# Pseudopotential Bethe-Salpeter calculations for shallow-core x-ray
absorption near-edge structures: excitonic effects in $\alpha$-Al2O3

M. Laura Urquiza  LSI, CNRS, CEA/DRF/IRAMIS, École Polytechnique, Institut
Polytechnique de Paris, F-91120 Palaiseau, France  European Theoretical
Spectroscopy Facility (ETSF)  Matteo Gatti  LSI, CNRS, CEA/DRF/IRAMIS, École
Polytechnique, Institut Polytechnique de Paris, F-91120 Palaiseau, France
European Theoretical Spectroscopy Facility (ETSF)  Synchrotron SOLEIL, L’Orme
des Merisiers, Saint-Aubin, BP 48, F-91192 Gif-sur-Yvette, France  Francesco
Sottile  LSI, CNRS, CEA/DRF/IRAMIS, École Polytechnique, Institut
Polytechnique de Paris, F-91120 Palaiseau, France  European Theoretical
Spectroscopy Facility (ETSF)

###### Abstract

We present an ab initio description of optical and X-ray absorption
spectroscopies, in a unified formalism based on the pseudopotential plane-wave
method at the level of the Bethe-Salpeter Equation (BSE) within Green’s
functions theory. We show that norm-conserving pseudopotentials are very
reliable and accurate not only for valence, but also for semi-core electron
absorption spectra. In order to validate our approach, we compare BSE results
with two codes: EXC, based on pseudopotentials, and Exciting, an all-electron
full-potential code. We take corundum $\alpha$-Al2O3 as an example, a
prototypical system that presents strong electron-hole interaction in both
valence and core electron excitations. We analyze the optical, as well as the
L1 and L2,3 edges, in detail in terms of anisotropy, crystal local fields,
interference and excitonic effects. We conclude with a thorough inspection of
the origin and localization of bright and dark excitons.

##  I  Introduction

X-ray absorption spectroscopy (XAS) and optical absorption are complementary
techniques to determine materials properties. In optical absorption, valence
electrons are excited into unoccupied conduction states across the band gap
(or the Fermi energy in metals). Their excitations determine the color (or the
transparency) of materials and are crucial to many materials properties and
functionalities, spanning from optoelectronics to solar energy conversion and
storage. In XAS, promoted to unoccupied conduction bands are instead core
electrons, tightly bound to the nuclei. X-ray absorption near-edge structures
(XANES), also known as near-edge X-ray absorption fine structure (NEXAFS),
being element specific, is a probe of the atomic environment, giving
structural and chemical informationvan Bokhoven and Lamberti (2016). In the
simplest independent-particle picture, XANES spectra are proportional to the
unoccupied density of states, projected on the absorbing atom and the angular
momentum component that is selected by dipole selection rules, whereas optical
spectra can be interpreted on the basis of the joint density of states of
valence and conduction bands. In both spectroscopies, the interaction between
the excited electron and the hole left behind can strongly alter this
independent-particle picture. Indeed, the electron-hole attraction can give
rise to excitons, i.e bound electron-hole pairs, leading to a transfer of
spectral weight to lower energies in the spectra, including the formation of
sharp peaks at their onset.

Given the importance of XANES spectroscopy, several theoretical methods have
been developed to interpret the measured spectra in solids, taking care of
core-hole effects at different levels of approximationde Groot  _et al._
(2021). The most efficient approaches are, on one side, multiple scattering
methodsFujikawa (1983); Tyson  _et al._ (1992); Ahlers  _et al._ (1998); Rehr
and Albers (2000); Rehr  _et al._ (2009, 2010), and, on the other side,
multiplet modelsde Groot (2005); De Groot and Kotani (2008); Haverkort  _et
al._ (2012). While the former usually neglect the electronic interactions, the
latter are often semi-empirical (i.e., not entirely parameter-free) and
generally neglect solid-state effects, being a many-body solution of finite-
cluster models. Since the excitations of the core electrons are localised at
the absorbing atoms, delta-self-consistent-field ($\Delta$SCF) methods can be
also employed, nowadays usually within first-principles density-functional
theoryMo and Ching (2000); Gougoussis  _et al._ (2009); Taillefumier  _et al._
(2002); Bunău and Calandra (2013); Mazevet  _et al._ (2010); Hetényi  _et al._
(2004); Prendergast and Galli (2006); Gao  _et al._ (2009); Prentice  _et al._
(2020). The core-excited atom is treated as an impurity in a supercell
approach, and the presence of the core hole is taken into account in different
ways, from the Z+1 approximationHjalmarson  _et al._ (1981); Lie  _et al._
(1999) (the absorbing atom is assumed to have one additional nuclear charge),
to the half core-hole approximation Triguero  _et al._ (1998); Klein  _et al._
(2021) (also known as Slater’s transition-state method) or the full core-hole
approximation (the electron removed from the core is put at lowest conduction
band, or ionized). Alternatively, XANES excitation spectra can be directly
obtained within linear-response theoryRehr  _et al._ (2005); Liang  _et al._
(2017), which is the standard approach for valence excitations and optical
spectra as wellOnida  _et al._ (2002). In this case, two possible options are
time-dependent density-functional theoryBesley  _et al._ (2009); Bunău and
Joly (2012, 2012) (TDDFT) and the Bethe-Salpeter equationStrinati (1982,
1984); Shirley (1998); Carlisle  _et al._ (1999); Shirley (2000) (BSE) of
Green’s function theoryMartin  _et al._ (2016); Bechstedt (2014). Since TDDFT
lacks of efficient approximations for describing accurately excitonic effects
in solidsBotti  _et al._ (2007), the BSE, even though computationally more
expensive, is usually more reliableOnida  _et al._ (2002). In the present
work, the solution of the BSE will therefore be also our preferred choice to
simulate valence and shallow-core excitation spectra within the same
formalism.

