Chapter 5.19. Comparing XANES calculations with experiment

Great progress has been made in the computation of theoretical X-ray absorption fine-structure (XAFS) spectra since it was first performed in 1934 by Hartree and coworkers for GeCl₄ (Hartree et al., 1934). Historically, agreement with experiment has been much more satisfactory in the extended X-ray absorption fine-structure (EXAFS) region of XAFS spectra than in the X-ray absorption near-edge structure (XANES/NEXAFS) region, largely because of complications owing to large-angle multiple scattering in XANES and errors in approximations to the molecular potentials, which are more important at lower photoelectron energies. Much of the progress is due to major advances in theory and computational methods, which are reviewed elsewhere in this volume, and the exponential growth in affordable computing power since the 1970s. In parallel with these developments, there have been significant advances in the measurement of accurate spectra, which is essential for assessing the accuracy of the computations, owing to improvements in synchrotron sources, beamlines, energy analyzers, detectors and electronics.

Historically, most direct comparisons between XANES theory and experiment have been qualitative, or at best semi-quantitative, yet nevertheless have proven to be useful (see, for example, Van Nordstrand, 1967; Lytle, 1999; Koningsberger & Prins, 1988). Comparison between experiment and calculation has often consisted of the correlation of the numbers, positions and approximate heights or areas of calculated spectral features (spectral shapes) with the corresponding features in experimental spectra, rather than a direct least-squares comparison to the data, as in, for example, Binsted & Hasnain (1996), Benfatto et al. (2001) and Smolentsev et al. (2008). The correspondence between calculated and experimental peaks has in many cases been conjectural, and similarly for the assignment of experimental features to particular transitions. Such assignments are made more convincing when they are based on numerical calculations of electronic structure or other evidence. Theoretical calculations can be useful for understanding trends, even if they are not quantitatively accurate, in the sense of being able to quantitatively reproduce the experimental data to within the experimental uncertainties.

A fundamental question to be asked is what information one wishes to extract from the data. Some features and properties are of greater importance than others when answering specific scientific questions. The presence or absence of a spectral feature may be crucial to answer a scientific question for one system but not for another, and similarly for the shift of an edge position or the intensity of an absorption line. For example, the existence of a pre-edge transition in the K-edge absorption of copper complexes may indicate whether the copper is in the +1 or +2 oxidation state. The presence of a strong pre-edge transition in the K edge of metal oxyanions (for example CrO₄) indicates asymmetry under inversion of the metal-ion site, which may suggest tetrahedral symmetry, as opposed to square-planar or octahedral coordination. Average nearest-neighbour bond lengths can be estimated from the energy width of K- and L₁-edge steps. The area of an L_2,3-edge peak may be useful for determining the filling of d orbitals.

Despite the substantial progress, at this time we are not yet at the point where quantitatively accurate XANES spectra can be reliably computed ab initio for an arbitrary material. There are several reasons for this, which are described below. It is also difficult to objectively assess the average reliability of comparisons between theory and experiment in the literature because there is a natural but unfortunate tendency for only the most successful comparisons to be published, while the poorer comparisons are not. Nevertheless, decisive quantitative structural information is derivable from XANES for many materials, assisted by the insight that is provided by theoretical calculations. A recent review of work along these lines with applications to catalysis can be found in Guda et al. (2019).

Reliable comparison of calculated spectra with experimental data requires great care. Experimental data are subject to experimental errors and uncertainties which can distort or modify the spectra: thickness/particle-size effects, self-absorption effects in fluorescence, unknown variable instrumental backgrounds, energy-calibration errors and noise. These can and should be minimized in the experiment, but there are always residual effects (for example residual background errors, tails of the monochromator energy resolution function and tails of the detector resolution function) that must be accounted for when comparing precisely with theoretical calculations, especially when attempting to obtain structural information through direct fitting or other methods. Such experimental uncertainties tend to increase the number of free parameters that are required in any comparison with theory and to effectively decrease the available information content because of fitting-parameter correlation. Bayesian methods can be used to quantify this.

Theoretical calculations also are accompanied by errors, and they also require the introduction of parameters (for example muffin-tin overlap factors), even if they are not explicitly stated. The various theoretical programs that are available rely on different approximations and computational methods, and they have strengths and weaknesses in their abilities to calculate spectra for different kinds of systems. For example, programs may use density-functional theory (static or time-dependent) to calculate the exchange–correlation potential and may use linear combinations of localized orbitals or scattered waves (at both negative and positive energies) to represent quantum wavefunctions and/or Green's functions (see, for example, Rehr & Ankudinov, 2005). An overview of progress from the point of view of multiple-scattering theory is given in a recent book (Sébilleau et al., 2018). Some programs make the muffin-tin approximation to the potential, while others do not (Joly, 2001). The structure may be represented as a finite cluster of atoms, as a molecule with explicit bonds or as a periodic solid (with a core hole to break the translational symmetry). The incorporation of many-body effects at some level is generally important. Electron-correlation effects are important when calculating pre-edge spectra of partially filled shells in transition metals, and this requires alternative methods. Representative examples of efforts to better account for many-body effects, or multiplets, with application to L edges are given by Vinson et al. (2011), Stavitski & Groot (2010), Ikeno et al. (2011) and Haverkort et al. (2012). Quadrupole effects may also be important in the pre-edge region, and individual programs may or may not include such effects. It is essential to account for X-ray polarization in fully oriented (for example single-crystal) and partially oriented (for example nonrandomly oriented polycrystalline) samples. Accounting for the disorder that is due to thermal and zero-point motion, which can be important when computing XANES in some systems (Bunker, 1984, 2010), can be performed through an averaging process or other methods, but this is often neglected because of computational cost or a perceived lack of necessity.

Ideally, one would like to obtain quantitative agreement between theory and experiment to within the experimental and theoretical uncertainties, and use this to determine the structure. This generally requires additional knowledge or a set of assumptions about the structure. The information required to specify a fully unknown structure (for example the 3D positions of all atoms in the neighbourhood of the absorber) will generally exceed the information content in the data, but if appropriate (and explicitly stated) assumptions can be made (for example hypothetical crystal symmetries) then theoretical calculations can be very informative and even decisive. This approach amounts to hypothesis testing by distinguishing structure types and then determining the range of values of structural parameters that allow the hypothetical model to fit the data adequately.

For some physical systems and computational approaches the goal of direct structure determination can nearly be achieved, but at present there is no single approach that works well for all experimental systems. At this time, a software-toolkit approach appears to be most appropriate for the task. Fortunately, a broad range of computational tools are available, many of which are described elsewhere in this volume, and new ones are constantly being created. These programs ordinarily take hypothetical structures (for example atomic positions and atomic numbers) as input and possibly information about interatomic forces to model thermal and zero-point motion. Some of these computational tools have been wrapped in code that carries out the fitting of spectra; a venerable example is MXAN (Benfatto et al., 2001, 2024). This is essentially the same procedure in concept as nonlinear least-squares fitting to any other kind of data. Depending on the search algorithm that is employed, the usual concerns to do with avoiding trapping in local minima, managing fitting-parameter correlations and applying physical constraints are relevant. Genetic algorithms (Dimakis & Bunker, 2006) and other stochastic search methods have proven to be useful in this connection, allowing hands-off fitting and the inclusion of computed thermal effects.

There is considerable room for optimism in the use of XANES as a means of structure determination by employing increasingly sophisticated theoretical and computational models. We also can expect substantial growth in the application of various machine-learning methods to this project, both as a quantitative extension of qualitative fingerprinting and as a means of directly extracting quantitative information.