Review Article  Open Access
1s2p RIXS Calculations for 3d Transition Metal Ions in Octahedral Symmetry
Abstract
We present a series of 1s2p resonant inelastic Xray scattering (RIXS) calculations for 3d transition metal ions in octahedral symmetry covering each ground state between 3d^{0} and 3d^{9}. The calculations are performed in octahedral (O_{h}) symmetry using the crystal field multiplet theory. We discuss the crystal field effects and the selection rules with respect to the 1s2p RIXS preedge and compare their final state energies with the corresponding 2p Xray 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, Xray absorption spectroscopy (XAS) as well as Xray 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 highresolution measurements accessible to researchers worldwide. Especially, the second order resonant inelastic Xray scattering (RIXS) process, where the Xray emission spectra are measured as a function of the incident Xray 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 twodimensional 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 twodimensional 1s2p RIXS (1s XAS, 2p XES) and the onedimensional 2p XAS, are illustrated in Figure 1.
(a)
(b)
We limit our discussion of 1s2p RIXS to the preedge 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 preedge structure in Kedge 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 preedge 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 preedge structure and produce additionally the socalled nonlocal peaks [3–8] (e.g., TiO_{2} [3] and CrO_{2} [9]).
When there is no inversion symmetry, the quadrupole (1s → 3d) and dipole (1s → 4p) peaks can mix locally in the preedge leading to an additional dipole contribution to the quadrupole peaks. For example, distortions or vibrational excitations can break the local inversion symmetry of O_{h} leading to some local dipole character in the preedge [10–12]. In addition, the tail of the much stronger dipole character of the main edge (1s → 4p) overlaps, but the Xray 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 preedge 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)Highresolution wavelengthdispersive detection of the emission enables for the socalled highenergy resolution fluorescencedetected (HERFD) spectra. The advantage here is the effective suppression of the corehole lifetime broadening due to the twostep 1s XAS → 2p XES process.(2)The final state configuration 1s^{2}2p^{5}3d^{N + 1} is identical in 1s2p RIXS and 2p XAS, which enables to access “L_{2,3}edge information” with hard Xrays. 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 twostep 1s2p RIXS process with the direct 2p XAS (L_{2,3}edge) spectra for 3d transition metal compounds. Based on the crystal field theory (CFT), we calculated at least one spectrum for each 3d^{N} ground state, where . We aim to give an overview of the general structure of 1s2p RIXS for the 3d^{N} 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 spinorbit coupling with the other open shells. As we will see, this is important when comparing 1s2p RIXS with 2p XAS [13]. The consequences on the outcome of the spectra will be discussed throughout this paper.
The paper is organised as follows. In Section 2, we describe the theoretical background of the calculation of 1s2p RIXS and the resulting selection rules. In Section 3, we give the computational details. In Section 4, we analyse the 1s2p RIXS of three didactic cases in more detail, and in Section 5, we describe a series of the remaining 3d^{N} cases including a short discussion of related experiments.
2. Theory
The necessary background being the theoretical framework and the computational tools used in this study are described in the following.
2.1. 1s2p RIXS with KramersHeisenberg
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 wavelengthdispersive detector (crystal analyser) reaching in the hard Xray range subeV 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 corehole lifetime broadening which is related to the RIXS crosssection [14, 15].
The RIXS process is described as a twostep photonin photonout scheme in which the incident photons undergo an inelastic scattering around a core resonance followed by a characteristic Xray emission. The theoretical modelling of the 1s2p RIXS crosssection σ is achieved with the KramersHeisenberg 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 twostep process relates to the two transition operators, for the quadrupole 1s XAS and for the dipole 2p XES, describing the 1s2p RIXS crosssection σ as the transition from the initial state into the final states via all possible intermediate states of the system with energies E_{i}, E_{f}, and E_{n}, 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 r_{i} and charge Z, is the electronelectron 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 spinorbit interaction on every open shell. These first four terms, together with a given ground state , describe the isolated atom in each state [21].
The electronelectron 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 RussellSaunders coupling, which will be used throughout the present paper (Table 1). In the absence of spinorbit coupling, all terms with the same and have the same energy, giving an energy level that is fold degenerate. When spinorbit 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 socalled RusselSaunders term symbol , where X is representing the orbital momentum ; for values of 0, 1, 2, and 3, one writes S, P, D, and F, respectively [21, 22]. For example, a single electron in an s shell is given as , and a single electron in a p shell is represented as and .

