# 1s2p RIXS Calculations for 3d Transition Metal Ions in Octahedral Symmetry

Authors: Patric Zimmermann; Myrtille O. J. Y. Hunault; Frank M. F. de Groot
Journal: Journal of Spectroscopy
 (2018)

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## Abstract

We present a series of 1s2p resonant inelastic X-ray scattering (RIXS) calculations for 3d transition metal ions in octahedral symmetry covering each ground state between 3d0 and 3d9. The calculations are performed in octahedral (Oh) symmetry using the crystal field multiplet theory. We discuss the crystal field effects and the selection rules with respect to the 1s2p RIXS pre-edge and compare their final state energies with the corresponding 2p X-ray absorption spectrum (XAS). The calculations provide a detailed understanding of 1s2p RIXS and serve as a basis for the future analysis of experimental spectra and also as a starting point for calculations that add additional channels including the nonlocal peaks.


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## Body

## 1. Introduction

Throughout the past decades, X-rayabsorption spectroscopy (XAS) as well as X-ray emission spectroscopy (XES) has played an important role towards the study of the electronic structure of transition metal complexes. The advent of synchrotron light sources has made high flux and high-resolution measurements accessible to researchers worldwide. Especially, the second order resonant inelastic X-ray scattering (RIXS) process, where the X-ray emission spectra are measured as a function of the incident X-ray energy, has proven to be a valuable instrument with respect to the investigation of the electronic structure. Such resonant measurements are nowadays routinely performed at the synchrotrons around the world.Here, we will focus on 1s2p RIXS measurements at the K preedge of the 3d transition metals, where a 1s core electron is promoted into the 3d band (electric quadrupole 1s XAS) and the subsequentKα1,2 decays (electric dipole 2p XES) are observed, which yields a two-dimensional RIXS map. In the following, such spectra will be referred to as 1s2p RIXS. In 2p XAS on the other hand, an electron is promoted from the 2p shell via a dipole transition into the 3d level. Both types of spectra, the two-dimensional 1s2p RIXS (1s XAS, 2p XES) and the one-dimensional 2p XAS, are illustrated in Figure 1.Figure 1
Scheme (a) illustrates the different pathways in the two-step 1s2p RIXS process, composed of the 1s XAS (K pre-edge) and 2p XES (Kα decays) transitions and the direct 2p XAS (L2,3-edge). The term scheme (b) illustrates the analogy between the energy transfer ET in 1s2p RIXS and 2p XAS.
(a)
(b)We limit our discussion of 1s2p RIXS to the pre-edge excitonic states, and we will not discuss the 1s2p RIXS plane related to excitations at the main edge and at higher energies. The 1s2p RIXS plane related to excitations at the edge can be explained from a convolution of the 1s XAS spectral shape and the 1s2p nonresonant XES spectral shape [1, 2].Most transition metal compounds show a pre-edge structure in K-edge absorption (1s XAS) which relates to the local and nonlocal electronic structure as well as the symmetry of the system under study. Usually, the shape of the pre-edge is associated with the quadrupole transitions from the 1s shell into thelocal 3d orbitals.In the case of inversion symmetry, local 4p3d mixing is forbidden by symmetry, but a nonlocal mixing of the local 4p orbitals with the 3d orbitals of the neighbouring ligands (→nonlocal) can alter the pre-edge structure and produce additionally the so-called nonlocal peaks [3–8] (e.g., TiO2 [3] and CrO2 [9]).When there is no inversion symmetry, the quadrupole (1s → 3d) and dipole (1s → 4p) peaks can mix locally in the pre-edge leading to an additional dipole contribution to the quadrupole peaks. For example, distortions or vibrational excitations can break the local inversion symmetry of Oh leading to some local dipole character in the pre-edge [10–12]. In addition, the tail of the much stronger dipole character of the main edge (1s → 4p) overlaps, but the X-ray emission due to this tail absorption can be subtracted from the 1s2p RIXS spectra. Due to this, the exact quadrupole and dipole contributions in the K pre-edge can be difficult to distinguish in experimental spectra. An exact analysis of such cases would go beyond the scope of the present paper, but it will be mentioned when relevant.The motivation for 1s2p RIXS has its origin mainly in two aspects:(1)
High-resolution wavelength-dispersive detection of theKα emission enables for the so-called high-energy resolution fluorescence-detected (HERFD) spectra. The advantage here is the effective suppression of the core-hole lifetime broadening due to the two-step 1s XAS → 2p XES process.
(2)
The final state configuration 1s22p53dN + 1 is identical in 1s2p RIXS and 2p XAS, which enables to access “L2,3-edge information” with hard X-rays. However, it shall be noted that the different selection rules (quadrupole versus dipole) and the inclusion of an intermediate state in 1s2p RIXS can lead to significant differences in the spectra.In this context, we are comparing the two-step 1s2p RIXS process with the direct 2p XAS (L2,3-edge) spectra for 3d transition metal compounds. Based on the crystal field theory (CFT), we calculated at least one spectrum for each 3dN ground state, where N∈0,…,9. We aim to give an overview of the general structure of 1s2p RIXS for the 3dN transition metal series to illustrate their specific behaviour, but the calculated spectra shall also serve as a reference for future analysis.In 1s2p RIXS, the 1s core hole created in 1s XAS interacts only weakly with the valence electrons. Since the 1s shell has no orbital momentumL, there is no spin-orbit coupling with the other open shells. As we will see, this is important when comparing 1s2p RIXS with 2p XAS [13]. The consequences on the outcome of the spectra will be discussed throughout this paper.The paper is organised as follows. In Section2, we describe the theoretical background of the calculation of 1s2p RIXS and the resulting selection rules. In Section 3, we give the computational details. In Section 4, we analyse the 1s2p RIXS of three didactic cases in more detail, and in Section 5, we describe a series of the remaining 3dN cases including a short discussion of related experiments.


## 2. Theory

The necessary background being the theoretical framework and the computational tools used in this study are described in the following.


### 2.1. 1s2p RIXS with Kramers-Heisenberg

In 1s2p RIXS experiments, the incident energy is tuned around a resonance related to a 1s → 3d transition (1s XAS). This excitation is followed by a subsequent radiative 2p → 1s decay (2p XES). The emitted photons are detected with a wavelength-dispersive detector (crystal analyser) reaching in the hard X-ray range sub-eV resolutions. Often this translates into a bandwidth of less than the lifetime broadening of the spectral features under study. This is often referred to as theRIXS sharpening effect overcoming the core-hole lifetime broadening which is related to the RIXS cross-section [14, 15].The RIXS process is described as a two-step photon-in photon-out scheme in which the incident photons undergo an inelastic scattering around a core resonance followed by a characteristic X-ray emission. The theoretical modelling of the 1s2p RIXS cross-sectionσ is achieved with the Kramers-Heisenberg formula for the second order quantum process which includes the 1s XAS, 2p XES, and interference effects [14–19]:
(1)d2σdΩdω=∑f∑nfT^2nnT^1iEi−En+ℏΩ+iγn/22×γf/2πEi−Ef+ℏΩ−ℏω2+γf2/4,with incident photons of energy Ein=ℏΩ, the inelastically scattered photons of energy Eout=ℏω, and the energy transfer ET=ℏΩ−ℏω. The two-step process relates to the two transition operators, T^1 for the quadrupole 1s XAS and T^2 for the dipole 2p XES, describing the 1s2p RIXS cross-section σ as the transition from the initial state i into the final states f via all possible intermediate states n of the system with energies Ei, Ef, and En, respectively. And finally, γn and γf are the natural line widths of the intermediate and final states. This means that the 1s XAS is naturally broadened by the intermediate state lifetime γn and 2p XES is naturally broadened by the final state lifetime γf [15, 20].


### 2.2. Crystal Field Theory: Local Model

We recapitulate briefly the aspects of the framework of the crystal field theory that are important for the present paper. Further details can be found in other references [13, 21]. The initial, intermediate, and final states i, n, and f, respectively, are described each by the corresponding Hamiltonian ℋ:
(2)ℋ=ℋkin+ℋe/N+ℋe/e+ℋSO+ℋCF,where ℋkin=∑Npi2/2m is the kinetic energy of the electrons with the momentum p and mass m, ℋe/N=∑N−Ze2/ri is the electrostatic interaction of the electron i of charge e with the nucleus at radius ri and charge Z, ℋe/e=∑pairse2/rij is the electron-electron Coulomb interaction which is determined by the direct Coulomb repulsion and the Coulomb exchange interaction described by the Slater integrals Fk and Gk, respectively, and HSO=∑Nζrili⋅si is the spin-orbit interaction on every open shell. These first four terms, together with a given ground state ψ, describe the isolated atom in each state [21].The electron-electron interactions of an electronic configuration give rise to multiple term symbols. The term symbol indicates a total orbital momentℒ, total spin moment S, and total angular moment J , with ℒ−S≤J≤ℒ+S. This is the ℒ.S coupling scheme or Russell-Saunders coupling, which will be used throughout the present paper (Table 1). In the absence of spin-orbit coupling, all terms with the same ℒ and S have the same energy, giving an energy level that is 2ℒ+12S+1-fold degenerate. When spin-orbit coupling is included, ℒ and S lose their meaning and the terms are split in energy according to their J value, each with a degeneracy of 2J+1. One can assign to these states a so-called Russel-Saunders term symbol 2S+1XJ, where X is representing the orbital momentum ℒ; for values of 0, 1, 2, and 3, one writes S, P, D, and F, respectively [21, 22]. For example, a single electron in an s shell is given as 2S1/2, and a single electron in a p shell is represented as 2P1/2 and 2P3/2.Table 1
Russell-Saunders term symbols for the3dN free ion configurations giving the number of states, the corresponding number of energy levels, and the terms for the given configuration. Note that dN gives the same terms as d10−N. A more detailed list can be found in Core Level Spectroscopy of Solids [13] (p. 103).
Config
States
Energy levels
Ground term
Excited terms
3d1, 3d9
10
1
2D
—
3d2, 3d8
45
5
3F
2P, 1G, 1D, 1S
3d3, 3d7
120
8
4F
4P, 2H, 2G, 2F, 2·2D, 2P
3d4, 3d6
210
16
5D
3H, 3G, 2·3F, 3D, 2·3P, 1I, 2·1G, 1F, 2·1D, 2·1S
3d5
252
16
6S
4G, 4F, 4D, 4P, 2I, 2H, 2·2G, 2·2F, 3·2D, 2P, 2SThe term symbol describes the symmetry aspects, but it does not say anything about its relative energy [14]. The electron-electron repulsion and the spin-orbit coupling define the relative energy of the different terms within a configuration. Here, Hund’s rules offer a convenient way to determine the state lowest in energy (maxS, maxℒ, and maxJ if more than 1/2 is filled, otherwise minJ) [21].The total number of states per configuration is calculated via thebinomial coefficients [21]:
(3)3dN:10N=10!N!·10−N!,1s13dN+1:21·10N+1=2·10!N+1!·10−N+1!,2p53dN+1:61·10N+1=6·10!N+1!·10−N+1!.The crystal field multiplet HamiltonianℋCF extends the atomic Hamiltonian with an electrostatic field created by the neighbouring atoms in the solid state. Because a large range of systems consist of a transition metal ion surrounded by six neighbouring atoms, where these neighbours are positioned at each corner of an octahedron centred around the transition metal atom, we will focus on the case of the cubic crystal field. The neighbours form a so-called octahedral field, which belongs to the Oh point group. Thus, we discuss all ions throughout this paper in an octahedral symmetry (Oh). We are aware that this is not necessarily true for all 3d transitions metal ions (e.g., CrO2 (Cr4+) and TiO2 (Ti4+) have tetragonal (D4h) symmetry; Fe2O3 (Fe3+) has C3 symmetry). However, we make this simplification since we are aiming to present and discuss the elementary differences for each case in an analogue and comparable way.In a single-electron picture, in a spherical environment, the 3d orbitals are degenerate. In Oh, the cubic crystal field induces the splitting of the five 3d orbitals into two symmetrically different groups: (1) the 3-fold t2g orbitals (xy, xz, and yz), which point in between the six metal-ligand bonds and thus participate in π bondings with the surrounding ligands, and (2) the 2-fold eg orbitals (z2 and x2−y2), which point along the metal-ligand bonds and therefore relate to σ bondings. The splitting between the eg and t2g orbitals is defined by the strength of the crystal field and is described by the parameter 10Dq [13].The symmetry of the multielectronic term changes from spherical symmetry (O3) to octahedral symmetry (Oh) causing the ℒ term (S, P, D, etc.) to branch to an Oh irreducible representation (IRREP): S (O3) branches into an A1 (Oh) symmetry state, P (O3) branches into a T1 (Oh) symmetry state, and D (O3) branches into T2 (Oh) and E (Oh) symmetry states. Because the point group O3 (resp., Oh) contains the inversion, a parity information should be added for gerade or ungerade to the term, + or − for the atomic (spherical) terms, and g or u for the crystal field terms. The 3dN initial state and the intermediate state 1s13dN + 1 in 1s2p RIXS are gerade (g), and the final state 2p53dN + 1 is, due to the open 2p shell, ungerade. In the following, the parity of the spherical term will be omitted for simplicity. All the branchings are given in Table 2, which we will use extensively throughout this paper.Table 2
Branching of theS, ℒ, and J atomic terms in an octahedral crystal field (O3→Oh) [28] (p.262). The first columns can be either the orbital momentum, the total spin S, or the total angular momentum J. The g for gerade and the u for ungerade must be added to indicate the parity.
O3
→
Oh
S, ℒ, or J
→
Term symbol
0 (S)
→
A1
1 (P)
→
T1
2 (D)
→
E ⊕ T2
3 (F)
→
T1 ⊕ T2 ⊕ A2
4 (G)
→
A1 ⊕ E ⊕ T1 ⊕ T2
5 (H)
→
E ⊕ 2 T2 ⊕ T2
6 (I)
→
A1 ⊕ T1 ⊕ E ⊕ 2 T2 ⊕ A2
7
→
2T2 ⊕ E ⊕ 2 T2 ⊕ A2
8
→
A1 ⊕ 2 T1 ⊕ 2E ⊕ 2 T2
1/2
→
E1/2
3/4
→
F3/2
5/2
→
F3/2 ⊕ E5/2
7/2
→
E1/2 ⊕ F3/2 ⊕ E5/2
9/2
→
E1/2 ⊕ 2F3/2
11/2
→
E1/2 ⊕ 2F3/2 ⊕ E5/2
13/2
→
E1/2 ⊕ 2F3/2 ⊕ 2E5/2
15/2
→
E1/2 ⊕ 3F3/2 ⊕ E5/2The relative energies of the Oh IRREPs are calculated by adding the effect of the cubic crystal field 10Dq to the atomic state energies. The diagrams of the relative energies with respect to the cubic crystal field, often without spin-orbit coupling, are known as Tanabe-Sugano diagrams. More generally, we will call them energy level diagram (ELD). They will appear in the context of the quadrupole 1s XAS because the ELD for any intermediate state in 1s2p RIXS with a 1s13dN + 1 electron configuration is the same as the diagram for a 1s23dN + 1 configuration times the 2A2g term to include the 1s core hole.When spin-orbit coupling (SOC) and crystal field are both to be included, there are in principle two ways to derive the term symbols: (1) by first deriving the branching of the uncoupled atomic terms (withℒ and S) in Oh symmetry and then by deriving the spin-orbital coupling or (2) by first deriving the atomic J quantum number in the ℒ.S coupling scheme and then deriving the branching of the J value in the Oh point group. Both methods yield identical final results and the identical ground state for a given Hamiltonian (see, e.g., Figure 2).Figure 2
Initial state term schemes for Cu2+ (3d9) illustrating the effect of the 3d SOC ζ3d and Oh crystal field 10Dq and the resulting splittings of the atomic ground state term 2D.The order in which the derivation should be done is related to the relative weights of the spin-orbit coupling and crystal field terms in the Hamiltonian. In the initial state of 1s2p RIXS, the 3d spin-orbit couplingζ3d is small (a few tens of meV) compared to the usual values of crystal field splitting 10Dq (a few eV). As a result, in cases with neglectable spin-orbit interaction, the mixing between the spin and the orbital momenta is weak and thus ℒ and S can still be used to describe the initial state. Similarly, for the intermediate state, the 1s core hole with ℒ=0 does not have spin-orbit coupling. Therefore, in these cases, the ℒ.S coupling is achieved after the crystal field branching. (See Xu et al. [23] for details on ℒ.S coupling.)In the present paper, we are focusing on the 3d transition metal ions represented by a series of cases with a ground state electron configuration ranging from 3d0 to 3d9. Depending on the number of 3d electrons, the value of the total spin S varies between 0 (minimum) and 5/2 (maximum). The branching for each value of S in Oh is given in Table 2. The ℒ.S-coupled term, also called J value IRREP, in Oh symmetry is obtained from the direct product of the ℒ IRREP and the S IRREP. For this calculation, the direct product tables Tables 3, 4, and 5 can be used. Note that the total orbital multiplicity is always maintained, for example,
(4)T1g⏟Size=3⊗T2g⏟Size=3⏟Totalsize:3∗3=9=A2g⏟Size=1⊕Eg⏟Size=2⊕T1g⏟Size=3⊕T2g⏟Size=3⏟Totalsize:1+2+3+3=9.Table 3
Direct products of representations (product table) inOh symmetry.
Oh
A1g
A2g
Eg
T1g
T2g
A1g
A1g
A2g
Eg
T1g
T2g
A2g
A1g
Eg
T2g
T1g
Eg
A1g ⊕ A2g ⊕ Eg
T1g ⊕ T2g
T1g ⊕ T2g
T1g
A1g ⊕ Eg ⊕ T1g ⊕ T2g
A2g ⊕ Eg ⊕ T1g ⊕ T2g
T2g
A1g ⊕ Eg ⊕ T1g ⊕ T2gTable 4
Direct products of representations (cont.) inOh symmetry (dipole).
Oh
A1u
A2u
Eu
T1u
T2u
A1g
A1u
A2u
Eu
T1u
T2u
A2g
A2u
A1u
Eu
T2u
T1u
Eg
Eu
Eu
A1u ⊕ A2u ⊕ Eu
T1u ⊕ T2u
T1u ⊕ T2u
T1g
T1u
T2u
T1u ⊕ T2u
A1u ⊕ Eu ⊕ T1u ⊕ T2u
A2u ⊕ Eu ⊕ T1u ⊕ T2u
T2g
T2u
T1u
T1u ⊕ T2u
A2u ⊕ Eu ⊕ T1u ⊕ T2u
A1u ⊕ Eu ⊕ T1u ⊕ T2u
A1u
A1g
A2g
Eg
T1g
T2g
A2u
A1g
Eg
T2g
T1g
Eu
A1g ⊕ A2g ⊕ Eg
T1g ⊕ T2g
T1g ⊕ T2g
T1u
A1g ⊕ Eg ⊕ T1g ⊕ T2g
A2g ⊕ Eg ⊕ T1g ⊕ T2g
T2u
A1g ⊕ Eg ⊕ T1g ⊕ T2gTable 5
Direct products of representations (cont.) in Oh symmetry (SOC). Note that in this table, the parity (g: gerade; u: ungerade) was omitted here as the table can be used for either case. For accurate labelling, the parity has to be added to the term symbol, where gerade times gerade equals gerade and gerade times ungerade equals an ungerade term symbol.
Oh
E1/2
E5/2
F3/2
A1
E1/2
E5/2
F3/2
A2
E5/2
E1/2
F3/2
E
F3/2
F3/2
E1/2 ⊕ E5/2 ⊕ F3/2
T1
E1/2 ⊕ F3/2
E1/2 ⊕ F3/2
E1/2 ⊕ E5/2 ⊕ 2F3/2
T2
E5/2 ⊕ F3/2
E1/2 ⊕ F3/2
E1/2 ⊕ E5/2 ⊕ 2F3/2
E1/2
A1 ⊕ T1
A2 ⊕ T2
E ⊕ T1 ⊕ T2
E5/2
A1 ⊕ T1
E ⊕ T1 ⊕ T2
F3/2
A1 ⊕ A2 ⊕ E ⊕ 2 T1 ⊕ T2 ⊕ T2On the contrary, in the final state configuration, the strong spin-orbit couplingζ2p of the 2d hole (a few eV) is dominant. This induces a strong mixing between the spin and orbital momenta and resulting in ℒ and S not being good quantum numbers anymore. Instead, the total angular momentum J must be used to describe the final states. The effect of the Oh crystal field is added as the corresponding branching of the atomic J value IRREP. The branching for each value of J is also given in Table 2. The obtained IRREPs are either A1u, T1u, Eu, T2u, or A2u for integer S or J. And they are E1/2u, F3/2u, and E3/2u for half-integer values of S or J.As we will see across the 3dN series, the information of ℒ, S, and J is important for the description of the ground state of the initial state electron configuration when comparing the one-step 2p XAS with the two-step 1s2p RIXS selection rules (see Section 2.3). Therefore, in an Oh crystal field, we will indicate the uncoupled atomic term (with ℒ and S) along with the total angular momentum J value IRREP using the following notation: 2S+1ℒ [J].We illustrate this here with an example for Ni2+ (3d8) with the atomic ground state term 3F (O3).
(5)3F→Oh3A2g+3T1g+3T2g.Note that the spin multiplicity should be kept and is part of the term symbol because spin-orbit coupling is not yet included. The3A2g symmetry represents in this case the state being the lowest in energy, and it is therefore the ground state term. In the next step, we derive the total angular momentum J IRREPs in the ℒ.S coupling scheme (S=1→⊗T1g) for the three terms derived in (5):
(6)3A2g⊗T1g=T2gsplittingofthe3A2gtermdueto3dSOC,(7)3T1g⊗T1g=A1g+Eg+T1g+T2gsplittingofthe3T1gtermdueto3dSOC,(8)3T2g⊗T1g=A2g+Eg+T1g+T2gsplittingofthe3T2gtermdueto3dSOC.In (6), we find the T2g total angular momentum IRREP as the ground state for the example 3d8 in an octahedral crystal field. To summarise all this information in a condensed form, we write
(9)3A2gT2g.This translates as aterm without SOC withSandℒ[term with SOC usingJ].In other words, in (9), the term 3A2g is before the inclusion of SOC, where the spin multiplicity is 3 and the orbital momentum is given as A2g. And the T2g term is the symmetry after spin-orbit interaction has been included (also identified as J value IRREP).Finally, we note that charge transfer effects (e.g., the interaction with a3dN−1L¯ configuration) are neglected but their relative importance will be discussed where appropriate throughout the 3dN series.


### 2.3. Selection Rules

The selection rules are the constrains that define the possibility of a transition between two states with a given transition operator. In other words, for 1s2p RIXS, the selection rules are the conditions for which the matrix elementsfT^2n and nT^1i in the Kramers-Heisenberg equation (1) are nonzero. The operators T^1 and T^2 describe the interaction of the photon with matter. They are defined by the interaction Hamiltonian p⋅A, where p is the momentum of the photon and A is the vector potential (e.g., the amplitude) of the electromagnetic field of the light. The electromagnetic interaction term is ε⋅peikr, where ε is the polarisation of the photon and k is the propagation vector of the photon. In a multipole expansion, the eikr term can be decomposed in a Taylor series (the selection rules can be derived from the decomposition of the electron dipole/quadrupole transition matrix element into angular and radial parts using the Wigner-Eckart theorem [13, 24], where the triangular relations of the symbol determine the selection rules) which leads to the electric dipole and the electric quadrupole terms [25]. Further details can be found in Core Level Spectroscopy of Solids [13] and elsewhere [16, 24, 25].Due to the electric dipole operator being the first-order term of this decomposition, the result is that theJ quantum number can only change by a value of 0 or 1. Thus, the electric dipole selection rule translates to ΔJ=+1,0,or−1. Because the electric quadrupole operator is the second-order term of this decomposition, the resulting selection rule translates to ΔJ=−2,−1,0,+1,or+2.However, when the spin-orbit coupling is neglected (or weak), the orbital momentumℒ and the spin momentum S can still be considered good quantum numbers. In this case, the transitions are spin-conserving (ΔS=0) and only ℒ changes. Due to the fact that light carries an orbital momentum of ℒ=1, this implies for electric dipole transitions Δℒ=+1 or −1. For the second-order term, the electric quadrupole transitions, this implies Δℒ=+2,0, or −2 [21].As a result, the transition operatorT^1 of the 1s XAS absorption step promoting an electron from 1s to 3d (Δℒ=+2) is an electric quadrupole term, and this step is forbidden as an electric dipole transition. On the contrary, the transition operator T^2 for the 2p XES decay from 2p to 1s (Δℒ=−1) is an electric dipole term. Similarly, the transition operator T^ of the 2p XAS absorption step from 2p to 3d (Δℒ=+1) is also an electric dipole term.These selection rules, defined in spherical symmetry, translate into the point group symmetry of the absorbing ion. In cubic symmetry (Oh), the selection rules of each operator depend on its symmetry properties. The electric dipole operator (first-rank tensor) behaves as the J=1 IRREP (O3) and branches into T1u in Oh. The electric quadrupole operator (second-rank tensor) behaves as a J=2 IRREP (O3) and branches in Oh into T2g⊕Eg (see Table 2). The transition matrix element between two states i,f with the IRREPs Γi and Γf and the transition operator T^ is nonzero if the direct product Γi⊗ΓT^ contains Γf, where ΓT^ is the IRREP of the transition operator. In other words, the final state IRREPs Γf accessible via the transition operator T^ are given by the direct product Γi⊗ΓT^. The selection rules are derived from the direct product tables for the Oh point group (see Tables 3, 4, and 5) and give the symmetry of the accessible states.The selection rules will be commented in more detail for each case of the 3dN series throughout this paper. Magnetic and natural dichroisms, which are a property from the crystal and not from the point group of the absorbing transition metal ion [26], will not be discussed here.


## 2.1. 1s2p RIXS with Kramers-Heisenberg

In 1s2p RIXS experiments, the incident energy is tuned around a resonance related to a 1s → 3d transition (1s XAS). This excitation is followed by a subsequent radiative 2p → 1s decay (2p XES). The emitted photons are detected with a wavelength-dispersive detector (crystal analyser) reaching in the hard X-ray range sub-eV resolutions. Often this translates into a bandwidth of less than the lifetime broadening of the spectral features under study. This is often referred to as theRIXS sharpening effect overcoming the core-hole lifetime broadening which is related to the RIXS cross-section [14, 15].The RIXS process is described as a two-step photon-in photon-out scheme in which the incident photons undergo an inelastic scattering around a core resonance followed by a characteristic X-ray emission. The theoretical modelling of the 1s2p RIXS cross-sectionσ is achieved with the Kramers-Heisenberg formula for the second order quantum process which includes the 1s XAS, 2p XES, and interference effects [14–19]:
(1)d2σdΩdω=∑f∑nfT^2nnT^1iEi−En+ℏΩ+iγn/22×γf/2πEi−Ef+ℏΩ−ℏω2+γf2/4,with incident photons of energy Ein=ℏΩ, the inelastically scattered photons of energy Eout=ℏω, and the energy transfer ET=ℏΩ−ℏω. The two-step process relates to the two transition operators, T^1 for the quadrupole 1s XAS and T^2 for the dipole 2p XES, describing the 1s2p RIXS cross-section σ as the transition from the initial state i into the final states f via all possible intermediate states n of the system with energies Ei, Ef, and En, respectively. And finally, γn and γf are the natural line widths of the intermediate and final states. This means that the 1s XAS is naturally broadened by the intermediate state lifetime γn and 2p XES is naturally broadened by the final state lifetime γf [15, 20].


## 2.2. Crystal Field Theory: Local Model

We recapitulate briefly the aspects of the framework of the crystal field theory that are important for the present paper. Further details can be found in other references [13, 21]. The initial, intermediate, and final states i, n, and f, respectively, are described each by the corresponding Hamiltonian ℋ:
(2)ℋ=ℋkin+ℋe/N+ℋe/e+ℋSO+ℋCF,where ℋkin=∑Npi2/2m is the kinetic energy of the electrons with the momentum p and mass m, ℋe/N=∑N−Ze2/ri is the electrostatic interaction of the electron i of charge e with the nucleus at radius ri and charge Z, ℋe/e=∑pairse2/rij is the electron-electron Coulomb interaction which is determined by the direct Coulomb repulsion and the Coulomb exchange interaction described by the Slater integrals Fk and Gk, respectively, and HSO=∑Nζrili⋅si is the spin-orbit interaction on every open shell. These first four terms, together with a given ground state ψ, describe the isolated atom in each state [21].The electron-electron interactions of an electronic configuration give rise to multiple term symbols. The term symbol indicates a total orbital momentℒ, total spin moment S, and total angular moment J , with ℒ−S≤J≤ℒ+S. This is the ℒ.S coupling scheme or Russell-Saunders coupling, which will be used throughout the present paper (Table 1). In the absence of spin-orbit coupling, all terms with the same ℒ and S have the same energy, giving an energy level that is 2ℒ+12S+1-fold degenerate. When spin-orbit coupling is included, ℒ and S lose their meaning and the terms are split in energy according to their J value, each with a degeneracy of 2J+1. One can assign to these states a so-called Russel-Saunders term symbol 2S+1XJ, where X is representing the orbital momentum ℒ; for values of 0, 1, 2, and 3, one writes S, P, D, and F, respectively [21, 22]. For example, a single electron in an s shell is given as 2S1/2, and a single electron in a p shell is represented as 2P1/2 and 2P3/2.Table 1
Russell-Saunders term symbols for the3dN free ion configurations giving the number of states, the corresponding number of energy levels, and the terms for the given configuration. Note that dN gives the same terms as d10−N. A more detailed list can be found in Core Level Spectroscopy of Solids [13] (p. 103).
Config
States
Energy levels
Ground term
Excited terms
3d1, 3d9
10
1
2D
—
3d2, 3d8
45
5
3F
2P, 1G, 1D, 1S
3d3, 3d7
120
8
4F
4P, 2H, 2G, 2F, 2·2D, 2P
3d4, 3d6
210
16
5D
3H, 3G, 2·3F, 3D, 2·3P, 1I, 2·1G, 1F, 2·1D, 2·1S
3d5
252
16
6S
4G, 4F, 4D, 4P, 2I, 2H, 2·2G, 2·2F, 3·2D, 2P, 2SThe term symbol describes the symmetry aspects, but it does not say anything about its relative energy [14]. The electron-electron repulsion and the spin-orbit coupling define the relative energy of the different terms within a configuration. Here, Hund’s rules offer a convenient way to determine the state lowest in energy (maxS, maxℒ, and maxJ if more than 1/2 is filled, otherwise minJ) [21].The total number of states per configuration is calculated via thebinomial coefficients [21]:
(3)3dN:10N=10!N!·10−N!,1s13dN+1:21·10N+1=2·10!N+1!·10−N+1!,2p53dN+1:61·10N+1=6·10!N+1!·10−N+1!.The crystal field multiplet HamiltonianℋCF extends the atomic Hamiltonian with an electrostatic field created by the neighbouring atoms in the solid state. Because a large range of systems consist of a transition metal ion surrounded by six neighbouring atoms, where these neighbours are positioned at each corner of an octahedron centred around the transition metal atom, we will focus on the case of the cubic crystal field. The neighbours form a so-called octahedral field, which belongs to the Oh point group. Thus, we discuss all ions throughout this paper in an octahedral symmetry (Oh). We are aware that this is not necessarily true for all 3d transitions metal ions (e.g., CrO2 (Cr4+) and TiO2 (Ti4+) have tetragonal (D4h) symmetry; Fe2O3 (Fe3+) has C3 symmetry). However, we make this simplification since we are aiming to present and discuss the elementary differences for each case in an analogue and comparable way.In a single-electron picture, in a spherical environment, the 3d orbitals are degenerate. In Oh, the cubic crystal field induces the splitting of the five 3d orbitals into two symmetrically different groups: (1) the 3-fold t2g orbitals (xy, xz, and yz), which point in between the six metal-ligand bonds and thus participate in π bondings with the surrounding ligands, and (2) the 2-fold eg orbitals (z2 and x2−y2), which point along the metal-ligand bonds and therefore relate to σ bondings. The splitting between the eg and t2g orbitals is defined by the strength of the crystal field and is described by the parameter 10Dq [13].The symmetry of the multielectronic term changes from spherical symmetry (O3) to octahedral symmetry (Oh) causing the ℒ term (S, P, D, etc.) to branch to an Oh irreducible representation (IRREP): S (O3) branches into an A1 (Oh) symmetry state, P (O3) branches into a T1 (Oh) symmetry state, and D (O3) branches into T2 (Oh) and E (Oh) symmetry states. Because the point group O3 (resp., Oh) contains the inversion, a parity information should be added for gerade or ungerade to the term, + or − for the atomic (spherical) terms, and g or u for the crystal field terms. The 3dN initial state and the intermediate state 1s13dN + 1 in 1s2p RIXS are gerade (g), and the final state 2p53dN + 1 is, due to the open 2p shell, ungerade. In the following, the parity of the spherical term will be omitted for simplicity. All the branchings are given in Table 2, which we will use extensively throughout this paper.Table 2
Branching of theS, ℒ, and J atomic terms in an octahedral crystal field (O3→Oh) [28] (p.262). The first columns can be either the orbital momentum, the total spin S, or the total angular momentum J. The g for gerade and the u for ungerade must be added to indicate the parity.
O3
→
Oh
S, ℒ, or J
→
Term symbol
0 (S)
→
A1
1 (P)
→
T1
2 (D)
→
E ⊕ T2
3 (F)
→
T1 ⊕ T2 ⊕ A2
4 (G)
→
A1 ⊕ E ⊕ T1 ⊕ T2
5 (H)
→
E ⊕ 2 T2 ⊕ T2
6 (I)
→
A1 ⊕ T1 ⊕ E ⊕ 2 T2 ⊕ A2
7
→
2T2 ⊕ E ⊕ 2 T2 ⊕ A2
8
→
A1 ⊕ 2 T1 ⊕ 2E ⊕ 2 T2
1/2
→
E1/2
3/4
→
F3/2
5/2
→
F3/2 ⊕ E5/2
7/2
→
E1/2 ⊕ F3/2 ⊕ E5/2
9/2
→
E1/2 ⊕ 2F3/2
11/2
→
E1/2 ⊕ 2F3/2 ⊕ E5/2
13/2
→
E1/2 ⊕ 2F3/2 ⊕ 2E5/2
15/2
→
E1/2 ⊕ 3F3/2 ⊕ E5/2The relative energies of the Oh IRREPs are calculated by adding the effect of the cubic crystal field 10Dq to the atomic state energies. The diagrams of the relative energies with respect to the cubic crystal field, often without spin-orbit coupling, are known as Tanabe-Sugano diagrams. More generally, we will call them energy level diagram (ELD). They will appear in the context of the quadrupole 1s XAS because the ELD for any intermediate state in 1s2p RIXS with a 1s13dN + 1 electron configuration is the same as the diagram for a 1s23dN + 1 configuration times the 2A2g term to include the 1s core hole.When spin-orbit coupling (SOC) and crystal field are both to be included, there are in principle two ways to derive the term symbols: (1) by first deriving the branching of the uncoupled atomic terms (withℒ and S) in Oh symmetry and then by deriving the spin-orbital coupling or (2) by first deriving the atomic J quantum number in the ℒ.S coupling scheme and then deriving the branching of the J value in the Oh point group. Both methods yield identical final results and the identical ground state for a given Hamiltonian (see, e.g., Figure 2).Figure 2
Initial state term schemes for Cu2+ (3d9) illustrating the effect of the 3d SOC ζ3d and Oh crystal field 10Dq and the resulting splittings of the atomic ground state term 2D.The order in which the derivation should be done is related to the relative weights of the spin-orbit coupling and crystal field terms in the Hamiltonian. In the initial state of 1s2p RIXS, the 3d spin-orbit couplingζ3d is small (a few tens of meV) compared to the usual values of crystal field splitting 10Dq (a few eV). As a result, in cases with neglectable spin-orbit interaction, the mixing between the spin and the orbital momenta is weak and thus ℒ and S can still be used to describe the initial state. Similarly, for the intermediate state, the 1s core hole with ℒ=0 does not have spin-orbit coupling. Therefore, in these cases, the ℒ.S coupling is achieved after the crystal field branching. (See Xu et al. [23] for details on ℒ.S coupling.)In the present paper, we are focusing on the 3d transition metal ions represented by a series of cases with a ground state electron configuration ranging from 3d0 to 3d9. Depending on the number of 3d electrons, the value of the total spin S varies between 0 (minimum) and 5/2 (maximum). The branching for each value of S in Oh is given in Table 2. The ℒ.S-coupled term, also called J value IRREP, in Oh symmetry is obtained from the direct product of the ℒ IRREP and the S IRREP. For this calculation, the direct product tables Tables 3, 4, and 5 can be used. Note that the total orbital multiplicity is always maintained, for example,
(4)T1g⏟Size=3⊗T2g⏟Size=3⏟Totalsize:3∗3=9=A2g⏟Size=1⊕Eg⏟Size=2⊕T1g⏟Size=3⊕T2g⏟Size=3⏟Totalsize:1+2+3+3=9.Table 3
Direct products of representations (product table) inOh symmetry.
Oh
A1g
A2g
Eg
T1g
T2g
A1g
A1g
A2g
Eg
T1g
T2g
A2g
A1g
Eg
T2g
T1g
Eg
A1g ⊕ A2g ⊕ Eg
T1g ⊕ T2g
T1g ⊕ T2g
T1g
A1g ⊕ Eg ⊕ T1g ⊕ T2g
A2g ⊕ Eg ⊕ T1g ⊕ T2g
T2g
A1g ⊕ Eg ⊕ T1g ⊕ T2gTable 4
Direct products of representations (cont.) inOh symmetry (dipole).
Oh
A1u
A2u
Eu
T1u
T2u
A1g
A1u
A2u
Eu
T1u
T2u
A2g
A2u
A1u
Eu
T2u
T1u
Eg
Eu
Eu
A1u ⊕ A2u ⊕ Eu
T1u ⊕ T2u
T1u ⊕ T2u
T1g
T1u
T2u
T1u ⊕ T2u
A1u ⊕ Eu ⊕ T1u ⊕ T2u
A2u ⊕ Eu ⊕ T1u ⊕ T2u
T2g
T2u
T1u
T1u ⊕ T2u
A2u ⊕ Eu ⊕ T1u ⊕ T2u
A1u ⊕ Eu ⊕ T1u ⊕ T2u
A1u
A1g
A2g
Eg
T1g
T2g
A2u
A1g
Eg
T2g
T1g
Eu
A1g ⊕ A2g ⊕ Eg
T1g ⊕ T2g
T1g ⊕ T2g
T1u
A1g ⊕ Eg ⊕ T1g ⊕ T2g
A2g ⊕ Eg ⊕ T1g ⊕ T2g
T2u
A1g ⊕ Eg ⊕ T1g ⊕ T2gTable 5
Direct products of representations (cont.) in Oh symmetry (SOC). Note that in this table, the parity (g: gerade; u: ungerade) was omitted here as the table can be used for either case. For accurate labelling, the parity has to be added to the term symbol, where gerade times gerade equals gerade and gerade times ungerade equals an ungerade term symbol.
Oh
E1/2
E5/2
F3/2
A1
E1/2
E5/2
F3/2
A2
E5/2
E1/2
F3/2
E
F3/2
F3/2
E1/2 ⊕ E5/2 ⊕ F3/2
T1
E1/2 ⊕ F3/2
E1/2 ⊕ F3/2
E1/2 ⊕ E5/2 ⊕ 2F3/2
T2
E5/2 ⊕ F3/2
E1/2 ⊕ F3/2
E1/2 ⊕ E5/2 ⊕ 2F3/2
E1/2
A1 ⊕ T1
A2 ⊕ T2
E ⊕ T1 ⊕ T2
E5/2
A1 ⊕ T1
E ⊕ T1 ⊕ T2
F3/2
A1 ⊕ A2 ⊕ E ⊕ 2 T1 ⊕ T2 ⊕ T2On the contrary, in the final state configuration, the strong spin-orbit couplingζ2p of the 2d hole (a few eV) is dominant. This induces a strong mixing between the spin and orbital momenta and resulting in ℒ and S not being good quantum numbers anymore. Instead, the total angular momentum J must be used to describe the final states. The effect of the Oh crystal field is added as the corresponding branching of the atomic J value IRREP. The branching for each value of J is also given in Table 2. The obtained IRREPs are either A1u, T1u, Eu, T2u, or A2u for integer S or J. And they are E1/2u, F3/2u, and E3/2u for half-integer values of S or J.As we will see across the 3dN series, the information of ℒ, S, and J is important for the description of the ground state of the initial state electron configuration when comparing the one-step 2p XAS with the two-step 1s2p RIXS selection rules (see Section 2.3). Therefore, in an Oh crystal field, we will indicate the uncoupled atomic term (with ℒ and S) along with the total angular momentum J value IRREP using the following notation: 2S+1ℒ [J].We illustrate this here with an example for Ni2+ (3d8) with the atomic ground state term 3F (O3).
(5)3F→Oh3A2g+3T1g+3T2g.Note that the spin multiplicity should be kept and is part of the term symbol because spin-orbit coupling is not yet included. The3A2g symmetry represents in this case the state being the lowest in energy, and it is therefore the ground state term. In the next step, we derive the total angular momentum J IRREPs in the ℒ.S coupling scheme (S=1→⊗T1g) for the three terms derived in (5):
(6)3A2g⊗T1g=T2gsplittingofthe3A2gtermdueto3dSOC,(7)3T1g⊗T1g=A1g+Eg+T1g+T2gsplittingofthe3T1gtermdueto3dSOC,(8)3T2g⊗T1g=A2g+Eg+T1g+T2gsplittingofthe3T2gtermdueto3dSOC.In (6), we find the T2g total angular momentum IRREP as the ground state for the example 3d8 in an octahedral crystal field. To summarise all this information in a condensed form, we write
(9)3A2gT2g.This translates as aterm without SOC withSandℒ[term with SOC usingJ].In other words, in (9), the term 3A2g is before the inclusion of SOC, where the spin multiplicity is 3 and the orbital momentum is given as A2g. And the T2g term is the symmetry after spin-orbit interaction has been included (also identified as J value IRREP).Finally, we note that charge transfer effects (e.g., the interaction with a3dN−1L¯ configuration) are neglected but their relative importance will be discussed where appropriate throughout the 3dN series.