In the simulation of core excitation spectra, the intuitive technique to
represent the single-particle wave functions are all-electron methods. They
explicitly deal with core electrons in extended materials by partitioning the
space into interstitial and muffin-tin (MT) regions, where wave functions are
described differently according to their localisation degreeWills  _et al._
(2010); Andersen (1975); Sjöstedt  _et al._ (2000); Madsen  _et al._ (2001).
Instead, methods that are based on plane-wave expansions cannot deal
explicitly with the quickly oscillatory behavior of core electrons, tightly
localised near the nuclei, which are instead generally taken into account
effectively through the design of suitable pseudopotentialsPayne  _et al._
(1992). Plane-wave methods are computationally cheaper and new theoretical
developments are easier to implement in plane-waves computer codes. Moreover,
the separation between core electrons, kept frozen, and valence electrons,
treated explicitly, is often not rigid. Between valence and deep core
electrons, there are often also shallow core (or semicore) electrons, which in
the pseudopotential framework can be in principle also treated as valence
electrons, although at a price of higher computational cost. However, in all
the cases, the pseudopotential formalism also introduces an important
approximation, requiring a pseudization of the valence wave functions near the
nuclei that make them smoother and node free. In the recent past, much work
has been devoted to assess pseudopotential calculations for excited-state
properties with respect to all-electron methods, notably for self-energy
calculations of quasiparticle band structure energiesKu and Eguiluz (2002);
Delaney  _et al._ (2004); Tiago  _et al._ (2004); van Schilfgaarde  _et al._
(2006a); Friedrich  _et al._ (2006); Gómez-Abal  _et al._ (2008); Luppi  _et
al._ (2008); Klimeš  _et al._ (2014). In the present work, we directly address
the question of the validity of the pseudopotential approximation for XANES
spectra of shallow-core edges (i.e., for electron binding energies smaller
than $\sim$180 eV), investigating the limits of use of pseudo wave functions
for shallow core states in many-body BSE calculations. It is clear that the
description of deep core levels will be always out of reach for plane-wave
basis methods. However, the high plane-wave cutoff required by semicore states
can be now alleviated by the new generation of ultrasoft norm-conserving
pseudopotentialsHamann (2013). Besides the promised lower computational cost
for shallower core levels, an advantage of pseudopotential plane-wave
calculations with respect to all-electron methods is that they do not make any
hypothesis concerning the localisation of the core hole inside the muffin
tin111The same hypothesis is made when the core orbitals are obtained from a
calculation of the isolated atomShirley (2004); Blöchl (1994); Unzog  _et al._
(2022)..

In particular, here we investigate the effects of the electron-hole
interactions on the optical absorption and shallow-core XANES spectra of
alumina. $\alpha$-Al2O3 is a wide-gap insulator, with many possible
applications as a structural ceramic (e.g. as a replacement to SiO2 gate oxide
technology) and optical material (also thanks to the high-damage threshold for
UV laser applications), and a prototypical system to investigate core-hole
effects in XANES spectroscopyFrench (1990); French  _et al._ (1994); Tanaka
and Adachi (1996); Cabaret  _et al._ (1996); Ildefonse  _et al._ (1998); Mo
and Ching (2000); van Bokhoven  _et al._ (2001).

The article is organised as follows. After a short description of the employed
methodology in Sec. II, comprising a review of the theoretical background
(Sec. II.1) and a summary of the computational details (Sec. II.2), Sec. III
presents the results of the calculations together with their analysis. In Sec.
III.2 pseudopotential calculations are assessed with respect to all-electron
benchmarks for both optical and Al L2,3 XANES spectra, while Sec. III.3
contains a discussion on the issue of the core-hole localisation in the muffin
tin for the Al L1 XANES spectrum. Sec. III.4 compares the calculated spectra
with available experiments and analyses the effects of the electron-hole
interactions on the spectra. Finally, Sec. IV draws the conclusions
summarizing the results of the work.

##  II  Methodology

###  II.1  Theoretical background

In the framework of Green’s function theoryMartin  _et al._ (2016), the Bethe-
Salpeter equation (BSE) yields the density response function from the solution
of a Dyson-like equation for the two-particle correlation functionStrinati
(1988). In the GW approximation (GWA) to the self-energyHedin (1965), with a
statically screened Coulomb interaction $W$, the BSE takes the form of an
excitonic HamiltonianOnida  _et al._ (2002) in the basis $|vc{\bf k}\rangle$
of transitions between occupied $v{\bf k}$ and unoccupied bands $c{\bf k}$
(i.e., uncorrelated electron-hole pairs):

$\langle vc{\bf k}|H_{\rm exc}|v^{\prime}c^{\prime}{\bf
k}^{\prime}\rangle=E_{vc{\bf
k}}\delta_{vv^{\prime}}\delta_{cc^{\prime}}\delta_{{\bf k}{\bf
k}^{\prime}}+\langle vc{\bf k}|\bar{v}_{c}-W|v^{\prime}c^{\prime}{\bf
k}^{\prime}\rangle.$ (1)

Here $E_{vc{\bf k}}=E_{c\mathbf{k}}-E_{v\mathbf{k}}$ are the interband
transition energies calculated in the GWA, while $\bar{v}_{c}$ is the Coulomb
interaction without its macroscopic component (i.e., the component ${\bf G}=0$
in reciprocal space). The statically screened Coulomb interaction
$W=\epsilon^{-1}v_{c}$ is usually calculated adopting the random-phase
approximation (RPA) for the inverse dielectric function $\epsilon^{-1}$.

The GWA-BSE is nowadays the state-of-the-art approach for the simulation,
interpretation and prediction of optical spectra in solidsAlbrecht  _et al._
(1998); Benedict  _et al._ (1998); Rohlfing and Louie (2000); Bechstedt
(2014); Martin  _et al._ (2016), and is more and more used also for the
simulation of core-level excitation spectraVinson  _et al._ (2011); Vinson and
Rehr (2012); Gilmore  _et al._ (2015, 2021); Geondzhian and Gilmore (2018);
Dashwood  _et al._ (2021); Vinson (2022); Olovsson  _et al._ (2009a, b, 2011);
Vorwerk  _et al._ (2017, 2019, 2020); Laskowski and Blaha (2010); Yao  _et
al._ (2022); de Groot  _et al._ (2021); Vorwerk  _et al._ (2022); Unzog  _et
al._ (2022). A great advantage of theory with respect to experiments is the
possibility to separately suppress (or activate) the various interactions at
play in the materials, which allows one to single out their specific effect on
the spectra and the materials properties. By setting to zero the two electron-
hole interactions, $\bar{v}_{c}$ and $-W$, the excitonic Hamiltonian (1)
reduces to a diagonal matrix and corresponds to the independent-particle
approximation (IPA). By switching on the electron-hole exchange interaction
$\bar{v}_{c}$ in Eq. (1), one retrieves the RPA. With respect to the IPA, the
RPA includes the so-called crystal local field effects. They are related to
the inhomogeneous charge response of materials through the induced microscopic
Hartree potentials counteracting the external perturbations. Finally, by also
switching on the electron-hole direct interaction $-W$, the full BSE (1)
describes excitonic effects, which are due to the electron-hole
attraction.222There is also the possibility to include $-W$ and exclude
$\bar{v}_{c}$, which corresponds to the description of spin-triplet
excitations. The electron-hole interactions contributing to the off-diagonal
matrix elements of the BSE (1) give rise to a mixing of the independent-
particle transitions, which is formally obtained from the solution of the
eigenvalue equation for the excitonic hamiltonian: $H_{\rm
exc}A_{\lambda}=E_{\lambda}A_{\lambda}$.