The term symbol describes the symmetry aspects, but it does not say anything about its relative energy [14]. The electronelectron repulsion and the spinorbit 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 socalled octahedral field, which belongs to the O_{h} point group. Thus, we discuss all ions throughout this paper in an octahedral symmetry (O_{h}). We are aware that this is not necessarily true for all 3d transitions metal ions (e.g., CrO_{2} (Cr^{4+}) and TiO_{2} (Ti^{4+}) have tetragonal (D_{4h}) symmetry; Fe_{2}O_{3} (Fe^{3+}) has C_{3} symmetry). However, we make this simplification since we are aiming to present and discuss the elementary differences for each case in an analogue and comparable way.
In a singleelectron picture, in a spherical environment, the 3d orbitals are degenerate. In O_{h}, the cubic crystal field induces the splitting of the five 3d orbitals into two symmetrically different groups: (1) the 3fold orbitals (, , and ), which point in between the six metalligand bonds and thus participate in bondings with the surrounding ligands, and (2) the 2fold orbitals ( and ), which point along the metalligand 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 (O_{3}) to octahedral symmetry (O_{h}) causing the term (S, P, D, etc.) to branch to an O_{h} irreducible representation (IRREP): S (O_{3}) branches into an A_{1} (O_{h}) symmetry state, P (O_{3}) branches into a T_{1} (O_{h}) symmetry state, and D (O_{3}) branches into T_{2} (O_{h}) and E (O_{h}) symmetry states. Because the point group O_{3} (resp., O_{h}) 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 3d^{N} initial state and the intermediate state 1s^{1}3d^{N + 1} in 1s2p RIXS are gerade (g), and the final state 2p^{5}3d^{N + 1} is, due to the open 2p shell, ungerade. In the following, the parity of the spherical term will be omitted for simplicity. All the branchings are given in Table 2, which we will use extensively throughout this paper.

The relative energies of the O_{h} IRREPs are calculated by adding the effect of the cubic crystal field 10Dq to the atomic state energies. The diagrams of the relative energies with respect to the cubic crystal field, often without spinorbit coupling, are known as TanabeSugano diagrams. More generally, we will call them energy level diagram (ELD). They will appear in the context of the quadrupole 1s XAS because the ELD for any intermediate state in 1s2p RIXS with a 1s^{1}3d^{N + 1} electron configuration is the same as the diagram for a 1s^{2}3d^{N + 1} configuration times the term to include the 1s core hole.
When spinorbit coupling (SOC) and crystal field are both to be included, there are in principle two ways to derive the term symbols: (1) by first deriving the branching of the uncoupled atomic terms (with and ) in O_{h} symmetry and then by deriving the spinorbital coupling or (2) by first deriving the atomic quantum number in the coupling scheme and then deriving the branching of the value in the O_{h} point group. Both methods yield identical final results and the identical ground state for a given Hamiltonian (see, e.g., Figure 2).
The order in which the derivation should be done is related to the relative weights of the spinorbit coupling and crystal field terms in the Hamiltonian. In the initial state of 1s2p RIXS, the 3d spinorbit 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 spinorbit 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 spinorbit coupling. Therefore, in these cases, the coupling is achieved after the crystal field branching. (See Xu et al. [23] for details on coupling.)
In the present paper, we are focusing on the 3d transition metal ions represented by a series of cases with a ground state electron configuration ranging from 3d^{0} to 3d^{9}. Depending on the number of 3d electrons, the value of the total spin varies between 0 (minimum) and 5/2 (maximum). The branching for each value of in O_{h} is given in Table 2. The coupled term, also called value IRREP, in O_{h} 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,



On the contrary, in the final state configuration, the strong spinorbit coupling of the 2d hole (a few eV) is dominant. This induces a strong mixing between the spin and orbital momenta and resulting in and not being good quantum numbers anymore. Instead, the total angular momentum must be used to describe the final states. The effect of the O_{h} crystal field is added as the corresponding branching of the atomic value IRREP. The branching for each value of is also given in Table 2. The obtained IRREPs are either , , , , or for integer or . And they are , , and for halfinteger values of or .
