Chapter 6.9. FitIt

The local structure refinement from XANES is based on the minimization of the discrepancy between theoretical and experimental spectra by varying structural parameters. The structural transformations are performed automatically with the FitIt tools, which are typically sufficient for molecules. For crystals, some external programs, for example Atoms from the Demeter (IFEFFIT) package (Ravel & Newville, 2005; Newville & Ravel, 2024), can be used. When dealing with molecules, one has to define rigid groups and then translate or rotate them along different axes. For crystalline materials, one can vary the positions of atoms in the unit cell as well as the cell constants.

XANES fitting includes two steps: the interpolation of spectra and minimization of the discrepancy between theory and experiment. Firstly, we construct the interpolation of the spectrum as a function of structural parameters. This allows the prediction of XANES for any values of parameters within selected limits of variation. Calculations using external codes such as FEFF (Ankudinov et al., 1998; Rehr et al., 2010; Kas et al., 2024) and FDMNES (Joly, 2001; Bunău et al., 2024) are performed during this step. Plugins to generate the required input files and export the results are part of the FitIt package. The interface of FitIt proposes sets of parameters for calculations with several external programs. Initially, they are used to check the quality of interpolation and are therefore called control points. If they have to be included in the polynomial, then they become interpolation nodes.

As a second step, we minimize the discrepancy between interpolated spectra μ_i(E) and experimental spectra μ_exp(E) by varying the structural parameters. The mean-square deviation is used as the main criterion, The Chebyshev criterion is an alternative:Here, E₁ and E₂ are energy limits for the comparison of spectra. In many cases, the analysis involves two states of the material. One spectrum corresponds to the known structure, while the goal is to determine the structure in the second state. In this case, a fitting of the differential spectrum can be performed,Here, and are the experimental and theoretical spectra for the reference phase with known structure. To exclude the influence of nonstructural parameters on the results of structural refinement, we perform the analysis in two steps. Firstly, we test the method of XANES calculations using the reference phase of the material or a closely related system with known structure. We choose the best method of calculation and select optimal values for the nonstructural parameters that give the best agreement with the experiment. As a second step, we switch to the system with unknown structure and vary the structure. All nonstructural parameters are fixed to their optimal values during structural refinement.

If the experimental input is a large series of spectra then it is decomposed using PCA (Smolentsev, Guilera et al., 2009) into a few components and their weights. Such a decomposition procedure can have some degrees of freedom. The corresponding PCA parameters can vary along with the structure during minimization of the discrepancy between theory and experiment.

Multidimensional interpolation of XANES is used to minimize the number of required calculations of XANES for different sets of structural parameters (Smolentsev & Soldatov, 2006). The following expansion of the spectrum is used:Here, p₁, p₂ … p_n are the initial values of structural parameters and δp₁, δp₂ … δp_n are the deviations from their initial values. The spectra calculated with an external program, μ(E, p^k), and interpolated XANES, μ_i(E, p^k), are equal for interpolation nodes p^k. Therefore, energy-dependent coefficients A_n(E) and B_mn(E) are obtained by solving the system of equationsAn iterative procedure is used to check and improve the polynomial. It is based on the idea that the higher order cross-terms are negligible if they are the product of any previously neglected term and another parameter. This procedure, together with a few examples, is described in greater detail in Smolentsev & Soldatov (2006). It has been implemented in the interface of FitIt for the construction of the interpolation.

Principal component analysis (Malinowski, 1991) is used to analyze large series of experimental spectra. Each spectrum in the series is described as a linear combination of several components with unknown weights which are dependent on the number of the spectrum in the series. Some or all of the spectroscopic components can be unknown. The goal of this method is to determine all statistically significant spectra that are mixed in the time-resolved series and their weights for each spectrum in the series.

The large matrix of measured spectra μ_ij is represented aswhere i is the energy point index, s_ik is spectroscopic component number k and w_kj is the weight of component k for the measured spectrum number j. Mathematically, such a decomposition does not have a unique solution: for any matrix T_pk that can be inverted,is another solution.

Therefore, the search for a physically meaningful solution can be performed in two steps. Firstly, singular value decomposition is used to find any (mathematical) solutionwhere the elements of the diagonal matrix l_km are sorted in a decreasing order and w_kj = l_kmv_mj.

Quantitative criteria (Malinowski, 1991) are available to determine the number of statistically meaningful components from the analysis of elements of l_km,where r is the number of energy points and c is the number of time points. The minima of IE and IND correspond to the number of observable components.

