, and R is the absorber–scatterer (i.e., metal–ligand) distance. Averaging over all atoms and introducing the disorder parameter s2 then gives the EXAFS equation for a single absorber–scatterer pair, as wðkÞ ¼
X Nf ðkÞ 2 2 e2k s sin ½2kR þ dðkÞ s kR2
[5]
The contributions of each parameter to the EXAFS data are best illustrated through examples. Figure 7 shows the dependence of the EXAFS data on the metal–ligand distance (R) together with the Fourier transform (FT) of the data into R-space (i.e., A˚). The FT allows one to visualize the radial distribution of electron density with respect to the central absorbing atom. As the Fe–O bond is elongated from 2 to 3 to 4 A˚, the amplitude of the EXAFS decreases and the frequency increases (Figure 7, inset). This is more readily visualized by examining the FT of the data, which shows the intensity of a scattering contribution at a given distance from the photoabsorber. It should be noted that the FTs are not phase-shift corrected, and thus the distances cannot be read directly from the FT. In any case, this plot clearly indicates that the decrease in intensity as the Fe–O distance is increased. As illustrated by eqn [5], the data have a 1/R2 dependence demonstrating that this is a local probe of metrical structure. As also indicated by eqn [5], the EXAFS will have a linear dependence on the coordination number. As the coordination number (N) of a given backscatterer increases, the amplitude of the EXAFS signal and the intensity of the FT increase linearly. A similar effect is observed for a change in the disorder parameter (s2), where an increase in disorder decreases the amplitude of both the EXAFS and the FT, with an exponential dependence. The identity of the backscatterer will affect both the phase and amplitude of the EXAFS signal. Figure 8 shows the calculated FTs and EXAFS for Fe–O, Fe–S, and Fe–Fe vectors. For
X-Ray Spectroscopy
1.5
0.12
[Fe(lll)Cl4]
1-
Normalized intensity
[Fe(lll)(acac)3] 0.08
Normalized intensity (X 1000)
0.1
431
0 0
[Fe(lll)(tpfp)Cl]
0.06
0.04
1
0.5
0.02
0 7020
7030
7040
7050
7060
7070
7080
Energy (eV)
(a)
0 7085
7090
7090
7095
7100
7105
7110
7115
7120
7110
7115
7120
Energy (eV)
(b) 4
0.12
3.5
[Fe(lll)(CN)6]
3-
Normalized intensity
[Fe(lll)(tacn)2] 0.08
Normalized intensity (X 1000)
0.1
3+ 1+
[Fe(lll)(tpp)(ImH)2]
0.06
0.04
3
2.5
2
1.5
1 0.02 0.5
0 7020
(c)
7030
7040
7050
7060
7070
7080
0 7085
7090
Energy (eV)
(d)
7090
7095
7100
7105
Energy (eV)
Figure 6 Fe Kb main line XES spectrum for high spin ferric complexes (a). Valence-to-core region for the complexes shown in (a) (b). Fe Kb main line XES spectrum for low-spin ferric complexes (c). Valence-to-core region for the complexes shown in (c) (d). Adapted from Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727.
higher atomic number backscatters, the FT intensity increases and the amplitude envelope of the EXAFS peaks at a higher k-value. This allows EXAFS to be used to determine the identity of the atoms ligated to an absorber. However, it should be noted that similar backscatterers will have similar phase and amplitude profiles; thus while EXAFS can distinguish oxygen from sulfur, it cannot distinguish oxygen from nitrogen. In general for first-row atoms, atomic numbers of Z 1 cannot be distinguished. Moving down the periodic table, the required change in Z becomes larger, as the change in one electron becomes a smaller and smaller perturbation on the change in overall electron density surrounding the scatterer. In fitting EXAFS data, the goal is to determine the sum of all the absorber–scatterer interactions that make up the total EXAFS signal. Using the EXAFS equation, this may be described
as a sum of damped sine waves in k-space. The radial distribution of backscatterers with respect to the absorber is most readily visualized by examination of the FT.
9.16.3
Experimental Methods
A brief overview of the experimental setup and sample requirements for XAS and XES measurements is described. Typically both XAS and XES measurements utilize intense tunable X-ray sources provided by synchrotron facilities. The experimental setup varies depending on the energy region of interest, with <2 keV measurements requiring ultra-high vacuum condition, 2–5 keV measurements requiring a helium environment, and > 5 keV measurements being performed in air.
432
X-Ray Spectroscopy
0.4
2Å 3Å 4Å EXAFS * k3
FT magnitude
0.3
0.2
2
4
6
8
10
12
14
-1
k (Å )
0.1
0 0
1
2
3
4
5
6
R (Å) Figure 7 Calculated FTs and EXAFS (inset) for an Fe–O interaction as it is elongated from 2 to 3 to 4 A˚.