As we will see across the 3d^{N} series, the information of , , and is important for the description of the ground state of the initial state electron configuration when comparing the onestep 2p XAS with the twostep 1s2p RIXS selection rules (see Section 2.3). Therefore, in an O_{h} 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 Ni^{2+} (3d^{8}) with the atomic ground state term ^{3}F (O_{3}).
Note that the spin multiplicity should be kept and is part of the term symbol because spinorbit coupling is not yet included. The symmetry represents in this case the state being the lowest in energy, and it is therefore the ground state term. In the next step, we derive the total angular momentum IRREPs in the coupling scheme () for the three terms derived in (5):
In (6), we find the total angular momentum IRREP as the ground state for the example 3d^{8} in an octahedral crystal field. To summarise all this information in a condensed form, we write
This translates as a term without SOC with and [term with SOC using ].
In other words, in (9), the term is before the inclusion of SOC, where the spin multiplicity is 3 and the orbital momentum is given as . And the term is the symmetry after spinorbit interaction has been included (also identified as value IRREP).
Finally, we note that charge transfer effects (e.g., the interaction with a configuration) are neglected but their relative importance will be discussed where appropriate throughout the 3d^{N} series.
2.3. Selection Rules
The selection rules are the constrains that define the possibility of a transition between two states with a given transition operator. In other words, for 1s2p RIXS, the selection rules are the conditions for which the matrix elements and in the KramersHeisenberg 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 WignerEckart 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 firstorder 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 secondorder term of this decomposition, the resulting selection rule translates to .
However, when the spinorbit 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 spinconserving () and only changes. Due to the fact that light carries an orbital momentum of , this implies for electric dipole transitions or −1. For the secondorder term, the electric quadrupole transitions, this implies , or −2 [21].
As a result, the transition operator of the 1s XAS absorption step promoting an electron from 1s to 3d () is an electric quadrupole term, and this step is forbidden as an electric dipole transition. On the contrary, the transition operator for the 2p XES decay from 2p to 1s () is an electric dipole term. Similarly, the transition operator of the 2p XAS absorption step from 2p to 3d () is also an electric dipole term.
These selection rules, defined in spherical symmetry, translate into the point group symmetry of the absorbing ion. In cubic symmetry (O_{h}), the selection rules of each operator depend on its symmetry properties. The electric dipole operator (firstrank tensor) behaves as the IRREP (O_{3}) and branches into in O_{h}. The electric quadrupole operator (secondrank tensor) behaves as a IRREP (O_{3}) and branches in into (see Table 2). The transition matrix element between two states with the IRREPs and and the transition operator is nonzero if the direct product contains , where is the IRREP of the transition operator. In other words, the final state IRREPs accessible via the transition operator are given by the direct product . The selection rules are derived from the direct product tables for the O_{h} 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 3d^{N} series throughout this paper. Magnetic and natural dichroisms, which are a property from the crystal and not from the point group of the absorbing transition metal ion [26], will not be discussed here.
3. Calculations and Computational Details
All calculations are done using the framework of the crystal field multiplet theory, which is a multielectronic, semiempirical approach initially developed by Thole et al. [27] and further established by Butler and Cowan [28, 29]. It takes into account all the 3d3d, 1s3d, and 2p3d electronic Coulomb interactions, as well as the spinorbit 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 manybody 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 HartreeFock values. The latter has empirically proven to correspond to the actual atomic values. For the spinorbit coupling parameters , the HartreeFock calculated values are used. All calculations are done in octahedral symmetry (O_{h}), 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 O_{h} 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.

All RIXS intensities are given as calculated by Quanty. They reflect the total absorption as the sum over the five quadrupole basis components and the three dipole emission polarisations and thus imply an isotropic spectrum. When needed, one can in fact also compute the scattered photons as the percentage of the incident beam in dependency of a given experimental setup. The 2p XAS spectra are scaled for the best comparison for each case. The energies E_{in} and E_{out} are given by Quanty relative to the centre of gravity for each of the three states in 1s2p RIXS, being the 3d^{N}, 1s^{1}3d^{N + 1}, and 2p^{5}3d^{N + 1} configurations, respectively. Thus, also the position of the energy transfer is based on the centre of gravity and E_{T} 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 spinorbit interactions as well as crystal field effects can induce a mixing of different states, leading to complex multielectronic states involving many microstates. For example, a 3d^{5} configuration has in total 252 microstates, but only microstates form in high spin the atomic multielectronic ground state ^{6}S (O_{3}) (Table 7).