As a second step, the mathematical solution is transformed into a physically meaningful solution. A priori knowledge about XANES and a system can be used. For instance, all of the spectra have to be positive and normalized, the weights should be in the interval from zero to one and for many applications they should have only one maximum. We have demonstrated (Smolentsev, Guilera et al., 2009) that using only such limitations (the corresponding algorithms are known as self-modelling curve resolution; Jiang et al., 2004; Maeder, 1987; Manne et al., 1999) is not sufficient to find a unique solution for XANES. Therefore, in FitIt we add an additional criterion, the similarity to the theoretical spectrum, to determine the components. The model for calculations can have some parameters. Therefore, minimization is performed with simultaneous variation of elements of the matrix T_pk and structural parameters. Other codes such as Demeter (Ravel & Newville, 2005), WinXAS (Ressler, 1998) and XANDA (Klementiev, 2012, 2024) use experimental spectra of reference compounds to determine the elements of T_pk, which transforms the mathematical solution into a physical solution.

The structural refinement based on XANES employing non-muffin-tin potentials has been demonstrated for the first time using Ni(acacR)₂ as an example, where R is a para tertiary butylbenzyl group attached to acetylacetonate in the 3-position (Smolentsev et al., 2007). The calculations were performed for the part of the complex shown in Fig. 1. The muffin-tin approximation is widely used in calculations of XANES within the full multiple-scattering theory, for example in FEFF9 (Rehr et al., 2010; Kas et al., 2024) and MXAN (Benfatto et al., 2001, 2024), even if there is a full-potential version of the full multiple scattering (Hatada et al., 2009). The finite-difference method implemented in FDMNES (Joly, 2001; Bunău et al., 2024) uses the non-muffin-tin potential and has been combined with FitIt.

Figure 1 Structural model of Ni(acacR)2 used for XANES fitting. Reproduced from Smolentsev et al. (2007).

Different approaches for XANES calculations have been tested for Ni(acacR)₂: full multiple scattering with the muffin-tin potential (FEFF8), the finite-difference method with the muffin-tin potential (FDMNES) and the non-muffin-tin finite-difference method (FDMNES). The use of the non-muffin-tin approach is essential to reproduce the experimental spectrum (Smolentsev et al., 2007). Within the muffin-tin approximation, the potential is spherically symmetric inside atomic spheres and is constant in the interstitial regions between atoms. Two types of non-muffin-tin effects are possible: (i) asymmetry inside atomic spheres, for example owing to the influence of strong covalent bonds, and (ii) variable interstitial potential. We found that the potential in the interstitial region between the atoms Ni—O—C₁—C₂ is not constant. This area is close to the absorbing atom and is relevant to the XANES calculations.

The parameters that were fitted during the structural refinement are shown in Fig. 1. The limits of variation were Ni—O distance, 1.76–1.92 Å; O—C₁ distance, 1.22–1.38 Å; C₁—C₂ distance, 1.30–1.46 Å; O—Ni—O angle, 80–100°. All of them significantly influence the shape of the XANES (Fig. 2 and Table 1). These distances change the position of maximum B and the minimum at the energy above 8380 eV (compare spectra 3, 4 and 5). The angle influences the relative position of maxima A and B and affects the intensity of feature C (compare spectra 1 and 2).

Figure 2 Theoretical Ni K edge XANES of Ni(acacR)2 calculated for the different structural models. The trace numbering corresponds to the sets of parameters in Table 1. Reproduced from Smolentsev et al. (2007).

The applied interpolation polynomial isIt has 21 interpolation nodes. Thus, only 21 calculations using the time-consuming finite-difference method are required to predict the shape of the theoretical spectrum for any values of parameters within the selected limits of variation.

The best-fit spectrum reproduces all of the features of the experiment (Fig. 3). The values of the structural parameters that correspond to this spectrum are Ni—O distance 1.83 Å, O—C₁ distance 1.28 Å, C₁—C₂ distance 1.39 Å and O—Ni—O angle 93°. These values are close to those determined usingX-ray diffraction for the related nickel complex (Cotton & Wise, 1966), EXAFS data (Feiters et al., 2005) and geometry optimization with density-functional theory (Smolentsev et al., 2007). If the full multiple scattering with the muffin-tin approximation is used for structural refinement, then the best-fit values (Ni—O distance 1.94 Å, O—C₁ distance 1.28 Å, C₁—C₂ distance 1.43 Å and O—Ni—O angle 102°) are far from the expected values. This confirms that in the present case the use of the non-muffin-tin potential is essential to precisely determine the structure from XANES.

Figure 3 Comparison of the experimental Ni K-edge XANES (solid line), interpolated (dashed blue line) and FDM-calculated (dotted red line) spectra for the best fit of parameters. Reproduced from Smolentsev et al. (2007).