## 2.3. Selection Rules

The selection rules are the constrains that define the possibility of a transition between two states with a given transition operator. In other words, for 1s2p RIXS, the selection rules are the conditions for which the matrix elementsfT^2n and nT^1i in the Kramers-Heisenberg equation (1) are nonzero. The operators T^1 and T^2 describe the interaction of the photon with matter. They are defined by the interaction Hamiltonian p⋅A, where p is the momentum of the photon and A is the vector potential (e.g., the amplitude) of the electromagnetic field of the light. The electromagnetic interaction term is ε⋅peikr, where ε is the polarisation of the photon and k is the propagation vector of the photon. In a multipole expansion, the eikr term can be decomposed in a Taylor series (the selection rules can be derived from the decomposition of the electron dipole/quadrupole transition matrix element into angular and radial parts using the Wigner-Eckart theorem [13, 24], where the triangular relations of the symbol determine the selection rules) which leads to the electric dipole and the electric quadrupole terms [25]. Further details can be found in Core Level Spectroscopy of Solids [13] and elsewhere [16, 24, 25].Due to the electric dipole operator being the first-order term of this decomposition, the result is that theJ quantum number can only change by a value of 0 or 1. Thus, the electric dipole selection rule translates to ΔJ=+1,0,or−1. Because the electric quadrupole operator is the second-order term of this decomposition, the resulting selection rule translates to ΔJ=−2,−1,0,+1,or+2.However, when the spin-orbit coupling is neglected (or weak), the orbital momentumℒ and the spin momentum S can still be considered good quantum numbers. In this case, the transitions are spin-conserving (ΔS=0) and only ℒ changes. Due to the fact that light carries an orbital momentum of ℒ=1, this implies for electric dipole transitions Δℒ=+1 or −1. For the second-order term, the electric quadrupole transitions, this implies Δℒ=+2,0, or −2 [21].As a result, the transition operatorT^1 of the 1s XAS absorption step promoting an electron from 1s to 3d (Δℒ=+2) is an electric quadrupole term, and this step is forbidden as an electric dipole transition. On the contrary, the transition operator T^2 for the 2p XES decay from 2p to 1s (Δℒ=−1) is an electric dipole term. Similarly, the transition operator T^ of the 2p XAS absorption step from 2p to 3d (Δℒ=+1) is also an electric dipole term.These selection rules, defined in spherical symmetry, translate into the point group symmetry of the absorbing ion. In cubic symmetry (Oh), the selection rules of each operator depend on its symmetry properties. The electric dipole operator (first-rank tensor) behaves as the J=1 IRREP (O3) and branches into T1u in Oh. The electric quadrupole operator (second-rank tensor) behaves as a J=2 IRREP (O3) and branches in Oh into T2g⊕Eg (see Table 2). The transition matrix element between two states i,f with the IRREPs Γi and Γf and the transition operator T^ is nonzero if the direct product Γi⊗ΓT^ contains Γf, where ΓT^ is the IRREP of the transition operator. In other words, the final state IRREPs Γf accessible via the transition operator T^ are given by the direct product Γi⊗ΓT^. The selection rules are derived from the direct product tables for the Oh point group (see Tables 3, 4, and 5) and give the symmetry of the accessible states.The selection rules will be commented in more detail for each case of the 3dN series throughout this paper. Magnetic and natural dichroisms, which are a property from the crystal and not from the point group of the absorbing transition metal ion [26], will not be discussed here.


## 3. Calculations and Computational Details

All calculations are done using the framework of the crystal field multiplet theory, which is a multielectronic, semiempirical approach initially developed by Thole et al. [27] and further established by Butler and Cowan [28, 29]. It takes into account all the 3d-3d, 1s-3d, and 2p-3d electronic Coulomb interactions, as well as the spin-orbit coupling ζ on every open shell of the absorbing atom (e.g., 2p SOC ζ2p and 3d SOC ζ3d in 1s2p RIXS).The 2p XAS and 1s2p RIXS spectra were calculated using Quanty which uses second quantisation and the Lanczos recursion method for the exact diagonalisation and Green functions to calculate the spectra [30–32]. This method enables to avoid the explicit calculation of the intermediate and final states, which are only defined with their respective Hamiltonian. Each of these many-body states is described by a linear combination of Slater determinants [31]. The atomic electronic interactions are parametrised by the Slater integrals, which are reduced to 80% of the Hartree-Fock values. The latter has empirically proven to correspond to the actual atomic values. For the spin-orbit coupling parameters ζ2p,ζ3d, the Hartree-Fock calculated values are used. All calculations are done in octahedral symmetry (Oh), and the crystal field is defined by the crystal field parameter 10Dq. The same value is used for the initial, intermediate, and final states. The influence of the crystal field splitting is investigated by varying the parameter 10Dq from 0 eV up to a few eV. The symmetry IRREPs of the labels of the final states in the Oh point group are obtained using the CTM4XAS program [33]. (Table 6 shows a general list of different notations to translate between them.) The natural broadenings γn=0.5eV and γf=0.25eV for the intermediate state and final state, respectively, were applied.Table 6
Summary of the different notations inOh symmetry. T/S/K: Tanabe/Sugano/Kamimura, though Sugano also uses a variation of this notation as shown. The CTM4XAS program uses mostly the Butler notation, though the .ora-file translation is listed for ease of use.
Altmann
Mulliken
T/S/K
Sugano
Koster/Bethe
Butler
CTM4XAS
A1
A1
A1
A1
Γ1
0
0
A2
A2
A2
A2
Γ2
0~
^0
E
E
E
E
Γ3
2
2
T1
T1
T1
T1
Γ4
1
1
T2
T2
T2
T2
Γ5
1~
^1
E1/2
E’
E or E1
E1/2
Γ6
1/2
SO
F3/2
U’
G
G3/2
Γ8
3/2
S1
E5/2
E”
E’ or E2”
E5/2
Γ7
1/2~
^S0All RIXS intensities are given as calculated by Quanty. They reflect the total absorption as the sum over the five quadrupole basis components and the three dipole emission polarisations and thus imply an isotropic spectrum. When needed, one can in fact also compute the scattered photons as the percentage of the incident beam in dependency of a given experimental setup. The 2p XAS spectra are scaled for the best comparison for each case. The energiesEin and Eout are given by Quanty relative to the centre of gravity for each of the three states in 1s2p RIXS, being the 3dN, 1s13dN + 1, and 2p53dN + 1 configurations, respectively. Thus, also the ET=0eV position of the energy transfer ET=Ein−Eout is based on the centre of gravity and ET does not reflect the real energy transfer.More details on the method can be found elsewhere [13, 30–32, 34] and the references therein. The calculations are performed for a temperature of T=300K as described in the following.


### 3.1.T=300K Approximation: Boltzmann Distribution

The multielectronic ground state at ambient condition is a linear combination of an ensemble ofN microstates ψi with i∈0,…,N−1. In general, the Coulomb and spin-orbit interactions as well as crystal field effects can induce a mixing of different states, leading to complex multielectronic states involving many microstates. For example, a 3d5 configuration has in total 252 microstates, but only N=6 microstates form in high spin the atomic multielectronic ground state 6S (O3) (Table 7).Table 7
Summary of the ground state terms, number of microstates, and crystal field values (10Dq) for all ions used in our calculations.
GS
Ion
Ground state terms
Number of MS
Total GS
10Dq
3dN
O3
Oh
HS
LS
10N
HS
LS
HS
HS
LS
3d03d13d23d33d43d53d63d73d83d9
Ti4+Ti3+Cr4+Cr3+Mn3+Fe3+Fe2+Co2+Ni2+Cu2+
1S2D3F4F5D6S5D4F3F2D
1A1gA1g2T2gF3/2g3T1gEg4A2gF3/2g5EgA1g6A1gF3/2g5T2gT2g4T1gE1/2g3A2gT2g2EgF3/2g
3T1gA1g2T2gE5/2g1A1gA1g2EgF3/2g
1694106151234
9614
110451202102522101204510
0 eV & 2.1 eV1.5 eV2.1 eV1.5 eV1.5 eV1.5 eV0.9 eV0.9 eV0.9 eV0 eV & 3.0 eV
3.9 eV3.9 eV3.0 eV3.0 eVHowever, at absolute zero (T=0K), formally, only the state ψ0 lowest in energy is populated. Thus, the calculated spectra result from a linear combination of each contributing state, weighted according to their respective population, because for temperatures T > 0 K higher states are also partially occupied. The population is described with the Boltzmann distribution.It gives the population probabilitypi for each state ψi in dependency of its relative energy Ei−E0, with E0=Eψ0 being the energy of the lowest microstate ψ0, and absolute temperature T. It is given (with Boltzmann constant kB) as
(10)pi=1aexp−Ei−E0kBT,with
(11)a=∑l=0Nexp−El−E0kBT.In the present case, the eigenvalues calculated with Quanty fromℋψ=Eψ give the energies Ei for each microstate ψi, which are used in (10) and (11) to compute the coefficients pi for the Boltzmann linear combination of each multielectronic ground state.For example, Ti3+ (3d1) has the atomic ground state 2D with N=10 microstates. In other words, the single 3d electron can be in 10 different states. In an Oh crystal field, this ground state branches without 3d spin-orbit interaction into two crystal field terms: 2D→Oh2T2g⊕2Eg.The term2T2g represents the multielectronic ground state in an Oh crystal field, formed as a linear combination of N=6 microstates. For 3d1, the 2T2g term is the lowest in energy as can be seen in the single electronic picture: t2g1eg0, as opposed to the 2Eg with t2g0eg1 in 4 possible microstates.Furthermore, it should be noted that theN microstates forming the multielectronic ground state are a priori not degenerate. For Ti3+, the lowest 10 microstates are in fact partially degenerate. With 3d spin-orbit interaction included, the microstates are spread in a 4 : 2 : 4 ratio, corresponding to the terms F3/2, E5/2, and F3/2, respectively. In other words, the lowest four microstates ψ0, ψ1, ψ2, and ψ3 are degenerate in energy (E0=E1=E2=E3), and the two states ψ4 and ψ5 are degenerate (E4=E5), as well as the remaining 4 microstates ψ6, ψ7, ψ8, and ψ9 (E6=E7=E8=E9). Hence, the 10 microstates, which are related to the atomic multielectronic ground state 2D (O3) and are split by the crystal field and spin-orbit interaction, are spread over three different energy levels. However, the six microstates forming the 2T2g (Oh) term are only spread over two different energies (see Figure 3).Figure 3
Initial state term schemes of Ti3+ (3d1) illustrating the effect of the 3d SOC and an Oh crystal field 10Dq on the splittings of the atomic ground state term 2D.Note, if not stated otherwise, we only use the lowest crystal field ground state for the Boltzmann linear combination (e.g.,2T2g for Ti3+); the next multielectronic term higher in energy (e.g., 2Eg for Ti3+) is neglected. This approximation relates to the fact that in general the second multielectronic term has a relatively small contribution in most cases.


## 3.1.T=300K Approximation: Boltzmann Distribution

The multielectronic ground state at ambient condition is a linear combination of an ensemble ofN microstates ψi with i∈0,…,N−1. In general, the Coulomb and spin-orbit interactions as well as crystal field effects can induce a mixing of different states, leading to complex multielectronic states involving many microstates. For example, a 3d5 configuration has in total 252 microstates, but only N=6 microstates form in high spin the atomic multielectronic ground state 6S (O3) (Table 7).Table 7
Summary of the ground state terms, number of microstates, and crystal field values (10Dq) for all ions used in our calculations.
GS
Ion
Ground state terms
Number of MS
Total GS
10Dq
3dN
O3
Oh
HS
LS
10N
HS
LS
HS
HS
LS
3d03d13d23d33d43d53d63d73d83d9
Ti4+Ti3+Cr4+Cr3+Mn3+Fe3+Fe2+Co2+Ni2+Cu2+
1S2D3F4F5D6S5D4F3F2D
1A1gA1g2T2gF3/2g3T1gEg4A2gF3/2g5EgA1g6A1gF3/2g5T2gT2g4T1gE1/2g3A2gT2g2EgF3/2g
3T1gA1g2T2gE5/2g1A1gA1g2EgF3/2g
1694106151234
9614
110451202102522101204510
0 eV & 2.1 eV1.5 eV2.1 eV1.5 eV1.5 eV1.5 eV0.9 eV0.9 eV0.9 eV0 eV & 3.0 eV
3.9 eV3.9 eV3.0 eV3.0 eVHowever, at absolute zero (T=0K), formally, only the state ψ0 lowest in energy is populated. Thus, the calculated spectra result from a linear combination of each contributing state, weighted according to their respective population, because for temperatures T > 0 K higher states are also partially occupied. The population is described with the Boltzmann distribution.It gives the population probabilitypi for each state ψi in dependency of its relative energy Ei−E0, with E0=Eψ0 being the energy of the lowest microstate ψ0, and absolute temperature T. It is given (with Boltzmann constant kB) as
(10)pi=1aexp−Ei−E0kBT,with
(11)a=∑l=0Nexp−El−E0kBT.In the present case, the eigenvalues calculated with Quanty fromℋψ=Eψ give the energies Ei for each microstate ψi, which are used in (10) and (11) to compute the coefficients pi for the Boltzmann linear combination of each multielectronic ground state.For example, Ti3+ (3d1) has the atomic ground state 2D with N=10 microstates. In other words, the single 3d electron can be in 10 different states. In an Oh crystal field, this ground state branches without 3d spin-orbit interaction into two crystal field terms: 2D→Oh2T2g⊕2Eg.The term2T2g represents the multielectronic ground state in an Oh crystal field, formed as a linear combination of N=6 microstates. For 3d1, the 2T2g term is the lowest in energy as can be seen in the single electronic picture: t2g1eg0, as opposed to the 2Eg with t2g0eg1 in 4 possible microstates.Furthermore, it should be noted that theN microstates forming the multielectronic ground state are a priori not degenerate. For Ti3+, the lowest 10 microstates are in fact partially degenerate. With 3d spin-orbit interaction included, the microstates are spread in a 4 : 2 : 4 ratio, corresponding to the terms F3/2, E5/2, and F3/2, respectively. In other words, the lowest four microstates ψ0, ψ1, ψ2, and ψ3 are degenerate in energy (E0=E1=E2=E3), and the two states ψ4 and ψ5 are degenerate (E4=E5), as well as the remaining 4 microstates ψ6, ψ7, ψ8, and ψ9 (E6=E7=E8=E9). Hence, the 10 microstates, which are related to the atomic multielectronic ground state 2D (O3) and are split by the crystal field and spin-orbit interaction, are spread over three different energy levels. However, the six microstates forming the 2T2g (Oh) term are only spread over two different energies (see Figure 3).Figure 3
Initial state term schemes of Ti3+ (3d1) illustrating the effect of the 3d SOC and an Oh crystal field 10Dq on the splittings of the atomic ground state term 2D.Note, if not stated otherwise, we only use the lowest crystal field ground state for the Boltzmann linear combination (e.g.,2T2g for Ti3+); the next multielectronic term higher in energy (e.g., 2Eg for Ti3+) is neglected. This approximation relates to the fact that in general the second multielectronic term has a relatively small contribution in most cases.


## 4. Didactic Examples

In this section, we discuss three model systems as didactic examples in detail before the results for the remaining 3dN series (with N∈1,…,7) which are discussed in a more summarising manner (Section 5). Here, we will emphasise the elementary differences between the atomic (O3) and the crystal field (Oh) cases to illustrate the effects with respect to the selection rules. Furthermore, we compare the results due to the different possible pathways in the direct one-step 2p XAS and the two-step 1s2p RIXS and the corresponding spectra.As didactical examples, we use the following three cases:(i)
3d9 Cu2+: single open shell with only a single hole in each state
(ii)
3d0 Ti4+: two peaks in the 1s XAS split by the crystal field 10Dq
(iii)
3d8 Ni2+: a common model system often used in educationWe aim to assign the symmetry labels to the relevant peaks in the 1s2p RIXS spectra and compare the 2p XES to the well-understood L2,3-edge (2p XAS) spectra [35].


### 4.1. 3d9: Divalent Copper Cu2+—Crystal Field Effects Altering the Selection Rules

In this part, we investigate the elementary differences between 1s2p RIXS and 2p XAS for a divalent copper ion Cu2+ with a 3d9 ground state. Each successive step of the 1s2p RIXS process has an electronic configuration with only one open shell and a single hole in the 3d, 1s, and 2p shells, respectively.
(12)We first describe the atomic case of an isolated ion without any crystal field. The electronic configuration of the initial state of Cu2+ is 1s22p63d9. In spherical symmetry (O3), the 3d spin-orbit coupling (SOC) ζ3d induces the splitting of the 10-fold degenerate ground state 2D (10 microstates) into two multielectronic states 2D5/2 (O3) and 2D3/2 (O3), defined by J=5/2 (ground state) and J=3/2, respectively. These two states are separated in energy by ΔE=5/2ζ3d. The electronic configuration of the 1s2p RIXS intermediate state of Cu2+ is 1s12p63d10, which corresponds to the 2-fold degenerate state 2S1/2 (O3) with J=1/2. The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Cu2+ is 1s22p53d10, which splits due to the 2p spin-orbit coupling ζ2p into J=3/2 (lowest energy state) and J=1/2 separated by 3/2ζ2p. This is summarised in the scheme in Figure 4.Figure 4
Atomic case: term scheme (with SOC; no crystal field) comparing the two-step (1s XAS, 2p XES) 1s2p RIXS and the one-step 2p XAS pathways for Cu2+. The sum over all J multiplicities 2J+1 yields the total number of microstates per configuration.Since the ground state of the initial state isJ=5/2, the electric dipole absorption operator (2p XAS) enables only to reach the J=3/2 final state (ΔJ=−1). In 1s2p RIXS, the selection rules of the electric quadrupole absorption for 1s XAS (ΔJ=−2) enable to reach the intermediate state J=1/2. The subsequent electric dipole emission (2p XES) enables to reach both J=1/2 and J=3/2 final states. As a result, for an isolated Cu2+ ion in spherical symmetry, the 2p XAS shows only one peak, while the 1s2p RIXS shows two peaks. The resulting spectra are shown in Figure 5.Figure 5
Calculated atomic 1s2p RIXS map with 2p XES projection for an isolated Cu2+ (3d9). Notice the 2p XAS (L2,3-edge) (dashed red) has only one peak, because only the final state with J=3/2 can be populated. See also Figure 7.3 in Core Level Spectroscopy of Solids [13].In the vertical direction, the Kα1 and Kα2 decays (2p XES) appear as two separate peaks. These two peaks are separated by the 2p SOC ζ2p≈13.5eV, corresponding to the two resonant emission transitions from the intermediate state with J=1/2 to the final states with J=1/2,3/2.


#### 4.1.1. Considering a Cubic (Octahedral) Crystal Field

When the absorbing ion is embedded in a solid state, one has to take into account the crystal field potential created by the surrounding atoms. In the following, we will assume an octahedral (Oh) crystal field described by the parameter 10Dq. In a single electron picture, this crystal field induces the splitting of the 3d orbitals into t2g and eg. In the multielectronic formalism (neglecting the 3d spin-orbit coupling in first approximation), the 2D term of the initial state branches in the Oh point group into the 2Eg (t2g6eg3) and 2T2g (t2g5eg4) terms. Thus, the multielectronic ground state (at T=0K) is 2Eg (Oh). The intermediate state term 2S (O3) branches in an octahedral crystal field into 2A1g (Oh) symmetry.In the Oh point group, the electric quadrupole 1s XAS operator has T2g and Eg symmetries. We derive the selection rules from the product table (Table 3) for the first step of the 1s2p RIXS process.
(13)2Eg⊗T2g=2T1g⊕2T2g,2Eg⏟1s23d9⊗Eg⏟1sXAS=2A1g⊕2A2g⊕2Eg⏟Accessiblestates.From (13), it becomes obvious that many symmetries are in principle accessible via 1s XAS from the multielectronic ground state symmetry 2Eg (Oh). In spite of the fact that many symmetries are in principle reachable via a quadrupole 1s XAS transition starting from the ground state symmetry 2Eg (Oh), the electronic structure of a Cu2+ ion only offers 2A1g (Oh) symmetry leading to a single peak in the 1s XAS projection.While spin-orbit coupling was neglected in this last step, it must be included in the description of the final state with the 2p core hole to understand the complete 1s2p RIXS process and to be able to compare it with the 2p XAS. Using theℒ.S-coupled terms obtained above in the atomic case, we derive their respective branchings in the octahedral crystal field (Oh).The initial ground state2D5/2 (O3) branches in octahedral symmetry (Oh) into E5/2g⊕F3/2g (Oh). The ground state at T=0K is 2EgF3/2g (Oh). The effects of the 3d spin-orbit coupling ζ3d and the Oh crystal field parameter 10Dq are illustrated in Figure 2.The intermediate state term2S1/2 (O3) branches in an octahedral crystal field (Oh) into E1/2g (Oh). The final states 2P3/2 (O3) and 2P1/2 (O3) branch into F3/2u (Oh) and E1/2u (Oh), respectively. In octahedral symmetry, the electric dipole operator has T1u symmetry and thus, using the product table (Table 5), we find the reachable final states in the subsequent electric dipole emission (2p XES):
(14)E1/2g⊗T1u=E1/2u⊕F3/2u.The direct 2p XAS starts from theF3/2g ground state yielding with the T1u IRREP the reachable final state terms:
(15)F3/2g⊗T1u=E1/2u⊕E5/2u⊕2F3/2u.Altogether, this demonstrates that the crystal field enables both final states to be probed by 2p XAS and 1s2p RIXS. This is summarised in the scheme in Figure6 and the corresponding calculated spectrum is shown in Figure 7.Figure 6
Multielectronic term scheme for 3d9 Cu2+ comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC S=1/2→⊗E1/2 is included, the spin multiplicity is omitted.Figure 7
Calculated 1s2p RIXS map for 10Dq = 3.0 eV with 2p XES projection for Cu2+ (3d9) for 10Dq = 3.0 eV. The 2p XES appears identical to the 2p XAS. The mixing induced by the crystal field symmetry breakdown enables to populate the J=1/2 and J=3/2 final states in direct 2p XAS.The RIXS projections for various values of 10Dq show almost no difference in both directions (XAS and XES). It is noteworthy that the 2p XAS (L2,3-edge) approaches the two peaks in the 2p XES projections for large 10Dq values. This is due to the crystal field (10Dq) affecting the spin-orbit interaction altering the 2p XAS selection rules.In other words, the L2-edge probes in 2p XAS the amount of J=3/2 character in the ground state. In the atomic case (10Dq = 0.0 eV) with spherical symmetry (O3) and at absolute zero T=0K, the ion is pure J=5/2 (D5/2). In octahedral Oh symmetry, on the other hand, an increasing crystal field (scaled via 10Dq) mixes more and more J=3/2 character from the D3/2 into the ground state, resulting in a continuous visible increase of the L2 peak in 2p XAS.And finally, it can be seen that the single-particle limit with two peaks is reached in both cases, 1s2p RIXS and 2p XAS, respectively. The intensity ratio of the two peaks of 2 : 1 is given by the degeneracy of the2P1/2 (O3) and 2P3/2 (O3) final states.


#### 4.1.2. 1s2p RIXS Experiments of 3d9 Systems

Experimental 1s2p RIXS spectra of CuO have been published by Hayashi et al. [2]. They show the 1s2p X-ray emission spectra from excitation energies before the K-edge, through the edge to the continuum. As such, they observe the transition from resonances in the Lorentzian tails to nonresonant 1s2p XES. At the excitation energy at the pre-edge, the two-peaked 1s2p RIXS spectrum is visible, in addition to the background from the edge [2].


### 4.2. 3d0: Tetravalent Titanium Ti4+—1s XAS Peaks Split by 10Dq

In the following, we discuss the differences between 2p XAS and 1s2p RIXS for the case of tetravalent titanium (Ti4+). Here, the initial state configuration 1s22p63d0 has no partially filled shells. This is interesting because the resulting selection rules are straightforward: the nature of the probed final states reflects the nature and symmetry of the transition operators. The transition into the intermediate state promotes a 1s electron into the 3d band yielding a 1s12p63d1 electron configuration. The final state in 1s2p RIXS and 2p XAS is in this case 1s22p53d1.
(16)The 3d0 case enables to describe the effect of the crystal field on the intermediate state, the interferences in the 1s2p RIXS process, and the more complex multielectronic effects in the final state. It provides an extent to L2,3-edge considerations of 3d0 ions previously described in [35].


#### 4.2.1. Atomic Case with Spin-Orbit Coupling

We start again with the case of an isolated Ti4+ ion (3d0) where the effect of the solid state, that is, the crystal field, is neglected. The initial state configuration of Ti4+ is 1s22p63d0. Since all shells are full, S=0 and ℒ=0. The ℒ.S-coupled total angular momentum is thus J=0, and the initial state symmetry is the totally symmetric term 1S0.The electronic configuration of the intermediate state in 1s2p RIXS of Ti4+ is 1s12p63d1, corresponding to the total orbital angular momentum ℒ=2 and total spin angular momentum S=0 or S=1 giving the Russel-Saunders terms 1D and 3D (2⋅10=20 microstates). Considering the spin-orbit interaction in the ℒ.S coupling scheme, one obtains J=2 for the 1D term and J=1,2,and3 for the 3D term (Figure 8). The splitting of these four terms (1D2, 3D1, 3D2, and 3D3) is defined by the 3d SOC ζ3d and exchange interaction Gsd2, which are both small (32 meV and 46 meV, resp.). The calculated energy splitting is ΔE=92meV and is beyond the reach of current experimental resolution and not resolved in our calculations discussed here.Figure 8
Splitting of the intermediate state spin singlet and triplet terms1,3D due to 3d SOC ζ3d for a Ti4+ ion into D1, D2, and D3. States accessible in 1s XAS (J=2) are shown in blue. However, the existing terms with J=1 or 3 cannot be populated and will therefore not contribute to the spectrum.The electric quadrupole transition of the 1s XAS enables to reach the intermediate states withJ=2, though here 3d SOC ζ3d is small and transitions to the spin triplet term 3D2 will be weak, such that the 1s XAS is dominated by the transition in the spin singlet term 1D2.The final state electronic configuration is 1s22p53d1 with two partially filled shells that have to be accounted for: the 2p and the 3d shell. The total orbital angular moment ℒ of the final state electronic configuration is ℒ=1,2,and3. The total spin angular moment S of the final state electronic configuration is S=0,1. This gives in the atomic case the spin singlet and triplet terms 1,3P, 1,3D, and 1,3F.As previously described by de Groot et al. [35], when spin-orbit interaction is neglected, the selection rules for the electric dipole transition in 2p XAS with ΔS=0 and Δℒ=±1 allow only to reach the singlet 1P term from the 1S initial state, leading to one peak only (the L2-edge). The L3-edge, corresponding to transitions into the spin triplet term 3P, would have zero intensity. But in fact the strong 2p spin-orbit coupling ζ2p in the final state (J is used) leads to three peaks, two of them forming the L2- and L3-edges, and a third weak peak related to triplet transitions.The derivation of the total angular momentumJ of the final state electronic configuration in the ℒ.S coupling scheme yields the term symbols illustrated in Figure 9.Figure 9
Splitting of the final state terms1,3P, 1,3D, and 1,3F due to spin-orbit coupling for Ti4+.The direct electric dipole transition in 2p XAS withΔJ=0,±1 (J=0→J=0 forbidden) starts for Ti4+ from the total symmetric term 1S0, which allows to reach all final states with J=1 (P1 and D1, marked with a red circle in Figure 9). The L2- and L3-edges are separated by the 2p spin-orbit coupling ζ2p corresponding to transitions into the 1P1 (L2-edge) and 3D1 (L3-edge) terms. More precisely, from the squared matrix elements, we find that the second peak is dominated by 3D1 contribution of 60%, adding 36% of 1P1 and 4% of 3P1 character. The small third peak in the 2p XAS, at ET≈−5eV in Figure 10, is related to transitions into the 3P1 terms. Though the resolution chosen in our calculations does not reveal them as individual peaks, the direct Coulomb and exchange terms are not negligible. As discussed by de Groot et al. [35], they lead to a splitting of the L2,3-edge into three absorption lines including a redistribution of the intensities.Figure 10
Calculated 1s2p RIXS map with 2p XES projection and the 2p XAS (L2,3-edge) for an isolated Ti4+ (3d0) ion in O3 symmetry.In the 1s2p RIXS process, the 2p XES decays withΔJ=0,±1 starting from the intermediate states with J=2 (D2) and enables to reach the final states P1,2, D1,2,3, and F2,3 with J=1,2,and3 (terms shown in green Figure 9). Therefore, all J values except 0 and 4 are reachable.With this, one can draw for Ti4+ the atomic term scheme with spin-orbit coupling for the two-step 1s2p RIXS and the one-step 2p XAS process as shown in Figure 11.Figure 11
Atomic term scheme with spin-orbit coupling for Ti4+ (3d0) comparing the two-step (1s XAS, 2p XES) 1s2p RIXS and the one-step 2p XAS pathways. The two 1s XAS transitions are separated by the small 3d SOC, Coulomb, and exchange interactions; hence, the splitting is small. The direct dipole transitions in 2p XAS can only populate the final states with J=1, while the dipole decays in 2p XES can reach final states with J=1,2,and3. However, terms with J=0 or 4 cannot be populated in either case and thus do not contribute to the spectrum. Nonaccessible terms are omitted for clarity.The intermediate state is the key element when comparing the direct 2p XAS and 1s2p RIXS. It enables to access additional terms in the 1s2p RIXS final state, adding multiple visible peaks in the energy transfer direction as shown in Figure10. But in both cases, J=0 and 4 are not possible to reach.In summary, so far, we have discussed the effect of the spin-orbit coupling in the atomic case in spherical symmetry (O3), but neglecting any crystal field. With the above, one can describe the 1s XAS with only one visible peak due to transitions into the 1D2 intermediate state term. The 3d spin-orbit coupling ζ3d and exchange interaction Gsd lead to small splittings which are not visible in this plot. Furthermore, the small 3d SOC may induce some mixing adding weak transitions into the 3D2 term.The splitting of the 2p XAS is dominated by the large 2p SOCζ2p separating the L2-edge from the L3-edge, for example, the spin singlet and the triplet states. The 2p XES on the other hand consists of transitions from the intermediate state with J=2 IRREP (mostly 1D2) into the final states with J=1,2,and3 IRREPs resulting multiple visible peaks in the energy transfer direction. All final state terms are also split by the exchange and direct Coulomb interactions, Gpd and Fpd, respectively.


#### 4.2.2. Considering an Oh Crystal Field

We now consider the effect of an octahedral Oh crystal field. The totally symmetric initial state term 1S0 branches in Oh into 1A1gA1g. Formal consideration of an Oh crystal field and 3d spin-orbit interaction leads to the branching of the intermediate state terms 1D and 3D into eight terms. The scheme in Figure 12 illustrates these branchings of the terms due to an Oh crystal field and the 3d spin-orbit interaction ζ3d (S=0→⊗A1g and S=1→⊗T1g).Figure 12
Branching of the intermediate state terms1D and 3D of Ti4+ (3d0) due to 3d SOC and an Oh CF. Symmetries accessible in the quadrupole 1s XAS step (GS → IS) are shown in blue.In spite of several available intermediate state terms, starting from the initial state1A1g and using the T2g and Eg IRREPs, representing the quadrupole 1s XAS operator, and the product table (Table 3) reflecting the orbital momentum selection rules, one finds the accessible intermediate state symmetries.
(17)1A1g⊗T2g=1T2g,1A1g⏟1s23d0⊗Eg⏟1sXAS=1Eg⏟Accessibleterms.In other words, from thetotally symmetric ground state A1g (Oh), only the Eg (Oh) and T2g (Oh) symmetries can be reached in the intermediate state via a quadrupole 1s XAS transition. Hence, for sufficiently large values of 10Dq, relative to the broadenings used in a calculation (or the experimental resolution in a measurement), two distinct peaks can be observed in the 1s XAS projection (Figure 13). Because the 3d spin-orbit coupling ζ3d is small, the transitions to the spin singlet intermediate states are dominating (i.e., matrix elements are the largest), while the transitions via the J=2 IRREP from triplet 3D should be weak. The 1s XAS is dominated by transitions to the Eg and T2g terms from the 1D2.Figure 13
Calculated 1s XAS projections of a 1s2p RIXS for Ti4+ (3d0) scaling the octahedral crystal field parameter 10Dq from 0.1 eV to 3.0 eV with an energy level diagram as overlay. The splitting of the 3d level into the two peaks T2g and Eg due to the crystal field is visible.In an Oh crystal field, the final state IRREPs with J=0,1,2,3 and 4 (see Figure 11) branch, respectively, into the ℒ.S-coupled terms: A1u, T1u, Eu⊕T2u, A2u⊕T1u⊕T2u, and A1u⊕Eu⊕T1u⊕T2u. The 2p XES in 1s2p RIXS from the two intermediate state terms T2g and Eg via the electric dipole operator, described by the T1u IRREP, leads to
(18)T2g⊗T1u=A2u⊕Eu⊕T1u⊕T2u,Eg⏟1s13d1⊗T1u⏟2pXES=T1u⊕T2u⏟Accessiblefinalstates.It is noteworthy here that only the intermediate state withT2g symmetry can access in the 2p XES decay the A2u and Eu final state symmetries. The T1u and T2u final state terms are however accessible from both T2g and Eg intermediate state symmetries. Furthermore, we note that the intermediate state symmetry A1u is not reachable as the final state symmetry and will therefore not contribute to the 2p XES spectrum.On the other hand, in 2p XAS, the possible final states must haveT1u symmetry; otherwise, the transition matrix element equals zero. As can be seen from the branchings of J (O3) in and octahedral crystal field (Table 2), the T1u IRREP can be reached not only from the J=1 IRREP but also from the J=3 and J=4 IRREPs.This results in only seven possibleT1u IRREPs in Oh symmetry, or in other words, only seven final states are reachable among the 60 available. The 2p XAS spectrum consists, in principle, of seven lines, as can be seen in Figure 14, where the calculated spectra for Ti4+ (3d0) for a crystal-field splitting of 10Dq = 2.1 eV are shown. While only spin singlet states would be reached in the absence of 2p SOC, the consideration of the strong spin-orbit coupling in the final state induces a mixing of the spin singlet and triplet states. This weakens the spin selection rule ΔS=0 allowing also transitions into former spin triplet states (e.g., 3T2u, 3T1u, 3Eu, and 3A2u).Figure 14
(a) Calculation of the 1s2p RIXS spectra for Ti4+ (3d0) with 10Dq=2.1eV separated into the quadrupole components Ylm (with m=0, 1 and l=2) in spherical symmetry. They translate into the T2g and Eg IRREPs describing the quadrupole 1s XAS transitions. The term scheme insets illustrate the corresponding pathways. (b) The corresponding 2p XES projections onto the energy transfer axis for the two quadrupole components and the calculated 2p XAS (L2,3-edge).
(a)
(b)This enables to draw the electronic term scheme in Figure15 illustrating the different pathways in 1s2p RIXS and 2p XAS for 3d0 Ti4+ in Oh symmetry. The accessible 1s2p RIXS final state terms are given in (18), while, as discussed above, the 2p XAS can only reach final states with T1u symmetry (A1g⊗T1u=T1u).Figure 15
Multielectronic term scheme comparing the 2p XAS and the 1s2p RIXS pathways. The two quadrupole 1s XAS transitions withT2g and Eg symmetry lead to two different intermediate state symmetries. The subsequent dipole decays (2p XES) with T1u symmetry yield multiple 1s2p RIXS final state symmetries. The direct 2p XAS (L2,3-edge) transition with T1u symmetry can on the other hand only access the T1u symmetry in the final state. Noncontributing states are omitted for clarity.Similarly to the 3d9 case (Cu2+) discussed in Section 4.1, comparing the final states, it becomes clear once again that the key elements are the intermediate states from which the 2p XES decays and thus enables to reach more final states when compared to the direct 2p XAS.This overall results in the 1s2p RIXS maps shown in Figure14. The total RIXS map is decomposed into the Eg and T2g quadrupole absorption components. The selective absorption towards each intermediate state is obtained. Experimentally, this dichroic behaviour has been observed for Ti4+ in cubic oxides such as the prototypal SrTiO3 [36]. The 1s2p RIXS spectra calculated for each of the two absorption peaks are shown (Figure 14(a)), and the 60 final states are labelled with their total angular IRREP (Figure 14(b)). This enables to see that both, the 1s2p RIXS and 2p XAS, probe the T1u final state symmetry. We further observe that within the four main peaks in 2p XAS, the first peaks of the L3 (at ET≈−3.5eV) and of the L2 (at ET≈2eV) are stronger via the T2g absorption channel of the 1s2p RIXS than the Eg and vice-versa: the second peaks of the L3 (ET≈−1eV) and of the L2 (at ET≈4.5eV) are stronger via the Eg absorption channel of the 1s2p RIXS than via the T2g. This can be understood in a single electron picture with the crystal field splitting of the final state: the lower states of the L3-edge contain the 3d single electron in the t2g orbitals while the higher energy states have the 3d electron in the eg orbitals and similarly for the L2-edge [37].