However, at absolute zero (), formally, only the state lowest in energy is populated. Thus, the calculated spectra result from a linear combination of each contributing state, weighted according to their respective population, because for temperatures T > 0 K higher states are also partially occupied. The population is described with the Boltzmann distribution.
It gives the population probability p_{i} for each state in dependency of its relative energy , with being the energy of the lowest microstate , and absolute temperature T. It is given (with Boltzmann constant k_{B}) 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, Ti^{3+} (3d^{1}) has the atomic ground state ^{2}D with microstates. In other words, the single 3d electron can be in 10 different states. In an O_{h} crystal field, this ground state branches without 3d spinorbit interaction into two crystal field terms: .
The term represents the multielectronic ground state in an O_{h} crystal field, formed as a linear combination of microstates. For 3d^{1}, the term is the lowest in energy as can be seen in the single electronic picture: , as opposed to the with in 4 possible microstates.
Furthermore, it should be noted that the microstates forming the multielectronic ground state are a priori not degenerate. For Ti^{3+}, the lowest 10 microstates are in fact partially degenerate. With 3d spinorbit interaction included, the microstates are spread in a 4 : 2 : 4 ratio, corresponding to the terms F_{3/2}, E_{5/2}, and F_{3/2}, respectively. In other words, the lowest four microstates , , , and are degenerate in energy (), and the two states and are degenerate (), as well as the remaining 4 microstates , , , and (). Hence, the 10 microstates, which are related to the atomic multielectronic ground state ^{2}D (O_{3}) and are split by the crystal field and spinorbit interaction, are spread over three different energy levels. However, the six microstates forming the (O_{h}) term are only spread over two different energies (see Figure 3).
Note, if not stated otherwise, we only use the lowest crystal field ground state for the Boltzmann linear combination (e.g., for Ti^{3+}); the next multielectronic term higher in energy (e.g., for Ti^{3+}) is neglected. This approximation relates to the fact that in general the second multielectronic term has a relatively small contribution in most cases.
4. Didactic Examples
In this section, we discuss three model systems as didactic examples in detail before the results for the remaining 3d^{N} series (with ) which are discussed in a more summarising manner (Section 5). Here, we will emphasise the elementary differences between the atomic (O_{3}) and the crystal field (O_{h}) 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 onestep 2p XAS and the twostep 1s2p RIXS and the corresponding spectra.
As didactical examples, we use the following three cases: (i)3d^{9} Cu^{2+}: single open shell with only a single hole in each state(ii)3d^{0} Ti^{4+}: two peaks in the 1s XAS split by the crystal field 10Dq(iii)3d^{8} Ni^{2+}: a common model system often used in education
We aim to assign the symmetry labels to the relevant peaks in the 1s2p RIXS spectra and compare the 2p XES to the wellunderstood L_{2,3}edge (2p XAS) spectra [35].
4.1. 3d^{9}: Divalent Copper Cu^{2+}—Crystal Field Effects Altering the Selection Rules
In this part, we investigate the elementary differences between 1s2p RIXS and 2p XAS for a divalent copper ion Cu^{2+} with a 3d^{9} ground state. Each successive step of the 1s2p RIXS process has an electronic configuration with only one open shell and a single hole in the 3d, 1s, and 2p shells, respectively.
We first describe the atomic case of an isolated ion without any crystal field. The electronic configuration of the initial state of Cu^{2+} is . In spherical symmetry (O_{3}), the 3d spinorbit coupling (SOC) induces the splitting of the 10fold degenerate ground state ^{2}D (10 microstates) into two multielectronic states (O_{3}) and (O_{3}), defined by (ground state) and , respectively. These two states are separated in energy by . The electronic configuration of the 1s2p RIXS intermediate state of Cu^{2+} is , which corresponds to the 2fold degenerate state (O_{3}) with . The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Cu^{2+} is , which splits due to the 2p spinorbit coupling into (lowest energy state) and separated by . This is summarised in the scheme in Figure 4.
Since the ground state of the initial state is , the electric dipole absorption operator (2p XAS) enables only to reach the final state (). In 1s2p RIXS, the selection rules of the electric quadrupole absorption for 1s XAS () enable to reach the intermediate state . The subsequent electric dipole emission (2p XES) enables to reach both and final states. As a result, for an isolated Cu^{2+} ion in spherical symmetry, the 2p XAS shows only one peak, while the 1s2p RIXS shows two peaks. The resulting spectra are shown in Figure 5.
