Journal of Spectroscopy
Volume 2018 (2018), Article ID 3618463, 50 pages
https://doi.org/10.1155/2018/3618463

Review Article

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

Patric Zimmermann, Myrtille O. J. Y. Hunault, and Frank M. F. de Groot

Debye Institute of Nanomaterial Science, Utrecht University, 3584 CA Utrecht, Netherlands

Correspondence should be addressed to Patric Zimmermann; p.zimmermann@uu.nl and Frank M. F. de Groot; F.M.F.deGroot@uu.nl

Received 11 November 2017; Accepted 27 December 2017; Published 24 June 2018

Academic Editor: Maria Carmen Yebra-Biurrun

Copyright © 2018 Patric Zimmermann et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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)

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 ( decays) transitions and the direct 2p XAS (L2,3-edge). The term scheme (b) illustrates the analogy between the energy transfer in 1s2p RIXS and 2p XAS.

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., 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 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 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 . 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 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 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 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 Ei, Ef, and En, 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].

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 ri 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 .

Table 1: Russell-Saunders term symbols for the free ion configurations giving the number of states, the corresponding number of energy levels, and the terms for the given configuration. Note that gives the same terms as . 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, ²S

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 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 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 10Dq [13].

The symmetry of the multielectronic term changes from spherical symmetry (O₃) to octahedral symmetry (Oh) causing the 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 the , , and atomic terms in an octahedral crystal field () [28] (p.262). The first columns can be either the orbital momentum, the total spin , or the total angular momentum . The g for gerade and the u for ungerade must be added to indicate the parity.

O₃
→
Oh

, , or
→
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 ⊕ E5/2

The 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 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 Oh 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 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 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 coupling 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 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 3d⁰ to 3d⁹. 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 Oh is given in Table 2. The -coupled term, also called value IRREP, in Oh 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,

Table 3: Direct products of representations (product table) in 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 ⊕ T2g

Table 4: Direct products of representations (cont.) in 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 ⊕ T2g

Table 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 ⊕ T₂

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 Oh 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 3dN 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 Oh crystal field, we will indicate the uncoupled atomic term (with and ) along with the total angular momentum value IRREP using the following notation: [].

We illustrate this here with an example for Ni²⁺ (3d⁸) with the atomic ground state term ³F (O₃).

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 3d⁸ 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 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 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 (Oh), the selection rules of each operator depend on its symmetry properties. The electric dipole operator (first-rank tensor) behaves as the IRREP (O₃) and branches into in Oh. The electric quadrupole operator (second-rank tensor) behaves as a IRREP (O₃) 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 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 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 (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 and for the intermediate state and final state, respectively, were applied.

Table 6: Summary of the different notations in 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

E
E
E
E
Γ₃
2
2

T₁
T₁
T₁
T₁
Γ₄
1
1

T₂
T₂
T₂
T₂
Γ₅

^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

^S0

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 Ein 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 position of the energy transfer 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 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 3d⁵ configuration has in total 252 microstates, but only 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

O₃
Oh
HS
LS

HS
LS

HS
HS
LS

1
6
9
4
10
6
15
12
3
4
9
6
1
4
1
10
45
120
210
252
210
120
45
10
0 eV & 2.1 eV
1.5 eV
2.1 eV
1.5 eV
1.5 eV
1.5 eV
0.9 eV
0.9 eV
0.9 eV
0 eV & 3.0 eV
3.9 eV
3.9 eV
3.0 eV
3.0 eV

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 pi 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 kB) as

with

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, Ti³⁺ (3d¹) has the atomic ground state ²D with 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: .

The term represents the multielectronic ground state in an Oh crystal field, formed as a linear combination of microstates. For 3d¹, 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 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 , , , 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 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 (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., for Ti³⁺); the next multielectronic term higher in energy (e.g., 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 ) 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 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 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.

We first describe the atomic case of an isolated ion without any crystal field. The electronic configuration of the initial state of Cu²⁺ is . In spherical symmetry (O₃), the 3d spin-orbit coupling (SOC) induces the splitting of the 10-fold degenerate ground state ²D (10 microstates) into two multielectronic states (O₃) and (O₃), defined by (ground state) and , respectively. These two states are separated in energy by . The electronic configuration of the 1s2p RIXS intermediate state of Cu²⁺ is , which corresponds to the 2-fold degenerate state (O₃) with . The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Cu²⁺ 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.

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 multiplicities yields the total number of microstates per configuration.