#### 4.2.3. 1s2p RIXS Experiments of 3d0 Systems

Experimental 1s XAS spectra (e.g., HERFD) of titanium oxides in the solid state often show more than two peaks in the pre-edge, which contradicts the result shown in Figure13. The reason is that there are additional dipole transitions at energies overlapping with the quadrupole prepeaks. These dipole peaks are related to so-called nonlocal peaks, where the 4p states from the absorbing ion mix with the 3d states of the neighbouring ions [3]. In addition, we note that for high valent ions such as Ti4+, charge transfer is important for the detailed description of the 2p XAS and 1s2p RIXS spectral shapes [37].Experimental 1s2p RIXS experiments have been published on TiO2 by Glatzel et al. [38] and Kas et al. [39]. The 1s2p RIXS planes show the quadrupole pre-edges and the nonlocal peaks. The 1s2p RIXS cross-section at the first quadrupole peak shows three peaks in qualitative agreement with the t2g calculation in Figure 14. Bagger et al. analysed the quadrupole and dipole RIXS separately [1]. We note that the ionic limit as calculated in Figure 14 is likely to be reached by 1s2p RIXS experiments at the K-edge on divalent calcium systems (Ca2+), including octahedral CaO and cubic CaF2.


### 4.3. 3d8: Divalent Nickel Ni2+—Single Peak in 1s XAS

In this part, we investigate the elementary differences between 2p XAS and 1s2p RIXS for a divalent nickel ion (Ni2+, 3d8). As we will show, the electronic configurations of the different steps of 1s2p RIXS present strong analogies with the case of Ti4+; the main difference being the nature of the initial state. We start again with the atomic case by deriving the J values as IRREPs.
(19)The electronic configuration of the initial state of Ni2+ is 1s22p63d8 with two holes in the 3d shell. The multielectronic interactions lead to 45 states spread across the terms 3F, 3P, 1G, 1D, and 1S [13]. In spherical symmetry (O3), the atomic ground state is 3F (ℒ=3 and S=1). The 3d spin-orbit coupling ζ3d≈0.1eV induces the splitting of the 3F term into the three terms:
(20)3F→3dSOC3F2,3F3,3F4.From those three IRREPs withJ=2,3,and4, the 3F4 term with J=4 is the ground state term symbol.The electronic configuration of the intermediate state in 1s2p RIXS of Ni2+ is 1s12p63d9 (one hole in the 1s shell and one hole in the 3d shell) which translates into the two terms 1D and 3D (2⋅10=20 microstates). It is noteworthy that this is identical to the intermediate state of Ti4+ described above (Section 4.2) due to the electron-hole equivalency. The 3d spin-orbit interaction splits the two terms into the 1D2 (for S=0) and 3D1, 3D2, and 3D3 (for S=1) Russel-Saunders terms. The resulting splittings distribute the four terms in energy over ΔE≈0.25eV for which the 1s XAS will appear only as a single peak due to the resolution chosen in our calculations, analogue to the Ti4+ atomic case.The difference in the 1s XAS step in 1s2p RIXS between the 3d8 Ni2+ and 3d0 Ti4+ cases arises from the spin multiplicity of the initial state. In both cases, the electric quadrupole absorption leads to a J=2 intermediate state. However, because of the small 3d spin-orbit coupling (ζ3d=0.1eV), the spin multiplicity is mostly conserved in the transition and the absorption from the spin triplet initial state of Ni2+ leads to the 3D2 state. For Ti4+ on the other hand, mainly, the 1D2 term is reached.The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Ni2+ is 1s22p53d9. This is again strictly analogue to the final state of Ti4+. The final state configuration corresponds to several atomic terms (1,3P, 1,3D, and 1,3F) which are split further due to the strong 2p spin-orbit coupling ζ2p=11.5eV. The total symmetry final state terms are represented with integer J values ranging from J=0 to J=4. The detailed splittings are given above in Figure 9 in Section 4.2.Only the final state terms withJ=1,2,and3 are formally accessible in the 1s2p RIXS from the intermediate state with J=2 via 2p XES decays (ΔJ=0,±1). The direct 2p XAS (with ΔJ=0,±1) on the other hand can only reach the final state terms with J=3 or J=4 from the ground state with J=4. Altogether, the selection rules for the atomic case can be summarised as
(21)This is also illustrated in the combined term scheme shown in Figure16.Figure 16
Atomic term scheme with spin-orbit coupling for Ni2+ comparing the two-step (1s XAS, 2p XES) 1s2p RIXS and the one-step 2p XAS pathways. Here, it should be noted that the two IS terms are only separated by the small 3d SOC ζ3d. Note that the final state is similar to Ti4+. Nonaccessible terms (e.g., final states with J=0) are omitted for clarity.The calculated 1s2p RIXS of an isolated Ni2+ ion (O3) is compared to the 2p XAS in Figure 17. The transitions are labelled with the dominant contribution of the corresponding final state terms. The four peaks in the 2p XAS (three in the L3 and one in the L2) correspond to the J=3 and J=4 IRREPs. It is noteworthy here that the 3F4 final state term is special in the sense, that it can be accessed only in 2p XAS, but its population via 1s2p RIXS is forbidden. Hence, in 2p XAS, the peak at ET≈−4eV is related to the J=4 IRREP (dominated by 3F4), while in 1s2p RIXS the peak at ET≈−4eV in the 2p XES decay relates to transitions into the final state IRREP J=2 (mostly 1D2). In other words, though it seems that the same peak at ET≈−4eV appears in the 2p XAS as well as in the 2p XES decay in 1s2p RIXS, it is in each case related to a different final state IRREP.Figure 17
Calculated 1s2p RIXS for an isolated Ni2+ ion (3d8) representing the atomic case. The projection of the total 1s2p RIXS map onto the energy transfer axis is shown together with the 2p XAS (L2,3-edge) in dashed red and the corresponding stick spectrum.Closer inspection of the sticks reveals that the3F3 term does not appear as a dominant contribution, but instead, the 1F3 term appears twice. This is due to the fact that the atomic terms lose their meaning when spin-orbit interaction is included. Then J=3 becomes the identifying IRREP which is formally a linear combination of 1F3, 3D3, and 3F3. Here, it happens to be the case that the 1F3 contribution is dominant in two J=3 IRREPs.As expected, theJ=0 IRREP (dominated by the 3P0 term), as well as the J=1 IRREP (3P1 and 3D1 terms), has no contribution to the 2p XAS spectrum. Furthermore, also, transitions into the J=2 IRREPs (P2, F2, and F2 terms) are not appearing in 2p XAS.In the 2p XES decay, on the other hand, theJ=2 IRREPs have a much stronger contribution to the spectrum. Those peaks, related to the final state IRREPs J=2, show in this example the largest difference between the 2p XAS and 2p XES as can be seen in Figure 17.


#### 4.3.1. Considering an Oh Crystal Field

The crystal field splitting in Oh symmetry of the atomic terms of the initial state configuration of Ni2+ (1s22p63d8) is well-known and described with the Tanabe-Sugano diagram. The ground state atomic term symbol 3F splits into 3A2g, 3T2g, and 3T1g and gives 3A2g as the octahedral crystal field ground state. The inclusion of the 3d spin-orbit coupling (S=1→⊗T1g) translates the ground state term as 3A2g⊗T1g=T2g. Thus, the crystal field ground state of 3d8 Ni2+ including 3d spin-orbit interaction is 3A2gT2g (Oh).The crystal field splitting of the 1s2p RIXS intermediate state of Ni2+ is analogue to Ti4+ as the 1,3D terms split into 1,3Eg⊕1,3T2g. From the ground state symmetry 3A2g (neglecting 3d SOC), the 1s XAS IRREPs T2g and Eg enable to reach only 3T1g and 3Eg intermediate state symmetries:
(22)3A2g→⊗T2g3T1g,(23)3A2g→⊗Eg1sXAS3Eg.In other words, the intermediate state term3T2g cannot be reached by the electric quadrupole 1s XAS absorption. Hence, only the 1s XAS IRREP Eg leads to an existing intermediate state symmetry: 3Eg. This is illustrated in Figure 18. As a result, the 1s XAS projection of the total 1s2p RIXS map will consist only of transitions into the 3Eg intermediate state symmetry, because the quadrupole 1s XAS transition into the 3T2g intermediate state term is forbidden. We note that the spin singlet and triplet intermediate states are only separated by a few meV by the 3d spin-orbit, Coulomb, and exchange interactions (e.g., ζ3d≈0.1eV). Therefore, the 1s XAS transitions appear as a single peak due to the resolution chosen in our calculations.Figure 18
1s XAS projections of the calculated 1s2p RIXS map for Ni2+ (3d8) showing only one peak as the T2g term is not reachable. The red lines indicate the energy levels of the intermediate state.The single peak observed in the 1s XAS step for Ni2+ (3d8) is the first important difference with Ti4+ (3d0), which arises from the different nature of the initial state term. This can be further understood in a single electron picture: since the two holes of the initial state of Ni2+ are in the eg orbitals, the excited 1s electron can only go to the empty eg orbitals (3Eg intermediate state). On the contrary, in the case of Ti4+ (Section 4.2), both eg and t2g orbitals are empty leaving both Eg and T2g intermediate state symmetries accessible.When the crystal field (10Dq) is large with respect to the other electronic interactions, such as the 3d spin-orbit interaction ζ3d, the 1s XAS step of the 1s2p RIXS can be described well with the noncoupled IRREPs (ℒ and S are good quantum numbers). However, when the crystal field is weak or when the electronic interactions are stronger, such as the ζ2d spin-orbit, it is necessary to use the ℒ.S-coupled terms. The spin-orbit coupling translates the intermediate state terms into the same total angular symmetry terms as for Ti4+ (see Figure 12). From this, it is evident that the intermediate state offers four different symmetries A2g, Eg, T1g, and T2g from which the 2p XES decays can occur.The branching in the octahedral symmetry of the final state of Ni2+ is again analogue as Ti4+ (see Section 4.2). Those terms translate in an Oh crystal field to the ungerade symmetries: A1u, A2u, Eu, T1u, and T2u. The dipole 2p XES decays in 1s2p RIXS from the intermediate state terms give
(24)Eg→⊗T1uT1u⊕T2u,(25)T1g→⊗T1uA1u⊕Eu⊕T1u⊕T2u,(26)T2g→⊗T1u2pXESA2u⊕Eu⊕T1u⊕T2u⏟AccessibleFSterms.From that, we find that in 2p XES all final state symmetries can be reached. However, for 2p XAS, the dipole transition (T1u) starts from the T2g ground state.
(27)T2g→⊗T1u2pXASA2u⊕Eu⊕T1u⊕T2u⏟AccessibleFSterms.From that, we find that the reachable 2p XAS final states can haveA2u, Eu, T1u, or T2u symmetry. In other words, all intermediate terms, except those with A1u, can be reached in 2p XAS. This is identical to the third case of the RIXS as described above in (26). However, the difference between the transition matrix elements of XAS and RIXS can result in different intensities.The comparison between the 2p XAS and 1s2p RIXS selection rules is summarised in the scheme in Figure19.Figure 19
Multielectronic term scheme forNi2+ 3d8 comparing the 1s2p RIXS and direct 2p XAS pathways. Populating the T2g intermediate state would require a two-electron transition in 3d8.Comparing this case with the case ofTi4+ in an Oh crystal field enables to highlight the crucial influence of the ground state symmetry.We conclude with the calculated 1s2p RIXS maps and a comparison with the corresponding 2p XAS as shown in Figure20.Figure 20
Calculated 1s2p RIXS for the ion Ni2+ (3d8) for 10Dq = 0.9 eV. As can be seen in the map reflecting the quadrupole IRREP T2g (top row, left), the contribution to the spectrum is nonzero because of the small 3d spin-orbit interaction leading to a weak mixing of the states. However, its contribution is about 186 times weaker and thus neglectable in first approximation.The calculations reveal two aspects: first, we notice the direct 2p XAS and the 2p XES final state spectra appear to have a similar appearance. This can be explained with the term scheme in Figure19 illustrating that the 2p XES and the direct 2p XAS probe similar final state terms. Second, we note that the intensity of the contribution to the 1s2p RIXS via the quadrupole T2g IRREP is two orders of magnitude weaker than the contribution via Eg 1s XAS IRREP. According to the strict ℒ=±2 selection rule (e.g., when neglecting 3d spin-orbit coupling), no transitions arise from the T2g absorption. However, the 3d spin-orbit interaction ζ3d is in fact nonzero, though small. Hence, when looking at the total angular momentum IRREPs of the intermediate states (e.g., the terms including SOC) and the 1s XAS selection rules shown in (22), it appears that some intermediate states can be probed by the T2g operator. In other words, the small but nonzero 3d spin-orbit coupling ζ3d induces some mixing such that the contributions to the spectrum via the 1s XAS IRREP T2g will be nonzero but weak. A direct comparison of the two 1s XAS transitions with T2g and Eg symmetries shows that the T2g spectrum multiplied by 186 is almost identical to the Eg spectrum (Figure 20).


#### 4.3.2. 1s2p RIXS Experiments of 3d8 Systems

Experimental 1s2p RIXS spectra of 3d8 NiF2 and molecular Ni2+ complexes have been published by Glatzel et al. [14, 40]. Within the resolution of the measurement, the NiF2 spectrum is exactly reproduced by the crystal field calculation [14].


## 4.1. 3d9: Divalent Copper Cu2+—Crystal Field Effects Altering the Selection Rules

In this part, we investigate the elementary differences between 1s2p RIXS and 2p XAS for a divalent copper ion Cu2+ with a 3d9 ground state. Each successive step of the 1s2p RIXS process has an electronic configuration with only one open shell and a single hole in the 3d, 1s, and 2p shells, respectively.
(12)We first describe the atomic case of an isolated ion without any crystal field. The electronic configuration of the initial state of Cu2+ is 1s22p63d9. In spherical symmetry (O3), the 3d spin-orbit coupling (SOC) ζ3d induces the splitting of the 10-fold degenerate ground state 2D (10 microstates) into two multielectronic states 2D5/2 (O3) and 2D3/2 (O3), defined by J=5/2 (ground state) and J=3/2, respectively. These two states are separated in energy by ΔE=5/2ζ3d. The electronic configuration of the 1s2p RIXS intermediate state of Cu2+ is 1s12p63d10, which corresponds to the 2-fold degenerate state 2S1/2 (O3) with J=1/2. The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Cu2+ is 1s22p53d10, which splits due to the 2p spin-orbit coupling ζ2p into J=3/2 (lowest energy state) and J=1/2 separated by 3/2ζ2p. This is summarised in the scheme in Figure 4.Figure 4
Atomic case: term scheme (with SOC; no crystal field) comparing the two-step (1s XAS, 2p XES) 1s2p RIXS and the one-step 2p XAS pathways for Cu2+. The sum over all J multiplicities 2J+1 yields the total number of microstates per configuration.Since the ground state of the initial state isJ=5/2, the electric dipole absorption operator (2p XAS) enables only to reach the J=3/2 final state (ΔJ=−1). In 1s2p RIXS, the selection rules of the electric quadrupole absorption for 1s XAS (ΔJ=−2) enable to reach the intermediate state J=1/2. The subsequent electric dipole emission (2p XES) enables to reach both J=1/2 and J=3/2 final states. As a result, for an isolated Cu2+ ion in spherical symmetry, the 2p XAS shows only one peak, while the 1s2p RIXS shows two peaks. The resulting spectra are shown in Figure 5.Figure 5
Calculated atomic 1s2p RIXS map with 2p XES projection for an isolated Cu2+ (3d9). Notice the 2p XAS (L2,3-edge) (dashed red) has only one peak, because only the final state with J=3/2 can be populated. See also Figure 7.3 in Core Level Spectroscopy of Solids [13].In the vertical direction, the Kα1 and Kα2 decays (2p XES) appear as two separate peaks. These two peaks are separated by the 2p SOC ζ2p≈13.5eV, corresponding to the two resonant emission transitions from the intermediate state with J=1/2 to the final states with J=1/2,3/2.


### 4.1.1. Considering a Cubic (Octahedral) Crystal Field

When the absorbing ion is embedded in a solid state, one has to take into account the crystal field potential created by the surrounding atoms. In the following, we will assume an octahedral (Oh) crystal field described by the parameter 10Dq. In a single electron picture, this crystal field induces the splitting of the 3d orbitals into t2g and eg. In the multielectronic formalism (neglecting the 3d spin-orbit coupling in first approximation), the 2D term of the initial state branches in the Oh point group into the 2Eg (t2g6eg3) and 2T2g (t2g5eg4) terms. Thus, the multielectronic ground state (at T=0K) is 2Eg (Oh). The intermediate state term 2S (O3) branches in an octahedral crystal field into 2A1g (Oh) symmetry.In the Oh point group, the electric quadrupole 1s XAS operator has T2g and Eg symmetries. We derive the selection rules from the product table (Table 3) for the first step of the 1s2p RIXS process.
(13)2Eg⊗T2g=2T1g⊕2T2g,2Eg⏟1s23d9⊗Eg⏟1sXAS=2A1g⊕2A2g⊕2Eg⏟Accessiblestates.From (13), it becomes obvious that many symmetries are in principle accessible via 1s XAS from the multielectronic ground state symmetry 2Eg (Oh). In spite of the fact that many symmetries are in principle reachable via a quadrupole 1s XAS transition starting from the ground state symmetry 2Eg (Oh), the electronic structure of a Cu2+ ion only offers 2A1g (Oh) symmetry leading to a single peak in the 1s XAS projection.While spin-orbit coupling was neglected in this last step, it must be included in the description of the final state with the 2p core hole to understand the complete 1s2p RIXS process and to be able to compare it with the 2p XAS. Using theℒ.S-coupled terms obtained above in the atomic case, we derive their respective branchings in the octahedral crystal field (Oh).The initial ground state2D5/2 (O3) branches in octahedral symmetry (Oh) into E5/2g⊕F3/2g (Oh). The ground state at T=0K is 2EgF3/2g (Oh). The effects of the 3d spin-orbit coupling ζ3d and the Oh crystal field parameter 10Dq are illustrated in Figure 2.The intermediate state term2S1/2 (O3) branches in an octahedral crystal field (Oh) into E1/2g (Oh). The final states 2P3/2 (O3) and 2P1/2 (O3) branch into F3/2u (Oh) and E1/2u (Oh), respectively. In octahedral symmetry, the electric dipole operator has T1u symmetry and thus, using the product table (Table 5), we find the reachable final states in the subsequent electric dipole emission (2p XES):
(14)E1/2g⊗T1u=E1/2u⊕F3/2u.The direct 2p XAS starts from theF3/2g ground state yielding with the T1u IRREP the reachable final state terms:
(15)F3/2g⊗T1u=E1/2u⊕E5/2u⊕2F3/2u.Altogether, this demonstrates that the crystal field enables both final states to be probed by 2p XAS and 1s2p RIXS. This is summarised in the scheme in Figure6 and the corresponding calculated spectrum is shown in Figure 7.Figure 6
Multielectronic term scheme for 3d9 Cu2+ comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC S=1/2→⊗E1/2 is included, the spin multiplicity is omitted.Figure 7
Calculated 1s2p RIXS map for 10Dq = 3.0 eV with 2p XES projection for Cu2+ (3d9) for 10Dq = 3.0 eV. The 2p XES appears identical to the 2p XAS. The mixing induced by the crystal field symmetry breakdown enables to populate the J=1/2 and J=3/2 final states in direct 2p XAS.The RIXS projections for various values of 10Dq show almost no difference in both directions (XAS and XES). It is noteworthy that the 2p XAS (L2,3-edge) approaches the two peaks in the 2p XES projections for large 10Dq values. This is due to the crystal field (10Dq) affecting the spin-orbit interaction altering the 2p XAS selection rules.In other words, the L2-edge probes in 2p XAS the amount of J=3/2 character in the ground state. In the atomic case (10Dq = 0.0 eV) with spherical symmetry (O3) and at absolute zero T=0K, the ion is pure J=5/2 (D5/2). In octahedral Oh symmetry, on the other hand, an increasing crystal field (scaled via 10Dq) mixes more and more J=3/2 character from the D3/2 into the ground state, resulting in a continuous visible increase of the L2 peak in 2p XAS.And finally, it can be seen that the single-particle limit with two peaks is reached in both cases, 1s2p RIXS and 2p XAS, respectively. The intensity ratio of the two peaks of 2 : 1 is given by the degeneracy of the2P1/2 (O3) and 2P3/2 (O3) final states.


### 4.1.2. 1s2p RIXS Experiments of 3d9 Systems

Experimental 1s2p RIXS spectra of CuO have been published by Hayashi et al. [2]. They show the 1s2p X-ray emission spectra from excitation energies before the K-edge, through the edge to the continuum. As such, they observe the transition from resonances in the Lorentzian tails to nonresonant 1s2p XES. At the excitation energy at the pre-edge, the two-peaked 1s2p RIXS spectrum is visible, in addition to the background from the edge [2].


## 4.1.1. Considering a Cubic (Octahedral) Crystal Field

When the absorbing ion is embedded in a solid state, one has to take into account the crystal field potential created by the surrounding atoms. In the following, we will assume an octahedral (Oh) crystal field described by the parameter 10Dq. In a single electron picture, this crystal field induces the splitting of the 3d orbitals into t2g and eg. In the multielectronic formalism (neglecting the 3d spin-orbit coupling in first approximation), the 2D term of the initial state branches in the Oh point group into the 2Eg (t2g6eg3) and 2T2g (t2g5eg4) terms. Thus, the multielectronic ground state (at T=0K) is 2Eg (Oh). The intermediate state term 2S (O3) branches in an octahedral crystal field into 2A1g (Oh) symmetry.In the Oh point group, the electric quadrupole 1s XAS operator has T2g and Eg symmetries. We derive the selection rules from the product table (Table 3) for the first step of the 1s2p RIXS process.
(13)2Eg⊗T2g=2T1g⊕2T2g,2Eg⏟1s23d9⊗Eg⏟1sXAS=2A1g⊕2A2g⊕2Eg⏟Accessiblestates.From (13), it becomes obvious that many symmetries are in principle accessible via 1s XAS from the multielectronic ground state symmetry 2Eg (Oh). In spite of the fact that many symmetries are in principle reachable via a quadrupole 1s XAS transition starting from the ground state symmetry 2Eg (Oh), the electronic structure of a Cu2+ ion only offers 2A1g (Oh) symmetry leading to a single peak in the 1s XAS projection.While spin-orbit coupling was neglected in this last step, it must be included in the description of the final state with the 2p core hole to understand the complete 1s2p RIXS process and to be able to compare it with the 2p XAS. Using theℒ.S-coupled terms obtained above in the atomic case, we derive their respective branchings in the octahedral crystal field (Oh).The initial ground state2D5/2 (O3) branches in octahedral symmetry (Oh) into E5/2g⊕F3/2g (Oh). The ground state at T=0K is 2EgF3/2g (Oh). The effects of the 3d spin-orbit coupling ζ3d and the Oh crystal field parameter 10Dq are illustrated in Figure 2.The intermediate state term2S1/2 (O3) branches in an octahedral crystal field (Oh) into E1/2g (Oh). The final states 2P3/2 (O3) and 2P1/2 (O3) branch into F3/2u (Oh) and E1/2u (Oh), respectively. In octahedral symmetry, the electric dipole operator has T1u symmetry and thus, using the product table (Table 5), we find the reachable final states in the subsequent electric dipole emission (2p XES):
(14)E1/2g⊗T1u=E1/2u⊕F3/2u.The direct 2p XAS starts from theF3/2g ground state yielding with the T1u IRREP the reachable final state terms:
(15)F3/2g⊗T1u=E1/2u⊕E5/2u⊕2F3/2u.Altogether, this demonstrates that the crystal field enables both final states to be probed by 2p XAS and 1s2p RIXS. This is summarised in the scheme in Figure6 and the corresponding calculated spectrum is shown in Figure 7.Figure 6
Multielectronic term scheme for 3d9 Cu2+ comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC S=1/2→⊗E1/2 is included, the spin multiplicity is omitted.Figure 7
Calculated 1s2p RIXS map for 10Dq = 3.0 eV with 2p XES projection for Cu2+ (3d9) for 10Dq = 3.0 eV. The 2p XES appears identical to the 2p XAS. The mixing induced by the crystal field symmetry breakdown enables to populate the J=1/2 and J=3/2 final states in direct 2p XAS.The RIXS projections for various values of 10Dq show almost no difference in both directions (XAS and XES). It is noteworthy that the 2p XAS (L2,3-edge) approaches the two peaks in the 2p XES projections for large 10Dq values. This is due to the crystal field (10Dq) affecting the spin-orbit interaction altering the 2p XAS selection rules.In other words, the L2-edge probes in 2p XAS the amount of J=3/2 character in the ground state. In the atomic case (10Dq = 0.0 eV) with spherical symmetry (O3) and at absolute zero T=0K, the ion is pure J=5/2 (D5/2). In octahedral Oh symmetry, on the other hand, an increasing crystal field (scaled via 10Dq) mixes more and more J=3/2 character from the D3/2 into the ground state, resulting in a continuous visible increase of the L2 peak in 2p XAS.And finally, it can be seen that the single-particle limit with two peaks is reached in both cases, 1s2p RIXS and 2p XAS, respectively. The intensity ratio of the two peaks of 2 : 1 is given by the degeneracy of the2P1/2 (O3) and 2P3/2 (O3) final states.


## 4.1.2. 1s2p RIXS Experiments of 3d9 Systems

Experimental 1s2p RIXS spectra of CuO have been published by Hayashi et al. [2]. They show the 1s2p X-ray emission spectra from excitation energies before the K-edge, through the edge to the continuum. As such, they observe the transition from resonances in the Lorentzian tails to nonresonant 1s2p XES. At the excitation energy at the pre-edge, the two-peaked 1s2p RIXS spectrum is visible, in addition to the background from the edge [2].


## 4.2. 3d0: Tetravalent Titanium Ti4+—1s XAS Peaks Split by 10Dq

In the following, we discuss the differences between 2p XAS and 1s2p RIXS for the case of tetravalent titanium (Ti4+). Here, the initial state configuration 1s22p63d0 has no partially filled shells. This is interesting because the resulting selection rules are straightforward: the nature of the probed final states reflects the nature and symmetry of the transition operators. The transition into the intermediate state promotes a 1s electron into the 3d band yielding a 1s12p63d1 electron configuration. The final state in 1s2p RIXS and 2p XAS is in this case 1s22p53d1.
(16)The 3d0 case enables to describe the effect of the crystal field on the intermediate state, the interferences in the 1s2p RIXS process, and the more complex multielectronic effects in the final state. It provides an extent to L2,3-edge considerations of 3d0 ions previously described in [35].


### 4.2.1. Atomic Case with Spin-Orbit Coupling

We start again with the case of an isolated Ti4+ ion (3d0) where the effect of the solid state, that is, the crystal field, is neglected. The initial state configuration of Ti4+ is 1s22p63d0. Since all shells are full, S=0 and ℒ=0. The ℒ.S-coupled total angular momentum is thus J=0, and the initial state symmetry is the totally symmetric term 1S0.The electronic configuration of the intermediate state in 1s2p RIXS of Ti4+ is 1s12p63d1, corresponding to the total orbital angular momentum ℒ=2 and total spin angular momentum S=0 or S=1 giving the Russel-Saunders terms 1D and 3D (2⋅10=20 microstates). Considering the spin-orbit interaction in the ℒ.S coupling scheme, one obtains J=2 for the 1D term and J=1,2,and3 for the 3D term (Figure 8). The splitting of these four terms (1D2, 3D1, 3D2, and 3D3) is defined by the 3d SOC ζ3d and exchange interaction Gsd2, which are both small (32 meV and 46 meV, resp.). The calculated energy splitting is ΔE=92meV and is beyond the reach of current experimental resolution and not resolved in our calculations discussed here.Figure 8
Splitting of the intermediate state spin singlet and triplet terms1,3D due to 3d SOC ζ3d for a Ti4+ ion into D1, D2, and D3. States accessible in 1s XAS (J=2) are shown in blue. However, the existing terms with J=1 or 3 cannot be populated and will therefore not contribute to the spectrum.The electric quadrupole transition of the 1s XAS enables to reach the intermediate states withJ=2, though here 3d SOC ζ3d is small and transitions to the spin triplet term 3D2 will be weak, such that the 1s XAS is dominated by the transition in the spin singlet term 1D2.The final state electronic configuration is 1s22p53d1 with two partially filled shells that have to be accounted for: the 2p and the 3d shell. The total orbital angular moment ℒ of the final state electronic configuration is ℒ=1,2,and3. The total spin angular moment S of the final state electronic configuration is S=0,1. This gives in the atomic case the spin singlet and triplet terms 1,3P, 1,3D, and 1,3F.As previously described by de Groot et al. [35], when spin-orbit interaction is neglected, the selection rules for the electric dipole transition in 2p XAS with ΔS=0 and Δℒ=±1 allow only to reach the singlet 1P term from the 1S initial state, leading to one peak only (the L2-edge). The L3-edge, corresponding to transitions into the spin triplet term 3P, would have zero intensity. But in fact the strong 2p spin-orbit coupling ζ2p in the final state (J is used) leads to three peaks, two of them forming the L2- and L3-edges, and a third weak peak related to triplet transitions.The derivation of the total angular momentumJ of the final state electronic configuration in the ℒ.S coupling scheme yields the term symbols illustrated in Figure 9.Figure 9
Splitting of the final state terms1,3P, 1,3D, and 1,3F due to spin-orbit coupling for Ti4+.The direct electric dipole transition in 2p XAS withΔJ=0,±1 (J=0→J=0 forbidden) starts for Ti4+ from the total symmetric term 1S0, which allows to reach all final states with J=1 (P1 and D1, marked with a red circle in Figure 9). The L2- and L3-edges are separated by the 2p spin-orbit coupling ζ2p corresponding to transitions into the 1P1 (L2-edge) and 3D1 (L3-edge) terms. More precisely, from the squared matrix elements, we find that the second peak is dominated by 3D1 contribution of 60%, adding 36% of 1P1 and 4% of 3P1 character. The small third peak in the 2p XAS, at ET≈−5eV in Figure 10, is related to transitions into the 3P1 terms. Though the resolution chosen in our calculations does not reveal them as individual peaks, the direct Coulomb and exchange terms are not negligible. As discussed by de Groot et al. [35], they lead to a splitting of the L2,3-edge into three absorption lines including a redistribution of the intensities.Figure 10
Calculated 1s2p RIXS map with 2p XES projection and the 2p XAS (L2,3-edge) for an isolated Ti4+ (3d0) ion in O3 symmetry.In the 1s2p RIXS process, the 2p XES decays withΔJ=0,±1 starting from the intermediate states with J=2 (D2) and enables to reach the final states P1,2, D1,2,3, and F2,3 with J=1,2,and3 (terms shown in green Figure 9). Therefore, all J values except 0 and 4 are reachable.With this, one can draw for Ti4+ the atomic term scheme with spin-orbit coupling for the two-step 1s2p RIXS and the one-step 2p XAS process as shown in Figure 11.Figure 11
Atomic term scheme with spin-orbit coupling for Ti4+ (3d0) comparing the two-step (1s XAS, 2p XES) 1s2p RIXS and the one-step 2p XAS pathways. The two 1s XAS transitions are separated by the small 3d SOC, Coulomb, and exchange interactions; hence, the splitting is small. The direct dipole transitions in 2p XAS can only populate the final states with J=1, while the dipole decays in 2p XES can reach final states with J=1,2,and3. However, terms with J=0 or 4 cannot be populated in either case and thus do not contribute to the spectrum. Nonaccessible terms are omitted for clarity.The intermediate state is the key element when comparing the direct 2p XAS and 1s2p RIXS. It enables to access additional terms in the 1s2p RIXS final state, adding multiple visible peaks in the energy transfer direction as shown in Figure10. But in both cases, J=0 and 4 are not possible to reach.In summary, so far, we have discussed the effect of the spin-orbit coupling in the atomic case in spherical symmetry (O3), but neglecting any crystal field. With the above, one can describe the 1s XAS with only one visible peak due to transitions into the 1D2 intermediate state term. The 3d spin-orbit coupling ζ3d and exchange interaction Gsd lead to small splittings which are not visible in this plot. Furthermore, the small 3d SOC may induce some mixing adding weak transitions into the 3D2 term.The splitting of the 2p XAS is dominated by the large 2p SOCζ2p separating the L2-edge from the L3-edge, for example, the spin singlet and the triplet states. The 2p XES on the other hand consists of transitions from the intermediate state with J=2 IRREP (mostly 1D2) into the final states with J=1,2,and3 IRREPs resulting multiple visible peaks in the energy transfer direction. All final state terms are also split by the exchange and direct Coulomb interactions, Gpd and Fpd, respectively.


### 4.2.2. Considering an Oh Crystal Field

We now consider the effect of an octahedral Oh crystal field. The totally symmetric initial state term 1S0 branches in Oh into 1A1gA1g. Formal consideration of an Oh crystal field and 3d spin-orbit interaction leads to the branching of the intermediate state terms 1D and 3D into eight terms. The scheme in Figure 12 illustrates these branchings of the terms due to an Oh crystal field and the 3d spin-orbit interaction ζ3d (S=0→⊗A1g and S=1→⊗T1g).Figure 12
Branching of the intermediate state terms1D and 3D of Ti4+ (3d0) due to 3d SOC and an Oh CF. Symmetries accessible in the quadrupole 1s XAS step (GS → IS) are shown in blue.In spite of several available intermediate state terms, starting from the initial state1A1g and using the T2g and Eg IRREPs, representing the quadrupole 1s XAS operator, and the product table (Table 3) reflecting the orbital momentum selection rules, one finds the accessible intermediate state symmetries.
(17)1A1g⊗T2g=1T2g,1A1g⏟1s23d0⊗Eg⏟1sXAS=1Eg⏟Accessibleterms.In other words, from thetotally symmetric ground state A1g (Oh), only the Eg (Oh) and T2g (Oh) symmetries can be reached in the intermediate state via a quadrupole 1s XAS transition. Hence, for sufficiently large values of 10Dq, relative to the broadenings used in a calculation (or the experimental resolution in a measurement), two distinct peaks can be observed in the 1s XAS projection (Figure 13). Because the 3d spin-orbit coupling ζ3d is small, the transitions to the spin singlet intermediate states are dominating (i.e., matrix elements are the largest), while the transitions via the J=2 IRREP from triplet 3D should be weak. The 1s XAS is dominated by transitions to the Eg and T2g terms from the 1D2.Figure 13
Calculated 1s XAS projections of a 1s2p RIXS for Ti4+ (3d0) scaling the octahedral crystal field parameter 10Dq from 0.1 eV to 3.0 eV with an energy level diagram as overlay. The splitting of the 3d level into the two peaks T2g and Eg due to the crystal field is visible.In an Oh crystal field, the final state IRREPs with J=0,1,2,3 and 4 (see Figure 11) branch, respectively, into the ℒ.S-coupled terms: A1u, T1u, Eu⊕T2u, A2u⊕T1u⊕T2u, and A1u⊕Eu⊕T1u⊕T2u. The 2p XES in 1s2p RIXS from the two intermediate state terms T2g and Eg via the electric dipole operator, described by the T1u IRREP, leads to
(18)T2g⊗T1u=A2u⊕Eu⊕T1u⊕T2u,Eg⏟1s13d1⊗T1u⏟2pXES=T1u⊕T2u⏟Accessiblefinalstates.It is noteworthy here that only the intermediate state withT2g symmetry can access in the 2p XES decay the A2u and Eu final state symmetries. The T1u and T2u final state terms are however accessible from both T2g and Eg intermediate state symmetries. Furthermore, we note that the intermediate state symmetry A1u is not reachable as the final state symmetry and will therefore not contribute to the 2p XES spectrum.On the other hand, in 2p XAS, the possible final states must haveT1u symmetry; otherwise, the transition matrix element equals zero. As can be seen from the branchings of J (O3) in and octahedral crystal field (Table 2), the T1u IRREP can be reached not only from the J=1 IRREP but also from the J=3 and J=4 IRREPs.This results in only seven possibleT1u IRREPs in Oh symmetry, or in other words, only seven final states are reachable among the 60 available. The 2p XAS spectrum consists, in principle, of seven lines, as can be seen in Figure 14, where the calculated spectra for Ti4+ (3d0) for a crystal-field splitting of 10Dq = 2.1 eV are shown. While only spin singlet states would be reached in the absence of 2p SOC, the consideration of the strong spin-orbit coupling in the final state induces a mixing of the spin singlet and triplet states. This weakens the spin selection rule ΔS=0 allowing also transitions into former spin triplet states (e.g., 3T2u, 3T1u, 3Eu, and 3A2u).Figure 14
(a) Calculation of the 1s2p RIXS spectra for Ti4+ (3d0) with 10Dq=2.1eV separated into the quadrupole components Ylm (with m=0, 1 and l=2) in spherical symmetry. They translate into the T2g and Eg IRREPs describing the quadrupole 1s XAS transitions. The term scheme insets illustrate the corresponding pathways. (b) The corresponding 2p XES projections onto the energy transfer axis for the two quadrupole components and the calculated 2p XAS (L2,3-edge).
(a)
(b)This enables to draw the electronic term scheme in Figure15 illustrating the different pathways in 1s2p RIXS and 2p XAS for 3d0 Ti4+ in Oh symmetry. The accessible 1s2p RIXS final state terms are given in (18), while, as discussed above, the 2p XAS can only reach final states with T1u symmetry (A1g⊗T1u=T1u).Figure 15
Multielectronic term scheme comparing the 2p XAS and the 1s2p RIXS pathways. The two quadrupole 1s XAS transitions withT2g and Eg symmetry lead to two different intermediate state symmetries. The subsequent dipole decays (2p XES) with T1u symmetry yield multiple 1s2p RIXS final state symmetries. The direct 2p XAS (L2,3-edge) transition with T1u symmetry can on the other hand only access the T1u symmetry in the final state. Noncontributing states are omitted for clarity.Similarly to the 3d9 case (Cu2+) discussed in Section 4.1, comparing the final states, it becomes clear once again that the key elements are the intermediate states from which the 2p XES decays and thus enables to reach more final states when compared to the direct 2p XAS.This overall results in the 1s2p RIXS maps shown in Figure14. The total RIXS map is decomposed into the Eg and T2g quadrupole absorption components. The selective absorption towards each intermediate state is obtained. Experimentally, this dichroic behaviour has been observed for Ti4+ in cubic oxides such as the prototypal SrTiO3 [36]. The 1s2p RIXS spectra calculated for each of the two absorption peaks are shown (Figure 14(a)), and the 60 final states are labelled with their total angular IRREP (Figure 14(b)). This enables to see that both, the 1s2p RIXS and 2p XAS, probe the T1u final state symmetry. We further observe that within the four main peaks in 2p XAS, the first peaks of the L3 (at ET≈−3.5eV) and of the L2 (at ET≈2eV) are stronger via the T2g absorption channel of the 1s2p RIXS than the Eg and vice-versa: the second peaks of the L3 (ET≈−1eV) and of the L2 (at ET≈4.5eV) are stronger via the Eg absorption channel of the 1s2p RIXS than via the T2g. This can be understood in a single electron picture with the crystal field splitting of the final state: the lower states of the L3-edge contain the 3d single electron in the t2g orbitals while the higher energy states have the 3d electron in the eg orbitals and similarly for the L2-edge [37].


