research papers
FDMX: extended Xray absorption fine structure calculations using the finite difference method
^{a}School of Physics, University of Melbourne, Australia, ^{b}Université Grenoble Alpes, Institut NÉEL, F38042 Grenoble Cedex 9, France, and ^{c}CNRS, Institut NÉEL, F38042 Grenoble Cedex 9, France
^{*}Correspondence email: chantler@unimelb.edu.au
A new theoretical approach and computational package, FDMX, for general calculations of Xray absorption fine structure (XAFS) over an extended energy range within a fullpotential model is presented. The finalstate photoelectron wavefunction is calculated over an energydependent spatial mesh, allowing for a complete representation of all scattering paths. The electronic potentials and corresponding wavefunctions are subject to constraints based on physicality and selfconsistency, allowing for accurate absorption cross sections in the nearedge region, while higherenergy results are enabled by the implementation of effective Debye–Waller damping and new implementations of secondorder lifetime broadening. These include inelastic photoelectron scattering and, for the first time, plasmon excitation coupling. This is the first fullpotential package available that can calculate accurate spectra across a complete energy range within a single framework and without fitted parameters. Example spectra are provided for elemental Sn, rutile TiO_{2} and the FeO_{6} octahedron.
Keywords: Xray absorption; finite difference methods; fine structure; XAFS; FDMNES; FDMX.
1. Introduction
Xray absorption fine structure (XAFS) refers to the oscillations in the energydependent photoabsorption coefficient of a condensed matter system, commonly seen at energies up to a few hundred eV above an ionization edge. These oscillations are the result of selfinterference of photoelectron excitations, which have scattered elastically from one or more local atoms in the material. The precise form of an
spectrum is directly determined by the spatially dependent elastic and inelastic electron scattering coefficients of the material, and therefore is a function of the complex electronic potential within a critical region near the ionized atom.The most obvious determinant of this potential is the position of neighbouring atoms, and particularly their associated electron densities. This makes
an excellent probe of local molecular structure, not only for simple elements and solids but also for large compounds, aqueous samples, gases and amorphous materials. The physical structure around a central atom can be probed by tuning the Xray energy to be near an innershell for a specific element, and in this way coordination numbers, crystal groups and bond lengths can be measured routinely for materials which do not lend themselves to effective study with other crystallographic methods.The relationship between i.e. photoelectron energies less than 60 eV), the spectrum is highly sensitive to minor changes in the potential that may be associated with changes to ionization state, and bonding effects. This XANES region is also very sensitive to artificial structures that commonly occur in many theoretical models, meaning that to accurately quantify XANES spectra a fullpotential model such as the finite difference method (FDM) or linearized augmented planewave (LAPW) method is required.
and the local complex potential has many other significant consequences. At energies very close to the (Such methods have seen longstanding use in XANES calculations, but have been less popular for
analysis than multiplescattering approaches, which utilize approximated and effective potentials in order to consider photoelectron scattering on a pathdependent basis. Multiplescattering approaches are substantially more efficient than fullpotential models, and have advantages with respect to fitting and analysis, as the contribution to the spectrum of individual scattering paths can be readily quantified. Therefore, explicitly pathdependent effects such as thermal motion and, to a lesser extent, could be modelled more robustly. These issues tend to be more significant for higher photoelectron energies, and have meant that fullpotential modelling has mostly been restricted to XANES spectra.Recent advances, however, have demonstrated that the FDM can be utilized for highenergy calculations using appropriate implementation of thermal and lifetime broadening effects, valenceshell contributions, and a sufficiently highprecision representation of the electronic and exchangecorrelation potentials within a small spherical cluster (Chantler & Bourke, 2014a; Bourke & Chantler, 2010a). This has enabled the development of a new package, presented here, capable of robust computation of Xray absorption spectra ranging from below an up to photoelectron energies of several keV, corresponding to the highenergy atomlike photoabsorption limit.
2. The Finite Difference Method for XAFS
The FDM is a common mathematical procedure for evaluating the solution to differential equations by approximating local derivatives over a discretized grid. At each grid point, these derivatives are linked together via a series of linear equations, enabling the determination of function values at any point provided appropriate boundary conditions are chosen. With respect to XANES or calculations, this technique is employed to solve the Schrödinger equation, facilitating the determination of electron wavefunctions and subsequently the transition matrix elements for photoelectric ionization.
