International
Tables for Crystallography Volume I Xray absorption spectroscopy and related techniques Edited by C. T. Chantler, F. Boscherini and B. Bunker © International Union of Crystallography 2022 
International Tables for Crystallography (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720007491 Multiplet approaches in Xray absorption spectroscopy^{a}Department of Chemistry, Utrecht University, The Netherlands An overview is given of the use of multiplet approaches in Xray absorption spectroscopy. Crystal field multiplet calculations are adequate for the 2p Xray absorption spectra of 3d transitionmetal ions, in contrast to the 2p Xray photoemission spectra, which are dominated by chargetransfer effects due to corehole screening. In the case of 1s Xray absorption there are no multiplet effects and a singleparticle densityfunctional theory or twoparticle Bethe–Salpeter calculation is sufficient. Keywords: multiplet theory. 
In this contribution, I will give my view on the status of our understanding of the multiplet theory interpretation of Xray absorption spectral shapes. The main focus is on rare earths and 3d transitionmetal ions.
Firstly, the situations in which multiplet theory should be applied are explained. Atomic multiplet theory can be applied to the 3d Xray absorption spectroscopy (XAS) spectra of rare earths. Subsequently, crystal field multiplet theory and chargetransfer multiplet theory are described. An overview is given of computer codes that are based on multiplet theory. Finally, an overview is given of progress in firstprinciples multiplet calculations.
The application of multiplet theory to Xray absorption spectroscopy depends strongly on the nature of the system under study and also on the specific Xray absorption spectrum that is measured. In short, one can state that multiplet theory is only important when (i) you have an openshell system and (ii) the core hole contains an orbital moment.
In Xray absorption spectroscopy the Xray photon excites a core hole to an empty state, and to describe the XAS spectral shape the electronic structure effects of both the core hole and the empty state have to be described. The interactions that occur due to corehole creation include the following.

A crucial distinction can be made between core holes with an orbital moment, i.e. p or d core holes, and core holes without an orbital moment, in other words the 1s core hole or K edges. For 1s core holes there is no spin–orbit coupling and there are no higher order terms of the exchange interaction. This implies that the κ2, κ4 and κ6 effects disappear. The exchange interaction of the 1s core hole is in the range of a few meV and can be ignored. This only leaves the corehole potential and the related screening effects. An important description of the analysis of K edges is based on firstprinciples electronic structure methods, to which a core hole is added for the spectral calculation of the electronic structure in the presence of the core hole. This approach is used in densityfunctional theory (DFT)based programs, including FEFF (Kas et al., 2020), WIEN2k (Blaha, 2021) and QuantumEspresso. For molecular calculations the usual approach is to describe the corevalence excitation with timeresolved DFT, as used in programs such as ORCA and ADF.
In the case of a 2p/3p or a 3d/4d core hole, one has to include all interactions κ1–κ7 as listed above. Using a 2p core hole in a 3d transitionmetal ion as an example, these interactions will now be described in more detail (Figs. 1 and 2).
The combined effects due to interactions κ5, κ6 and κ7 are denoted as the multiplet effect. From the numbers given above, it can be seen that 4d systems have small multiplet effects and large 2p spin–orbit coupling, which brings the L_{2,3} edge of 4d systems close to the limit where multiplet effects can in a first approximation be neglected. For every system and Xray absorption edge the atomic spin–orbit and multiplet parameters can be calculated. If the multiplet parameters are small with respect to the lifetime broadening of the core hole, they can be neglected.
In the case of rareearth systems, the partly filled 4f states often behave as quasiatomic states. This implies that the 3d XAS and 4d XAS spectra of rare earths can be calculated with atomic multiplet theory. The crystal field effects are, at ∼50 meV, smaller than the lifetime broadening and as such are invisible in XAS. The trivalent rareearth ions behave ionically and are not visibly affected by chargetransfer effects. This is in contrast to tetravalent systems such as CeO_{2}, for which the ground state is a mixture of 4f^{0} and 4f^{1} configurations, implying that chargetransfer effects are very important.
