#### 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 3d^{0} and 3d^{9}. The calculations are performed in octahedral (O* _{h}*) 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.

#### 1. Introduction

Throughout the past decades, X-ray *absorption 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 subsequent 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.

**(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 the *local* 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., TiO_{2} [3] and CrO_{2} [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 O* _{h}* 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 the 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 1s^{2}2p^{5}3d^{N + 1} is identical in 1s2p RIXS and 2p XAS, which enables to access “L_{2,3}-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 (L_{2,3}-edge) spectra for 3d transition metal compounds. Based on the crystal field theory (CFT), we calculated at least one spectrum for each 3d* ^{N}* ground state, where . We aim to give an overview of the general structure of 1s2p RIXS for the 3d

*transition metal series to illustrate their specific behaviour, but the calculated spectra shall also serve as a reference for future analysis.*

^{N}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 momentum *L*, 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 Section 2, 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 3d* ^{N}* 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 the *RIXS 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-section *σ* 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]:
with incident photons of energy , the inelastically scattered photons of energy , and the energy transfer . The two-step process relates to the two transition operators, for the quadrupole 1s XAS and for the dipole 2p XES, describing the 1s2p RIXS cross-section *σ* as the transition from the initial state into the final states via all possible intermediate states of the system with energies *E _{i}*,

*E*, and

_{f}*E*, respectively. And finally, and 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 and 2p XES is naturally broadened by the final state lifetime [15, 20].

_{n}##### 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 , , and , respectively, are described each by the corresponding Hamiltonian :
where is the kinetic energy of the electrons with the momentum *p* and mass *m*, is the electrostatic interaction of the electron *i* of charge *e* with the nucleus at radius *r _{i}* and charge

*Z*, is the electron-electron Coulomb interaction which is determined by the direct Coulomb repulsion and the Coulomb exchange interaction described by the Slater integrals and , respectively, and is the spin-orbit interaction on every open shell. These first four terms, together with a given ground state , 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 moment , total spin moment , and total angular moment , with . This is the 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 and have the same energy, giving an energy level that is -fold degenerate. When spin-orbit coupling is included, and lose their meaning and the terms are split in energy according to their value, each with a degeneracy of . One can assign to these states a so-called Russel-Saunders term symbol , where *X* is representing the orbital momentum ; 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 , and a single electron in a p shell is represented as and .

The 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 (, , and if more than 1/2 is filled, otherwise ) [21].

The total number of states per configuration is calculated via the *binomial coefficients* [21]:

The crystal field multiplet Hamiltonian 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 O* _{h}* point group. Thus, we discuss all ions throughout this paper in an octahedral symmetry (O

*). We are aware that this is not necessarily true for all 3d transitions metal ions (e.g., CrO*

_{h}_{2}(Cr

^{4+}) and TiO

_{2}(Ti

^{4+}) have tetragonal (D

_{4h}) symmetry; Fe

_{2}O

_{3}(Fe

^{3+}) has C

_{3}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 O* _{h}*, the cubic crystal field induces the splitting of the five 3d orbitals into two symmetrically different groups: (1) the 3-fold orbitals (, , and ), which point in between the six metal-ligand bonds and thus participate in bondings with the surrounding ligands, and (2) the 2-fold orbitals ( and ), which point along the metal-ligand bonds and therefore relate to bondings. The splitting between the and orbitals is defined by the strength of the crystal field and is described by the parameter 10D

*q*[13].

The symmetry of the multielectronic term changes from spherical symmetry (O_{3}) to octahedral symmetry (O* _{h}*) causing the term (S, P, D, etc.) to branch to an O

*irreducible representation (IRREP): S (O*

_{h}_{3}) branches into an A

_{1}(O

*) symmetry state, P (O*

_{h}_{3}) branches into a T

_{1}(O

*) symmetry state, and D (O*

_{h}_{3}) branches into T

_{2}(O

*) and E (O*

_{h}*) symmetry states. Because the point group O*

_{h}_{3}(resp., O

*) contains the inversion, a parity information should be added for*

_{h}*gerade*or

*ungerade*to the term, + or − for the atomic (spherical) terms, and g or u for the crystal field terms. The 3d

*initial state and the intermediate state 1s*

^{N}^{1}3d

^{N + 1}in 1s2p RIXS are gerade (g), and the final state 2p

^{5}3d

^{N + 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.