In the vertical direction, the Kα_{1} and Kα_{2} decays (2p XES) appear as two separate peaks. These two peaks are separated by the 2p SOC , corresponding to the two resonant emission transitions from the intermediate state with to the final states with .
4.1.1. Considering a Cubic (Octahedral) Crystal Field
When the absorbing ion is embedded in a solid state, one has to take into account the crystal field potential created by the surrounding atoms. In the following, we will assume an octahedral (O_{h}) crystal field described by the parameter 10Dq. In a single electron picture, this crystal field induces the splitting of the 3d orbitals into and e_{g}. In the multielectronic formalism (neglecting the 3d spinorbit coupling in first approximation), the ^{2}D term of the initial state branches in the O_{h} point group into the ^{2}E_{g} () and () terms. Thus, the multielectronic ground state (at ) is ^{2}E_{g} (O_{h}). The intermediate state term ^{2}S (O_{3}) branches in an octahedral crystal field into (O_{h}) symmetry.
In the O_{h} point group, the electric quadrupole 1s XAS operator has and E_{g} 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 ^{2}E_{g} (O_{h}). In spite of the fact that many symmetries are in principle reachable via a quadrupole 1s XAS transition starting from the ground state symmetry ^{2}E_{g} (O_{h}), the electronic structure of a Cu^{2+} ion only offers (O_{h}) symmetry leading to a single peak in the 1s XAS projection.
While spinorbit coupling was neglected in this last step, it must be included in the description of the final state with the 2p core hole to understand the complete 1s2p RIXS process and to be able to compare it with the 2p XAS. Using the coupled terms obtained above in the atomic case, we derive their respective branchings in the octahedral crystal field (O_{h}).
The initial ground state (O_{3}) branches in octahedral symmetry (O_{h}) into (O_{h}). The ground state at is (O_{h}). The effects of the 3d spinorbit coupling and the O_{h} crystal field parameter 10Dq are illustrated in Figure 2.
The intermediate state term (O_{3}) branches in an octahedral crystal field (O_{h}) into (O_{h}). The final states (O_{3}) and (O_{3}) branch into (O_{h}) and (O_{h}), respectively. In octahedral symmetry, the electric dipole operator has symmetry and thus, using the product table (Table 5), we find the reachable final states in the subsequent electric dipole emission (2p XES):
The direct 2p XAS starts from the ground state yielding with the IRREP the reachable final state terms:
Altogether, this demonstrates that the crystal field enables both final states to be probed by 2p XAS and 1s2p RIXS. This is summarised in the scheme in Figure 6 and the corresponding calculated spectrum is shown in Figure 7.
The RIXS projections for various values of 10Dq show almost no difference in both directions (XAS and XES). It is noteworthy that the 2p XAS (L_{2,3}edge) approaches the two peaks in the 2p XES projections for large 10Dq values. This is due to the crystal field (10Dq) affecting the spinorbit interaction altering the 2p XAS selection rules.
In other words, the L_{2}edge probes in 2p XAS the amount of character in the ground state. In the atomic case (10Dq = 0.0 eV) with spherical symmetry (O_{3}) and at absolute zero , the ion is pure (D_{5/2}). In octahedral O_{h} symmetry, on the other hand, an increasing crystal field (scaled via 10Dq) mixes more and more character from the D_{3/2} into the ground state, resulting in a continuous visible increase of the L_{2} peak in 2p XAS.
And finally, it can be seen that the singleparticle limit with two peaks is reached in both cases, 1s2p RIXS and 2p XAS, respectively. The intensity ratio of the two peaks of 2 : 1 is given by the degeneracy of the (O_{3}) and (O_{3}) final states.
4.1.2. 1s2p RIXS Experiments of 3d^{9} Systems
Experimental 1s2p RIXS spectra of CuO have been published by Hayashi et al. [2]. They show the 1s2p Xray emission spectra from excitation energies before the Kedge, 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 preedge, the twopeaked 1s2p RIXS spectrum is visible, in addition to the background from the edge [2].