The structure of a crystalline compound was determined with FitIt for the first time using InAs under high pressure as an example (Smolentsev et al., 2006). A transition from the Fm3m to the Cmcm phase is observed when the pressure is increased from 14 to 19 GPa. The goal was to determine the structure of the high-pressure phase; the structure at 14 GPa was well known. For the Fm3m phase, calculations with the non-muffin-tin finite-difference method agree with the experiment (Fig. 4). The unit-cell constants for the Cmcm phase p₁ = a = c, p₂ = b and the internal cell parameter p₃ = δ were the fitted parameters. δ represents the displacement of atoms from their position in the Fm3m phase: (0, 3/4 − δ, 1/4) and (0, 1/4 − δ, 1/4). They varied within the limits 5.396–5.096 Å for p₁, 5.396–5.696 Å for p₂ and 0–0.1 for p₃. During the construction of the interpolation polynomial, it was found that the dependence of the spectrum on the internal parameter p₃ is strongly non­linear. Owing to the symmetry, even powers were used in the polynomial expansion for this parameter. Three expansion terms , and were required for precise interpolation within selected limits. The linear terms dominate in the interpolation as a function of the unit-cell constants. As a result, a quite complex polynomial has been obtained,The experimental spectral changes observed from comparison of the Fm3m and Cmcm phases are small. To increase the sensitivity of the fitting procedure, we have compared the experimental and theoretical differential spectra μ_Cmcm − μ_Fm3m and minimized the discrepancy between them. The best-fit results are presented in Fig. 4. The corresponding values of the parameters are a = c = 5.366 ± 0.025 Å, b = 5.411 ± 0.025 Å and δ = 0.025 ± 0.005. The possibility of refining the same set of structural parameters using Rietveld refinement of the X-ray diffraction data (Aquilanti et al., 2003) and XANES links X-ray spectroscopy and crystallography, and can be used for the cross-checking of results obtained by these methods in experimentally challenging cases such as high-pressure research.

Figure 4 Top: experimental As K-edge XANES spectra of InAs in the reference Fm3m phase at 14 GPa (solid line) and in the high-pressure (19 GPa) Cmcm phase (dashed line) are compared with theoretical simulations. Bottom: differences between the As K-edge XANES spectra of InAs in Cmcm and Fm3m high-pressure phases: dots, experimental data; solid line, calculations. The best-fit geometry is used for the Cmcm phase. Adapted from Smolentsev et al. (2006). Copyright IOP Publishing. Reproduced with permission. All rights reserved.

FitIt can also be used to determine the coordination shell distances. In particular, the local structure around dopants and defects in crystals can be investigated. ZnO doped with manganese represents research of this type (Smolentsev, Soldatov et al., 2009).

This example demonstrates the application of FitIt to the analysis of time-resolved XANES data. The multicomponent photocatalytic mixture for hydrogen production with a cobaloxime catalyst (Fig. 5) has been studied and details can be found in Smolentsev et al. (2015). The goal was to determine the structure of the intermediate state of the catalyst. A series of 200 spectra obtained using the pump–sequential probes method (Smolentsev et al., 2014) was the experimental input. Firstly, principal component analysis was applied to determine the number of independent spectral components in the data. We found that only two species contributed to this XANES series. One of them corresponds to the initial state of the catalyst, while the second is a transiently formed intermediate. As an additional benefit, principal component analysis allows extraction of the transient signal (the difference between the spectra of the intermediate and initial states) based on analysis of the whole series of spectra with the best possible signal-to-noise ratio. Taking into account that the spectrum of the initial state was known, the transient XANES (with arbitrary normalization) was extracted without any additional free parameters.

Figure 5 Structure of the cobaloxime catalyst Co(dmgBF2)2(CH3CN)2. C atoms, grey; N, blue; O, red; Co, magenta; B, green; F, yellow. The dmg2− ligands form an equatorial plane; solvent molecules coordinate axially.

FitIt in combination with full multiple-scattering XANES calculations using the self-consistent muffin-tin potential (FEFF8) has been used. The spectrum of the initial cobalt(II) state agrees with the experimental spectrum (Fig. 6, top). A cobalt(I) intermediate has been identified from the shift of the absorption edge to lower energies. Preliminary tests of a few models have shown (Smolentsev et al., 2014) that the structure of the cobalt(I) state is similar to that of the cobalt(II) state, but has (i) some displacements of cobalt out of the equatorial plane formed by the dmg²⁻ ligands, (ii) changes of the distances between the cobalt ion and solvent molecules and (iii) small changes in the bond lengths between the metal centre and dmg²⁻ ligands. These parameters varied during the fitting of the transient XANES. The best-fit results correspond to a model with a 0.08 Å displacement of cobalt out of the equatorial plane, Co—N(solvent) distances of 2.00 and 2.11 Å and a Co—N(dmg²⁻) distance of 1.88 Å. The best-fit transient spectrum is shown at the bottom of Fig. 6.

Figure 6 Top: experimental (solid line) and theoretical (dashed line) Co K-edge XANES of the initial cobalt(II) state of cobaloxime in acetonitrile. Bottom: transient signal corresponding to the formation of a cobalt(I) intermediate. The experimental (solid line) and theoretical (dashed line) calculations for the best-fit model are shown. Reproduced from Smolentsev et al. (2015).