The relative energies of the O* _{h}* IRREPs are calculated by adding the effect of the cubic crystal field 10D

*q*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 1s

^{1}3d

^{N + 1}electron configuration is the same as the diagram for a 1s

^{2}3d

^{N + 1}configuration times the 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 (with and ) in O* _{h}* symmetry and then by deriving the spin-orbital coupling or (2) by first deriving the atomic quantum number in the coupling scheme and then deriving the branching of the value in the O

*point group. Both methods yield identical final results and the identical ground state for a given Hamiltonian (see, e.g., Figure 2).*

_{h}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 coupling is small (a few tens of meV) compared to the usual values of crystal field splitting 10D*q* (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 and can still be used to describe the initial state. Similarly, for the intermediate state, the 1s core hole with does not have spin-orbit coupling. Therefore, in these cases, the coupling is achieved after the crystal field branching. (See Xu et al. [23] for details on 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 3d^{0} to 3d^{9}. Depending on the number of 3d electrons, the value of the total spin varies between 0 (minimum) and 5/2 (maximum). The branching for each value of in O* _{h}* is given in Table 2. The -coupled term, also called value IRREP, in O

*symmetry is obtained from the direct product of the IRREP and the 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,*

_{h}On the contrary, in the final state configuration, the strong spin-orbit coupling of the 2d hole (a few eV) is dominant. This induces a strong mixing between the spin and orbital momenta and resulting in and not being good quantum numbers anymore. Instead, the total angular momentum must be used to describe the final states. The effect of the O* _{h}* crystal field is added as the corresponding branching of the atomic value IRREP. The branching for each value of is also given in Table 2. The obtained IRREPs are either , , , , or for integer or . And they are , , and for half-integer values of or .

As we will see across the 3d* ^{N}* series, the information of , , and 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 O

*crystal field, we will indicate the uncoupled atomic term (with and ) along with the total angular momentum value IRREP using the following notation: [].*

_{h}We illustrate this here with an example for Ni^{2+} (3d^{8}) with the atomic ground state term ^{3}F (O_{3}).

Note that the spin multiplicity should be kept and is part of the term symbol because spin-orbit coupling is not yet included. The 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 IRREPs in the coupling scheme () for the three terms derived in (5):

In (6), we find the total angular momentum IRREP as the ground state for the example 3d^{8} in an octahedral crystal field. To summarise all this information in a condensed form, we write

This translates as a *term without SOC with ** and ** [term with SOC using **]*.

In other words, in (9), the term is *before* the inclusion of SOC, where the spin multiplicity is 3 and the orbital momentum is given as . And the term is the symmetry *after* spin-orbit interaction has been included (also identified as value IRREP).

Finally, we note that charge transfer effects (e.g., the interaction with a configuration) are neglected but their relative importance will be discussed where appropriate throughout the 3d* ^{N}* 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 elements and in the Kramers-Heisenberg equation (1) are nonzero. The operators and describe the interaction of the photon with matter. They are defined by the interaction Hamiltonian , 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 , where is the polarisation of the photon and **k** is the propagation vector of the photon. In a multipole expansion, the 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 the quantum number can only change by a value of 0 or 1. Thus, the electric dipole selection rule translates to . Because the electric quadrupole operator is the second-order term of this decomposition, the resulting selection rule translates to .

However, when the spin-orbit coupling is neglected (or weak), the orbital momentum and the spin momentum can still be considered good quantum numbers. In this case, the transitions are spin-conserving () and only changes. Due to the fact that light carries an orbital momentum of , this implies for electric dipole transitions or −1. For the second-order term, the electric quadrupole transitions, this implies , or −2 [21].

As a result, the transition operator of the 1s XAS absorption step promoting an electron from 1s to 3d () is an electric quadrupole term, and this step is forbidden as an electric dipole transition. On the contrary, the transition operator for the 2p XES decay from 2p to 1s () is an electric dipole term. Similarly, the transition operator of the 2p XAS absorption step from 2p to 3d () 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 (O* _{h}*), the selection rules of each operator depend on its symmetry properties. The electric dipole operator (first-rank tensor) behaves as the IRREP (O

_{3}) and branches into in O

*. The electric quadrupole operator (second-rank tensor) behaves as a IRREP (O*

_{h}_{3}) and branches in into (see Table 2). The transition matrix element between two states with the IRREPs and and the transition operator is nonzero if the direct product contains , where is the IRREP of the transition operator. In other words, the final state IRREPs accessible via the transition operator are given by the direct product . The selection rules are derived from the direct product tables for the O