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 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 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 , 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 (Oh) crystal field described by the parameter 10Dq. In a single electron picture, this crystal field induces the splitting of the 3d orbitals into 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 () and () terms. Thus, the multielectronic ground state (at ) is ²Eg (Oh). The intermediate state term ²S (O₃) branches in an octahedral crystal field into (Oh) symmetry.

In the Oh point group, the electric quadrupole 1s XAS operator has and Eg symmetries. We derive the selection rules from the 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 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 (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 the -coupled terms obtained above in the atomic case, we derive their respective branchings in the octahedral crystal field (Oh).

The initial ground state (O₃) branches in octahedral symmetry (Oh) into (Oh). The ground state at is (Oh). The effects of the 3d spin-orbit coupling and the Oh crystal field parameter 10Dq are illustrated in Figure 2.

The intermediate state term (O₃) branches in an octahedral crystal field (Oh) into (Oh). The final states (O₃) and (O₃) branch into (Oh) and (Oh), 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):

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.

Figure 6: Multielectronic term scheme for 3d⁹ Cu²⁺ comparing the 1s2p RIXS and direct 2p XAS pathways. Where SOC 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 and 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 character in the ground state. In the atomic case (10Dq = 0.0 eV) with spherical symmetry (O₃) and at absolute zero , the ion is pure (D5/2). In octahedral Oh symmetry, on the other hand, an increasing crystal field (scaled via 10Dq) mixes more and more 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 the (O₃) and (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.

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, and . The -coupled total angular momentum is thus , 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 and total spin angular momentum or giving the Russel-Saunders terms ¹D and ³D ( microstates). Considering the spin-orbit interaction in the coupling scheme, one obtains for the ¹D term and for the ³D term (Figure 8). The splitting of these four terms (¹D₂, ³D₁, ³D₂, and ³D₃) 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.

Figure 8: Splitting of the intermediate state spin singlet and triplet terms 1,3D due to 3d SOC for a Ti⁴⁺ ion into D₁, D₂, and D₃. States accessible in 1s XAS () are shown in blue. However, the existing terms with 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 with , though here 3d SOC 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 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 1,3P, 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 and 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 in the final state ( 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 momentum of the final state electronic configuration in the coupling scheme yields the term symbols illustrated in Figure 9.

Figure 9: Splitting of the final state terms 1,3P, 1,3D, and 1,3F due to spin-orbit coupling for Ti⁴⁺.

The direct electric dipole transition in 2p XAS with ( forbidden) starts for Ti⁴⁺ from the total symmetric term ¹S₀, which allows to reach all final states with (P₁ and D₁, marked with a red circle in Figure 9). The L₂- and L₃-edges are separated by the 2p spin-orbit coupling 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 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 with starting from the intermediate states with (D₂) and enables to reach the final states P1,2, D1,2,3, and F2,3 with (terms shown in green Figure 9). Therefore, all 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 , while the dipole decays in 2p XES can reach final states with . However, terms with 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 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 (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 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 SOC 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 IRREP (mostly ¹D₂) 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, 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 . 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 ( and ).

Figure 12: Branching of the intermediate state terms ¹D 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 state and using the 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.

In other words, from the totally symmetric ground state (Oh), only the Eg (Oh) and (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 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 ³D should be weak. The 1s XAS is dominated by transitions to the Eg and 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 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 Eg via the electric dipole operator, described by the IRREP, leads to

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 (O₃) 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 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 allowing also transitions into former spin triplet states (e.g., , , , and ).

(a)

(b)

Figure 14: (a) Calculation of the 1s2p RIXS spectra for Ti⁴⁺ (3d⁰) with separated into the quadrupole components (with , 1 and ) in spherical symmetry. They translate into the and 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).

This enables to draw the electronic term scheme in Figure 15 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 symmetry ().

Figure 15: Multielectronic term scheme comparing the 2p XAS and the 1s2p RIXS pathways. The two quadrupole 1s XAS transitions with and symmetry lead to two different intermediate state symmetries. The subsequent dipole decays (2p XES) with symmetry yield multiple 1s2p RIXS final state symmetries. The direct 2p XAS (L2,3-edge) transition with symmetry can on the other hand only access the 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 Figure 14. The total RIXS map is decomposed into the Eg and 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 final state symmetry. We further observe that within the four main peaks in 2p XAS, the first peaks of the L₃ (at ) and of the L₂ (at ) are stronger via the absorption channel of the 1s2p RIXS than the Eg and vice-versa: the second peaks of the L₃ () and of the L₂ (at ) are stronger via the Eg 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 L₃-edge contain the 3d single electron in the 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 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 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 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 values as IRREPs.