The first application of the FDM to this problem was in the Finite Difference Method for NearEdge Structure (FDMNES) package (Joly, 2001). This package has become one of the foremost computational tools for XANES analysis due to its physical representation of electronic potentials for lowenergy spectra, and robust applicability to finite molecular structures.
Here we present the Finite Difference Method for XAFS (FDMX) package, a significant enhancement built upon the original FDMNES in order to calculate extended spectra in a similarly robust and physical manner. As in FDMNES, FDMX calculations treat the problem of in a relatively general way, starting from the basic need to determine the optical transition matrix elements which, in the quadrupolar approximation, may be written as
where k is the photon wavevector polarized in the ∊ direction, and and are the initial and final states for the absorbing electron. The transition amplitudes are then summed to give the absorption σ following
where α is the finestructure constant, the energy of the incident photon, and E_{f},E_{g} the final and initial state energies. The initial state is that of an electron bound to an atomic core orbital, and is approximated via a relativistic Dirac–Slater model (Rosen & Ellis, 1975). The final state is calculated from the cluster potential using the FDM.
The FDM solves a large number of simultaneous linear equations linking the values of the wavefunctions , and potentials V_{i}, at points i in a defined grid in real space. The Laplacian operator needed to solve the Schrödinger equation is approximated using a fourthorder polynomial and, in the case of a cubic grid, may be written as
Here = + or −, and are the first and second nearest neighbouring grid points to i in the direction , and h is the distance between neighbouring grid points. A hexagonal mesh is also used to more efficiently model crystal groups with corresponding hexagonal symmetry. In that case, there are six first and six second neighbour points (instead of four) in the plane perpendicular to the threefold axis. The corresponding Laplacian operators are then multiplied by 2/3 compared with the cubic mesh. Of course, the operators for the neighbour points along the threefold axis are not modified. With our approximated Laplacian, we write the nonrelativistic Schrödinger equation for a nonmagnetic material in discretized form as
where again refers to wavefunction values at grid points neighbouring i. The values of the wavefunction are initially determined for all points i following a firstorder estimate of the potentials V_{i}. These are approximated by the Coulomb potential induced by the groundstate electron densities plus an exchangecorrelation potential, by default approximated using the Hedin–Lundqvist theory (Hedin & Lundqvist, 1971; Joly et al., 1999). Electron densities are initially approximated via an atomistic Dirac–Hartree–Fock algorithm, which is highly accurate for the majority of bound electrons contributing to the spectrum, particularly including the core of the absorbing atom. The wavefunctions and potentials are then iterated in order to converge to a selfconsistent result within the discretized regions of the spherical cluster. In this fashion the total electron number within the cluster is constrained to match the sum of groundstate contributions from each atom.
The code also provides the possibility of relativistic calculations which increase the complexity of the equations slightly with a spinorbit term depending on the gradient of the potential and doubling the basis with spin up and spin down wavefunction components. What is presented here therefore applies also to relativistic and magnetic materials.
The FDM algorithm is applied only in an interstitial region between atoms. In regions close to atomic cores, the shape of the potential demands prohibitively high grid densities, and so instead a model is used in which the potential is approximated as symmetric, and the wavefunction is evaluated as a sum of spherical harmonics. These wavefunctions are then matched via continuity conditions to the values of the wavefunction at the edge of the atomic cores, which are typically around 0.6 Å in radius. The code also optionally offers the possibility of using a nonspherical potential inside this sphere, giving a set of expansion of radial wavefunctions depending on both l and m quantum numbers. So far we have not found cases where this was necessary. In a similar manner, wavefunctions outside of the spherical cluster (usually centred at the absorbing atom) comprise Neumann and Bessel functions.
The radius of the spherical cluster used for the calculation is chosen such that contributions from atoms lying a greater distance from the absorber would be expected to be negligible. A typical choice for the cluster radius may be around 6–8 Å at low photoelectron energies, but for extended
these radii may be reduced due to the effects of and thermal motion.It is useful to define h as the distance between neighbouring grid points, the value of which ultimately determines the precision of the representation of the potential. For XANES calculations and FDMNES, it is usually sufficient to use a constant value for h of around 0.25 Å. In FDMX, however, h is reduced as a function of photoelectron energy in order to maintain convergence of the wavefunction across the entire spectrum.