*point group (see Tables 3, 4, and 5) and give the symmetry of the accessible states.*

_{h}The selection rules will be commented in more detail for each case of the 3d* ^{N}* 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 on every open shell of the absorbing atom (e.g., 2p SOC and 3d SOC 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 , the Hartree-Fock calculated values are used. All calculations are done in octahedral symmetry (O* _{h}*), and the crystal field is defined by the crystal field parameter 10D

*q*. 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 10D

*q*from 0 eV up to a few eV. The symmetry IRREPs of the labels of the final states in the O

*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 and for the intermediate state and final state, respectively, were applied.*

_{h}All 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 energies *E*_{in} and *E*_{out} are given by Quanty relative to the centre of gravity for each of the three states in 1s2p RIXS, being the 3d* ^{N}*, 1s

^{1}3d

^{N + 1}, and 2p

^{5}3d

^{N + 1}configurations, respectively. Thus, also the position of the energy transfer is based on the centre of gravity and

*E*does not reflect the real energy transfer.

_{T}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 as described in the following.

##### 3.1. Approximation: Boltzmann Distribution

The multielectronic ground state at ambient condition is a linear combination of an ensemble of microstates with . 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 3d^{5} configuration has in total 252 microstates, but only microstates form in high spin the atomic multielectronic ground state ^{6}S (O_{3}) (Table 7).

However, at absolute zero (), formally, only the state 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 probability *p _{i}* for each state in dependency of its relative energy , with being the energy of the lowest microstate , and absolute temperature

*T*. It is given (with Boltzmann constant

*k*) as with

_{B}In the present case, the eigenvalues calculated with Quanty from give the energies for each microstate , which are used in (10) and (11) to compute the coefficients for the Boltzmann linear combination of each multielectronic ground state.

For example, Ti^{3+} (3d^{1}) has the atomic ground state ^{2}D with microstates. In other words, the single 3d electron can be in 10 different states. In an O* _{h}* crystal field, this ground state branches without 3d spin-orbit interaction into two crystal field terms: .

The term represents the multielectronic ground state in an O* _{h}* crystal field, formed as a linear combination of microstates. For 3d

^{1}, the term is the lowest in energy as can be seen in the single electronic picture: , as opposed to the with in 4 possible microstates.

Furthermore, it should be noted that the microstates forming the multielectronic ground state are a priori not degenerate. For Ti^{3+}, 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 F_{3/2}, E_{5/2}, and F_{3/2}, respectively. In other words, the lowest four microstates , , , and are degenerate in energy (), and the two states and are degenerate (), as well as the remaining 4 microstates , , , and (). Hence, the 10 microstates, which are related to the atomic multielectronic ground state ^{2}D (O_{3}) and are split by the crystal field and spin-orbit interaction, are spread over three different energy levels. However, the six microstates forming the (O* _{h}*) term are only spread over two different energies (see Figure 3).

Note, if not stated otherwise, we only use the lowest crystal field ground state for the Boltzmann linear combination (e.g., for Ti^{3+}); the next multielectronic term higher in energy (e.g., for Ti^{3+}) 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 3d* ^{N}* series (with ) which are discussed in a more summarising manner (Section 5). Here, we will emphasise the elementary differences between the atomic (O

_{3}) and the crystal field (O

*) 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.*

_{h}As didactical examples, we use the following three cases:
(i)3d^{9} Cu^{2+}: single open shell with only a single hole in each state(ii)3d^{0} Ti^{4+}: two peaks in the 1s XAS split by the crystal field 10D*q*(iii)3d^{8} Ni^{2+}: a common model system often used in education

We aim to assign the symmetry labels to the relevant peaks in the 1s2p RIXS spectra and compare the 2p XES to the well-understood L_{2,3}-edge (2p XAS) spectra [35].

##### 4.1. 3d^{9}: Divalent Copper Cu^{2+}—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 Cu^{2+} with a 3d^{9} 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.

We first describe the atomic case of an isolated ion without any crystal field. The electronic configuration of the initial state of Cu^{2+} is . In spherical symmetry (O_{3}), the 3d spin-orbit coupling (SOC) induces the splitting of the 10-fold degenerate ground state ^{2}D (10 microstates) into two multielectronic states (O_{3}) and (O_{3}), defined by (ground state) and , respectively. These two states are separated in energy by . The electronic configuration of the 1s2p RIXS intermediate state of Cu^{2+} is , which corresponds to the 2-fold degenerate state (O_{3}) with . The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Cu^{2+} is , which splits due to the 2p spin-orbit coupling into (lowest energy state) and separated by . This is summarised in the scheme in Figure 4.

