Chapter 6.5. EXCURVE

X-ray absorption spectroscopy (XAS), including X-ray absorption near-edge structure (XANES) and extended X-ray absorption fine structure (EXAFS or XAFS), took off in the 1970s as a technique to study the structures of materials that were not amenable to single-crystal X-ray diffraction. The availability of synchrotrons as a source of brilliant tuneable X-rays coincided with the notion (Sayers et al., 1971) that Fourier transformation of the spectrum yields a radial distribution function that gives the distances of atoms from the absorber atom (see Section 2). An established theoretical framework based on low-energy electron diffraction (Pendry, 1974) served as inspiration for exact theories (Lee & Pendry, 1975; Ashley & Doniach, 1975). Early EXAFS was interpreted using the plane-wave approximation (Teo & Joy, 1981; Koningsberger & Prins, 1988; Stern et al., 1975; Lee & Beni, 1977; Stern, 1974; Citrin et al., 1976; Teo et al., 1977) in either a Fourier-fitting or curve-fitting approach. These neglected the computationally more complex curvature of the electron wave, giving poor agreement between simulation and experiment in the low-energy region. Nevertheless, these early results are reliable insofar as they deal with strong backscatterers in systems where only single scattering is important.

EXCURVE was a joint effort of theoreticians and programmers led by John Pendry at the first dedicated second-generation synchrotron storage ring, the Synchrotron Radiation Source at the Daresbury Laboratory, UK, and its user community. The first interactive version of the program was EXINT (Gurman, 1980). The incorporation of fast spherical wave theory into EXINT was a major advance because it used exact curved-wave theory with an efficient angle-averaging procedure: the fast spherical wave method (Gurman et al., 1984). Further development by Binsted led to a command-based program, which was named EXCURVE by E. Pantos. Before this there were very few examples of EXAFS simulations that did not use the plane-wave approximation: exact calculations of, for example, As₂O₃, were computationally demanding (Gurman & Pettifer, 1979). Subsequent versions of EXCURVE included multiple scattering (up to third order; EXCURV86; Gurman et al., 1986; subsequently extended to fifth order), the small-atom approximation (EXCURV88; Gurman, 1988; Rehr & Albers, 1990), constrained refinement (EXCURV90) and restrained refinement (EXCURV92; Binsted et al., 1992). EXCURV98 included multiple cluster methods, point-group symmetry, 3D refinement, amino-acid and ligand databases, and anharmonicity. Options for modelling surfaces (including exact polarization effects), commands for visualizing clusters and ligands, and whole-spectrum XAS analysis (Binsted & Hasnain, 1996) are also available in EXCURVE.

In the 1990s, the alternative EXAFS programs FEFF (Rehr et al., 1991; Zabinsky et al., 1995; Kas et al., 2024) and GNXAS (Filipponi & Di Cicco, 1995; Filipponi et al., 2024) became available. These are based, like EXCURVE, on Fermi's golden rule and electron Green's functions and take the curvature of the electron wave into account, but differ in the details of their subsequent treatments (Ravel, 2016). A fruitful dialogue with the developers of FEFF led to the incorporation into EXCURVE of the Rehr–Albers approach for multiple scattering and the Hedin–Lundqvist potential and exchange. EXCURV98 was made available as DL_EXCURV using the DL Visualize (DLV) graphical interface (Tomic et al., 2004). EXCURVE has unique features, such as constrained and restrained refinement (Binsted et al., 1992) and its combination with powder diffraction (Binsted et al., 1996). It has been used for automated analysis of bioXAFS data in ABRA (Automated BioXAS Refinement and Analysis; Wellenreuther et al., 2010) and continues to serve a distinct group of users. A crude bibliometric approach (Fig. 1) suggests that the fraction of the total EXAFS output analysed with exact curved-wave theory among available software remains remarkably and disappointingly low.

Figure 1 Number of publications on EXAFS (white; topic EXAFS, scaled by a factor of 0.20), EXCURVE (black; citation of Gurman et al., 1984), FEFF (dark grey; citation of Rehr et al., 1991) and GNXAS (light grey; citation of Filipponi & Di Cicco, 1995) in the years 1971–2020. Source: Clarivate Web of Science (April 2020).