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 ( and ). The 3d spin-orbit coupling induces the splitting of the ³F term into the three terms:

From those three IRREPs with , the ³F₄ term with 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 ( 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 ) and ³D₁, ³D₂, and ³D₃ (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 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 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 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 . 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.

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 . Note that the final state is similar to Ti⁴⁺. Nonaccessible terms (e.g., final states with ) 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 and 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 is related to the IRREP (dominated by ³F₄), while in 1s2p RIXS the peak at in the 2p XES decay relates to transitions into the final state IRREP (mostly ¹D₂). 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.

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 the ³F₃ 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 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 IRREPs.

As expected, the IRREP (dominated by the ³P₀ term), as well as the IRREP (³P₁ and ³D₁ terms), has no contribution to the 2p XAS spectrum. Furthermore, also, transitions into the IRREPs (P₂, F₂, and F₂ 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 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 , , 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 3d⁸ Ni²⁺ including 3d spin-orbit interaction is (Oh).

The crystal field splitting of the 1s2p RIXS intermediate state of Ni²⁺ is analogue to Ti⁴⁺ as the 1,3D terms split into . From the ground state symmetry (neglecting 3d SOC), the 1s XAS IRREPs and Eg enable to reach only and ³Eg intermediate state symmetries:

In other words, the intermediate state term 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 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.

Figure 18: 1s XAS projections of the calculated 1s2p RIXS map for Ni²⁺ (3d⁸) showing only one peak as the 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 orbitals are empty leaving both Eg and 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 , 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 Ti⁴⁺ (see Figure 12). From this, it is evident that the intermediate state offers four different symmetries , Eg, , and 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: , , Eu, , and . The dipole 2p XES decays in 1s2p RIXS from the intermediate state terms give

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.

Figure 19: Multielectronic term scheme for comparing the 1s2p RIXS and direct 2p XAS pathways. Populating the intermediate state would require a two-electron transition in .

Comparing this case with the case of 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 Figure 20.

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 (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 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 Eg 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 Eg symmetries shows that the 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 () 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 (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 “ per valency” approximation for transition metal oxides. To illustrate the differences, we additionally select for to 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 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. 3d¹ 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.

As shown below, the 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 : the atomic ground state term is ²D, which splits into and by the 3d spin-orbit coupling (see Table 1). In the Oh crystal field, the ²D term splits into and .

The difference with (3d⁹) is that the ground state term of () is (6 microstates). When considering the 3d spin-orbit coupling , this term splits into the two total symmetry irreducible representations (IRREPs) (mixing of and ) and (from term), where 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 (Figure 2). Furthermore, we note that due to the small 3d SOC in the initial state (), the splitting of the ground state 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 K, Boltzmann population of the (4-fold degenerate) is 86% and 14% for the (2-fold degenerate). The contribution of the second term is thus small but not entirely negligible.

The intermediate state electron configuration ( microstates) is equivalent to the initial state of coupled to the 1s core hole [13]. The atomic terms are , , 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 Table 2, one finds the branchings of the atomic states in an Oh crystal field. The resulting Oh term symbols for a configuration are , , , , and . The energy splitting of these Oh crystal field terms is qualitatively similar to the Tanabe-Sugano diagram of . In the particular case of , 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 () 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 ground state symmetry can reach , Eg, , and symmetries, but the 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.

This shows, in an Oh crystal field, starting from the ground state, the 1s XAS operators and Eg enable to reach all even terms except the 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 (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 (Section 4) demonstrated the necessity to consider the -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 ():

The result shows that all three odd total symmetric IRREPS , , and are thus reachable in the intermediate state. In other words, all three odd IRREPs are contributing to the 2p XES decays.

The final state configuration contains microstates due to the hole in the 2p shell. The atomic term symbols span from S to H with spin doublets and quartets and values ranging from 1/2 to 11/2 [13]. In an Oh crystal field, these terms branch into the three IRREPs , , and (Table 2).

The 2p XES dipole transition IRREP 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 terms and are to be considered due to the small splitting. This leads with the dipole transition IRREP also to all three final state symmetries:

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 Figure 22.

Figure 22: Multielectronic term scheme for () 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 Figure 23 demonstrate that the RIXS intermediate state enables to select some specific final states.