4.2. 3d^{0}: Tetravalent Titanium Ti^{4+}—1s XAS Peaks Split by 10Dq
In the following, we discuss the differences between 2p XAS and 1s2p RIXS for the case of tetravalent titanium (Ti^{4+}). Here, the initial state configuration 1s^{2}2p^{6}3d^{0} has no partially filled shells. This is interesting because the resulting selection rules are straightforward: the nature of the probed final states reflects the nature and symmetry of the transition operators. The transition into the intermediate state promotes a 1s electron into the 3d band yielding a 1s^{1}2p^{6}3d^{1} electron configuration. The final state in 1s2p RIXS and 2p XAS is in this case 1s^{2}2p^{5}3d^{1}.
The 3d^{0} case enables to describe the effect of the crystal field on the intermediate state, the interferences in the 1s2p RIXS process, and the more complex multielectronic effects in the final state. It provides an extent to L_{2,3}edge considerations of 3d^{0} ions previously described in [35].
4.2.1. Atomic Case with SpinOrbit Coupling
We start again with the case of an isolated Ti^{4+} ion (3d^{0}) where the effect of the solid state, that is, the crystal field, is neglected. The initial state configuration of Ti^{4+} is 1s^{2}2p^{6}3d^{0}. Since all shells are full, and . The coupled total angular momentum is thus , and the initial state symmetry is the totally symmetric term ^{1}S_{0}.
The electronic configuration of the intermediate state in 1s2p RIXS of Ti^{4+} is 1s^{1}2p^{6}3d^{1}, corresponding to the total orbital angular momentum and total spin angular momentum or giving the RusselSaunders terms ^{1}D and ^{3}D ( microstates). Considering the spinorbit interaction in the coupling scheme, one obtains for the ^{1}D term and for the ^{3}D term (Figure 8). The splitting of these four terms (^{1}D_{2}, ^{3}D_{1}, ^{3}D_{2}, and ^{3}D_{3}) is defined by the 3d SOC and exchange interaction , which are both small (32 meV and 46 meV, resp.). The calculated energy splitting is and is beyond the reach of current experimental resolution and not resolved in our calculations discussed here.
The electric quadrupole transition of the 1s XAS enables to reach the intermediate states with , though here 3d SOC is small and transitions to the spin triplet term ^{3}D_{2} will be weak, such that the 1s XAS is dominated by the transition in the spin singlet term ^{1}D_{2}.
The final state electronic configuration is 1s^{2}2p^{5}3d^{1} with two partially filled shells that have to be accounted for: the 2p and the 3d shell. The total orbital angular moment of the final state electronic configuration is . The total spin angular moment of the final state electronic configuration is . This gives in the atomic case the spin singlet and triplet terms ^{1,3}P, ^{1,3}D, and ^{1,3}F.
As previously described by de Groot et al. [35], when spinorbit interaction is neglected, the selection rules for the electric dipole transition in 2p XAS with and allow only to reach the singlet ^{1}P term from the ^{1}S initial state, leading to one peak only (the L_{2}edge). The L_{3}edge, corresponding to transitions into the spin triplet term ^{3}P, would have zero intensity. But in fact the strong 2p spinorbit coupling in the final state ( is used) leads to three peaks, two of them forming the L_{2} and L_{3}edges, and a third weak peak related to triplet transitions.
The derivation of the total angular momentum of the final state electronic configuration in the coupling scheme yields the term symbols illustrated in Figure 9.
The direct electric dipole transition in 2p XAS with ( forbidden) starts for Ti^{4+} from the total symmetric term ^{1}S_{0}, which allows to reach all final states with (P_{1} and D_{1}, marked with a red circle in Figure 9). The L_{2} and L_{3}edges are separated by the 2p spinorbit coupling corresponding to transitions into the ^{1}P_{1} (L_{2}edge) and ^{3}D_{1} (L_{3}edge) terms. More precisely, from the squared matrix elements, we find that the second peak is dominated by ^{3}D_{1} contribution of 60%, adding 36% of ^{1}P_{1} and 4% of ^{3}P_{1} character. The small third peak in the 2p XAS, at in Figure 10, is related to transitions into the ^{3}P_{1} terms. Though the resolution chosen in our calculations does not reveal them as individual peaks, the direct Coulomb and exchange terms are not negligible. As discussed by de Groot et al. [35], they lead to a splitting of the L_{2,3}edge into three absorption lines including a redistribution of the intensities.
In the 1s2p RIXS process, the 2p XES decays with starting from the intermediate states with (D_{2}) and enables to reach the final states P_{1,2}, D_{1,2,3}, and F_{2,3} with (terms shown in green Figure 9). Therefore, all values except 0 and 4 are reachable.
