# Core-hole effect in the one-particle approximation revisited from density functional theory

Vincent Mauchamp, Michel Jaouen, Peter Schattschneider

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Vincent Mauchamp, Michel Jaouen, Peter Schattschneider. Core-hole effect in the one-particle approximation revisited from density functional theory. Physical Review B, 2009, 79 (23), pp.235106. 10.1103/PhysRevB.79.235106. hal-05487961

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# Core-hole effect in the one-particle approximation revisited from Density Functional Theory.

Vincent Mauchamp* and Michel Jaouen

Laboratoire de Physique des Matériaux (PHYMAT), CNRS UMR 6630,
SP2MI - Boulevard 3, Téléport 2 - BP 30179, 86962 Futuroscope Chasseneuil Cedex, France

Peter Schattschneider

Institut for Solid States Physics, Vienna University of Technology,
Wiedner Haupstrasse 8-10, A-1040 Wien, Austria

(Dated: April 20, 2009)

The strength of the core-hole effect in simulations of Electron Energy-Loss Near Edge Structures (ELNES) or X-ray Absorption Near Edge Structures (XANES) is investigated using ab initio calculations based on the Density Functional Theory. Calculations were performed using the WIEN2k and FEFF codes at different edges for the following model compounds: rutile TiO₂, MgO and for some transition metals (Ti, Fe, Co, Cu, Zn). We demonstrate that although a core-hole is always present in any core-level spectroscopy experiment, its effect is not always observable. To observe or not a core-hole effect in any experimental situation can, however, be predicted from the examination of the material's ground state electronic structure, especially of the states just above the Fermi level. Indeed, it is shown that the two important criteria governing the core-hole strength at a given edge depend firstly on the localization and on the character of the first empty states of the material under investigation, and secondly on the nature of the compensating charge necessary to maintain the crystal neutrality. These criteria are expected to give a good description of the core-hole effect in the one-particle approximation. However, nothing can be inferred as for the good/bad agreement between calculations and experiments for more complex cases such as excitations of delocalized semi-core states or of atomic-like multiplet structures.

PACS numbers:

Keywords: 61.10.Ht, 68.37.Lp, 79.20.Uv

## I. INTRODUCTION

Fine structures observed on spectra obtained from Electron Energy-Loss Spectroscopy (EELS)¹ or X-Ray Absorption Spectroscopy (XAS)² experiments can often be interpreted successfully thanks to well established theoretical methods.³,⁴ For instance, first principle calculations have been used for chemical bonding analysis,⁵,⁶ for studying electronic structure anisotropy,⁷,⁸ alloying effects,⁹ or local distorsion at interfaces.¹⁰ In condensed matter physics, such simulations are often performed by using electronic structure calculations based on the Density Functional Theory (DFT).¹¹ Within this framework, the fundamental quantity, the charge density, is calculated from one-particle wave functions (the Kohn-Sham orbitals - KS) obtained self consistently through the calculation of a potential that includes all correlation effects occurring inside the solid. However, the DFT being a ground state theory, one is always facing the problem of the impact of the core-hole created during the excitation process on the electronic structure and thus on the so predicted fine structures.

In the non-relativistic first Born approximation, and ignoring dynamical diffraction effects,¹² the inelastic scattering EELS cross section is given by:¹³

$$ \sigma(\omega)_{EELS} = \sigma_{Ru} \int S(\mathbf{q}, \omega) d\mathbf{q} \quad (1) $$

where the index Ru refers to the electron-electron

Rutherford scattering cross section. Despite being out of the scope of the present paper, note that Eq. (1) also describes Non-Resonant Inelastic X-Ray Scattering (NRIXS)¹⁴ if we replace $\sigma_{Ru}$ by the Thomson scattering cross section $\sigma_{Th}$. Therefore, EELS and NRIXS only differ by the coupling of the probe (EELS: electrons, NRIXS: photons) interacting with the system, but both carry the same informations (see Ref. 15 and references therein).

The Dynamical Form (or Structure) Factor (DFF) $S(\mathbf{q}, \omega)$ describes the excitation of the system from its initial state $|\Psi_i\rangle$ to the final states $|\Psi_f\rangle$. It is a function of the momentum transfer $\mathbf{q}$ and of the energy $\omega$ involved in the transition. If $|\Psi_i\rangle$ and $|\Psi_f\rangle$ are approximated to one-particle states, the DFF can be expressed in terms of the Fermi golden rule, e.g. a matrix element of a transition operator $\hat{A}$ between these two states. When considering a small momentum transfer, $\hat{A}$ can be expressed in terms of a dipole operator.²,¹⁶ Using atomic units (a.u.), it is then given by:

$$ S(\mathbf{q}, \omega) = \sum_f |\langle\Psi_i|\mathbf{q}\cdot\mathbf{r}|\Psi_f\rangle|^2 \delta(\omega - E_f + E_i) \quad (2) $$

Concerning XAS, using the above mentioned approximations, the absorption cross section can also be expressed in terms of the Fermi golden rule:¹⁷
$$ \sigma(\omega)_{XAS} = 4\pi^2\alpha\omega \sum_f |\langle\Psi_i|\varepsilon.\mathbf{r}|\Psi_f\rangle|^2 \delta(\omega - E_f + E_i) \quad (3) $$

where $\varepsilon$ is the electric field polarization vector and $\alpha$ the fine structure constant. The momentum $\mathbf{q}$ in an EELS experiment playing the same role as the polarization vector $\varepsilon$ in XAS, both spectroscopies can be interpreted using similar theories and give comparable results.18,19 Within the previously mentioned approximations, Electron Energy-Loss Near Edge Structures (ELNES) or X-Ray Absorption Near Edge Structures (XANES) spectra can be, to first order, correlated to the unoccupied density of states of the excited atom projected on the proper symmetries allowed by the dipole selection rules.20 It is worth mentioning that this formalism is no longer valid when many-body effects (core-hole photoelectron interaction, local-field effects due to the screening of the incoming field) require to go beyond the one-particle approximation, i.e. when $|\Psi_i\rangle$ and $|\Psi_f\rangle$ are not one-particle states. These two-particle interactions need to be treated using other theoretical schemes like the Bethe-Salpeter equation (BSE),21 or a combined BSE-TDDFT (time-dependent DFT) approach.22

It follows from equations 2 and 3 that the main task when modelling an experimental spectrum is the proper calculation of the KS orbitals $|\Psi_i\rangle$ and $|\Psi_f\rangle$; the important question is to know if one has to consider or not the contribution of the positive charge induced by the hole. Indeed, this perturbation can strongly modify the potential of the excited atom: the valence/conduction states can then be contracted and shifted in energy.23 The projected density of states can thus change and the overall shape of the spectrum be strongly modified even leading to excitonic peaks in the ELNES or XANES. Such modifications are generally well reproduced by DFT calculations that explicitly include the core-hole. This is illustrated by Moreau et al. in the case of the carbon K edge in lamellar $C_3N_4$ compounds,24 Jaouen et al. for the simulation of the boron K edge in hexagonal boron nitride25 or Jiang et al. when calculating the lithium and oxygen K edges in $Li_2O$.26

It is often believed that in insulators and semiconductors, the core-hole should be systematically included into the calculation since the valence electrons do not have the freedom to screen it.4 This is confirmed by studies of $Li_2O$, Si, MgO or $SiO_2$ where the systematic inclusion of the core-hole effect is necessary to reach satisfactory agreement with the experimental spectra.26,27 Counter-examples can however be found in the literature: no core-hole effect is observed at the titanium or oxygen K edges in rutile $TiO_2$,7,28 or at the oxygen K edge in NiO which are both semiconductors or insulators.29 As a corollary of this assumption, it is widely believed that no core-hole is needed for describing XANES or ELNES spectra related to pure metals because its screening by the valence electrons is efficient enough. Such an assumption is confirmed by studies of the lithium K edge in metal-

lic systems like pure lithium or $LiMn_2O_4$.30 However, this argument does not hold from a fundamental point of view when considering self-consistent field (SCF) approaches since the screening cloud should, by definition of SCF, come out naturally. This paradox is illustrated by the fact that the core-hole effect was found to be important at the platinum metal $L_{2,3}$ and aluminium metal K edges.31 A partial core-hole was also needed to properly simulate the copper $L_{2,3}$ edges,32 but the physics underlying such an ‘ad hoc’ treatment remains unclear.