Since the ground state of the initial state is , the electric dipole absorption operator (2p XAS) enables only to reach the final state (). In 1s2p RIXS, the selection rules of the electric quadrupole absorption for 1s XAS () enable to reach the intermediate state . The subsequent electric dipole emission (2p XES) enables to reach both and final states. As a result, for an isolated Cu^{2+} 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.

In the vertical direction, the K*α*_{1} and K*α*_{2} decays (2p XES) appear as two separate peaks. These two peaks are separated by the 2p SOC , corresponding to the two resonant emission transitions from the intermediate state with to the final states with .

###### 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 (O* _{h}*) crystal field described by the parameter 10D

*q*. In a single electron picture, this crystal field induces the splitting of the 3d orbitals into and

*e*. In the multielectronic formalism (neglecting the 3d spin-orbit coupling in first approximation), the

_{g}^{2}D term of the initial state branches in the O

*point group into the*

_{h}^{2}E

*() and () terms. Thus, the multielectronic ground state (at ) is*

_{g}^{2}E

*(O*

_{g}*). The intermediate state term*

_{h}^{2}S (O

_{3}) branches in an octahedral crystal field into (O

*) symmetry.*

_{h}In the O* _{h}* point group, the electric quadrupole 1s XAS operator has and E

*symmetries. We derive the selection rules from the*

_{g}*product table*(Table 3) for the first step of the 1s2p RIXS process.

From (13), it becomes obvious that many symmetries are in principle accessible via 1s XAS from the multielectronic ground state symmetry ^{2}E* _{g}* (O

*). In spite of the fact that many symmetries are in principle reachable via a quadrupole 1s XAS transition starting from the ground state symmetry*

_{h}^{2}E

*(O*

_{g}*), the electronic structure of a Cu*

_{h}^{2+}ion only offers (O

*) symmetry leading to a single peak in the 1s XAS projection.*

_{h}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 the -coupled terms obtained above in the atomic case, we derive their respective branchings in the octahedral crystal field (O* _{h}*).

The initial ground state (O_{3}) branches in octahedral symmetry (O* _{h}*) into (O

*). The ground state at is (O*

_{h}*). The effects of the 3d spin-orbit coupling and the O*

_{h}*crystal field parameter 10D*

_{h}*q*are illustrated in Figure 2.

The intermediate state term (O_{3}) branches in an octahedral crystal field (O* _{h}*) into (O

*). The final states (O*

_{h}_{3}) and (O

_{3}) branch into (O

*) and (O*

_{h}*), respectively. In octahedral symmetry, the electric dipole operator has symmetry and thus, using the product table (Table 5), we find the reachable final states in the subsequent electric dipole emission (2p XES):*

_{h}The direct 2p XAS starts from the ground state yielding with the IRREP the reachable final state terms:

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 Figure 6 and the corresponding calculated spectrum is shown in Figure 7.

The RIXS projections for various values of 10D*q* show almost no difference in both directions (XAS and XES). It is noteworthy that the 2p XAS (L_{2,3}-edge) approaches the two peaks in the 2p XES projections for large 10D*q* values. This is due to the crystal field (10D*q*) affecting the spin-orbit interaction altering the 2p XAS selection rules.

In other words, the L_{2}-edge probes in 2p XAS the amount of character in the ground state. In the atomic case (10D*q* = 0.0 eV) with spherical symmetry (O_{3}) and at absolute zero , the ion is pure (D_{5/2}). In octahedral O* _{h}* symmetry, on the other hand, an increasing crystal field (scaled via 10D

*q*) mixes more and more character from the D

_{3/2}into the ground state, resulting in a continuous visible increase of the L

_{2}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 the (O_{3}) and (O_{3}) final states.

###### 4.1.2. 1s2p RIXS Experiments of 3d^{9} 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. 3d^{0}: Tetravalent Titanium Ti^{4+}—1s XAS Peaks Split by 10D*q*

In the following, we discuss the differences between 2p XAS and 1s2p RIXS for the case of tetravalent titanium (Ti^{4+}). Here, the initial state configuration 1s^{2}2p^{6}3d^{0} 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 1s^{1}2p^{6}3d^{1} electron configuration. The final state in 1s2p RIXS and 2p XAS is in this case 1s^{2}2p^{5}3d^{1}.

