Chapter 6.16. MXAN: a method for the quantitative structural analysis of the XANES energy region

X-ray absorption spectroscopy (XAS) is a powerful method for obtaining both electronic and structural information on the absorbing atom site of different types of matter from biological systems to condensed materials. The low-energy part of the XAS spectrum, from the rising edge up to a few hundreds of electronvolts, the so-called X-ray absorption near-edge structure (XANES) region, is extremely rich in both electronic and structural information, from the oxidation state to the structural details of the absorbing site (overall symmetry, distances and angles; see Benfatto & Meneghini, 2015, and references therein). In principle, an almost complete quantitative recovery of the geometrical structure up to 6–7 Å (Monesi et al., 2005) from the absorber can be achieved from this part of the experimental spectrum if a suitable scheme of fitting is used such as those available for the extended X-ray absorption fine structure (EXAFS) part of the spectrum. For a long time the fitting of XANES data has been an aim of users of this technique, especially in the cases of limited k-range experimental data where a standard EXAFS analysis cannot easily be performed.

A further advantage of using the XANES data lies in the limited effects of the atomic thermal disorder. This can be easily seen by the consideration that any signal associated with the nth multiple-scattering (MS) event can be written as a sinusoidal function with an argument given by kR_tot + F(k, R_i, …, R_n), where R_tot is the total length of the MS path of order n and the F function depends on the three-dimensional geometry of this path. As a consequence, the associated thermal damping factor always contains a term such as exp(−k²σ²) coming from the R_tot part. This is the dominant term and is a kind of Debye–Waller damping factor that is almost equal to 1 in the low-energy part of the spectrum, i.e. for small k values (Benfatto et al., 1989).

The possibility of performing quantitative XANES analysis to obtain a structural determination of an unknown compound can be relevant in many scientific fields, such as very dilute systems, biological systems where the low signal-to-noise ratio and the weak scattering power of the light elements limit the k range of the available experimental data, local investigation of materials under extreme conditions and recently the analysis of time-dependent data from metastable systems with lifetimes of a few picoseconds or even less.

However, the quantitative analysis of XANES spectra presents some difficulties, mainly due to the theoretical approximations needed in the treatment of the potential and the more time-consuming algorithms to calculate the absorbing cross section in the framework of the full multiple-scattering approach (Tyson et al., 1992; Rehr & Albers, 2000). For these reasons, `XANES analysis' is still considered a `qualitative' technique that is used to help in standard EXAFS studies.

Some years ago, Benfatto and Della Longa proposed a fitting procedure, MXAN (MINUIT XANES), based on a full MS theory (Benfatto & Della Longa, 2001), which could extract local structural information around the absorbing atom from experimental XANES data. Since then, the MXAN method has been successfully used for the analyses of many known and unknown systems, yielding structural geometries and metrics comparable to X-ray diffraction and/or EXAFS results (Della Longa et al., 2001; Arcovito et al., 2007; Sarangi et al., 2012).

In this chapter, we present a review of the MXAN method, also describing the new possibilities that are now available in the latest version of the program, such as the analysis of XANES data from time-dependent and structurally disordered systems. An application to the problem of the solvation shell of chlorine in solution is also presented with details.

The method is based on the comparison between experimental data and many theoretical calculations performed by varying selected structural parameters starting from a well defined initial geometrical configuration around the absorber. The calculation of XANES spectra is performed within the so-called full MS approach, i.e. the inverse of the scattering path operator is computed exactly, avoiding any a priori selection of the relevant MS paths (Tyson et al., 1992; Rehr & Albers, 2000). The fit procedure is performed in energy space without the use of any Fourier transform algorithm; polarized spectra can be easily analysed because the calculation is performed using the full MS approach. Details and the definition of the error function R_sq can be found in Benfatto & Della Longa (2001).

A typical fit involves an experimental energy range of about 150–200 eV from the rising edge; applications to several test cases indicates that the best-fit solution is quite stable and is independent of the starting conditions.

