ALTERNATIVE STRATEGIES FOR XAFS DATA ANALYSIS USING FEFF AND MATHEMATICA

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1 ALTERNATIVE STRATEGIES FOR XAFS DATA ANALYSIS USING FEFF AND MATHEMATICA 7 Grant Bunker Professor of Physics Illinois Institute of Technology Chicago, IL ABSTRACT The FEFF series of programs for the calculation of x-ray absorption spectra has had a transformative impact on EXAFS analysis because of its accuracy, flexibility, and portability. Its primary use has been in supporting a path by path analysis of experimental data using auxiliary programs such as IFEFFIT, Artemis, SixPack, etc. In this paper several alternative strategies for XAFS analysis that combine FEFF and Mathematica 7 are described. Executable programs are available from the author. INTRODUCTION It has been about 40 years since Stern, Sayers, and Lytle (SSL) presented a seminal series of papers (Sayers et al., 1971; Stern, 1974; Lytle et al., 1975; Stern et al., 1975) that demonstrated that Extended X-ray Absorption Fine Structure (EXAFS) spectra can be physically understood and how it can be used to study the local structure in materials. SSL derived the canonical EXAFS equation in a simple formulation that is still widely used as a guide to interpreting EXAFS spectra. This formalism made several simplifying approximations, among them that of single-scattering: the photo-excited electron was assumed to scatter only once from each neighboring atom. The single-scattering theory of SSL was soon generalized (Lee and Pendry, 1975; Ashley and Doniach, 1975) in order to account for multiple scattering processes, which were experimentally known to be important, particularly for linear and near-linear geometries. However this sophistication came at the cost of much greater mathematical complexity and diminished intuitiveness of the formalism. At that time the theoretically calculated scattering phase shifts and amplitudes were not sufficiently accurate for use in accurately modeling experimental data without the use of ad-hoc empirical corrections. Fortunately the single scattering model permitted the use of experimentally measured amplitudes and phases derived from known standard compounds when analyzing the EXAFS of unknown structures. However this empirical approach was severely limited in cases where multiple scattering could not be neglected. More recently these limitations have been largely eliminated because of marked improvements in theory and the growing availability of inexpensive computational power. SSL pointed out that many-body effects are essential to understanding EXAFS: the photoelectron range is limited to a few Ångstroms by a combination of the core-hole lifetime and the electron mean free path. Over the last two decades sophisticated methods of computing many-body effects have been incorporated into the theoretical programs, and efficient schemes for calculating the multiple scattering expansion in terms of real-space Greens functions were developed. Foremost among these programs are the FEFF series of programs from the group of J.J. Rehr at the University of Washington

2 This document was presented at the Denver X-ray Conference (DXC) on Applications of X-ray Analysis. Sponsored by the International Centre for Diffraction Data (ICDD). This document is provided by ICDD in cooperation with the authors and presenters of the DXC for the express purpose of educating the scientific community. All copyrights for the document are retained by ICDD. Usage is restricted for the purposes of education and scientific research. DXC Website ICDD Website -

3 (Ankudinov et al., 1998; Rehr and Albers, 2000). It is remarkable that the EXAFS, including the multiple scattering and many-body effects, can be expressed in a form isomorphic to the original SSL single scattering equation, with the important difference that the sum over scatterers is replaced by a sum over scattering paths; the complex scattering amplitude F(k) is replaced by effective Feff(k,r), the r dependence stemming from inclusion of spherical wave effects. This is the origin of the name ``FEFF''. FEFF has provided a robust platform for calculation of XAFS, mostly using the scattering path expansion. Modeling on a path by path basis is flexible, but it is not without some problems. The purpose of this paper is to point out some of these problems, and to indicate solutions to them which combine the power of FEFF with the power of Mathematica. These approaches have been implemented by the author, and executable Mathematica programs (Bunker, 2010) can be downloaded from or obtained by request from the author. Mathematica 7 Mathematica (Wolfram Research, 2008) is a powerful and flexible programming language and integrated environment for technical computing that has been continuously developed for more than two decades by a large cadre of mathematicians, scientists, and computer programmers working under the guidance of Stephen Wolfram. It transparently combines symbolic computing, numerical computing, 2D and 3D graphics, robust statistics, and other functionality that is too extensive to describe here. The most recent version, v7, also allows one to trivially program interactive simulations, and provides a much more interactive user interface than previous versions the reader may have encountered. FEFF and Mathematica complement each other: they are both powerful and flexible in different ways. FEFF can calculate x-ray absorption spectra for a vast variety of structures, but it does not process or analyze data. Mathematica can analyze data, with suitable code (see examples), but it knows nothing about XAFS. It also provides robust statistics and confidence interval estimation. The disadvantage of using Mathematica (as opposed to Python or other interpreted languages) is that it is not freely available to everyone, as is open-source software. Although many universities have site licenses for Mathematica, smaller institutions and small businesses may find it prohibitive. It also is subject to some export restrictions. Other options exist, however. If a user merely wishes to execute Mathematica programs, a run-time-only version of the Mathematica environment (Mathematica Player Pro) is available at a much lower cost than the full development system. Finally, the ideas implemented here in Mathematica often can be re-coded into forms that are supported by open source platforms. The symbolic processing program Maxima (derived from Macsyma) is available on an open source basis as part of the Sage computing environment.