### 4.2.3. 1s2p RIXS Experiments of 3d0 Systems

Experimental 1s XAS spectra (e.g., HERFD) of titanium oxides in the solid state often show more than two peaks in the pre-edge, which contradicts the result shown in Figure13. The reason is that there are additional dipole transitions at energies overlapping with the quadrupole prepeaks. These dipole peaks are related to so-called nonlocal peaks, where the 4p states from the absorbing ion mix with the 3d states of the neighbouring ions [3]. In addition, we note that for high valent ions such as Ti4+, charge transfer is important for the detailed description of the 2p XAS and 1s2p RIXS spectral shapes [37].Experimental 1s2p RIXS experiments have been published on TiO2 by Glatzel et al. [38] and Kas et al. [39]. The 1s2p RIXS planes show the quadrupole pre-edges and the nonlocal peaks. The 1s2p RIXS cross-section at the first quadrupole peak shows three peaks in qualitative agreement with the t2g calculation in Figure 14. Bagger et al. analysed the quadrupole and dipole RIXS separately [1]. We note that the ionic limit as calculated in Figure 14 is likely to be reached by 1s2p RIXS experiments at the K-edge on divalent calcium systems (Ca2+), including octahedral CaO and cubic CaF2.


## 4.2.1. Atomic Case with Spin-Orbit Coupling

We start again with the case of an isolated Ti4+ ion (3d0) where the effect of the solid state, that is, the crystal field, is neglected. The initial state configuration of Ti4+ is 1s22p63d0. Since all shells are full, S=0 and ℒ=0. The ℒ.S-coupled total angular momentum is thus J=0, and the initial state symmetry is the totally symmetric term 1S0.The electronic configuration of the intermediate state in 1s2p RIXS of Ti4+ is 1s12p63d1, corresponding to the total orbital angular momentum ℒ=2 and total spin angular momentum S=0 or S=1 giving the Russel-Saunders terms 1D and 3D (2⋅10=20 microstates). Considering the spin-orbit interaction in the ℒ.S coupling scheme, one obtains J=2 for the 1D term and J=1,2,and3 for the 3D term (Figure 8). The splitting of these four terms (1D2, 3D1, 3D2, and 3D3) is defined by the 3d SOC ζ3d and exchange interaction Gsd2, which are both small (32 meV and 46 meV, resp.). The calculated energy splitting is ΔE=92meV and is beyond the reach of current experimental resolution and not resolved in our calculations discussed here.Figure 8
Splitting of the intermediate state spin singlet and triplet terms1,3D due to 3d SOC ζ3d for a Ti4+ ion into D1, D2, and D3. States accessible in 1s XAS (J=2) are shown in blue. However, the existing terms with J=1 or 3 cannot be populated and will therefore not contribute to the spectrum.The electric quadrupole transition of the 1s XAS enables to reach the intermediate states withJ=2, though here 3d SOC ζ3d is small and transitions to the spin triplet term 3D2 will be weak, such that the 1s XAS is dominated by the transition in the spin singlet term 1D2.The final state electronic configuration is 1s22p53d1 with two partially filled shells that have to be accounted for: the 2p and the 3d shell. The total orbital angular moment ℒ of the final state electronic configuration is ℒ=1,2,and3. The total spin angular moment S of the final state electronic configuration is S=0,1. This gives in the atomic case the spin singlet and triplet terms 1,3P, 1,3D, and 1,3F.As previously described by de Groot et al. [35], when spin-orbit interaction is neglected, the selection rules for the electric dipole transition in 2p XAS with ΔS=0 and Δℒ=±1 allow only to reach the singlet 1P term from the 1S initial state, leading to one peak only (the L2-edge). The L3-edge, corresponding to transitions into the spin triplet term 3P, would have zero intensity. But in fact the strong 2p spin-orbit coupling ζ2p in the final state (J is used) leads to three peaks, two of them forming the L2- and L3-edges, and a third weak peak related to triplet transitions.The derivation of the total angular momentumJ of the final state electronic configuration in the ℒ.S coupling scheme yields the term symbols illustrated in Figure 9.Figure 9
Splitting of the final state terms1,3P, 1,3D, and 1,3F due to spin-orbit coupling for Ti4+.The direct electric dipole transition in 2p XAS withΔJ=0,±1 (J=0→J=0 forbidden) starts for Ti4+ from the total symmetric term 1S0, which allows to reach all final states with J=1 (P1 and D1, marked with a red circle in Figure 9). The L2- and L3-edges are separated by the 2p spin-orbit coupling ζ2p corresponding to transitions into the 1P1 (L2-edge) and 3D1 (L3-edge) terms. More precisely, from the squared matrix elements, we find that the second peak is dominated by 3D1 contribution of 60%, adding 36% of 1P1 and 4% of 3P1 character. The small third peak in the 2p XAS, at ET≈−5eV in Figure 10, is related to transitions into the 3P1 terms. Though the resolution chosen in our calculations does not reveal them as individual peaks, the direct Coulomb and exchange terms are not negligible. As discussed by de Groot et al. [35], they lead to a splitting of the L2,3-edge into three absorption lines including a redistribution of the intensities.Figure 10
Calculated 1s2p RIXS map with 2p XES projection and the 2p XAS (L2,3-edge) for an isolated Ti4+ (3d0) ion in O3 symmetry.In the 1s2p RIXS process, the 2p XES decays withΔJ=0,±1 starting from the intermediate states with J=2 (D2) and enables to reach the final states P1,2, D1,2,3, and F2,3 with J=1,2,and3 (terms shown in green Figure 9). Therefore, all J values except 0 and 4 are reachable.With this, one can draw for Ti4+ the atomic term scheme with spin-orbit coupling for the two-step 1s2p RIXS and the one-step 2p XAS process as shown in Figure 11.Figure 11
Atomic term scheme with spin-orbit coupling for Ti4+ (3d0) comparing the two-step (1s XAS, 2p XES) 1s2p RIXS and the one-step 2p XAS pathways. The two 1s XAS transitions are separated by the small 3d SOC, Coulomb, and exchange interactions; hence, the splitting is small. The direct dipole transitions in 2p XAS can only populate the final states with J=1, while the dipole decays in 2p XES can reach final states with J=1,2,and3. However, terms with J=0 or 4 cannot be populated in either case and thus do not contribute to the spectrum. Nonaccessible terms are omitted for clarity.The intermediate state is the key element when comparing the direct 2p XAS and 1s2p RIXS. It enables to access additional terms in the 1s2p RIXS final state, adding multiple visible peaks in the energy transfer direction as shown in Figure10. But in both cases, J=0 and 4 are not possible to reach.In summary, so far, we have discussed the effect of the spin-orbit coupling in the atomic case in spherical symmetry (O3), but neglecting any crystal field. With the above, one can describe the 1s XAS with only one visible peak due to transitions into the 1D2 intermediate state term. The 3d spin-orbit coupling ζ3d and exchange interaction Gsd lead to small splittings which are not visible in this plot. Furthermore, the small 3d SOC may induce some mixing adding weak transitions into the 3D2 term.The splitting of the 2p XAS is dominated by the large 2p SOCζ2p separating the L2-edge from the L3-edge, for example, the spin singlet and the triplet states. The 2p XES on the other hand consists of transitions from the intermediate state with J=2 IRREP (mostly 1D2) into the final states with J=1,2,and3 IRREPs resulting multiple visible peaks in the energy transfer direction. All final state terms are also split by the exchange and direct Coulomb interactions, Gpd and Fpd, respectively.


## 4.2.2. Considering an Oh Crystal Field

We now consider the effect of an octahedral Oh crystal field. The totally symmetric initial state term 1S0 branches in Oh into 1A1gA1g. Formal consideration of an Oh crystal field and 3d spin-orbit interaction leads to the branching of the intermediate state terms 1D and 3D into eight terms. The scheme in Figure 12 illustrates these branchings of the terms due to an Oh crystal field and the 3d spin-orbit interaction ζ3d (S=0→⊗A1g and S=1→⊗T1g).Figure 12
Branching of the intermediate state terms1D and 3D of Ti4+ (3d0) due to 3d SOC and an Oh CF. Symmetries accessible in the quadrupole 1s XAS step (GS → IS) are shown in blue.In spite of several available intermediate state terms, starting from the initial state1A1g and using the T2g and Eg IRREPs, representing the quadrupole 1s XAS operator, and the product table (Table 3) reflecting the orbital momentum selection rules, one finds the accessible intermediate state symmetries.
(17)1A1g⊗T2g=1T2g,1A1g⏟1s23d0⊗Eg⏟1sXAS=1Eg⏟Accessibleterms.In other words, from thetotally symmetric ground state A1g (Oh), only the Eg (Oh) and T2g (Oh) symmetries can be reached in the intermediate state via a quadrupole 1s XAS transition. Hence, for sufficiently large values of 10Dq, relative to the broadenings used in a calculation (or the experimental resolution in a measurement), two distinct peaks can be observed in the 1s XAS projection (Figure 13). Because the 3d spin-orbit coupling ζ3d is small, the transitions to the spin singlet intermediate states are dominating (i.e., matrix elements are the largest), while the transitions via the J=2 IRREP from triplet 3D should be weak. The 1s XAS is dominated by transitions to the Eg and T2g terms from the 1D2.Figure 13
Calculated 1s XAS projections of a 1s2p RIXS for Ti4+ (3d0) scaling the octahedral crystal field parameter 10Dq from 0.1 eV to 3.0 eV with an energy level diagram as overlay. The splitting of the 3d level into the two peaks T2g and Eg due to the crystal field is visible.In an Oh crystal field, the final state IRREPs with J=0,1,2,3 and 4 (see Figure 11) branch, respectively, into the ℒ.S-coupled terms: A1u, T1u, Eu⊕T2u, A2u⊕T1u⊕T2u, and A1u⊕Eu⊕T1u⊕T2u. The 2p XES in 1s2p RIXS from the two intermediate state terms T2g and Eg via the electric dipole operator, described by the T1u IRREP, leads to
(18)T2g⊗T1u=A2u⊕Eu⊕T1u⊕T2u,Eg⏟1s13d1⊗T1u⏟2pXES=T1u⊕T2u⏟Accessiblefinalstates.It is noteworthy here that only the intermediate state withT2g symmetry can access in the 2p XES decay the A2u and Eu final state symmetries. The T1u and T2u final state terms are however accessible from both T2g and Eg intermediate state symmetries. Furthermore, we note that the intermediate state symmetry A1u is not reachable as the final state symmetry and will therefore not contribute to the 2p XES spectrum.On the other hand, in 2p XAS, the possible final states must haveT1u symmetry; otherwise, the transition matrix element equals zero. As can be seen from the branchings of J (O3) in and octahedral crystal field (Table 2), the T1u IRREP can be reached not only from the J=1 IRREP but also from the J=3 and J=4 IRREPs.This results in only seven possibleT1u IRREPs in Oh symmetry, or in other words, only seven final states are reachable among the 60 available. The 2p XAS spectrum consists, in principle, of seven lines, as can be seen in Figure 14, where the calculated spectra for Ti4+ (3d0) for a crystal-field splitting of 10Dq = 2.1 eV are shown. While only spin singlet states would be reached in the absence of 2p SOC, the consideration of the strong spin-orbit coupling in the final state induces a mixing of the spin singlet and triplet states. This weakens the spin selection rule ΔS=0 allowing also transitions into former spin triplet states (e.g., 3T2u, 3T1u, 3Eu, and 3A2u).Figure 14
(a) Calculation of the 1s2p RIXS spectra for Ti4+ (3d0) with 10Dq=2.1eV separated into the quadrupole components Ylm (with m=0, 1 and l=2) in spherical symmetry. They translate into the T2g and Eg IRREPs describing the quadrupole 1s XAS transitions. The term scheme insets illustrate the corresponding pathways. (b) The corresponding 2p XES projections onto the energy transfer axis for the two quadrupole components and the calculated 2p XAS (L2,3-edge).
(a)
(b)This enables to draw the electronic term scheme in Figure15 illustrating the different pathways in 1s2p RIXS and 2p XAS for 3d0 Ti4+ in Oh symmetry. The accessible 1s2p RIXS final state terms are given in (18), while, as discussed above, the 2p XAS can only reach final states with T1u symmetry (A1g⊗T1u=T1u).Figure 15
Multielectronic term scheme comparing the 2p XAS and the 1s2p RIXS pathways. The two quadrupole 1s XAS transitions withT2g and Eg symmetry lead to two different intermediate state symmetries. The subsequent dipole decays (2p XES) with T1u symmetry yield multiple 1s2p RIXS final state symmetries. The direct 2p XAS (L2,3-edge) transition with T1u symmetry can on the other hand only access the T1u symmetry in the final state. Noncontributing states are omitted for clarity.Similarly to the 3d9 case (Cu2+) discussed in Section 4.1, comparing the final states, it becomes clear once again that the key elements are the intermediate states from which the 2p XES decays and thus enables to reach more final states when compared to the direct 2p XAS.This overall results in the 1s2p RIXS maps shown in Figure14. The total RIXS map is decomposed into the Eg and T2g quadrupole absorption components. The selective absorption towards each intermediate state is obtained. Experimentally, this dichroic behaviour has been observed for Ti4+ in cubic oxides such as the prototypal SrTiO3 [36]. The 1s2p RIXS spectra calculated for each of the two absorption peaks are shown (Figure 14(a)), and the 60 final states are labelled with their total angular IRREP (Figure 14(b)). This enables to see that both, the 1s2p RIXS and 2p XAS, probe the T1u final state symmetry. We further observe that within the four main peaks in 2p XAS, the first peaks of the L3 (at ET≈−3.5eV) and of the L2 (at ET≈2eV) are stronger via the T2g absorption channel of the 1s2p RIXS than the Eg and vice-versa: the second peaks of the L3 (ET≈−1eV) and of the L2 (at ET≈4.5eV) are stronger via the Eg absorption channel of the 1s2p RIXS than via the T2g. This can be understood in a single electron picture with the crystal field splitting of the final state: the lower states of the L3-edge contain the 3d single electron in the t2g orbitals while the higher energy states have the 3d electron in the eg orbitals and similarly for the L2-edge [37].


## 4.2.3. 1s2p RIXS Experiments of 3d0 Systems

Experimental 1s XAS spectra (e.g., HERFD) of titanium oxides in the solid state often show more than two peaks in the pre-edge, which contradicts the result shown in Figure13. The reason is that there are additional dipole transitions at energies overlapping with the quadrupole prepeaks. These dipole peaks are related to so-called nonlocal peaks, where the 4p states from the absorbing ion mix with the 3d states of the neighbouring ions [3]. In addition, we note that for high valent ions such as Ti4+, charge transfer is important for the detailed description of the 2p XAS and 1s2p RIXS spectral shapes [37].Experimental 1s2p RIXS experiments have been published on TiO2 by Glatzel et al. [38] and Kas et al. [39]. The 1s2p RIXS planes show the quadrupole pre-edges and the nonlocal peaks. The 1s2p RIXS cross-section at the first quadrupole peak shows three peaks in qualitative agreement with the t2g calculation in Figure 14. Bagger et al. analysed the quadrupole and dipole RIXS separately [1]. We note that the ionic limit as calculated in Figure 14 is likely to be reached by 1s2p RIXS experiments at the K-edge on divalent calcium systems (Ca2+), including octahedral CaO and cubic CaF2.


## 4.3. 3d8: Divalent Nickel Ni2+—Single Peak in 1s XAS

In this part, we investigate the elementary differences between 2p XAS and 1s2p RIXS for a divalent nickel ion (Ni2+, 3d8). As we will show, the electronic configurations of the different steps of 1s2p RIXS present strong analogies with the case of Ti4+; the main difference being the nature of the initial state. We start again with the atomic case by deriving the J values as IRREPs.
(19)The electronic configuration of the initial state of Ni2+ is 1s22p63d8 with two holes in the 3d shell. The multielectronic interactions lead to 45 states spread across the terms 3F, 3P, 1G, 1D, and 1S [13]. In spherical symmetry (O3), the atomic ground state is 3F (ℒ=3 and S=1). The 3d spin-orbit coupling ζ3d≈0.1eV induces the splitting of the 3F term into the three terms:
(20)3F→3dSOC3F2,3F3,3F4.From those three IRREPs withJ=2,3,and4, the 3F4 term with J=4 is the ground state term symbol.The electronic configuration of the intermediate state in 1s2p RIXS of Ni2+ is 1s12p63d9 (one hole in the 1s shell and one hole in the 3d shell) which translates into the two terms 1D and 3D (2⋅10=20 microstates). It is noteworthy that this is identical to the intermediate state of Ti4+ described above (Section 4.2) due to the electron-hole equivalency. The 3d spin-orbit interaction splits the two terms into the 1D2 (for S=0) and 3D1, 3D2, and 3D3 (for S=1) Russel-Saunders terms. The resulting splittings distribute the four terms in energy over ΔE≈0.25eV for which the 1s XAS will appear only as a single peak due to the resolution chosen in our calculations, analogue to the Ti4+ atomic case.The difference in the 1s XAS step in 1s2p RIXS between the 3d8 Ni2+ and 3d0 Ti4+ cases arises from the spin multiplicity of the initial state. In both cases, the electric quadrupole absorption leads to a J=2 intermediate state. However, because of the small 3d spin-orbit coupling (ζ3d=0.1eV), the spin multiplicity is mostly conserved in the transition and the absorption from the spin triplet initial state of Ni2+ leads to the 3D2 state. For Ti4+ on the other hand, mainly, the 1D2 term is reached.The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Ni2+ is 1s22p53d9. This is again strictly analogue to the final state of Ti4+. The final state configuration corresponds to several atomic terms (1,3P, 1,3D, and 1,3F) which are split further due to the strong 2p spin-orbit coupling ζ2p=11.5eV. The total symmetry final state terms are represented with integer J values ranging from J=0 to J=4. The detailed splittings are given above in Figure 9 in Section 4.2.Only the final state terms withJ=1,2,and3 are formally accessible in the 1s2p RIXS from the intermediate state with J=2 via 2p XES decays (ΔJ=0,±1). The direct 2p XAS (with ΔJ=0,±1) on the other hand can only reach the final state terms with J=3 or J=4 from the ground state with J=4. Altogether, the selection rules for the atomic case can be summarised as
(21)This is also illustrated in the combined term scheme shown in Figure16.Figure 16
Atomic term scheme with spin-orbit coupling for Ni2+ comparing the two-step (1s XAS, 2p XES) 1s2p RIXS and the one-step 2p XAS pathways. Here, it should be noted that the two IS terms are only separated by the small 3d SOC ζ3d. Note that the final state is similar to Ti4+. Nonaccessible terms (e.g., final states with J=0) are omitted for clarity.The calculated 1s2p RIXS of an isolated Ni2+ ion (O3) is compared to the 2p XAS in Figure 17. The transitions are labelled with the dominant contribution of the corresponding final state terms. The four peaks in the 2p XAS (three in the L3 and one in the L2) correspond to the J=3 and J=4 IRREPs. It is noteworthy here that the 3F4 final state term is special in the sense, that it can be accessed only in 2p XAS, but its population via 1s2p RIXS is forbidden. Hence, in 2p XAS, the peak at ET≈−4eV is related to the J=4 IRREP (dominated by 3F4), while in 1s2p RIXS the peak at ET≈−4eV in the 2p XES decay relates to transitions into the final state IRREP J=2 (mostly 1D2). In other words, though it seems that the same peak at ET≈−4eV appears in the 2p XAS as well as in the 2p XES decay in 1s2p RIXS, it is in each case related to a different final state IRREP.Figure 17
Calculated 1s2p RIXS for an isolated Ni2+ ion (3d8) representing the atomic case. The projection of the total 1s2p RIXS map onto the energy transfer axis is shown together with the 2p XAS (L2,3-edge) in dashed red and the corresponding stick spectrum.Closer inspection of the sticks reveals that the3F3 term does not appear as a dominant contribution, but instead, the 1F3 term appears twice. This is due to the fact that the atomic terms lose their meaning when spin-orbit interaction is included. Then J=3 becomes the identifying IRREP which is formally a linear combination of 1F3, 3D3, and 3F3. Here, it happens to be the case that the 1F3 contribution is dominant in two J=3 IRREPs.As expected, theJ=0 IRREP (dominated by the 3P0 term), as well as the J=1 IRREP (3P1 and 3D1 terms), has no contribution to the 2p XAS spectrum. Furthermore, also, transitions into the J=2 IRREPs (P2, F2, and F2 terms) are not appearing in 2p XAS.In the 2p XES decay, on the other hand, theJ=2 IRREPs have a much stronger contribution to the spectrum. Those peaks, related to the final state IRREPs J=2, show in this example the largest difference between the 2p XAS and 2p XES as can be seen in Figure 17.


### 4.3.1. Considering an Oh Crystal Field

The crystal field splitting in Oh symmetry of the atomic terms of the initial state configuration of Ni2+ (1s22p63d8) is well-known and described with the Tanabe-Sugano diagram. The ground state atomic term symbol 3F splits into 3A2g, 3T2g, and 3T1g and gives 3A2g as the octahedral crystal field ground state. The inclusion of the 3d spin-orbit coupling (S=1→⊗T1g) translates the ground state term as 3A2g⊗T1g=T2g. Thus, the crystal field ground state of 3d8 Ni2+ including 3d spin-orbit interaction is 3A2gT2g (Oh).The crystal field splitting of the 1s2p RIXS intermediate state of Ni2+ is analogue to Ti4+ as the 1,3D terms split into 1,3Eg⊕1,3T2g. From the ground state symmetry 3A2g (neglecting 3d SOC), the 1s XAS IRREPs T2g and Eg enable to reach only 3T1g and 3Eg intermediate state symmetries:
(22)3A2g→⊗T2g3T1g,(23)3A2g→⊗Eg1sXAS3Eg.In other words, the intermediate state term3T2g cannot be reached by the electric quadrupole 1s XAS absorption. Hence, only the 1s XAS IRREP Eg leads to an existing intermediate state symmetry: 3Eg. This is illustrated in Figure 18. As a result, the 1s XAS projection of the total 1s2p RIXS map will consist only of transitions into the 3Eg intermediate state symmetry, because the quadrupole 1s XAS transition into the 3T2g intermediate state term is forbidden. We note that the spin singlet and triplet intermediate states are only separated by a few meV by the 3d spin-orbit, Coulomb, and exchange interactions (e.g., ζ3d≈0.1eV). Therefore, the 1s XAS transitions appear as a single peak due to the resolution chosen in our calculations.Figure 18
1s XAS projections of the calculated 1s2p RIXS map for Ni2+ (3d8) showing only one peak as the T2g term is not reachable. The red lines indicate the energy levels of the intermediate state.The single peak observed in the 1s XAS step for Ni2+ (3d8) is the first important difference with Ti4+ (3d0), which arises from the different nature of the initial state term. This can be further understood in a single electron picture: since the two holes of the initial state of Ni2+ are in the eg orbitals, the excited 1s electron can only go to the empty eg orbitals (3Eg intermediate state). On the contrary, in the case of Ti4+ (Section 4.2), both eg and t2g orbitals are empty leaving both Eg and T2g intermediate state symmetries accessible.When the crystal field (10Dq) is large with respect to the other electronic interactions, such as the 3d spin-orbit interaction ζ3d, the 1s XAS step of the 1s2p RIXS can be described well with the noncoupled IRREPs (ℒ and S are good quantum numbers). However, when the crystal field is weak or when the electronic interactions are stronger, such as the ζ2d spin-orbit, it is necessary to use the ℒ.S-coupled terms. The spin-orbit coupling translates the intermediate state terms into the same total angular symmetry terms as for Ti4+ (see Figure 12). From this, it is evident that the intermediate state offers four different symmetries A2g, Eg, T1g, and T2g from which the 2p XES decays can occur.The branching in the octahedral symmetry of the final state of Ni2+ is again analogue as Ti4+ (see Section 4.2). Those terms translate in an Oh crystal field to the ungerade symmetries: A1u, A2u, Eu, T1u, and T2u. The dipole 2p XES decays in 1s2p RIXS from the intermediate state terms give
(24)Eg→⊗T1uT1u⊕T2u,(25)T1g→⊗T1uA1u⊕Eu⊕T1u⊕T2u,(26)T2g→⊗T1u2pXESA2u⊕Eu⊕T1u⊕T2u⏟AccessibleFSterms.From that, we find that in 2p XES all final state symmetries can be reached. However, for 2p XAS, the dipole transition (T1u) starts from the T2g ground state.
(27)T2g→⊗T1u2pXASA2u⊕Eu⊕T1u⊕T2u⏟AccessibleFSterms.From that, we find that the reachable 2p XAS final states can haveA2u, Eu, T1u, or T2u symmetry. In other words, all intermediate terms, except those with A1u, can be reached in 2p XAS. This is identical to the third case of the RIXS as described above in (26). However, the difference between the transition matrix elements of XAS and RIXS can result in different intensities.The comparison between the 2p XAS and 1s2p RIXS selection rules is summarised in the scheme in Figure19.Figure 19
Multielectronic term scheme forNi2+ 3d8 comparing the 1s2p RIXS and direct 2p XAS pathways. Populating the T2g intermediate state would require a two-electron transition in 3d8.Comparing this case with the case ofTi4+ in an Oh crystal field enables to highlight the crucial influence of the ground state symmetry.We conclude with the calculated 1s2p RIXS maps and a comparison with the corresponding 2p XAS as shown in Figure20.Figure 20
Calculated 1s2p RIXS for the ion Ni2+ (3d8) for 10Dq = 0.9 eV. As can be seen in the map reflecting the quadrupole IRREP T2g (top row, left), the contribution to the spectrum is nonzero because of the small 3d spin-orbit interaction leading to a weak mixing of the states. However, its contribution is about 186 times weaker and thus neglectable in first approximation.The calculations reveal two aspects: first, we notice the direct 2p XAS and the 2p XES final state spectra appear to have a similar appearance. This can be explained with the term scheme in Figure19 illustrating that the 2p XES and the direct 2p XAS probe similar final state terms. Second, we note that the intensity of the contribution to the 1s2p RIXS via the quadrupole T2g IRREP is two orders of magnitude weaker than the contribution via Eg 1s XAS IRREP. According to the strict ℒ=±2 selection rule (e.g., when neglecting 3d spin-orbit coupling), no transitions arise from the T2g absorption. However, the 3d spin-orbit interaction ζ3d is in fact nonzero, though small. Hence, when looking at the total angular momentum IRREPs of the intermediate states (e.g., the terms including SOC) and the 1s XAS selection rules shown in (22), it appears that some intermediate states can be probed by the T2g operator. In other words, the small but nonzero 3d spin-orbit coupling ζ3d induces some mixing such that the contributions to the spectrum via the 1s XAS IRREP T2g will be nonzero but weak. A direct comparison of the two 1s XAS transitions with T2g and Eg symmetries shows that the T2g spectrum multiplied by 186 is almost identical to the Eg spectrum (Figure 20).


### 4.3.2. 1s2p RIXS Experiments of 3d8 Systems

Experimental 1s2p RIXS spectra of 3d8 NiF2 and molecular Ni2+ complexes have been published by Glatzel et al. [14, 40]. Within the resolution of the measurement, the NiF2 spectrum is exactly reproduced by the crystal field calculation [14].


## 4.3.1. Considering an Oh Crystal Field

The crystal field splitting in Oh symmetry of the atomic terms of the initial state configuration of Ni2+ (1s22p63d8) is well-known and described with the Tanabe-Sugano diagram. The ground state atomic term symbol 3F splits into 3A2g, 3T2g, and 3T1g and gives 3A2g as the octahedral crystal field ground state. The inclusion of the 3d spin-orbit coupling (S=1→⊗T1g) translates the ground state term as 3A2g⊗T1g=T2g. Thus, the crystal field ground state of 3d8 Ni2+ including 3d spin-orbit interaction is 3A2gT2g (Oh).The crystal field splitting of the 1s2p RIXS intermediate state of Ni2+ is analogue to Ti4+ as the 1,3D terms split into 1,3Eg⊕1,3T2g. From the ground state symmetry 3A2g (neglecting 3d SOC), the 1s XAS IRREPs T2g and Eg enable to reach only 3T1g and 3Eg intermediate state symmetries:
(22)3A2g→⊗T2g3T1g,(23)3A2g→⊗Eg1sXAS3Eg.In other words, the intermediate state term3T2g cannot be reached by the electric quadrupole 1s XAS absorption. Hence, only the 1s XAS IRREP Eg leads to an existing intermediate state symmetry: 3Eg. This is illustrated in Figure 18. As a result, the 1s XAS projection of the total 1s2p RIXS map will consist only of transitions into the 3Eg intermediate state symmetry, because the quadrupole 1s XAS transition into the 3T2g intermediate state term is forbidden. We note that the spin singlet and triplet intermediate states are only separated by a few meV by the 3d spin-orbit, Coulomb, and exchange interactions (e.g., ζ3d≈0.1eV). Therefore, the 1s XAS transitions appear as a single peak due to the resolution chosen in our calculations.Figure 18
1s XAS projections of the calculated 1s2p RIXS map for Ni2+ (3d8) showing only one peak as the T2g term is not reachable. The red lines indicate the energy levels of the intermediate state.The single peak observed in the 1s XAS step for Ni2+ (3d8) is the first important difference with Ti4+ (3d0), which arises from the different nature of the initial state term. This can be further understood in a single electron picture: since the two holes of the initial state of Ni2+ are in the eg orbitals, the excited 1s electron can only go to the empty eg orbitals (3Eg intermediate state). On the contrary, in the case of Ti4+ (Section 4.2), both eg and t2g orbitals are empty leaving both Eg and T2g intermediate state symmetries accessible.When the crystal field (10Dq) is large with respect to the other electronic interactions, such as the 3d spin-orbit interaction ζ3d, the 1s XAS step of the 1s2p RIXS can be described well with the noncoupled IRREPs (ℒ and S are good quantum numbers). However, when the crystal field is weak or when the electronic interactions are stronger, such as the ζ2d spin-orbit, it is necessary to use the ℒ.S-coupled terms. The spin-orbit coupling translates the intermediate state terms into the same total angular symmetry terms as for Ti4+ (see Figure 12). From this, it is evident that the intermediate state offers four different symmetries A2g, Eg, T1g, and T2g from which the 2p XES decays can occur.The branching in the octahedral symmetry of the final state of Ni2+ is again analogue as Ti4+ (see Section 4.2). Those terms translate in an Oh crystal field to the ungerade symmetries: A1u, A2u, Eu, T1u, and T2u. The dipole 2p XES decays in 1s2p RIXS from the intermediate state terms give
(24)Eg→⊗T1uT1u⊕T2u,(25)T1g→⊗T1uA1u⊕Eu⊕T1u⊕T2u,(26)T2g→⊗T1u2pXESA2u⊕Eu⊕T1u⊕T2u⏟AccessibleFSterms.From that, we find that in 2p XES all final state symmetries can be reached. However, for 2p XAS, the dipole transition (T1u) starts from the T2g ground state.
(27)T2g→⊗T1u2pXASA2u⊕Eu⊕T1u⊕T2u⏟AccessibleFSterms.From that, we find that the reachable 2p XAS final states can haveA2u, Eu, T1u, or T2u symmetry. In other words, all intermediate terms, except those with A1u, can be reached in 2p XAS. This is identical to the third case of the RIXS as described above in (26). However, the difference between the transition matrix elements of XAS and RIXS can result in different intensities.The comparison between the 2p XAS and 1s2p RIXS selection rules is summarised in the scheme in Figure19.Figure 19
Multielectronic term scheme forNi2+ 3d8 comparing the 1s2p RIXS and direct 2p XAS pathways. Populating the T2g intermediate state would require a two-electron transition in 3d8.Comparing this case with the case ofTi4+ in an Oh crystal field enables to highlight the crucial influence of the ground state symmetry.We conclude with the calculated 1s2p RIXS maps and a comparison with the corresponding 2p XAS as shown in Figure20.Figure 20
Calculated 1s2p RIXS for the ion Ni2+ (3d8) for 10Dq = 0.9 eV. As can be seen in the map reflecting the quadrupole IRREP T2g (top row, left), the contribution to the spectrum is nonzero because of the small 3d spin-orbit interaction leading to a weak mixing of the states. However, its contribution is about 186 times weaker and thus neglectable in first approximation.The calculations reveal two aspects: first, we notice the direct 2p XAS and the 2p XES final state spectra appear to have a similar appearance. This can be explained with the term scheme in Figure19 illustrating that the 2p XES and the direct 2p XAS probe similar final state terms. Second, we note that the intensity of the contribution to the 1s2p RIXS via the quadrupole T2g IRREP is two orders of magnitude weaker than the contribution via Eg 1s XAS IRREP. According to the strict ℒ=±2 selection rule (e.g., when neglecting 3d spin-orbit coupling), no transitions arise from the T2g absorption. However, the 3d spin-orbit interaction ζ3d is in fact nonzero, though small. Hence, when looking at the total angular momentum IRREPs of the intermediate states (e.g., the terms including SOC) and the 1s XAS selection rules shown in (22), it appears that some intermediate states can be probed by the T2g operator. In other words, the small but nonzero 3d spin-orbit coupling ζ3d induces some mixing such that the contributions to the spectrum via the 1s XAS IRREP T2g will be nonzero but weak. A direct comparison of the two 1s XAS transitions with T2g and Eg symmetries shows that the T2g spectrum multiplied by 186 is almost identical to the Eg spectrum (Figure 20).


## 4.3.2. 1s2p RIXS Experiments of 3d8 Systems

Experimental 1s2p RIXS spectra of 3d8 NiF2 and molecular Ni2+ complexes have been published by Glatzel et al. [14, 40]. Within the resolution of the measurement, the NiF2 spectrum is exactly reproduced by the crystal field calculation [14].


## 5. 1s2p RIXS for Other 3dN Configurations

In this section, the remaining 3dN configurations (N∈1,…,7) are discussed in a more condensed manner. Furthermore, for the systems with a 3d4, 3d5, 3d6, and 3d7 ground state, a high spin (HS) and a low spin (LS) case is presented. This is due to the fact that the crystal field splitting energy (10Dq) and the pairing energy in those cases are competing. In other words, for sufficiently large values of 10Dq, the ground state changes from high spin to low spin. Subsequently, this affects the possible transitions within those systems and hence the resulting spectra will have a different appearance [13].For each ion in high spin, we are choosing a 10Dq value matching roughly the empirically found “0.5−0.6eV per valency” approximation for transition metal oxides. To illustrate the differences, we additionally select for 3d4 to 3d7 a sufficiently large value of 10Dq to obtain the corresponding spectra for a low spin configuration. The used crystal field values (10Dq) are summarised in Table 7.As we have seen already throughout the didactic cases in Section4, the 2p XES decays are always described with the dipole IRREP T1u. As this will be used in the following sections in a rather repetitive manner, in (28), (29), (30), (31), (32), (33), (34), and (35), we are summarising all the reachable final state symmetries for any given intermediate state symmetry (see direct product Table 4).
(28)A1g⊗T1u=T1u,(29)A2g⊗T1u=T2u,(30)Eg⊗T1u=T1u⊕T2u,(31)T1g⊗T1u=A1u⊕Eu⊕T1u⊕T2u,(32)T2g⊗T1u=A2u⊕Eu⊕T1u⊕T2u,(33)E1/2g⊗T1u=E1/2u⊕F3/2u,(34)E5/2g⊗T1u=E5/2u⊕F3/2u,(35)F3/2g⊗T1u=E1/2u⊕2F3/2u⊕E5/2u.As can be seen above, the first five equations, (28), (29), (30), (31), and (32), relate to intermediate states with an even number of electrons in open shells (e.g., integer S value). The bottom three equations (33), (34), and (35) on the other hand relate to intermediate state symmetries with an electron configuration having an odd number of electrons in open shells (e.g. half-integer S value). In the following, we prefer to refer to these equations instead of repeating them every time in each of the following cases.


