Chapter 6.8. The FEFF code

FEFF is a widely used all-electron real-space multiple-scattering Green's function code for simulating a variety of X-ray spectroscopies. However, the code also calculates self-consistent atomic, ground-state and excited-state electronic structure (Rehr & Albers, 2000; Rehr et al., 2009, 2010). The code is highly automated and is applicable to both periodic and aperiodic systems of up to several hundred atoms, throughout the periodic table, which can be defined by a set of atomic coordinates and their atomic numbers. The calculations carried out by FEFF include the dominant many-body effects in the spectra including inelastic mean free paths and vibrational damping. FEFF is named for the effective scattering amplitude f_eff in EXAFS, and is probably best known for its use in extended X-ray absorption fine-structure (EXAFS) calculations, particularly within various analysis codes such as DEMETER (Ravel & Newville, 2005) and LARCH (Newville, 2013; Newville & Ravel, 2024). In addition, FEFF can be used to simulate many other spectroscopies including X-ray absorption near-edge structure (XANES), X-ray emission spectroscopy (XES), electron energy loss spectra (EELS), nonresonant and resonant inelastic X-ray scattering (NRIXS and RIXS) and several others.

The FEFF calculations are broken into several independent steps or modules, which are used sequentially to calculate different properties of the system. The major steps of the calculation are as follows:

(i) Dirac–Fock atomic calculation.

(ii) Potentials and densities.

(iii) Embedded atomic cross sections and phase shifts.

(iv) Calculation of the multiple-scattering Green's function.

(v) Spectral calculations.

Each of these steps calculates different ingredients, for example total potentials, densities, scattering phase shifts etc., which are then used to calculate the final spectrum. In this section we describe each step of the calculation, noting the main approximations and how the ingredients produced are incorporated into the overall calculation.

The first step of a FEFF simulation is a calculation of the atomic properties of each of the unique atoms in the structure specified by the user by solving the relativistic Dirac equation within the Dirac–Fock approximation (Ankudinov et al., 1996; Desclaux, 1975). Of particular importance are the total atomic energies, atomic densities and the occupied spinor wavefunctions. In addition an extra calculation is performed for the absorbing atom, which includes a core hole in a given core level (K, L₁ etc.). The total atomic energies are used to approximate the edge energy for the system, and naturally include spin–orbit coupling effects beyond the perturbative method. The edge energy is approximated by taking the difference between the total energies of the absorbing atom with and without a core hole, which allows an approximation of the relaxation energy. Atomic electron densities are used in the second step to produce the muffin-tin potentials, while the core-level wavefunctions are used in the calculation of the embedded atomic cross sections.

The second step is to calculate the potentials used to describe the scattering of the valence electrons and the photoelectron. This is performed by self-consistently solving the Dirac equation within the local density approximation (LDA) using the real-space multiple-scattering Green's function (RSGF) approach (Ankudinov et al., 1998) and the final-state rule (FSR) for the core hole. In particular, the total LDA potential v(r) can be calculated given the nuclear Coulomb potentials and the total electron density ρ(r), which is related to the Green's function byHere, μ is the chemical potential and is constrained by the total number of electrons, . The Green's function in turn can be calculated given the total LDA potential; thus, a looping procedure is used to calculate the densities, potentials, the Green's function and back to densities until the density is self-consistent, i.e. the same (within a specified tolerance) for two consecutive iterations. The self-consistency loop starts with an approximate density based on overlapped atomic densities. The FEFF code then uses the spherical muffin-tin approximation; while this approximation is highly efficient, it can lead to substantial errors close to the edge. Fig. 1 shows a flow chart of the self-consistency loop. For more details, see Rehr et al. (2024).

Figure 1 Flow diagram of FEFF's self-consistency loop.