1.6
O S EXAFS * k3
FT Magnitude
Fe
1.2
higher resolution, it is worth commenting on the process of fluorescence-detected XAS in more detail. After the excitation of a core electron, the core hole will be refilled by an electron from a higher-energy shell and a fluorescent photon will be produced. This assumes that the number of emitted Ka-photons is proportional to the number of photons absorbed, which is a reasonable assumption in the dilute limit. As a standard solid-state detector will have an E/DE of 50, no fine structure is observed on the emission lines. XES measurements also use fluorescence detection, but rely on high-resolution Bragg optics spectrometers (E/DE of 5000 or greater) to spectrally analyze the fluorescence, which is then detected by a standard solid-state detector. Unlike XAS, XES typically utilizes a fixed-energy incident beam with a photon energy well above the absorption of the absorber. The spherical crystal spectrometer is then scanned in energy in a y–2y relationship defined by a Rowland circle. Using a Von Hamos geometry with a cylindrical analyzer together with a positionsensitive detector, dispersive XES measurements are possible. Variable energy XES which scans through the incident energy is also possible, and is more commonly referred to as resonant XES or resonant-inelastic x-ray scattering. Sample requirements for XAS and XES measurements in the hard X-ray range are similar. Samples can be run in almost any form, but are most typically run as dilute solids or solutions. Typically concentrations of 1 mM or greater in the absorber of interest are needed. Required volumes will depend on the individual beam line and the size of beam spot, though sample volumes of 50–150 microliters are common.
9.16.4
0.8 5
10 k (Å-1)
0.4
0 0
1
2
3
4
Theoretical Calculation of X-Ray Spectra
15
5
6
R (Å) Figure 8 Calculated FTs and EXAFS (inset) for Fe–O, Fe–S, and Fe–Fe interactions all at a distance of 2.5 A˚.
Currently, most XAS measurements are performed by scanning the energy of the incident X-ray beam from just below the ionization threshold of the absorbing atom to well above the edge. This is achieved using either a crystal monochromator or a grating (at lower energies) for energy selection through Bragg reflection. A standard XAS scan spans an energy range of 100 eV (to over 1000 eV for EXAFS). At high energies (> 2 keV), the incident-beam intensity is measured using a gas-filled ionization chamber, whereas at lower energies it is often more convenient to measure total electron yield from a standard (such as a metal grid). Absorption by the sample is then measured either directly by transmission or indirectly by fluorescence or electron yield. For fluorescence measurements, most often a solid-state Ge or Si detector is used. As XES measurements are enabled by the ability to detect fluorescence at
In recent years it has become increasingly commonplace to supplement XAS and XES measurements with quantum chemical calculations. The mostly used method is DFT.22 Hence the focus of the description will be on DFT-based methods. In this section, the scope and limitations of the available techniques are briefly described. Since the chapter cannot be comprehensive, we focus on molecular systems. Molecular techniques can also beneficially be used to study defects in solids. However, for calculations on extended solid-state systems and infinitelattice compounds, special techniques are in use but are outside the scope of this chapter.
9.16.4.1
General Considerations
X-Ray spectroscopic techniques involve the excitation of coreelectrons from deep-lying core-orbitals into semi-occupied or unoccupied valence orbitals. Both bound final states and continuum final states are accessed. In addition, strong core-level SOC effects can prominently influence the spectra shapes. The requirements on the theoretical methods that are able to deal with these challenges are very different. Hence, it is customary to divide the discussion into three separate areas dealing with the calculation of transition energies and intensities, SOC effects, and continuum states. We start by considering the excitation of a core s-electron into a semi-occupied or empty valence orbital. In this case, there are no important effects of SOC and the final state is
X-Ray Spectroscopy
bound. The overall goal of the theoretical calculations is the prediction of: (a) transition energies, (b) transition intensities, and (c) the correct number of spectral features. While (c) appears to be a trivial requirement, it is among the most difficult objectives to meet.
9.16.4.1.1 Transition energies In calculating transition energies, it is important to determine beforehand whether one is aiming at accurate, absolute, or relative transition energies. ‘Relative’ is, in this context, understood as the transitions occurring in one compound or analogous transitions occurring in a series of compounds. The accurate calculation of absolute transition energies, on the other hand, is a formidable challenge. To succeed in this, undertaking requires to accurately capture all differential energy contributions between the ground state and the core-hole excited state. Within DFT, this first of all involves the proper description of core orbitals in an all-electron treatment. Many contemporary calculations are based on effective core potentials (ECPs), which require additional approximations and assumptions in the treatment of core-level excitation events. Unfortunately, contemporary DFT potentials fall short of being accurate in the core region (as well as in the outer region of an atom or molecule) and hence core-orbital energies deviate from their exact Kohn–Sham values by dozens if not hundreds of eV. Secondly, the proper incorporation of kinematic (scalar) relativistic effects into the treatment is required. These spin-free relativistic effects are known to preferentially stabilize s- and p-orbitals and destabilize d- and f-electrons.23 The lower the orbital energies, the larger will be the associated effects. Hence, scalar relativistic effects will make important contributions to the energies of, say, s!d excitations. Unfortunately, contemporary methods to treat relativistic effects, such as the zerothorder regular approximation (ZORA)24 for relativistic effects, do not treat the core orbitals properly and overstabilize them.24 To some extent, the introduced errors tend to cancel with the inherent errors in the DFT potentials. However, such a cancelation is not systematic or reliable. Third, the treatment should properly capture the differential correlation energies of the states, which is not a simple task as the correlation energies of, say, (s2dn) configurations and (s1dnþ1) configurations in transition-metal complexes may be very different. Fourth, the relaxation of the electronic structure due to the creation of a core hole must be treated accurately and this is, again, anything but a simple undertaking. It is probably fair to state that the systematically accurate calculation of absolute transition energies is not feasible, except when using the most advanced theoretical methods (based on correlated wavefunction theory) on the smallest systems (e.g., atoms, ions, and very small molecules). It will be argued below that the calculation of transition energies that carry large, but highly systematic errors is much more feasible and leads to very useful predictions, provided that consistent computational protocols are employed.