### 5.1. 3d1 Ground State System, for Example, Ti3+

In the previous cases, the splitting of the ground state electronic terms was leading to a single term, for example,2EgF3/2g (Oh) for Cu2+ (3d9) or 3A2gT2g (Oh) for Ni2+ (3d8). In the following, we discuss the 3d1 ground state configuration representing for instance the case of a Ti3+ ion.
(36)As shown below, the3d1 case is an interesting transition towards the other transition metal ions, as it enables to explore two aspects in the RIXS: (i) the influence of the spin-orbit splitting of the ground state leading to the contribution of several terms for T > 0 K and (ii) the effect of the multielectronic interactions of the 3d shell in the intermediate and final states of 1s2p RIXS. The atomic case will not be detailed, and we will only discuss the influence of the Oh crystal field.The initial state electronic configuration 1s22p63d1 only contains a single electron in the 3d shell. It is noteworthy that this electronic configuration is analogue to the initial state of Cu2+: the atomic ground state term is 2D, which splits into D5/2 and D3/2 by the 3d spin-orbit coupling (see Table 1). In the Oh crystal field, the 2D term splits into 2T2gt2g1eg0 and 2Egt2g0eg1.The difference withCu2+ (3d9) is that the ground state term of Ti3+ (3d1) is 2T2g (6 microstates). When considering the 3d spin-orbit coupling ζ3d, this term splits into the two total symmetry irreducible representations (IRREPs) F3/2g (mixing of D5/2 and D3/2) and E5/2g (from D5/2 term), where F3/2g is the lowest. This is summarised in Figure 3, which shows the splittings of the ground state Russel-Saunders term 2D with SOC and an Oh crystal field. It should be compared with Cu2+ (Figure 2). Furthermore, we note that due to the small 3d SOC in the initial state (ζ3d=19meV), the splitting of the ground state 2T2g is only 28 meV. Thus, for temperatures T > 0 K, the second IRREP should also be included according to the Boltzmann population. For instance, at T=300 K, Boltzmann population of the F3/2g (4-fold degenerate) is 86% and 14% for the E5/2g (2-fold degenerate). The contribution of the second term E5/2g is thus small but not entirely negligible.The intermediate state electron configuration1s13d2 (2⋅45=90 microstates) is equivalent to the initial state of Ni2+ coupled to the 1s core hole [13]. The atomic terms are 2,4F, 2,4P, 2G, and 2S (Table 1). Because the Coulomb exchange Gsd is small, the splitting between the doublet and the quartet spin states is small and beyond the accessible energy resolution.Using Table2, one finds the branchings of the atomic states in an Oh crystal field. The resulting Oh term symbols for a 1s13d2 configuration are 2,4A2g, 2,4T2g, 2,4T1g, 2A1g, and 2Eg. The energy splitting of these Oh crystal field terms is qualitatively similar to the Tanabe-Sugano diagram of 3d2. In the particular case of Ti3+, the result of the calculation is given in Figure 21 where the 1s XAS projections are shown together with the crystal field terms assigned.Figure 21
Influence of the Oh crystal field on the 1s XAS projection of the calculated 1s2p RIXS spectra for Ti3+ (3d1) with ELD overlay. A comparison to the calculated 1s XAS projections now shows on the one hand that neglecting 3d SOC in this step is a valid approximation. And on the other hand, it confirms that a quadrupole transition from the 2T2g ground state symmetry can reach A1g, Eg, T1g, and T2g symmetries, but the A2g symmetry does apparently not contribute to the spectrum.Because the 3d spin-orbit coupling is weak, one can neglect it in first approximation; the spin conservation leads to the 1s XAS being dominated by transitions into the spin doublet states, while the matrix elements of the transition to the spin quartet terms are much smaller. In the following, we will therefore focus on the doublet spin states.(37)2T2g→⊗T2g2A1g⊕2Eg⊕2T1g⊕2T2g,(38)2T2g⏟GS→⊗Eg1sXAS2T1g⊕2T2g⏟ReachableISterms.This shows, in an Oh crystal field, starting from the 2T2g ground state, the 1s XAS operators T2g and Eg enable to reach all even terms except the 2A2g IRREP. This results in numerous multiplet states reached in the intermediate state of 1s2p RIXS. The 1s XAS projections in Figure 21 show the splitting of the peaks due to the scaling of the crystal field parameter 10Dq. We note that the 2A2g (2F) does not get any intensity in the absorption step of 1s2p RIXS in agreement with the 1s XAS selection rules as found in the expressions given in (37) and (38).The previous didactic examples (Section4) demonstrated the necessity to consider the ℒ.S-coupled terms to describe the 2p XES transitions to the final state in 1s2p RIXS and the 2p XAS. In the intermediate state, the Oh total symmetry IRREPs of the doublet spin states are obtained by coupling the reachable orbital IRREPs with the spin IRREP (S=1/2→⊗E1/2g):
(39)A1g⊗E1/2g=E1/2g,Eg⊗E1/2g=F3/2g,T1g⊗E1/2g=E1/2g⊕F3/2g,T2g⊗E1/2g=E5/2g⊕F3/2g.The result shows that all three odd total symmetric IRREPSE1/2g, F3/2g, and E5/2g are thus reachable in the intermediate state. In other words, all three odd IRREPs are contributing to the 2p XES decays.The final state configuration1s22p53d2 contains 6⋅45=270 microstates due to the hole in the 2p shell. The atomic term symbols span from S to H with spin doublets and quartets and J values ranging from 1/2 to 11/2 [13]. In an Oh crystal field, these terms branch into the three IRREPs E1/2u, F3/2u, and E5/2u (Table 2).The 2p XES dipole transition IRREPT1u gives the transitions from the three intermediate state symmetries into all three final state symmetries as given in (33), (34), and (35).For the direct 2p XAS, as mentioned above, the two ground state termsF3/2g and E5/2g are to be considered due to the small splitting. This leads with the dipole transition IRREP T1u also to all three final state symmetries:
(40)E5/2g→⊗T1uE5/2u⊕F3/2uF3/2g→⊗T1u2pXASE1/2u⊕2F3/2u⊕E5/2u.It becomes obvious that the selection rules enable to reach all three final state IRREPs in both cases: 2p XAS and 1s2p RIXS. This is summarised in the combined term scheme shown in Figure22.Figure 22
Multielectronic term scheme forTi3+ (3d1) comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted as its meaning is lost in that case.Although it seems that all the final state symmetries can be reached in both cases, 2p XAS and the 2p XES in 1s2p RIXS, the calculated spectra shown in Figure23 demonstrate that the RIXS intermediate state enables to select some specific final states.Figure 23
Calculated 1s2p RIXS spectrum forTi3+ (3d1) for 10Dq=1.5eV shown together with some selected CIE slices. The stick spectrum at the bottom illustrates the distribution of the three final state terms E1/2u, F3/2u, and E5/2u with respect to the emissions.According to the combined term scheme in Figure22, the intermediate states T1g, T2g, and Eg enable to reach all final state symmetries. The calculated 1s2p RIXS together with some CIE slices is shown in Figure 23. However, due to the first term 2T1g corresponding to the excitation of the 1s electron into the t2g orbitals, the final states probed resonantly with RIXS are mainly the lower energy part of the states, corresponding to the “t2g” part of the spectra. The intensity arises mainly from the T2g absorption operator. The 2T2g intermediate state selects higher energy final states equally probed by the T2g and Eg 1s XAS operators.The second term2T1g (from 2P atomic term symbol) corresponds to the absorption to the eg orbitals and is consequently probed by the Eg absorption operator and corresponds to the “eg” part of the final states. The intermediate state 2A1g (from 2S atomic term symbol) only gives a very weak absorption via the T2g quadrupole operator in agreement with the selection rules as given in (37).This intermediate state only decays toE1/2u (see, e.g., the final state peak at ET≈8.3eV in Figure 23) and F3/2g (at ET≈2−3eV). Such intermediate state selection is identical for the L3- and L2-edges. This is similar to the results obtained for Ti4+ where the “t2g” and “eg” parts of the finals states are selected with the intermediate state of RIXS.


#### 5.1.1. 1s2p RIXS Experiments of 3d1 Systems

Experimental 1s2p RIXS spectra of3d1V4+ ions in TiO2 have been measured by Bordage [41]. However, the experimental resolution does not allow to compare the data in detail with the calculations as shown in Figure 23.


### 5.2. 3d2 Ground State System, for Example, Cr4+

In this part, we present the elementary differences between 2p XAS and 1s2p RIXS for the case of tetravalent chromium (Cr4+). First, we gather the relevant information analogue to the previous cases which are then summarised in a combining term scheme illustrating the transition paths in 1s2p RIXS (1s XAS, 2p XES) and the direct 2p XAS. The electronic configurations involved here are
(41)In the following, we will discuss this case with respect to the influence of the SOC in the intermediate state leading to a significant increase of the possible 1s2p RIXS final states. The atomic case will not be detailed, and we will only discuss the influence of the Oh crystal field for 10Dq=2.1eV.The initial state electron configuration1s22p63d2 corresponds to the atomic ground state term 3F (O3) with 45 microstates. It translates into an Oh crystal field (without 3d SOC) into a 3T1g ground state symmetry with 9 microstates. By including 3d SOC ζ3d, we find the further splittings of the 3T1g ground state term (S=1→⊗T1g):
(42)3F4O3→CF3T1gOh→SOCA1g⊕Eg⊕T1g⊕T2gOh.From these terms, Eg lies lowest in energy; hence, the ground state is 3T1gEg (Oh).The intermediate state configuration1s13d3 with 2⋅120=240 microstates translates into the atomic term symbols 3,5F, 3,5P, 3H, 3G, 3D, and 3P (Table 1). Using Table 2 one finds the branchings of the atomic states in an Oh crystal field. The resulting Oh term symbols are 3,5A2g, 3,5T2g, 3,5T1g, 3A1g, 3A2g, 3Eg, 3T1g, and 3T2g. However, the Coulomb exchange Gsd is small; hence, the splitting between the triplet and the quintet spin states is small and beyond the accessible energy resolution. The energy splittings of those terms are qualitatively similar to the Tanabe-Sugano diagram of a 3d3 electron configuration.Because the 3d SOC is weak, the spin is conserved leading to the 1s XAS transitions being dominated by transitions into the spin triplet states, while the matrix elements of the transition to the spin quintet terms are weak. Thus, we will focus on the spin triplet symmetries.(43)3T1g→⊗T2gA32g⊕3Eg⊕3T1g⊕3T2g,(44)3T1g→⊗Eg1sXAS3T1g⊕3T2g⏟AccessibleISterms.As can be seen from the results found in (43) and (44), starting from a 3T1g ground state, all but the 3A1g intermediate state symmetries can be reached via the quadrupole 1s XAS. The 1s XAS projections of the 1s2p RIXS for 10Dq ranging from 0.1 eV to 3.0 eV are shown in Figure 24.Figure 24
Effect of the Oh crystal field on the 1s XAS projections of the calculated 1s2p RIXS spectra for Cr4+ (3d2). The corresponding ELD overlay illustrates the distribution of the reachable intermediate state terms contributing in 1s XAS.For the 2p XES decays in 1s2p RIXS, however, the inclusion of SOC is again required. Using theT1g IRREP (S=1→⊗T1g) and the direct product table (Table 3), one finds the intermediate state symmetries including 3d SOC ζ3d as also shown in the combined term scheme in Figure 25.
(45)3A2g⊗T1g=T2g,3Eg⊗T1g=T1g⊕T2g,3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g,3T2g⊗T1g=A2g⊕Eg⊕T1g⊕T2g.Figure 25
Multielectronic term scheme forCr4+ (3d2) comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted as its meaning is lost in that case.The subsequent 2p XES starts from any of the intermediate state symmetries found in expression (45). The possible 2p XES final state symmetries are all even symmetries as given in expressions (28), (29), (30), (31), and (32).The final state (1s22p53d3) contains 6⋅120=720 microstates due to the hole in the 2p shell spanning J from 0 to 7. In an Oh crystal field, these terms branch into the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. In other words, all existing FS symmetries can be reached in the 2p XES.Also for 2p XAS, spin-orbit interaction is crucial. Thus, starting from the3T1gEg (Oh) ground state and using the T1u IRREP, we find that only T1u and T2u IRREPs are reached, which makes a drastic difference between 2p XAS and 1s2p RIXS. The selection rules for a 3d2 ground state system are summarised in Figure 25 showing all reachable symmetries.According to the electric quadrupole selection rules derived in (43), the A2g and Eg intermediate state symmetries are only reachable via the quadrupole T2g IRREP. This is confirmed in Figure 26 (RIXS maps) where the low energy peak at Ein=−2.25eV appears only in the RIXS map corresponding to the T2g IRREP. The t2g and eg components appear well separated with (i) the t2g components probed by the T2g absorption giving two intense and narrow peaks at ET≈−6.5eV and (ii) the eg components probed by the Eg absorption leading to a larger group of final states spread in energy between ET=−5eV and 0 eV. Furthermore, it can be noted that the X-ray emission spectra (2p XES) are here overall very rich. The corresponding sticks in Figure 26 illustrate how dense the reachable final states are distributed. We observe that the 2p XAS and 1s2p RIXS differs mainly in the low energy final states: the 1s2p RIXS enhances the final states below ET≈−2.5eV to which the 2p XAS transition matrix elements are weak. We note that the 1s2p RIXS spectra which are the closest to the 2p XAS spectrum are those obtained at the higher energy transfers at ET>0eV.Figure 26
Calculated 1s2p RIXS spectrum forCr4+ (3d2) for 10Dq=2.1eV. Note that the A2g intermediate state symmetry is in agreement with the selection rules only populated via the T2g 1s XAS IRREP. The slices have been scaled to a comparable intensity to enhance their character better. The bottom plot compares the integrated incident energy (IIE) 2p XES with the 2p XAS (L2,3-edge).Considering now the quantitative influence of the natural broadening as the principal limit of any experimental spectrum, it becomes clear that an experimental separation of the final states is impossible. However, high-resolution RIXS measurements enable to extract CIE slices similar to the illustrated examples shown in Figure26.And finally, in Figure27, we show the 2p XES projections of the calculated 1s2p RIXS to illustrate its evolution when the crystal field value 10Dq is scaled from 10Dq=0.1eV,…,3eV.Figure 27
Effect of the Oh crystal field on the 2p XES projections of the calculated 1s2p RIXS spectra for Cr4+ (3d2).


#### 5.2.1. 1s2p RIXS Experiments of 3d2 Systems

Experimental 1s2p RIXS spectra of 3d2 CrO2 have been published by Zimmermann et al. [9]. Analysis of the magnetic circular dichroism (MCD) of the 1s2p RIXS revealed that the pre-edge structure that is visible in the 1s2p RIXS plane is dominated by a nonlocal contribution. The MCD data reveals the real quadrupole peak, but due to its low intensity and limited resolution, it does not allow to compare the quadrupole peak to the spectra calculated in Figure 26.


### 5.3. 3d3 Ground State System, for Example, Cr3+

The electronic configurations involved for a3d3 ion are summarised below:
(46)For3d3 systems such as Cr3+, the initial state configuration 1s22p63d3 (120 microstates) translates in Oh to a 4A2g ground state (4 microstates). With 3d SOC (S=3/2→⊗F3/2g), the ground state of a 3d2 ion is thus 4A2gF3/2g (Oh) (Table 5).The intermediate state terms (420 microstates) contain spin sextets, quartets, and doublets and orbital momentum ranging from S to I. They split in the Oh crystal field as the Tanabe-Sugano diagram of 3d4 systems. Those terms, neglecting 3d spin-orbit interaction, branch into multiple Oh terms:
(47)6,4,2T2g,6,4,2Eg,4,2T1g,4,2A2g,and4,2A1gFor 1s XAS, the weak spin-orbit coupling in the 3d shell results in spin-conserving transitions from the ground state4A2g to the spin-quartet intermediate states with quadrupole IRREPs 4T1g and 4Eg:
(48)4A2g→⊗T2g4T1g,4A2g→⊗Eg1sXAS4Eg.This illustrates, though many symmetries exist in the intermediate state, only the transitions into theT1g and Eg symmetries contribute to the spectrum as can be seen in Figure 28.Figure 28
1s XAS projections of a calculated 1s2p RIXS spectrum for Cr3+ (3d3) with ELD.Again, as detailed throughout the examples above for the 2p XES decays, consideration of 3d SOC is again required. Hence, we give the splitting of the two reachable intermediate state terms4T1g and 4Eg due to SOC (S=3/2→⊗F3/2):
(49)4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g,4Eg⊗F3/2g=E1/2g⊕E5/2g⊕F3/2g.Here, it is noteworthy that both IS terms (4T1g and 4Eg) are branching into the same three symmetries: E1/2, E5/2, and F3/2.The final state (1s22p53d4) contains 6⋅210=1260 microstates due to the hole in the 2p shell. The atomic term symbols, obtained by coupling the 3d4 atomic terms with 2P, span from S to K with spin doublets, quartets, and sextets and J values from 1/2 to 15/2. A detailed list can be found in the book Core Level Spectroscopy of Solids [13] (p. 103). The important information is that these J values branch in Oh into the three odd IRREPs E1/2u, F3/2u, andE5/2u.As derived in (49), the 2p XES decays in 1s2p RIXS start from either of the three intermediate state symmetries E1/2g, E5/2g, and F3/2g and can reach E1/2u, E5/2u, and F3/2u final state IRREPs. The transitions are again given in (33), (34), and (35).The direct 2p XAS final state terms are similar though it starts from theF3/2g ground state symmetry.
(50)F3/2g→⊗T1u2pXASE1/2u⊕E5/2u⊕2F3/2u.Altogether, the reachable final state terms are the same in 1s2p RIXS and the direct 2p XAS, the selection rules, and the corresponding transitions yield different results.The selection rules are summarised in the term scheme shown in Figure29.Figure 29
Multielectronic term scheme forCr3+ (3d3) comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted as its meaning is lost in that case.The calculated 1s2p RIXS maps and some CIE slices together with the corresponding 1s XAS forCr3+ (3d3) for a crystal field of 10Dq=1.5eV are shown in Figure 30. As can be seen from the corresponding RIXS maps, from the A2g ground state symmetry, the quadrupole T2g IRREP populates only states with T1g symmetry, while the quadrupole Eg IRREP populates only intermediate states with Eg symmetry. The 4T1g (4H) intermediate state symmetry corresponds to the electronic configuration with four 3d electrons in the t2g orbitals, and as a result, the corresponding final states (between ET≈−5eV and −2.5eV) are reached in the RIXS. These states correspond to spin-quartet final states and are weak in 2p XAS. The first 4Eg IRREP intermediate state (Ein≈−0.6eV) leads to the final states between ET≈−5eV and −2.5eV, which correspond to the eg part of the final states with (t2g)3(eg)1. The other 4Eg intermediate state symmetries at Ein≈2.35eV and 3.5eV lead to higher energy final states, where it is the RIXS spectrum of 4Eg (ET≈3.5eV) that resembles more to 2p XAS. This is consistent with the fact that the atomic term of this intermediate state is similar to the atomic term of the ground state (4F) leading to similar 1s2p RIXS and 2p XAS spectra.Figure 30
Calculated 1s2p RIXS spectrum for3d3Cr3+ for 10Dq=1.5eV.


#### 5.3.1. 1s2p RIXS Experiments of 3d3 Systems

Examples for 3d3 systems are Cr3+ and Mn4+. Experimental 1s2p RIXS spectra of a number of Cr3+ 3d3 systems have been published by Frommer et al. [42]. Thomas et al. (unpublished) compare Cr(III)(acac) and Cr2O3 with 1s2p RIXS crystal field multiplet calculations, equivalent to Figure 30. Cr(III)acac is well reproduced by the crystal field multiplet calculations. In the case of Cr2O3, additional intensity due to the nonlocal peak overlaps with the second pre-edge peaks in the 1s XAS. Experimental 1s2p RIXS spectra of MnO2 have been published by Glatzel et al. [43]. Mn4+ is a covalent system dominated by charge transfer. As such, the 1s2p RIXS is dominated by fluorescence and the 1s2p RIXS resonances calculated here are not visible as separate features.


### 5.4. 3d4 Ground State System, for Example, Mn3+

The electronic configurations involved in this example are summarised below:(51)This is the first case in which the crystal field splitting energy (10Dq) and the pairing energy are competing. Thus, as shortly discussed in the introduction notes for Section 5, the ground state can have a high spin or low spin configuration.


#### 5.4.1. 3d4 System in High Spin (Quintet Ground State)

The initial state electronic configuration is1s23d4 (210 microstates) having in high spin the atomic ground state term 5D. In the Oh crystal field, the 5D ground state splits into the 5Egt2g2eg2 and 5T2gt2g3eg1 terms, where 5Eg is the lowest in energy. The inclusion of spin-orbit interaction S=2→⊗Eg⊕T2g yields the following splittings in the initial state:
(52)5Eg⊗Eg⊕T2g=A1g⊕A2g⊕Eg⊕T1g⊕T2g.The ground state5Eg splits due to 3d SOC into several symmetries from which the term A1g is the lowest in energy; hence, the ground state is 5EgA1g (Oh).The intermediate state with 1s13d5 (2⋅252=504 microstates) corresponds to the atomic terms of 3d5 coupled with the 1s core hole. This gives spin multiplicities ranging from 1 to 7. Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d5. Because the 1s XAS is spin-conserving, only quintet intermediate states are reached in the absorption from the quintet high spin ground state (5Eg). The atomic quintet terms are 5S, 5D, 5F, 5G, and 5P which correspond to the crystal field terms 5A1g, 5A2g, 5Eg, 5T1g, and 5T2g.Starting from the5Eg ground state term, Table 3 gives the reachable intermediate state symmetries:
(53)5Eg→⊗T2g5T1g⊕5T2g,(54)5Eg⏟GS→⊗Eg1sXAS5A1g⊕5A2g⊕5Eg⏟AccessibleISinhighspin.The result is that all quintet intermediate state symmetries can be reached in 1s XAS. This is confirmed in Figure31 where in high spin only terms with a quintet component contribute to the spectrum. Furthermore, it can be seen that the high spin configuration is maintained up to a crystal field of 10Dq≈2.8eV.Figure 31
1s XAS projection of a calculated 1s2p RIXS map forMn3+ (3d4) with the corresponding ELD. The high spin to low spin transition at approximately 10Dq≈2.85eV can be recognised. In fact, there are two HS → LS transitions: in the ground and the intermediate state.The inclusion of the 3d spin-orbit interactionS=2→⊗E⊕T2 yields the corresponding splittings in the intermediate state:
(55)5A1g⊗Eg⊕T2g=Eg⊕T2g,5A2g⊗Eg⊕T2g=Eg⊕T1g,5Eg⊗Eg⊕T2g=A1g⊕A2g⊕Eg⊕T1g⊕T2g,5T1g⊗Eg⊕T2g=T1g⊕T2g⊕A2g⊕Eg⊕T1g⊕T2g,5T2g⏟ISterms⊗Eg⊕T2g=T1g⊕T2g⊕A1g⊕Eg⊕T1g⊕T2g⏟IStermsincluding3dSOC.Subsequently, the 2p XES decays can start from any of these terms.The final state (1s22p53d5) contains 6⋅252=1512 microstates due to the hole in the 2p shell spanning J from 0 to 8. In an Oh crystal field, these terms branch into the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. In other words, all even symmetries exist in the final state.The reachable final state terms in 2p XES are found again with the dipole IRREPT1u as summarised in (28), (29), (30), (31), and (32). In short, all existing final state symmetries are also reachable in the 2p XES decays in 1s2p RIXS as illustrated in Figure 32.Figure 32
Multielectronic term scheme forMn3+ (3d4) in high spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.The situation is very different for the direct 2p XAS. Starting from theA1g total symmetry IRREP term and using the T1u dipole IRREP, only final states with T1u symmetry are reachable.The calculated 1s2p RIXS, separated into the contributions via the quadrupole IRREPT2g and Eg, together with some slices is shown in Figure 33.Figure 33
Calculated 1s2p RIXS map forMn3+ (3d4) with a crystal field for 10Dq=1.5eV reflecting the high spin case. As can be seen by the two RIXS maps (top row, left and middle), the contributions are in agreement with the predicted symmetries found in (51) and (52).A comparison of the RIXS maps in Figure33 with the results found in expressions (53) and (54) confirms the 1s XAS selection rules. As predicted in expression (53), the quadrupole IRREP T2g populates the intermediate state symmetries 5T1g and 5T2g. While expression (54) showed for the quadrupole IRREP Eg, only the intermediate state symmetries 5A1g, 5A2g, and 5Eg can be reached.


#### 5.4.2. 3d4 in Low Spin (Triplet Ground State)

For large crystal field energies (10Dq), the initial state electronic configuration 1s23d4 has a low spin (LS) ground state. The atomic Russel-Saunders term 3D translates into a 3T1g ground state with 9 microstates. The inclusion of 3d SOC (S=1→⊗T1g) yields the initial state splittings:
(56)3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g.The ground state3T1g splits due to 3d SOC into several symmetries from which the term A1g is the lowest in energy; hence, the low spin ground state is 3T1gA1g (Oh).The intermediate state electron configuration1s13d5 is the same as described above in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the initial state (3T1g) being a spin-triplet, the spin-conserving 1s XAS only reaches spin-triplet intermediate state symmetries:
(57)3T1g→⊗T2g3A2g⊕3Eg⊕3T1g⊕3T2g,3T1g⏟GS→⊗Eg1sXAS3T1g⊕3T2g⏟AccessiblelowspinIS.Here, all intermediate state terms, exceptA1g, are reachable in 1s XAS. The exclusion of the orbital symmetry term A1g, due to the different ground state term, is a first notable difference with respect to the high spin case. The corresponding 1s XAS projections for the low spin case are shown in Figure 34.Figure 34
1s XAS projection of a calculated 1s2p RIXS map forMn3+ (3d4) in low spin.Applying the 3d SOC (S=1→⊗T1g) to the reachable intermediate state terms yields all even total angular momentum IRREPs.
(58)3A2g⊗T1g=T2g,3Eg⊗T1g=T1g⊕T2g,3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g,3T2g⊗T1g=A2g⊕Eg⊕T1g⊕T2g.The final states (1s22p53d5) are again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u.For the 2p XES decay, we find again from expressions (28), (29), (30), (31), and (32) that all even symmetries (A1u, A2u, Eu, T1u, and T2u) can be reached. In 2p XAS, starting from the A1g total symmetry term and using the T1u IRREP, we find that only T1u final state IRREPs can be reached. The selection rules are summarised in the combined term scheme in Figure 35.Figure 35
Multielectronic term scheme forMn3+ (3d4) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.An example calculation for the low spin case (10Dq=3.9eV) of Mn3+ (3d4) is shown in Figure 36.Figure 36
Calculation of 1s2p RIXS and 2p XAS forMn3+ (3d4) for 10Dq=3.9eV (low spin).This example is the first case here in which the crystal field energy (10Dq) and the pairing energy are competing to induce a low spin configuration for sufficiently large values of 10Dq. In the present case of Mn3+ (3d4), we see the high spin to low spin transition at around 10Dq≈2.85eV as is observable in the 1s XAS and 2p XES projections (Figures 31 and 37, resp.).Figure 37
2p XES projections of a calculated 1s2p RIXS spectrum forMn3+ (3d4). Note the HS-LS transition at approximately 10Dq≈2.85eV. In fact, there are two HS → LS transitions, in the GS and in the IS, which are not necessarily occurring at the same 10Dq value.One difference between the low spin and high spinMn3+ is the spin state of the ground state. The result is that in the HS, the 1s XAS absorption reaches the spin-quintet IS while in the LS case, it reaches the spin-triplet IS. This results in different final states being probed regardless of the effect of the crystal field. In the low spin case, the intermediate state is separating the two quadrupole components T2g and Eg (Figure 26 top row). Although the projected 1s2p RIXS and the 2p XAS present close intensities across the final states, the calculations show that by choosing the 1s2p RIXS incident energy Ein, one can select specific intermediate states.We observe that the quadrupoleT2g IREEP can reach only one 3T2g (3I) intermediate state symmetry, while it cannot be reached with the quadrupole Eg operator. This is consistent with the corresponding electron configuration of the 3T2g IRREP in the 3d shell: t2g5eg0.The quadrupole Eg IRREP enables to reach only the intermediate state spin-triplets at higher energy. Consequently, it appears that in the case of low spin Mn3+, depending on the RIXS intermediate states that are probed, one can enhance either the transitions into the t2g orbitals of the final states or the transitions into the eg orbitals.


#### 5.4.3. 1s2p RIXS Experiments of 3d4 Systems

Experimental 1s2p RIXS spectra of a number ofMn3+ 3d4 systems have been published by Glatzel et al. [43] for Mn2O3 and also for a series of molecular complexes that serve as reference systems for photosystem II. The pre-edge 1s2p RIXS resonances can be separated well from the main peak. The 1s2p RIXS plane of Mn(III)acac [43] is similar to the calculated plane in Figure 36.


### 5.5. 3d5 Ground State System, for Example, Fe3+

In this part, we present the elementary differences for the case of anFe3+ ion (3d5) (Mn2+ analogue). First, we gather the relevant information analogue to the previous cases which are again summarised in a combining term scheme illustrating the transition paths in 1s2p RIXS (1s XAS, 2p XES) and 2p XAS. The electronic configurations involved in this case are summarised below:
(59)Also in3d5 ground state systems like Fe3+ or Mn2+, high spin to low spin transitions can be observed. Thus, for large values of 10Dq, the ground state changes from 6A1g for t2g3eg2 to 2T2g for t2g5eg0.


#### 5.5.1. Fe3+ (3d5) in High Spin (Sextet Ground State)

The initial state configuration1s23d5 (252 microstates) has in high spin the ground state term 6S (Table 1) branching in Oh into a 6A1g symmetry (6 microstates). The spin-orbit interaction S=5/2→⊗F3/2g⊕E5/2g yields the following splittings of the ground state term:
(60)6A1g⊗F3/2g⊕E5/2g=F3/2g⊕E5/2g.Here, the termF3/2g lies the lowest in energy; thus, the ground state is 6A1gF3/2g (Oh). However, the splitting is small (few meV) and thus also the E52g term would contribute weakly in a measurement.The intermediate state with1s13d6 (2⋅210=420 microstates) gives the atomic terms with spin multiplicity from 0 to 6 and orbital momentum from S to I. The Oh crystal field splits these terms analogue to the classic Tanabe-Sugano diagram of 3d6. In 1s XAS, due to the weak 3d SOC starting from the 6A1g ground state, only sextet spin terms 6T2g and 6Eg arising from the only atomic sextet spin term 6D are reached (Table 3):
(61)6A1g→⊗T2g6T2g,6A1g⏟GS→⊗Eg1sXAS6Eg⏟ReachableIS.The corresponding 1s XAS projections for a3d5 system with 10Dq between 0.1 eV and 4.2 eV are shown in Figure 38. It is noteworthy that this situation presents some similarities with the case of 3d0 (Ti4+). In both cases, the excited 1s electron can go into either the t2g orbitals or the eg orbitals. The drastic difference is that for 3d5, all spin-up orbitals are full in the total symmetric IRREP 6A1g (Oh).Figure 38
1s XAS projections forFe3+ (3d5) with the corresponding ELD. Notice the high spin to low spin transitions in the ground and intermediate states between 10Dq≈3.0eV and 3.3 eV.For the 2p XES decay, spin-orbit interaction is again crucial; hence, we include 3d spin-orbit couplingS=5/2→⊗F3/2g⊕E5/2g for the reachable intermediate state terms:
(62)6T2g⊗F3/2g⊕E5/2g=E1/2g⊕E5/2g⊕2F3/2g⊕E1/2g⊕F3/2g,6Eg⏟ReachableIS⊗F3/2g⊕E5/2g=E1/2g⊕E5/2g⊕F3/2g⊕F3/2g.Due to the spin (S=5/2) and the resulting splittings, all three odd IRREPs (E1/2g, E5/2g, and F3/2g) appear more than once. Thus, the2p XES can decay from any of the three IRREPs E1/2g, E5/2g, or F3/2g. As given in (33), (34), and (35), the dipole IRREP T1u also enables to reach all three odd symmetries E1/2u, E5/2u, and 2F3/2u in the final state.The direct 2p XAS on the other hand starts from theF3/2g ground state term; hence, the reachable final state symmetries are identical to those found in (35). In other words, the direct 2p XAS reaches the same final states as the 2p XES decays from the F3/2g intermediate state symmetry. The resulting overall selection rules for 3d5 in high spin are summarised in the combined term scheme in Figure 39.Figure 39
Multielectronic term scheme forFe3+ 3d5 high spin comparing the1s2p RIXS and direct 2p XAS pathways. Where SOC is included (S=5/2), the spin multiplicity is omitted.The calculated 1s2p RIXS maps shown in Figure40 confirm the selection rules as summarised above. The two RIXS maps corresponding to the two quadrupole IRREPs T2g and Eg (Figure 40 top row, left and middle) confirm that the intermediate states are probed selectively according to the quadrupole transition operator. This means that the eg and t2g components of the final state can, via the intermediate state, be selectively probed in 1s2p RIXS.Figure 40
Calculated 1s2p RIXS spectrum forFe3+ (3d5) for 10Dq=1.5eV (high spin).This example is interesting, because the 1s XAS transitions are identical to theTi4+ (3d0) case. The ground state is totally symmetric with A1g; thus, the quadrupole IRREPs T2g and Eg populate the corresponding intermediate symmetries (T2g and Eg). However, there are two important differences: (i) for the high spin configuration of 3d5, all spin-up orbitals are full; thus, only the spin-down electrons can populate the remaining holes in the 3d shell. And (ii) the final state electron configuration 2p53d6 of Fe3+ has, due to the six electrons in the 3d orbitals, 6⋅210=1260 microstates. This translates into a very rich and dense stick spectrum shown below the slices in Figure 40.Finally, we note that the matrix elements of the 2p XAS appear very similar to those of the resonant 2p XES decays occurring from the6Eg intermediate state, while the total 2p XES shown as integrated incident energy (IIE) shows notable differences when compared to the 2p XAS due to the population from the intermediate state symmetries.


#### 5.5.2. Fe3+ (3d5) in Low Spin (Doublet Ground State)

The initial state electron configuration3d5 translates in low spin to t2g5eg0 with the atomic ground state term 2D (10 microstates). In Oh, this translates into a 2T2g ground state term with 6 microstates. The inclusion of 3d spin-orbit interaction S=1/2→⊗E1/2g yields the following splitting:
(63)2T2g⊗E1/2g=E5/2g⊕F3/2g.Here, the termE5/2g lies the lowest in energy; thus, the ground state is 2T2gE5/2g (Oh).The intermediate state configuration1s13d6 is still the same as in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the ground state being a spin doublet (2T2g), the spin-conserving 1s XAS only reaches spin doublet intermediate state IRREPs (Table 3):
(64)2T2g→⊗T2g2A1g⊕2Eg⊕2T1g⊕2T2g,2T2g⏟GS→⊗Eg1sXAS2T1g⊕2T2g⏟ReachableISlowspin.This means that from the2T2g ground state the quadrupole 1s XAS, as shown in Figure 41, can reach all even intermediate state symmetries, except any A2g IRREPs.Figure 41
Low spin 1s XAS projections (10Dq=3.3eV to 4.2eV) for Fe3+ (3d5) with ELD. In the single electron picture the 3d shell has either t2g6eg0 for 2A1g or t2g5eg1 for 4T1g and 4T2g, respectively.The reachable intermediate state terms split due to the 3d spin-orbit interactionS=1/2→⊗E1/2g:
(65)2A1g⊗E1/2g=E1/2g,2Eg⊗E1/2g=F3/2g,2T1g⊗E1/2g=E1/2g⊕F3/2g,2T2g⊗E1/2g=E5/2g⊕F3/2g.We find all three half-valued terms from which the decays in 1s2p RIXS can occur.The final state (1s22p53d6) is again identical to the high spin case regardless of the crystal field effects, offering the three total symmetry IRREPs E1/2u, E5/2u, andF3/2u. As given in (33), (34), and (35), the decay paths are formally the same as in the high spin case reaching all three odd symmetry terms. The differences in the spectra therefore relate to the difference of the intermediate state population via the first 1s XAS step. Again, the intermediate state plays the decisive role.For 2p XAS, starting from theE5/2g ground state, only two final state terms are reachable:
(66)E5/2g⏟GS→⊗T1u2pXASE1/2u⊕F3/2u⏟AccessibleFSterms.The results of the selection rules as discussed above are summarised in the combined term scheme shown in Figure42.Figure 42
Multielectronic term scheme forFe3+ (3d5) low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is considered, the spin multiplicity was omitted.The calculated 1s2p RIXS map for a low spin case (10Dq=3.9eV) is given in Figure 43.Figure 43
Calculated 1s2p RIXS spectrum forFe3+ (3d5) for 10Dq=3.9eV (low spin).In the left RIXS map in Figure43(a), it appears that the quadrupole IRREP T2g enables to reach only the first fully symmetric 2A1g intermediate state symmetry. This term corresponds in the single electron picture to a t2g6eg0 configuration of the 3d shell where all the t2g orbitals occupied. The spin doublet emerges solely from the core hole in the 1s shell.The 2p XES decay reaches two final states corresponding to thet2g orbital full and a hole in the 2p shell, relating to one in the L3 and one in the L2. In the multielectronic picture, these two peaks correspond to the 2A1g⊗2P final states (see Figures 43 and 44). It is noteworthy that this peak is not reached in the L2-edge via the 2p XAS. This can be explained in the single electron picture of the initial state, which contains only one hole in the t2g orbitals. The L2-edge does not get a peak because it would require a spin-flip to fill the t2g hole with an electron from the 2p shell. In 1s2p RIXS, these two peaks arise only from the decay of the 2p shell electron into the 1s shell, which is possible for both L3 and L2. The higher energy range of the final states is reached via the intermediate state probed by the quadrupole Eg IRREP (1s XAS operator).Figure 44
2p XES projections forFe3+ (3d5) for 10Dq=0.1eV to 4.2eV. The high spin to low spin transition between 10Dq≈3.0eV and 3.3 eV also induces visible changes in the 2p XES. The most notable is the appearance of the two peaks related to the first 2A1g intermediate state symmetry around ET<−5eV and ET≈7eV.


#### 5.5.3. 1s2p RIXS Experiments of 3d5 Systems

Examples of3d5 systems include Mn2+ and Fe3+. Experimental 1s2p RIXS spectra of MnO have been published by Kas et al. [39]. Compared with the calculations in Figure 40, the intensity ratio of the T2g and the Eg peak is inverted in the experiment, with the Eg peak at [6540, 638] eV having the largest intensity. The same observation can be made for the Fe3+ systems Fe2O3 and iron impurities in MgO [14]. The tetrahedral Fe3+ contains a single 1s2p RIXS feature, reproduced by the calculations. 1s2p RIXS of a series of iron minerals has been published by Vercamer in his PhD thesis [44]. The 1s2p RIXS spectrum of octahedral low spin compounds Fe(III)(tacn) and Fe(III)cyanide has been measured and reproduced in multiplet calculations by Lundberg et al. [45].