Because the 3d XAS spectra of trivalent rareearth systems can be described well with atomic multiplet theory, this implies that their spectral shape is exactly the same for every system. This also holds for the shape (but not the magnitude) of Xray magnetic circulardichroism (XMCD) spectra. The 3d XAS spectra (Thole et al., 1985) and XMCD spectra (Goedkoop et al., 1988) have been calculated. In the case of the actinides, the 5f crystal field is larger than for the rare earths, but is still relatively small (<0.3 eV), so that it can be ignored in the first approximation. In other words, actinide spectra also show a large correspondence to the atomic multiplet calculation related to the groundstate configuration, provided that chargetransfer effects are also small.
The 2p XAS spectra of 3d transitionmetal ions are dominated by the crystal field effects on the openshell 3d electrons. The combination of atomic multiplet theory and the crystal field effect yields crystal field multiplet theory. This is the analogue of crystal field theory for optical spectroscopy with the addition of the interactions due to the core hole, i.e. both the corehole spin–orbit coupling and the multiplet effects.
The reason that one can use crystal field multiplet theory for 2p XAS is that the experiment involves a transition that keeps the system locally neutral. Using Mn^{2+} as an example, the 3d^{5} ground state is excited to a 2p^{5}3d^{6} final state. Because this final state has the same charge as the ground state, in a first approximation the neighbouring atoms do not notice that the 2p core electron has been excited to a 3d state. In other words, the 2p XAS experiment is a selfscreened experiment. This also implies that the chargetransfer effects in 2p XAS are very small. Only when the ground state has to be described by two or more configurations can one observe the effects of charge transfer in 2p XAS. The details of crystal field multiplet theory have been described in a number of reviews and books (de Groot, 2001, 2005; van der Laan, 2006; de Groot & Kotani, 2008; van Veenendaal, 2015).
In systems that have to be described with two (or more) configurations in the ground state, it is important to take the screening effects due to charge transfer into account. An ion with large chargetransfer effects is Cu^{3+}. Its ground state can be described as a linear combination of 3d^{8} and 3d^{9}L. The combination of screening and multiplet effects generates a spectral shape in which the 2p^{5}3d^{9} and 2p^{5}3d^{10}L configurations are clearly distinguishable (Hu et al., 1998; Fig. 3).

The nickel 2p XAS spectrum indicating the effects of charge transfer (κ4) and the combined multiplet effects (κ5,6,7), also including the crystal field effect. The sticks are the actual result of the calculation and they are treated with Lorentzian and Gaussian broadening. 
The advantage of chargetransfer multiplet theory is that one can also describe corelevel Xray photoelectron spectroscopy (XPS) spectra, which are always dominated by strong screening effects as one excites the core electron out of the system. As such, chargetransfer multiplet theory can describe both XAS and XPS from a unified description (Okada & Kotani, 1992; Fig. 4).

The finalstate configurations in the 2p XAS calculation of an Mn^{2+} ion. In 2p XAS the chargetransfer configurations do not change their relative energy, implying a selfscreened excitation. In 2p XPS the electron is excited out of the system and the two chargetransfer configurations change their order in energy because the 2p^{5}3d^{6}L configuration is pulled down by the corehole potential, implying large screening effects. 
The crystal field multiplet model has been used to explain the 3p XAS spectra of transitionmetal systems with a localized transition from 3d^{N} to 3p^{5}3d^{N+1} (Shin et al., 1981). The chargetransfer model has been used to explain the screening effects in Xray photoemission (Kotani & Toyozawa, 1974). The crystal field multiplet model and its combination into the chargetransfer multiplet model has been developed by Theo Thole, Akio Kotani, Gerrit van der Laan, George Sawatzky and coworkers (Thole et al., 1988; Groot et al., 1990a,b; van der Laan, 1991; van der Laan & Kirkman, 1991; Okada & Kotani, 1992). The multiplet program from Theo Thole has been used in the CTM4XAS interface. CTM4XAS can calculate the XAS, XPS, Xray emission spectroscopy (XES) and resonant inelastic Xray scattering (RIXS) spectra of transitionmetal systems and rare earths (Stavitsky & de Groot, 2010). An alternative interface is provided by the Missing interface (Gusmeroli, 2006). The multiplet code from Theo Thole essentially used a single metal ion, where all other effects were included as effective electric and magnetic fields, with charge transfer being described to a delocalized d wavefunction. Alternative codes were developed based on an MO_{6} cluster (Tanaka & Jo, 1994; Crocombette & Jollet, 1996; FernándezRodríguez et al., 2015). The CTM4XAS software also includes an option to switch from Theo Thole's programs to the Quanty program from Maurits Haverkort, which can also be run as a chargetransfer multiplet program (Haverkort et al., 2012). Chargetransfer multiplet theory is very successful in describing the 2p XAS spectra of 3d systems, but it is partly based on empirical parameters, for example the value of the crystal field splitting.
