The FDMNES code

Bunău, O.; Ramos, A. Y.; Joly, Y.

doi:10.1107/S1574870720003304

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International
Tables for
Crystallography
Volume I
X-ray absorption spectroscopy and related techniques
Edited by C. T. Chantler, F. Boscherini and B. Bunker

International Tables for Crystallography (2024). Vol. I. ch. 6.6, pp. 752-757
https://doi.org/10.1107/S1574870720003304

Chapter 6.6. The FDMNES code

O. Bunău,^a A. Y. Ramos^a and Y. Joly^a ^*

^aUniversité Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France
Correspondence e-mail: [email protected]

The FDMNES code is user-friendly ab initio software that allows the simulation of X-ray absorption spectroscopy as well as X-ray resonant and nonresonant scattering spectroscopies. Here, the theoretical components of the code, the calculation steps and a typical example of its application are briefly presented. One of the main characteristics of the code is that two different techniques can be independently used to solve the electronic structure. The full-potential finite-difference method precisely applies for free shape potentials, while full multiple-scattering theory gives less precise but faster calculations. Calculations can be applied to all classes of materials. Especially adapted for the low-energy range of the spectra around the absorption range, its relativistic self-consistent density functional theory (DFT) approach makes it efficient for the K edges of all elements and the L_2,3 edges of heavy elements. A time-dependent DFT extension expands the scope of the software to other edges.

Keywords: FDMNES.

1. Introduction

FDMNES (Finite Difference Method Near Edge Structure) is a code which allows ab initio simulation of X-ray spectroscopies involving absorption or scattering processes at energies near the characteristic absorption edges. It is a user-friendly tool, the objective of which is to make the interpretation of X-ray absorption spectroscopy (XAS)-related experiments as easy as possible.¹ As it is especially efficient close to the edge, the code can calculate spectra involving one-photon processes such as X-ray absorption near-edge structure (XANES) and X-ray emission spectroscopy (XES), as well as polarization-dependent processes such as X-ray magnetic circular dichroism (XMCD), X-ray magnetic linear dichroism (XMLD), X-ray natural linear dichroism (XNLD) and X-ray natural circular dichroism (XNCD). It can also calculate spectra involving two-photon processes such as resonant elastic X-ray scattering (REXS; also called RXD or DANES) as well as nonresonant inelastic X-ray scattering (NRIXS or Raman X-ray scattering).

Since it essentially uses density functional theory (DFT) methods, the code is most suitable for application to the calculation of K and L₁ edges, but it is also applicable to the L_2,3 edges of heavy elements. With the other edges often being sensitive to multi-electronic phenomena during the transition process, other theoretical approaches are more appropriate than DFT, which is a basically a ground-state theory. Nevertheless, the code architecture is general and it is not rare to obtain satisfactory agreement with data even at these edges. Moreover, a recent development based on the time-dependent DFT (TDDFT; Bunău & Joly, 2012 ) approach improves the results for L_2,3 edges of the first 3d transition elements. The first versions of the code date from the end of the 1990s (Joly, 1997 , 2001 ; Joly et al., 1999 ), but since then many improvements and extensions have been included.

Concerning the simulated system, no assumption on the potential is made and the code is fully relativistic. Consequently, virtually all systems can be calculated, including bulk crystals, liquids, small molecules and large metalloproteins (Joly, 2003 ). From the point of view of chemical species, all elements ranging from hydrogen to the heavier actinides are allowed in the calculation.

In the next section we describe the current theoretical framework of FDMNES, while Section 3 details the various calculation steps and practical use of the code.