The MXAN method is based on the standard MS theoretical approach within the muffin-tin (MT) approximation for the shape of the potential and the so-called `extended continuum' scheme to calculate both the continuum and the bound part of the XAS spectrum. It also uses the concept of complex optical potential based on the local density approximation of the self-energy of the excited photoelectron (Tyson et al., 1992; Rehr & Albers, 2000). The total charge density needed to calculate the whole potential is derived by superimposing atomic self-consistent Hartree–Fock charges derived using neutral or non-neutral atoms.

In the MT approximation it is necessary to define the radii of the spheres surrounding all of the atoms used in the calculation; they are chosen according to the Norman criterion, with some percentage of overlap between the MT spheres. The potential is recalculated at each step of the minimization procedure, keeping the overlapping factor fixed. This parameter controls all of the MT radii and it can be considered as a free parameter of the theory. At the same time it is necessary to define the constant interstitial potential. In the `extended continuum' scheme, the `outer sphere' is not used and the interstitial potential can also be considered as a free parameter of the theory. Both the overlapping factor and interstitial potential can be optimized during the fit procedure and, in the last version of the program, they are inside the structural loop in order to minimize the computing time and to calculate the statistical correlations with the geometrical parameters. It turns out they are small in most cases, with a very weak influence on the structural determination. Clearly, overcoming the MT approximation (Natoli et al., 1986; Hatada et al., 2007) and the use of a self-consistent field (SCF) potential (Ankudinov et al., 1998) is the best way to eliminate the arbitrariness that the use of such free parameters introduces into the calculation. However, the introduction of the non-MT corrections and the use of the SCF potential in a fitting procedure in which the geometrical structure changes at each step of computation is complicated and quite time-consuming. There is also the risk of stabilizing incorrect electronic configurations when the geometry is still far from reality, with the possibility of increasing the likelihood of finding of false minima.

On the other hand, the optimization of both the interstitial potential and the MT radii can be a way to mimic the non-MT corrections and the use of an SCF potential for the whole cluster, giving a good agreement between theory and experiment and an accurate structural recovery. The reader can refer to Benfatto et al. (2005) and Benfatto & Della Longa (2009) for technical details.

The self-energy is calculated in the framework of the Hedin–Lundqvist (HL) scheme (Hedin & Lundqvist, 1970; Kas et al., 2007). The use of the full complex HL potential introduces, in most cases, a significant overdamping at low energies, especially in the case of covalent molecular systems. For this reason, in MXAN we have developed a phenomenological approach to calculate the inelastic losses based on the convolution of the theoretical spectrum, calculated by using only the real part of the HL potential, with a suitable Lorentzian function having an energy-dependent width of the form Γ_tot(E) = Γ_c + Γ_mfp(E), where Γ_c is the core-hole lifetime and the energy-dependent term represents all of the intrinsic and extrinsic inelastic processes (Benfatto et al., 2005). The Γ_mfp(E) function is zero below an onset energy E_s that corresponds in the extended systems to the plasmon excitation energy, and begins to increase from a value A_s following the universal form of the mean free path in solids (Müller et al., 1982). Their numerical values are derived at each step of the fit on the basis of a Monte Carlo thermal annealing procedure. The experimental resolution is taken into account by a further convolution with an energy-independent Gaussian function. MXAN has a total of four parameters that completely control the damping procedure. The user chooses whether to use them or to keep them fixed to some value. Many applications to test cases and unknown systems have shown the reliability of this type of phenomenological approach.

Recently, the MXAN code has been modified to allow the fitting of difference spectra, i.e. signals from the difference of two XANES data sets. It is thus possible to analyse differential transient XAS data, which consist of the difference between the transmission spectra of an unexcited and a laser-excited sample. This approach greatly increases the sensitivity of the data to small changes, and at the same time reduces the influence of systematic errors in the experiment and in the calculation. In this way it is possible to analyse in detail spectra related to atomic configurations having lifetimes of as short as a few picoseconds or even less (Gawelda et al., 2007).