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compared to molecular dimensions (a few Angstro¨ms). Thus, one has to recall that the interaction between light and matter is based on a multipole expansion that is usually terminated after the first term, which is represented by the electric-dipole operator. However, a precise treatment shows that the form of the electric-dipole operator that arises naturally from the treatment is the dipole-velocity form, while in practice the dipolelength operator is commonly employed.25 In the limit of exact molecular Eigen states, the two forms are equivalent, but in an approximate treatment they are not. Fortunately, this seems to be of limited relevance for the calculation of X-ray spectra because the dipole length and velocity forms yield almost indistinguishable results. The next terms in the multipole expansion are represented by the electric quadrupole and magnetic-dipole intensity mechanisms. While both are expected to be about four orders of magnitude smaller than the electric-dipole mechanism, it is important that they follow different selection rules. Thus, the requirement for an allowed electric-quadrupole transition is that the direct product of the initial and final states with the six components of the quadrupole operator (transforming as xx, yy, zz, xy, xz, or yz) contain the totally symmetric representation.26 Magnetic dipole transitions are allowed when the direct product with the molecular-rotation operators (Rx, Ry, and Rz) contain the totally symmetric representation. In practice, electricquadrupole transitions have clearly been observed in XAS spectra and can be experimentally distinguished from electricdipole transitions by the angular dependence of the transition moment in measurement on oriented crystals.27 Evidence for dominant magnetic-dipole mechanisms has not been obtained to the best of our knowledge.
9.16.4.1.3 Spin–orbit coupling Large SOC effects arise if the excitation does not arise from a core-level s-orbital but from a core-level p-orbital. The in-state SOC of the p5 configuration leads to very large splittings and a pronounced multiplet structure in the spectra. This is difficult to model with good accuracy theoretically. In particular, multiplet treatments in DFT are scarcely developed and the most common approach is based on heavily parameterized quasi-atomic multiplet calculations28,29 as will be further elaborated below.
9.16.4.1.4 Continuum states Excitations into the continuum of unbound states cannot be described with traditional techniques of quantum chemistry. Here the language of scattering theory must employed, that is largely a domain of physics. The necessary techniques are so different from the apparatus used to calculate the pre-edge region of XAS spectra that even their cursory description is outside of the range of this chapter. Most research groups involved in EXAFS employ the FEFF code.30 The underlying theory is described in the literature.30–35
9.16.4.1.5 The rising edge 9.16.4.1.2 Transition intensities The calculation of transition intensities in core-level spectra is somewhat more involved than for valence-to-valence transitions in molecular systems. The underlying reason is that the wavelength of the employed radiation is no longer large
The rising edge region of XAS spectra represents weakly-bound states close to the continuum. These states are difficult to calculate with either molecular, solid state, or scattering techniques. Since the focus of this chapter is X-ray spectroscopy of molecular systems, we focus below on the calculation of the
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X-Ray Spectroscopy
pre-edge region of XAS spectra and the valence-to-core region of XES spectra.
with ð IÞ
Akl ¼
9.16.4.2
1 5
Ck ¼ oI
Time-Dependent DFT
The protocol outlined above is applied below to ligand and metal K-edge spectra and is therefore described briefly. In DFT linear-response theory, the transition frequencies o are calculated from the nonstandard eigenvalue-type equation:22 A B X 1 0 X ¼o [6] B A Y 0 1 Y
[7]
f XC jb cHF jckb Bia, jb ¼ ia~
[8]
And the effective two-electron exchange-correlation operator 1 f~XC ðr1 ; r2 Þ ¼ r12 þ
d2 EXC drðr1 Þdrðr2 Þ
[9]
For some two-electron operator g(r1,r2), the integrals are defined as ðð pqgrs ¼ cp ðr1 Þcq ðr1 Þgðr1 ; r2 Þcr ðr2 Þcs ðr2 Þdr1 dr2 [10] Labels i,j refer to occupied spin–orbitals of the Kohn–Sham solution, a,b to unoccupied orbitals (p, q, r, s to general spin orbitals), ep are the orbital energies, and cHF is the fraction of the Hartree–Fock exchange in the density functional. EXC is the exchange correlation functional, r(r1) the ground-state electron density at position r1, and r12 is the interelectronic distance. The solution of eqn [6] yields the transition frequencies o and transition amplitudes Xai, Yai from which one can calculate dipole- and quadrupole-transition moments as X ^ a ðXia þ Yia Þ iD [11] D¼ ia
Q¼
X
^ a ðXia þ Yia Þ iQ
[12]
ia
^ k ¼ rk D ^ kl ¼ rk rl 1r 2 dkl Q 3
[13] [14]