2. Main FDMNES theoretical ingredients

2.1. Transition process

The transition cross section between a core and an unoccupied state (φ_i and φ_f, respectively) is determined by $[\left\langle\varphi _{\rm i}\left|\hat{o}\right|\varphi_{\rm f}\right\rangle]$ , where $[\hat{o}]$ is the operator taking account of the electromagnetic field–matter interaction. FDMNES is specially efficient in calculating and comparing all of the possible transition channels resulting from its expansion, including the electric and magnetic parts (Joly & Grenier, 2015 ), $[\hat{o} = \boldvarepsilon\cdot{\bf r}\left[1+{{i} \over {2}}{\bf k}\cdot{\bf r}-{{1} \over {6}}({\bf k}\cdot{\bf r})^{2}\right]+c_{m}{\bf k}\times\boldvarepsilon\cdot ({\bf L}+2{\bf S})+ \ldots. \eqno (1)]$ The three first terms represent the electric dipole (E1), electric quadrupole (E2) and electric octupole (E3) transition channels, respectively. The remaining term is the magnetic dipole (M1) transition channel; higher terms are negligible in practice. $[\boldvarepsilon]$ and k are the photon polarization and wavevector, respectively, and r is the position vector. c_m = $[\hbar/[2m({\cal E}_{\rm f}-{\cal E}_{\rm i})]]$ , S and L are the spin and orbital angular momentum operators, respectively, m is the electron mass, and $[{\cal E}_{\rm i}]$ and $[{\cal E}_{\rm f}]$ are the core and photoelectron state energies, respectively.

Using this operator, the golden rule gives the absorption cross section formula $[\sigma(\omega) = 4\pi^{2}\alpha \hbar\omega\textstyle \sum \limits_{\rm f,i}|\langle\varphi _{\rm i}|\hat{o}|\varphi _{\rm f}\rangle|^2\delta({\cal E}_{\rm f}-{\cal E}_{\rm i}-\hbar\omega), \eqno (2)]$ where ℏω is the photon energy, δ is the density of state and α is the fine-structure constant. In the one-electron approximation, it involves the sum of the individual transition probabilities from the initial state to the final state of energy $[{\cal E}_{\rm f} = {\cal E}_{\rm i}+\hbar\omega]$ .

It is worth noting that the NRIXS formula is very similar to the XANES formula. Indeed, its dynamic structure factor is also proportional to $[\textstyle\sum _{\rm f,i}|\langle\varphi_{\rm i}|\hat{o}|\varphi_{\rm f}\rangle|^{2}]$ . The difference is that in this case the operator is not related to the polarization but to the scattering vector q by $[\hat{o}]$ = exp(iq · r).

It is easy to extend a code calculating XANES to REXS. Indeed, the formula to obtain the resonant form factor is given by (Grenier & Joly, 2014 ) $[f^{\prime}(\omega)-if^{\prime\prime}(\omega)\cong m\omega^{2}\lim_{\eta\rightarrow 0}{\textstyle \sum \limits_{\rm f,i}}{{\langle\varphi_{\rm i}|\hat{o}_{\rm out}^{*}|\varphi_{\rm f}\rangle\langle\varphi_{\rm f}|\hat{o}_{\rm in}|\varphi _{\rm i}\rangle} \over {\hbar\omega-({\cal E}_{\rm f}-{\cal E}_{\rm i})+i\eta}}, \eqno (3)]$ where the two transition operators $[\hat{o}_{\rm in}]$ and $[\hat{o}_{\rm out}]$ correspond to the incoming and outgoing polarizations and wavevectors. To obtain the diffracted intensity, one then has to add the nonresonant contributions (Thomson and magnetic), sum over the different atoms in the unit cell and consider their phase factor to obtain the total structure factor as in conventional Bragg diffraction.

2.2. Symmetries

Symmetrization allows the calculation time to be significantly reduced. Thus, FDMNES exploits both space-group and magnetic crystal group symmetry to determine the equivalent and nonequivalent atoms in the unit cell for 3D materials. The signal is only calculated for the prototypical atoms embedded in their surrounding cluster. The total signal is then obtained by summing firstly all of the equivalent atoms by using the corresponding symmetry operations and secondly the nonequivalent atoms. Any relative energy shifts resulting from different oxidation states are also automatically taken into account. The code equally exploits the point-group symmetry (possibly magnetic) in the cluster in which the DFT calculation is performed.

2.3. The tensor formalism

X-ray absorption and scattering phenomena are generally anisotropic, i.e. the cross section depends on the orientation of the beam polarization with respect to the sample. Therefore, the physical origin and the azimuthal dependence of the spectra can be described in terms of tensor algebra.