3. Thermal effects
FDMX implements thermal effects via a Debye–Waller formalism based on the correlated Debye model of Beni & Platzman (1976). For explicitly pathdependent theories, it is common to parameterize thermal motion via an array of pathdependent effective isotropic thermal parameters (ITPs) , corresponding to the meansquare relative displacements between the photoabsorbing atom and its nearby neighbours. Higherorder ITPs may also be used for multiplescattering paths when the scattering atoms are not arranged colinearly (Rehr et al., 2009). However, it is not currently practicable for any model to treat these parameters independently, and so an effective ITP is used, which may be based on the dominant scattering path. For most structures, this approximates to the shortest or most degenerate scattering path. The data can certainly be sensitive to multiple ITPs from dominant paths.
The ITP for any given scattering path is given in terms of the mean square relative displacements of the absorbing atom, , its neighbour atom and the displacement correlation function , following
u_{0} is the instantaneous displacement of the photoabsorbing atom from its equilibrium position, u_{j} is the same for some neighbouring atom j, and R_{j} is a unit vector pointing from the photoabsorber to atom j. The displacement of atoms within the absorbing material arises due to the propagation of resonant phonons of energy , where is the phonon momentum and λ is the polarization index. In the special case of a monoatomic crystal, one can write the mean square relative thermal displacement in terms of these phonon resonances following (Beni & Platzman, 1976)
where m is the atom mass, k_{B} the is a unit vector in the direction of polarization, and T is the temperature of the material. A value for T may be provided by the user of FDMX, or will otherwise default to 298 K. In general, determination of the phonon spectrum can be performed explicitly via experiment (Fornasini et al., 2004) or theory (Vila et al., 2007), or may be estimated using an Einstein, Debye or more complex model. In FDMX, a Debye model is used, which leads to the following expression for the ITP (Greegor & Lytle, 1979),
, and q_{D} are, respectively, the Debye temperature, frequency and wavenumber. These parameters are related by
where V is the mean volume per atom in the material. The D_{n} parameters are definite integrals given by
This formalism enables a sensible estimate of the ITP of the dominant scattering path j, and hence a sufficiently accurate value for the effective ITP in many applications. In the highenergy limit, for a monoatomic cubic structure, it has been shown to approach the correct experimental value from crystallography (Tantau et al., 2015). The ITP is implemented within FDMX by a direct adjustment to the calculated absorption following
where, from standard k is the photoelectron wavenumber, is the total and is the associated atomlike background contribution from the ionized shell, discussed in §5. For nearedge studies and complex molecular structures, typically provides a good estimate for the thermal contributions despite approximations in the derivation of its value. Errors in the final spectrum arising from these approximations tend to be quite small. In particular, at high photoelectron energies the impact of inelastic electronic scattering processes tends to be more significant, while errors from the thermal contribution in the XANES region are typically much smaller than those that are introduced by the use of an incomplete or heavily approximated electronic potential (Glover et al., 2007).
notation,Nevertheless, it is possible, and may often be appropriate, for the user to provide their own values for or for for particular cases, as is common with other theory packages. This can be done using the DWfactor or TDebye keywords. Such extra inputs are particularly recommended for lowtemperature cases where zeropoint motion may be significant, or for samples with high static disorder, which similarly may be modelled via an additive contribution to the effective ITP.
4. Inelastic photoelectron scattering
While it is the elastic (coherent) scattering of the excited photoelectron that is responsible for localized region near the absorbing atom (Bourke & Chantler, 2010b). The rate of determines the effective size of the region being probed by and in this application is most conveniently quantified in terms of the electron inelastic (IMFP). The IMFP is formally defined as the mean distance travelled by an electron of a given energy between successive inelastic collisions (Powell & Jablonski, 2009).
spectra, it is the presence of that imparts its most useful property, a preferential measurement of the electronic and physical structure in aMuch progress has been made in recent years in the development of theoretical determinations for the IMFP, particularly in terms of optical data models (Sorini et al., 2006; Denton et al., 2008; Bourke & Chantler, 2012; Da et al., 2014). These models utilize various algorithms to generalize known optical scattering data into electron scattering data, typically within statistical and independent oscillator approximations (Tung et al., 1979), and originally gained popular use with the advent of the Penn algorithm (Penn, 1987). FDMX implements a new approach, but follows a more general formalism including coupling between electron excitation channels. The approach, known as the coupledplasmon model, is presented in full elsewhere (Bourke & Chantler, 2015), and briefly summarized here.