Another argument that can be found in the literature is that the core-hole is better screened when the density of valence states is highly localized on the excited atom.33 This argument was addressed to explain the weakness of the core-hole effect at the carbon K edge in transition metal carbides.34 However, several studies report that the core-hole effect is very strong at the oxygen K edge in MgO or $Li_2O$ whereas the valence bands in these compounds mainly have an oxygen p character, or at the silicon K edge in pure silicon.26,27,35,36,37 A last argument is given by De Groot et al.38 The absence of a core-hole effect at the oxygen K edge for transition metal oxides is justified by the fact that the first empty states being essentially localized on the transition metal site, they are weakly affected by the core-hole on the oxygen 1s orbital.

To include or not the core-hole effect into the calculation of ELNES or XANES spectra is thus a recurrent question for two reasons: the interpretations of experimental spectra can be severely impaired if this effect is not properly taken into account and its incorporation is not always an easy task to handle. In fact, for electronic structure codes based on the Bloch waves formalism, to include a core-hole requires to built a supercell which drastically increases the number of atoms and thus the computation resources needed for the calculation.39,40

In short, no general rule can be deduced from all above cited examples, and the justifications for the core-hole strength at a given edge seem to vary from one system to another. The primary aim of this paper is to give a clear interpretation of the core-hole effect using DFT calculations and a general understanding of the underlying mechanisms governing its strength at a given edge. Let us emphasize that the present study is realized assuming the final-state rule, any discussion of the validity of which is beyond the scope of this paper. To do so, the WIEN2k (and its TELNES2 extension)41 and FEFF17 codes were used to compute the ELNES (or XANES) spectra for various systems. Three model systems were studied: rutile $TiO_2$ (r-TiO$_2$), MgO and pure 3d transition metals. The first two ones were selected since they were widely studied in the literature and, although being both oxides, they exhibit very different behaviour with respect to the core-hole effect. We expect these two systems to be representative of the two extreme cases that can be encountered in such compounds. Concerning transition metals, they also exhibit a wide range of behaviour allowing for a better understanding of the core-hole effect in metals. The response
of the electronic structure to the creation of the core-hole
has been investigated under three aspects: the compen-
sation charge mechanism occurring inside the solid due
to the creation of the core-hole, its impact on the excited
atom projected local density of states (LDOS), and the
influence of the core-hole depth on these modifications.

The paper is organized as follows. The first part deals
with the computational methods. After that, results con-
cerning the electronic structure modifications induced by
the core-hole are given. In the last part, the results are
discussed: two rules are obtained concerning the local-
ization and the symmetry of the levels perturbed by the
core-hole. These rules are illustrated by simulations re-
lated to the three above mentioned model systems.

## II. COMPUTATIONAL METHODS

### A. WIEN2k

WIEN2k calculations are based on a full potential
band-structure approach: valence or conduction KS wave
functions are described as Bloch states. Wave func-
tions (as well as potentials) are described in the whole
space using a muffin-tin (MT) approach consisting of
non-overlapping spheres centered at each atomic site, no
assumption are made in the interstitial region about the
shape of the potentials: WIEN2k is thus a full-potential
code. Inside MT spheres, the KS wave functions are de-
veloped on a linearized augmented plane wave (LAPW)⁴²
basis set or an augmented plane wave (APW) basis set
combined with local orbitals (APW+lo).⁴³ This second
approach was used in the present paper. Plane waves are
used in the interstitial space. The core states, needed for
the calculation of the matrix elements of equations 2 and
3 are calculated with a relativistic atomic code.⁴⁴,⁴⁵ Cal-
culations were performed within the Generalized Gradi-
ent Approximation using the Perdew-Burke-Ernzerhof's
potential for exchange and correlation.⁴⁶ ELNES and
XANES calculations were performed using the TELNES2
extension of WIEN2k. All the parameters used for the
calculations are given in table VI of the Appendix.

Two different approaches can be used to take the core-
hole into account in the SCF calculation: removing a
given core electron of the excited atom or replacing the
excited atom by its right neighbour in the periodic table
(the so called Z+1 approximation). In the present pa-
per, the first approach was systematically used excepted
for Zn where the Z+1 approximation has been consid-
ered. Due to periodic boundary conditions, 2*2*2 super-
cells were considered systematically in order to minimize
mutual interactions between adjacent holes.³⁹ Although
any residual interaction cannot be excluded, such
a supercell size is typical of WIEN2k calculations
including core-holes: see reference 37 for MgO,
reference 26 for Li₂O, and references 34 and 54
concerning transition metals. Moreover, given
the good agreement between WIEN2k and FEFF

calculations, we are confident that adjacent holes
interactions are of minor importance with respect
to the here studied effects.

### B. FEFF

The FEFF code¹⁷ (version 8.4) can be described as
a real-space Korringa-Kohn-Rostoker⁴⁷,⁴⁸ (KKR) ap-
proach. Being a Green's function method, it doesn't
need to calculate the KS orbitals. Instead, FEFF uses as
basis scattering states to represent the Green's function
around each scattering center: it is therefore a real-space
full multiple-scattering (RSFMS) method. One advan-
tage of such a real-space approach is that FEFF allows
to link the fine structures observed in an experimental
spectrum to the geometry and to the chemical structure
of the probed material which is not restricted to be pe-
riodic: non periodic systems can be studied alike. FEFF
approximates the total electronic potential for the ini-
tial and final states with spherically averaged MT poten-
tials centered on each atom. In this approximation, the
potential is constant and flat in the interstitial region
between the spheres: it is one of the main differences
with WIEN2k that includes non-spherical corrections as
noted above. However, non-spherical corrections to the
potentials are roughly approximated by allowing the MT
spheres to overlap (we always use the default value: over-
lap 15%). Furthermore, WIEN2k and FEFF differ in the
treatment of the excited final states. In the former, a
fully screened core-hole and a GGA exchange-correlation
potential are used, while in the later a central atom with a
screened core-hole and a GW self-energy Σ(E) are consid-
ered. This last is analogous to the exchange-correlation
potential used for the ground state potential calculation.
Even if it has been shown that it is possible to go beyond
this approximation,⁴⁹ for all here presented results we
have used the single plasmon-pole model of Hedin and
Lundquist⁵⁰ implemented in FEFF for calculating this
self-energy. To estimate the weight of the core-hole effect
into calculated XANES or ELNES spectra, we have used
the possibility offered by FEFF to switch-off the core-hole
(NOHOLE card, see FEFF documentation).⁵¹ It also ex-
ists other many-body effects that cause additional inelas-
tic losses and affect the magnitude of the fine structures
close to an edge onset.⁵² They are traduced through an
overall many-body amplitude reduction factor noted S₀²
that depends weakly on the energy and that is gener-
ally kept constant.² We have use S₀²=0.9, a value usually
close to those accepted for pure metals. **Being a cluster
based method, FEFF needs to define the cluster
size to be used to match the experimental data.**
The optimum parameters we have obtained using
an *s p d* basis set are gathered in Table VII of
the Appendix. Furthermore, to compare with ex-
perimental spectra, all FEFF hereafter presented
XANES spectra related to pure metals have been
convoluted with a Gaussian (FWHM: 1 eV).
### III. RESULTS

#### A. Charge compensation in the crystal

It is well documented that the Z+1 approximation and the explicit account of a core-hole give rather similar results.²⁷ However, concerning the second method, one has to think about conserving the neutrality of the crystal (it is not ionized during a EELS or XAS experiment). In WIEN2k, the generally used approach is to introduce a background negative charge spread over the unit cell at the beginning of the calculation. It is then interesting to see how the overall charge density is reorganized due to the presence of the hole and of this background charge. We have thus investigated this reorganization for a core-hole put either on an oxygen 1s or on a titanium 2p₃/₂ state in r-TiO₂ and for a 1s core-hole created either on an oxygen or on a magnesium atom in MgO. Results obtained with WIEN2k are given in Table I for r-TiO₂ and MgO. In this table, the total and valence charges on the excited atom (labelled with a star) are compared to those calculated for the same atom in the ground state (GS) configuration. The decomposition of the valence charge on the different angular quantum numbers is also given.

