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 (2022). Vol. I. Early view chapter
https://doi.org/10.1107/S1574870720007491

Multiplet approaches in X-ray absorption spectroscopy

Frank de Groota*

aDepartment of Chemistry, Utrecht University, The Netherlands
Correspondence e-mail: f.m.f.degroot@uu.nl

An overview is given of the use of multiplet approaches in X-ray absorption spectroscopy. Crystal field multiplet calculations are adequate for the 2p X-ray absorption spectra of 3d transition-metal ions, in contrast to the 2p X-ray photoemission spectra, which are dominated by charge-transfer effects due to core-hole screening. In the case of 1s X-ray absorption there are no multiplet effects and a single-particle density-functional theory or two-particle Bethe–Salpeter calculation is sufficient.

Keywords: multiplet theory.

1. Introduction

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.

2. When should one apply multiplet theory?

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.

  • κ1: the core-hole potential.

  • κ2: the core-hole lifetime and its associated broadening.

  • κ3: the core-hole spin–orbit coupling.

  • κ4: core-hole-induced screening effects, including charge transfer and shake states.

  • κ5: the core hole–valence hole exchange interaction.

  • κ6: the higher order term of the core hole–valence hole exchange interaction.

  • κ7: core hole–valence hole multipole interactions.

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[link]), WIEN2k (Blaha, 2021[link]) 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[link] and 2[link]).

  • κ1. The core-hole potential shifts the valence states to lower energy, thereby creating the option of excitonic states. The energy shift has no effect on the spectral shape, and because the exact excitation energy cannot be calculated with an accuracy of better than 1.0 eV, the theoretical spectral shape is usually shifted to align with the experiment.

    [Figure 1]

    Figure 1

    The copper 2p XAS spectrum indicating the effects of the binding energy affected by the core-hole potential (κ1), the 2p lifetime broadening (κ2) and the 2p spin–orbit splitting (κ3).

    [Figure 2]

    Figure 2

    The cerium 3d XAS spectrum indicating the effects of 3d spin–orbit splitting (κ3), charge transfer (κ4) and the combined multiplet effects (κ5,6,7).

  • κ2. Core holes have lifetimes of the order of femtoseconds. Using Heisenberg's uncertainty relation, this yields lifetime broadenings that are approximately 0.4 eV for 2p core holes in 3d systems and 1.5 eV for 1s core holes in 3d systems.

  • κ3. The core-hole spin–orbit coupling creates two separated features in the X-ray absorption spectrum, historically having the names L3 and L2 edges for 2p core holes. Without the effects mentioned in κ4 and κ5 and without the inclusion of the ground-state spin–orbit coupling, the L3 and L2 spectra relate to a 2p3/2 core hole and a 2p1/2 core hole, respectively. This implies that they have an integral ratio of 2:1 and also implies that the spin sum rule will be valid. The 2p core-hole coupling is 11.5 eV for 3d8 Ni2+ and 107 eV for 4d8 Pd2+.

  • κ4. Core-hole-induced screening, including charge-transfer and shake states. The screening of the core hole can be different for different empty states; for example, localized (d and f) states will have a tendency to more strongly screen the core hole. This modifies their energy position and occupation with respect to the ground state. In the case of localized (d and f) states these state-specific screening effects can give rise to separate features in the spectral shapes known as charge-transfer peaks. The 3d XAS spectrum of CeO2 is a prototype example (Butorin et al., 1996[link]).

  • κ5. The core hole–valence hole exchange interaction, in other words the direct spin–spin coupling of the core spin and the valence spins, is described in multiplet theory with the [G^{1}_{pd}] Slater integral. [G^{1}_{pd}] is 4.6 eV for 3d8 Ni2+ and 1.5 eV for 4d8 Pd2+; in other words, the 2p3d Slater integrals are three times larger than the 2p4d Slater integrals.

  • κ6. The higher order term of the core hole–valence hole exchange interaction is described with the [G^{3}_{pd}] Slater integral. [G^{3}_{pd}] is 2.6 eV for 3d8 Ni2+ and 0.88 eV for 4d8 Pd2+.

  • κ7. The core hole (2p)–valence hole (3d) multipole interactions are described with the [F^{2}_{pd}] Slater integral. [F^{2}_{pd}] is 6.2 eV for 3d8 Ni2+ and 1.8 eV for 4d8 Pd2+.