With this, one can draw for Ti^{4+} the atomic term scheme with spinorbit coupling for the twostep 1s2p RIXS and the onestep 2p XAS process as shown in Figure 11.
The intermediate state is the key element when comparing the direct 2p XAS and 1s2p RIXS. It enables to access additional terms in the 1s2p RIXS final state, adding multiple visible peaks in the energy transfer direction as shown in Figure 10. But in both cases, and 4 are not possible to reach.
In summary, so far, we have discussed the effect of the spinorbit coupling in the atomic case in spherical symmetry (O_{3}), but neglecting any crystal field. With the above, one can describe the 1s XAS with only one visible peak due to transitions into the ^{1}D_{2} intermediate state term. The 3d spinorbit coupling and exchange interaction G_{sd} lead to small splittings which are not visible in this plot. Furthermore, the small 3d SOC may induce some mixing adding weak transitions into the ^{3}D_{2} term.
The splitting of the 2p XAS is dominated by the large 2p SOC separating the L_{2}edge from the L_{3}edge, for example, the spin singlet and the triplet states. The 2p XES on the other hand consists of transitions from the intermediate state with IRREP (mostly ^{1}D_{2}) into the final states with IRREPs resulting multiple visible peaks in the energy transfer direction. All final state terms are also split by the exchange and direct Coulomb interactions, G_{pd} and F_{pd}, respectively.
4.2.2. Considering an O_{h} Crystal Field
We now consider the effect of an octahedral O_{h} crystal field. The totally symmetric initial state term ^{1}S_{0} branches in O_{h} into . Formal consideration of an O_{h} crystal field and 3d spinorbit interaction leads to the branching of the intermediate state terms ^{1}D and ^{3}D into eight terms. The scheme in Figure 12 illustrates these branchings of the terms due to an O_{h} crystal field and the 3d spinorbit interaction ( and ).
In spite of several available intermediate state terms, starting from the initial state and using the and E_{g} IRREPs, representing the quadrupole 1s XAS operator, and the product table (Table 3) reflecting the orbital momentum selection rules, one finds the accessible intermediate state symmetries.
In other words, from the totally symmetric ground state (O_{h}), only the E_{g} (O_{h}) and (O_{h}) 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 spinorbit coupling is small, the transitions to the spin singlet intermediate states are dominating (i.e., matrix elements are the largest), while the transitions via the IRREP from triplet ^{3}D should be weak. The 1s XAS is dominated by transitions to the E_{g} and terms from the ^{1}D_{2}.
In an O_{h} crystal field, the final state IRREPs with and 4 (see Figure 11) branch, respectively, into the coupled terms: , , , , and . The 2p XES in 1s2p RIXS from the two intermediate state terms and E_{g} 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 (O_{3}) in and octahedral crystal field (Table 2), the IRREP can be reached not only from the IRREP but also from the and IRREPs.
This results in only seven possible IRREPs in O_{h} symmetry, or in other words, only seven final states are reachable among the 60 available. The 2p XAS spectrum consists, in principle, of seven lines, as can be seen in Figure 14, where the calculated spectra for Ti^{4+} (3d^{0}) for a crystalfield 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 spinorbit coupling in the final state induces a mixing of the spin singlet and triplet states. This weakens the spin selection rule allowing also transitions into former spin triplet states (e.g., , , , and ).
(a)
(b)
This enables to draw the electronic term scheme in Figure 15 illustrating the different pathways in 1s2p RIXS and 2p XAS for 3d^{0} Ti^{4+} in O_{h} symmetry. The accessible 1s2p RIXS final state terms are given in (18), while, as discussed above, the 2p XAS can only reach final states with symmetry ().
Similarly to the 3d^{9} case (Cu^{2+}) discussed in Section 4.1, comparing the final states, it becomes clear once again that the key elements are the intermediate states from which the 2p XES decays and thus enables to reach more final states when compared to the direct 2p XAS.