Figure 23: Calculated 1s2p RIXS spectrum for () for shown together with some selected CIE slices. The stick spectrum at the bottom illustrates the distribution of the three final state terms , , and with respect to the emissions.

According to the combined term scheme in Figure 22, the intermediate states , , 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 corresponding to the excitation of the 1s electron into the orbitals, the final states probed resonantly with RIXS are mainly the lower energy part of the states, corresponding to the “” part of the spectra. The intensity arises mainly from the absorption operator. The intermediate state selects higher energy final states equally probed by the and Eg 1s XAS operators.

The second term (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 (from ²S atomic term symbol) only gives a very weak absorption via the quadrupole operator in agreement with the selection rules as given in (37).

This intermediate state only decays to (see, e.g., the final state peak at in Figure 23) and (at ). Such intermediate state selection is identical for the L₃- and L₂-edges. This is similar to the results obtained for where the “” 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 of 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,

In this part, we present the elementary differences between 2p XAS and 1s2p RIXS for the case of tetravalent chromium (). 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

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 .

The initial state electron configuration corresponds to the atomic ground state term ³F (O₃) with 45 microstates. It translates into an Oh crystal field (without 3d SOC) into a ground state symmetry with 9 microstates. By including 3d SOC , we find the further splittings of the ground state term ():

From these terms, Eg lies lowest in energy; hence, the ground state is (Oh).

The intermediate state configuration with microstates translates into the atomic term symbols , , 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 , , , , , , , and . However, the Coulomb exchange 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 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.

As can be seen from the results found in (43) and (44), starting from a ground state, all but the 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 (). 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 the IRREP () and the direct product table (Table 3), one finds the intermediate state symmetries including 3d SOC as also shown in the combined term scheme in Figure 25.

Figure 25: Multielectronic term scheme for () 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 () contains microstates due to the hole in the 2p shell spanning from 0 to 7. In an crystal field, these terms branch into the five total symmetry IRREPs , , Eu, , and . 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 the (Oh) ground state and using the IRREP, we find that only and 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 and Eg intermediate state symmetries are only reachable via the quadrupole IRREP. This is confirmed in Figure 26 (RIXS maps) where the low energy peak at appears only in the RIXS map corresponding to the IRREP. The and eg components appear well separated with (i) the components probed by the absorption giving two intense and narrow peaks at and (ii) the eg components probed by the Eg absorption leading to a larger group of final states spread in energy between 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 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 .

Figure 26: Calculated 1s2p RIXS spectrum for () for . Note that the intermediate state symmetry is in agreement with the selection rules only populated via the 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 Figure 26.

And finally, in Figure 27, we show the 2p XES projections of the calculated 1s2p RIXS to illustrate its evolution when the crystal field value 10Dq is scaled from .

Figure 27: Effect of the Oh crystal field on the 2p XES projections of the calculated 1s2p RIXS spectra for ().

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 a ion are summarised below:

For systems such as , the initial state configuration (120 microstates) translates in Oh to a ground state (4 microstates). With 3d SOC (), the ground state of a ion is thus (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 systems. Those terms, neglecting 3d spin-orbit interaction, branch into multiple Oh terms:

For 1s XAS, the weak spin-orbit coupling in the 3d shell results in spin-conserving transitions from the ground state to the spin-quartet intermediate states with quadrupole IRREPs and :

This illustrates, though many symmetries exist in the intermediate state, only the transitions into the 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 terms and due to SOC ():

Here, it is noteworthy that both IS terms ( and ) are branching into the same three symmetries: , , and .

The final state () contains microstates due to the hole in the 2p shell. The atomic term symbols, obtained by coupling the atomic terms with ²P, span from S to K with spin doublets, quartets, and sextets and 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 values branch in Oh into the three odd IRREPs , , and.

As derived in (49), the 2p XES decays in 1s2p RIXS start from either of the three intermediate state symmetries , , and and can reach , , and 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 the ground state symmetry.

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 Figure 29.

Figure 29: Multielectronic term scheme for () 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 for () for a crystal field of are shown in Figure 30. As can be seen from the corresponding RIXS maps, from the ground state symmetry, the quadrupole IRREP populates only states with symmetry, while the quadrupole Eg IRREP populates only intermediate states with Eg symmetry. The (⁴H) intermediate state symmetry corresponds to the electronic configuration with four 3d electrons in the orbitals, and as a result, the corresponding final states (between and ) are reached in the RIXS. These states correspond to spin-quartet final states and are weak in 2p XAS. The first IRREP intermediate state () leads to the final states between and , which correspond to the eg part of the final states with ()³(eg)¹. The other intermediate state symmetries at and lead to higher energy final states, where it is the RIXS spectrum of () 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 for for .