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 L2,3 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.

3. Atomic multiplet theory

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[link]) and XMCD spectra (Goedkoop et al., 1988[link]) 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.

4. Crystal field multiplet theory

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[link], 2005[link]; van der Laan, 2006[link]; de Groot & Kotani, 2008[link]; van Veenendaal, 2015[link]).

5. Charge-transfer multiplet theory

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[link]; Fig. 3[link]).

[Figure 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[link]; Fig. 4[link]).

[Figure 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.

6. Semi-empirical multiplet codes

The crystal field multiplet model has been used to explain the 3p XAS spectra of transition-metal systems with a localized transition from 3dN to 3p53dN+1 (Shin et al., 1981[link]). The charge-transfer model has been used to explain the screening effects in X-ray photoemission (Kotani & Toyozawa, 1974[link]). 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[link]; Groot et al., 1990a[link],b[link]; van der Laan, 1991[link]; van der Laan & Kirkman, 1991[link]; Okada & Kotani, 1992[link]). 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[link]). An alternative interface is provided by the Missing interface (Gusmeroli, 2006[link]). 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[link]; Crocombette & Jollet, 1996[link]; Fernández-Rodríguez et al., 2015[link]). 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[link]). 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.

7. First-principles multiplet codes

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.

7.1. Cluster multiplets on solid-state electronic structure

Maurits Haverkort has developed an approach in which in the first step of the calculation the ground-state 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 real-space cluster involving the transition-metal 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 first-principles calculations or its usage as a semi-empirical 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[link]). An equivalent line of approach has been pursued by Uozumi and Hariki, where the emphasis in the publications is on X-ray photoemission, but the program is for 2p XAS equivalent to the Quanty approach (Hariki et al., 2017[link]).

7.2. Cluster multiplets on cluster electronic structure calculations

The DFT-CI model has been applied to transition-metal oxides (Ikeno et al., 2006[link], 2011[link]; Ikeno, 2016[link]). The multiplet effects are treated correctly, but the method still has some limitations with regard to the treatment of charge-transfer effects. A similar approach based on the ADF code has been developed (Ramanantoanina & Daul, 2017[link]). The MultiX program allows crystal field multiplet calculations based on a real-space cluster around the absorbing transition-metal ion (Uldry et al., 2012[link]).

7.3. Restricted active-space calculations

Starting from quantum-chemistry codes, a number of first-principles calculational programs are being developed based on restricted active-space (RAS) methods. The RAS method is an ab initio method based on the multiconfigurational self-consistent field approach. The general approach is the complete active-space (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[link]), Uppsala (Pinjari et al., 2014[link]), Groningen (Klooster, 2015[link]) and Rostock (Preusse et al., 2016[link]). An overview of the efficiency of RAS calculations has been reported for iron complexes (Pinjari et al., 2016[link]). A method based on double excitation within CI has been used for Fe L edges (Otero et al., 2009[link]). The group in Müllheim is pursuing a RAS approach from the starting point of a restricted-open-shell configuration interaction with singles (ROCIS; Roemelt et al., 2013[link]; Maganas et al., 2014[link]).

7.4. Bethe–Salpeter and time-dependent DFT

Bethe–Salpeter (BSE) approaches explicitly describe single-hole single-electron excitations (Shirley, 2005[link]; Vinson et al., 2011[link]). An extended BSE calculation for 2p XAS has been incorporated into the WIEN2k code (Laskowski & Blaha, 2010[link]), and Kruger has developed the multi-channel multiple-scattering (MCMS) method to calculate the 2p XAS spectrum of TiO2 (Krüger, 2010[link]). A problem with the BSE approach is that it does not include the multiplet interactions correctly for open-shell systems. The group in Trieste calculated molecules and solids using time-dependent DFT (TD-DFT) within the ADF code (Stener et al., 2003[link]; Fronzoni et al., 2009[link]). Similarly to the BSE approach, in TD-DFT the multiplet effects are also not incorporated exactly. TD-DFT essentially treats single-hole single-electron excitations and neglects excitations that involve two electrons to make a transition. Due to the very large multiplet interactions, such multi-electron interactions are very important in 2p XAS.

8. Concluding remarks

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.

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