The absorption spectra, expressed both in the optical and XANES regimes by the
imaginary part of the macroscopic dielectric function,
$\textrm{Im}\epsilon_{M}(\omega)$, in the long wavelength limit $\mathbf{q}\to
0$,in the so-called Tamm-Dancoff approximation can be directly written in
terms of eigenvectors $A_{\lambda}$ and eigenvalues $E_{\lambda}$ of the BSE
Hamiltonian (1) as:

$\textrm{Im}\epsilon_{M}(\omega)=\lim_{\mathbf{q}\to 0}\frac{8\pi^{2}}{\Omega
q^{2}}\sum_{\lambda}\left|\sum_{vc{\bf k}}A_{\lambda}^{vc{\bf
k}}\tilde{\rho}_{vc{\bf k}}(\mathbf{q})\right|^{2}\delta(\omega-E_{\lambda}),$
(2)

where $\Omega$ is the crystal volume, and $\tilde{\rho}_{vc{\bf
k}}(\mathbf{q})=\int\varphi^{*}_{v\mathbf{k}-\mathbf{q}}(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}\varphi_{c\mathbf{k}}(\mathbf{r})d\mathbf{r}$
are the independent-particle oscillator strengths. Here the single-particle
orbitals $\varphi_{i}$ are usually Kohn-Sham orbitals. If the exciton energy
$E_{\lambda}$ is smaller than the smallest independent-particle transition
energy $E_{vc{\bf k}}$, the exciton $\lambda$ is said to be bound: the
difference between $E_{vc{\bf k}}$ and $E_{\lambda}$ is its binding energy.

The contribution of each exciton $\lambda$ to the spectrum can be analysed by
introducing the cumulative function:

$S_{\lambda}(\omega)=\lim_{\mathbf{q}\rightarrow 0}\frac{8\pi}{\Omega
q^{2}}\;\left|\sum_{vc{\bf k}}^{{E_{vc{\bf k}}<\omega}}A_{\lambda}^{vc{\bf
k}}\tilde{\rho}_{vc{\bf k}}(\mathbf{q})\right|^{2}.$ (3)

Since the eigenvectors $A_{\lambda}$ and the oscillator strengths
$\tilde{\rho}(\mathbf{q})$ are both complex quantities, the cumulative
function (3) is not a monotonic function of $\omega$. The limit
$S_{\lambda}(\omega\to\infty)$ is the oscillator strength of the exciton
$\lambda$ in the absorption spectrum. If it is negligibly small, the exciton
is said to be dark, otherwise it is called a bright exciton, for it
contributes to the spectrum. Even in the $\mathbf{q}\to 0$, the oscillator
strengths $\tilde{\rho}(\mathbf{q})$ depends on the direction of $\mathbf{q}$,
so each exciton $\lambda$ can at the same time be a bright exciton in one
polarization direction and dark in another.

Finally, the investigation of the electron-hole correlation function for each
exciton $\lambda$,

$\Psi_{\lambda}(\mathbf{r}_{h},\mathbf{r}_{e})=\sum_{vc{\bf
k}}A_{\lambda}^{vc{\bf
k}}\phi^{*}_{v\mathbf{k}}(\mathbf{r}_{h})\phi_{c\mathbf{k}}(\mathbf{r}_{e}),$
(4)

gives information about the localisation in real space of the electron-hole
pair, which results from the electron-hole attraction. Assuming that the hole
is in a specific position $\mathbf{r}_{h}=\mathbf{r}_{h}^{0}$, one can
visualize the corresponding density distribution of the electron
$|\Psi_{\lambda}(\mathbf{r}_{h}^{0},\mathbf{r}_{e})|^{2}$.

###  II.2  Computational details

Figure 1:  Convergence of BSE absorption spectra with the number of unoccupied
conduction bands (cb). (Left) Optical spectrum. (Middle) XANES at L2,3 edge.
(Right) XANES at L1 edge.

We have performed calculations using both a pseudopotential (PP) plane-wave
method and a full-potential all-electron (AE) linearized augmented plane-wave
method. AE calculations have been done in particular to assess the validity of
PP calculations for the core-level excitations (see Sec. III.2). The converged
BSE absorption spectra and their analysis (see Sec. III.4) have been then
obtained in the PP framework. In the pseudopotential case, we have used the
Abinit codeGonze  _et al._ (2016) for the ground-state and screening
calculations, and the EXC codeReining  _et al._ for the BSE calculations. In
the all-electron case, we have used the Exciting codeGulans  _et al._ (2014)
for obtaining all the benchmark results.

The Kohn-Sham ground-state calculations have been performed within the local
density approximationKohn and Sham (1965) (LDA).

We have employed norm-conserving Troullier-Martins Troullier and Martins
(1991) (TM) and optimized norm-conserving Vanderbilt Hamann (2013); van Setten
_et al._ (2018) (ONCVPSP) pseudopotentials. In particular, for the absorption
spectra a special TM pseudopotentialZhou  _et al._ (2020) treating also Al
$2s$ and $2p$ states as valence electrons has been used. Calculation with the
ONCVPSP pseudopotential converged with 42 Hartree cutoff of the plane-wave
expansion, while the hard TM pseudopotential required 320 Hartree.

The statically screened Coulomb interaction $W$ has been obtained (within the
RPA) with the ONCVPSP pseudopotential (without Al $2s$ and $2p$ core levels),
including 100 bands, and with a cutoff of 8 and 14.7 Hartree for the Kohn-Sham
wave functions for the optical and shallow-core excitations, respectively. The
size of the screening matrix in the plane-wave basis was 6 Hartree for the
optical and 8 Hartree for the core spectrum. We have verified that, contrary
to calculations of the screened interaction for other materials like
siliconLuppi  _et al._ (2008) or simple metalsSturm  _et al._ (1990); Quong
and Eguiluz (1993); Zhou  _et al._ (2018), the effect of core polarization is
negligible in $\alpha$-Al2O3.