As an example of the potential of MXAN, we report here in brief the study of the structural changes of the iron(II) tris(bipyridine) [Fe^II(byp)₃]²⁺ complex induced by ultrashort pulse excitation (Gawelda et al., 2007). Photoexcitation of [Fe^II(byp)₃]²⁺ by UV–visible light populates the singlet metal-to-ligand charge-transfer (MLCT) states, and is followed by a cascade of intersystem crossing steps through MLCT and ligand-field states, which brings the system to its lowest-lying high-spin (HS) quintet state within <1 ps. At room temperature, this state relaxes to the low-spin (LS) ground state in about 0.6 ns. Through the study of the picosecond Fe K edge XAS data of the HS state, it is possible to determine the structural changes that occur on going from the LS to the HS state. These changes are extracted from a fit of the experimental difference XAS spectra without a priori assumptions.

The laser-pump–X-ray-probe experiments were performed on the micro-XAS beamline of the Swiss Light Source and the sample was in aqueous solution at 25 mM. Details of the experiment are described in Gawelda et al. (2007). The MXAN analysis starts from the fit of the ground-state LS XANES data using the X-ray diffraction (XRD) determination of the crystal as the initial geometry. In the inset of Fig. 1 we report the comparison between the experimental data at the Fe K edge and the calculation (full red line) at the best-fit conditions. The fit has been made allowing changes in the Fe—N bond length; the atoms in the pyridine rings are linked together during the movement. The agreement between theory and experiment is quite good over the whole energy range, and the best-fit conditions correspond to a bond length of R_Fe—N = 2.00 ± 0.02 Å. The XRD structural determination reports a distance of R_Fe—N = 1.967 ± 0.006 Å. The fit of the LS XAS data is the starting point of the structural analysis of the transient XAS data of the HS state. In Fig. 1 we report the comparison between the experimental difference spectrum (HS − LS XAS data) recorded 50 ps after laser excitation and the MXAN calculations (full red line) under the best-fit conditions. There is a very good agreement in the whole energy range, and the best-fit conditions correspond to an increase of the Fe—N bond length, ΔR_Fe—N = 0.20 ± 0.05 Å. This value is in good agreement with density-functional-theory calculations (Lawson Daku et al., 2005).

Figure 1 Comparison between the difference XANES experimental spectrum of [FeII(byp)3]2+ at the Fe K edge (blue points) and the theoretical calculation (red line) at the best-fit condition. In the inset, we report the comparison between the experimental XANES data of the ground LS state (blue points) and the MXAN calculation at the best-fit conditions (full red line).

The problem of how to carry out an ab initio quantitative treatment of geometrical disorder in the XANES energy region is still an open question. The only attempt to solve this problem has been made by use of the so-called `augmented space formalism' (Benfatto, 1995). This approach needs to define the distribution functions of the stochastic variables, for example bond distances, but severe difficulties exist in writing these functions for structural disorder that differs from the Gaussian function. A possibility for overcoming such difficulties is to combine molecular-dynamics (MD) and MXAN simulations (D'Angelo et al., 2006). The proper configurational average spectrum is obtained by averaging thousands of spectra generated from distinct MD snapshots. Each snapshot is used to generate the XANES signal associated with the corresponding instantaneous geometry, and the average theoretical spectrum is obtained by summing all of the instantaneous spectra and dividing by the total number of MD snapshots used. Typically, a snapshot is taken every 500 fs starting from the time at which the system is considered to be equilibrated. The convergence of the average is achieved when the residual function, defined as the difference between the incremental N and N − 1 averaged spectra, is less than 10⁻⁵.

This `dynamic MXAN' analysis (hereafter called D-MXAN) is very sensitive to the conformational sampling, as demonstrated by its applications to several different systems (D'Angelo et al., 2010; Chillemi et al., 2016). An example is given below.