4 DATA REDUCTION XAFS data are normally collected as absorption or fluorescence spectra as a function of energy. These are significant only modulo a constant scale factor and slowly varying background. To extract the EXAFS oscillations, chi(k), a standard sequence of operations is normally done: choosing E0, normalize to unit edge step, subtract slowly varying background, weight the data with a suitable power of k to make it into a smooth wave packet (to improve separation of Fourier peaks), and Fourier transform. The author has posted ( example code to automatically carry out all of these operations in a reasonably robust manner (using iterative Fourier Filtering). This sequence of operations is implemented in less than one page of Mathematica code (Bunker, 2010). PATH BY PATH FITTING Popular programs such as IFEFFIT, which is used as a computational engine by Athena, Artemis, (Ravel and Newville, 2005) and other programs, do fit data, on a path by path basis. This approach requires the user to specify path lengths and Debye Waller factors, and possibly other parameters such as energy shifts and cumulants, for every important path. More precisely these are expressed as deviations from a reference structure. The problems is there may be far too many parameters to realistically fit, because the information content in the XAFS data is limited to about 2/Pi*deltaK*deltaR, where DeltaK and DeltaR are the ranges in K and R space over which the data are substantial. In practice this may limit the number of parameters that it is justifiable to float to The strategy taken by IFEFFIT is to express the parameters for each path (called "numerical path parameters") in terms of a much smaller number of fit parameters. It is up to the user to figure out the correspondence between them, and describe that correspondence mathematically ("math expressions") in a Fortran-like syntax in the IFEFFIT input file. If there are dozens or hundreds of paths this can be both very tedious and prone to error. AUTOMATED GENERATION OF MATH EXPRESSIONS FOR IFEFFIT Mathematica 7 can be used to automate the generation of such math expressions as shown in the simple example below. In four lines of code the program defines the structure in terms of parameters a,b,c; generates an interactive plot of the structure, with sliders to change the parameters; and defines a function that, given sequence of scatterers, returns a symbolic expression for total the path length, conveniently in Fortran form as required by IFEFFIT.

5 Figure 1 - Screen shot from Mathematica 7 program demonstrating the automatic generation of Math Expressions for input to IFEFFIT. Complete code is shown as well as example output for the scattering paths and The function R can quickly generate math expressions for any path.

6 Furthermore, Mathematica provides a convenient function Splice["filename"], which replaces any Mathematica code that it finds in the text file filename contained between delimiters <* *> with the result of its evaluation. This functionality can be used to greatly simplify the generation of input files for IFEFFIT-based programs by splicing appropriate Fortran code into a suitable template file. AUTOMATIC KERNEL GENERATION FOR REGULARIZATION In systems for which the single scattering approximation can be used, the mapping from the pair distribution function g(r) to chi(k) is a linear one, i.e. it is represented as a linear integral equation of the Fredholm type. Sampling on finite grid reduces the problem to a set of linear equations. Unfortunately the inverse mapping from chi(k) -> g(r) is unstable; rendering it stable requires use of regularization methods (or similar). The earliest implementations of this were by Tikhonov regularization (Babanov et al. 1981a; Babanov et al. 1981b; Vasin and Ageev (1995); Yang and Bunker 1996; Khelashvili and Bunker 1999; Babanov, Yu.A., Vasin, V.V., Ageev, A.L., Ershov, N.V. (1981)). Subsequent work demonstrated some advantages to iterative Landweber regularization (Khelashvili 2001; Rossberg and Funke 2010). Both approaches require a kernel A(k,r), which essentially pre-computes the single scattering chi(k) for a specific absorberscatterer pair for all relevant distances r. Fortunately a library of such kernels can be generated for various absorber-scatterer pairs, so that effort has to be done only once, if there is sufficient chemical transferability. Mathematica can be used to generate such kernels automatically, by generating a sequence of input files corresponding to different interatomic distances, executing FEFF, importing the results, combining all of the data into a single array, adding descriptive header fields, and exporting the result. Mathematica 7 code to automatically build kernels is available from the author; concise example code to implement iterative Tikhonov regularization is posted on BEYOND PATH BY PATH FITTING Despite the flexibility of the path by path fitting approach, it is not without some problems. There are an infinite number of multiple scattering paths, and FEFF does not calculate contributions from all of them, only the ones that it estimates are important. The importance of a path depends strongly on molecular geometry, particularly when forward scattering (focussing) is important, which occurs when three or more atoms are in a linear or near-linear configuration. When optimizing the hypothetical geometry during fitting it may happen that a path that was considered negligible in the reference geometry becomes important in a modified hypothetical configuration. In the path by path approach in which deviations from a reference structure are considered assumed to be small, such a situation could give rise to serious errors. Another approximation that is usually made but may be inadequate in some circumstances, is accounting for disorder solely in terms of Debye-Waller factors for each path. Although this approximation suffices for bond length variations (particularly when cumulants are included), it does not properly account for a configurational average