After the self-consistent potentials have been found, we can proceed with the third step, which is to calculate the embedded-atom cross sections and phase shifts. For this step, the Dirac equation is solved one more time for each embedded atom, i.e. for each self-consistently defined muffin-tin potential. The cross sections are given in terms of the regular and irregular solutions of the Dirac equation at a particular energy, and the matrix elements and embedded-atom cross sections can then be calculated. The scattering phase shifts are determined by matching these solutions to spherical Bessel functions (the free-space solutions) at the muffin-tin radius. The dipole cross sections are proportional to the atomic background absorption for the material, similar to that defined experimentally for EXAFS analysis, while the phase shifts define the scattering T-matrices (in an angular momentum, site basis) for each atom i with angular momentum componentswhich can be used to calculate the scattering amplitude of the photoelectron from the atoms. Additional photoelectron lifetime effects are also included in this final solution of the embedded-atom Dirac equation in the form of an energy-dependent self-energy, which defines the EXAFS inelastic mean free path.

These scattering matrices can thus be used (along with an expansion of the free-electron Green's function) to calculate the multiple-scattering Green's function either by path expansion or full multiple scattering by matrix inversion. The path method is very efficient and has the benefit that it can be directly correlated to the EXAFS equation, which is used for analysis purposes. In particular, the path-expansion approach is used to define the effective parameters, i.e. scattering amplitudes, phase shifts and inelastic mean free paths, to allow the use of theoretical standards. For example, the effective scattering amplitude f_eff is where the FEFF code gets its name. For near-edge (XANES) calculations, the path expansion fails to converge and thus the full-multiple scattering (FMS) method is used. The FMS approach sums all paths for a small cluster of approximately 150 atoms analytically by performing a matrix inverse, i.e.where T is the scattering matrix and G⁰ is the free Green's function matrix, in which the on-site diagonal terms are set to zero.

Here, we discuss recent developments in the treatment of the key many-body effects in deep-core excitation spectra. From a many-pole representation of the dielectric function and the GW approximation (see Section 3.1), we obtain photoelectron self-energies, inelastic mean free paths (IMFPs) and contributions from multi-electron excitations. Next, as an improvement on the FSR, core-hole effects can be treated via a random phase approximation (RPA) calculation of the local response function (Takimoto et al., 2007), which is then used to calculate the screened core-hole interaction. Finally, from calculations of the dynamical matrix and a many-pole Lanczos algorithm, we obtain phonon spectra and DW factors. By iterating these steps, we then obtain improved calculations of optical constants from the UV to X-ray energies. All of the above procedures have been introduced into calculations of excited-state spectra based on the real-space Green's function formalism described in the previous section (Rehr & Albers, 2000; Rehr & Ankudinov, 2005).

In addition to improved theoretical methods, FEFF9 has a number of features which make it more user-friendly compared with previous versions. The most notable of these are (i) a new GUI, (ii) improved memory management and (iii) the ability to run within a cloud-computing cluster. The code is now accessible to a wide variety of users (from beginners to expert users) within a number of operating systems.

In addition, the FEFF9 code has been reformatted to conform to the Fortran95 standard. The Fortran95 standard allows improved memory management compared with Fortran77, which is the language that was used for previous versions of FEFF. In particular, users will no longer need to recompile the code in order to increase the cluster size or the angular momentum cutoff.

The availability of large, virtualized pools of computing resources such as those offered by Amazon Web Services provides a new and possibly advantageous computing paradigm for scientific research. We have recently demonstrated the feasibility of scientific computation using these cloud-computing resources as an alternative to traditional tools (Jorissen et al., 2012) and have shown that cloud computing can provide convenient access to reliable, high-performance clusters without the need to purchase and maintain sophisticated hardware. As part of the FEFF project, we have developed a set of tools which makes Amazon's Elastic Compute Cloud (EC2) behave like a virtual homogeneous computer cluster. They consist of a set of Bash scripts that perform the basic tasks involved in efficiently interacting with a virtual cluster, including: launching, connecting to, transferring files to and from, monitoring and terminating such clusters.

In this section, we briefly discuss the capabilities of the FEFF code, including the available spectra and advanced approximations that can be made. The different kinds of spectroscopies that can currently be simulated with FEFF include:

(i) XAS (XANES/EXAFS): X-ray absorption spectra including near-edge and extended;

(ii) XES: X-ray emission spectra;

(iii) XMCD: X-ray magnetic circular dichroism;

(iv) DAS (DAFS/DANES): diffraction anomalous scattering and fine structure;

(v) EELS: electron energy-loss spectra;

(vi) NRIXS: nonresonant inelastic X-ray scattering;

(vii) RIXS: resonant inelastic X-ray scattering; and

(viii) Compton scattering.