5.3.1. 1s2p RIXS Experiments of 3d³ Systems

Examples for 3d³ systems are and . Experimental 1s2p RIXS spectra of a number of 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]. 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:

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 is (210 microstates) having in high spin the atomic ground state term . In the crystal field, the ground state splits into the and terms, where is the lowest in energy. The inclusion of spin-orbit interaction yields the following splittings in the initial state:

The ground state splits due to SOC into several symmetries from which the term is the lowest in energy; hence, the ground state is ().

The intermediate state with 1s¹3d⁵ ( microstates) corresponds to the atomic terms of 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 crystal field which are analogue to the Tanabe-Sugano diagram of . Because the 1s XAS is spin-conserving, only quintet intermediate states are reached in the absorption from the quintet high spin ground state (). The atomic quintet terms are , , , , and which correspond to the crystal field terms , , , , and .

Starting from the ground state term, Table 3 gives the reachable intermediate state symmetries:

The result is that all quintet intermediate state symmetries can be reached in 1s XAS. This is confirmed in Figure 31 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 .

Figure 31: 1s XAS projection of a calculated 1s2p RIXS map for () with the corresponding ELD. The high spin to low spin transition at approximately 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 interaction yields the corresponding splittings in the intermediate state:

Subsequently, the 2p XES decays can start from any of these terms.

The final state () contains microstates due to the hole in the 2p shell spanning from 0 to 8. In an Oh crystal field, these terms branch into the five total symmetry IRREPs , , Eu, , and . 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 IRREP 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 for () 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 the total symmetry IRREP term and using the dipole IRREP, only final states with symmetry are reachable.

The calculated 1s2p RIXS, separated into the contributions via the quadrupole IRREP and Eg, together with some slices is shown in Figure 33.

Figure 33: Calculated 1s2p RIXS map for () with a crystal field for 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 Figure 33 with the results found in expressions (53) and (54) confirms the 1s XAS selection rules. As predicted in expression (53), the quadrupole IRREP populates the intermediate state symmetries and . While expression (54) showed for the quadrupole IRREP Eg, only the intermediate state symmetries , , and can be reached.

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

For large crystal field energies (10Dq), the initial state electronic configuration has a low spin (LS) ground state. The atomic Russel-Saunders term ³D translates into a ground state with 9 microstates. The inclusion of 3d SOC () yields the initial state splittings:

The ground state splits due to 3d SOC into several symmetries from which the term is the lowest in energy; hence, the low spin ground state is (Oh).

The intermediate state electron configuration 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 () being a spin-triplet, the spin-conserving 1s XAS only reaches spin-triplet intermediate state symmetries:

Here, all intermediate state terms, except , are reachable in 1s XAS. The exclusion of the orbital symmetry term , 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 for () in low spin.

Applying the 3d SOC () to the reachable intermediate state terms yields all even total angular momentum IRREPs.

The final states () are again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs , , , , and .

For the 2p XES decay, we find again from expressions (28), (29), (30), (31), and (32) that all even symmetries (, , Eu, , and ) can be reached. In 2p XAS, starting from the total symmetry term and using the IRREP, we find that only final state IRREPs can be reached. The selection rules are summarised in the combined term scheme in Figure 35.

Figure 35: Multielectronic term scheme for (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 () of () is shown in Figure 36.

Figure 36: Calculation of 1s2p RIXS and 2p XAS for () for (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 (), we see the high spin to low spin transition at around 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 for (3d⁴). Note the HS-LS transition at approximately . 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 spin 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 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 , one can select specific intermediate states.

We observe that the quadrupole IREEP can reach only one (³I) intermediate state symmetry, while it cannot be reached with the quadrupole Eg operator. This is consistent with the corresponding electron configuration of the IRREP in the 3d shell: .

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 , depending on the RIXS intermediate states that are probed, one can enhance either the transitions into the 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 of 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 an ion () ( 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:

Also in ground state systems like or , high spin to low spin transitions can be observed. Thus, for large values of 10Dq, the ground state changes from for to for .

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

The initial state configuration (252 microstates) has in high spin the ground state term ⁶S (Table 1) branching in Oh into a symmetry (6 microstates). The spin-orbit interaction yields the following splittings of the ground state term:

Here, the term lies the lowest in energy; thus, the ground state is (Oh). However, the splitting is small (few meV) and thus also the term would contribute weakly in a measurement.

