In this contribution, I will give my view on the status of our understanding of the multiplet theory interpretation of X-ray absorption spectral shapes. The main focus is on rare earths and 3d transition-metal ions.

Firstly, the situations in which multiplet theory should be applied are explained. Atomic multiplet theory can be applied to the 3d X-ray absorption spectroscopy (XAS) spectra of rare earths. Subsequently, crystal field multiplet theory and charge-transfer 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 first-principles multiplet calculations.

The application of multiplet theory to X-ray absorption spectroscopy depends strongly on the nature of the system under study and also on the specific X-ray absorption spectrum that is measured. In short, one can state that multiplet theory is only important when (i) you have an open-shell system and (ii) the core hole contains an orbital moment.

In X-ray absorption spectroscopy the X-ray 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 core-hole 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 core-hole potential and the related screening effects. An important description of the analysis of K edges is based on first-principles 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 density-functional theory (DFT)-based programs, including FEFF (Kas et al., 2020), WIEN2k (Blaha, 2021) and Quantum-Espresso. For molecular calculations the usual approach is to describe the core-valence excitation with time-resolved 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 transition-metal 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 X-ray 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 rare-earth systems, the partly filled 4f states often behave as quasi-atomic 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 rare-earth ions behave ionically and are not visibly affected by charge-transfer effects. This is in contrast to tetravalent systems such as CeO₂, for which the ground state is a mixture of 4f⁰ and 4f¹ configurations, implying that charge-transfer effects are very important.

Because the 3d XAS spectra of trivalent rare-earth 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 X-ray magnetic circular-dichroism (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 ground-state configuration, provided that charge-transfer effects are also small.

The 2p XAS spectra of 3d transition-metal ions are dominated by the crystal field effects on the open-shell 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 core-hole 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 Mn²⁺ as an example, the 3d⁵ ground state is excited to a 2p⁵3d⁶ 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 self-screened experiment. This also implies that the charge-transfer 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 charge-transfer effects is Cu³⁺. Its ground state can be described as a linear combination of 3d⁸ and 3d⁹L. The combination of screening and multiplet effects generates a spectral shape in which the 2p⁵3d⁹ and 2p⁵3d¹⁰L configurations are clearly distinguishable (Hu et al., 1998; Fig. 3).

Figure 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 charge-transfer multiplet theory is that one can also describe core-level X-ray photoelectron spectroscopy (XPS) spectra, which are always dominated by strong screening effects as one excites the core electron out of the system. As such, charge-transfer multiplet theory can describe both XAS and XPS from a unified description (Okada & Kotani, 1992; Fig. 4).

Figure 4 The final-state configurations in the 2p XAS calculation of an Mn2+ ion. In 2p XAS the charge-transfer configurations do not change their relative energy, implying a self-screened excitation. In 2p XPS the electron is excited out of the system and the two charge-transfer configurations change their order in energy because the 2p53d6L configuration is pulled down by the core-hole potential, implying large screening effects.

The crystal field multiplet model has been used to explain the 3p XAS spectra of transition-metal systems with a localized transition from 3d^N to 3p⁵3d^N+1 (Shin et al., 1981). The charge-transfer model has been used to explain the screening effects in X-ray photoemission (Kotani & Toyozawa, 1974). The crystal field multiplet model and its combination into the charge-transfer 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, X-ray emission spectroscopy (XES) and resonant inelastic X-ray scattering (RIXS) spectra of transition-metal 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 MO₆ cluster (Tanaka & Jo, 1994; Crocombette & Jollet, 1996; Fernández-Rodríguez 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 charge-transfer multiplet program (Haverkort et al., 2012). Charge-transfer 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) first-principles multiplet codes have been developed, in which a number of approaches are being pursued.

The present situation can be characterized by the presence of (i) semi-empirical multiplet codes that can calculate 2p XAS spectra quickly and (ii) the development of a series of alternative methods for first-principles 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 ground-state electronic structure codes, either based on real-space or reciprocal-space calculations. The main emphasis is likely to be on an extension towards resonance experiments, including resonant inelastic X-ray scattering, coherent excitations and the time evolution of the X-ray excitations.