There is a great interest in studying the solvation sphere of halides, mainly chlorine, iodine and bromine, as many recent reviews indicate. Ion solvation is an important study and impacts many different fields from chemistry to biology. Here, we present an application of the D-MXAN method to the study of the solvation sphere of chlorine in water solution via analysis of the Cl K edge. A more extended study is presented in Antalek et al. (2016), in which it was shown that combined MXAN and EXAFS analysis indicates that water molecules around chlorine are found in two discrete structurally coherent but axially asymmetric shells. The reader is referred to that article for experimental and analysis details.

The D-MXAN method is very sensitive to the conformational sampling. The MD calculations are performed with GROMOS version 54A (Schmidt et al., 2007). In order to demonstrate this sensitivity, we show in Fig. 2 the comparison between the experimental data (Fig. 2a, blue line) and three calculations obtained with different MD parameters: (1) Lennard–Jones parameters for the chlorine–water interaction obtained using the SPC/E water model (Berendsen et al., 1987; black curve), (2) the same as (1) with the inclusion of the Encad-shift option that decreases the Coulomb potential (Levitt, 1983; red curve) and (3) the same as (2) with only selected frames (see below; green curve). We tested several Lennard–Jones parameters for the chlorine–water interaction; the best choice was the MD (1) set, which gives similar results to that developed by Reif & Hünenberger (2011). The reproduction of the experimental XANES signal is not very good (black dots in Fig. 2a), with an R_sq of 16.3. Most of the discrepancies are confined to the first 20 eV, an energy region very sensitive to the structural details of the solvation spheres. The use of the Encad-shift option [MD (2); red dots in Fig. 2a] produces an improvement between experiment and calculation, with an R_sq value that decreases to 14.3. A better refinement was obtained by selecting the MD frames that have an R_th, defined asof less than 7 × 10⁻⁴, where m is the total number of theoretical energy points, is the theoretical cross section at the ith energy point for a given MD snapshot and is the theoretical cross section at the same energy point obtained using the geometrical structure at the static best fit.

Figure 2 (a) K-edge XAS spectrum of dissolved chloride is shown as a blue continuous line; the K-edge XAS calculated using D-MXAN from snapshots derived from the molecular-dynamics simulation are reported for MD conditions (1), (2) and (3) (see text) as black, red and green dotted lines. (b) Cl–O radial distribution functions (RDFs) for the first 15 water molecules obtained from MD conditions (1), (2) and (3) shown as black, red and green continuous lines. The coordination numbers N for the three MD conditions are shown as dashed lines.

This procedure produces a strong increase in the agreement between theory and experiment [MD (3); green dots in Fig. 2a], with the R_sq value decreasing to 7.1. The EXAFS part is still well reproduced and we begin to have the two-peak structure with the correct intensity in the very low energy region. To better understand the reason for such a decrease, in Fig. 2(b) we report a comparison between the Cl–O radial distribution functions (RDFs) obtained with three MD options. This plot clearly shows that the better reproduction of the experimental XANES signal is due to a slightly more compressed hydration structure in the region between 3.1 and 4.6 Å, while the structure is identical before and after this distance range. A small increment in coordination number is already obtained with the inclusion of the Encad-shift option (compare the red and black dashed lines in Fig. 2b), but the greatest improvement in the XANES analysis is clearly obtained by an even larger increase in coordination number (compare the green and red dashed lines in Fig. 2b).

In this chapter, we have presented in detail the main ideas behind the MXAN package and two recent applications. MXAN still remains the only package that allows a quantitative analysis of the XANES energy region to derive structural information using a full multiple-scattering procedure. A different approach based on a multidimensional interpolation of XANES calculations obtained using any type of method has been proposed in the literature (Smolentsev et al., 2007).

MXAN was presented in the literature in 2001 and since then has been applied by many groups to the analysis of data from many different samples from biology to solid-state physics. It is almost impossible to summarize all of the obtained results here. For this reason, we have decided to show a new possibility for the use of the MXAN package, a possibility that arises from combination of the program with molecular-dynamics calculations, the D-MXAN analysis. This allows the user to go beyond the `static' view of the standard fitting procedure and derive some dynamical information that is important for characterizing the system under study. At the same time, this method allows the validity of the MD procedures to be tested by direct comparison with experimental XAS data.