The intermediate state with ( 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 . In 1s XAS, due to the weak 3d SOC starting from the ground state, only sextet spin terms and arising from the only atomic sextet spin term ⁶D are reached (Table 3):

The corresponding 1s XAS projections for a 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 (). In both cases, the excited 1s electron can go into either the orbitals or the orbitals. The drastic difference is that for , all spin-up orbitals are full in the total symmetric IRREP ().

Figure 38: 1s XAS projections for () with the corresponding ELD. Notice the high spin to low spin transitions in the ground and intermediate states between and 3.3 eV.

For the 2p XES decay, spin-orbit interaction is again crucial; hence, we include 3d spin-orbit coupling for the reachable intermediate state terms:

Due to the spin () and the resulting splittings, all three odd IRREPs (, , and ) appear more than once. Thus, the2p XES can decay from any of the three IRREPs , , or . As given in (33), (34), and (35), the dipole IRREP also enables to reach all three odd symmetries , , and in the final state.

The direct 2p XAS on the other hand starts from the 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 intermediate state symmetry. The resulting overall selection rules for in high spin are summarised in the combined term scheme in Figure 39.

Figure 39: Multielectronic term scheme for high spin comparing the1s2p RIXS and direct 2p XAS pathways. Where SOC is included (), the spin multiplicity is omitted.

The calculated 1s2p RIXS maps shown in Figure 40 confirm the selection rules as summarised above. The two RIXS maps corresponding to the two quadrupole IRREPs and (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 and components of the final state can, via the intermediate state, be selectively probed in 1s2p RIXS.

Figure 40: Calculated 1s2p RIXS spectrum for () for (high spin).

This example is interesting, because the 1s XAS transitions are identical to the () case. The ground state is totally symmetric with ; thus, the quadrupole IRREPs and populate the corresponding intermediate symmetries ( and ). However, there are two important differences: (i) for the high spin configuration of , 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 of has, due to the six electrons in the 3d orbitals, 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 the 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 configuration translates in low spin to with the atomic ground state term (10 microstates). In Oh, this translates into a ground state term with 6 microstates. The inclusion of 3d spin-orbit interaction yields the following splitting:

Here, the term lies the lowest in energy; thus, the ground state is (Oh).

The intermediate state configuration 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 (), the spin-conserving 1s XAS only reaches spin doublet intermediate state IRREPs (Table 3):

This means that from the ground state the quadrupole 1s XAS, as shown in Figure 41, can reach all even intermediate state symmetries, except any IRREPs.

Figure 41: Low spin 1s XAS projections ( to ) for () with ELD. In the single electron picture the 3d shell has either for or for and , respectively.

The reachable intermediate state terms split due to the 3d spin-orbit interaction :

We find all three half-valued terms from which the decays in 1s2p RIXS can occur.

The final state () is again identical to the high spin case regardless of the crystal field effects, offering the three total symmetry IRREPs , , and. 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 the ground state, only two final state terms are reachable:

The results of the selection rules as discussed above are summarised in the combined term scheme shown in Figure 42.

Figure 42: Multielectronic term scheme for () 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 () is given in Figure 43.

Figure 43: Calculated 1s2p RIXS spectrum for () for (low spin).

In the left RIXS map in Figure 43(a), it appears that the quadrupole IRREP enables to reach only the first fully symmetric intermediate state symmetry. This term corresponds in the single electron picture to a configuration of the 3d shell where all the 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 the 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 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 orbitals. The L₂-edge does not get a peak because it would require a spin-flip to fill the 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 IRREP (1s XAS operator).

Figure 44: 2p XES projections for () for to . The high spin to low spin transition between 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 intermediate state symmetry around and .

5.5.3. 1s2p RIXS Experiments of 3d⁵ Systems

Examples of systems include and . 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 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 systems Fe₂O₃ and iron impurities in MgO [14]. The tetrahedral 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,

In this part, we present the elementary differences between 1s XAS and 1s2p RIXS for the case of (; 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:

As in the previous case, the crystal field splitting energy (10Dq) and the pairing energy are in competition in a configuration. Thus, for sufficiently high values of 10Dq, the ground state changes from high spin for to low spin for .

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

The initial state electronic configuration is ( microstates) with the atomic ground state term ⁵D (25 microstates). In an Oh crystal field, the ⁵D term splits into for and for . The inclusion of spin-orbit interaction yields the following splittings:

Here, the term lies the lowest in energy; thus, the ground state is (Oh).