Within the first Born approximation, the et al., 2012), commonly known as the energy loss function (ELF), i.e. ELF = = . As such, the probability of an energetic electron undergoing a scattering event and depositing energy and momentum into a material is proportional to the integral of this loss function over all applicable energy and momenta for that electron. We therefore express the electron IMFP, , as (Tanuma et al., 1991)
for stimulated electronic transitions is proportional to the imaginary part of the negative inverse of the of the material = + (Nikjoowhere E is the electron energy, E_{F} the a_{0} the and the momentum limits are determined kinematically. Therefore it is only necessary to determine the electron ELF for the material, which can be done via a generalization of the optical ELF, , corresponding to the lowmomentum limit. This definition is due to the relative insignificance of the photon momentum compared with a propagating electron, and allows us to generalize equation (12) in terms of some externally determined optical ELF (Bourke & Chantler, 2015),
The dielectric function is a Mermin function (Mermin, 1970), and this expression can uniquely describe the evolution of electronic excitation channels with increasing momentum. It includes a broadening term , the inverse of the lifetime of excitations, and in this formalism is determined selfconsistently in terms of the electron IMFP and excitation group velocity . Θ is a Heaviside step function, δ a positive infinitesimal, and N is an iteration index, which typically enables the formula to achieve convergence when N is set to a value higher than 3.
This approach is robust, selfconsistent and produces IMFP values in better agreement with experiment than other techniques (Bourke & Chantler, 2015), but requires an input of , corresponding to the optical loss function of the material. This spectrum can be determined experimentally, or via density functional packages such as WIEN2k (Blaha et al., 2001; AmbroschDraxl & Sofo, 2006). For highly accurate analysis of XANES spectra, or for complex molecular structures, it is recommended that FDMX users provide values for this function where possible, using the ELFin input keyword. This enables a fully selfconsistent implementation of effects, including plasmon coupling effects, not currently available in other packages.
Users can also provide IMFP data via the IMFPin keyword. In the absence of optical ELF or IMFP data, FDMX will utilize tabulated data from Tanuma et al. (2011) where available, which approximately corresponds to a firstorder implementation of equation (13) (i.e. with N = 1). If data are not available, an IMFP estimate will be made according to the TPP2M equation (Tanuma et al., 2011). These latter estimates are less robust in the XANES region, below around 60 eV, however as with the thermal processes tend to contribute less error than the use of an incomplete or approximated potential, as is common with other packages.
5. Lifetime broadening and background absorption
The inelastic photoelectron scattering contributes a broadening function to the
spectrum, corresponding to the photoelectron lifetime = . It is therefore simply related to the electron IMFP followingIn addition to the photoelectron lifetime broadening, the corehole relaxation also contributes a broadening . Here we adapt an adiabatic approximation, and so consider only an energyindependent value for the relaxation lifetime = . As these processes both possess Lorentzian line widths, we can sum them directly to obtain the energydependent Lorentzian broadening of the spectrum,
Values for the hole relaxation broadening may be provided by the user with the Gamma_hole keyword, but are otherwise included by default from the tabulations of Scofield and Kostroun et al. for Z = 21–50 (Scofield, 1969; Kostroun et al., 1971), and from Bambynek et al. for Z = 51–100 (Bambynek et al., 1972). Errors in spectra associated with the corehole lifetime are typically observable only for very low photoelectron energies, below 10 eV. Note that, especially when simulating highresolution experiments in fluorescence mode (HERFD, RIXS), the user can optionally reduce the value.
With the oscillatory component of the spectrum determined, inclusive of broadening, thermal and static disorder contributions, the atomlike components of the spectrum must be properly quantified in order to isolate the Kshell contribution, and the contribution from less strongly bound electrons (i.e. the L shells, M shells, valence electrons etc.).
spectrum for robust analysis. This includes the background absorption both in terms of theThe background Kshell contribution to the spectrum, , is estimated via explicit calculations of atomic spectra within the FDMX package. These are performed with small clusters, approximately 1.5 Å in radius, and extremely high grid density in order to attain highly accurate background functions for all possible absorbing elements with Z = 21–92. These functions are tabulated and output as part of standard FDMX calculations, allowing users to readily extract the oscillatory part of the spectrum for analysis. This method is highly advantageous over traditional spline techniques as it does not introduce artificial lowfrequency structures into the spectrum. In some extreme cases, however, care must be taken within a few eV of the due to the potential for solidstate bonding structure to significantly affect the and thus affect the extracted for k 1.5.