### 5.6. 3d6 Ground State System, for Example, Fe2+

In this part, we present the elementary differences between 1s XAS and 1s2p RIXS for the case ofFe2+ (3d6; Co3+ analogue). As in the previous case(s), first, we gather the relevant informations which are summarised in a combining term scheme illustrating the transition paths in 1s2p RIXS (1s XAS, 2p XES) and 1s XAS. The electronic configurations involved are summarised below:
(67)As in the previous case, the crystal field splitting energy (10Dq) and the pairing energy are in competition in a 3d6 configuration. Thus, for sufficiently high values of 10Dq, the ground state changes from high spin 5T2g for t2g4eg2 to low spin 1A1g for t2g6eg0.


#### 5.6.1. Fe2+ (3d6) in High Spin (Quintet Ground State)

The initial state electronic configuration is3d6 (210 microstates) with the atomic ground state term 5D (25 microstates). In an Oh crystal field, the 5D term splits into 5T2g for t2g4eg2 and 5Eg for t2g3eg3. The inclusion of spin-orbit interaction S=2→⊗Eg⊕T2g yields the following splittings:
(68)5T2g⊗Eg⊕T2g=A1g⊕Eg⊕2T1g⊕2T2g.Here, the termT2g lies the lowest in energy; thus, the ground state is 5T2gT2g (Oh).The intermediate state electron configuration1s13d73d7 (2⋅120=240 microstates) yields the atomic terms 1,3,5F, 1,3,5P, 1,3D, 1,3G, and 1,3H (Table 1). Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d7. Because the 1s XAS is spin-conserving, only quintet intermediate states are reached in the absorption, corresponding to the crystal field terms derived from 5F and 5P: 5A2g, 5T2g, 5T1g, and 5T1g, respectively. For 1s XAS, starting from the 5T2g ground state term, the reachable high spin intermediate state terms are
(69)5T2g→⊗T2g5A1g⊕5Eg⊕5T1g⊕5T2g,5T2g⏟GS→⊗Eg1sXAS5T1g⊕5T2g⏟ReachableIShighspin.The reachable intermediate state terms ofFe2+ (3d6) in high spin are therefore 5T1g and 5T2g. In other words, though the 5A1g and 5Eg symmetries are formally reachable in 1s XAS, the intermediate state does not offer this IRREP to be populated. Furthermore, though an 5A2g IRREP exists, the quadrupole selection rules do not allow it to be reachable in 1s XAS (see Figure 45). However, the small 3d SOC may induce some mixing enabling also weak transitions into this IRREP.Figure 45
1s XASprojections of the calculated 1s2p RIXSspectrum forFe2+ (3d6) showing the high spin to low spin transition just above 10Dq≈1.8eV.Adding SOCS=2→⊗Eg⊕T2g to the reachable IS terms leads to the splittings in the IS:
(70)5A1g⊗E⊕T2=E⊕T2,5T1g⊗Eg⊕T2g=T1g⊕T2g⊕A2g⊕Eg⊕T1g⊕T2g,5T2g⊗Eg⊕T2g=T1g⊕T2g⊕A1g⊕Eg⊕T1g⊕T2g.We find the decays in 1s2p RIXS can occur from all integer symmetries (A1g, A2g, Eg, T1g, andT2g).The final state electron configuration1s22p53d7 contains 6⋅120=720 microstates due to the hole in the 2p shell spanning J from 0 to 7. In an Oh crystal field, these terms branch into the five integer total symmetry IRREPs A1g, A2g, Eg, T1g, andT2g. The2p XES decays can occur from any of the populated intermediate state terms; thus, the accessible final state terms are again derived with the dipole T1u IRREP as given in (28), (29), (30), (31), and (32). In short, all “ungerade” integer IRREPs are reachable in the final state of 2p XES.In 2p XAS, starting from theT2g total symmetry IRREP term and using the T1u IRREP, the reachable final states are again identical to the 2p XES final state term reachable from the T2g intermediate state IRREP as given (32).Altogether, this enables to draw the term scheme forFe2+ (3d6) in high spin as shown in Figure 46.Figure 46
Multielectronic term scheme forFe2+ (3d6) (high spin) comparing the 1s2p RIXS and 2p XAS pathways. Where SOC is considered, the spin multiplicity is omitted as its meaning is lost.The result of the calculation for high spinFe2+ (3d6) for 10Dq=0.9eV is shown in Figure 47.Figure 47
Calculated 1s2p RIXS spectra and slices forFe2+ (3d6) for 10Dq=0.9eV (high spin).We note that the quadrupoleT2g IRREP (left RIXS map in Figure 47(a)) is dominated by transitions into the 5T2g intermediate state symmetry, while the quadrupole Eg IRREP (middle RIXS map in Figure 47(a)) leads mostly into the T1g intermediate state IRREPs. As predicted above, the 1s XAS spectra gain their intensity only from two intermediate state symmetries, the 5T1g and 5T2g IRREPs. The 5A2g intermediate state symmetry does not contribute to the spectrum as can be seen in Figure 45.


#### 5.6.2. Fe2+ (3d6) in Low Spin (Singlet Ground State)

For a sufficiently large crystal field splitting (10Dq), a 3d6 electronic configuration results in a low spin configuration with all t2g orbitals fully occupied (t2g6eg0). This corresponds to the totally symmetric ground state term 1A1g (1 microstate) derived from the atomic term 1I (Table 1). The inclusion of spin-orbit interaction S=0→⊗A1g gives the total ground state as 1A1gA1g (Oh).The intermediate state electron configuration 1s13d7 is still the same as in the high spin case. However, for high crystal fields (10Dq), the order of the intermediate state energy levels is different in low spin. In the low spin case, the initial state being a spin singlet, the spin-conserving 1s XAS will only reach spin singlet intermediate states (Table 3):
(71)1A1g→⊗T2g1T2g,1A1g⏟GS→⊗Eg1sXAS1Eg⏟ReachableIS.Although there are1T2g and 1Eg IRREPs existing in the intermediate state, only the 1Eg derived from the 1G atomic term can be reached. This can be understood from the Δℒ=±2 quadrupole selection rule, which allows only to reach the 1G atomic term (ℒ=4) from the 1I atomic ground state term (ℒ=6). The quadrupole 1s XAS formally can, according to the selection rules, only reach the 1T2g and 1Eg IRREPs from this term. But finally, the transition to the 1T2g intermediate state symmetry would imply a two-electron transition from t2g6eg0 to t2g5eg2, which has a very low probability. Hence, only the 1Eg IRREP from the 1G atomic term is populated thus adding intensity to the 1s XAS spectrum as visible in Figure 48.Figure 48
1s XAS projections of the calculated 1s2p RIXS spectrum forFe2+ (3d6) showing low spin part above 10Dq≈1.8eV only. It reveals only the Eg symmetry is giving a significant contribution.For the 2p XES decays in 1s2p RIXS, spin-orbit interaction in the intermediate state must be consideredS=0→⊗A1g leading the identities:
(72)1Eg⏟reachableIS⊗A1g=Eg.The final state electron configuration1s22p53d7 is again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. The 2p XES decaying from the populated intermediate state term Eg can reach with the dipole IRREP T1u the final state terms T1u and T2u as given in (30).On the contrary, the 2p XAS, starting from the total symmetricA1g ground state symmetry, can reach only the T1u final state IRREP.This again enables altogether to draw the combined term scheme comparing the two-step 1s2p RIXS with the direct dipole 2p XAS pathways in Figure49.Figure 49
Multielectronic term scheme forFe2+ (3d6) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. We note its similarity to the Ti4+ (3d0) case shown in Figure 15.The resulting calculated 1s2p RIXS forFe2+ (3d6) is shown in Figure 50. As already discussed, the 1s2p RIXS is dominated by the Eg intermediate state symmetry. As can be seen, the quadrupole T2g IRREP has overall a negligible contribution to the total RIXS. From the term scheme in Figure 49, one can furthermore derive that the difference between the integrated 2p XES (IIE) and the direct 2p XAS emerges from the T2u final state IRREP which can only be reached in 1s2p RIXS.Figure 50
Calculated 1s2p RIXS spectrum forFe2+ (3d6) for 10Dq=3.0eV (low spin).


#### 5.6.3. 1s2p RIXS Experiments of3d6 Systems

3d6 systems include Fe2+ and Co3+. A large number of experimental 1s2p RIXS spectra of Fe2+ systems have been published. Vercamer measured a large series of Fe2+ minerals in his PhD thesis [44]. The tetrahedral and octahedral Fe2+ oxides/minerals have been studied and reproduced by multiplet calculations [5, 17, 46]. Guo et al. [47] studied low spin Fe2+ cyanide complexes. Lundberg et al. [45] compared these experiments to the tacn complex and reproduced the spectra from multiplet calculations. Leidel et al. [48] studied iron hydrogenase complexes. Vanko et al. [49] studied the time evolution of the 1s2p RIXS planes. The static spectra of the low spin systems Feterpy22+ and Febipy32+ and the laser-excited spectrum of Feterpy22+ show a partial transition to a high spin ground state as is confirmed in the calculations.In case ofCo3+, all measured systems were low spin 1A1g. Vanko et al. [7] compared an isolated Co3+ complex with a series of cobalt oxides. They nicely show the presence of the nonlocal peaks as a function of the Co-O-Co angle.


### 5.7. 3d7 Ground State System, for Example, Co2+

In this part, we present the elementary differences between 1s2p RIXS and 2p XAS for the case ofCo2+ (3d7). First, we gather the relevant information analogue to the previous cases which are again summarised in the combining term scheme in Figure 39 term scheme illustrating the transition paths involved (Figures 51 and 52). The electronic configurations involved are summarised below:
(73)Figure 51
Multielectronic term scheme for Co2+ (3d7) in high spin comparing the 1s2p RIXS and direct 2p XAS pathways. Also, the spin multiplicity was omitted where the spin-orbit interaction is included.Figure 52
Multielectronic term scheme for aCo2+ ion (3d7) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.Also, a 3d7 system encounters, for sufficiently high values of 10Dq, a change of the ground state from 4T2g for t2g5eg2 (high spin) to 2Eg for t2g6eg1 (low spin).


#### 5.7.1. Co2+ (3d7) in High Spin (Quartet Ground State)

For the initial state configuration3d7 (120 microstates), the high spin ground state term symbol is 4F (Table 1), which branches in Oh into 4T1g, 4T2g, and 4A2g, where the 4T1g term with 12 microstates is the lowest in energy. The inclusion in spin-orbit interaction (S=3/2→⊗F3/2g) yields the subsequent splitting of 4T1g term:
(74)4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2F3/2g.The ground state4T1g splits due to 3d SOC into the three odd symmetries E1/2g, E5/2g, and F3/2g from which the term E1/2g is lowest in energy; hence, the combined ground state is 4T1gE1/2g (Oh).The intermediate state electron configuration is1s13d8 (2⋅45=90 microstates). The atomic terms are 2,4F, 2,4P, 2S, 2D, and 2G (Table 1). Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d8. The spin-conserving 1s XAS reaches only spin quartet intermediate states, which corresponds to the crystal field terms issued from 4F and 4P: 4A2g, 4T2g, 4T1g, and 4T1g, respectively. Starting from the high spin ground state term 4T1g, the two quadrupole IRREPs give the reachable intermediate state terms:
(75)4T1g→⊗T2g4A2g⊕4Eg⊕4T1g⊕4T2g,(76)4T1g⏟GS→⊗Eg1sXAS4T1g⊕4T2g⏟AccessiblelowspinIS.The reachable 1s XAS intermediate state terms in high spin are therefore4A2g, 4T1g, and 4T2g. Here, it shall be noticed that an Eg symmetry would be reachable by the quadrupole selection rules, but the intermediate state does not offer any (quartet) 4Eg symmetries. The 1s XAS projections displayed in Figure 53 confirm these findings.Figure 53
1s XAS projections of a calculated 1s2p RIXS spectrum forCo2+ (3d7) with ELD. It confirms that in high spin only the quartet terms 4,2F and 4,2P contribute to the 1s XAS. The high spin to low spin transition appears at approximately 10Dq≈2.2eV.Applying the 3d spin-orbit interaction (S=3/2→⊗F3/2g) to the three reachable terms yields the splittings in the intermediate state:
(77)4A2g⊗F3/2g=F3/2g,4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g,4T2g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g.The result is that all three half-valued IRREPs appear more than once and hence the 2p XES can decay from any of these terms.The final state (1s22p53d8) contains ⋅6⋅45=270 microstates due to the hole in the 2p shell spanning J from 1/2 to 11/2. In an Oh crystal field, these terms branch into the three total symmetry IRREPs E1/2u, E5/2u, and F3/2u. The 2p XES starts from either E1/2g, E5/2g, or F3/2g, and we find ((33), (34), and (35)) that all three odd terms are also reachable in the 1s2p RIXS final state.The 2p XAS on the other hand, starting from theE1/2g ground state term, can only reach the E1/2u and F3/2u final state terms (Table 4).A summary of the findings above is shown below as a combined term scheme in Figure51.We conclude this part with the corresponding calculations of the 1s2p RIXS forCo2+ (3d7) for a crystal field energy of 10Dq=0.9eV which is shown in Figure 54.Figure 54
Calculated 1s2p RIXS spectrum forCo2+ (3d7) for 10Dq=0.9eV (high spin).It can be noted that the RIXS map for the quadrupoleT2g IRREP (Figure 54(a), left) is dominated by the 1s XAS transitions into the 4A2g intermediate state symmetry which corresponds in the single electron picture to a t2g6eg2 configuration. However, as given in expression (75), it has also contributions into the two T1g intermediate state symmetries though they are weak with respect to the A2g contribution. The stick spectrum here is not as dense as in some of the cases discussed above, and this enables to identify also some of the reachable final state terms. For example, the strong peak at ET≈−4.5eV relating to the decay from the 4A2g intermediate state symmetry corresponds to the two final state terms E1/2u and F3/2u. This can be understood considering that the 4A2g branches due to SOC into a F3/2g intermediate state symmetry, which can reach the above two final state terms via the dipole T1u IRREP.The quadrupole IRREPEg on the other hand shows only contributions into the 4T1g and 4T2g intermediate state symmetries as predicted in expression (76).And finally, it should be noted that the 1s XAS projections in Figure53 show with increasing crystal field energy 10Dq the branching of the 4F into the three IRREPs 4A2, 4T2g, and 4T1g, while the atomic term 4P translates in an Oh crystal field only into a 4T1g symmetry.


#### 5.7.2. Co2+ (3d7) in Low Spin (Doublet Ground State)

At high crystal field energies, the3d7 initial state gives the low spin ground state term 2Eg (4 microstates) arising from the atomic term 2F which corresponds to the electronic configuration t2g6eg1. The inclusion of spin-orbit interaction (S=1/2→⊗E1/2g) yields the subsequent splittings:
(78)2Eg⊗E1/2g=F3/2g.Hence, the combined ground state is2EgF3/2g (Oh).The intermediate state configuration 1s13d8 is still the same as in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the initial state being a spin doublet, the spin-conserving 1s XAS only reaches spin doublet intermediate state terms (Table 3), which correspond to the crystal field terms issued from 2S, 2P, 2D, 2F, and 2G and 2A1g, 2A2g, 2Eg, 2T1g, and 2T2g. The quadrupole 1s XAS starts from the low spin ground state term 2Eg revealing the reachable terms:
(79)2Eg→⊗T2g2T1g⊕2T2g,2Eg⏟GS→⊗Eg1sXAS2A1g⊕2A2g⊕2Eg⏟ReachableIS.Thus, according to the quadrupole selection rules, from the2Eg ground state, the 1s XAS transitions can reach all existing spin doublet IRREPs in the intermediate state. However, because in the low spin initial state all t2g orbitals are occupied, 1s XAS transitions can only go into the eg orbitals via the quadrupole Eg IRREP. The transitions into the t2g orbitals with the quadrupole T2g IRREP are less probable as they require a two-electron transition. In other words, the matrix elements of the quadrupole Eg IRREP are larger than those of the quadrupole T2g IRREP. The resulting 1s XAS projections for the low spin case are shown in Figure 55.Figure 55
1s XAS projections of the calculated 1s2p RIXS maps for aCo2+ ion (3d7) with 10Dq=2.4eV to 3.3eV (low spin only).For the 1p XES decays, the inclusion of 3d SOC (S=1/2→⊗E1/2g) is again required, leading to the following splittings of the five reachable terms in the intermediate state:
(80)2A1g⊗E1/2g=E1/2g,2A2g⊗E12g=E5/2g,2Eg⊗E1/2g=F3/2g,2T1g⊗E1/2g=E1/2g⊕F3/2g,2T2g⊗E1/2g=E5/2g⊕F3/2g.Hence, the 2p XES decays in 1s2p RIXS can occur from any of these three total angular momentum IRREPsE1/2g, E5/2g, or F3/2g.The final state electron configuration1s22p53d8 is again identical to the high spin case regardless of the crystal field effects, offering the three total angular momentum IRREPs E1/2u,E5/2u, and F3/2u. For the 2p XES decaying from the populated intermediate state terms, using the T1u IRREP (Table 4) again reveals that all three final state IRREPs are also reachable. Furthermore, in 2p XAS, starting from the F3/2g ground state symmetry, also, all three final state IRREPs are reachable (Table 4).The results according to the selection rules as discussed above are again summarised in a combined term scheme as displayed in Figure52.Again, we conclude with an example calculation for aCo2+ ion (3d7 ground state configuration) with 10Dq=3.0eV reflecting the low spin case as shown in Figure 56.Figure 56
Calculated 1s2p RIXS spectrum forCo2+ (3d7) for 10Dq=3.0eV (low spin).It is noteworthy that the quadrupole IRREPT2g here is very weak (10−3) with respect to the Eg IRREP. This can again be explained with the very small matrix elements for the two-electron transition that is required to create a hole in the t2g orbitals, because the quadrupole IRREP T2g populates the intermediate state symmetries T1g and T2g, which both require a hole in the t2g level.This is also confirmed by the total RIXS map (Figure56(a), right) showing that the overall result is dominated by the transitions into the three intermediate state symmetries 2A2g, 2Eg, and 2A1g, while the 2T1g and 2T2g intermediate state symmetries are negligible due to the reason given above.However, as predicted by the selection rules for the quadrupole IRREPS,T2g and Eg as shown in Figure 56 confirm the predicted reachable terms as found in expression (79).


### 5.8. 1s2p RIXS Experiments of3d7 Systems

A300meV resolution 1s2p RIXS experiment on CoO was published by Kurian et al. [50]. The increased resolution revealed new details in the pre-edge fine structure, and it was possible to study the effect of interference effects on the 1s2p RIXS plane.


## 5.1. 3d1 Ground State System, for Example, Ti3+

In the previous cases, the splitting of the ground state electronic terms was leading to a single term, for example,2EgF3/2g (Oh) for Cu2+ (3d9) or 3A2gT2g (Oh) for Ni2+ (3d8). In the following, we discuss the 3d1 ground state configuration representing for instance the case of a Ti3+ ion.
(36)As shown below, the3d1 case is an interesting transition towards the other transition metal ions, as it enables to explore two aspects in the RIXS: (i) the influence of the spin-orbit splitting of the ground state leading to the contribution of several terms for T > 0 K and (ii) the effect of the multielectronic interactions of the 3d shell in the intermediate and final states of 1s2p RIXS. The atomic case will not be detailed, and we will only discuss the influence of the Oh crystal field.The initial state electronic configuration 1s22p63d1 only contains a single electron in the 3d shell. It is noteworthy that this electronic configuration is analogue to the initial state of Cu2+: the atomic ground state term is 2D, which splits into D5/2 and D3/2 by the 3d spin-orbit coupling (see Table 1). In the Oh crystal field, the 2D term splits into 2T2gt2g1eg0 and 2Egt2g0eg1.The difference withCu2+ (3d9) is that the ground state term of Ti3+ (3d1) is 2T2g (6 microstates). When considering the 3d spin-orbit coupling ζ3d, this term splits into the two total symmetry irreducible representations (IRREPs) F3/2g (mixing of D5/2 and D3/2) and E5/2g (from D5/2 term), where F3/2g is the lowest. This is summarised in Figure 3, which shows the splittings of the ground state Russel-Saunders term 2D with SOC and an Oh crystal field. It should be compared with Cu2+ (Figure 2). Furthermore, we note that due to the small 3d SOC in the initial state (ζ3d=19meV), the splitting of the ground state 2T2g is only 28 meV. Thus, for temperatures T > 0 K, the second IRREP should also be included according to the Boltzmann population. For instance, at T=300 K, Boltzmann population of the F3/2g (4-fold degenerate) is 86% and 14% for the E5/2g (2-fold degenerate). The contribution of the second term E5/2g is thus small but not entirely negligible.The intermediate state electron configuration1s13d2 (2⋅45=90 microstates) is equivalent to the initial state of Ni2+ coupled to the 1s core hole [13]. The atomic terms are 2,4F, 2,4P, 2G, and 2S (Table 1). Because the Coulomb exchange Gsd is small, the splitting between the doublet and the quartet spin states is small and beyond the accessible energy resolution.Using Table2, one finds the branchings of the atomic states in an Oh crystal field. The resulting Oh term symbols for a 1s13d2 configuration are 2,4A2g, 2,4T2g, 2,4T1g, 2A1g, and 2Eg. The energy splitting of these Oh crystal field terms is qualitatively similar to the Tanabe-Sugano diagram of 3d2. In the particular case of Ti3+, the result of the calculation is given in Figure 21 where the 1s XAS projections are shown together with the crystal field terms assigned.Figure 21
Influence of the Oh crystal field on the 1s XAS projection of the calculated 1s2p RIXS spectra for Ti3+ (3d1) with ELD overlay. A comparison to the calculated 1s XAS projections now shows on the one hand that neglecting 3d SOC in this step is a valid approximation. And on the other hand, it confirms that a quadrupole transition from the 2T2g ground state symmetry can reach A1g, Eg, T1g, and T2g symmetries, but the A2g symmetry does apparently not contribute to the spectrum.Because the 3d spin-orbit coupling is weak, one can neglect it in first approximation; the spin conservation leads to the 1s XAS being dominated by transitions into the spin doublet states, while the matrix elements of the transition to the spin quartet terms are much smaller. In the following, we will therefore focus on the doublet spin states.(37)2T2g→⊗T2g2A1g⊕2Eg⊕2T1g⊕2T2g,(38)2T2g⏟GS→⊗Eg1sXAS2T1g⊕2T2g⏟ReachableISterms.This shows, in an Oh crystal field, starting from the 2T2g ground state, the 1s XAS operators T2g and Eg enable to reach all even terms except the 2A2g IRREP. This results in numerous multiplet states reached in the intermediate state of 1s2p RIXS. The 1s XAS projections in Figure 21 show the splitting of the peaks due to the scaling of the crystal field parameter 10Dq. We note that the 2A2g (2F) does not get any intensity in the absorption step of 1s2p RIXS in agreement with the 1s XAS selection rules as found in the expressions given in (37) and (38).The previous didactic examples (Section4) demonstrated the necessity to consider the ℒ.S-coupled terms to describe the 2p XES transitions to the final state in 1s2p RIXS and the 2p XAS. In the intermediate state, the Oh total symmetry IRREPs of the doublet spin states are obtained by coupling the reachable orbital IRREPs with the spin IRREP (S=1/2→⊗E1/2g):
(39)A1g⊗E1/2g=E1/2g,Eg⊗E1/2g=F3/2g,T1g⊗E1/2g=E1/2g⊕F3/2g,T2g⊗E1/2g=E5/2g⊕F3/2g.The result shows that all three odd total symmetric IRREPSE1/2g, F3/2g, and E5/2g are thus reachable in the intermediate state. In other words, all three odd IRREPs are contributing to the 2p XES decays.The final state configuration1s22p53d2 contains 6⋅45=270 microstates due to the hole in the 2p shell. The atomic term symbols span from S to H with spin doublets and quartets and J values ranging from 1/2 to 11/2 [13]. In an Oh crystal field, these terms branch into the three IRREPs E1/2u, F3/2u, and E5/2u (Table 2).The 2p XES dipole transition IRREPT1u gives the transitions from the three intermediate state symmetries into all three final state symmetries as given in (33), (34), and (35).For the direct 2p XAS, as mentioned above, the two ground state termsF3/2g and E5/2g are to be considered due to the small splitting. This leads with the dipole transition IRREP T1u also to all three final state symmetries:
(40)E5/2g→⊗T1uE5/2u⊕F3/2uF3/2g→⊗T1u2pXASE1/2u⊕2F3/2u⊕E5/2u.It becomes obvious that the selection rules enable to reach all three final state IRREPs in both cases: 2p XAS and 1s2p RIXS. This is summarised in the combined term scheme shown in Figure22.Figure 22
Multielectronic term scheme forTi3+ (3d1) comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted as its meaning is lost in that case.Although it seems that all the final state symmetries can be reached in both cases, 2p XAS and the 2p XES in 1s2p RIXS, the calculated spectra shown in Figure23 demonstrate that the RIXS intermediate state enables to select some specific final states.Figure 23
Calculated 1s2p RIXS spectrum forTi3+ (3d1) for 10Dq=1.5eV shown together with some selected CIE slices. The stick spectrum at the bottom illustrates the distribution of the three final state terms E1/2u, F3/2u, and E5/2u with respect to the emissions.According to the combined term scheme in Figure22, the intermediate states T1g, T2g, and Eg enable to reach all final state symmetries. The calculated 1s2p RIXS together with some CIE slices is shown in Figure 23. However, due to the first term 2T1g corresponding to the excitation of the 1s electron into the t2g orbitals, the final states probed resonantly with RIXS are mainly the lower energy part of the states, corresponding to the “t2g” part of the spectra. The intensity arises mainly from the T2g absorption operator. The 2T2g intermediate state selects higher energy final states equally probed by the T2g and Eg 1s XAS operators.The second term2T1g (from 2P atomic term symbol) corresponds to the absorption to the eg orbitals and is consequently probed by the Eg absorption operator and corresponds to the “eg” part of the final states. The intermediate state 2A1g (from 2S atomic term symbol) only gives a very weak absorption via the T2g quadrupole operator in agreement with the selection rules as given in (37).This intermediate state only decays toE1/2u (see, e.g., the final state peak at ET≈8.3eV in Figure 23) and F3/2g (at ET≈2−3eV). Such intermediate state selection is identical for the L3- and L2-edges. This is similar to the results obtained for Ti4+ where the “t2g” and “eg” parts of the finals states are selected with the intermediate state of RIXS.


### 5.1.1. 1s2p RIXS Experiments of 3d1 Systems

Experimental 1s2p RIXS spectra of3d1V4+ ions in TiO2 have been measured by Bordage [41]. However, the experimental resolution does not allow to compare the data in detail with the calculations as shown in Figure 23.


## 5.1.1. 1s2p RIXS Experiments of 3d1 Systems

Experimental 1s2p RIXS spectra of3d1V4+ ions in TiO2 have been measured by Bordage [41]. However, the experimental resolution does not allow to compare the data in detail with the calculations as shown in Figure 23.


## 5.2. 3d2 Ground State System, for Example, Cr4+

In this part, we present the elementary differences between 2p XAS and 1s2p RIXS for the case of tetravalent chromium (Cr4+). First, we gather the relevant information analogue to the previous cases which are then summarised in a combining term scheme illustrating the transition paths in 1s2p RIXS (1s XAS, 2p XES) and the direct 2p XAS. The electronic configurations involved here are
(41)In the following, we will discuss this case with respect to the influence of the SOC in the intermediate state leading to a significant increase of the possible 1s2p RIXS final states. The atomic case will not be detailed, and we will only discuss the influence of the Oh crystal field for 10Dq=2.1eV.The initial state electron configuration1s22p63d2 corresponds to the atomic ground state term 3F (O3) with 45 microstates. It translates into an Oh crystal field (without 3d SOC) into a 3T1g ground state symmetry with 9 microstates. By including 3d SOC ζ3d, we find the further splittings of the 3T1g ground state term (S=1→⊗T1g):
(42)3F4O3→CF3T1gOh→SOCA1g⊕Eg⊕T1g⊕T2gOh.From these terms, Eg lies lowest in energy; hence, the ground state is 3T1gEg (Oh).The intermediate state configuration1s13d3 with 2⋅120=240 microstates translates into the atomic term symbols 3,5F, 3,5P, 3H, 3G, 3D, and 3P (Table 1). Using Table 2 one finds the branchings of the atomic states in an Oh crystal field. The resulting Oh term symbols are 3,5A2g, 3,5T2g, 3,5T1g, 3A1g, 3A2g, 3Eg, 3T1g, and 3T2g. However, the Coulomb exchange Gsd is small; hence, the splitting between the triplet and the quintet spin states is small and beyond the accessible energy resolution. The energy splittings of those terms are qualitatively similar to the Tanabe-Sugano diagram of a 3d3 electron configuration.Because the 3d SOC is weak, the spin is conserved leading to the 1s XAS transitions being dominated by transitions into the spin triplet states, while the matrix elements of the transition to the spin quintet terms are weak. Thus, we will focus on the spin triplet symmetries.(43)3T1g→⊗T2gA32g⊕3Eg⊕3T1g⊕3T2g,(44)3T1g→⊗Eg1sXAS3T1g⊕3T2g⏟AccessibleISterms.As can be seen from the results found in (43) and (44), starting from a 3T1g ground state, all but the 3A1g intermediate state symmetries can be reached via the quadrupole 1s XAS. The 1s XAS projections of the 1s2p RIXS for 10Dq ranging from 0.1 eV to 3.0 eV are shown in Figure 24.Figure 24
Effect of the Oh crystal field on the 1s XAS projections of the calculated 1s2p RIXS spectra for Cr4+ (3d2). The corresponding ELD overlay illustrates the distribution of the reachable intermediate state terms contributing in 1s XAS.For the 2p XES decays in 1s2p RIXS, however, the inclusion of SOC is again required. Using theT1g IRREP (S=1→⊗T1g) and the direct product table (Table 3), one finds the intermediate state symmetries including 3d SOC ζ3d as also shown in the combined term scheme in Figure 25.
(45)3A2g⊗T1g=T2g,3Eg⊗T1g=T1g⊕T2g,3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g,3T2g⊗T1g=A2g⊕Eg⊕T1g⊕T2g.Figure 25
Multielectronic term scheme forCr4+ (3d2) comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted as its meaning is lost in that case.The subsequent 2p XES starts from any of the intermediate state symmetries found in expression (45). The possible 2p XES final state symmetries are all even symmetries as given in expressions (28), (29), (30), (31), and (32).The final state (1s22p53d3) contains 6⋅120=720 microstates due to the hole in the 2p shell spanning J from 0 to 7. In an Oh crystal field, these terms branch into the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. In other words, all existing FS symmetries can be reached in the 2p XES.Also for 2p XAS, spin-orbit interaction is crucial. Thus, starting from the3T1gEg (Oh) ground state and using the T1u IRREP, we find that only T1u and T2u IRREPs are reached, which makes a drastic difference between 2p XAS and 1s2p RIXS. The selection rules for a 3d2 ground state system are summarised in Figure 25 showing all reachable symmetries.According to the electric quadrupole selection rules derived in (43), the A2g and Eg intermediate state symmetries are only reachable via the quadrupole T2g IRREP. This is confirmed in Figure 26 (RIXS maps) where the low energy peak at Ein=−2.25eV appears only in the RIXS map corresponding to the T2g IRREP. The t2g and eg components appear well separated with (i) the t2g components probed by the T2g absorption giving two intense and narrow peaks at ET≈−6.5eV and (ii) the eg components probed by the Eg absorption leading to a larger group of final states spread in energy between ET=−5eV and 0 eV. Furthermore, it can be noted that the X-ray emission spectra (2p XES) are here overall very rich. The corresponding sticks in Figure 26 illustrate how dense the reachable final states are distributed. We observe that the 2p XAS and 1s2p RIXS differs mainly in the low energy final states: the 1s2p RIXS enhances the final states below ET≈−2.5eV to which the 2p XAS transition matrix elements are weak. We note that the 1s2p RIXS spectra which are the closest to the 2p XAS spectrum are those obtained at the higher energy transfers at ET>0eV.Figure 26
Calculated 1s2p RIXS spectrum forCr4+ (3d2) for 10Dq=2.1eV. Note that the A2g intermediate state symmetry is in agreement with the selection rules only populated via the T2g 1s XAS IRREP. The slices have been scaled to a comparable intensity to enhance their character better. The bottom plot compares the integrated incident energy (IIE) 2p XES with the 2p XAS (L2,3-edge).Considering now the quantitative influence of the natural broadening as the principal limit of any experimental spectrum, it becomes clear that an experimental separation of the final states is impossible. However, high-resolution RIXS measurements enable to extract CIE slices similar to the illustrated examples shown in Figure26.And finally, in Figure27, we show the 2p XES projections of the calculated 1s2p RIXS to illustrate its evolution when the crystal field value 10Dq is scaled from 10Dq=0.1eV,…,3eV.Figure 27
Effect of the Oh crystal field on the 2p XES projections of the calculated 1s2p RIXS spectra for Cr4+ (3d2).


### 5.2.1. 1s2p RIXS Experiments of 3d2 Systems

Experimental 1s2p RIXS spectra of 3d2 CrO2 have been published by Zimmermann et al. [9]. Analysis of the magnetic circular dichroism (MCD) of the 1s2p RIXS revealed that the pre-edge structure that is visible in the 1s2p RIXS plane is dominated by a nonlocal contribution. The MCD data reveals the real quadrupole peak, but due to its low intensity and limited resolution, it does not allow to compare the quadrupole peak to the spectra calculated in Figure 26.


## 5.2.1. 1s2p RIXS Experiments of 3d2 Systems

Experimental 1s2p RIXS spectra of 3d2 CrO2 have been published by Zimmermann et al. [9]. Analysis of the magnetic circular dichroism (MCD) of the 1s2p RIXS revealed that the pre-edge structure that is visible in the 1s2p RIXS plane is dominated by a nonlocal contribution. The MCD data reveals the real quadrupole peak, but due to its low intensity and limited resolution, it does not allow to compare the quadrupole peak to the spectra calculated in Figure 26.


## 5.3. 3d3 Ground State System, for Example, Cr3+

The electronic configurations involved for a3d3 ion are summarised below:
(46)For3d3 systems such as Cr3+, the initial state configuration 1s22p63d3 (120 microstates) translates in Oh to a 4A2g ground state (4 microstates). With 3d SOC (S=3/2→⊗F3/2g), the ground state of a 3d2 ion is thus 4A2gF3/2g (Oh) (Table 5).The intermediate state terms (420 microstates) contain spin sextets, quartets, and doublets and orbital momentum ranging from S to I. They split in the Oh crystal field as the Tanabe-Sugano diagram of 3d4 systems. Those terms, neglecting 3d spin-orbit interaction, branch into multiple Oh terms:
(47)6,4,2T2g,6,4,2Eg,4,2T1g,4,2A2g,and4,2A1gFor 1s XAS, the weak spin-orbit coupling in the 3d shell results in spin-conserving transitions from the ground state4A2g to the spin-quartet intermediate states with quadrupole IRREPs 4T1g and 4Eg:
(48)4A2g→⊗T2g4T1g,4A2g→⊗Eg1sXAS4Eg.This illustrates, though many symmetries exist in the intermediate state, only the transitions into theT1g and Eg symmetries contribute to the spectrum as can be seen in Figure 28.Figure 28
1s XAS projections of a calculated 1s2p RIXS spectrum for Cr3+ (3d3) with ELD.Again, as detailed throughout the examples above for the 2p XES decays, consideration of 3d SOC is again required. Hence, we give the splitting of the two reachable intermediate state terms4T1g and 4Eg due to SOC (S=3/2→⊗F3/2):
(49)4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g,4Eg⊗F3/2g=E1/2g⊕E5/2g⊕F3/2g.Here, it is noteworthy that both IS terms (4T1g and 4Eg) are branching into the same three symmetries: E1/2, E5/2, and F3/2.The final state (1s22p53d4) contains 6⋅210=1260 microstates due to the hole in the 2p shell. The atomic term symbols, obtained by coupling the 3d4 atomic terms with 2P, span from S to K with spin doublets, quartets, and sextets and J values from 1/2 to 15/2. A detailed list can be found in the book Core Level Spectroscopy of Solids [13] (p. 103). The important information is that these J values branch in Oh into the three odd IRREPs E1/2u, F3/2u, andE5/2u.As derived in (49), the 2p XES decays in 1s2p RIXS start from either of the three intermediate state symmetries E1/2g, E5/2g, and F3/2g and can reach E1/2u, E5/2u, and F3/2u final state IRREPs. The transitions are again given in (33), (34), and (35).The direct 2p XAS final state terms are similar though it starts from theF3/2g ground state symmetry.
(50)F3/2g→⊗T1u2pXASE1/2u⊕E5/2u⊕2F3/2u.Altogether, the reachable final state terms are the same in 1s2p RIXS and the direct 2p XAS, the selection rules, and the corresponding transitions yield different results.The selection rules are summarised in the term scheme shown in Figure29.Figure 29
Multielectronic term scheme forCr3+ (3d3) comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted as its meaning is lost in that case.The calculated 1s2p RIXS maps and some CIE slices together with the corresponding 1s XAS forCr3+ (3d3) for a crystal field of 10Dq=1.5eV are shown in Figure 30. As can be seen from the corresponding RIXS maps, from the A2g ground state symmetry, the quadrupole T2g IRREP populates only states with T1g symmetry, while the quadrupole Eg IRREP populates only intermediate states with Eg symmetry. The 4T1g (4H) intermediate state symmetry corresponds to the electronic configuration with four 3d electrons in the t2g orbitals, and as a result, the corresponding final states (between ET≈−5eV and −2.5eV) are reached in the RIXS. These states correspond to spin-quartet final states and are weak in 2p XAS. The first 4Eg IRREP intermediate state (Ein≈−0.6eV) leads to the final states between ET≈−5eV and −2.5eV, which correspond to the eg part of the final states with (t2g)3(eg)1. The other 4Eg intermediate state symmetries at Ein≈2.35eV and 3.5eV lead to higher energy final states, where it is the RIXS spectrum of 4Eg (ET≈3.5eV) that resembles more to 2p XAS. This is consistent with the fact that the atomic term of this intermediate state is similar to the atomic term of the ground state (4F) leading to similar 1s2p RIXS and 2p XAS spectra.Figure 30
Calculated 1s2p RIXS spectrum for3d3Cr3+ for 10Dq=1.5eV.


### 5.3.1. 1s2p RIXS Experiments of 3d3 Systems

Examples for 3d3 systems are Cr3+ and Mn4+. Experimental 1s2p RIXS spectra of a number of Cr3+ 3d3 systems have been published by Frommer et al. [42]. Thomas et al. (unpublished) compare Cr(III)(acac) and Cr2O3 with 1s2p RIXS crystal field multiplet calculations, equivalent to Figure 30. Cr(III)acac is well reproduced by the crystal field multiplet calculations. In the case of Cr2O3, additional intensity due to the nonlocal peak overlaps with the second pre-edge peaks in the 1s XAS. Experimental 1s2p RIXS spectra of MnO2 have been published by Glatzel et al. [43]. Mn4+ is a covalent system dominated by charge transfer. As such, the 1s2p RIXS is dominated by fluorescence and the 1s2p RIXS resonances calculated here are not visible as separate features.