The matrix elements for a given initial state i, on an absorbing atom j, in equations (2) and (3) can be written equivalently using the sample tensors $[{\cal T}]$ and the polarization tensors $[{\cal P}]$ , $[\displaystyle \textstyle \sum \limits_{\rm f}\langle\varphi_{\rm i}^{j}|\hat{o}_{\rm out}^{*}|\varphi_{\rm f}\rangle\langle\varphi_{\rm f}|\hat{o}_{\rm in}|\varphi_{\rm i}^{j}\rangle = \textstyle \sum \limits_{{IO}}{\cal P}_{IO}{\cal T}_{IO,{\rm i}}^{j}({\cal E}_{\rm f}), \eqno (4)]$ where I, O = E1, E2, E3, M1 correspond to one of the terms in the mutipolar expansion (1 ) for the incoming (I) and outgoing (O) electromagnetic field, respectively. $[{\cal T}]$ and $[{\cal P}]$ may be expressed in both Cartesian or spherical basis. The formulation above is general, with crossed IO terms (for example E1E2) for all of the spectrocopies, with the simplification $[\hat{o}_{\rm in} = \hat{o}_{\rm out}]$ in XANES and NRIXS.

The sample tensor is obtained by summing the individual tensors of the atoms j, at position R_j, in the unit cell for periodic systems or in the entire cluster for molecules. In the case of REXS a phase factor must be considered in the summation, $[{\cal T}_{IO,{\rm i}} = \textstyle \sum \limits_{j}\exp(i{\bf q}\cdot{\bf R}_{j}){\cal T}_{IO,{\rm i}}^{j}, \eqno (5)]$ while for XANES (or XES) exp(iq · R_j) is omitted, yielding the crystal absorption tensor.

The great advantage of the tensor description is that it provides a unified description of absorption σ(ω) and scattering f(ω) spectroscopies and enables polarization studies, as for example in the azimuthal scan in REXS, at practically no expense. Moreover, some of the tensor elements disappear for particular combinations of crystal symmetry and polarization conditions. They are thus the signature of fundamental properties of the material. For instance, E1E2 is null if the local structure of the absorbing atoms exhibits a point group including the inversion operation. It is equally null in X-ray absorption if the unit cell contains an inversion centre or when the polarization is linear.

The FDMNES code is extremely powerful with respect to analysis of the transition tensor. The calculation may be performed by restraining the tensor elements to a specific rank, which enables the identification of the origin of each structure in the spectrum.

The code uses Cartesian tensors (Joly et al., 2007 ), but on demand output using expansion in spherical tensors is also possible. The advantage is that these tensors are directly interpretable in terms of physical properties such as charge, magnetic moment, toroidal moments or even the magnetic monopole (Di Matteo et al., 2005 ).

2.4. DFT simulation

2.4.1. The potential

A DFT simulation first needs a potential (Martin, 2004 ). To begin with, one performs a Dirac–Fock calculation of atomic densities and energies for all of the chemical species of the atoms in the calculation cluster. By default and at the K edge, for the absorbing atom one assumes an excited electronic configuration with a core hole and an extra electron placed on the first available unoccupied level. This is the so-called final state rule.

The electrostatic potential is obtained by solving the Poisson equation, the source term of which takes the sum of all of the individual self-consistent charge densities. Spin-polarized local exchange and correlation effects are taken into account as suggested by Hedin & Lundqvist (1971 ) and Von Barth & Hedin (1972 ) or by Perdew & Wang (1992 ).

2.4.2. Multiple-scattering theory and finite-difference method

The evaluation of φ_f (or $[\textstyle \sum_{\rm f}|\varphi_{\rm f}\rangle\langle\varphi_{\rm f}|]$ ) in equations (2) and (3) is performed in FDMNES according to two different DFT techniques: the multiple-scattering theory (MST) and the finite-difference method (FDM), with the code being named after the latter. MST was first used by Natoli et al. (1980 ) for XANES simulations and subsequently became the most used one-electron approach. MST and FDM are explained in Part 2 of this volume of International Tables for Crystallography. In brief, they are real-space techniques applied to a cluster approach surrounding the absorbing atom. The calculation volume is limited by a sphere, the radius of which typically ranges between 4 and 7 Å. The advantage of FDM over MST is that it solves the electronic states without limitations on the shape of the potential. MST is faster but is often less precise because it uses the muffin-tin approximation, in which the potential is taken as spherically symmetric around the atoms and constant between them. Thanks to Guda et al. (2015 ), the FDM approach has been made considerably less demanding in terms of CPU time than previously. The availability of both techniques within the same code offers common ground for the comparison of their respective performances, which is essential for comparing the two theoretical approaches.