The intermediate state electron configuration ( microstates) yields the atomic terms , , , , and (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: , , , and , respectively. For 1s XAS, starting from the ground state term, the reachable high spin intermediate state terms are

The reachable intermediate state terms of () in high spin are therefore and . In other words, though the and symmetries are formally reachable in 1s XAS, the intermediate state does not offer this IRREP to be populated. Furthermore, though an 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 for () showing the high spin to low spin transition just above .

Adding SOC to the reachable IS terms leads to the splittings in the IS:

We find the decays in 1s2p RIXS can occur from all integer symmetries (, , , , and).

The final state electron configuration contains microstates due to the hole in the 2p shell spanning from 0 to 7. In an Oh crystal field, these terms branch into the five integer total symmetry IRREPs , , , , and. 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 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 the total symmetry IRREP term and using the IRREP, the reachable final states are again identical to the 2p XES final state term reachable from the intermediate state IRREP as given (32).

Altogether, this enables to draw the term scheme for (3d⁶) in high spin as shown in Figure 46.

Figure 46: Multielectronic term scheme for () (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 spin () for is shown in Figure 47.

Figure 47: Calculated 1s2p RIXS spectra and slices for () for (high spin).

We note that the quadrupole IRREP (left RIXS map in Figure 47(a)) is dominated by transitions into the intermediate state symmetry, while the quadrupole Eg IRREP (middle RIXS map in Figure 47(a)) leads mostly into the intermediate state IRREPs. As predicted above, the 1s XAS spectra gain their intensity only from two intermediate state symmetries, the and IRREPs. The 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 (), a 3d⁶ electronic configuration results in a low spin configuration with all orbitals fully occupied (). This corresponds to the totally symmetric ground state term (1 microstate) derived from the atomic term ¹I (Table 1). The inclusion of spin-orbit interaction gives the total ground state as (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):

Although there are and IRREPs existing in the intermediate state, only the derived from the ¹G atomic term can be reached. This can be understood from the quadrupole selection rule, which allows only to reach the ¹G atomic term () from the atomic ground state term (). The quadrupole 1s XAS formally can, according to the selection rules, only reach the and IRREPs from this term. But finally, the transition to the intermediate state symmetry would imply a two-electron transition from to , which has a very low probability. Hence, only the 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 for () showing low spin part above 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 considered leading the identities:

The final state electron configuration is again identical to the high spin case regardless of the crystal field effects, offering the five total symmetry IRREPs , , , , and . The 2p XES decaying from the populated intermediate state term can reach with the dipole IRREP the final state terms and as given in (30).

On the contrary, the 2p XAS, starting from the total symmetric ground state symmetry, can reach only the 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 Figure 49.

Figure 49: Multielectronic term scheme for (3d⁶) in low spin comparing the 1s2p RIXS and direct 2p XAS pathways. We note its similarity to the (3d⁰) case shown in Figure 15.

The resulting calculated 1s2p RIXS for (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 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 final state IRREP which can only be reached in 1s2p RIXS.

Figure 50: Calculated 1s2p RIXS spectrum for () for (low spin).

5.6.3. 1s2p RIXS Experiments of Systems

3d⁶ systems include and . A large number of experimental 1s2p RIXS spectra of systems have been published. Vercamer measured a large series of minerals in his PhD thesis [44]. The tetrahedral and octahedral oxides/minerals have been studied and reproduced by multiplet calculations [5, 17, 46]. Guo et al. [47] studied low spin 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 and and the laser-excited spectrum of show a partial transition to a high spin ground state as is confirmed in the calculations.

In case of , all measured systems were low spin . Vanko et al. [7] compared an isolated 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,

In this part, we present the elementary differences between 1s2p RIXS and 2p XAS for the case of (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:

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 a ion () 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 for (high spin) to for (low spin).

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

For the initial state configuration (120 microstates), the high spin ground state term symbol is ⁴F (Table 1), which branches in Oh into , , and , where the term with 12 microstates is the lowest in energy. The inclusion in spin-orbit interaction () yields the subsequent splitting of term:

The ground state splits due to 3d SOC into the three odd symmetries , , and from which the term is lowest in energy; hence, the combined ground state is (Oh).