7 over scattering angles. The scattering factors are strongly dependent on scattering angle. Thermal motion and static disorder can introduce substantial variations in scattering angle, and the resulting spectra can be sensitive to such variations (Alberding and Crozier, 1983). This potential problem is a consequence of excessive reliance on a single reference structure. A simple remedy for this is to perform a FEFF calculation for each hypothetical configuration generated in the fitting process. FEFF then will estimate the importance of each scattering path for every hypothetical structure. This approach is time consuming, but it does eliminate a class of potential problems. Furthermore it obviates the need to define math expressions for each path. The structure can be described in terms of a small set of meaningful variables, as is the basic notion of IFEFFIT (Newville, 2001), but the tedious and time consuming description in terms of numerical path parameters becomes unnecessary. This approach also makes possible the determination of some structural information by fitting the XANES. An example of this approach is described by Dimakis and Bunker (2009a). Heme groups in hemoproteins contain 25 or more non-hydrogen atoms. Because of strong multiplescattering effects, they are all important, and floating the interatomic distances, bond angles, and dihedral angles naively would require far too many parameters. However the conformational distortions that occur in proteins are generally low-energy ones, which suggests that describing the distortions in terms of a handful of low frequency normal modes may suffice. This parameterization was proposed by Jentzen et al (1995) and was termed Normal Coordinate Structural Decomposition. Typical deformation is shown in figure 2. This approach was used to parameterize the structure for the purposes of XAFS fitting by Dimakis and Bunker (2009a). Figure 2: heme coordinates and typical low energy deformation, from two perspectives. These are readily animated within Mathematica 7.

8 For small structures (e.g. 10 atoms) structures each function evaluation takes seconds, and a full fitting run may take hours on a typical 2010-era desktop computer. Performing full multiple scattering calculations of XANES for a large cluster of atoms can take many minutes or hours. If a cluster of computers were employed, with the independent function evaluations (each a FEFF job) running on separate computers, this could reduce the fitting time to minutes, or for difficult XANES calculations, to hours. It is also possible to combine other programs (e.g. for full-potential XANES calculations) in the same manner, to complement FEFF. AB INITIO DEBYE WALLER FACTORS It has been shown (Dimakis and Bunker 1998; Dimakis et al. 1999; Dimakis and Bunker 2001, 2002, 2004, 2005; Bunker et al. 2005; Dimakis and Bunker 2006; Dimakis et al. 2008, 2009b) that DFT (Density Functional Theory) can provide useful estimates of thermal Debye Waller factors in metalloproteins as a function of geometry and temperature. In this approach the DWFs are computed for each multiple scattering path, and tabulated for use in EXAFS data analysis. This requires a good deal of computational effort but it only has to be done once. These vibrational calculations are combined with FEFF's electronic scattering calculations in a robust fitting procedure using the Differential Evolution fitting algorithm (Storn and Price, 1997). This provides a robust "hands-free" approach to data analysis for metalloproteins (Dimakis and Bunker 2004; Bunker et al, 2005). Mathematica code to implement differential evolution algorithm is available by request from the author. An alternative approach Poiarkova and Rehr (1999) uses dynamical matrices produced by DFT calculations, and either the equation of motion method or recursion method, both of which are implemented within FEFF8, to calculate DWFs for each path. In this case the difficulty is automatically recomputing the dynamical matrices as a function of geometry, because the force constants depend on the geometry, which changes during the fitting process. In principle doing this could accomplished by calling an external program in a fitting algorithm encoded in Mathematica, or another language, but to our knowledge this has not yet been done. CONCLUSION We have described several alternatives to the path by path fitting approach that avoid several basic pitfalls in XAFS data analysis, are less time consuming for the data analyst, and less sensitive to user error. Although the "holistic" fitting approach is more computeintensive, the use of parallel computing can make it practical on a routine basis. When implementing these alternative approaches Mathematica 7 has been found to be a uniquely powerful complementary tool to FEFF8.

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