In the all-electron results, the ground-state calculations were performed
using a plane wave cutoff, $R_{\rm MT}|\mathbf{G}+\mathbf{k}|_{\rm max}$, of
18 Hartree and muffin-tin (MT) spheres $R_{\rm MT}$ of 2$a_{0}$ and
1.45$a_{0}$ for aluminum and oxygen, respectively. The RPA screening was
obtained with 100 conduction bands and a cutoff in the matrix size of 5
Hartree (maintaining the same cutoff of the ground state for the plane waves).

The GW band structure has been approximated within a scissor correction model.
The LDA conduction bands have been rigidly shifted upwards by 2.64 eV, which
corresponds to the band gap correction obtained within the perturbative G0W0
scheme by Marinopoulos and GrüningMarinopoulos and Grüning (2011).

The BSE calculations for the absorption spectra have been performed with
shifted $\mathbf{k}$-point grids (i.e., not containing high-symmetry
$\mathbf{k}$ points), which allowed for a quicker convergence of the
spectraBenedict  _et al._ (1998). The optical absorption spectrum converged
with a 10$\times$10$\times$10 $\mathbf{k}$-point grid, while the XANES spectra
at the Al L2,3 and L1 edges converged with a 8$\times$8$\times$8
$\mathbf{k}$-point grid. The exciton analysis and plot have been instead done
with a smaller $\Gamma$-centered 4$\times$4$\times$4 $\mathbf{k}$-point grid.

The BSE spectra for the optical spectrum or the XANES spectra at the Al L2,3
and L1 edges had a different convergence rate with respect to the number of
empty bands considered in the BSE hamiltonian. Fig. 1 shows their convergence
study (carried out here with a reduced number of $\mathbf{k}$ points in a
$\Gamma$-centered 2$\times$2$\times$2 $\mathbf{k}$-point grid). While the
optical spectrum (left panel) quickly converges with the number of empty
bands, the XANES spectra (middle and right panels) require many more empty
bands, also to converge the lowest energy peak. In the converged spectra,
obtained with many more $\mathbf{k}$ points, this slow convergence is
partially attenuated by the fact that the spectra become smoother. The optical
absorption spectra have been thus obtained with 12 valence bands and 12
unoccupied bands. The XANES spectra at the L2,3 and L1 edges included all the
corresponding core levels together with 30 unoccupied bands. A 0.1 eV
Lorentzian broadening has been applied to the spectra.

In the all-electron BSE calculations, we considered the same parameters used
in the calculation of the screening: 9 Hartree for the wavefunction cutoff and
5 Hartree to describe the electron-hole terms. In the pseudopotential BSE
calculations, we have used a 30 Hartree cutoff for the Kohn-Sham wavefunctions
expansion and 7.3 Hartree for the plane-wave representation of the electron-
hole interactions. We note that, as usual (see e.g.Lorin  _et al._ (2021)),
the plane-wave cutoffs for the BSE matrixelements can be significantly reduced
with respect to the high cutoff needed for the ground-state calculation.
Therefore, even for pseudopotential BSE calculations of shallow-core
excitations, the limiting factor remains the large size of the BSE hamiltonian
(1) in extended systems, which is given by the number of electron-hole
transitions (i.e., the number of occupied bands $\times$ the number of
unoccupied bands $\times$ the number of $\mathbf{k}$ points in the full
Brillouin zone).

##  III  Results

###  III.1  Crystal and electronic structure of $\alpha$-Al2O3

Figure 2:  Primitive rhombohedral unit cell of the crystal structure of
$\alpha$-Al2O3. Red and grey balls represent O and Al atoms, respectively. The
Al atoms are aligned along the cartesian $z$ axis, which is the vertical
direction in the figure, while the O atoms belong to the $xy$ planes
perpendicular to it.

The crystal structure of corundum $\alpha$-Al2O3 is trigonal (see Fig. 2). In
the primitive rhombohedral unit cell (space group R$\bar{3}$c, number 167)
there are two formula units. The corundum structure can also be viewed as a
hexagonal cell containing six formula units with alternate layers of Al and O
atoms in planes perpendicular to the hexagonal $c_{H}$ axis. In the
$\alpha$-Al2O3 structure all Al atoms occupy octahedral sites coordinated with
6 O atoms, which form two equilateral triangles located respectively slightly
above and below each Al atom along the $c_{H}$ direction.

We adopted the experimental lattice parameters from Ref. Newnham and Haan
(1962): $a_{H}=b_{H}$ = 4.7589 Å and $c_{H}$ = 12.991 Å in the hexagonal unit
cell, which corresponds to $a_{R}$ = 5.128 Å and $\alpha=55.287^{\circ}$ in
the rhombohedral primitive cell. In the reference frame used in the
simulations, the hexagonal $c_{H}$ axis is aligned along the cartesian $z$
axis, which is the vertical direction in Fig. 2.

Figure 3:  (Left) LDA Kohn-Sham band structure of $\alpha$-Al2O3. The top of
the valence band has been set to zero. Density of states projected on (middle)
O and (right) Al atoms, resolved in the angular components: $s$ (red), $p$
(blue) and $d$ (green).

The left panel of Fig. 3 shows the Kohn-Sham LDA band structure along a high
symmetry path in the first Brillouin zone, together with the projected density
of states on the O (middle panel) and Al (right panel) atoms. $\alpha$-Al2O3
has a direct bandgap at the $\Gamma$ point, which amounts to 6.21 eV in the
LDA. This value is in very good agreement with the result of Ref. Mackrodt
_et al._ , 2019 obtained with the same experimental lattice parameters.
CalculationsAhuja  _et al._ (2004); Marinopoulos and Grüning (2011); Santos
_et al._ (2015); Mackrodt  _et al._ (2019) that adopt a crystal structure
optimised within the LDA, rather than the experimental one, instead obtain
larger band gaps. In particular, the difference with respect to Ref.
Marinopoulos and Grüning, 2011 is 0.51 eV. We refer to Ref. Santos  _et al._ ,
2015 for a detailed analysis of the dependence on the band gap on the lattice
parameters. As usual, the Kohn-Sham band gap underestimates the experimental
fundamental gap, estimated to be 9.57 eV from temperature-dependent vacuum
ultraviolet (VUV) spectroscopyFrench  _et al._ (1994) and 9.6 eV from
conductivity measurementsWill  _et al._ (1992).