EXCURVE calculates the structure in the absorption coefficient for a photon of energy E within a single-particle formalism using the dipole approximation (Stern, 1988),where N_α is the number of atoms per unit volume, ρ(E_f) is the density of final states, |i〉 is the initial core state with energy E_i, |f〉 is the final state with energy E_f = E_i + hω, where hω is the photon energy and z is the form of the dipole operator for the unpolarized case, with e the electric vector of the photon and r the position with respect to the atomic nucleus. The initial state is the core state of energy E_c in the unperturbed atom and the final state is a photoelectron wavefunction which includes the effect of scattering by surrounding atoms. μ(E) is related by equation (2) to μ₀, the absorption coefficient due to the absorbing atom alone, which is normally, but not necessarily, removed by background subtraction, isolating the oscillatory component χ,

The calculation of χ follows the general method of Lee & Pendry (1975) where, for amorphous or polycrystalline samples, the scattering contributions for a specific final state l are of the form (the real part of the expression is taken)where Z is a sum over terms for single, double scattering etc. The final term defines the change in phase due to the passage of the photoelectron in and out of the central atom potential. Z is usually calculated using the angle-averaged `fast spherical wave' method (Gurman et al., 1984, 1986), in which each single-scattering atom contribution is given bywhere the lth element of T isThe Hankel functions h depend on the interatomic distances, and the matrices T on the scattering phase shifts δ_l for each atom. T is diagonal as the phase shifts are calculated from a spherically symmetric potential. The 3J coefficient is that of Brink & Satchler (1968), also given by Zare (1988). The l sums are strongly restricted by rules on angular momentum coupling.

Where a site may be statistically occupied, as in a metal alloy, it is possible to define a `mixed-site atom' and use this to generate the T matrix. For the more general case requiring polarization dependence, the angular part of the dipole matrix elements are required and the calculation is less efficient, unless the less accurate `small-atom' option (Gurman, 1988) is used. In general, for initial states of p or d symmetry, only the dominant transition l → l + 1 is calculated; however, it is also possible to include the l → l − 1 term provided that the atomic contributions are calculated at the same time as the phase shifts. Extrinsic losses, mainly due to two-electron transitions, are approximated by a constant amplitude factor.

2.3. Treatment of disorder

If many-body effects are ignored, thermal and static disorder in EXAFS can be described in terms of pair distribution functions for each leg of a scattering path. In the harmonic approximation disorder is described approximately by a Debye–Waller factor, given by ), where the mean-square variation in path length is given in units of Ų (Beni & Platzman, 1976; Binsted et al., 2005).

For a single leg of a scattering path, in one dimension and assuming isotropic motion of individual atoms a and b, the contribution to is given by the mean-square relative displacement,where is the mean-square displacement of atom a and C_ab is a displacement correlation function. The are related to the crystallographic isotropic atomic temperature factor B_iso byPhotoelectron propagation is orders of magnitude faster than thermal motion, so when a leg appears in a scattering path n times, the contribution to is n² times that of an isolated leg. Thus, for a single scattering path (where the in and out legs are the same) in one dimension, and assuming isotropic atomic motion, is given byThe single-scattering Debye–Waller factor is thenwhere A_j is a refinable parameter which characterizes each shell j.

For multiple scattering paths, it is necessary to take the angle between the legs of the path into account. In EXCURVE it is assumed that is given by

The α_a are the angles between the legs of the scattering path which meet at atom a (i.e. 0° for backscattering, 180° for forward scattering and, for example, 126° for the Zn—N₁—C₂ angle in Fig. 2, bottom left) and Ce_a is an effective correlation term that applies to atom a. n is the number of times that an atom occurs in a leg. Thus, for a linear chain of atoms 0–a–b–c–b–a–0 (where 0 is the central atom), will be identical to that of a single-scattering path 0–c–0. For the path 0–a–0–a–0–a–0, will be 25/4 times that of the value for the single-scattering path 0–a–0. When the path is nonlinear, however, the effective correlation terms for each atom, Ce_n, depend on the and C_nm of all of the other atoms m in the path. The mean-square displacement terms and correlation terms C_ab in multiple-scattering paths are not treated as separate variables in EXCURVE, but are derived by partitioning the A_j parameters which define each shell for the atoms involved in the scattering path, thus avoiding an unwarranted increase in refinable parameters. The terms are therefore approximated using a sum over all the A_j values for atoms in the path, weighted according to the angles between the atoms (Binsted et al., 2005).

Figure 2 k3-weighted Zn K EXAFS (top), phase-shift-corrected Fourier transforms (middle, modulus and imaginary parts included) and averaged geometries from crystallography (bottom) of Zn–imidazole complexes. (a) Zn(imidazole)4 diperchlorate, (b) Zn(imidazole)2 bisacetate: experimental (blue grey) with simulation (red) based on the crystal structures (Bear et al., 1975; Horrocks et al., 1982). Adapted from Feiters et al. (2003) and Feiters & Meyer-Klaucke (2013).