where the sum r is the position operator of the electron with components (k ¼ x, y, z, r ¼ (rx2 þ ry2 þ rz2)1/2). The calculation of transition-dipole moments is straightforward, but the calculation of quadrupole-transition moments presents some difficulty, since the results depend on the choice of origin. Since the quadrupole transition-moment mechanism arises from a series expansion, it seems natural to choose the origin such that the expansion yields optimum convergence. As discussed in detail by DeBeer George et al.16, it is possible to readjust the origin for each transition (I) from the solution of a simple linear equation system as: AðIÞ R ¼ CðIÞ
[15]
X
ðIÞ
ðIÞ
Qkl Dl
l
2 3
4 ðIÞ ðIÞ o I Dk Dl 15
X
ðIÞ
eklm Ml DðmIÞ
[16] [17]
lm
Here eklm is the Levi–Civitta tensor and Ml is the magnetic transition-dipole moment. This procedure was followed throughout the present work. Neglecting the B-matrix amounts to the so-called Tamm–Dancoff (TDA) approximation. One then simply solves AX ¼ oX
With the supermatrices f XC jb cHF ijab Aia, jb ¼ ðea ei Þdai, jb þ ia~
ð IÞ 8 oI D 2 dkl 15
[18]
which resembles the configuration interaction with a single excitation approach. The key point in solving eqn [6] is to project the solution onto the manifold of core-excited single excitations. This obviously neglects the coupling of the coreexcited K-edge states to L- and M-edge, as well as valenceexcited states. The plausibility of this approximation has been discussed.17 It is most important to emphasize that due to the factors discussed above, the calculations do not yield accurate absolute transition energies. The errors are large but very systematic such that they can be compensated for by calibration. Such a calibration involves the comparison of theory and experiment for a set of ‘test molecules’. Obviously, the resulting calibration curve must be done for each element separately and depends on the basis set and the specific-density functional used in the calculation and on whether relativistic effects have been included in the treatment or not. It is therefore necessary to employ a consistent protocol. For example, a recent proposal involves the BP functional, the def2-TZVP basis set, and the ZORA treatment for relativistic effects.15
9.16.4.3
Ligand K-Edge Spectra
Ligand K-edge absorption spectra are obtained by exciting a ligand core-electron into semi-occupied or empty valence orbitals. In practice, this is best done at the chlorine, sulfur, and phosphorous edges. The most salient points are readily appreciated by taking a system with a single hole as an example.4 The starting point is the frozen orbital approximation to the initial and final states. The initial state is a single Slater determinant: jIi ¼ jc1 c1 . . . ck ck . . . cn cn cSOMO j
[19]
and the final state is the one where an electron is promoted from a ligand orbital with 1s character, ck, to the singly occupied molecular orbital (SOMO) jFk i ¼ jc1 c1 . . . ck cSOMO . . . cn cn cSOMO j
[20]
In the dipole-length approximation to the transition-dipole operator and using the Fermi golden rule, an electric-dipole transition has intensity proportional to: 2
! [21] Dk / cSOMO r ck
X-Ray Spectroscopy
where A sums over nuclei and i over electrons and ZA is the ! ! charge of atom A. R A is the position of the A0 th nucleus and r denotes an electron position. The ground-state singly occupied orbital, cSOMO, is written pffiffiffiffiffiffiffiffiffiffiffiffiffiffi cSOMO ¼ 1 a2 jMi ajLi [22] where the overlap between the metal-centered part of the SOMO, |Mi, and the ligand-centered part, |Li, is neglected. Therefore 100 a2 is the percentage ligand character in the SOMO which is distributed over the individual ligands that contribute to the SOMO and 100 (1 a2) is the percentage metal character in the SOMO. The ligand part will in general be a linear combination of valence ns and np orbitals jLi ¼ kjnsi þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 k2 jnpi
[23]
with hybridization coefficient k. The donor MO, ck, is assumed to be a pure ligand 1s orbital (see below). The square of the transition-dipole moment becomes D E2 1 2 ! 2 2 c [24] SOMO jr jck ¼ 3 a 1 k hr inp, 1s
435
instrumental for establishing whether the ligand behaves in a noninnocent way or not.5
9.16.4.4
Metal K-Edge Spectra
The main difference between metal- and ligand K-edge spectra in terms of the computational protocol is the intensity mechanism that is operative. At the metal K-edge, the dominating transitions in the pre-edge region are of metal-1s!metal-nd character. These transitions are electric quadrupole allowed and electric-dipole forbidden. Dipole character is induced in these transitions due to mixing of d- and p-orbitals at the metal, which, in turn, is induced by deviations from centrosymmetry. These are fairly subtle effects since the p-mixings are small and effectively compete with the quadrupole-intensity mechanism. These factors were all systematically addressed in a study dealing with iron K-edge spectra.17 It was found that the agreement between theory and experiment is also fairly good (Figure 10). Transition energies are typically computed to an accuracy of better than 0.5 eV and transition intensities correlate with experiment with a correlation coefficient R ¼ 0.98–0.99.17
with
hr inp, 1s ¼ Radnp jrjRad1s
[25]