Contributions to the , 2000). These tabulations are calculated via the FFAST package, which uses the multiconfigurational Dirac–Hartree–Fock technique for the selfconsistent evaluation of relativistic atomic wavefunctions, and a local density approximation (LDA) for the treatment of the Coulomb and exchangecorrelation potentials. The photoelectron absorption from FFAST is typically accurate to well within 1% for energies that are not close to an which is almost always the case for the background contribution from other electrons. Contributions from photon scattering are not included in FDMX, so that the output is strictly the photoabsorption, rather than total attenuation.
from less strongly bound electrons are incorporated using atomic form factor tabulations (Chantler, 1995A final contribution of background absorption is sometimes necessary in order to directly compare theoretical results with experiment spectra on an absolute scale. This is due to the edgejump discrepancy, sometimes called the triangle effect (Tantau et al., 2015), that is currently common to all photoabsorption computations (Chantler & Bourke, 2014a). By default this contribution is not included in FDMX, but the user may add an additional background absorption using the Expntl or Victoreen keywords. The contribution will then be added following
where is the total photoabsorption coefficient and is a smooth function of the form = Aexp(EE_{0}/B) if Expntl is used (Chantler & Bourke, 2010), or = A(E_{0}^{ 3}/E^{ 3}) + B(E_{0}^{ 4}/E^{ 4}) if Victoreen is used (Victoreen, 1949). In both cases, E_{0} is the absolute edge energy and A and B are parameters.
These functions allow a smooth scaling of the background absorption without artificial oscillatory structures as with a cubic spline, and tend to produce quite good results even for materials where the offset is large (see, for example, Fig. 1). When an exponential function is used, A controls the edge jump scaling and B controls the rate of convergence to the atomic result. For a Victoreen function, A+B gives the scaling and the ratio A/B gives the rate of convergence. Typically for a Victoreen function, a negative value is required for A.
6. Computational convergence
One of the key limitations of the previous implementation of the FDM for the calculation of finalstate wavefunctions was the inability to sufficiently sample the potential at high energies within a feasible computational framework. Essentially, the grid density required for convergence became increasingly large at high energies, leading inevitably to either unstable results or impractical calculations (Bourke et al., 2007).
FDMX addresses this issue in a number of ways. Firstly, the grid density now defaults to a variable parameter that increases stepwise as the energy is increased, in order to ensure a proper representation of the electronic potential, and acceptable convergence of the finalstate wavefunction. Typically, this can be interpreted as a representation that results in errors in the calculated absorption of 0.1% or less prior to the introduction of broadening contributions. The grid density, and the rate at which it changes, is adjustable using the adimp keyword in the event that satisfactory convergence is not achieved for a particular molecule. Secondly, the radius of the computational cluster is also changed with increasing energy, in order to ensure that the number of grid points considered remains practical.
The reduction of the cluster size, which may be controlled with the Radius keyword, may result in the loss of outer coordination shells from the calculation at sufficiently high photoelectron energies. In rare cases this may result in the loss of oscillatory structure, and so the user must be conscious of this potential drawback. However, in the vast majority of cases, the oscillatory structure will remain unchanged due to the broadening contribution from the electron IMFP, which will increasingly filter contributions from the more distant neighbouring atoms (Bourke et al., 2007). Therefore it is typically safe, and substantially more efficient, to use default values to define the form of the cluster.
In addition to these measures, the most recent implementations of the FDM, including FDMX, feature improved functionality through the use of more efficient data management (Glover et al., 2007) and the employment of the MUMPS Fortran libraries for the manipulation of large data arrays (Amestoy et al., 2001). The MUMPSrelated features have been recently implemented for FDM calculations by Guda, Soldatov et al., and enable calculations of absorption spectra up to 30 times faster, and with substantially less memory use, than was previously possible (Guda et al., 2015; Amestoy et al., 2006). These improvements, coupled with the dynamic cluster parameters, allow routine determinations of extended spectra up to the smooth atomlike region within the space of an hour for simple materials with high symmetry, or a few hours for arbitrary molecular structures.