In this section, we briefly discuss the Java-based graphical user interface JFEFF (Rehr et al., 2010), which automates the FEFF calculations (Fig. 2). This GUI is self-explanatory and easy to use, greatly facilitating all aspects of carrying out FEFF calculations on various systems from laptops to mainframes. Notable features of JFEFF including a plotting window to show results of the calculations and the structure used. JFEFF executes the modules in FEFF, and can import, write and change specified input. JFEFF also allows users to quickly view crystal or molecular structures using Jmol, plot spectral output and local densities of states, and compare simulated data with experimental data. The GUI consists of a main panel which controls the most important options in FEFF, such as the description of the system, the type of spectroscopy and the modules of the code which should be run. Additional options are grouped into separate tabs, with one tab for each module (or step) of the FEFF code, with common and advanced options separated on each tab. All options have sensible defaults which depend on the type of spectroscopy. This allows users to focus on specifying only the property of interest and the system. The system can be specified by importing atomic coordinates in x, y, z format, by importing crystallographic information files (CIFs) or by hand. For example, a user who wants to calculate XANES can download the crystal or molecular structure from a database, import it into JFEFF, set the spectrum to XANES and run the calculations. In order facilitate the process, pop-up help is available for each input option. JFEFF runs on most commonly used operating systems including Windows, MacOS and Linux. Finally, for more advanced users, JFEFF can be used to run FEFF in parallel locally, on a remote computing cluster or on the Amazon cloud (Jorissen et al., 2012).

Figure 2 The Java-based GUI JFEFF. The main panel, output of the run, Jmol structure viewer and a plot of the final spectrum are shown.

Calculations based on the standard quasiparticle theory and the RSGF formalism now make a general treatment of XAS possible, encompassing XANES and EXAFS, as well as a number of other X-ray spectroscopies. These calculations typically rely on simplified models to deal with many-body effects. However, recent theoretical advances have led to efficient, ab initio approaches for calculations of the key many-body damping factors in these core-level X-ray spectroscopies.

In summary, our calculations of many-body effects are made possible by three key developments: (i) an extension of our RSGF code for full-spectrum calculations, (ii) an efficient many-pole representation of the dielectric function and (iii) an efficient Lanczos approach for phonon spectra and DW factors. These developments yield significantly improved self-energies compared with plasmon-pole models, as well as quantitative IMFPs and losses due to multi-electron excitations. All of these many-body effects are important in calculations of optical constants over a broad spectrum, and their inclusion yields improved amplitudes and phases from the UV to X-ray energies (Prange et al., 2009). The results presented show that the new approaches for calculating inelastic losses, self-energies and Debye–Waller factors yield significant improvements in amplitude and phase compared with the semi-empirical approaches, both for near-edge spectra (XANES; Fig. 3) and EXAFS (Newville et al., 2009). Our approach includes solid-state effects (for example edge shifts, fine structure and temperature-dependent DW factors) and is applicable to general aperiodic materials. Thus, our optical constants complement and can potentially replace empirical tables (Hagemann et al., 1975; Henke et al., 1993; Palik, 1998; Chantler, 1995; Elam et al., 2002) or atomic models (Cromer & Liberman, 1970) for many applications.

Figure 3 Polarization-dependent XANES of ZnO as calculated within the many-pole (solid) and plasmon-pole approximations in versions FEFF84 (blue dots) and FEFF90 (green dashes) compared with experimental results (Ney et al., 2008). The X-ray linear dichroism (XLD) is also shown.

The FEFF9 code is capable of calculating a variety of spectra not available in FEFF8 including NRIXS, EELS, RIXS etc. and comes with the JFEFF GUI to facilitate user input and control. Additional improvements, documentation and future updates can be found at http://feff-project.org/ .