where ‘Rad’ denotes the radial part of the orbitals. The expression has a straightforward interpretation – the pre-edge intensity increases as the ligand character the SOMO increases (a2), as the ligand in question makes a larger contribution to the total 2 ) and as the p-character ligand character in the SOMO (cSOMO 2 of this contribution increases (1 kSOMO ). In absolute terms, for a particular ligand atom, the factor 1 =3 hr i2np, 1s is the intrinsic intensity of an atomic 1s!npx,y,z transition. Its value does, however, not reflect a real atomic transition because the radial wavefunctions distort in the molecule according to the atoms’ environment. Since 1 =3 hr i2np, 1s depends on the molecular environment, Solomon and coworkers have found it necessary to determine its value for different types of bonding situations (e.g., for sulfur ligands: R–S, R–S–S–R, S2 etc.) separately before values of a2 can be determined from the areas under the experimental XAS absorption curves. However, neither 2 2 1 =3 hr inp, 1s itself nor the value of a is a physical observable. Only their product is directly related to the oscillator strength which is an observable. We have therefore chosen a different approach and directly calculated the physically observable quantity, for example, the oscillator strengths of the observed transitions using a simple protocol outlined above. The TD–DFT protocol outlined above was successfully applied to a substantial number of sulfur and chlorine containing transition-metal complexes and was found to be surprisingly successful. Transition energies are typically predicted within 0.2–0.3 eV and the correlation between calculated oscillator strength and properly normalized experimental intensities is on the order of 10%. An example for the comparison between theory and experiment is given in Figure 9. Following the detailed study of Ray et al.,5 DeBeer, Wieghardt, and coworkers have studied a number of transition-metal dithiolene complexes where the comparison between calculated and experimental sulfur K-edge spectra was
9.16.4.5
Metal L-Edge Spectra
The same TD–DFT protocol can obviously not be applied to the calculation of metal L-edge spectra as it (a) does not contain any SOC effects and (b) does not properly treat multiplet splittings. The first shortcoming could be fixed by resorting to twocomponent TD–DFT methods. However, this approach can still not be successful because the multiplet effects are not properly accounted for. The term ‘multiplet effects’ describe the fact that a number of final states of different energy arise from a given electronic configuration. If the number of unpaired electrons in a given electronic configuration exceeds the value of 2S, where S is the total spin of the target state, then several spin couplings arise, all of which lead to a given total spin S. In a similar way, the orbital angular momenta couple to different total values of L. In atoms and ions this behavior is well understood and summarized in the Russell–Saunders (LS) coupling scheme. The problem of transferring this scheme to molecules is that the different Russell– Saunders-like multiplet states are not single determinants, even if the ground state is well described by a single Slater determinant. The necessary determinants involved in properly building multiplet states may well be double- or triply excited relative to the electronic ground state. Hence, they are not accessible by TD–DFT that only incorporates single excitations. In fact, a DFT based theory that would properly include these effects has apparently not been developed. In principle, one can of course resort to wavefunction-based configuration interaction methods in which multiplet effects, spin–couplings, and higher excitations can be incorporated explicitly. These methods suffer from two defects: (a) the highcomputational cost of the calculations and (b) the poor quality of the Hartree–Fock–type reference wavefunction from which these calculations must be started. Thus, enormous levels of configuration interaction would be necessary to capture the physics of the problem correctly. These extensive calculations are not feasible and would also be of questionable quality because of the size-consistency problems that are inherent in configuration
436
X-Ray Spectroscopy
2.0
Normalized absorption
D2d-[CuCl4]2D4h-[CuCl4]21.5
[NiCl4]2[CoCl4]2-
1.0
[Fe(II)Cl4]2[Fe(III)Cl4]1[TiCl4]
0
0.5
0.0 2818
2819
2820
2821
2822
2823
2824
Energy (eV)
Calculated pre-edge intensity
6 5 4 3 2 1 0 2733
2734
2735
2736
2737
2738
2739
Energy (eV) Figure 9 Comparison of experimental (top) Cl K-pre-edge XAS data to the calculated spectra (bottom). Adapted from DeBeer George, S.; Petrenko, T.; Neese, F. Inorg. Chim. Acta 2008, 361, 965–972.
0.20
1+,0,1-,23-,4Calculated quadrupole + dipole contributions
Calculated pre-edge enery (eV)
7115
7114
7113
7112
7111
7110 7110
[FeCl4]1-
0.15
0.05
0.00 7111
7112
7113
7114
Experimental pre-edge energy (eV)
7115
[FeCl4]2-
0.10
[Fe(salen)Cl]
[Fe(acac)3] [FeCl6]3[Fe(CN)6]3[FeCl6]4[Fe(prpep)2]+ [Fe(CN)6]4- [Fe(prpep)2]
0
5
10
15
20
25
Experimental intensity
Figure 10 Correlation of experimental and theoretical transition energies (left) and intensities (right). Adapted from DeBeer George, S.; Petrenko, T.; Neese, F. J. Phys. Chem. A 2008, 112, 12936–12943.