The intermediate state electron configuration is ( microstates). The atomic terms are , , , , 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 . The spin-conserving 1s XAS reaches only spin quartet intermediate states, which corresponds to the crystal field terms issued from and ⁴P: , , , and , respectively. Starting from the high spin ground state term , the two quadrupole IRREPs give the reachable intermediate state terms:

The reachable 1s XAS intermediate state terms in high spin are therefore , , and . Here, it shall be noticed that an symmetry would be reachable by the quadrupole selection rules, but the intermediate state does not offer any (quartet) symmetries. The 1s XAS projections displayed in Figure 53 confirm these findings.

Figure 53: 1s XAS projections of a calculated 1s2p RIXS spectrum for () 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 .

Applying the 3d spin-orbit interaction () to the three reachable terms yields the splittings in the intermediate state:

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 microstates due to the hole in the 2p shell spanning from 1/2 to 11/2. In an Oh crystal field, these terms branch into the three total symmetry IRREPs , , and . The 2p XES starts from either , , or , 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 the ground state term, can only reach the and final state terms (Table 4).

A summary of the findings above is shown below as a combined term scheme in Figure 51.

We conclude this part with the corresponding calculations of the 1s2p RIXS for () for a crystal field energy of which is shown in Figure 54.

Figure 54: Calculated 1s2p RIXS spectrum for () for (high spin).

It can be noted that the RIXS map for the quadrupole IRREP (Figure 54(a), left) is dominated by the 1s XAS transitions into the intermediate state symmetry which corresponds in the single electron picture to a configuration. However, as given in expression (75), it has also contributions into the two intermediate state symmetries though they are weak with respect to the 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 relating to the decay from the intermediate state symmetry corresponds to the two final state terms and . This can be understood considering that the branches due to SOC into a intermediate state symmetry, which can reach the above two final state terms via the dipole IRREP.

The quadrupole IRREP on the other hand shows only contributions into the and intermediate state symmetries as predicted in expression (76).

And finally, it should be noted that the 1s XAS projections in Figure 53 show with increasing crystal field energy the branching of the into the three IRREPs , , and , while the atomic term translates in an crystal field only into a symmetry.

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

At high crystal field energies, the initial state gives the low spin ground state term ( microstates) arising from the atomic term which corresponds to the electronic configuration . The inclusion of spin-orbit interaction () yields the subsequent splittings:

Hence, the combined ground state is (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 , , , , and and , , , , and . The quadrupole 1s XAS starts from the low spin ground state term revealing the reachable terms:

Thus, according to the quadrupole selection rules, from the 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 orbitals are occupied, 1s XAS transitions can only go into the orbitals via the quadrupole IRREP. The transitions into the orbitals with the quadrupole IRREP are less probable as they require a two-electron transition. In other words, the matrix elements of the quadrupole IRREP are larger than those of the quadrupole 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 a ion (3d⁷) with to (low spin only).

For the 1p XES decays, the inclusion of 3d SOC () is again required, leading to the following splittings of the five reachable terms in the intermediate state:

Hence, the 2p XES decays in 1s2p RIXS can occur from any of these three total angular momentum IRREPs , , or .

The final state electron configuration is again identical to the high spin case regardless of the crystal field effects, offering the three total angular momentum IRREPs ,, and . For the 2p XES decaying from the populated intermediate state terms, using the IRREP (Table 4) again reveals that all three final state IRREPs are also reachable. Furthermore, in 2p XAS, starting from the 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 Figure 52.

Again, we conclude with an example calculation for a ion ( ground state configuration) with reflecting the low spin case as shown in Figure 56.

Figure 56: Calculated 1s2p RIXS spectrum for () for (low spin).

It is noteworthy that the quadrupole IRREP here is very weak () with respect to the 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 orbitals, because the quadrupole IRREP populates the intermediate state symmetries and , which both require a hole in the level.

This is also confirmed by the total RIXS map (Figure 56(a), right) showing that the overall result is dominated by the transitions into the three intermediate state symmetries , , and , while the and intermediate state symmetries are negligible due to the reason given above.

However, as predicted by the selection rules for the quadrupole IRREPS, and as shown in Figure 56 confirm the predicted reachable terms as found in expression (79).

5.8. 1s2p RIXS Experiments of Systems

A resolution 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 the (), (), and (), 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., ) 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 every ground state case (with ). 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., or SOC) are scaled. For the crystal field parameter , 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 the , , or 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 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 the 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 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 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.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this article.

Acknowledgments

The authors wish to acknowledge Maurits Harverkort (Heidelberg) and Robert Green (UBC) for their support on the Quanty software and answering all questions always quickly in a very detailed manner. For financial support, they acknowledge the European Research Council (ERC advanced Grant 340279).

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