The calculation is not linearized in the sense that the multiple-scattering amplitude in the MST case, and the wavefunctions in FDM, are fully evaluated at all energy points in a chosen grid along the spectra. A very wide energy range can thus be calculated without loss of accuracy. On demand, simulations can be nonmagnetic, scalar including nonrelativistic magnetism or fully relativistic, i.e. including spin–orbit scattering. In this last case we follow Wood & Boring (1978 ) in explicitly solving only the large component orbital instead of the four determined in the Dirac–Slater calculations. This reformulation gives a couple of Schrödinger-like equations that are closely akin to, but are an improvement upon, the Pauli equation.

2.4.3. Self-consistency

Before the main signal calculation, a self-consistent procedure (Bunău & Joly, 2009 ) can be used, on demand, to obtain a better potential. As in the main step, calculations with or without the core hole are possible. Not only does the self-consistent calculation output a highly accurate scattering potential, but it also enables automatic calculation of the energy-cutoff level. This feature is essential for the calculation of spectra, since the spectroscopies, except for XES, only probe the unoccupied electronic states.

2.4.4. Hubbard corrections

The mono-electronic description of correlated electronic levels may be improved by breaking the rotational symmetry of the exchange correlation potential corresponding to the correlated states. In practice, this is performed by adding an orbital-dependent term to the exchange-correlation potential for these orbitals. This correction is generally applied for the d orbitals of transition elements or the f orbitals of the lanthanides. FDMNES implements the original scheme proposed by Dudarev et al. (1998 ), with a slight modification, $[\Delta V_{mm^{\prime}}^{\sigma\sigma^{\prime}} = (U-J)\left({{1} \over {2}}\delta_{mm^{\prime}}\delta_{\sigma\sigma^{\prime}}-\rho_{mm^{\prime}}^{\sigma\sigma^{\prime}}\right), \eqno (6)]$ where $[\Delta V_{mm^{\prime}}^{\sigma\sigma^{\prime}}]$ is the orbital (m, m′) and spin (σ, σ′) dependent Hubbard potential and $[\rho_{mm^{\prime}}^{\sigma\sigma^{\prime}}]$ is the occupation matrix. U and J represent the spherically averaged on-site repulsion and exchange. FDMNES treats U − J as a calculation parameter. It has been applied, for example, in the XMCD study of rare-earth zinc compounds (Galéra et al., 2008 ).

2.5. Time-dependent density-functional theory

FDMNES can partially include multi-electronic effects through time-dependent extension of DFT in the limits of linear response. Following the original idea of Schwitalla & Ebert (1998 ), the tensor element in equation (4) is modified in order to allow crossing channels between the individual core-valence state transitions, a pure multi-electronic feature. The linear response function (or susceptibility) χ is then not diagonal in the initial states i and i′. The relationship between χ and its mono-electronic counterpart χ₀ is given by the kernel K: χ = (1 − χ₀K)⁻¹χ₀.

In TDDFT one makes an ansatz on the kernel K. FDMNES implements the local field approximation $[K_{\sigma\sigma^{\prime}}({\bf r},{\bf r}^{\prime}) = {{2} \over {|{\bf r}-{\bf r}^{\prime}|}}+f_{\sigma\sigma^{\prime}}({\bf r})\delta({\bf r}-{\bf r}^{\prime}),]$ in which the first term is the Coulomb repulsion and the second term is a local exchange term. A detailed description of the TDFFT implementation in FDMNES can be found in Bunău & Joly (2012 ). This method has proven to improve the agreement and more specifically the L₂/L₃ ratio of the first half of the transition elements.