Where one of the atoms in a path is highly polarizable, or one of the atoms in a lattice occupies an off-site position, the harmonic approximation may no longer be valid and it is then necessary to describe a pair distribution in terms of a cumulant expansion (Tranquada & Ingalls, 1983; Binsted et al., 2005). Using a one-dimensional expansion in the plane-wave approximation, this generates a phase term generated by the third cumulant (skewness) and an amplitude term given by the fourth cumulant (kurtosity). In addition, it gives an additional phase term due to the second cumulant (variance) that was disregarded in the initial description. The phase terms can be described as an apparent interatomic distance (r′) in the Hankel functions h(kr′). In terms of the cumulants of the interatomic distance, σ₂σ₃σ₄, this givesand an amplitude termIf an additional option to include the effect of isotropic motion in three dimensions is included, the effective distance is further modified,When using the cumulant expansion, shell parameters B_j are equivalent to 10³σ₃ and C_j are equivalent to 10⁶σ₄. There is also an option for evaluating the integral over the pair distribution function numerically.

The usual input for EXAFS simulations is an experiment which has been background-subtracted and is usually presented as χ(k), with the energy axis converted from energy (E, in eV) to wavevector (k, in Å⁻¹), usingwhere m_e is the mass of the electron, E₀ is the edge energy and is h/2π. This choice of x axis has the advantage of showing the EXAFS oscillations as sinusoids. To offset the damping of the oscillations, the fine structure is often k³-weighted, so that the oscillations have a near-constant amplitude over the k range. The main ingredients for interpretation of EXAFS are the structural model and the phase shifts.

Simulations require a starting-model structure, which is typically based on prior chemical understanding of the system under study. Structural models can be built in EXCURVE in a number of ways: (i) shell by shell, defining atom type, occupancy, distance to absorber and the Debye–Waller parameter; (ii) by generating coordinates from shell parameters by defining the point symmetry of the cluster or from tabulated simple crystal structures (for example f.c.c., NaCl, CaF₂); or (iii) by reading atomic coordinates from databases, including the Protein Data Bank (PDB) or using EXCURVE's built-in Ligand Database, which is based on the crystallo­graphic refinement program PROLSQ (Hendrickson, 1985). Multiple scattering is calculated in EXCURVE from `units', which typically contain one group of the structural model, or whole clusters, involving all atoms involved in the structural model, for example the imidazole group. It is also possible to read a PDB file, or build a structure in three dimensions, and assign it to a cluster. In this case the EXAFS, including multiple scattering, can be calculated in three dimensions and will include inter-unit multiple scattering as well as intra-unit multiple scattering terms.

A numerical indication of the quality of a simulation is the `fit index', which is a measure of the difference (φ_EXAFS) between the experimental and simulated spectrum over the whole data range; alternative indications of the goodness of fit are χ² (Bunker et al., 1991) and the R factor (Binsted et al., 1992). For refinement guided by ideal geometric parameters, the fit index φ is defined as the sum of the contributions owing to deviations between experiment and theory (φ_EXAFS) and between refined and the idealized bond distances (φ_dist) and bond angles (φ_angle), respectively, for which the respective weightings w_EXAFS, w_dist and w_angle can be adjusted,

For every shell of backscattering atoms a reasonable choice is made of atom type and of the three most important parameters: the absorber–scatterer distance r, the shell occupancy N and the Debye–Waller parameter. EXAFS is calculated for a reasonably chosen ΔE₀ (threshold energy, EF). The target for refinement is to seek agreement with the experimental EXAFS as defined by φ_EXAFS. Although it is good practice to check whether there is also good agreement with the real (modulus) as well as the imaginary component of the Fourier transform, it should be clear that no Fourier filtering is involved, and the fit index is only calculated for the EXAFS.

Simulations to find the best fit to the experimental data are performed by a process of `iterative refinement'. It is desirable to achieve an absolute minimum in the multidimensional parameter/fit-index space and false minima should be avoided by performing refinement from a different set of starting parameters. The results can be checked against the semi-empirical bond-valence sum approach (Thorp, 1992). It is also possible to produce a contour map of the fit index for a pair of parameters.