X-Ray Spectroscopy
interaction treatments. While these problems are all severe, they may be surmountable in the foreseeable future. An alternative approach that has seen much use is the parametric multiplet approach as prominently exercised by Sawatzky, de Groot, and coworkers.28,29,36–39 In these approaches, one essentially performs an atomic configuration interaction calculation that leads to the correct LS multiplets but perturbs these multiplets with ligand-field splittings and charge-transfer interactions. The necessary parameters involve metal–ligand covalencies, which lead to anisotropy in the ligand field and the electron–electron interaction parameters. Together with the ligand-field and charge-transfer parameters that enter the model, there are a substantial number of empirical parameters that enter the calculations. These parameters are not straightforward to predict from DFT or ab initio calculations, which means that the main domain of application of the multiplet model is the fitting of known spectra. Good agreement with experiment can be achieved in this way, as shown in Figure11, which shows the calculated and experimental L-edge spectra of a low-spin Fe(III) model complex.40
9.16.4.6
X-Ray Emission
A physically rigorous modeling of the XES experiment would involve a calculation of the emission process following the core-level ionization. This is a fairly involved task as it will require the treatment of the electronic relaxation following the ionization event together with possible spin recouplings to give spin- and space-symmetry adapted final states. In practice, it has become evident that a much simpler approach is successful that is described below.21 The total number of photons emitted per unit time IIF that correspond to the transition of an initial core-ionized or coreexcited state |Ii to the final state |Fi in the limit of low concentration of the absorbing species and small path length of the incident beam is given by IIF ¼ WGI DN
AIF DI
2
720
725
The emission rate corresponding to the transition of state |Ii to state |Fi is given by the Einstein coefficient AIF which is proportional to the emission oscillator strength fIF AIF ¼ 3fIF gel
[28]
where gel is the classical radiative decay rate of the single-electron oscillator at frequency oIF which is equal to the transition frequency between the initial and final states (CGS units) gel ¼
2e2 o2IF 3me c3
[29]
Here, me is the mass of the electron, e is the electron charge, D is the speed of light, DI is the total decay rate of the coreexcited state |Ii which is the sum of the radiative (DIR) and nonradiative (DIN) decay rates R N DI ¼ D I þ DI X DRI ¼ AIF0
[30]
F0
According to eqns [26]–[30], the relative intensities of different transitions in the XES spectrum can be written as o2 fIF IRIF ¼ X IF 2 oIF fIF F
[31]
0
730
4
715
[27]
where L(ocm,oIF,G,s) is a lineshape function which, in general, depends on homogeneous (G) and inhomogeneous (s) broadening parameters. From eqn [7], it follows that Ðthe calculated XES spectrum has an integrated area of unity ( IR(oem) ¼ 1), and thus can be directly compared with the normalized experimental XES spectrum. The emission oscillator strength fIF is related to the absorption oscillator fFI strength as fIF ¼ fFI. The oscillator strength can be expanded as a power series in terms of the dimensionless fine-structure constant a (¼1/137.03599). The first term in this expansion represents the electric-dipole oscillator strength (fIFed). For orientationally averaged transition rates, the term linear in a is zero, and the next leading term a2 can be written as the sum of the electric quadrupole (fIFeq) and magnetic dipole (fIFmd) oscillator strengths:21,41
6
710
WGI ¼ sGI ðo0 ÞI0
[26]
8
705
where DN is the total number of absorbing species in the irradiated volume, and WGI is the transition rate between the ground state |Gi and state |Ii (capital letters are used for many electron states, lowercase letters for orbitals). WGI is related to the photoionization cross section sGI(o0) at the incident photon frequency o0 as
Then the normalized XES spectrum IIR(oem) corresponding to emission from state |Ii is X IRI ðoem Þ ¼ IRIF Lðocm ; oIF G; sÞ [32]
10
0 700
437
Energy (eV) Figure 11 Comparison of calculated (dashed line) and experimental (full line) L-edge X-ray absorption spectra of [Fe(tacn)2]3þ. Reprinted with permission from Wasinger, E. C.; de Groot, F. M. F.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 2003, 125, 12894–12906. Copyright 2003 American Chemical Society.
F0
eq
fIF ¼ fIFed þ fIFmd þ fIF
[33]
The mathematical form of the electric-dipole, magneticdipole, and electric-quadrupole contributions to this oscillator strength are the same as described above for the absorption process.