## 5.3.1. 1s2p RIXS Experiments of 3d3 Systems

Examples for 3d3 systems are Cr3+ and Mn4+. Experimental 1s2p RIXS spectra of a number of Cr3+ 3d3 systems have been published by Frommer et al. [42]. Thomas et al. (unpublished) compare Cr(III)(acac) and Cr2O3 with 1s2p RIXS crystal field multiplet calculations, equivalent to Figure 30. Cr(III)acac is well reproduced by the crystal field multiplet calculations. In the case of Cr2O3, additional intensity due to the nonlocal peak overlaps with the second pre-edge peaks in the 1s XAS. Experimental 1s2p RIXS spectra of MnO2 have been published by Glatzel et al. [43]. Mn4+ is a covalent system dominated by charge transfer. As such, the 1s2p RIXS is dominated by fluorescence and the 1s2p RIXS resonances calculated here are not visible as separate features.


## 5.4. 3d4 Ground State System, for Example, Mn3+

The electronic configurations involved in this example are summarised below:(51)This is the first case in which the crystal field splitting energy (10Dq) and the pairing energy are competing. Thus, as shortly discussed in the introduction notes for Section 5, the ground state can have a high spin or low spin configuration.


### 5.4.1. 3d4 System in High Spin (Quintet Ground State)

The initial state electronic configuration is1s23d4 (210 microstates) having in high spin the atomic ground state term 5D. In the Oh crystal field, the 5D ground state splits into the 5Egt2g2eg2 and 5T2gt2g3eg1 terms, where 5Eg is the lowest in energy. The inclusion of spin-orbit interaction S=2→⊗Eg⊕T2g yields the following splittings in the initial state:
(52)5Eg⊗Eg⊕T2g=A1g⊕A2g⊕Eg⊕T1g⊕T2g.The ground state5Eg splits due to 3d SOC into several symmetries from which the term A1g is the lowest in energy; hence, the ground state is 5EgA1g (Oh).The intermediate state with 1s13d5 (2⋅252=504 microstates) corresponds to the atomic terms of 3d5 coupled with the 1s core hole. This gives spin multiplicities ranging from 1 to 7. Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d5. Because the 1s XAS is spin-conserving, only quintet intermediate states are reached in the absorption from the quintet high spin ground state (5Eg). The atomic quintet terms are 5S, 5D, 5F, 5G, and 5P which correspond to the crystal field terms 5A1g, 5A2g, 5Eg, 5T1g, and 5T2g.Starting from the5Eg ground state term, Table 3 gives the reachable intermediate state symmetries:
(53)5Eg→⊗T2g5T1g⊕5T2g,(54)5Eg⏟GS→⊗Eg1sXAS5A1g⊕5A2g⊕5Eg⏟AccessibleISinhighspin.The result is that all quintet intermediate state symmetries can be reached in 1s XAS. This is confirmed in Figure31 where in high spin only terms with a quintet component contribute to the spectrum. Furthermore, it can be seen that the high spin configuration is maintained up to a crystal field of 10Dq≈2.8eV.Figure 31
1s XAS projection of a calculated 1s2p RIXS map forMn3+ (3d4) with the corresponding ELD. The high spin to low spin transition at approximately 10Dq≈2.85eV can be recognised. In fact, there are two HS → LS transitions: in the ground and the intermediate state.The inclusion of the 3d spin-orbit interactionS=2→⊗E⊕T2 yields the corresponding splittings in the intermediate state:
(55)5A1g⊗Eg⊕T2g=Eg⊕T2g,5A2g⊗Eg⊕T2g=Eg⊕T1g,5Eg⊗Eg⊕T2g=A1g⊕A2g⊕Eg⊕T1g⊕T2g,5T1g⊗Eg⊕T2g=T1g⊕T2g⊕A2g⊕Eg⊕T1g⊕T2g,5T2g⏟ISterms⊗Eg⊕T2g=T1g⊕T2g⊕A1g⊕Eg⊕T1g⊕T2g⏟IStermsincluding3dSOC.Subsequently, the 2p XES decays can start from any of these terms.The final state (1s22p53d5) contains 6⋅252=1512 microstates due to the hole in the 2p shell spanning J from 0 to 8. In an Oh crystal field, these terms branch into the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. In other words, all even symmetries exist in the final state.The reachable final state terms in 2p XES are found again with the dipole IRREPT1u as summarised in (28), (29), (30), (31), and (32). In short, all existing final state symmetries are also reachable in the 2p XES decays in 1s2p RIXS as illustrated in Figure 32.Figure 32
Multielectronic term scheme forMn3+ (3d4) in high spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.The situation is very different for the direct 2p XAS. Starting from theA1g total symmetry IRREP term and using the T1u dipole IRREP, only final states with T1u symmetry are reachable.The calculated 1s2p RIXS, separated into the contributions via the quadrupole IRREPT2g and Eg, together with some slices is shown in Figure 33.Figure 33
Calculated 1s2p RIXS map forMn3+ (3d4) with a crystal field for 10Dq=1.5eV reflecting the high spin case. As can be seen by the two RIXS maps (top row, left and middle), the contributions are in agreement with the predicted symmetries found in (51) and (52).A comparison of the RIXS maps in Figure33 with the results found in expressions (53) and (54) confirms the 1s XAS selection rules. As predicted in expression (53), the quadrupole IRREP T2g populates the intermediate state symmetries 5T1g and 5T2g. While expression (54) showed for the quadrupole IRREP Eg, only the intermediate state symmetries 5A1g, 5A2g, and 5Eg can be reached.


### 5.4.2. 3d4 in Low Spin (Triplet Ground State)

For large crystal field energies (10Dq), the initial state electronic configuration 1s23d4 has a low spin (LS) ground state. The atomic Russel-Saunders term 3D translates into a 3T1g ground state with 9 microstates. The inclusion of 3d SOC (S=1→⊗T1g) yields the initial state splittings:
(56)3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g.The ground state3T1g splits due to 3d SOC into several symmetries from which the term A1g is the lowest in energy; hence, the low spin ground state is 3T1gA1g (Oh).The intermediate state electron configuration1s13d5 is the same as described above in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the initial state (3T1g) being a spin-triplet, the spin-conserving 1s XAS only reaches spin-triplet intermediate state symmetries:
(57)3T1g→⊗T2g3A2g⊕3Eg⊕3T1g⊕3T2g,3T1g⏟GS→⊗Eg1sXAS3T1g⊕3T2g⏟AccessiblelowspinIS.Here, all intermediate state terms, exceptA1g, are reachable in 1s XAS. The exclusion of the orbital symmetry term A1g, due to the different ground state term, is a first notable difference with respect to the high spin case. The corresponding 1s XAS projections for the low spin case are shown in Figure 34.Figure 34
1s XAS projection of a calculated 1s2p RIXS map forMn3+ (3d4) in low spin.Applying the 3d SOC (S=1→⊗T1g) to the reachable intermediate state terms yields all even total angular momentum IRREPs.
(58)3A2g⊗T1g=T2g,3Eg⊗T1g=T1g⊕T2g,3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g,3T2g⊗T1g=A2g⊕Eg⊕T1g⊕T2g.The final states (1s22p53d5) are again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u.For the 2p XES decay, we find again from expressions (28), (29), (30), (31), and (32) that all even symmetries (A1u, A2u, Eu, T1u, and T2u) can be reached. In 2p XAS, starting from the A1g total symmetry term and using the T1u IRREP, we find that only T1u final state IRREPs can be reached. The selection rules are summarised in the combined term scheme in Figure 35.Figure 35
Multielectronic term scheme forMn3+ (3d4) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.An example calculation for the low spin case (10Dq=3.9eV) of Mn3+ (3d4) is shown in Figure 36.Figure 36
Calculation of 1s2p RIXS and 2p XAS forMn3+ (3d4) for 10Dq=3.9eV (low spin).This example is the first case here in which the crystal field energy (10Dq) and the pairing energy are competing to induce a low spin configuration for sufficiently large values of 10Dq. In the present case of Mn3+ (3d4), we see the high spin to low spin transition at around 10Dq≈2.85eV as is observable in the 1s XAS and 2p XES projections (Figures 31 and 37, resp.).Figure 37
2p XES projections of a calculated 1s2p RIXS spectrum forMn3+ (3d4). Note the HS-LS transition at approximately 10Dq≈2.85eV. In fact, there are two HS → LS transitions, in the GS and in the IS, which are not necessarily occurring at the same 10Dq value.One difference between the low spin and high spinMn3+ is the spin state of the ground state. The result is that in the HS, the 1s XAS absorption reaches the spin-quintet IS while in the LS case, it reaches the spin-triplet IS. This results in different final states being probed regardless of the effect of the crystal field. In the low spin case, the intermediate state is separating the two quadrupole components T2g and Eg (Figure 26 top row). Although the projected 1s2p RIXS and the 2p XAS present close intensities across the final states, the calculations show that by choosing the 1s2p RIXS incident energy Ein, one can select specific intermediate states.We observe that the quadrupoleT2g IREEP can reach only one 3T2g (3I) intermediate state symmetry, while it cannot be reached with the quadrupole Eg operator. This is consistent with the corresponding electron configuration of the 3T2g IRREP in the 3d shell: t2g5eg0.The quadrupole Eg IRREP enables to reach only the intermediate state spin-triplets at higher energy. Consequently, it appears that in the case of low spin Mn3+, depending on the RIXS intermediate states that are probed, one can enhance either the transitions into the t2g orbitals of the final states or the transitions into the eg orbitals.


### 5.4.3. 1s2p RIXS Experiments of 3d4 Systems

Experimental 1s2p RIXS spectra of a number ofMn3+ 3d4 systems have been published by Glatzel et al. [43] for Mn2O3 and also for a series of molecular complexes that serve as reference systems for photosystem II. The pre-edge 1s2p RIXS resonances can be separated well from the main peak. The 1s2p RIXS plane of Mn(III)acac [43] is similar to the calculated plane in Figure 36.


## 5.4.1. 3d4 System in High Spin (Quintet Ground State)

The initial state electronic configuration is1s23d4 (210 microstates) having in high spin the atomic ground state term 5D. In the Oh crystal field, the 5D ground state splits into the 5Egt2g2eg2 and 5T2gt2g3eg1 terms, where 5Eg is the lowest in energy. The inclusion of spin-orbit interaction S=2→⊗Eg⊕T2g yields the following splittings in the initial state:
(52)5Eg⊗Eg⊕T2g=A1g⊕A2g⊕Eg⊕T1g⊕T2g.The ground state5Eg splits due to 3d SOC into several symmetries from which the term A1g is the lowest in energy; hence, the ground state is 5EgA1g (Oh).The intermediate state with 1s13d5 (2⋅252=504 microstates) corresponds to the atomic terms of 3d5 coupled with the 1s core hole. This gives spin multiplicities ranging from 1 to 7. Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d5. Because the 1s XAS is spin-conserving, only quintet intermediate states are reached in the absorption from the quintet high spin ground state (5Eg). The atomic quintet terms are 5S, 5D, 5F, 5G, and 5P which correspond to the crystal field terms 5A1g, 5A2g, 5Eg, 5T1g, and 5T2g.Starting from the5Eg ground state term, Table 3 gives the reachable intermediate state symmetries:
(53)5Eg→⊗T2g5T1g⊕5T2g,(54)5Eg⏟GS→⊗Eg1sXAS5A1g⊕5A2g⊕5Eg⏟AccessibleISinhighspin.The result is that all quintet intermediate state symmetries can be reached in 1s XAS. This is confirmed in Figure31 where in high spin only terms with a quintet component contribute to the spectrum. Furthermore, it can be seen that the high spin configuration is maintained up to a crystal field of 10Dq≈2.8eV.Figure 31
1s XAS projection of a calculated 1s2p RIXS map forMn3+ (3d4) with the corresponding ELD. The high spin to low spin transition at approximately 10Dq≈2.85eV can be recognised. In fact, there are two HS → LS transitions: in the ground and the intermediate state.The inclusion of the 3d spin-orbit interactionS=2→⊗E⊕T2 yields the corresponding splittings in the intermediate state:
(55)5A1g⊗Eg⊕T2g=Eg⊕T2g,5A2g⊗Eg⊕T2g=Eg⊕T1g,5Eg⊗Eg⊕T2g=A1g⊕A2g⊕Eg⊕T1g⊕T2g,5T1g⊗Eg⊕T2g=T1g⊕T2g⊕A2g⊕Eg⊕T1g⊕T2g,5T2g⏟ISterms⊗Eg⊕T2g=T1g⊕T2g⊕A1g⊕Eg⊕T1g⊕T2g⏟IStermsincluding3dSOC.Subsequently, the 2p XES decays can start from any of these terms.The final state (1s22p53d5) contains 6⋅252=1512 microstates due to the hole in the 2p shell spanning J from 0 to 8. In an Oh crystal field, these terms branch into the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. In other words, all even symmetries exist in the final state.The reachable final state terms in 2p XES are found again with the dipole IRREPT1u as summarised in (28), (29), (30), (31), and (32). In short, all existing final state symmetries are also reachable in the 2p XES decays in 1s2p RIXS as illustrated in Figure 32.Figure 32
Multielectronic term scheme forMn3+ (3d4) in high spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.The situation is very different for the direct 2p XAS. Starting from theA1g total symmetry IRREP term and using the T1u dipole IRREP, only final states with T1u symmetry are reachable.The calculated 1s2p RIXS, separated into the contributions via the quadrupole IRREPT2g and Eg, together with some slices is shown in Figure 33.Figure 33
Calculated 1s2p RIXS map forMn3+ (3d4) with a crystal field for 10Dq=1.5eV reflecting the high spin case. As can be seen by the two RIXS maps (top row, left and middle), the contributions are in agreement with the predicted symmetries found in (51) and (52).A comparison of the RIXS maps in Figure33 with the results found in expressions (53) and (54) confirms the 1s XAS selection rules. As predicted in expression (53), the quadrupole IRREP T2g populates the intermediate state symmetries 5T1g and 5T2g. While expression (54) showed for the quadrupole IRREP Eg, only the intermediate state symmetries 5A1g, 5A2g, and 5Eg can be reached.


## 5.4.2. 3d4 in Low Spin (Triplet Ground State)

For large crystal field energies (10Dq), the initial state electronic configuration 1s23d4 has a low spin (LS) ground state. The atomic Russel-Saunders term 3D translates into a 3T1g ground state with 9 microstates. The inclusion of 3d SOC (S=1→⊗T1g) yields the initial state splittings:
(56)3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g.The ground state3T1g splits due to 3d SOC into several symmetries from which the term A1g is the lowest in energy; hence, the low spin ground state is 3T1gA1g (Oh).The intermediate state electron configuration1s13d5 is the same as described above in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the initial state (3T1g) being a spin-triplet, the spin-conserving 1s XAS only reaches spin-triplet intermediate state symmetries:
(57)3T1g→⊗T2g3A2g⊕3Eg⊕3T1g⊕3T2g,3T1g⏟GS→⊗Eg1sXAS3T1g⊕3T2g⏟AccessiblelowspinIS.Here, all intermediate state terms, exceptA1g, are reachable in 1s XAS. The exclusion of the orbital symmetry term A1g, due to the different ground state term, is a first notable difference with respect to the high spin case. The corresponding 1s XAS projections for the low spin case are shown in Figure 34.Figure 34
1s XAS projection of a calculated 1s2p RIXS map forMn3+ (3d4) in low spin.Applying the 3d SOC (S=1→⊗T1g) to the reachable intermediate state terms yields all even total angular momentum IRREPs.
(58)3A2g⊗T1g=T2g,3Eg⊗T1g=T1g⊕T2g,3T1g⊗T1g=A1g⊕Eg⊕T1g⊕T2g,3T2g⊗T1g=A2g⊕Eg⊕T1g⊕T2g.The final states (1s22p53d5) are again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u.For the 2p XES decay, we find again from expressions (28), (29), (30), (31), and (32) that all even symmetries (A1u, A2u, Eu, T1u, and T2u) can be reached. In 2p XAS, starting from the A1g total symmetry term and using the T1u IRREP, we find that only T1u final state IRREPs can be reached. The selection rules are summarised in the combined term scheme in Figure 35.Figure 35
Multielectronic term scheme forMn3+ (3d4) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.An example calculation for the low spin case (10Dq=3.9eV) of Mn3+ (3d4) is shown in Figure 36.Figure 36
Calculation of 1s2p RIXS and 2p XAS forMn3+ (3d4) for 10Dq=3.9eV (low spin).This example is the first case here in which the crystal field energy (10Dq) and the pairing energy are competing to induce a low spin configuration for sufficiently large values of 10Dq. In the present case of Mn3+ (3d4), we see the high spin to low spin transition at around 10Dq≈2.85eV as is observable in the 1s XAS and 2p XES projections (Figures 31 and 37, resp.).Figure 37
2p XES projections of a calculated 1s2p RIXS spectrum forMn3+ (3d4). Note the HS-LS transition at approximately 10Dq≈2.85eV. In fact, there are two HS → LS transitions, in the GS and in the IS, which are not necessarily occurring at the same 10Dq value.One difference between the low spin and high spinMn3+ is the spin state of the ground state. The result is that in the HS, the 1s XAS absorption reaches the spin-quintet IS while in the LS case, it reaches the spin-triplet IS. This results in different final states being probed regardless of the effect of the crystal field. In the low spin case, the intermediate state is separating the two quadrupole components T2g and Eg (Figure 26 top row). Although the projected 1s2p RIXS and the 2p XAS present close intensities across the final states, the calculations show that by choosing the 1s2p RIXS incident energy Ein, one can select specific intermediate states.We observe that the quadrupoleT2g IREEP can reach only one 3T2g (3I) intermediate state symmetry, while it cannot be reached with the quadrupole Eg operator. This is consistent with the corresponding electron configuration of the 3T2g IRREP in the 3d shell: t2g5eg0.The quadrupole Eg IRREP enables to reach only the intermediate state spin-triplets at higher energy. Consequently, it appears that in the case of low spin Mn3+, depending on the RIXS intermediate states that are probed, one can enhance either the transitions into the t2g orbitals of the final states or the transitions into the eg orbitals.


## 5.4.3. 1s2p RIXS Experiments of 3d4 Systems

Experimental 1s2p RIXS spectra of a number ofMn3+ 3d4 systems have been published by Glatzel et al. [43] for Mn2O3 and also for a series of molecular complexes that serve as reference systems for photosystem II. The pre-edge 1s2p RIXS resonances can be separated well from the main peak. The 1s2p RIXS plane of Mn(III)acac [43] is similar to the calculated plane in Figure 36.


## 5.5. 3d5 Ground State System, for Example, Fe3+

In this part, we present the elementary differences for the case of anFe3+ ion (3d5) (Mn2+ analogue). First, we gather the relevant information analogue to the previous cases which are again summarised in a combining term scheme illustrating the transition paths in 1s2p RIXS (1s XAS, 2p XES) and 2p XAS. The electronic configurations involved in this case are summarised below:
(59)Also in3d5 ground state systems like Fe3+ or Mn2+, high spin to low spin transitions can be observed. Thus, for large values of 10Dq, the ground state changes from 6A1g for t2g3eg2 to 2T2g for t2g5eg0.


### 5.5.1. Fe3+ (3d5) in High Spin (Sextet Ground State)

The initial state configuration1s23d5 (252 microstates) has in high spin the ground state term 6S (Table 1) branching in Oh into a 6A1g symmetry (6 microstates). The spin-orbit interaction S=5/2→⊗F3/2g⊕E5/2g yields the following splittings of the ground state term:
(60)6A1g⊗F3/2g⊕E5/2g=F3/2g⊕E5/2g.Here, the termF3/2g lies the lowest in energy; thus, the ground state is 6A1gF3/2g (Oh). However, the splitting is small (few meV) and thus also the E52g term would contribute weakly in a measurement.The intermediate state with1s13d6 (2⋅210=420 microstates) gives the atomic terms with spin multiplicity from 0 to 6 and orbital momentum from S to I. The Oh crystal field splits these terms analogue to the classic Tanabe-Sugano diagram of 3d6. In 1s XAS, due to the weak 3d SOC starting from the 6A1g ground state, only sextet spin terms 6T2g and 6Eg arising from the only atomic sextet spin term 6D are reached (Table 3):
(61)6A1g→⊗T2g6T2g,6A1g⏟GS→⊗Eg1sXAS6Eg⏟ReachableIS.The corresponding 1s XAS projections for a3d5 system with 10Dq between 0.1 eV and 4.2 eV are shown in Figure 38. It is noteworthy that this situation presents some similarities with the case of 3d0 (Ti4+). In both cases, the excited 1s electron can go into either the t2g orbitals or the eg orbitals. The drastic difference is that for 3d5, all spin-up orbitals are full in the total symmetric IRREP 6A1g (Oh).Figure 38
1s XAS projections forFe3+ (3d5) with the corresponding ELD. Notice the high spin to low spin transitions in the ground and intermediate states between 10Dq≈3.0eV and 3.3 eV.For the 2p XES decay, spin-orbit interaction is again crucial; hence, we include 3d spin-orbit couplingS=5/2→⊗F3/2g⊕E5/2g for the reachable intermediate state terms:
(62)6T2g⊗F3/2g⊕E5/2g=E1/2g⊕E5/2g⊕2F3/2g⊕E1/2g⊕F3/2g,6Eg⏟ReachableIS⊗F3/2g⊕E5/2g=E1/2g⊕E5/2g⊕F3/2g⊕F3/2g.Due to the spin (S=5/2) and the resulting splittings, all three odd IRREPs (E1/2g, E5/2g, and F3/2g) appear more than once. Thus, the2p XES can decay from any of the three IRREPs E1/2g, E5/2g, or F3/2g. As given in (33), (34), and (35), the dipole IRREP T1u also enables to reach all three odd symmetries E1/2u, E5/2u, and 2F3/2u in the final state.The direct 2p XAS on the other hand starts from theF3/2g ground state term; hence, the reachable final state symmetries are identical to those found in (35). In other words, the direct 2p XAS reaches the same final states as the 2p XES decays from the F3/2g intermediate state symmetry. The resulting overall selection rules for 3d5 in high spin are summarised in the combined term scheme in Figure 39.Figure 39
Multielectronic term scheme forFe3+ 3d5 high spin comparing the1s2p RIXS and direct 2p XAS pathways. Where SOC is included (S=5/2), the spin multiplicity is omitted.The calculated 1s2p RIXS maps shown in Figure40 confirm the selection rules as summarised above. The two RIXS maps corresponding to the two quadrupole IRREPs T2g and Eg (Figure 40 top row, left and middle) confirm that the intermediate states are probed selectively according to the quadrupole transition operator. This means that the eg and t2g components of the final state can, via the intermediate state, be selectively probed in 1s2p RIXS.Figure 40
Calculated 1s2p RIXS spectrum forFe3+ (3d5) for 10Dq=1.5eV (high spin).This example is interesting, because the 1s XAS transitions are identical to theTi4+ (3d0) case. The ground state is totally symmetric with A1g; thus, the quadrupole IRREPs T2g and Eg populate the corresponding intermediate symmetries (T2g and Eg). However, there are two important differences: (i) for the high spin configuration of 3d5, all spin-up orbitals are full; thus, only the spin-down electrons can populate the remaining holes in the 3d shell. And (ii) the final state electron configuration 2p53d6 of Fe3+ has, due to the six electrons in the 3d orbitals, 6⋅210=1260 microstates. This translates into a very rich and dense stick spectrum shown below the slices in Figure 40.Finally, we note that the matrix elements of the 2p XAS appear very similar to those of the resonant 2p XES decays occurring from the6Eg intermediate state, while the total 2p XES shown as integrated incident energy (IIE) shows notable differences when compared to the 2p XAS due to the population from the intermediate state symmetries.


### 5.5.2. Fe3+ (3d5) in Low Spin (Doublet Ground State)

The initial state electron configuration3d5 translates in low spin to t2g5eg0 with the atomic ground state term 2D (10 microstates). In Oh, this translates into a 2T2g ground state term with 6 microstates. The inclusion of 3d spin-orbit interaction S=1/2→⊗E1/2g yields the following splitting:
(63)2T2g⊗E1/2g=E5/2g⊕F3/2g.Here, the termE5/2g lies the lowest in energy; thus, the ground state is 2T2gE5/2g (Oh).The intermediate state configuration1s13d6 is still the same as in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the ground state being a spin doublet (2T2g), the spin-conserving 1s XAS only reaches spin doublet intermediate state IRREPs (Table 3):
(64)2T2g→⊗T2g2A1g⊕2Eg⊕2T1g⊕2T2g,2T2g⏟GS→⊗Eg1sXAS2T1g⊕2T2g⏟ReachableISlowspin.This means that from the2T2g ground state the quadrupole 1s XAS, as shown in Figure 41, can reach all even intermediate state symmetries, except any A2g IRREPs.Figure 41
Low spin 1s XAS projections (10Dq=3.3eV to 4.2eV) for Fe3+ (3d5) with ELD. In the single electron picture the 3d shell has either t2g6eg0 for 2A1g or t2g5eg1 for 4T1g and 4T2g, respectively.The reachable intermediate state terms split due to the 3d spin-orbit interactionS=1/2→⊗E1/2g:
(65)2A1g⊗E1/2g=E1/2g,2Eg⊗E1/2g=F3/2g,2T1g⊗E1/2g=E1/2g⊕F3/2g,2T2g⊗E1/2g=E5/2g⊕F3/2g.We find all three half-valued terms from which the decays in 1s2p RIXS can occur.The final state (1s22p53d6) is again identical to the high spin case regardless of the crystal field effects, offering the three total symmetry IRREPs E1/2u, E5/2u, andF3/2u. As given in (33), (34), and (35), the decay paths are formally the same as in the high spin case reaching all three odd symmetry terms. The differences in the spectra therefore relate to the difference of the intermediate state population via the first 1s XAS step. Again, the intermediate state plays the decisive role.For 2p XAS, starting from theE5/2g ground state, only two final state terms are reachable:
(66)E5/2g⏟GS→⊗T1u2pXASE1/2u⊕F3/2u⏟AccessibleFSterms.The results of the selection rules as discussed above are summarised in the combined term scheme shown in Figure42.Figure 42
Multielectronic term scheme forFe3+ (3d5) low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is considered, the spin multiplicity was omitted.The calculated 1s2p RIXS map for a low spin case (10Dq=3.9eV) is given in Figure 43.Figure 43
Calculated 1s2p RIXS spectrum forFe3+ (3d5) for 10Dq=3.9eV (low spin).In the left RIXS map in Figure43(a), it appears that the quadrupole IRREP T2g enables to reach only the first fully symmetric 2A1g intermediate state symmetry. This term corresponds in the single electron picture to a t2g6eg0 configuration of the 3d shell where all the t2g orbitals occupied. The spin doublet emerges solely from the core hole in the 1s shell.The 2p XES decay reaches two final states corresponding to thet2g orbital full and a hole in the 2p shell, relating to one in the L3 and one in the L2. In the multielectronic picture, these two peaks correspond to the 2A1g⊗2P final states (see Figures 43 and 44). It is noteworthy that this peak is not reached in the L2-edge via the 2p XAS. This can be explained in the single electron picture of the initial state, which contains only one hole in the t2g orbitals. The L2-edge does not get a peak because it would require a spin-flip to fill the t2g hole with an electron from the 2p shell. In 1s2p RIXS, these two peaks arise only from the decay of the 2p shell electron into the 1s shell, which is possible for both L3 and L2. The higher energy range of the final states is reached via the intermediate state probed by the quadrupole Eg IRREP (1s XAS operator).Figure 44
2p XES projections forFe3+ (3d5) for 10Dq=0.1eV to 4.2eV. The high spin to low spin transition between 10Dq≈3.0eV and 3.3 eV also induces visible changes in the 2p XES. The most notable is the appearance of the two peaks related to the first 2A1g intermediate state symmetry around ET<−5eV and ET≈7eV.


### 5.5.3. 1s2p RIXS Experiments of 3d5 Systems

Examples of3d5 systems include Mn2+ and Fe3+. Experimental 1s2p RIXS spectra of MnO have been published by Kas et al. [39]. Compared with the calculations in Figure 40, the intensity ratio of the T2g and the Eg peak is inverted in the experiment, with the Eg peak at [6540, 638] eV having the largest intensity. The same observation can be made for the Fe3+ systems Fe2O3 and iron impurities in MgO [14]. The tetrahedral Fe3+ contains a single 1s2p RIXS feature, reproduced by the calculations. 1s2p RIXS of a series of iron minerals has been published by Vercamer in his PhD thesis [44]. The 1s2p RIXS spectrum of octahedral low spin compounds Fe(III)(tacn) and Fe(III)cyanide has been measured and reproduced in multiplet calculations by Lundberg et al. [45].


## 5.5.1. Fe3+ (3d5) in High Spin (Sextet Ground State)

The initial state configuration1s23d5 (252 microstates) has in high spin the ground state term 6S (Table 1) branching in Oh into a 6A1g symmetry (6 microstates). The spin-orbit interaction S=5/2→⊗F3/2g⊕E5/2g yields the following splittings of the ground state term:
(60)6A1g⊗F3/2g⊕E5/2g=F3/2g⊕E5/2g.Here, the termF3/2g lies the lowest in energy; thus, the ground state is 6A1gF3/2g (Oh). However, the splitting is small (few meV) and thus also the E52g term would contribute weakly in a measurement.The intermediate state with1s13d6 (2⋅210=420 microstates) gives the atomic terms with spin multiplicity from 0 to 6 and orbital momentum from S to I. The Oh crystal field splits these terms analogue to the classic Tanabe-Sugano diagram of 3d6. In 1s XAS, due to the weak 3d SOC starting from the 6A1g ground state, only sextet spin terms 6T2g and 6Eg arising from the only atomic sextet spin term 6D are reached (Table 3):
(61)6A1g→⊗T2g6T2g,6A1g⏟GS→⊗Eg1sXAS6Eg⏟ReachableIS.The corresponding 1s XAS projections for a3d5 system with 10Dq between 0.1 eV and 4.2 eV are shown in Figure 38. It is noteworthy that this situation presents some similarities with the case of 3d0 (Ti4+). In both cases, the excited 1s electron can go into either the t2g orbitals or the eg orbitals. The drastic difference is that for 3d5, all spin-up orbitals are full in the total symmetric IRREP 6A1g (Oh).Figure 38
1s XAS projections forFe3+ (3d5) with the corresponding ELD. Notice the high spin to low spin transitions in the ground and intermediate states between 10Dq≈3.0eV and 3.3 eV.For the 2p XES decay, spin-orbit interaction is again crucial; hence, we include 3d spin-orbit couplingS=5/2→⊗F3/2g⊕E5/2g for the reachable intermediate state terms:
(62)6T2g⊗F3/2g⊕E5/2g=E1/2g⊕E5/2g⊕2F3/2g⊕E1/2g⊕F3/2g,6Eg⏟ReachableIS⊗F3/2g⊕E5/2g=E1/2g⊕E5/2g⊕F3/2g⊕F3/2g.Due to the spin (S=5/2) and the resulting splittings, all three odd IRREPs (E1/2g, E5/2g, and F3/2g) appear more than once. Thus, the2p XES can decay from any of the three IRREPs E1/2g, E5/2g, or F3/2g. As given in (33), (34), and (35), the dipole IRREP T1u also enables to reach all three odd symmetries E1/2u, E5/2u, and 2F3/2u in the final state.The direct 2p XAS on the other hand starts from theF3/2g ground state term; hence, the reachable final state symmetries are identical to those found in (35). In other words, the direct 2p XAS reaches the same final states as the 2p XES decays from the F3/2g intermediate state symmetry. The resulting overall selection rules for 3d5 in high spin are summarised in the combined term scheme in Figure 39.Figure 39
Multielectronic term scheme forFe3+ 3d5 high spin comparing the1s2p RIXS and direct 2p XAS pathways. Where SOC is included (S=5/2), the spin multiplicity is omitted.The calculated 1s2p RIXS maps shown in Figure40 confirm the selection rules as summarised above. The two RIXS maps corresponding to the two quadrupole IRREPs T2g and Eg (Figure 40 top row, left and middle) confirm that the intermediate states are probed selectively according to the quadrupole transition operator. This means that the eg and t2g components of the final state can, via the intermediate state, be selectively probed in 1s2p RIXS.Figure 40
Calculated 1s2p RIXS spectrum forFe3+ (3d5) for 10Dq=1.5eV (high spin).This example is interesting, because the 1s XAS transitions are identical to theTi4+ (3d0) case. The ground state is totally symmetric with A1g; thus, the quadrupole IRREPs T2g and Eg populate the corresponding intermediate symmetries (T2g and Eg). However, there are two important differences: (i) for the high spin configuration of 3d5, all spin-up orbitals are full; thus, only the spin-down electrons can populate the remaining holes in the 3d shell. And (ii) the final state electron configuration 2p53d6 of Fe3+ has, due to the six electrons in the 3d orbitals, 6⋅210=1260 microstates. This translates into a very rich and dense stick spectrum shown below the slices in Figure 40.Finally, we note that the matrix elements of the 2p XAS appear very similar to those of the resonant 2p XES decays occurring from the6Eg intermediate state, while the total 2p XES shown as integrated incident energy (IIE) shows notable differences when compared to the 2p XAS due to the population from the intermediate state symmetries.


## 5.5.2. Fe3+ (3d5) in Low Spin (Doublet Ground State)

The initial state electron configuration3d5 translates in low spin to t2g5eg0 with the atomic ground state term 2D (10 microstates). In Oh, this translates into a 2T2g ground state term with 6 microstates. The inclusion of 3d spin-orbit interaction S=1/2→⊗E1/2g yields the following splitting:
(63)2T2g⊗E1/2g=E5/2g⊕F3/2g.Here, the termE5/2g lies the lowest in energy; thus, the ground state is 2T2gE5/2g (Oh).The intermediate state configuration1s13d6 is still the same as in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the ground state being a spin doublet (2T2g), the spin-conserving 1s XAS only reaches spin doublet intermediate state IRREPs (Table 3):
(64)2T2g→⊗T2g2A1g⊕2Eg⊕2T1g⊕2T2g,2T2g⏟GS→⊗Eg1sXAS2T1g⊕2T2g⏟ReachableISlowspin.This means that from the2T2g ground state the quadrupole 1s XAS, as shown in Figure 41, can reach all even intermediate state symmetries, except any A2g IRREPs.Figure 41
Low spin 1s XAS projections (10Dq=3.3eV to 4.2eV) for Fe3+ (3d5) with ELD. In the single electron picture the 3d shell has either t2g6eg0 for 2A1g or t2g5eg1 for 4T1g and 4T2g, respectively.The reachable intermediate state terms split due to the 3d spin-orbit interactionS=1/2→⊗E1/2g:
(65)2A1g⊗E1/2g=E1/2g,2Eg⊗E1/2g=F3/2g,2T1g⊗E1/2g=E1/2g⊕F3/2g,2T2g⊗E1/2g=E5/2g⊕F3/2g.We find all three half-valued terms from which the decays in 1s2p RIXS can occur.The final state (1s22p53d6) is again identical to the high spin case regardless of the crystal field effects, offering the three total symmetry IRREPs E1/2u, E5/2u, andF3/2u. As given in (33), (34), and (35), the decay paths are formally the same as in the high spin case reaching all three odd symmetry terms. The differences in the spectra therefore relate to the difference of the intermediate state population via the first 1s XAS step. Again, the intermediate state plays the decisive role.For 2p XAS, starting from theE5/2g ground state, only two final state terms are reachable:
(66)E5/2g⏟GS→⊗T1u2pXASE1/2u⊕F3/2u⏟AccessibleFSterms.The results of the selection rules as discussed above are summarised in the combined term scheme shown in Figure42.Figure 42
Multielectronic term scheme forFe3+ (3d5) low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is considered, the spin multiplicity was omitted.The calculated 1s2p RIXS map for a low spin case (10Dq=3.9eV) is given in Figure 43.Figure 43
Calculated 1s2p RIXS spectrum forFe3+ (3d5) for 10Dq=3.9eV (low spin).In the left RIXS map in Figure43(a), it appears that the quadrupole IRREP T2g enables to reach only the first fully symmetric 2A1g intermediate state symmetry. This term corresponds in the single electron picture to a t2g6eg0 configuration of the 3d shell where all the t2g orbitals occupied. The spin doublet emerges solely from the core hole in the 1s shell.The 2p XES decay reaches two final states corresponding to thet2g orbital full and a hole in the 2p shell, relating to one in the L3 and one in the L2. In the multielectronic picture, these two peaks correspond to the 2A1g⊗2P final states (see Figures 43 and 44). It is noteworthy that this peak is not reached in the L2-edge via the 2p XAS. This can be explained in the single electron picture of the initial state, which contains only one hole in the t2g orbitals. The L2-edge does not get a peak because it would require a spin-flip to fill the t2g hole with an electron from the 2p shell. In 1s2p RIXS, these two peaks arise only from the decay of the 2p shell electron into the 1s shell, which is possible for both L3 and L2. The higher energy range of the final states is reached via the intermediate state probed by the quadrupole Eg IRREP (1s XAS operator).Figure 44
2p XES projections forFe3+ (3d5) for 10Dq=0.1eV to 4.2eV. The high spin to low spin transition between 10Dq≈3.0eV and 3.3 eV also induces visible changes in the 2p XES. The most notable is the appearance of the two peaks related to the first 2A1g intermediate state symmetry around ET<−5eV and ET≈7eV.


## 5.5.3. 1s2p RIXS Experiments of 3d5 Systems

Examples of3d5 systems include Mn2+ and Fe3+. Experimental 1s2p RIXS spectra of MnO have been published by Kas et al. [39]. Compared with the calculations in Figure 40, the intensity ratio of the T2g and the Eg peak is inverted in the experiment, with the Eg peak at [6540, 638] eV having the largest intensity. The same observation can be made for the Fe3+ systems Fe2O3 and iron impurities in MgO [14]. The tetrahedral Fe3+ contains a single 1s2p RIXS feature, reproduced by the calculations. 1s2p RIXS of a series of iron minerals has been published by Vercamer in his PhD thesis [44]. The 1s2p RIXS spectrum of octahedral low spin compounds Fe(III)(tacn) and Fe(III)cyanide has been measured and reproduced in multiplet calculations by Lundberg et al. [45].