2.6. About REXS and NRIXS

The REXS technique (Joly, Di Matteo et al., 2012 ) is highly sensitive to charge-ordering phenomena, as shown in magnetite below the Verwey transition (Nazarenko et al., 2006 ), or to magnetic ordering, as seen in V₂O₃ (Joly et al., 2004 ). Sensitivity to orbital ordering is more difficult to observe independently of the associated crystallographic distortion. It has been seen, for example, in NdMg, where a peculiar magnetic ordering induces a quadrupolar electronic ordering (Bunău et al., 2010 ). FDMNES is thus written in order to make the REXS simulation as easy as possible for the user. Nonresonant contributions are automatically calculated and added to the resonant part. Space-group and magnetic group symmetries are fully exploited in order to calculate the diffracted intensities with all the possibilities for incoming and outgoing polarizations. REXS data can be very sensitive to self-absorption phenomena and sometimes also to birefringence. FDMNES is able to take these effects into account and thus to accurately interpret spectral line shapes and scattering-dependent polarization changes, as shown in recent work on CuO (Joly, Collins et al., 2012 ).

NRIXS (as well as electron energy-loss spectroscopy) depends on the q direction. At large scattering angles a monopolar term appears, obeying the δl = 0 selection rule. Thus, this spectroscopy not only offers the possibility of studying very low energy edges, but also probes other projections of the electronic state around the absorbing atom. These phenomena are also calculated by the FDMNES code (Joly et al., 2017 ).

2.7. Broadening and lifetime

Equations (2) and (3) neglect the effect of the core-hole lifetime and photoelectron inelastic scattering. These effects are taken into account using a Lorentzian broadening for XANES with a photoelectron energy-dependent width $[\Gamma({\cal E}_{\rm f}) = \Gamma_{\rm H}+\Gamma _{\rm f}({\cal E}_{\rm f})]$ . Γ_H is simply chosen according to Krause & Oliver (1979 ), but smaller values can be used to interpret high energy resolution experiments. The energy-dependent part is a signature of inelastic electron-scattering phenomena. It is set according to an empirical arctangent model similar to the Seah–Dench formalism (Seah & Dench, 1979 ). More precisely, Γ is given by $[\Gamma = \Gamma_{\rm H}+\Gamma_{\rm m}\left\{{{1} \over {2}}+{{1} \over {\pi}}\arctan\left[{{\pi} \over {3}}{{\Gamma_{\rm m}} \over {E_{\rm w}}} (e-1/e^{2} )\right]\right\}, \eqno (7)]$ in which $[e = ({\cal E}_{\rm f}-E_{\rm F})/E_{\rm p}]$ . Γ_m corresponds to Γ_f at infinity. E_w and E_p are the width and the energy position of the centre of the arctangent. E_p is typically the plasmon energy value. E_F is the Fermi level. In practice, most often only Γ_m is modified from the default values.

Note that applying a broadening after a calculation is nearly equivalent to the use of a complex energy in the main calculation. This procedure has the advantage of being fast and of giving the possibility of easily testing different broadenings. The work of Bourke et al. (2016 ) on the FDMX code improves this part.

For REXS, we simply replace η with $[\Gamma({\cal E}_{\rm f})]$ in (3).

3. Description of the FDMNES calculation

A simplified FDMNES flowchart is given Fig. 1 , where one can see the different steps described in the previous section. FDMNES can also fit sample parameters (structural or related to electronic populations) to the measured spectrum or set of spectra (for REXS). Independent calculations are run for distinct values of the parameters and then optimized using reliability factors (Joly et al., 1999 ) between the theoretical spectra and the experimental spectra. This procedure must be used with care when simulating XANES, limiting the number of unknowns. For REXS, very often one can simulate and simultaneously fit a large number of spectra and consequently confidently handle more parameters. This was performed, for instance, to determine the complex charge ordering and geometrical distortion in magnetite below the Verwey transition in the Cc space group (Joly et al., 2008 ).

Figure 1

Flowchart of FDMNES.