438
X-Ray Spectroscopy
The one-electron approximation consists of employing a set of frozen ground-state orbitals for the calculations. These are conveniently taken to be Kohn–Sham DFT orbitals. Associated with the Kohn–Sham procedure is the ground-state Kohn– Sham determinant jGi ¼ ji . . . j . . . nj
[34]
In the one-electron approach, the initial core-excited and final states are constructed as jIi ¼ ja . . . j . . . nj
[35]
jFi ¼ ja . . . i . . . nj
[36]
For the specific application at hand, orbital i is a coreorbital, for example, the metal 1s-derived orbital of a given absorber atom, a is an arbitrary unoccupied orbital (or a continuum unbound one-electron level), and j is another semicore or valence orbital of the same atom. Rather than trying to obtain accurate total energies for all of these states, the simple approximation is adopted for the emission energy
EsIF ¼ ħosIF ¼ esj esi
This is of course fairly simplistic. In terms of time-dependent DFT, it can be shown that the orbital energy difference is a welldefined approximation to the state energy difference if no Hartree–Fock exchange is present in the DFT potential.22 Calculations indicate that DFT in conjunction with the oneelectron approach does not provide an accurate modeling of the Kb main line. This is presumably due to: (a) a general underestimation of the core-level spin-polarization and (b) the lack of a proper treatment of the multiplet structure giving rise to the actual Kb main line peaks. On the other hand, the one-electron approach has been found to be very successful in the modeling of the Kb2,5 region, representing valence-to-core transitions. A comparison between calculated and measured spectra for a series of Fe(II) and Fe(III) complexes is shown in Figure 12 and demonstrates that the main features of the spectrum are all faithfully reproduced by the calculations. Thus it appears that this region mainly reflects the ground-state electronic structure. The same comments concerning the actual calibration procedure apply
80
1.5
[Fe(III)Cl4]
10
70
[Fe(III)(acac)3]
0
[Fe(III)(tpfp)Cl]
60
Calculated intensity
Normalized intensity (X 1000)
[37]
1
0.5
50 40 30 20 10
0 7085
7090
7095
7100
7105
7110
7115
Energy (eV)
(a)
0 7085
7120
7090
7095
7100
7105
7110
7115
7120
7110
7115
7120
Energy (eV)
(b) 150
4 3+
[Fe(III)(tacn)2]
3
3-
[Fe(III)(CN)6]
[Fe(III)(tpp)(ImH)2]
1+
Calculated intensity
Normalized intensity (X 1000)
3.5
2.5 2 1.5
100
50
1 0.5 0 7085
(c)
7090
7095
7100
7105
Energy (eV)
7110
7115
0 7085
7120
(d)
7090
7095
7100
7105
Energy (eV)
Figure 12 Experimental valence-to-core spectra for high-spin ferric complexes (a) and the corresponding calculated spectra (b). Experimental valence-to-core spectra for low-spin ferric complexes (c) and the corresponding calculated spectra (d). A 2.5 eV broadening and a constant shift of 182.5 eV have been to all calculated spectra. Reprinted with permission from Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727. Copyright 2010 American Chemical Society.
X-Ray Spectroscopy
as in the case of the TD–DFT-based XAS calculations. In the calibration study by Lee et al., 21 it was found, however, that the errors in the calculated transition energies are somewhat larger than in the case of XAS and amount to about 1 eV. The reasons for this increase in the error are not fully understood and may involve the limitations of the one-electron approach but also uncertainties in the experimental calibration procedure.
9.16.5
Conclusion
We hope that this chapter has demonstrated the complementarity of XAS, XES, and EXAFS. When the results of these techniques are viewed together, one can obtain a fairly detailed picture of the geometric and electronic structure of the investigated systems. Most importantly, the techniques are highly element specific and such highly selective measurements can even be performed on complex systems. In order to employ synchrotron-based techniques, it is not necessary to use isotopically enriched samples and it is also not necessary for the species to be paramagnetic. All of these advantages have added to the growing popularity of synchrotron-based X-ray spectroscopic techniques over the past decades in several research fields ranging from material sciences over biological chemistry to heterogeneous and homogenous catalyses. In this chapter, we have provided a cursory overview of the experimental and theoretical techniques that are presently in large-scale use. We hope that it became evident or at least plausible that one can take the analysis of the information content of the experimental data to the next higher level through the combination with quantum chemical calculations. These calculations can now efficiently be performed with user-friendly programs and are no longer confined to ‘expert territory.’ Together with the increasing availability of bright beam lines and ever improving X-ray detectors at the various synchrotrons worldwide, there appears to be every reason to be optimistic about the future of synchrotron-based spectroscopic techniques.
References 1. Koningsberger, D. C.; Prins, R. X-ray Absorption, Principles, Applications Techniques of EXAFS, SEXAFS, and XANES. Wiley: New York, 1988. 2. Glatzel, P.; Bergmann, U. Coord. Chem. Rev. 2005, 249, 65–95. 3. Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 1990, 112, 1643–1645. 4. Neese, F.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. Inorg. Chem. 1999, 38, 4854–4860. 5. Ray, K.; DeBeer George, S.; Solomon, E. I.; Wieghardt, K.; Neese, F. Chem. Eur. J. 2007, 13, 2783–2797. 6. Ray, K.; Petrenko, T.; Wieghardt, K.; Neese, F. Dalton Trans. 2007, 1552–1566. 7. Roe, A. L.; Schneider, D. J.; Mayer, R. J.; Pyrz, J. W.; Widom, J.; Que, L. J. Am. Chem. Soc. 1984, 106, 1676–1681. 8. Westre, T. E.; Kennepohl, P.; DeWitt, J. G.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 1997, 119, 6297–6314. 9. DuBois, J. L.; Mukherjee, P.; Collier, A. M.; Mayer, J. M.; Solomon, E. I.; Hedman, B.; Stack, T. D. P.; Hodgson, K. O. J. Am. Chem. Soc. 1997, 119, 8578–8579.