This overall results in the 1s2p RIXS maps shown in Figure 14. The total RIXS map is decomposed into the E_{g} and quadrupole absorption components. The selective absorption towards each intermediate state is obtained. Experimentally, this dichroic behaviour has been observed for Ti^{4+} in cubic oxides such as the prototypal SrTiO_{3} [36]. The 1s2p RIXS spectra calculated for each of the two absorption peaks are shown (Figure 14(a)), and the 60 final states are labelled with their total angular IRREP (Figure 14(b)). This enables to see that both, the 1s2p RIXS and 2p XAS, probe the final state symmetry. We further observe that within the four main peaks in 2p XAS, the first peaks of the L_{3} (at ) and of the L_{2} (at ) are stronger via the absorption channel of the 1s2p RIXS than the E_{g} and viceversa: the second peaks of the L_{3} () and of the L_{2} (at ) are stronger via the E_{g} absorption channel of the 1s2p RIXS than via the . This can be understood in a single electron picture with the crystal field splitting of the final state: the lower states of the L_{3}edge contain the 3d single electron in the orbitals while the higher energy states have the 3d electron in the e_{g} orbitals and similarly for the L_{2}edge [37].
4.2.3. 1s2p RIXS Experiments of 3d^{0} Systems
Experimental 1s XAS spectra (e.g., HERFD) of titanium oxides in the solid state often show more than two peaks in the preedge, 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 socalled nonlocal peaks, where the 4p states from the absorbing ion mix with the 3d states of the neighbouring ions [3]. In addition, we note that for high valent ions such as Ti^{4+}, charge transfer is important for the detailed description of the 2p XAS and 1s2p RIXS spectral shapes [37].
Experimental 1s2p RIXS experiments have been published on TiO_{2} by Glatzel et al. [38] and Kas et al. [39]. The 1s2p RIXS planes show the quadrupole preedges and the nonlocal peaks. The 1s2p RIXS crosssection 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 Kedge on divalent calcium systems (Ca^{2+}), including octahedral CaO and cubic CaF_{2}.
4.3. 3d^{8}: Divalent Nickel Ni^{2+}—Single Peak in 1s XAS
In this part, we investigate the elementary differences between 2p XAS and 1s2p RIXS for a divalent nickel ion (Ni^{2+}, 3d^{8}). As we will show, the electronic configurations of the different steps of 1s2p RIXS present strong analogies with the case of Ti^{4+}; the main difference being the nature of the initial state. We start again with the atomic case by deriving the values as IRREPs.
The electronic configuration of the initial state of Ni^{2+} is 1s^{2}2p^{6}3d^{8} with two holes in the 3d shell. The multielectronic interactions lead to 45 states spread across the terms ^{3}F, ^{3}P, ^{1}G, ^{1}D, and ^{1}S [13]. In spherical symmetry (O_{3}), the atomic ground state is ^{3}F ( and ). The 3d spinorbit coupling induces the splitting of the ^{3}F term into the three terms:
From those three IRREPs with , the ^{3}F_{4} term with is the ground state term symbol.
The electronic configuration of the intermediate state in 1s2p RIXS of Ni^{2+} is 1s^{1}2p^{6}3d^{9} (one hole in the 1s shell and one hole in the 3d shell) which translates into the two terms ^{1}D and ^{3}D ( microstates). It is noteworthy that this is identical to the intermediate state of Ti^{4+} described above (Section 4.2) due to the electronhole equivalency. The 3d spinorbit interaction splits the two terms into the ^{1}D_{2} (for ) and ^{3}D_{1}, ^{3}D_{2}, and ^{3}D_{3} (for ) RusselSaunders terms. The resulting splittings distribute the four terms in energy over for which the 1s XAS will appear only as a single peak due to the resolution chosen in our calculations, analogue to the Ti^{4+} atomic case.
The difference in the 1s XAS step in 1s2p RIXS between the 3d^{8} Ni^{2+} and 3d^{0} Ti^{4+} cases arises from the spin multiplicity of the initial state. In both cases, the electric quadrupole absorption leads to a intermediate state. However, because of the small 3d spinorbit coupling (), the spin multiplicity is mostly conserved in the transition and the absorption from the spin triplet initial state of Ni^{2+} leads to the ^{3}D_{2} state. For Ti^{4+} on the other hand, mainly, the ^{1}D_{2} term is reached.
The electronic configuration of the final state in 2p XAS and 1s2p RIXS of Ni^{2+} is 1s^{2}2p^{5}3d^{9}. This is again strictly analogue to the final state of Ti^{4+}. The final state configuration corresponds to several atomic terms (^{1,3}P, ^{1,3}D, and ^{1,3}F) which are split further due to the strong 2p spinorbit 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