7. Example spectra
To demonstrate the use of the new package, we present example spectra for elemental tin, the mineral rutile (TiO_{2}) and the octahedron FeO_{6}. In all cases, default options were used for physical parameters including electron IMFP, thermal effects and background absorption. Computational parameters were varied in some cases in order to ensure convergence, and are given in Table 1. For TiO_{2} and FeO_{6}, the results are compared with those using the multiplescattering technique, implemented within the same package and also within FDMNES.

7.1. Metallic tin, Sn
We firstly consider elemental tin, the attenuation spectrum for which is shown in Fig. 1. It is compared here with high absolute accuracy experimental data using the standard βtin allotrope (de Jonge et al., 2007).
The resulting XANES and FDMX calculations for solid metals (Chantler & Bourke, 2014a; Tantau et al., 2015). The only significant difference appears in the height of the peak at 83 eV, most likely due to an overestimate of the electron IMFP in the default data (Bourke & Chantler, 2012). A full computation of the IMFP using the selfconsistent model of equation (13) provides more accurate results, and is an optional (and unique) feature of FDMX. This result using the FDM is similar to that using a full multiplescattering (FMS) approach within the same package (including the additional developments of IMFP, thermal broadening and holewidths as discussed above), as approximations in the electronic potential are less critical for calculations involving elemental solids.
spectra are in very strong agreement with the experimental data, as has previously been shown with prototypeThe dotted red curve in Fig. 1 shows the calculated spectrum with an added exponential scaling using the parameters A = 0.18 and B = 780 eV. A similar background is obtainable for elemental Sn using a Victoreen function with parameters A = −4.95 and B = 5.13. Direct comparison with experimental spectra, and subsequent quantitative analysis, is therefore possible using a simple background function that does not oscillate, even over an extended energy range. This is in contrast with the common cubic spline functions usually used in alternative packages. With this background, we find that the discrepancy from experiment is 63.5 over the energy range spanning −10 eV to 500 eV relative to the edge. This is an excellent result given the high absolute accuracy of the experimental spectrum (typically of order 0.1%), lack of optimization of physical parameters, and range of energies that include the edge, XANES and regions. The average discrepancy between theory and experiment over this range is 0.77%.
7.2. Rutile, TiO_{2}
Fig. 2 shows the calculated spectra for the mineral rutile, TiO_{2}. In this example we make a direct comparison between the result using the fullpotential FDM calculation and an equivalent FMS calculation also performed within the FDMX package. As with Sn, default settings are used for all physical parameters for both the FDM and FMS models, while the computational parameters are varied in accordance with Table 1. For the FMS case, grid density parameters are not applicable as a muffintin approximation is used, corresponding to a constant potential in the interstitial region.
The spectra resulting from the two models are not convergent for this material even for the majority of the _{2}, this meant that the grid spacing h needed to be reduced below its default value. The clear conclusion is that the FDMX package and approach has advantage in nearedge and central regions in avoiding mathematical artefacts due to flat interstitial potentials and discontinuities.
region, with acceptable agreement only apparent after 400 eV, if at all. As the calculations were performed with identical systems within the same package and using the same parameters, all of the observed differences are necessarily due to the approximated form of the potential in the FMS calculation. This FMS potential follows a muffintin form, requiring a constant value in the interstitial region, leading to discontinuities in the gradient of the potential and, as this is a compound, in its value at the interface between the regions of the calculation. Strictly speaking, the FDM potential is also approximated due to the finite density of grid points; however, care has been taken to ensure convergence has been reached. For TiOThis result is highly significant in terms of the choice of model for et al., 2012), and is, for example, almost three times greater than the absolute discrepancy seen in the previous section between theory and experiment for Sn.
calculations with compound materials, especially those with molecular or cluster sites with significant variations of The mean variation between the two spectra is 2.0% between −10 eV and 500 eV. This is significantly higher than the absolute accuracies now obtainable even for complex molecules using precision techniques (ChantlerIn the XANES region, the differences between models are particularly strong, as is expected due to the strong impact of discontinuities in the shape and value of the electronic potential on the reflection of the lowenergy electron wavefunction. Indeed, for energies below 60 eV, the difference between FDM and FMS spectra is 4.6%. These discontinuities can therefore lead to incorrect conclusions regarding the broadening mechanisms and longrange bonding environment in the studied material.