The 6 bands located between -19 eV and -15.9 eV are the O $2s$ states, while
the upper 18 valence bands, starting at $\sim$ -7 eV, are mostly due to O $2p$
states, partially hybridised with Al states. The valence bands are quite flat
along the entire path. The bottom conduction band consists of Al $3s$
hybridised with O $3s$ at the $\Gamma$ point and also with O $2p$ elsewhere,
showing a strong dispersion around the $\Gamma$ point. The higher conduction
bands have mainly Al $3p$ and $3d$ character, also hybridised with O states.
This overview of the electronic properties confirms the intermediate covalent-
ionic nature of the chemical bond in $\alpha$-Al2O3.

Finally, the Al $2p$ and $2s$ core levels (not shown in Fig. 3) in LDA are
located 61.7 eV and 99.4 eV below the top valence, which, as usual, largely
underestimates the experimental valuesCrist (2004) of 70.7 eV and 115.6 eV,
respectively. The calculations do not include the spin-orbit coupling, so the
$2p_{1/2}$ and $2p_{3/2}$ levels are not split. In all cases, we have verified
that pseudopotential and all-electron calculations give the same band
structures and core-level energies.

###  III.2  All-electron benchmark

Figure 4:  Comparison of absorption spectra calculated with pseudopotential
(red lines) and all-electron (blue lines) methods, using an unshifted $8\times
8\times 8$ $\mathbf{k}$-point grid, (left panels) in the independent particle
approximation (IPA), (middle panels) in the random-phase approximation (RPA),
(right panels) from the full Bethe-Salpeter equation (BSE). (Upper panels)
Optical spectra (with 12 valence bands and 20 conduction bands). (Bottom
panels) XANES spectra at Al L${}_{2_{3}}$ edge (with 12 core levels and 12
conduction bands).

One of the main goals of this work is to demonstrate that shallow core spectra
can be calculated with high accuracy using the pseudopotential (PP)
approximation. The importance of this objective is underlined by the many
works in the same spirit Willand  _et al._ (2013); Lejaeghere  _et al._
(2014); Prandini  _et al._ (2018); Lejaeghere  _et al._ (2016). However, at
variance with previous works that concern tests on ground-state properties,
mostly related to total-energy calculations, here we aim at a much more
stringent test, which involves occupied (both valence and semi-core) and
unoccupied states. The latter could be in particular affected by the presence
of ghost states Gonze  _et al._ (1991), which could jeopardize completely the
excitation spectrum, while leaving unaffected a total energy calculation.
Therefore, in order to validate the optical and core spectra calculated with
PPs, we benchmark the results with full-potential all-electron (AE)
calculations, considered as a gold-standard method for solving DFT in extended
systems Puschnig and Ambrosch-Draxl (2002); Gulans  _et al._ (2014). In order
to perform this comparison properly, for both optical and L2,3 edge absorption
spectra the same choice of valence electrons is made in the two calculations,
and the number of plane wave was converged consistently in the two cases.

The valence and L2,3 spectra obtained at different levels of approximations,
IPA, RPA and BSE, are shown in the top and bottom panels of Fig. 4,
respectively. We can make several observations: i) The results of the left
panels of Fig.4 show that the pseudopotential approximation reproduces the
all-electron spectra with excellent accuracy within the IPA. ii) For the RPA
spectrum (central panels) we find a similar result. This is in part related to
the fact that local fields effects are not important in the energy ranges
considered. iii) Finally, also the BSE calculations with the two approaches
are in very good agreement. Recent comparisonsUnzog  _et al._ (2022) between
all-electron and projected augmented wave method approaches, for instance,
present much bigger discrepancies than our results appearing in the right
panels of Fig.4. The origin of this residual difference lies in the different
treatment between the two codes of the integrable singularity of the diagonal
matrix elements of $W$ in (1), calculated in reciprocal space, when
$\mathbf{k}-\mathbf{k}^{\prime}=\mathbf{q}=0$ and the reciprocal-lattice
vectors are $\mathbf{G}=\mathbf{G}^{\prime}=0$. We note that the different
treatment of this singularity was already mentioned also recently in a
comparison among different GW codes Rangel  _et al._ (2020). This singularity
is, in fact, eliminated, by evaluating the integral

$-\frac{4\pi}{\Omega}\epsilon^{-1}_{\mathbf{G}=0,\mathbf{G}^{\prime}=0}(\mathbf{q}\to
0)\frac{1}{(2\pi)^{3}}\int_{\Omega_{\mathbf{q}=0}}d\mathbf{q}\frac{1}{q^{2}},$

where $\Omega_{\mathbf{q}=0}=\Omega_{BZ}/N_{k}$. In order to carry out,
numerically or analytically, the integral, one has to define the shape for the
little volume $\Omega_{\mathbf{q}=0}$ around the origin of the Brillouin zone
and, in anisotropic materials, choose the $\mathbf{q}\to 0$ direction in order
to evaluate the inverse dielectric function $\epsilon^{-1}(\mathbf{q}\to 0)$.
The details about how this integral is performed are in Ref.Albrecht (1999)
and Refs. Puschnig (2002); Freysoldt  _et al._ (2008), for EXC and Exciting,
respectively. If we exclude this singular contribution, the two BSE results
become superposed, as in the IPA case. In addition, this contribution vanishes
(more or less rapidly according to the kind of excitonFuchs  _et al._ (2008))
in the convergency with $\mathbf{k}$ points. Fig. 5 indeed shows that the
differences in the spectra obtained with the two codes tend to vanish with
increasing number of $\mathbf{k}$ points. Most importantly for the scope of
the present work, we find that the differences between the PP and AE codes
remain always of the same order of magnitude for both valence and shallow-core
spectra. Therefore, in summary, we can safely conclude that the benchmarks
with the all-electron approach show that pseudopotential calculations for
optical and XANES spectroscopies (with semi-core states) are reliable and
accurate.

Figure 5:  Convergence of BSE absorption spectra calculated with
pseudopotential (solid lines) and all-electron (dot-dashed lines) methods
(with 2 conduction and 2 valence bands), for increasing number of $\mathbf{k}$
points ($\Gamma$-centered grids with 8, 10 and 12 divisions for bottom,
central and top panel, respectively).