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10. Aliaga-Alcalde, N.; Mienert, B.; Bill, E.; Wieghardt, K.; George, S. D.; Neese, F. Angew. Chem. Int. Ed. 2005, 44, 2908–2912. 11. Berry, J. F.; Bill, E.; Bothe, E.; DeBeer George, S.; Mienert, B.; Neese, F.; Wieghardt, K. Science 2006, 312, 1937–1941. 12. de Oliveira, F. T.; Chanda, A.; Banerjee, D.; Shan, X.; Mondal, S.; Que, L., Jr.; Bominaar, E. L.; Mu¨nck, E.; Collins, T. J. Science 2007, 315, 835–838. 13. Rohde, J.-U.; In, J.-H.; Lim, M. H.; Brennessel, W. W.; Bukowski, M. R.; Stubna, A.; Mu¨nck, E.; Nam, W.; Que, L., Jr. Science 2003, 299, 1037–1039. 14. DuBois, J. L.; Mukherjee, P.; Stack, T. D. P.; Hedman, B.; Solomon, E. I.; Hodgson, K. O. J. Am. Chem. Soc. 2000, 122, 5775–5787. 15. DeBeer George, S.; Neese, F. Inorg. Chem. 2009, 49, 1849–1853. 16. DeBeer George, S.; Petrenko, T.; Neese, F. Inorg. Chim. Acta 2008, 361, 965–972. 17. DeBeer George, S.; Petrenko, T.; Neese, F. J. Phys. Chem. A 2008, 112, 12936–12943. 18. de Groot, F. M. F. Coord. Chem. Rev. 2005, 249, 31–63. 19. DeBeer George, S.; Metz, M.; Szilagyi, R. K.; Wang, H. X.; Cramer, S. P.; Lu, Y.; Tolman, W. B.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 2001, 123, 5757–5767. 20. George, S. J.; Lowery, M. D.; Solomon, E. I.; Cramer, S. P. J. Am. Chem. Soc. 1993, 115, 2968–2969. 21. Lee, N.; Petrenko, T.; Bergmann, U.; Neese, F.; DeBeer, S. J. Am. Chem. Soc. 2010, 132, 9715–9727. 22. Neese, F. Coord. Chem. Rev. 2009, 253, 526–563. 23. Hess, B. A.; Marian, C. M. In Computational Molecular Spectroscopy; Jensen, P., Bunker, P. R., Eds.; Wiley: New York, 2000; p 169ff. 24. van Lenthe, E.; Snijders, J. G.; Baerends, E. J. J. Chem. Phys. 1996, 105, 6505–6516. 25. Steinfeld, J. I. Molecules and Radiation. Harper and Row: New York, 1979. 26. Cotton, F. A. Chemical Applications of Group Theory. Wiley Interscience: New York, 1990. 27. Solomon, E. I. Comments Inorg. Chem. 1984, 3, 227. 28. Glatzel, P.; Bergmann, U.; de Groot, F.; Cramer, S. P. Abst. Pap. Am. Chem. Soc. 2002, 224, 025-INOR. 29. Degroot, F. M. F. J Electron Spectros Relat Phenom 1993, 62, 111–130. 30. Ankudinov, A. L.; Ravel, B.; Rehr, J. J.; Conradson, S. D. Phys. Rev. B 1998, 58, 7565–7576. 31. Ankudinov, A. L.; Conradson, S. D.; de Leon, J. M.; Rehr, J. J. Physical Review B 1998, 57, 7518–7525. 32. Rehr, J. J.; Ankudinov, A.; Zabinsky, S. I. Catal. Today 1998, 39, 263–269. 33. Ankudinov, A. L.; Rehr, J. J.; Low, J. J.; Bare, S. R. Top. Catal 2002, 18, 3–7. 34. Ankudinov, A. L.; Bouldin, C. E.; Rehr, J. J.; Sims, J.; Hung, H. Physical Review B 2002, 65, 104107. 35. Ankudinov, A. L.; Rehr, J. J.; Low, J. J.; Bare, S. R. J. Chem. Phys. 2002, 116, 1911–1919. 36. Esteva, J. M.; Karnatak, R. C.; Fuggle, J. C.; Sawatzky, G. A. Phys. Rev. Lett. 1983, 50, 910–913. 37. Kotani, A.; Ogasawara, H.; Okada, K.; Thole, B. T.; Sawatzky, G. A. Physical Review B 1989, 40, 65–73. 38. Vanderlaan, G.; Thole, B. T.; Sawatzky, G. A.; Verdaguer, M. Physical Review B 1988, 37, 6587–6589. 39. Hocking, R. K.; Wasinger, E. C.; Yan, Y. L.; de Groot, F. M. F.; Walker, F. A.; Hodgson, K. O.; Hedman, B.; Solomon, E. I. J. Am. Chem. Soc. 2007, 129, 113–125. 40. Wasinger, E. C.; de Groot, F. M. F.; Hedman, B.; Hodgson, K. O.; Solomon, E. I. J. Am. Chem. Soc. 2003, 125, 12894–12906. 41. Berry, J. F.; DeBeer George, S.; Neese, F. Phys. Chem. Chem. Phys. 2008, 10, 4361–4374. http://www.cec.mpg.de – Max-Plank-Institut fu¨r Chemische Energiekonversion.
Relevant Website http://www.cec.mpg.de – Max-Plank-Institut fu¨r Chemische Energiekonversion.