The 3d^{0} 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 L_{2,3}-edge considerations of 3d^{0} ions previously described in [35].

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

We start again with the case of an isolated Ti^{4+} ion (3d^{0}) where the effect of the solid state, that is, the crystal field, is neglected. The initial state configuration of Ti^{4+} is 1s^{2}2p^{6}3d^{0}. Since all shells are full, and . The -coupled total angular momentum is thus , and the initial state symmetry is the *totally symmetric* term ^{1}S_{0}.

The electronic configuration of the intermediate state in 1s2p RIXS of Ti^{4+} is 1s^{1}2p^{6}3d^{1}, corresponding to the total orbital angular momentum and total spin angular momentum or giving the Russel-Saunders terms ^{1}D and ^{3}D ( microstates). Considering the spin-orbit interaction in the coupling scheme, one obtains for the ^{1}D term and for the ^{3}D term (Figure 8). The splitting of these four terms (^{1}D_{2}, ^{3}D_{1}, ^{3}D_{2}, and ^{3}D_{3}) is defined by the 3d SOC and exchange interaction , which are both small (32 meV and 46 meV, resp.). The calculated energy splitting is and is beyond the reach of current experimental resolution and not resolved in our calculations discussed here.

The electric quadrupole transition of the 1s XAS enables to reach the intermediate states with , though here 3d SOC is small and transitions to the spin triplet term ^{3}D_{2} will be weak, such that the 1s XAS is dominated by the transition in the spin singlet term ^{1}D_{2}.

The final state electronic configuration is 1s^{2}2p^{5}3d^{1} with two partially filled shells that have to be accounted for: the 2p and the 3d shell. The total orbital angular moment of the final state electronic configuration is . The total spin angular moment of the final state electronic configuration is . This gives in the atomic case the spin singlet and triplet terms ^{1,3}P, ^{1,3}D, and ^{1,3}F.

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 and allow only to reach the singlet ^{1}P term from the ^{1}S initial state, leading to one peak only (the L_{2}-edge). The L_{3}-edge, corresponding to transitions into the spin triplet term ^{3}P, would have zero intensity. But in fact the strong 2p spin-orbit coupling in the final state ( is used) leads to three peaks, two of them forming the L_{2}- and L_{3}-edges, and a third weak peak related to triplet transitions.

The derivation of the total angular momentum of the final state electronic configuration in the coupling scheme yields the term symbols illustrated in Figure 9.

The direct electric dipole transition in 2p XAS with ( forbidden) starts for Ti^{4+} from the total symmetric term ^{1}S_{0}, which allows to reach all final states with (P_{1} and D_{1}, marked with a red circle in Figure 9). The L_{2}- and L_{3}-edges are separated by the 2p spin-orbit coupling corresponding to transitions into the ^{1}P_{1} (L_{2}-edge) and ^{3}D_{1} (L_{3}-edge) terms. More precisely, from the squared matrix elements, we find that the second peak is dominated by ^{3}D_{1} contribution of 60%, adding 36% of ^{1}P_{1} and 4% of ^{3}P_{1} character. The small third peak in the 2p XAS, at in Figure 10, is related to transitions into the ^{3}P_{1} 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 L_{2,3}-edge into three absorption lines including a redistribution of the intensities.

In the 1s2p RIXS process, the 2p XES decays with starting from the intermediate states with (D_{2}) and enables to reach the final states P_{1,2}, D_{1,2,3}, and F_{2,3} with (terms shown in green Figure 9). Therefore, all values except 0 and 4 are reachable.

With this, one can draw for Ti^{4+} 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.

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 Figure 10. But in both cases, 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 (O_{3}), but neglecting any crystal field. With the above, one can describe the 1s XAS with only one visible peak due to transitions into the ^{1}D_{2} intermediate state term. The 3d spin-orbit coupling and exchange interaction G_{sd} 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 ^{3}D_{2} term.

The splitting of the 2p XAS is dominated by the large 2p SOC separating the L_{2}-edge from the L_{3}-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 IRREP (mostly ^{1}D_{2}) into the final states with IRREPs resulting multiple visible peaks in the energy transfer direction. All final state terms are also split by the exchange and direct Coulomb interactions, G_{pd} and F_{pd}, respectively.