## 5.6. 3d6 Ground State System, for Example, Fe2+

In this part, we present the elementary differences between 1s XAS and 1s2p RIXS for the case ofFe2+ (3d6; Co3+ analogue). As in the previous case(s), first, we gather the relevant informations which are summarised in a combining term scheme illustrating the transition paths in 1s2p RIXS (1s XAS, 2p XES) and 1s XAS. The electronic configurations involved are summarised below:
(67)As in the previous case, the crystal field splitting energy (10Dq) and the pairing energy are in competition in a 3d6 configuration. Thus, for sufficiently high values of 10Dq, the ground state changes from high spin 5T2g for t2g4eg2 to low spin 1A1g for t2g6eg0.


### 5.6.1. Fe2+ (3d6) in High Spin (Quintet Ground State)

The initial state electronic configuration is3d6 (210 microstates) with the atomic ground state term 5D (25 microstates). In an Oh crystal field, the 5D term splits into 5T2g for t2g4eg2 and 5Eg for t2g3eg3. The inclusion of spin-orbit interaction S=2→⊗Eg⊕T2g yields the following splittings:
(68)5T2g⊗Eg⊕T2g=A1g⊕Eg⊕2T1g⊕2T2g.Here, the termT2g lies the lowest in energy; thus, the ground state is 5T2gT2g (Oh).The intermediate state electron configuration1s13d73d7 (2⋅120=240 microstates) yields the atomic terms 1,3,5F, 1,3,5P, 1,3D, 1,3G, and 1,3H (Table 1). Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d7. Because the 1s XAS is spin-conserving, only quintet intermediate states are reached in the absorption, corresponding to the crystal field terms derived from 5F and 5P: 5A2g, 5T2g, 5T1g, and 5T1g, respectively. For 1s XAS, starting from the 5T2g ground state term, the reachable high spin intermediate state terms are
(69)5T2g→⊗T2g5A1g⊕5Eg⊕5T1g⊕5T2g,5T2g⏟GS→⊗Eg1sXAS5T1g⊕5T2g⏟ReachableIShighspin.The reachable intermediate state terms ofFe2+ (3d6) in high spin are therefore 5T1g and 5T2g. In other words, though the 5A1g and 5Eg symmetries are formally reachable in 1s XAS, the intermediate state does not offer this IRREP to be populated. Furthermore, though an 5A2g IRREP exists, the quadrupole selection rules do not allow it to be reachable in 1s XAS (see Figure 45). However, the small 3d SOC may induce some mixing enabling also weak transitions into this IRREP.Figure 45
1s XASprojections of the calculated 1s2p RIXSspectrum forFe2+ (3d6) showing the high spin to low spin transition just above 10Dq≈1.8eV.Adding SOCS=2→⊗Eg⊕T2g to the reachable IS terms leads to the splittings in the IS:
(70)5A1g⊗E⊕T2=E⊕T2,5T1g⊗Eg⊕T2g=T1g⊕T2g⊕A2g⊕Eg⊕T1g⊕T2g,5T2g⊗Eg⊕T2g=T1g⊕T2g⊕A1g⊕Eg⊕T1g⊕T2g.We find the decays in 1s2p RIXS can occur from all integer symmetries (A1g, A2g, Eg, T1g, andT2g).The final state electron configuration1s22p53d7 contains 6⋅120=720 microstates due to the hole in the 2p shell spanning J from 0 to 7. In an Oh crystal field, these terms branch into the five integer total symmetry IRREPs A1g, A2g, Eg, T1g, andT2g. The2p XES decays can occur from any of the populated intermediate state terms; thus, the accessible final state terms are again derived with the dipole T1u IRREP as given in (28), (29), (30), (31), and (32). In short, all “ungerade” integer IRREPs are reachable in the final state of 2p XES.In 2p XAS, starting from theT2g total symmetry IRREP term and using the T1u IRREP, the reachable final states are again identical to the 2p XES final state term reachable from the T2g intermediate state IRREP as given (32).Altogether, this enables to draw the term scheme forFe2+ (3d6) in high spin as shown in Figure 46.Figure 46
Multielectronic term scheme forFe2+ (3d6) (high spin) comparing the 1s2p RIXS and 2p XAS pathways. Where SOC is considered, the spin multiplicity is omitted as its meaning is lost.The result of the calculation for high spinFe2+ (3d6) for 10Dq=0.9eV is shown in Figure 47.Figure 47
Calculated 1s2p RIXS spectra and slices forFe2+ (3d6) for 10Dq=0.9eV (high spin).We note that the quadrupoleT2g IRREP (left RIXS map in Figure 47(a)) is dominated by transitions into the 5T2g intermediate state symmetry, while the quadrupole Eg IRREP (middle RIXS map in Figure 47(a)) leads mostly into the T1g intermediate state IRREPs. As predicted above, the 1s XAS spectra gain their intensity only from two intermediate state symmetries, the 5T1g and 5T2g IRREPs. The 5A2g intermediate state symmetry does not contribute to the spectrum as can be seen in Figure 45.


### 5.6.2. Fe2+ (3d6) in Low Spin (Singlet Ground State)

For a sufficiently large crystal field splitting (10Dq), a 3d6 electronic configuration results in a low spin configuration with all t2g orbitals fully occupied (t2g6eg0). This corresponds to the totally symmetric ground state term 1A1g (1 microstate) derived from the atomic term 1I (Table 1). The inclusion of spin-orbit interaction S=0→⊗A1g gives the total ground state as 1A1gA1g (Oh).The intermediate state electron configuration 1s13d7 is still the same as in the high spin case. However, for high crystal fields (10Dq), the order of the intermediate state energy levels is different in low spin. In the low spin case, the initial state being a spin singlet, the spin-conserving 1s XAS will only reach spin singlet intermediate states (Table 3):
(71)1A1g→⊗T2g1T2g,1A1g⏟GS→⊗Eg1sXAS1Eg⏟ReachableIS.Although there are1T2g and 1Eg IRREPs existing in the intermediate state, only the 1Eg derived from the 1G atomic term can be reached. This can be understood from the Δℒ=±2 quadrupole selection rule, which allows only to reach the 1G atomic term (ℒ=4) from the 1I atomic ground state term (ℒ=6). The quadrupole 1s XAS formally can, according to the selection rules, only reach the 1T2g and 1Eg IRREPs from this term. But finally, the transition to the 1T2g intermediate state symmetry would imply a two-electron transition from t2g6eg0 to t2g5eg2, which has a very low probability. Hence, only the 1Eg IRREP from the 1G atomic term is populated thus adding intensity to the 1s XAS spectrum as visible in Figure 48.Figure 48
1s XAS projections of the calculated 1s2p RIXS spectrum forFe2+ (3d6) showing low spin part above 10Dq≈1.8eV only. It reveals only the Eg symmetry is giving a significant contribution.For the 2p XES decays in 1s2p RIXS, spin-orbit interaction in the intermediate state must be consideredS=0→⊗A1g leading the identities:
(72)1Eg⏟reachableIS⊗A1g=Eg.The final state electron configuration1s22p53d7 is again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. The 2p XES decaying from the populated intermediate state term Eg can reach with the dipole IRREP T1u the final state terms T1u and T2u as given in (30).On the contrary, the 2p XAS, starting from the total symmetricA1g ground state symmetry, can reach only the T1u final state IRREP.This again enables altogether to draw the combined term scheme comparing the two-step 1s2p RIXS with the direct dipole 2p XAS pathways in Figure49.Figure 49
Multielectronic term scheme forFe2+ (3d6) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. We note its similarity to the Ti4+ (3d0) case shown in Figure 15.The resulting calculated 1s2p RIXS forFe2+ (3d6) is shown in Figure 50. As already discussed, the 1s2p RIXS is dominated by the Eg intermediate state symmetry. As can be seen, the quadrupole T2g IRREP has overall a negligible contribution to the total RIXS. From the term scheme in Figure 49, one can furthermore derive that the difference between the integrated 2p XES (IIE) and the direct 2p XAS emerges from the T2u final state IRREP which can only be reached in 1s2p RIXS.Figure 50
Calculated 1s2p RIXS spectrum forFe2+ (3d6) for 10Dq=3.0eV (low spin).


### 5.6.3. 1s2p RIXS Experiments of3d6 Systems

3d6 systems include Fe2+ and Co3+. A large number of experimental 1s2p RIXS spectra of Fe2+ systems have been published. Vercamer measured a large series of Fe2+ minerals in his PhD thesis [44]. The tetrahedral and octahedral Fe2+ oxides/minerals have been studied and reproduced by multiplet calculations [5, 17, 46]. Guo et al. [47] studied low spin Fe2+ cyanide complexes. Lundberg et al. [45] compared these experiments to the tacn complex and reproduced the spectra from multiplet calculations. Leidel et al. [48] studied iron hydrogenase complexes. Vanko et al. [49] studied the time evolution of the 1s2p RIXS planes. The static spectra of the low spin systems Feterpy22+ and Febipy32+ and the laser-excited spectrum of Feterpy22+ show a partial transition to a high spin ground state as is confirmed in the calculations.In case ofCo3+, all measured systems were low spin 1A1g. Vanko et al. [7] compared an isolated Co3+ complex with a series of cobalt oxides. They nicely show the presence of the nonlocal peaks as a function of the Co-O-Co angle.


## 5.6.1. Fe2+ (3d6) in High Spin (Quintet Ground State)

The initial state electronic configuration is3d6 (210 microstates) with the atomic ground state term 5D (25 microstates). In an Oh crystal field, the 5D term splits into 5T2g for t2g4eg2 and 5Eg for t2g3eg3. The inclusion of spin-orbit interaction S=2→⊗Eg⊕T2g yields the following splittings:
(68)5T2g⊗Eg⊕T2g=A1g⊕Eg⊕2T1g⊕2T2g.Here, the termT2g lies the lowest in energy; thus, the ground state is 5T2gT2g (Oh).The intermediate state electron configuration1s13d73d7 (2⋅120=240 microstates) yields the atomic terms 1,3,5F, 1,3,5P, 1,3D, 1,3G, and 1,3H (Table 1). Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d7. Because the 1s XAS is spin-conserving, only quintet intermediate states are reached in the absorption, corresponding to the crystal field terms derived from 5F and 5P: 5A2g, 5T2g, 5T1g, and 5T1g, respectively. For 1s XAS, starting from the 5T2g ground state term, the reachable high spin intermediate state terms are
(69)5T2g→⊗T2g5A1g⊕5Eg⊕5T1g⊕5T2g,5T2g⏟GS→⊗Eg1sXAS5T1g⊕5T2g⏟ReachableIShighspin.The reachable intermediate state terms ofFe2+ (3d6) in high spin are therefore 5T1g and 5T2g. In other words, though the 5A1g and 5Eg symmetries are formally reachable in 1s XAS, the intermediate state does not offer this IRREP to be populated. Furthermore, though an 5A2g IRREP exists, the quadrupole selection rules do not allow it to be reachable in 1s XAS (see Figure 45). However, the small 3d SOC may induce some mixing enabling also weak transitions into this IRREP.Figure 45
1s XASprojections of the calculated 1s2p RIXSspectrum forFe2+ (3d6) showing the high spin to low spin transition just above 10Dq≈1.8eV.Adding SOCS=2→⊗Eg⊕T2g to the reachable IS terms leads to the splittings in the IS:
(70)5A1g⊗E⊕T2=E⊕T2,5T1g⊗Eg⊕T2g=T1g⊕T2g⊕A2g⊕Eg⊕T1g⊕T2g,5T2g⊗Eg⊕T2g=T1g⊕T2g⊕A1g⊕Eg⊕T1g⊕T2g.We find the decays in 1s2p RIXS can occur from all integer symmetries (A1g, A2g, Eg, T1g, andT2g).The final state electron configuration1s22p53d7 contains 6⋅120=720 microstates due to the hole in the 2p shell spanning J from 0 to 7. In an Oh crystal field, these terms branch into the five integer total symmetry IRREPs A1g, A2g, Eg, T1g, andT2g. The2p XES decays can occur from any of the populated intermediate state terms; thus, the accessible final state terms are again derived with the dipole T1u IRREP as given in (28), (29), (30), (31), and (32). In short, all “ungerade” integer IRREPs are reachable in the final state of 2p XES.In 2p XAS, starting from theT2g total symmetry IRREP term and using the T1u IRREP, the reachable final states are again identical to the 2p XES final state term reachable from the T2g intermediate state IRREP as given (32).Altogether, this enables to draw the term scheme forFe2+ (3d6) in high spin as shown in Figure 46.Figure 46
Multielectronic term scheme forFe2+ (3d6) (high spin) comparing the 1s2p RIXS and 2p XAS pathways. Where SOC is considered, the spin multiplicity is omitted as its meaning is lost.The result of the calculation for high spinFe2+ (3d6) for 10Dq=0.9eV is shown in Figure 47.Figure 47
Calculated 1s2p RIXS spectra and slices forFe2+ (3d6) for 10Dq=0.9eV (high spin).We note that the quadrupoleT2g IRREP (left RIXS map in Figure 47(a)) is dominated by transitions into the 5T2g intermediate state symmetry, while the quadrupole Eg IRREP (middle RIXS map in Figure 47(a)) leads mostly into the T1g intermediate state IRREPs. As predicted above, the 1s XAS spectra gain their intensity only from two intermediate state symmetries, the 5T1g and 5T2g IRREPs. The 5A2g intermediate state symmetry does not contribute to the spectrum as can be seen in Figure 45.


## 5.6.2. Fe2+ (3d6) in Low Spin (Singlet Ground State)

For a sufficiently large crystal field splitting (10Dq), a 3d6 electronic configuration results in a low spin configuration with all t2g orbitals fully occupied (t2g6eg0). This corresponds to the totally symmetric ground state term 1A1g (1 microstate) derived from the atomic term 1I (Table 1). The inclusion of spin-orbit interaction S=0→⊗A1g gives the total ground state as 1A1gA1g (Oh).The intermediate state electron configuration 1s13d7 is still the same as in the high spin case. However, for high crystal fields (10Dq), the order of the intermediate state energy levels is different in low spin. In the low spin case, the initial state being a spin singlet, the spin-conserving 1s XAS will only reach spin singlet intermediate states (Table 3):
(71)1A1g→⊗T2g1T2g,1A1g⏟GS→⊗Eg1sXAS1Eg⏟ReachableIS.Although there are1T2g and 1Eg IRREPs existing in the intermediate state, only the 1Eg derived from the 1G atomic term can be reached. This can be understood from the Δℒ=±2 quadrupole selection rule, which allows only to reach the 1G atomic term (ℒ=4) from the 1I atomic ground state term (ℒ=6). The quadrupole 1s XAS formally can, according to the selection rules, only reach the 1T2g and 1Eg IRREPs from this term. But finally, the transition to the 1T2g intermediate state symmetry would imply a two-electron transition from t2g6eg0 to t2g5eg2, which has a very low probability. Hence, only the 1Eg IRREP from the 1G atomic term is populated thus adding intensity to the 1s XAS spectrum as visible in Figure 48.Figure 48
1s XAS projections of the calculated 1s2p RIXS spectrum forFe2+ (3d6) showing low spin part above 10Dq≈1.8eV only. It reveals only the Eg symmetry is giving a significant contribution.For the 2p XES decays in 1s2p RIXS, spin-orbit interaction in the intermediate state must be consideredS=0→⊗A1g leading the identities:
(72)1Eg⏟reachableIS⊗A1g=Eg.The final state electron configuration1s22p53d7 is again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs A1u, A2u, Eu, T1u, and T2u. The 2p XES decaying from the populated intermediate state term Eg can reach with the dipole IRREP T1u the final state terms T1u and T2u as given in (30).On the contrary, the 2p XAS, starting from the total symmetricA1g ground state symmetry, can reach only the T1u final state IRREP.This again enables altogether to draw the combined term scheme comparing the two-step 1s2p RIXS with the direct dipole 2p XAS pathways in Figure49.Figure 49
Multielectronic term scheme forFe2+ (3d6) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. We note its similarity to the Ti4+ (3d0) case shown in Figure 15.The resulting calculated 1s2p RIXS forFe2+ (3d6) is shown in Figure 50. As already discussed, the 1s2p RIXS is dominated by the Eg intermediate state symmetry. As can be seen, the quadrupole T2g IRREP has overall a negligible contribution to the total RIXS. From the term scheme in Figure 49, one can furthermore derive that the difference between the integrated 2p XES (IIE) and the direct 2p XAS emerges from the T2u final state IRREP which can only be reached in 1s2p RIXS.Figure 50
Calculated 1s2p RIXS spectrum forFe2+ (3d6) for 10Dq=3.0eV (low spin).


## 5.6.3. 1s2p RIXS Experiments of3d6 Systems

3d6 systems include Fe2+ and Co3+. A large number of experimental 1s2p RIXS spectra of Fe2+ systems have been published. Vercamer measured a large series of Fe2+ minerals in his PhD thesis [44]. The tetrahedral and octahedral Fe2+ oxides/minerals have been studied and reproduced by multiplet calculations [5, 17, 46]. Guo et al. [47] studied low spin Fe2+ cyanide complexes. Lundberg et al. [45] compared these experiments to the tacn complex and reproduced the spectra from multiplet calculations. Leidel et al. [48] studied iron hydrogenase complexes. Vanko et al. [49] studied the time evolution of the 1s2p RIXS planes. The static spectra of the low spin systems Feterpy22+ and Febipy32+ and the laser-excited spectrum of Feterpy22+ show a partial transition to a high spin ground state as is confirmed in the calculations.In case ofCo3+, all measured systems were low spin 1A1g. Vanko et al. [7] compared an isolated Co3+ complex with a series of cobalt oxides. They nicely show the presence of the nonlocal peaks as a function of the Co-O-Co angle.


## 5.7. 3d7 Ground State System, for Example, Co2+

In this part, we present the elementary differences between 1s2p RIXS and 2p XAS for the case ofCo2+ (3d7). First, we gather the relevant information analogue to the previous cases which are again summarised in the combining term scheme in Figure 39 term scheme illustrating the transition paths involved (Figures 51 and 52). The electronic configurations involved are summarised below:
(73)Figure 51
Multielectronic term scheme for Co2+ (3d7) in high spin comparing the 1s2p RIXS and direct 2p XAS pathways. Also, the spin multiplicity was omitted where the spin-orbit interaction is included.Figure 52
Multielectronic term scheme for aCo2+ ion (3d7) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC is included, the spin multiplicity is omitted.Also, a 3d7 system encounters, for sufficiently high values of 10Dq, a change of the ground state from 4T2g for t2g5eg2 (high spin) to 2Eg for t2g6eg1 (low spin).


### 5.7.1. Co2+ (3d7) in High Spin (Quartet Ground State)

For the initial state configuration3d7 (120 microstates), the high spin ground state term symbol is 4F (Table 1), which branches in Oh into 4T1g, 4T2g, and 4A2g, where the 4T1g term with 12 microstates is the lowest in energy. The inclusion in spin-orbit interaction (S=3/2→⊗F3/2g) yields the subsequent splitting of 4T1g term:
(74)4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2F3/2g.The ground state4T1g splits due to 3d SOC into the three odd symmetries E1/2g, E5/2g, and F3/2g from which the term E1/2g is lowest in energy; hence, the combined ground state is 4T1gE1/2g (Oh).The intermediate state electron configuration is1s13d8 (2⋅45=90 microstates). The atomic terms are 2,4F, 2,4P, 2S, 2D, and 2G (Table 1). Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d8. The spin-conserving 1s XAS reaches only spin quartet intermediate states, which corresponds to the crystal field terms issued from 4F and 4P: 4A2g, 4T2g, 4T1g, and 4T1g, respectively. Starting from the high spin ground state term 4T1g, the two quadrupole IRREPs give the reachable intermediate state terms:
(75)4T1g→⊗T2g4A2g⊕4Eg⊕4T1g⊕4T2g,(76)4T1g⏟GS→⊗Eg1sXAS4T1g⊕4T2g⏟AccessiblelowspinIS.The reachable 1s XAS intermediate state terms in high spin are therefore4A2g, 4T1g, and 4T2g. Here, it shall be noticed that an Eg symmetry would be reachable by the quadrupole selection rules, but the intermediate state does not offer any (quartet) 4Eg symmetries. The 1s XAS projections displayed in Figure 53 confirm these findings.Figure 53
1s XAS projections of a calculated 1s2p RIXS spectrum forCo2+ (3d7) with ELD. It confirms that in high spin only the quartet terms 4,2F and 4,2P contribute to the 1s XAS. The high spin to low spin transition appears at approximately 10Dq≈2.2eV.Applying the 3d spin-orbit interaction (S=3/2→⊗F3/2g) to the three reachable terms yields the splittings in the intermediate state:
(77)4A2g⊗F3/2g=F3/2g,4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g,4T2g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g.The result is that all three half-valued IRREPs appear more than once and hence the 2p XES can decay from any of these terms.The final state (1s22p53d8) contains ⋅6⋅45=270 microstates due to the hole in the 2p shell spanning J from 1/2 to 11/2. In an Oh crystal field, these terms branch into the three total symmetry IRREPs E1/2u, E5/2u, and F3/2u. The 2p XES starts from either E1/2g, E5/2g, or F3/2g, and we find ((33), (34), and (35)) that all three odd terms are also reachable in the 1s2p RIXS final state.The 2p XAS on the other hand, starting from theE1/2g ground state term, can only reach the E1/2u and F3/2u final state terms (Table 4).A summary of the findings above is shown below as a combined term scheme in Figure51.We conclude this part with the corresponding calculations of the 1s2p RIXS forCo2+ (3d7) for a crystal field energy of 10Dq=0.9eV which is shown in Figure 54.Figure 54
Calculated 1s2p RIXS spectrum forCo2+ (3d7) for 10Dq=0.9eV (high spin).It can be noted that the RIXS map for the quadrupoleT2g IRREP (Figure 54(a), left) is dominated by the 1s XAS transitions into the 4A2g intermediate state symmetry which corresponds in the single electron picture to a t2g6eg2 configuration. However, as given in expression (75), it has also contributions into the two T1g intermediate state symmetries though they are weak with respect to the A2g contribution. The stick spectrum here is not as dense as in some of the cases discussed above, and this enables to identify also some of the reachable final state terms. For example, the strong peak at ET≈−4.5eV relating to the decay from the 4A2g intermediate state symmetry corresponds to the two final state terms E1/2u and F3/2u. This can be understood considering that the 4A2g branches due to SOC into a F3/2g intermediate state symmetry, which can reach the above two final state terms via the dipole T1u IRREP.The quadrupole IRREPEg on the other hand shows only contributions into the 4T1g and 4T2g intermediate state symmetries as predicted in expression (76).And finally, it should be noted that the 1s XAS projections in Figure53 show with increasing crystal field energy 10Dq the branching of the 4F into the three IRREPs 4A2, 4T2g, and 4T1g, while the atomic term 4P translates in an Oh crystal field only into a 4T1g symmetry.


### 5.7.2. Co2+ (3d7) in Low Spin (Doublet Ground State)

At high crystal field energies, the3d7 initial state gives the low spin ground state term 2Eg (4 microstates) arising from the atomic term 2F which corresponds to the electronic configuration t2g6eg1. The inclusion of spin-orbit interaction (S=1/2→⊗E1/2g) yields the subsequent splittings:
(78)2Eg⊗E1/2g=F3/2g.Hence, the combined ground state is2EgF3/2g (Oh).The intermediate state configuration 1s13d8 is still the same as in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the initial state being a spin doublet, the spin-conserving 1s XAS only reaches spin doublet intermediate state terms (Table 3), which correspond to the crystal field terms issued from 2S, 2P, 2D, 2F, and 2G and 2A1g, 2A2g, 2Eg, 2T1g, and 2T2g. The quadrupole 1s XAS starts from the low spin ground state term 2Eg revealing the reachable terms:
(79)2Eg→⊗T2g2T1g⊕2T2g,2Eg⏟GS→⊗Eg1sXAS2A1g⊕2A2g⊕2Eg⏟ReachableIS.Thus, according to the quadrupole selection rules, from the2Eg ground state, the 1s XAS transitions can reach all existing spin doublet IRREPs in the intermediate state. However, because in the low spin initial state all t2g orbitals are occupied, 1s XAS transitions can only go into the eg orbitals via the quadrupole Eg IRREP. The transitions into the t2g orbitals with the quadrupole T2g IRREP are less probable as they require a two-electron transition. In other words, the matrix elements of the quadrupole Eg IRREP are larger than those of the quadrupole T2g IRREP. The resulting 1s XAS projections for the low spin case are shown in Figure 55.Figure 55
1s XAS projections of the calculated 1s2p RIXS maps for aCo2+ ion (3d7) with 10Dq=2.4eV to 3.3eV (low spin only).For the 1p XES decays, the inclusion of 3d SOC (S=1/2→⊗E1/2g) is again required, leading to the following splittings of the five reachable terms in the intermediate state:
(80)2A1g⊗E1/2g=E1/2g,2A2g⊗E12g=E5/2g,2Eg⊗E1/2g=F3/2g,2T1g⊗E1/2g=E1/2g⊕F3/2g,2T2g⊗E1/2g=E5/2g⊕F3/2g.Hence, the 2p XES decays in 1s2p RIXS can occur from any of these three total angular momentum IRREPsE1/2g, E5/2g, or F3/2g.The final state electron configuration1s22p53d8 is again identical to the high spin case regardless of the crystal field effects, offering the three total angular momentum IRREPs E1/2u,E5/2u, and F3/2u. For the 2p XES decaying from the populated intermediate state terms, using the T1u IRREP (Table 4) again reveals that all three final state IRREPs are also reachable. Furthermore, in 2p XAS, starting from the F3/2g ground state symmetry, also, all three final state IRREPs are reachable (Table 4).The results according to the selection rules as discussed above are again summarised in a combined term scheme as displayed in Figure52.Again, we conclude with an example calculation for aCo2+ ion (3d7 ground state configuration) with 10Dq=3.0eV reflecting the low spin case as shown in Figure 56.Figure 56
Calculated 1s2p RIXS spectrum forCo2+ (3d7) for 10Dq=3.0eV (low spin).It is noteworthy that the quadrupole IRREPT2g here is very weak (10−3) with respect to the Eg IRREP. This can again be explained with the very small matrix elements for the two-electron transition that is required to create a hole in the t2g orbitals, because the quadrupole IRREP T2g populates the intermediate state symmetries T1g and T2g, which both require a hole in the t2g level.This is also confirmed by the total RIXS map (Figure56(a), right) showing that the overall result is dominated by the transitions into the three intermediate state symmetries 2A2g, 2Eg, and 2A1g, while the 2T1g and 2T2g intermediate state symmetries are negligible due to the reason given above.However, as predicted by the selection rules for the quadrupole IRREPS,T2g and Eg as shown in Figure 56 confirm the predicted reachable terms as found in expression (79).


## 5.7.1. Co2+ (3d7) in High Spin (Quartet Ground State)

For the initial state configuration3d7 (120 microstates), the high spin ground state term symbol is 4F (Table 1), which branches in Oh into 4T1g, 4T2g, and 4A2g, where the 4T1g term with 12 microstates is the lowest in energy. The inclusion in spin-orbit interaction (S=3/2→⊗F3/2g) yields the subsequent splitting of 4T1g term:
(74)4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2F3/2g.The ground state4T1g splits due to 3d SOC into the three odd symmetries E1/2g, E5/2g, and F3/2g from which the term E1/2g is lowest in energy; hence, the combined ground state is 4T1gE1/2g (Oh).The intermediate state electron configuration is1s13d8 (2⋅45=90 microstates). The atomic terms are 2,4F, 2,4P, 2S, 2D, and 2G (Table 1). Using Table 2, one finds the branchings of the atomic states in an Oh crystal field which are analogue to the Tanabe-Sugano diagram of 3d8. The spin-conserving 1s XAS reaches only spin quartet intermediate states, which corresponds to the crystal field terms issued from 4F and 4P: 4A2g, 4T2g, 4T1g, and 4T1g, respectively. Starting from the high spin ground state term 4T1g, the two quadrupole IRREPs give the reachable intermediate state terms:
(75)4T1g→⊗T2g4A2g⊕4Eg⊕4T1g⊕4T2g,(76)4T1g⏟GS→⊗Eg1sXAS4T1g⊕4T2g⏟AccessiblelowspinIS.The reachable 1s XAS intermediate state terms in high spin are therefore4A2g, 4T1g, and 4T2g. Here, it shall be noticed that an Eg symmetry would be reachable by the quadrupole selection rules, but the intermediate state does not offer any (quartet) 4Eg symmetries. The 1s XAS projections displayed in Figure 53 confirm these findings.Figure 53
1s XAS projections of a calculated 1s2p RIXS spectrum forCo2+ (3d7) with ELD. It confirms that in high spin only the quartet terms 4,2F and 4,2P contribute to the 1s XAS. The high spin to low spin transition appears at approximately 10Dq≈2.2eV.Applying the 3d spin-orbit interaction (S=3/2→⊗F3/2g) to the three reachable terms yields the splittings in the intermediate state:
(77)4A2g⊗F3/2g=F3/2g,4T1g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g,4T2g⊗F3/2g=E1/2g⊕E5/2g⊕2⋅F3/2g.The result is that all three half-valued IRREPs appear more than once and hence the 2p XES can decay from any of these terms.The final state (1s22p53d8) contains ⋅6⋅45=270 microstates due to the hole in the 2p shell spanning J from 1/2 to 11/2. In an Oh crystal field, these terms branch into the three total symmetry IRREPs E1/2u, E5/2u, and F3/2u. The 2p XES starts from either E1/2g, E5/2g, or F3/2g, and we find ((33), (34), and (35)) that all three odd terms are also reachable in the 1s2p RIXS final state.The 2p XAS on the other hand, starting from theE1/2g ground state term, can only reach the E1/2u and F3/2u final state terms (Table 4).A summary of the findings above is shown below as a combined term scheme in Figure51.We conclude this part with the corresponding calculations of the 1s2p RIXS forCo2+ (3d7) for a crystal field energy of 10Dq=0.9eV which is shown in Figure 54.Figure 54
Calculated 1s2p RIXS spectrum forCo2+ (3d7) for 10Dq=0.9eV (high spin).It can be noted that the RIXS map for the quadrupoleT2g IRREP (Figure 54(a), left) is dominated by the 1s XAS transitions into the 4A2g intermediate state symmetry which corresponds in the single electron picture to a t2g6eg2 configuration. However, as given in expression (75), it has also contributions into the two T1g intermediate state symmetries though they are weak with respect to the A2g contribution. The stick spectrum here is not as dense as in some of the cases discussed above, and this enables to identify also some of the reachable final state terms. For example, the strong peak at ET≈−4.5eV relating to the decay from the 4A2g intermediate state symmetry corresponds to the two final state terms E1/2u and F3/2u. This can be understood considering that the 4A2g branches due to SOC into a F3/2g intermediate state symmetry, which can reach the above two final state terms via the dipole T1u IRREP.The quadrupole IRREPEg on the other hand shows only contributions into the 4T1g and 4T2g intermediate state symmetries as predicted in expression (76).And finally, it should be noted that the 1s XAS projections in Figure53 show with increasing crystal field energy 10Dq the branching of the 4F into the three IRREPs 4A2, 4T2g, and 4T1g, while the atomic term 4P translates in an Oh crystal field only into a 4T1g symmetry.


## 5.7.2. Co2+ (3d7) in Low Spin (Doublet Ground State)

At high crystal field energies, the3d7 initial state gives the low spin ground state term 2Eg (4 microstates) arising from the atomic term 2F which corresponds to the electronic configuration t2g6eg1. The inclusion of spin-orbit interaction (S=1/2→⊗E1/2g) yields the subsequent splittings:
(78)2Eg⊗E1/2g=F3/2g.Hence, the combined ground state is2EgF3/2g (Oh).The intermediate state configuration 1s13d8 is still the same as in the high spin case; however, the higher crystal field results in different energies for the intermediate state energy levels. In the low spin case, the initial state being a spin doublet, the spin-conserving 1s XAS only reaches spin doublet intermediate state terms (Table 3), which correspond to the crystal field terms issued from 2S, 2P, 2D, 2F, and 2G and 2A1g, 2A2g, 2Eg, 2T1g, and 2T2g. The quadrupole 1s XAS starts from the low spin ground state term 2Eg revealing the reachable terms:
(79)2Eg→⊗T2g2T1g⊕2T2g,2Eg⏟GS→⊗Eg1sXAS2A1g⊕2A2g⊕2Eg⏟ReachableIS.Thus, according to the quadrupole selection rules, from the2Eg ground state, the 1s XAS transitions can reach all existing spin doublet IRREPs in the intermediate state. However, because in the low spin initial state all t2g orbitals are occupied, 1s XAS transitions can only go into the eg orbitals via the quadrupole Eg IRREP. The transitions into the t2g orbitals with the quadrupole T2g IRREP are less probable as they require a two-electron transition. In other words, the matrix elements of the quadrupole Eg IRREP are larger than those of the quadrupole T2g IRREP. The resulting 1s XAS projections for the low spin case are shown in Figure 55.Figure 55
1s XAS projections of the calculated 1s2p RIXS maps for aCo2+ ion (3d7) with 10Dq=2.4eV to 3.3eV (low spin only).For the 1p XES decays, the inclusion of 3d SOC (S=1/2→⊗E1/2g) is again required, leading to the following splittings of the five reachable terms in the intermediate state:
(80)2A1g⊗E1/2g=E1/2g,2A2g⊗E12g=E5/2g,2Eg⊗E1/2g=F3/2g,2T1g⊗E1/2g=E1/2g⊕F3/2g,2T2g⊗E1/2g=E5/2g⊕F3/2g.Hence, the 2p XES decays in 1s2p RIXS can occur from any of these three total angular momentum IRREPsE1/2g, E5/2g, or F3/2g.The final state electron configuration1s22p53d8 is again identical to the high spin case regardless of the crystal field effects, offering the three total angular momentum IRREPs E1/2u,E5/2u, and F3/2u. For the 2p XES decaying from the populated intermediate state terms, using the T1u IRREP (Table 4) again reveals that all three final state IRREPs are also reachable. Furthermore, in 2p XAS, starting from the F3/2g ground state symmetry, also, all three final state IRREPs are reachable (Table 4).The results according to the selection rules as discussed above are again summarised in a combined term scheme as displayed in Figure52.Again, we conclude with an example calculation for aCo2+ ion (3d7 ground state configuration) with 10Dq=3.0eV reflecting the low spin case as shown in Figure 56.Figure 56
Calculated 1s2p RIXS spectrum forCo2+ (3d7) for 10Dq=3.0eV (low spin).It is noteworthy that the quadrupole IRREPT2g here is very weak (10−3) with respect to the Eg IRREP. This can again be explained with the very small matrix elements for the two-electron transition that is required to create a hole in the t2g orbitals, because the quadrupole IRREP T2g populates the intermediate state symmetries T1g and T2g, which both require a hole in the t2g level.This is also confirmed by the total RIXS map (Figure56(a), right) showing that the overall result is dominated by the transitions into the three intermediate state symmetries 2A2g, 2Eg, and 2A1g, while the 2T1g and 2T2g intermediate state symmetries are negligible due to the reason given above.However, as predicted by the selection rules for the quadrupole IRREPS,T2g and Eg as shown in Figure 56 confirm the predicted reachable terms as found in expression (79).


## 5.8. 1s2p RIXS Experiments of3d7 Systems

A300meV resolution 1s2p RIXS experiment on CoO was published by Kurian et al. [50]. The increased resolution revealed new details in the pre-edge fine structure, and it was possible to study the effect of interference effects on the 1s2p RIXS plane.


## 6. Concluding Remarks

We have presented first three didactic cases, being theCu2+ (3d9), Ti4+ (3d0), and Ni2+ (3d8), respectively, for which detailed descriptions were given. For those cases, we included also an atomic calculation without a crystal field to illustrate the effect an octahedral crystal field has. The didactic cases were chosen as these cases have a low number of microstates involved (e.g., having an almost full or almost empty 3d shell). Though the effort would be significantly larger, when needed, one can further separate the 2p XES transitions for the more complex cases discussed in Section 5. For example, calculations with an unrealistic low natural broadening (e.g., 1−2meV) would enable to do a detailed fingerprint comparison with the stick spectra allowing to assign the peaks.Altogether, we illustrated the effects of an octahedral crystal field and the selection rules for an example ion for every3dN ground state case (with N=0,…,9). Here, we provide a few selected examples of calculations. For an even better illustration, one can for example combine the calculated RIXS maps or combine many slices into short movie clips where one or more parameters (e.g., 10Dq or SOC) are scaled. For the crystal field parameter 10Dq, such scaling would relate to the corresponding Tanabe-Sugano diagram which is also shown throughout this paper for the 1s XAS projections.Another important aspect is the mixing of the states induced by spin-orbit or exchange interactions, as well as the crystal field. Although we have assigned labels to many states involved in the transitions, those labels represent only the dominant contribution to each state. In reality, most states are not pure in their nature as it was also mentioned for several cases throughout this paper. As can be seen for example in theCr3+, Mn3+, or Fe3+ cases, the stick spectra show a very dense distribution of states. To give a detailed insight into the structure of the spectra, we have used a rather small Lorentzian broadening, while any Gaussian broadening was omitted for all calculations. Hence, when a realistic natural broadening is assumed and an experimental broadening is also considered, one can hardly separate the transitions visible in one of the complex spectra. Due to the often very small splittings in many cases, the states appear as a band rather than discrete states. The difference between the final states probed in 1s2p RIXS versus 2p XAS is emphasised for each 3dN case.The computational effort is for the complex cases quite time-consuming, especially when aiming for high-resolution spectra which are needed when the broadenings are small. However, this effort is justified as it helps to get a better understanding of RIXS spectra and they can in some cases be also useful to determine for example the spin state (high spin versus low spin).With increasing computational power being available in the future, also Monte Carlo-like calculations can become a favourable approach, as one can for example calculate “all RIXS spectra” for a sufficient section of the parameter space and then select the best fit. However, it is safe to assume that for most ions such calculations will yield multiple results making the combination with additional information, especially for more complex cases, necessary. Although we show the importance of the multielectronic excitations in the 1s2p RIXS, the emerging first-principle estimation of the crystal field parameters will open new opportunities for the modelling of experimental data.For each case of the3dN series, we have discussed the 1s2p RIXS crystal field calculation in the light of the available experimental data. With few exceptions, 1s2p RIXS experiments are to date performed with an experimental resolution of 500meV or more. This resolution does not allow a competitive comparison with 2p XAS regarding the details of the electronic structure parameters as discussed here. The constant improvements of the experimental resolution in hard X-ray spectroscopy enable to acquire a better experimental spectrum where natural broadenings are approached as the ultimate limit. An overall experimental resolution of 200meV would be necessary to improve upon the determination of the parameters. Tetravalent systems are dominated by nonlocal dipole transitions that overwhelm the quadrupole pre-edges. This makes the analysis of 1s2p RIXS complicated. The best systems to study the details of the 1s2p RIXS resonances are ionic divalent systems.



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