Figure 6:  Absorption spectra at the L1 calculated with EXC (pseudopotential
code) and Exciting (all-electron code). All the calculations are performed
using a $\Gamma$-centered $8\times 8\times 8$ grid of $\mathbf{k}$ points and
30 unoccupied bands. In EXC we include the 4 $2s$ levels corresponding to the
4 Al atoms, while in Exciting we include only one $2s$ level (i.e., the $2s$
state on the Al atom where the core hole is created). For this reason, the
spectra of Exciting are multiplied $\times$ 4.

###  III.3  Interference effects at the L1 edge

The comparison between all-electron and pseudopotential approximation is more
delicate for the L1 edge, since the electrons are treated differently in the
two codes. While Exciting includes the 2s states of Al inside the muffin-tin,
in EXC they are considered as valence and treated with plane-waves.

One of the limitations of the linearized augmented-plane-wave (LAPW) method is
that it could give a wrong description of semicore states when they are
considered inside the muffin-tin (MT) sphere, but they overlap significantly
with valence electrons or are too extended to be entirely contained inside the
MT Goedecker (1993); Gulans  _et al._ (2014). In order to overcome this
problem, local orbitals are included to complement the basis. However, the
quality of this basis set depends on the choice of energy parameters Singh
(1991); Gulans  _et al._ (2014). In addition, there could be some interference
effects that play an important role, and are not obviously included when
considering the states inside the muffin-tin Vorwerk  _et al._ (2022). For all
these reasons, since we validated the pseudopotential approach for the valence
electrons (optical and L23 edge), we will use it to benchmark the L1 edge.

The absorption spectra calculated for the L1 edge using different levels of
approximations are shown in Fig. 6. Notice that in EXC, the 4 bands
corresponding to the 2s state of the 4 Al atoms need to be considered in order
to properly represent the electronic transitions, while in Exciting, only one
occupied level is considered, the 2s state of the Al atom where the core-hole
is sitting. Since there are 4 equivalent Al atoms in the cell, the overall
spectrum coming out of Exciting needs to be multiplied by 4, for a correct
comparison.

In all level of approximations, the pseudopotential and all-electron results
differ slightly (and more than in the optical or L2,3 edge cases), showing
that small interference effects among the Al atoms come to play. These
interferences are small in the system under study, for the Al atoms lie in
equivalent positions in the cell, but they are detectable. We have verified
that in other systems Vorwerk  _et al._ (2022) these effects can be
quantitative and qualitatively more important. While including these effects
is still feasible with Exciting (and all approaches that create a core-hole in
a specific position), by doing multiple calculations and generalizing Eq. (2),
interferences come up naturally in pseudopotential approaches, for all
electrons are treated on the same footing and belong to the whole system, not
just to one atom.

###  III.4  Optical and XANES spectra: valence and shallow core excitations

####  III.4.1 Comparison with experiments

Figure 7:  Comparison of theoretical results with experimental data from
Tomiki et al. Tomiki  _et al._ (1993) and French et al.French  _et al._ (1998)
for the optical absorption, and Weigel et al. Weigel  _et al._ (2008) for the
XANES at the L2,3 edge. The calculated spectra are obtained in the independent
particle approximation (IPA), green lines, in the random-phase approximation
(RPA), orange lines, and from the solution of the Bethe-Salpeter equation
(BSE), red lines. Optical absorption spectra for polarization in the (a) $xy$
plane and (b) in the $z$ direction: the calculated spectra have been
blueshifted by 0.7 eV. (c) Absorption spectra at the L2,3 edge in the $xy$
(solid lines) and $z$ (dot-dashed lines) polarizations compared to the
isotropic XANES experimental spectrumWeigel  _et al._ (2008), to which a
vertical offset has been added for improved clarity. (d) Absorption spectra at
the L1 edge in the $xy$ (solid lines) and $z$ (dot-dashed lines)
polarizations.

Fig. 7 compares the calculated absorption spectra,
$\text{Im}\epsilon_{M}(\omega)$, with experiment, for both the optical
absorption corresponding to valence excitations and the XANES spectrum of the
shallow-core excitations at the Al L2,3 edge. The same figure also displays
the results of the calculations at the Al L1 edge, where, to best of our
knowledge, no experimental XANES spectra are available for $\alpha$-Al2O3,
since this core level excitation is less commonly studied than the Al K
edgeCabaret  _et al._ (1996); Ildefonse  _et al._ (1998); van Bokhoven  _et
al._ (1999); Mizoguchi  _et al._ (2009). In all cases, the presence of sharp
and pronounced peaks at the onset of the BSE spectra (red lines), which are
absent in the RPA and IPA spectra (orange and green lines), is an evidence of
strong excitonic effects. Taking into account the electron-hole attraction in
the BSE is the key to bring the calculations in agreement with experiment.

As already discussed in Ref. Marinopoulos and Grüning, 2011, for the optical
absorption in the polarization direction perpendicular to the $z$ axis (i.e.
in the $xy$ plane), where two VUV spectroscopy experiments Tomiki  _et al._
(1993); French  _et al._ (1998) are available, there are large discrepancies
between the experimental spectra themselves [see Fig. 7(a)]. They agree on the
position of the absorption onset and the presence of a sharp peak at $\sim$
9.2 eV, while they largely differ in the intensities of the various spectral
features. Those differences can be attributed to the fact that both absorption
spectra have been obtained from measured reflectivity data using the Kramers-
Kroning relations, which introduces uncertainties in the
$\text{Im}\epsilon_{M}(\omega)$ spectra. The calculated optical spectra in
Fig. 7(a)-(b) have been blueshifted by 0.7 eV. This underestimation of the
onset of the absorption spectrum is a manifestation of the underestimation of
the band gap by the perturbative G0W0 approach, which is a systematic error
for large gap materialsvan Schilfgaarde  _et al._ (2006b). As a matter of
fact, the 2.64 eV scissor correction that we have employed here, which is
taken from the G0W0 calculation in Ref. Marinopoulos and Grüning, 2011,
underestimates the band gap correction to the LDA. The BSE calculation in Ref.
Marinopoulos and Grüning, 2011 is also in very good agreement with the present
result: the difference in the peak positions is actually due to the LDA band
gap difference (see Sec. III.1). The BSE spectrum in the $xy$ polarization
reproduces well the spectral shape measured by French et al.French  _et al._
(1998), while there are larger differences with the experimental spectra in
both polarizations measured by Tomiki et al.Tomiki  _et al._ (1993).