Focussing on the first two columns of Table I, one can see that either for the excited oxygen or the titanium one, the positive charge induced on the atom by the core-hole is totally compensated at the end of the SCF loop. The background charge is entirely relocalized on the excited atom (see second column of table I: the valence charge is systematically increased by one unit with respect to the ground state configuration). Focussing on the decomposition of the valence charge on the different orbital symmetries, one observes that the extra charge is relocalised in a given atomic orbital of *p* symmetry concerning the oxygen atom and *d* symmetry for the titanium one. In short, in r-TiO₂ the charge compensation is made in the first empty atomic levels of the excited atom. From this point of view, the valence band of the excited atom is comparable to that of its next neighbour on the periodic table, a fact that justifies the equivalence with the Z+1 approximation. It should be mentioned that the valence charge of the first neighbour of the excited atom (not shown in table I) is not modified by the presence of the core-hole.

This compensating charge mechanism is very close to the one used in the FEFF code. In this approach, the extra charge is relocalized in the muffin tin sphere (MTS) of the excited atom at the beginning of the calculation so that the electronic configuration obtained at the end of the SCF loop is very close to that found with WIEN2k. This can be seen in Table II where the results obtained with FEFF are collected: the extra charge is also mainly localized in the oxygen *p* LDOS for an oxygen 1s core-hole and in the titanium *d* bands when considering a titanium 2 *p*₃/₂ core-hole. It should however be noticed that charge values can differ from one code to the other, especially for the

titanium and magnesium atoms or the transition metals (see tables III and IV). One can find two reasons for such discrepancies. First of all, the separations between valence and core states are hold differently in these two codes what explains that the charge values per orbital quantum number can differ significantly. For instance, the magnesium 3s and 3p states are treated as valence states in WIEN2k and as core states in FEFF so that 6.307 *p* states are found for the magnesium GS configuration in table I and 0,709 in table II. Differences can also be observed when comparing the calculated valence charge differences between the GS and the excited configurations obtained with WIEN2k or FEFF. For instance, such a difference is found larger for the magnesium *p* states in FEFF (+0.438) with respect to WIEN2k (+0.175). Similar discrepancies are observed when focussing on the *p* states of the end-members of the transition metal series (see copper and zinc in tables III and IV). Such differencies can be attributed to the different descriptions of the potentials. WIEN2k is based on a full potential approach whereas FEFF use overlapped (15%) muffin tin potentials. For delocalized states such as the magnesium or the transition metals *p* states, this can lead to significant differences since a non-negligible part of the charge density is delocalized in the interstitial region. However, when considering localized states such as transition metals *d* states, the muffin tin approximation comes to be a good approximation so that WIEN2k and FEFF results are in much better agreement (compare the last columns of tables III and IV).

To conclude this discussion, it is worth mentioning that the overlap of the muffin tin spheres makes that FEFF double counts charge in the region where two MTS overlap and does not count charge at all in the region covered by any sphere.¹⁷ We have not test how this overlap influences the charge distributions we report in Tables II and IV. It should be added that R<sub>SCF</sub> (see table VII) also impacts the results due to surface states of the cluster: in fact we have verified that to increase R<sub>SCF</sub> beyond the values quoted in Table VII only affects the FEFF charge distributions at the margin.

Coming back to the charge compensation mechanism, it can be seen that the situation in MgO is rather different from that in r-TiO₂. Focussing first on WIEN2k calculations (Table I), one can see that the background charge is not totally relocalized on the excited atom at the end of the SCF loop. The charge compensation is higher on the oxygen atom than on the magnesium one, but there is still a lack of 0.148 electron on O*. The remaining charge is not delocalized on the
<table><caption>TABLE I: Projected total and valence charges inside the MTS obtained from WIEN2k for the oxygen or titanium atoms in r-TiO₂ (two first lines), and oxygen and magnesium in MgO (two last lines). In both cases, the excited (*) or ground state (GS) configurations are given for each atom. The decomposition of the valence charge on the different angular quantum numbers is also given. In brackets: difference between the excited and GS charges.</caption><thead><tr><th></th><th>Total Charge</th><th>Valence Charge</th><th>l=0 (s)</th><th>l=1 (p)</th><th>l=2 (d)</th></tr></thead><tbody><tr><td>O*</td><td>7.632 (+ 0.057)</td><td>6.632 (+ 1.057)</td><td>1.774 (+ 0.094)</td><td>4.832 (+ 0.96)</td><td>0.0181 (+ 0.0018)</td></tr><tr><td>O<sub>GS</sub></td><td>7.575</td><td>5.575</td><td>1.680</td><td>3.872</td><td>0.0163</td></tr><tr><td>Ti*</td><td>19.275 (+ 0.021)</td><td>10.275 (+ 1.021)</td><td>2.072 (+ 0.013)</td><td>5.972 (+ 0.062)</td><td>2.243 (+ 1.003)</td></tr><tr><td>Ti<sub>GS</sub></td><td>19.254</td><td>9.254</td><td>2.059</td><td>5.910</td><td>1.240</td></tr><tr><td>O*</td><td>8.115 (- 0.148)</td><td>7.115 (+ 0.852)</td><td>1.880 (+ 0.086)</td><td>5.218 (+ 0.763)</td><td>0.0144 (+ 0.036)</td></tr><tr><td>O<sub>GS</sub></td><td>8.263</td><td>6.263</td><td>1.794</td><td>4.455</td><td>0.0108</td></tr><tr><td>Mg*</td><td>9.719 (- 0.653)</td><td>8.720 (+ 0.348)</td><td>2.269 (+ 0.135)</td><td>6.307 (+ 0.175)</td><td>0.122 (+ 0.035)</td></tr><tr><td>Mg<sub>GS</sub></td><td>10.372</td><td>8.372</td><td>2.134</td><td>6.132</td><td>0.087</td></tr></tbody></table>

<table><caption>TABLE II: Decomposition of the valence charge in r-TiO₂ and MgO as in Table I obtained from FEFF. In brackets: difference between the excited and GS charges.</caption><thead><tr><th></th><th>l=0 (s)</th><th>l=1 (p)</th><th>l=2 (d)</th></tr></thead><tbody><tr><td>O*</td><td>1.876 (+ 0.036)</td><td>5.09 (+ 0.692)</td><td>0.054 (- 0.012)</td></tr><tr><td>O<sub>GS</sub></td><td>1.84</td><td>4.398</td><td>0.066</td></tr><tr><td>Ti*</td><td>0.404 (+ 0.007)</td><td>6.702 (+ 0.035)</td><td>3.490 (+ 1.154)</td></tr><tr><td>Ti<sub>GS</sub></td><td>0.397</td><td>6.667</td><td>2.336</td></tr><tr><td>O*</td><td>1.87 (+ 0.049)</td><td>5.108 (+ 0.569)</td><td>0.013 (- 0.002)</td></tr><tr><td>O<sub>G.S.</sub></td><td>1.821</td><td>4.539</td><td>0.015</td></tr><tr><td>Mg*</td><td>0.800 (+ 0.359)</td><td>1.147 (+ 0.438)</td><td>0.640 (+ 0.166)</td></tr><tr><td>Mg<sub>G.S.</sub></td><td>0.441</td><td>0.709</td><td>0.474</td></tr></tbody></table>

neighboring atoms: it is located into the interstitial region. This delocalization goes with a repartition of the compensating charge on different atomic orbitals. This is particularly visible for the magnesium atom where the + 0.348 electron is mainly shared between the *s* orbitals (0.135) and the *p* ones (+0.175). Such a difference for the charge compensation between r-TiO₂ and MgO can be understood when focussing on the projected DOS as well as on the localization of the electronic densities around the Fermi level. The LDOS for r-TiO₂ and MgO are plotted in Fig. 1, and 2D-projections of the electronic densities in the dense plane of both structures are presented in Fig. 2. Results presented in these two figures were obtained with WIEN2k.