According to the FDMNES flowchart, the calculation of the electronic structure is a first necessary step in the simulation of X-ray spectrocopies. Therefore, the projected density of states can be obtained on demand as a complementary output. This possibility has, for example, been extensively used by Manceau et al. (2015 ) in a study of the coordination of mercury in thiolates. In the same way, the different tensor analysis of the material and the resulting signals can be printed out.

Finally, FDMNES contains the FDMX extension (Bourke et al., 2016 ; Bourke, 2024 ) to perform EXAFS simulation using FDM.

In FDMNES, the only absolutely required inputs are (i) the photoelectron energy range with the energy steps, (ii) the cluster radius and (iii) a description of the crystal or molecular cluster, i.e. the chemical identity and position of all atoms. As stated above, the calculation fully exploits crystal symmetry according to the space-group information featured in International Tables for Crystallography and the unit-cell description can be made giving the standard space-group names. It is also possible to directly read CIF- or PDB-format crystal structure files.

By default, the XANES is calculated at the K edge of the first chemical element appearing in the input data file, in dipole approximation and using the FDM. Many optional keywords permit the default parameters (edge, absorber, …) to be changed, specific convolution parameters to be requested, a magnetic material to be defined and so on. So, the FDMNES input data file contains keywords and values as follows for the example of the Fe K edge in magnetite: [Scheme scheme1]

Keyword names are chosen to be intuitive (Filout, Range, Radius, Crystal, Spgroup, Convolution, …). This example asks for the simulation of the (4, 4, 4) reflection in σ-in σ-out polarization (keyword RXD) and considering self-absorption (keyword Self_abs). The simulation is self-consistent (keyword SCF), using the MST (keyword Green) and including the quadrupole transition (keyword Quadrupole); that is, E2E2 and E1E2. The output corresponding to this calculation is shown in Fig. 2 .

Figure 2

REXS (bottom) and XANES (top) at the Fe K edge in magnetite. REXS is calculated for the (4, 4, 4)_σσ peak without and with (corrected) self-absorption.

4. Conclusions

FDMNES is a DFT and TDDFT ab initio code to simulate different spectroscopies related to the X-ray absorption process. It uses two calculation techniques: MST, which is fast, and FDM, which is more precise as it avoids the muffin-tin approximation. Since its first publication, the code has been greatly improved following the demands and remarks of users and thanks to the help of many collaborators. Much work is still in progress; for example, the possibility of calculating crystal truncation rods in 2D diffraction in resonant conditions will soon be available. Progress in taking multi-electronic phenomena into account is also planned.

Acknowledgements

Calogero Natoli is especially thanked for his help in many stages of the development of the code. The work has also benefitted from discussions with and the helpful expertise of Jay Bourke, Christian Brouder, Delphine Cabaret, Christopher Chantler, Stephen Collins, Maurizio De Santis, Sergio Di Matteo, Vladimir Dmitrienko, José Goulon, Stéphane Grenier, Alexander Guda, Sergey Guda, Keisuke Hatada, Amélie Juhin, José-Emilio Lorenzo-Diaz, Stephen Lovesey, Denis Raoux, Elena Nazarenko, Elena Ovchinnikova, Hubert Renevier, Andrei Rogalev, Sergey Smolensev, Alexander Soldatov, Mikhail Soldatov and Rainer Wilcke.

References

Barth, U. & Hedin, L. (1972). J. Phys. C Solid State Phys. 5, 1629–1642.Google Scholar

Bourke, J. D. (2024). Int. Tables Crystallogr. I, ch. 6.7, 758–763 .Google Scholar

Bourke, J. D., Chantler, C. T. & Joly, Y. (2016). J. Synchrotron Rad. 23, 551–559.Google Scholar

Bunău, O., Galéra, R. M., Joly, Y., Amara, M., Luca, S. E. & Detlefs, C. (2010). Phys. Rev. B, 81, 144402.Google Scholar

Bunău, O. & Joly, Y. (2009). J. Phys. Condens. Matter, 21, 345501.Google Scholar

Bunău, O. & Joly, Y. (2012). Phys. Rev. B, 85, 155121.Google Scholar

Di Matteo, S., Joly, Y. & Natoli, C. R. (2005). Phys. Rev. B, 72, 144406.Google Scholar

Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J. & Sutton, A. P. (1998). Phys. Rev. B, 57, 1505–1509.Google Scholar

Galéra, R. M., Joly, Y., Rogalev, A. & Binggeli, N. (2008). J. Phys. Condens. Matter, 20, 395217.Google Scholar

Grenier, S. & Joly, Y. (2014). J. Phys. Conf. Ser. 519, 012001.Google Scholar

Guda, S. A., Guda, A. A., Soldatov, M. A., Lomachenko, K. A., Bugaev, A. L., Lamberti, C., Gawelda, W., Bressler, C., Smolentsev, G., Soldatov, A. V. & Joly, Y. (2015). J. Chem. Theory Comput. 11, 4512–4521.Google Scholar

Hedin, L. & Lundqvist, S. (1971). J. Phys. C Solid State Phys. 4, 2064–2083.Google Scholar

Joly, Y. (1997). J. Phys. IV Fr. 7, C2-11–C2-115.Google Scholar

Joly, Y. (2001). Phys. Rev. B, 63, 125120.Google Scholar

Joly, Y. (2003). J. Synchrotron Rad. 10, 58–63.Google Scholar

Joly, Y., Cabaret, D., Renevier, H. & Natoli, C. R. (1999). Phys. Rev. Lett. 82, 2398–2401.Google Scholar

Joly, Y., Cavallari, C., Guda, S. A. & Sahle, C. J. (2017). J. Chem. Theory Comput. 13, 2172–2177.Google Scholar

Joly, Y., Collins, S. P., Grenier, S., Tolentino, H. C. N. & De Santis, M. (2012). Phys. Rev. B, 86, 220101.Google Scholar

Joly, Y., Di Matteo, S. & Bunău, O. (2012). Eur. Phys. J. Spec. Top. 208, 21–38.Google Scholar

Joly, Y., Di Matteo, S. & Natoli, C. R. (2004). Phys. Rev. B, 69, 224401.Google Scholar

Joly, Y. & Grenier, S. (2015). X-ray Absorption and X-ray Emission Spectroscopy: Theory and Application., edited by J. A. Van Bokhoven & C. Lamberti, pp. 73–97. Chichester: John Wiley & Sons.Google Scholar

Joly, Y., Lorenzo, J. E., Nazarenko, E., Hodeau, J. L., Mannix, D. & Marin, C. (2008). Phys. Rev. B, 78, 134110.Google Scholar

Joly, Y., Nazarenko, E., Lorenzo, E., Di Matteo, S. & Natoli, C. R. (2007). AIP Conf. Proc. 882, 89–93.Google Scholar

Krause, M. O. & Oliver, J. H. (1979). J. Phys. Chem. Ref. Data, 8, 329–338.Google Scholar

Manceau, A., Lemouchi, C., Rovezzi, M., Lanson, M., Glatzel, P., Nagy, K. L., Gautier-Luneau, I., Joly, Y. & Enescu, M. (2015). Inorg. Chem. 54, 11776–11791.Google Scholar

Martin, R. (2004). Electronic Structure: Basic Theory and Practical Methods. Cambridge University Press.Google Scholar

Natoli, C. R., Misemer, D. K., Doniach, S. & Kutzler, F. W. (1980). Phys. Rev. A, 22, 1104–1108.Google Scholar

Nazarenko, E., Lorenzo, J. E., Joly, Y., Hodeau, J. L., Mannix, D. & Marin, C. (2006). Phys. Rev. Lett. 97, 056403.Google Scholar

Perdew, J. P. & Wang, Y. (1992). Phys. Rev. B, 45, 13244–13249.Google Scholar

Seah, M. P. & Dench, W. A. (1979). Surf. Interface Anal. 1, 2–11.Google Scholar

Schwitalla, J. & Ebert, H. (1998). Phys. Rev. Lett. 80, 4586–4589.Google Scholar

Wood, J. H. & Boring, A. M. (1978). Phys. Rev. B, 18, 2701–2711.Google Scholar

International Tables for Crystallography (2024). Vol. I. ch. 6.6, pp. 752-757
https://doi.org/10.1107/S1574870720003304