At the Al L2,3 edge, see Fig. 7(c), the calculated spectra have been
blueshifted by 9.75 eV, which matches well the needed correction to the LDA Al
$2p$ core level energy (see Sec. III.1). The calculations neglect the spin-
orbit coupling and therefore miss the splitting of the main peak into a
doublet separated by 0.47 eV in the high-resolution experimental XANES
spectrum from Ref. Weigel  _et al._ , 2008 (which also agrees well with
previous experiments O’Brien  _et al._ (1991, 1993); Tomiki  _et al._ (1993)).
In the spectra, the first, most prominent, excitonic peak is followed by a
series of lower intensity peaks. While the absolute intensity of the
experimental spectrumWeigel  _et al._ (2008) is arbitrary, the relative
intensity of the first and second peaks gives information about the
coordination number of Al and the nature of the chemical bond: a lower
symmetry enhances the intensity of the second peak. Moreover, a lower
coordination shifts the edge to lower energies, while higher bond ionicity
shifts the edge to higher energies van Bokhoven  _et al._ (2001); Weigel  _et
al._ (2008).

At the Al L1 edge there is no available experiment. Therefore, the curves in
Fig. 7(d) have been shifted by 19.5 eV, in order for the smallest independent-
particle transition energy, from the $2s$ band to the bottom-conduction band,
to match the experimental value of 125.2 eV, which corresponds to the sum of
the fundamental band gap plus the binding energy of the $2s$ states French
_et al._ (1994); Crist (2004) (see Sec. III.1). We find that the main
prominent excitonic peak in the BSE spectra is preceded by a pre-edge
structure, more evident in the $xy$ direction (solid lines). At the Al K edge,
which mainly probes the analogous $1s\to 3p$ transition, there has been much
work to explain the origin of a similar prepeak structureCabaret  _et al._
(1996); Mo and Ching (2000); Cabaret  _et al._ (2005); Cabaret and Brouder
(2009); Brouder  _et al._ (2010); Manuel  _et al._ (2012); Nemausat  _et al._
(2016); Delhommaye  _et al._ (2021), which has been finally interpreted in
terms of atomic vibrations enabling monopole transitions to unoccupied Al $3s$
states. In the present case, the calculations do not take into account the
coupling with atomic vibrations and nevertheless the BSE spectra show a
prepeak structure. This finding therefore calls for a detailed comparison with
other calculations including atomic vibrations and, possibly, experiments at
the Al L1 edge.

####  III.4.2 Anisotropy and local field effects

The $\alpha$-Al2O3 crystal is optically uniaxial. As shown by Fig. 7(a)-(b),
at the onset of the optical spectrum the anisotropy is rather small, while it
becomes larger for higher energy features. The lowest energy exciton is
visible along the $z$ polarization, while it is dark in the perpendicular $xy$
polarization. It is separated by $\sim$ 25 meV from a pair of degenerate
excitons that are visible in the perpendicular $xy$ direction and, conversely,
dark in the $z$ direction. Tomiki et al.Tomiki  _et al._ (1993) experimentally
determined a similar splitting of the exciton peaks in the two polarization
directions (35 meV at room temperature and 86 meV at 10 K). We find that the
binding energy of these excitons is of order of 0.3 eV, which is more than
twice the 0.13 eV value estimated from temperature-dependent VUV
spectroscopyFrench  _et al._ (1994). A similar splitting of the lowest energy
exciton occurs also at the L2,3 edgeTomiki  _et al._ (1993), where its binding
energy largely increases up to 1.6 eV. For the optical and the L2,3 cases,
both the lowest energy exciton in the BSE spectrum and the excitation at the
smallest independent-particle transition energy in the IPA spectrum have a
significant oscillator strength. Instead, at the L1 edge the lowest energy
transitions have a $2s\to 3s$ character and are dipole forbidden. We find that
the binding energy of the lowest dark exciton at the L1 edge is 1.2 eV. The
lowest bright excitons in the $z$ and $xy$ polarization directions are located
1.6 eV and 1.8 eV above it, respectively. They belong to the prepeak in the
spectrum. In this case, we define their binding energy as the difference with
respect to the corresponding first allowed transition in the IPA spectrum: it
amounts to 0.6 eV. The splitting of the main exciton peak in the two
polarizations is also the largest one at the L1 edge, being more than 0.2 eV.

By comparing the RPA and IPA optical spectra, orange and green lines in Fig.
7(a)-(b), respectively, we note that the effect of crystal local fields is
quite small for both polarizations, in contrast to typical layered van der
Waals materials like graphite, where local field effects are strong for the
polarization along the hexagonal axisMarinopoulos  _et al._ (2002).
Marinopoulos and GrüningMarinopoulos and Grüning (2011) also found that local
field effects are not essential to describe satisfactorily the low energy part
of the experimental spectra, whereas they become crucial for higher energies
(above 16 eV, not shown in Fig. 7), in correspondence to the excitation of the
more localised O $2s$ electrons. Indeed, the degree of electron localisation
directly correlates with the degree of charge inhomogeneity, which is a key
factor for the induced microscopic local fields. One may therefore expect that
the excitation spectra of shallow core levels, which are even more localised,
should be more affected by local field effects. This phenomenon has been in
fact observed for many shallow core levelsVast  _et al._ (2002); Dash  _et
al._ (2007); Huotari  _et al._ (2010); Cudazzo  _et al._ (2014); Ruotsalainen
_et al._ (2021). However, in $\alpha$-Al2O3 for both the L2,3 and L1 edges the
comparison of the absorption spectra calculated within the RPA and in the IPA
shows that local field effects are actually negligible333It is well known that
local field effects, expressed as electron-hole exchange interaction in the
BSE framework, are essential to get the correct branching ratios between L2
and L3 components, see e.g. Gilmore  _et al._ (2015); Vinson and Rehr (2012);
Ankudinov  _et al._ (2005). However, in the present case the neglect of spin-
orbit coupling does not allow us to resolve the two components. For
$\alpha$-Al2O3 an electron–hole exchange energy of 0.3 eV has been estimated
Weigel  _et al._ (2008); O’Brien  _et al._ (1991). (even weaker than in the
optical regime). We can understand this result by noticing that the intensity
of the L2,3 and L1 absorption spectra is one or two orders of magnitude
smaller than for the optical absorption. This large intensity difference
reflects the fact that Al $2p$ and $2s$ states are much less polarizable than
valence states. Therefore, even though their electronic charge is much more
localized and inhomogeneous, local fields associated to the excitations of Al
$2p$ and $2s$ are small because they are weakly polarizable, which also leads
to weak induced potentials.