Looking first at the valence bands, it can be observed that they have mainly an *O p* character in both materials. The charge densities corresponding to the last occupied levels represent thus, to a good approximation, the *O p* states in these two compounds. The main difference lies in the difference of spatial extension of these *O p* states: it is clear from the plotted densities that the *O p* charge density is much more contracted in r-TiO₂ than in MgO (see Fig. 2). This is not a question of bond distance since the Ti-O bond is smaller than the Mg-O one (1.946 Å and 2.108 Å, respectively). The extra charge reallocated in the *O p* band is thus more localized on the oxygen atom in r-TiO₂ than in MgO. Concerning the compensating charge on Mg or Ti, the difference is much more pronounced. Focussing on the conduction band in r-TiO₂, one can see that it has mainly a titanium *d* character (fig 1), the corresponding charge density being strongly local-

<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>FIG. 1: (Color) Total and projected DOS for r-TiO₂ (top) and MgO (bottom) obtained with WIEN2k.</figcaption></figure>
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 2: (Color) Electronic densities corresponding to the last occupied (left) and first unoccupied (right) bands projected in a (110) plane of TiO$_2$ (a) and (100) plane of MgO (b). The same values were used to plot the isolines.</figcaption></figure>

ized on the titanium atom (fig 2 (a)). The extra charge relocalized in the *d* states of the excited titanium atom is thus strongly localized on it. On the opposite, the first states of the conduction band in MgO have oxygen and magnesium *s* and *p* characters (fig 1). The compensating charge on the excited Mg atom will thus be spread on the *s* and *p* states and highly delocalized into the interstitial region. The situation here observed is equivalent to the one reported in the literature concerning the charge compensation in transition metal silicides.⁵³

The charge compensation mechanism we observe for r-TiO₂ and MgO is very general and can be extended to other materials. For example, Table III (results from WIEN2k) and IV (results from FEFF) show the charge compensation resulting from a 1s core-hole for some transition metals when going from titanium to zinc. Considering first the results obtained with WIEN2k (table III), one can see that the situation is very similar from titanium to cobalt : the extra charge goes into an empty *d* state. For copper and zinc, it is different since

the *d* band is nearly filled in the GS (9.290 and 9.819 *d* electrons for copper and zinc, respectively). The extra charge is thus spread over the remaining *d* states and the 4*s* and 4*p* orbitals. The *s* and *p* states being much more delocalized than the *d* ones, the compensating charge leaks out of the MTS and the valence charge compensation decreases. The evolution observed with FEFF is identical (see table III): one observes a decreasing *d* character (and an increasing *s* and *p* one) of the compensating charge when going from titanium to zinc. An equivalent mechanism was proposed in transition metals when considering the core-hole screening effect in the initial state of Auger emission.⁵⁴

## B. Modifications on the excited atom projected LDOS

The localization of the wave function also governs the way the electronic structure is relaxed in the presence of a core-hole on a given atom. This is very important for ELNES or XANES simulations since, for one-electron excitations, the fine structures observed experimentally are directly linked the unoccupied LDOS projected on the excited atom. Comparison between the oxygen *p* and titanium *d* LDOS calculated with WIEN2k considering or not a core-hole in r-TiO₂ are shown in Fig. 3. FEFF calculations, not shown here, give very similar results. Focussing first on the oxygen DOS, it is obvious that the weight of the excited oxygen *p* LDOS is transferred at the bottom of the valence band (between -8.8 and -6.8 eV) where intense sharp peaks are observed. This is a clear signature of the localization of the extra electron in the oxygen *p* band. On the contrary, the empty states are weakly affected by the core-hole: no extra peak is observed and the relative intensities of the two main peaks just above the Fermi level (between 2 and 8 eV) are very close to that of the ground state calculation. The core-hole effect at the oxygen K edge is thus weak.⁷,²⁸,⁵⁵ On the other hand, a core-hole on the titanium atom has almost no effect on the occupied titanium *d* LDOS as illustrated by Fig. 3. The only impact of the extra *d* electron is the growth of a small peak located at the bottom of the valence band. The empty titanium *d* LDOS is however much more affected by the core-hole: an intense localized peak appears just above the Fermi level. This can be viewed as the anti-bonding state corresponding the localized *d* electron. The core-hole effect at the titanium *L*₃ edge is thus expected to be very strong. The difference of core-hole effect between the oxygen *K* and the titanium *L*₃ edge in r-TiO₂ observed by Heiliger *et al.* is thus a direct consequence of the existence or not of such modifications on the oxygen *p* and titanium *d* LDOS.⁵⁵

The same study can be performed on the oxygen and magnesium *p* LDOS in MgO: results obtained with or without a 1s core-hole on each atom are presented in
<table><caption>TABLE III: Projected total and valence charges inside the MTS related to the excited (*) or ground state (GS) configurations obtained from WIEN2k for some transition metals. For each atom, the decomposition of the valence charge on the different angular quantum numbers is also given. In brackets: difference between the excited and GS charges.</caption><thead><tr><th></th><th>Total Charge</th><th>Valence Charge</th><th>l=0 (s)</th><th>l=1 (p)</th><th>l=2 (d)</th></tr></thead><tbody><tr><td>Ti<sup>*</sup></td><td>20.668 (+ 0.112)</td><td>11.668 (+ 1.112)</td><td>2.344 (+ 0.022)</td><td>6.267 (+ 0.042)</td><td>3.044 (+ 1.05)</td></tr><tr><td>Ti<sub>GS</sub></td><td>20.556</td><td>10.556</td><td>2.322</td><td>6.225</td><td>1.994</td></tr><tr><td>Fe<sup>*</sup></td><td>25.083 (+ 0.097)</td><td>14.084 (+ 1.096)</td><td>0.412 (+ 0.004)</td><td>6.411 (+ 0.006)</td><td>7.222 (+ 1.089)</td></tr><tr><td>Fe<sub>GS</sub></td><td>24.986</td><td>12.988</td><td>0.408</td><td>6.405</td><td>6.133</td></tr><tr><td>Co<sup>*</sup></td><td>26.130 (- 0.048)</td><td>15.235 (+ 1.055)</td><td>0.459 (+ 0.005)</td><td>6.466 (+ 0.011)</td><td>8.265 (+ 1.04)</td></tr><tr><td>Co<sub>GS</sub></td><td>26.178</td><td>14.180</td><td>0.454</td><td>6.455</td><td>7.225</td></tr><tr><td>Cu<sup>*</sup></td><td>28.085 (- 0.235)</td><td>17.085 (+ 0.766)</td><td>0.637 (+ 0.114)</td><td>6.607 (+ 0.137)</td><td>9.802 (+ 0.512)</td></tr><tr><td>Cu<sub>GS</sub></td><td>28.32</td><td>16.319</td><td>0.523</td><td>6.470</td><td>9.290</td></tr><tr><td>Zn<sup>*</sup></td><td>29.565 (+ 0.422)</td><td>11.566 (+ 0.422)</td><td>0.923 (+ 0.21)</td><td>0.692 (+ 0.099)</td><td>9.939 (+ 0.12)</td></tr><tr><td>Zn<sub>GS</sub></td><td>29.143</td><td>11.144</td><td>0.713</td><td>0.593</td><td>9.819</td></tr></tbody></table>