Over the last ten years a range of (partly) firstprinciples multiplet codes have been developed, in which a number of approaches are being pursued.
Maurits Haverkort has developed an approach in which in the first step of the calculation the groundstate electronic structure is calculated from an adequate electronic structure model, for example dynamical mean field theory. The results from this calculation are subsequently projected to a realspace cluster involving the transitionmetal ion where the core hole is excited. In the next step, a localized multiplet calculation is performed using the electronic structure parameters as derived for the ground state. For example, if the LDA+U method is used only U remains as an empirical parameter. This approach is the basis of the Quanty program that allows either coupling to firstprinciples calculations or its usage as a semiempirical multiplet code. Quanty contains very efficient numerical software based on Green's functions. This allows it to directly calculate the required spectral shapes without the need to calculate the exact final state (Haverkort et al., 2012). An equivalent line of approach has been pursued by Uozumi and Hariki, where the emphasis in the publications is on Xray photoemission, but the program is for 2p XAS equivalent to the Quanty approach (Hariki et al., 2017).
The DFTCI model has been applied to transitionmetal oxides (Ikeno et al., 2006, 2011; Ikeno, 2016). The multiplet effects are treated correctly, but the method still has some limitations with regard to the treatment of chargetransfer effects. A similar approach based on the ADF code has been developed (Ramanantoanina & Daul, 2017). The MultiX program allows crystal field multiplet calculations based on a realspace cluster around the absorbing transitionmetal ion (Uldry et al., 2012).
Starting from quantumchemistry codes, a number of firstprinciples calculational programs are being developed based on restricted activespace (RAS) methods. The RAS method is an ab initio method based on the multiconfigurational selfconsistent field approach. The general approach is the complete activespace (CAS) method, which involves a full configuration interaction (CI) calculation between selected `active' orbitals. In the RAS method the CI calculated is restricted to a number of excitations. RAS calculations have been published by the groups in Stockholm (Josefsson et al., 2012), Uppsala (Pinjari et al., 2014), Groningen (Klooster, 2015) and Rostock (Preusse et al., 2016). An overview of the efficiency of RAS calculations has been reported for iron complexes (Pinjari et al., 2016). A method based on double excitation within CI has been used for Fe L edges (Otero et al., 2009). The group in Müllheim is pursuing a RAS approach from the starting point of a restrictedopenshell configuration interaction with singles (ROCIS; Roemelt et al., 2013; Maganas et al., 2014).
Bethe–Salpeter (BSE) approaches explicitly describe singlehole singleelectron excitations (Shirley, 2005; Vinson et al., 2011). An extended BSE calculation for 2p XAS has been incorporated into the WIEN2k code (Laskowski & Blaha, 2010), and Kruger has developed the multichannel multiplescattering (MCMS) method to calculate the 2p XAS spectrum of TiO_{2} (Krüger, 2010). A problem with the BSE approach is that it does not include the multiplet interactions correctly for openshell systems. The group in Trieste calculated molecules and solids using timedependent DFT (TDDFT) within the ADF code (Stener et al., 2003; Fronzoni et al., 2009). Similarly to the BSE approach, in TDDFT the multiplet effects are also not incorporated exactly. TDDFT essentially treats singlehole singleelectron excitations and neglects excitations that involve two electrons to make a transition. Due to the very large multiplet interactions, such multielectron interactions are very important in 2p XAS.
The present situation can be characterized by the presence of (i) semiempirical multiplet codes that can calculate 2p XAS spectra quickly and (ii) the development of a series of alternative methods for firstprinciples multiplet calculations. In the near future I expect that this situation will develop in a way such that that the cluster multiplet codes could be coupled to a series of groundstate electronic structure codes, either based on realspace or reciprocalspace calculations. The main emphasis is likely to be on an extension towards resonance experiments, including resonant inelastic Xray scattering, coherent excitations and the time evolution of the Xray excitations.
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