###### 4.2.2. Considering an O_{h} Crystal Field

_{h}

We now consider the effect of an octahedral O* _{h}* crystal field. The totally symmetric initial state term

^{1}S

_{0}branches in O

*into . Formal consideration of an O*

_{h}*crystal field and 3d spin-orbit interaction leads to the branching of the intermediate state terms*

_{h}^{1}D and

^{3}D into eight terms. The scheme in Figure 12 illustrates these branchings of the terms due to an O

*crystal field and the 3d spin-orbit interaction ( and ).*

_{h}In spite of several available intermediate state terms, starting from the initial state and using the and E* _{g}* 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.

In other words, from the *totally symmetric* ground state (O* _{h}*), only the E

*(O*

_{g}*) and (O*

_{h}*) symmetries can be reached in the intermediate state via a quadrupole 1s XAS transition. Hence, for sufficiently large values of 10D*

_{h}*q*, 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 is small, the transitions to the spin singlet intermediate states are dominating (i.e., matrix elements are the largest), while the transitions via the IRREP from triplet

^{3}D should be weak. The 1s XAS is dominated by transitions to the E

*and terms from the*

_{g}^{1}D

_{2}.

In an O* _{h}* crystal field, the final state IRREPs with and 4 (see Figure 11) branch, respectively, into the -coupled terms: , , , , and . The 2p XES in 1s2p RIXS from the two intermediate state terms and E

*via the electric dipole operator, described by the IRREP, leads to*

_{g}It is noteworthy here that only the intermediate state with symmetry can access in the 2p XES decay the and final state symmetries. The and final state terms are however accessible from both and intermediate state symmetries. Furthermore, we note that the intermediate state symmetry 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 have symmetry; otherwise, the transition matrix element equals zero. As can be seen from the branchings of (O_{3}) in and octahedral crystal field (Table 2), the IRREP can be reached not only from the IRREP but also from the and IRREPs.

This results in only seven possible IRREPs in O* _{h}* 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 Ti

^{4+}(3d

^{0}) for a crystal-field splitting of 10D

*q*= 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 allowing also transitions into former spin triplet states (e.g., , , , and ).

**(a)**

**(b)**

This enables to draw the electronic term scheme in Figure 15 illustrating the different pathways in 1s2p RIXS and 2p XAS for 3d^{0} Ti^{4+} in O* _{h}* 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 symmetry ().

Similarly to the 3d^{9} case (Cu^{2+}) 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 Figure 14. The total RIXS map is decomposed into the E* _{g}* and quadrupole absorption components. The selective absorption towards each intermediate state is obtained. Experimentally, this dichroic behaviour has been observed for Ti

^{4+}in cubic oxides such as the prototypal SrTiO

_{3}[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 final state symmetry. We further observe that within the four main peaks in 2p XAS, the first peaks of the L

_{3}(at ) and of the L

_{2}(at ) are stronger via the absorption channel of the 1s2p RIXS than the E

*and vice-versa: the second peaks of the L*

_{g}_{3}() and of the L

_{2}(at ) are stronger via the E

*absorption channel of the 1s2p RIXS than via the . This can be understood in a single electron picture with the crystal field splitting of the final state: the lower states of the L*

_{g}_{3}-edge contain the 3d single electron in the orbitals while the higher energy states have the 3d electron in the

*e*orbitals and similarly for the L

_{g}_{2}-edge [37].

###### 4.2.3. 1s2p RIXS Experiments of 3d^{0} 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 Figure 13. 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 Ti^{4+}, 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 TiO_{2} 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 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 (Ca^{2+}), including octahedral CaO and cubic CaF_{2}.

##### 4.3. 3d^{8}: Divalent Nickel Ni^{2+}—Single Peak in 1s XAS

In this part, we investigate the elementary differences between 2p XAS and 1s2p RIXS for a divalent nickel ion (Ni^{2+}, 3d^{8}). As we will show, the electronic configurations of the different steps of 1s2p RIXS present strong analogies with the case of Ti^{4+}; the main difference being the nature of the initial state. We start again with the atomic case by deriving the values as IRREPs.

The electronic configuration of the initial state of Ni^{2+} is 1s^{2}2p^{6}3d^{8} with two holes in the 3d shell. The multielectronic interactions lead to 45 states spread across the terms ^{3}F, ^{3}P, ^{1}G, ^{1}D, and ^{1}S [13]. In spherical symmetry (O_{3}), the atomic ground state is ^{3}F ( and ). The 3d spin-orbit coupling induces the splitting of the ^{3}F term into the three terms:

From those three IRREPs with , the ^{3}F_{4} term with is the ground state term symbol.