<table><caption>TABLE IV: Same quantities as in Table III obtained from FEFF.</caption><thead><tr><th></th><th>Total Charge</th><th>Valence Charge</th><th>l=0 (s)</th><th>l=1 (p)</th><th>l=2 (d)</th></tr></thead><tbody><tr><td>Ti<sup>*</sup></td><td>22</td><td>11.18 (+ 1.18)</td><td>0.672 (+ 0.007)</td><td>6.856 (+ 0.0652)</td><td>3.652 (+ 1.111)</td></tr><tr><td>Ti<sub>GS</sub></td><td>22</td><td>10</td><td>0.665</td><td>6.794</td><td>2.541</td></tr><tr><td>Fe<sup>*</sup></td><td>26</td><td>9.037 (+ 1.037)</td><td>0.617 (- 0.0004)</td><td>0.770 (+ 0.006)</td><td>7.65 (+ 1.035)</td></tr><tr><td>Fe<sub>GS</sub></td><td>26</td><td>8</td><td>0.621</td><td>0.764</td><td>6.615</td></tr><tr><td>Co<sup>*</sup></td><td>27</td><td>10 (+ 1.0)</td><td>0.623 (- 0.002)</td><td>0.740 (+0.005)</td><td>8.638 (+ 0.998)</td></tr><tr><td>Co<sub>GS</sub></td><td>27</td><td>9</td><td>0.625</td><td>0.735</td><td>7.64</td></tr><tr><td>Cu<sup>*</sup></td><td>29</td><td>12.076 (+ 1.076)</td><td>0.911 (+ 0.218)</td><td>1.092 (+ 0.339)</td><td>10.073 (+ 0.519)</td></tr><tr><td>Cu<sub>GS</sub></td><td>29</td><td>11</td><td>0.693</td><td>0.753</td><td>9.554</td></tr><tr><td>Zn<sup>*</sup></td><td>30</td><td>13.164 (+ 1.164)</td><td>1.377 (+ 0.371)</td><td>1.544 (+ 0.58)</td><td>10.243 (+ 0.207)</td></tr><tr><td>Zn<sub>GS</sub></td><td>30</td><td>12</td><td>1.006</td><td>0.958</td><td>10.036</td></tr></tbody></table>

Fig. 4. The core-hole effect on the O *p* LDOS in the valence band (VB) is similar to the one observed for r-TiO₂. A strong localized peak appears below the VB (around - 8.4 eV) and the weight of the excited oxygen atom in this band is strongly reduced. However, contrary to r-TiO₂, the O *p* LDOS in the conduction band (CB) is also strongly modified. The intense peak located at 12.5 eV in the GS LDOS has disappeared and several new structures appear below 10 eV. As already reported in several studies, the core-hole must be explicitly taken into account to get simulations of the O *K* edge in good agreement with the experiments.²⁷,³⁵ The situation is similar for the magnesium *p* LDOS. A new structure is created at the bottom of the VB; however, it is less intense than the one observed in the O *p* LDOS and it is not clearly separated from the VB. This suggests that the core-hole effect is less intense on the magnesium states in the VB than on the oxygen ones. Considering now the CB, strong modifications are induced by the core-hole: the intense peak around 18 eV has disappeared and several important structures appear below 15 eV. As mentioned in several papers, the core-hole effect at the magnesium *K* edge is important.²⁷,³⁵,³⁷ The core-hole effects in r-TiO₂ and MgO are thus different: if both metallic sites are affected by the presence of a core-hole, this is not the case for the oxygen atom. The oxygen unoccupied *p* LDOS is not affected in r-TiO₂ whereas a clear

effect is observed in MgO.

The impact of a core-hole on a given wave function in the solid is strongly related to the spatial localization (Fig. 2) and to the weight of the atom that will be excited (e.g. which contains the core-hole) in the considered wave function. Indeed, the importance of the core-hole effect can be discussed qualitatively considering the perturbation ΔE induced by the creation of a core-hole on the eigenenergies of the system in its ground state. This can be achieved by considering the standard stationary perturbation theory applied to a given valence or conduction wave function,⁵⁶ the perturbation $\tilde{W}$ being mainly related to the difference of the atomic potentials felt by the electrons in the ground and excited states. This approach is well suited to Bloch states as obtained in WIEN2k since they are stationary solutions of the time independent Schrödinger equation for the solid.⁵⁷ For simplicity, let's consider a binary system AB in its GS with one atomic orbital $|\phi_A\rangle$ and $|\phi_B\rangle$ per atomic site. In that case, an eigenvector $|\Psi\rangle$ of eigenvalue $E_0$ of the AB system can be written as:

$$|\Psi\rangle = C_A|\phi_A\rangle + C_B|\phi_B\rangle \quad (4)$$

Assuming that the overlap integral $\langle\phi_A|\phi_B\rangle$ is negligible (e.g. $|\phi_A\rangle$ and $|\phi_B\rangle$ is a basis set), the normalization
<figure><img src="image_6.png" /><figcaption>FIG. 3: (Color) Oxygen p LDOS (top) and titanium d LDOS (bottom) in r-TiO₂ with (full lines) or without (dotted lines) a 1s core-hole.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIG. 4: (Color) Oxygen (top) and magnesium (bottom) p LDOS in MgO with (full lines) or without (dotted lines) a 1s core-hole.</figcaption></figure>

of $|\Psi\rangle$ imposes that:

$$|C_A|^2 + |C_B|^2 = 1 \qquad (5)$$

Let us now consider the effect of the perturbation $\hat{W}$ induced by the removal of a core electron from atom A on the eigenvalue $E_0$. To the first order, the total perturbation on this eigenvalue is given by:

$$\Delta E = E_n - E_0 = \langle \Psi | \hat{W} | \Psi \rangle \qquad (6)$$

$\Delta E$ can thus be expressed as a sum of three terms:

$$\Delta E = |C_A|^2 \langle \phi_A | \hat{W} | \phi_A \rangle + |C_B|^2 \langle \phi_B | \hat{W} | \phi_B \rangle + 2 \mathrm{Re}(C_A^* C_B \langle \phi_A | \hat{W} | \phi_B \rangle) \qquad (7)$$

It remains to know the values of the coefficients entering in Eq. 4-7. The coefficients $|C_{A,B}|^2$ can be estimated

from the DOS as follows. For each compound here studied, we divide the sum of the projected LDOS for a given atom (Ti, Mg or O) by the total DOS, restricting these summations to the MTS, e.g. we neglect the contribution of the interstitial region. The results obtained within this framework are plotted in Fig. 5. Here again, for sake of simplicity, only the results obtained from WIEN2k calculations are reported, but FEFF calculations give very similar results. The multiplicity of the atoms in the unit cell were taken into account for these calculations; it explains that the oxygen weight in r-TiO₂ cannot exceed 50 %. It can be seen from Fig. 5 that the VB in r-TiO₂ has predominantly an O character (like in MgO) whereas the first conduction states have a marked titanium character. In MgO, Mg and O atoms have nearly equal weight in the CB: this is the main difference between these two materials.

Let us apply equation 7 to a conduction state in r-TiO₂ just above the gap considering that atom A is a titanium
<figure><img src="image_8.png" /><figcaption>FIG. 5: Percentage of oxygen (dotted line) and titanium or magnesium (full line) character in the wave function for r-TiO₂ (top) and MgO (bottom).</figcaption></figure>

atom and B an oxygen one. In this case, $C_{Ti} \gg C_O$ from Fig. 5. If the perturbation (e.g. the core-hole) is localized on the titanium atom, all matrix elements involving $\tilde{W}$ and the atomic orbital $|\phi_O\rangle$ can be neglected because of the strong localization of $\tilde{W}$ on the titanium. Only the first matrix element in equation 7 remains non negligible. The coefficient $C_{Ti}$ being much higher than $C_O$, the perturbation on $|\Psi\rangle$ is then important. Considering now the core-hole on an oxygen atom, the only significant matrix element in $\Delta E$ is the second one in Eq. 7. However, the prefactor $|C_O|^2$ is then very small compared to unity (see Fig. 5) so that the global perturbation $\Delta E$ is also small: $|\Psi\rangle$ is then weakly affected by the perturbation. This just is what is observed on the LDOS including the core-hole reported in Fig. 3. In the first case, the core-hole being localized on the titanium atom (Fig. 3 bottom), the first conduction states are strongly affected by the core-hole and so is the titanium *d* LDOS. On the contrary, when the core-hole is localized on the

oxygen atom, the first conduction states are weakly affected so that the O *p* LDOS in the corresponding energy range is fairly insensitive to the core-hole (Fig. 3 top). The situation is reversed when considering states in the VB: $C_O$ is then much higher than $C_{Ti}$ and the effect of the perturbation on valence states is much more important when the core-hole is located on an oxygen atom than on a titanium one.