It is especially notable that the muffintin potential can impact
structures in both the XANES and regions of the spectrum. Although the impact is strongly materialdependent, it is critical for quantitative analysis that such errors be minimized, and therefore a fullpotential modelling is strongly advised.7.3. FeO_{6} octahedra
An octahedral structure FeO_{6} is modelled in Fig. 3, where we see that implementations of the FDM and FMS models within FDMX can both be useful for high energies. Over the energy range plotted, the difference between the two theoretical models is only 0.86%, which is partly aided by the relatively weak signal common to disordered structures without longrange order or highly degenerate scattering paths. Nevertheless, key differences between the predicted peaks between the two models are observed, which may prove significant for detailed analysis with highly accurate experimental data.
Most particularly, however, we are here interested in comparing the performance of the two models with experimental measurements in the XANES region. We therefore provide a closer view of the FeO_{6} spectra in Fig. 4. Here we observe a small offset in the edgejump, and an offset in the energy of the edge itself. While the former issue may be similar in nature to the effect seen in Sn, the latter issue may be related to small errors in the in the model predictions, approximations from the Hedin–Lundqvist exchangecorrelation potential, or properties of the lowenergy band structure of the material that cannot be interpreted within an atomistic model.
The forms of the spectra themselves are an important demonstration of the value of the use of a fullpotential model, which in this case is not apparent from a direct comparison of discrepancy values. Up to 70 eV, the FDM result deviates from experiment by an average of 6.9%, while the FMS deviates by only 5.6%. The theories deviate from one another in this range by an average of 2.0%. In addition to the effects already mentioned, some of the discrepancy with experiment is also due to the model itself, which in this case uses an explicit FeO_{6} octahedron, while the experimental data were a more complex structure, including a possible second oxygen shell and effects of the hydrogen atoms, with a formal charge of Fe^{2+} and recorded at the FAME beamline, ESRF (Testamale et al., 2009). Hence we expect to see chemical shifts, small amplitude changes and changes of structure associated with the nonnearest neighbours. In other words, this example is expected to require structural development before achieving perfect agreement with experimental data.
The FDM theory does, however, produce extra structures that are not predicted by the FMS theory. In particular, the FDM theory is able to replicate the characteristic peak at around 25 eV from the
seen in the experimental result. A peak of this relative magnitude and at this energy is unlikely to be introduced into an FMS calculation by any of the effects discussed, and is therefore an important indicator of the impact of fullpotential modelling.Sometimes such features can be obscured or distorted by approximations in the broadening mechanisms, and most particularly the photoelectron IMFP. An overestimate of the IMFP can lead to extraneous structures that in practice are broadened out in the experimental measurements. This can be expected for molecular structures that do not have a well defined optical ELF (Bourke & Chantler, 2012), and is most significant in the nearedge region (Bourke & Chantler, 2010b).
Similar difficulties exist within FMS theories, but may be overlooked as in this case such a model does not predict the small nearedge peaks in the first place. Broadening mechanisms may smooth out artificial structures from theoretical XANES and FDMX is critically important for the interpretation of nearedge structures, in addition to producing strong agreement with experiment in the regime. More work is surely needed in the development of these ideas further.
spectra, but cannot introduce the scattering paths required to reproduce observed experimental spectra. For this reason the fullpotential modelling of8. Conclusions
The FDMX package is the first fullpotential package capable of calculating robust and accurate spectra across all applicable energies within a single selfconsistent computation. Over the course of development of the package, this capacity has been shown to enable new extractions of fundamental physical properties from experimental spectra (Bourke & Chantler, 2010b; Tantau et al., 2015), which in turn have led to new insights in fundamental theory (Chantler & Bourke, 2014b; Bourke & Chantler, 2015).
Fullpotential modelling has long been considered necessary for the interpretation of XANES structures, particularly for compounds and molecular samples. This work demonstrates that the advantages of the FDMX approach, while strongly materialdependent, can also be significant for energies at least several hundreds of eV above the absorption edge.
FDMX is currently available by free download from the CNRS website (https://neel.cnrs.fr/spip.php?rubrique1007&;lang=en ) and from The University of Melbourne, School of Physics Xray Optics and Synchrotron Science Group (https://www.ph.unimelb.edu.au/~chantler/opticshome/softwarepackagedownloads.html#FDMX ).
Acknowledgements
The authors acknowledge the contributions of their collaborators N. A. Rae, J. L. Glover, L. J. Tantau and M. T. Islam in testing and assisting the development of ideas that lead to this work.
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