In single-scattering simulations, the parameters that are refined are ΔE₀ (EF), the distances r and the Debye–Waller parameters. An important decision to be made is on the possible inclusion of an extra shell to the fit; criteria to take this decision have been defined (Joyner et al., 1987). For multiple-scattering simulations, the angle A (absorber)–B (backscatterer)–R (remote backscatterer) can also be refined for every R (remote) shell. For unknown systems the occupancy of each shell might also be refined. Generally, the number of refined parameters should be kept as low as possible to avoid overinterpretation of the data; it should always be less than the number of independent data points N(ind), which depends on the k and r range that are fitted (Stern, 1993),

In multiple-scattering simulations, there is also a necessity to separately include the contributions of atoms in shells that may not be resolved in the Fourier transform, such as, for example, the two C atoms in the second shell of an imidazole ligand (Fig. 2) which have to be included separately with A–B–R angles of different signs.

In the case of heteroaromatic ligands, such as the haem or imidazole in biological systems (Fig. 2), a limit on the number of parameters used can be derived from the geometry of the multiple-scattering units. Parameters such as the distance of an imidazole, its rotation with respect to the metal–nitrogen bond and the distance of the metal with respect to the imidazole plane should be treated as real variables, while internal parameters, such as the distances between the atoms in the imidazole ring, should be constrained or restrained (Strange et al., 1987; Binsted et al., 1992).

In constrained refinement, the distances within the unit are fixed at ideal values imported from small-molecule crystallo­graphy, and the parameters refined are ΔE₀, the angle for any unit, and one r and one Debye–Waller factor for every shell. This approach can be too rigid for many simulations, as can be seen from the differences (up to 0.04 Å shorter) going from Zn(imidazole)₄ diperchlorate to Zn(imidazole)₂ bisacetate in Fig. 2. Therefore restrained refinement is often used, whereby restraints for distances and angles are applied and the metal–unit atom distances are allowed to vary freely in the refinement. In deviations from the restraints, a weighted penalty is added to the fit index to keep the unit close to ideal geometry (Binsted et al., 1992). The Debye–Waller factors generally increase with the distance of the shell to the metal ion, and applying the same value to shells at similar distances in a single unit limits the number of parameters and increases the observation:parameter ratio. Fig. 2 illustrates the accuracy of the EXAFS simulations in EXCURVE based on the crystallographic data, including the tilting of the imidazoles with respect to the Zn—N bond.

The introduction of constrained/restrained refinement procedures using idealized parameters was inspired by modelling methods in protein crystallography (Hendrickson & Konnert, 1980; Sussman et al., 1977; Engh & Huber, 1991), where such restraints are used for the refinement of structures against electron-density maps. 3D-EXAFS restrained refinements can be performed on metalloprotein crystal structure coordinates read by EXCURVE from the PDB. Examples include refining metal-site coordination to improved accuracy independent of the crystallographic resolution of azurin (Cheung et al., 2000) and vanadium-containing bromoperoxidase (Renirie et al., 2010). The combination of 3D-EXAFS and crystallography has also been used to improve the crystallographic modelling of N and O ligand atoms coordinated in the heavy-atom environments of the FeMo and VFe proteins of nitrogenase (Strange et al., 2003). It is also possible to simulate the EXAFS on the basis of multiple clusters where the element of interest occurs in very similar (different oxidation state) or very different (different physical phase) situations. Examples of the contributions of EXCURVE to biological applications of XAS are given in Strange (2024). EXCURVE has also found wide applications in materials, for example geology and environmental science (Helz et al., 1996; Farquhar et al., 2002; Little et al., 2014), and catalysis (Gruenert et al., 1994; Corker & Evans, 1994; Corker et al., 1996; Fiddy et al., 2007; Bauer et al., 2012; Santos et al., 2015; van Weerdenburg et al., 2015).

EXCURVE has attractive features for EXAFS specialists in different fields, such as the combination with powder diffraction for materials scientists and the ligand database and access to crystallographic data for molecular chemists and biochemists. The refinement is in k-space with no need for Fourier filtering. With the modern potential/phase-shift calculations, few parameters need to be adjusted in order to obtain good agreement between experiment and theory in the simulation of experimental data of a system for which the structure is already known.

On the availability: EXCURVE has moved from the CLRC/CCP3 context to Forge (https://ccpforge.cse.rl.ac.uk/gf/project/excurv/frs/ ) and is administrated by Barry Searle. The official site for DL-EXCURV is http://www.cse.scitech.ac.uk/cmg/EXCURV/ . The graphics in the standard version of the Daresbury Laboratory Visualize (DLV) package can be made to work on Linux or Cygwin.