This line of reasoning is transposable to MgO. Let us consider a state $|\Psi\rangle$ corresponding to a conduction state for which $|C_O|^2 \approx |C_{Mg}|^2$ (Fig. 5). In such a case, whatever the localization of the core-hole is, the matrix elements $\langle\phi_O|\tilde{W}|\phi_O\rangle$ or $\langle\phi_{Mg}|\tilde{W}|\phi_{Mg}\rangle$ are associated to a non-negligible prefactor $|C_O|^2$ or $|C_{Mg}|^2$. Thus, $\Delta E$ is significant in both cases: $|\Psi\rangle$ is then modified in any case and a core-hole effect is observed at both O *K* and Mg *K* edges. Considering equation 7, it is also obvious that the localization of the wave functions is a crucial parameter when evaluating the different matrix elements between the perturbation $\tilde{W}$ and the different atomic orbitals.

Even if such a model is oversimplified, nevertheless it emphasizes two important factors to consider when analyzing the importance of the core-hole effect on an EELS or XAS experimental spectrum: the weight of the wave function on the different atomic sites of the crystal and the localization of the wave functions.

### C. Influence of the core-hole depth

In this section, we address a third important criterion: the influence of the core-hole depth, i.e. the energy of the excited core level, on the perturbation. To do so, we compare the effect induced in r-TiO₂ by a titanium 1s or 2s core-hole to that obtained with a $2p_{3/2}$ one as considered before: results obtained in terms of charge compensation are given in Table V. Comparing the values obtained with these three core-holes, the observed compensation mechanism is very similar: the valence charge is always increased by one electron. However when focussing on the distribution of this charge on the different angular momenta, the 1s core-hole differs slightly from the two others: the charge is not entirely relocalized in the empty *d* states, but it is spread for 90% in the *d* states and for 10% in states of *s* and *p* symmetries. The same value of 0.9 extra *d* electron was found when using a full potential multiple scattering approach.⁵⁸ This is however a weak difference and the comparison of the (*s*+*p*) and *d* LDOS plotted in Fig. 6 show that whatever the core-hole depth is, its impact on the LDOS is rather similar. The titanium (*s*+*p*) LDOS is not affected by the core-hole and is found to be identical to that of a ground state calculation in all cases. Concerning the titanium *d* DOS, whatever the core-hole depth is, the previously mentioned strong modifications are observed: the strong extra peak above the Fermi level appears in the three cases. Fig. 6 illustrates an important point: only the bands where the extra charge is relocalized are affected by the core-hole.
<table><caption>TABLE V: WIEN2k projected total and valence charges inside the MTS of the titanium atom in r-TiO₂ considering either a 1s, 2s or 2p₃/₂ core-hole and the GS configuration.</caption><thead><tr><th></th><th>Total Charge</th><th>Valence Charge</th><th>l=0 (s)</th><th>l=1 (p)</th><th>l=2 (d)</th></tr></thead><tbody><tr><td>Ti<sup>*</sup><sub>1s</sub></td><td>19.275</td><td>10.275 (+ 1.021)</td><td>2.082 (+ 0.023)</td><td>6.003 (+ 0.093)</td><td>2.146 (+ 0.906)</td></tr><tr><td>Ti<sup>*</sup><sub>2s</sub></td><td>19.334</td><td>10.334 (+ 1.08)</td><td>2.071 (+ 0.012)</td><td>5.968 (+ 0.058)</td><td>2.251 (+ 1.011)</td></tr><tr><td>Ti<sup>*</sup><sub>2p<sub>3/2</sub></sub></td><td>19.331</td><td>10.332 (+ 1.078)</td><td>2.072 (+ 0.013)</td><td>5.972 (+ 0.062)</td><td>2.243 (+ 1.003)</td></tr><tr><td>Ti<sub>G.S.</sub></td><td>19.254</td><td>9.254</td><td>2.059</td><td>5.91</td><td>1.240</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>FIG. 6: (Color) Comparison of the projected titanium ($s+p$) (top) or titanium $d$ (bottom) LDOS in r-TiO₂ when considering the ground state or different core-holes.</figcaption></figure>

To have a more quantitative evaluation of the core-hole depth impact, one can have a look at the evolution of the screening properties of r-TiO₂ when considering the three kinds of core-holes. Within the linear response theory, the screening properties of a material, with respect to an external electromagnetic field, is given by

the square modulus of the complex dielectric function: $\epsilon(\mathbf{q}, \omega) \cdot \epsilon^*(\mathbf{q}, \omega)$.⁵⁹ The complex dielectric function of r-TiO₂ in its ground state and considering the three different core-holes were calculated using the OPTIC extension of WIEN2k (calculations are performed for $\mathbf{q}=0$).⁶⁰ DFT calculations were already proven to give a good description of the dielectric response of r-TiO₂ (at least concerning the static dielectric constant).⁶¹ For the calculation including the core-holes, 2*2*2 supercells were used. Results are presented in Fig. 7 for the first ten eV.

Above two eV, the main change in the screening function is the decrease of the intensity of the two strong peaks located around 3 and 7.5 eV. Whatever the localization of the core-hole is, the screening functions are very similar in this energy range. Below 2 eV, significant changes can be observed: the localization of the core-hole has an impact in this energy range. With a core-hole on the oxygen atom, the screening function is equivalent to that of the ground state calculation. This can be linked to the fact that the core-hole effect principally modifies the electronic structure below the VB and not around the Fermi level. When localizing it on a titanium atom, the screening is clearly increased. This can be related to the appearance of the strong localized peak observed just above the Fermi level in the titanium $d$ LDOS in Fig. 6. If the screening functions calculated with a 2s or 2p₃/₂ core-hole have similar intensities, it is slightly decreased when the core-hole is localized on the titanium 1s orbital. As stated a long time ago by Slater,⁶² a 1s core-hole is intrinsically more screened by the remaining electrons of the atom than a core-hole created in more external orbitals where a part of the core electrons becomes less efficient for its screening. As a consequence, the compensating charge on the excited atom is less localized and a small part is transferred from the $d$ to $s$ and $p$ orbitals as observed in Table V. The resulting screening is slightly reduced. When considering deep core levels, the core-hole depth has thus an impact on the screening properties of the material but as far as the LDOS are concerned, the variations are very small. The core-hole effect on ELNES or XANES spectra should then be very similar. The situation can be different when considering less tightly bound levels: Nufer et. al. have shown for instance that the core-hole effect is very different when considering
<figure><img src="image_10.png" /><figcaption>FIG. 7: (Color) Square modulus of the complex dielectric function of r-TiO₂ calculated (WIEN2k) for the ground state and for some excited configurations.</figcaption></figure>

either the aluminium *K* or *L*₁ edges in α-Al₂O₃.⁶³

IV. DISCUSSION

Results presented here evidence that the perturbation
due to a core-hole on a deep core level is localized on
the excited atom. The compensating charge, necessary
to maintain the neutrality of the crystal (fully relaxed
configuration), is more or less entirely relocalized on the
excited atom depending on the localization of the cor-
responding wave function. Considering such an atomic
process, the nature of the compensating charge can be
roughly predicted as corresponding to the first empty
atomic levels of the excited atom: mainly *p* character
for an oxygen atom, *s* and *p* character for a magnesium
atom, *d* character for the early transition metals with
increasing *s* and *p* contributions when going towards the
end members of the series. When considering ELNES or
XANES simulations, two criteria are found to be impor-
tant: the weight of the excited atom in the first states
above the Fermi level and the symmetry of the compen-
sating charge.