The electronic configuration of the intermediate state in 1s2p RIXS of Ni^{2+} is 1s^{1}2p^{6}3d^{9} (one hole in the 1s shell and one hole in the 3d shell) which translates into the two terms ^{1}D and ^{3}D ( microstates). It is noteworthy that this is identical to the intermediate state of Ti^{4+} described above (Section 4.2) due to the *electron-hole equivalency*. The 3d spin-orbit interaction splits the two terms into the ^{1}D_{2} (for ) and ^{3}D_{1}, ^{3}D_{2}, and ^{3}D_{3} (for ) Russel-Saunders terms. The resulting splittings distribute the four terms in energy over for which the 1s XAS will appear only as a single peak due to the resolution chosen in our calculations, analogue to the Ti^{4+} atomic case.

The difference in the 1s XAS step in 1s2p RIXS between the 3d^{8} Ni^{2+} and 3d^{0} Ti^{4+} cases arises from the spin multiplicity of the initial state. In both cases, the electric quadrupole absorption leads to a intermediate state. However, because of the small 3d spin-orbit coupling (), the spin multiplicity is mostly conserved in the transition and the absorption from the spin triplet initial state of Ni^{2+} leads to the ^{3}D_{2} state. For Ti^{4+} on the other hand, mainly, the ^{1}D_{2} term is reached.

The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Ni^{2+} is 1s^{2}2p^{5}3d^{9}. This is again strictly analogue to the final state of Ti^{4+}. The final state configuration corresponds to several atomic terms (^{1,3}P, ^{1,3}D, and ^{1,3}F) which are split further due to the strong 2p spin-orbit coupling . The total symmetry final state terms are represented with integer values ranging from to . The detailed splittings are given above in Figure 9 in Section 4.2.

Only the final state terms with are formally accessible in the 1s2p RIXS from the intermediate state with via 2p XES decays (). The direct 2p XAS (with ) on the other hand can only reach the final state terms with or from the ground state with . Altogether, the selection rules for the atomic case can be summarised as

This is also illustrated in the combined term scheme shown in Figure 16.

The calculated 1s2p RIXS of an isolated Ni^{2+} ion (O_{3}) 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 L_{3} and one in the L_{2}) correspond to the and IRREPs. It is noteworthy here that the ^{3}F_{4} 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 is related to the IRREP (dominated by ^{3}F_{4}), while in 1s2p RIXS the peak at in the 2p XES decay relates to transitions into the final state IRREP (mostly ^{1}D_{2}). In other words, though it seems that the same peak at 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.

Closer inspection of the sticks reveals that the ^{3}F_{3} term does not appear as a dominant contribution, but instead, the ^{1}F_{3} term appears twice. This is due to the fact that the atomic terms lose their meaning when spin-orbit interaction is included. Then becomes the identifying IRREP which is formally a linear combination of ^{1}F_{3}, ^{3}D_{3}, and ^{3}F_{3}. Here, it happens to be the case that the ^{1}F_{3} contribution is dominant in two IRREPs.

As expected, the IRREP (dominated by the ^{3}P_{0} term), as well as the IRREP (^{3}P_{1} and ^{3}D_{1} terms), has no contribution to the 2p XAS spectrum. Furthermore, also, transitions into the IRREPs (P_{2}, F_{2}, and F_{2} terms) are not appearing in 2p XAS.

In the 2p XES decay, on the other hand, the IRREPs have a much stronger contribution to the spectrum. Those peaks, related to the final state IRREPs , 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 O_{h} Crystal Field

_{h}

The crystal field splitting in O* _{h}* symmetry of the atomic terms of the initial state configuration of Ni

^{2+}(1s

^{2}2p

^{6}3d

^{8}) is well-known and described with the Tanabe-Sugano diagram. The ground state atomic term symbol

^{3}F splits into , , and and gives as the octahedral crystal field ground state. The inclusion of the 3d spin-orbit coupling () translates the ground state term as . Thus, the crystal field ground state of 3d

^{8}Ni

^{2+}including 3d spin-orbit interaction is (O

*).*

_{h}The crystal field splitting of the 1s2p RIXS intermediate state of Ni^{2+} is analogue to Ti^{4+} as the ^{1,3}D terms split into . From the ground state symmetry (neglecting 3d SOC), the 1s XAS IRREPs and E* _{g}* enable to reach only and