The first point is of primary interest since it results
that the impact of the core-hole effect is not linked to
the metallic or insulating nature of the probed sample.
The pertinent approach is to focus on the chemical nature
of the first empty states. Considering this argument, it
is clear that a core-hole effect is expected in all pure
compounds, including metals, what justifies the core-hole
effect observed at the platinum metal and copper L2,3
edges or at the aluminium metal K edge.

When focussing on complex compounds, this argu-
ment allows to understand on which atomic site will the
core-hole effect be important. For instance, considering

transition metal oxides, no core-hole effect is expected
(and observed) at the oxygen *K* edge.⁷,²⁸,²⁹ However, a
strong core-hole effect is observed at the oxygen *K* edge
in Li₂O.²⁶ This is not surprising when looking at the
lithium oxide density of states (see Ref. 26): the bottom
of the conduction band has clearly a strong oxygen *p*
character. Focussing on SrTiO₃, the LDOS calculations
of Orhan *et. al*. show that the bottom of the conduction
band has mainly titanium character, with no strontium
and a small oxygen contribution.⁶⁴ This explains the ab-
sence of a core-hole effect observed at the Sr *L*₃ edge,
the weak effect at the O *K* edge while a strong effect is
observed when considering the titanium *L*₃ edge.⁴⁰

The second crucial point is the nature of the compen-
sating charge since it governs which symmetry of the
wave function is perturbed by the presence of a core-hole.
This is illustrated in Fig. 8 and 9 where the titanium *K*,
*L*₁ and *L*₃ edges in r-TiO₂ calculated with WIEN2k and
FEFF are plotted. It is clear from these results that the
core-hole effect is much more important on the titanium
*L*₃ edge than on the two others. The *L*₃ white line is
shifted by 3 eV in the WIEN2k simulation and the rel-
ative intensities of the two fine structures at the top of
the edge are inverted whereas no effect can be observed
on the titanium Ti *L*₁ edge excepted a small shift of the
spectral weight in the WIEN2k simulation (no effect at
all is observed in the FEFF calculation). The titanium *K*
edge is a bit more sensitive to the core-hole than the Ti
*L*₁ edge, but the changes remain quite small, especially
when focussing on the FEFF calculation. It is a general
observation that the core-hole effect is always more pro-
nounced in the WIEN2k calculations than in the FEFF
ones. This might be due to the different formalisms con-
sidered in the two codes: a full potential approach is used
in WIEN2k whereas FEFF calculations are performed in
the muffin tin approximation.

The important core-hole effect observed at the Ti *L*₃ edge is correlated to the localization of the compensating charge in the titanium *d* states for a 2p₃/₂ core-hole (see Table V): the perturbation on these bands is then important and observed when probing the corresponding edge. Considering the Ti *L*₁ edge, the compensating charge being also localized in the titanium *d* bands, no effect is observed when probing the empty *p* density of states and therefore the core-hole effect is invisible at this edge. **The dipole selection rule plays an important role here by imposing the symmetry of the unoccupied probed states.** Finally the fact that the Ti *K* edge is more sensitive to the core-hole effect than the Ti *L*₁ one can be explained by the small part of the compensating charge introduced in the Ti (*s*,*p*) LDOS when a 1 *s* core-hole is created (see Table V).

Following the same line of reasoning and consider-
ing now Tables III and IV, one is able to predict a
weak/inexistent core-hole effect at the *K* edges of the
early transition metals and a more pronounced one when
going to copper and zinc. Such predictions are confirmed
by the examination of Fig. 10 and 11 were the results of
calculations obtained with WIEN2k and FEFF are pre-
sented. It is clear from Fig. 11 that the core-hole effect
for the first three metals is completely negligible and that
it must be included in the two last cases to match the
experimental spectra. Concerning the WIEN2k simula-
tions, conclusions are identical. However, one can note a
small decrease of the first peak's intensity at the titanium
*K* edge when considering a core-hole. Such a difference
remains of minor importance when compared to experi-
ment and would be reduced by increasing the broadening
applied to the simulated spectra. This importance of the
*d* band filling has already been mentioned by Tamura
*et. al.* to explain the observed core-hole effect at the
platinum *L*₂,₃ and aluminium *K* edges,³¹ and by Yuan
concerning the core-hole effect observed by Auger spec-
troscopy for transition metals.⁵⁴ The core-hole effect is
thus mainly driven by the modification of the population
of a given band. This is the reason why a strong core-hole
effect is observed at the early transition metals *L*₂,₃ edges
where the compensating charge is fully relocalized in the
*d* bands and both observed at the *L*₂,₃ and *K* edges for
copper and zinc where the compensating charge is spread
over the *s*, *p* and *d* states.

It can thus be concluded from these arguments that
if a core-hole is always present in a spectroscopic experi-
ment, its effect is not always observed because it depends
on the probed states. Two important criteria have been
emphasized (1) the spatial localization of the first empty
states, (2) the nature of the compensating charge. From
these two criteria, two general rules can be given to ex-
plain the presence or not of a strong core-hole effect at a
given edge when comparing experiments to calculations:

1. for complex compounds, the core-hole effect is ex-
pected to be prominent for the atom that majori-
tary participates to the first empty states (this is
the case of a titanium atom in r-TiO₂), and less
sensitive on the other ones (the oxygen in r-TiO₂).
When different atoms have an important weight,
the core-hole effect is expected to be important on
all of them (see the example of MgO). For pure
materials, the core-hole effect is always present.

2. for a given atom, only the LDOS having the same character as the compensating charge will be affected by the core-hole. This compensating charge can be predicted to fill the first empty *atomic levels* of the excited atom.

These simple rules allow roughly to predict the core-hole
effect strength at a given edge from the calculation point
of view. In the cases for which the DFT formalism is
known to be a good approximation (e.g. when the one-
particle approximation is verified), this should give the
justification for considering or not a core-hole into the
simulation. This is more satisfactory than the approach
that consists in performing two simulations and keeping
the one that best fits with the experiment. However,
when considering edges where the DFT is not the op-
timal formalism, these rules just predict the effect of a

core-hole on the calculated edge, but nothing can be said
*a priori* concerning the agreement or not with the exper-
iment. It is well known that the core-hole effect is im-
portant in DFT calculations of the transition metal L₂,₃
edges but generally ground state calculations give better
results when compared to experiment.⁶⁵ In this situation,
the screening of the core-hole is not well treated by the
DFT and one should consider more complex approaches
to obtain a better agreement with the experiment.⁶⁶ An-
other example is the lithium *K* edge in pure metallic
lithium. Here again a strong core-hole effect is predicted
from the calculation as it can be observed in Fig. 12,
but the ground state calculation is in much better agree-
ment with the experiment. Here again the validity of
the DFT is questionable since the lithium 1 *s* orbital is
a semi-core state and electron-hole interactions within
delocalized wave functions should be treated with more
complex formalisms.⁶⁷

Finally, let us remark that the present analysis is dif-
ferent from the one recently proposed by Gao et al.³³ who
study the core-hole effect in 1 *s* core-level spectroscopy of
the first-row elements. Compared with the experimental
data, these authors show that one must use in the cal-
culations a fully relaxed configuration (a core-hole and
an extra screening electron) to match the experimental
spectra related to these light elements. Indeed, it is ob-
served that the fine structures at the edge onset are re-
pulsed towards higher energies when using an unrelaxed
configuration (no core-hole). Gao et al. deduce from an
analysis of their results a relationship between the VB
charge population and the core-hole strength: larger dif-
ferences between spectra seem to occur for smaller VB
charge populations. However, as these authors note, the
screening effect of valence electron populations is only an
approximate index to rank the strength of the core-hole.
Indeed, it seems to be incompatible with what we found
at the *K* edges of pure Cu and Zn (Fig. 10 and 11). In
both cases, the core-hole must be included to match the
experimental data while it has a negligible effect for Ti,
Fe and Co (Fig. 10 and 11). This is in contradiction
with the results of Gao et al. since the VB populations
of the latter are smaller than those of the former that are
nearly filled.

V. CONCLUSION

The electronic structure relaxation mechanism
induced by a core-hole has been investigated
within the one-particle approximation using the
Density Functional Theory. Electronic structure cal-
culations as well as ELNES/XANES simulations have
been performed with two widely used codes: WIEN2k
and FEFF.

Electronic structure calculations show that whatever
edge is considered, the compensating charge introduced
into the calculation to keep the crystal neutrality is sys-
tematically relocalized in the first empty states of the
excited atom. In a highly ionic compound such as TiO₂, these states correspond to the first unoccupied levels of the *atomic* configuration for an isolated atom. The localization of these first empty states directly governs the spatial delocalization of this extra-charge. In complex compounds, the nature of these first empty states govern the importance of the core-hole effect: if strongly localized on a given atom, it can be inferred from the here presented results that the core-hole effect will be prominent on this atom and rather small on the other ones. If equally spread over several atoms of the structure, the core-hole effect should be important on all these atoms. Another crucial parameter is the nature of this compensating charge which imposes, in a first approximation, which band will be most affected in the crystal and thus rules which edge will be affected by the core-hole: it explains why a core-hole effect is observed at the titanium *L*₃ edge in r-TiO₂ and not at the *K* or *L*₁ ones. These results clearly show that a core-hole effect is always present in any core-level spectroscopy experiment but not always evidenced. Its importance is not driven by the metallic or insulating nature of the material, **as sometimes mentioned**, and only indirectly influenced by the density of valence electrons that would screen it. Our calculations however demonstrate the observation given by De Groot *et al.*³⁸ for transition metal oxides: the core-hole effect at the oxygen *K* edge in transition metal oxides is weak because the first empty states in the conduction band have a strong weight on the transition metal.

The influence of the core-hole depth on titanium in r-TiO₂ was also investigated and was shown to be negligible as far as the titanium LDOS is concerned. A more quantitative approach based on the calculation of the screening function shows however that a 1s core-hole is more screened than outer shells electrons. Finally, compared to FEFF, it is observed that a full potential approach like the one implemented in WIEN2k is generally more sensitive to differences between GS and excited configurations when the core-hole effect strength is weak.

The presented results give a good description of what should the core-hole effect be like when considering edges that can be well described within the one-particle approximation. When the DFT is not the optimal formalism for describing the edge of interest, such results only explain the importance of the core-hole effect in the calculation but nothing can be said concerning the agreement with the experiment. The *L*₂,₃ edges of transition metals or the lithium *K* edge in metallic lithium are good examples where a core-hole effect is expected, and observed, from DFT calculations. Nevertheless the agreement with the experiment is improved when considering a ground state calculation, a result clearly emphasizing the limits of DFT.

# VI. ACKNOWLEDGEMENTS

One of the authors (P.S.) thanks the University of Poitiers for hospitality, where part of this work was carried out during a stay as Associate Professor. Part of the WIEN2k calculations presented in this work were performed at the "Centre régional de Calcul Intensif des Pays de la Loire" financed by the French Research Ministry, the "Région Pays de la Loire" and the University of Nantes.

# VII. APPENDIX
<table><caption>TABLE VI: Main parameters used for the WIEN2k calculations. The numbers in the SCF k-point columns refer to the sampling of the full Brilloin zone. Muffin tin radii ($R_{MT}$) are given in bohr and $K_{max}$ defines the wave vector cutoff for the plane waves in the interstitial space.</caption><thead><tr><th></th><th>R<sub>MT</sub></th><th>R<sub>MT</sub>.K<sub>max</sub></th><th>k points (SCF)</th><th>k points (ELNES)</th></tr></thead><tbody><tr><td>MgO</td><td>Mg : 1.98<br>O : 1.98</td><td>8.0</td><td>7*7*7<br>supercell : 3*3*3</td><td></td></tr><tr><td>TiO<sub>2</sub></td><td>Ti : 1.94<br>O : 1.72</td><td>7.0</td><td>8*8*13<br>supercell : 6*6*4</td><td>6*6*9<br>supercell : 4*4*6</td></tr><tr><td>Ti</td><td>2.00</td><td>8.0</td><td>24*24*13<br>supercell : 12*12*6</td><td>8*8*4<br>supercell : 4*4*2</td></tr><tr><td>Fe</td><td>2.33</td><td>7.50</td><td>22*22*22<br>supercell : 11*11*11</td><td>8*8*8<br>supercell : 4*4*4</td></tr><tr><td>Co</td><td>2.34</td><td>7.50</td><td>28*28*15<br>supercell : 14*14*7</td><td>9*9*5<br>supercell : 4*4*2</td></tr><tr><td>Cu</td><td>2.40</td><td>7.50</td><td>18*18*18<br>supercell : 9*9*9</td><td>6*6*6<br>supercell : 3*3*3</td></tr><tr><td>Zn</td><td>2.50</td><td>7.50</td><td>25*25*12<br>supercell : 13*13*5</td><td>8*8*4<br>supercell : 4*4*2</td></tr><tr><td>Li</td><td>2.85</td><td>7.0</td><td>17*17*17<br>supercell : 8*8*8</td><td>9*9*9<br>supercell : 4*4*4</td></tr></tbody></table>

<table><caption>TABLE VII: Main parameters used for the FEFF calculations. $R_{SCF}$ refers to the size of the clusters used to obtain reliable potentials from the SCF loop while $R_{FMS}$ is the size of the clusters (including $N_{atoms}$) used for calculating the XANES spectra.</caption><thead><tr><th></th><th>Edge</th><th>R<sub>SCF</sub> (Å)</th><th>R<sub>FMS</sub> (Å)</th><th>N<sub>atoms</sub></th></tr></thead><tbody><tr><td rowspan="6">Pure Metals</td><td>Ti K</td><td>4</td><td>7</td><td>81</td></tr><tr><td>Fe K</td><td>4.1</td><td>6</td><td>65</td></tr><tr><td>Co K</td><td>4</td><td>6</td><td>81</td></tr><tr><td>Ni K</td><td>4</td><td>6.2</td><td>87</td></tr><tr><td>Cu K</td><td>4</td><td>8</td><td>177</td></tr><tr><td>Zn K</td><td>4</td><td>8</td><td>135</td></tr><tr><td rowspan="4">TiO<sub>2</sub></td><td>Ti K</td><td>4.6</td><td>7.9</td><td>197</td></tr><tr><td>O K</td><td>4.6</td><td>7.9</td><td>197</td></tr><tr><td>Ti L<sub>3</sub></td><td>4.6</td><td>7</td><td>137</td></tr><tr><td>Ti L<sub>1</sub></td><td>4.6</td><td>7</td><td>137</td></tr><tr><td rowspan="2">MgO</td><td>Mg K</td><td>4</td><td>6.7</td><td>147</td></tr><tr><td>OK</td><td>4</td><td>7.9</td><td>250</td></tr></tbody></table>
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>FIG. 8: (Color) Comparison of the Ti K, Ti L₁ and Ti L₃ edges in r-TiO₂ calculated (WIEN2k) with (dotted line with circles) or without (full line) a core-hole.</figcaption></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>FIG. 9: (Color) Comparison of the Ti K, Ti L₁ and Ti L₃ edges in r-TiO₂ calculated (FEFF) with (dotted line with circles) or without (full line) a core-hole.</figcaption></figure>
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>FIG. 10: (Color) Calculated (WIEN2k) K edges for Ti, Fe, Co, Cu and Zn with (dotted line with circles) or without (full line) a core-hole.</figcaption></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>

<figure><img src="image_25.png" /></figure>

<figure><img src="image_26.png" /><figcaption>FIG. 11: (Color) Calculated (FEFF) K edges for Ti, Fe, Co, Cu and Zn with (circles) and without (full line) a core-hole. Experimental spectra taken from the XANES data base (thick black line) are also plotted for comparison.</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>FIG. 12: (Color) Li K edge calculated (WIEN2k) with (bottom) or without (middle) a core-hole.³⁰ The experimental EELS spectrum (top) is taken from reference 30.</figcaption></figure>
* Electronic address: vincent.mauchamp@univ-poitiers.fr

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