^{3}E

*intermediate state symmetries:*

_{g}In other words, the intermediate state term cannot be reached by the electric quadrupole 1s XAS absorption. Hence, only the 1s XAS IRREP E* _{g}* leads to an existing intermediate state symmetry:

^{3}E

*. 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*

_{g}^{3}E

*intermediate state symmetry, because the quadrupole 1s XAS transition into the 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., ). Therefore, the 1s XAS transitions appear as a single peak due to the resolution chosen in our calculations.*

_{g}The single peak observed in the 1s XAS step for Ni^{2+} (3d^{8}) is the first important difference with Ti^{4+} (3d^{0}), 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 Ni^{2+} are in the *e _{g}* orbitals, the excited 1s electron can only go to the empty

*e*orbitals (

_{g}^{3}E

*intermediate state). On the contrary, in the case of Ti*

_{g}^{4+}(Section 4.2), both

*e*and orbitals are empty leaving both E

_{g}*and intermediate state symmetries accessible.*

_{g}When the crystal field (10D*q*) is large with respect to the other electronic interactions, such as the 3d spin-orbit interaction , the 1s XAS step of the 1s2p RIXS can be described well with the noncoupled IRREPs ( and are good quantum numbers). However, when the crystal field is weak or when the electronic interactions are stronger, such as the spin-orbit, it is necessary to use the -coupled terms. The spin-orbit coupling translates the intermediate state terms into the same total angular symmetry terms as for Ti^{4+} (see Figure 12). From this, it is evident that the intermediate state offers four different symmetries , E* _{g}*, , and from which the 2p XES decays can occur.

The branching in the octahedral symmetry of the final state of Ni^{2+} is again analogue as Ti^{4+} (see Section 4.2). Those terms translate in an O* _{h}* crystal field to the

*ungerade*symmetries: , , E

*, , and . The dipole 2p XES decays in 1s2p RIXS from the intermediate state terms give*

_{u}From that, we find that in 2p XES all final state symmetries can be reached. However, for 2p XAS, the dipole transition () starts from the ground state.

From that, we find that the reachable 2p XAS final states can have , , , or symmetry. In other words, all intermediate terms, except those with , 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 Figure 19.

Comparing this case with the case of in an O* _{h}* 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 Figure 20.

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 Figure 19 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 IRREP is two orders of magnitude weaker than the contribution via E* _{g}* 1s XAS IRREP. According to the strict selection rule (e.g., when neglecting 3d spin-orbit coupling), no transitions arise from the absorption. However, the 3d spin-orbit interaction 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 operator. In other words, the small but nonzero 3d spin-orbit coupling induces some mixing such that the contributions to the spectrum via the 1s XAS IRREP will be nonzero but weak. A direct comparison of the two 1s XAS transitions with and E

*symmetries shows that the spectrum multiplied by 186 is almost identical to the E*

_{g}*spectrum (Figure 20).*

_{g}###### 4.3.2. 1s2p RIXS Experiments of 3d^{8} Systems

Experimental 1s2p RIXS spectra of 3d^{8} NiF_{2} and molecular Ni^{2+} complexes have been published by Glatzel et al. [14, 40]. Within the resolution of the measurement, the NiF_{2} spectrum is exactly reproduced by the crystal field calculation [14].

#### 5. 1s2p RIXS for Other 3d^{N} Configurations

^{N}

In this section, the remaining 3d* ^{N}* configurations () are discussed in a more condensed manner. Furthermore, for the systems with a , , , and 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 (10D

*q*) and the pairing energy in those cases are competing. In other words, for sufficiently large values of 10D

*q*, 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 10D*q* value matching roughly the empirically found “ per valency” approximation for transition metal oxides. To illustrate the differences, we additionally select for to a sufficiently large value of 10D*q* to obtain the corresponding spectra for a low spin configuration. The used crystal field values (10D*q*) are summarised in Table 7.

As we have seen already throughout the didactic cases in Section 4, the 2p XES decays are always described with the dipole IRREP . 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).

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 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 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. 3d^{1} Ground State System, for Example,

In the previous cases, the splitting of the ground state electronic terms was leading to a single term, for example, () for () or () for (). In the following, we discuss the ground state configuration representing for instance the case of a ion.