Figure 8:  Contributions of independent transitions to the lowest energy
bright exciton intensity in the absorption spectra: (left) for the optical
spectrum; (middle) for the XANES at L2,3; (right) for the XANES at the L1
edge. The size of the circles is proportional to
$|\tilde{\rho}_{vc{\mathbf{k}}}A_{\lambda}^{vc{\mathbf{k}}}|$.

####  III.4.3 Analysis of excitonic effects

Excitonic effects in solids can be understood as the result of the mixing of
the independent-particle, vertical interband transitions at various
$\mathbf{k}$ points in the Brillouin zone, which are weighted by the excitonic
coefficients $A_{vc\mathbf{k}}^{\lambda}$, i.e., the eigenvectors of the
excitonic Hamiltonian (1). The analysis of the excitonic coefficients
therefore directly informs on the character of the exciton.

Fig. 8 represents, projected on the LDA band structure, the partial
contributions
$\left|A_{vc\mathbf{k}}^{\lambda}\tilde{\rho}_{vc\mathbf{k}}\right|$ to the
oscillator strength of the lowest energy bright excitons in the absorption
spectra of Fig. 7. Each independent-particle transition ${v\mathbf{k}\to
c\mathbf{k}}$ is represented by a pair of circles, one in the occupied band
$v$ and one in the unoccupied band $c$, whose size is proportional to the
value of the contribution. For the optical spectrum (left panel of Fig. 8), we
consider the exciton giving rise to the first peak in the absorption spectrum
in the $z$ polarization. Our analysis shows that the largest contribution
stems from the top-valence bottom-conduction transition at the $\Gamma$ point,
in correspondence to the direct band gap. The next $\mathbf{k}$ points along
the L$\Gamma$X line in the conduction band give a contribution that is already
10 times smaller. The others are even smaller. This is due to the fact that
for this exciton the top-valence bottom-conduction transition at the $\Gamma$
point has the predominant coefficient $A^{vc{\mathbf{k}}}_{\lambda}$, together
with a large single-particle oscillator strength $\tilde{\rho}_{vc\mathbf{k}}$
in the $z$ direction. Instead, the same $\tilde{\rho}_{vc\mathbf{k}}$ is
negligibly small in the $x$ or $y$ direction, explaining why the same exciton
is dark in the $xy$ plane.

For the L2,3 and L1 excitation spectra, all the $\mathbf{k}$ points for the
corresponding core levels are involved in the spectra, as one may expect from
the fact that the core levels are not dispersive. Also for first exciton peak
in the L2,3 XANES spectrum (middle panel of Fig. 8), the lowest conduction
band at the $\Gamma$ point gives the largest contribution, having a large Al
$3s$ character (see Sec. 3). However, in this case the other $\mathbf{k}$
points of the bottom conduction band and the higher conduction bands
significantly contribute to the spectrum as well. This illustrate the
deviation from a simple independent-particle picture of a Al $2p\to 3s$ atomic
transition, since many transitions are mixed together to produce the excitonic
peak at the onset of the L2,3 XANES spectrum.

For the L1 XANES spectrum (right panel of Fig. 8), we consider the first
bright exciton in the $z$ polarization direction, which belongs to the prepeak
in the spectrum in Fig. 7(d). Contrary to the other two cases, the bottom-
conduction band at the $\Gamma$ point gives no contribution, consistently with
the $2s\to 3s$ character of the transition, which is dipole forbidden. The
largest contributions are instead given by the $\mathbf{k}$ points along the
$\Gamma$T line of the bottom conduction band, which have $3p$ character as
well. Even in this case higher conduction bands contribute significantly to
the intensity of the excitonic prepeak.

The plot in Fig. 9 of the cumulative sums $S_{\lambda}(\omega)$, see Eq. (3),
as a function of the number of conduction bands explains the different
convergence behavior between the optical and L2,3 XANES spectra shown in Fig.
1. By increasing the number of conduction bands in the BSE Hamiltonian (1),
the largest possible independent-particle transition energy progressively
increases. Therefore, the curves for larger numbers of conduction bands extend
to higher energies. However, in the case of the optical spectrum (top panel),
the cumulative sum $S_{\lambda}(\omega)$ rapidly converges to the final
result. Already considering transition energies within 12 eV from the smallest
one and including 15 conduction bands in the BSE hamiltonian give a result of
the oscillator strength very close to 100%. Instead, in the case of the L2,3
edge (bottom panel), the range of transition energies needed to get close to
100% has to be much larger, of the order of 50 eV above the smallest
transition energy. Moreover, the various curves in the bottom panel of Fig. 9
do not overlap, as it is the case for the optical spectrum in the upper panel.
This behavior indicates that, at the L2,3 edge, interband transitions to
higher conduction bands in the BSE hamiltonian mix together with transitions
to lower conductions bands, which affects the behavior of the cumulative sum
$S_{\lambda}(\omega)$ also at lower energies. The reason of this strong mixing
is the fact that at the L2,3 edge there are many interband transitions with
similar weak intensity. This, in turns, explains why the convergence of the
XANES spectrum with the number of conduction bands is slow (see Fig. 1), and
requires extra care.

Figure 9:  Cumulative sums $S_{\lambda}(\omega)$ as a function of number of
conduction bands (cb) in the BSE hamiltonian for the lowest energy bright
exciton in the $z$ direction for (top panel) the optical spectrum (bottom
panel) and the XANES spectrum at the L2,3 edge. In each case, the zero of the
energy axis has been set to the smallest independent-particle transition
energy and $S_{\lambda}(\omega)$ has been normalised to its largest value.

The lowest-energy dark excitons, both in the optical spectrum and the L2,3
edge, have a cumulative sum $S_{\lambda}(\omega)$ that is always close to
zero. It means that all the independent-particle oscillator strengths
$\tilde{\rho}_{vc\mathbf{k}}$ are always small, indicating dipole forbidden
transitions. The situation is instead different for the lowest dark exciton at
the L1 edge. In this case, some transitions to the lowest conduction bands
have a weak but not zero contribution $|\tilde{\rho}_{vck}A_{\lambda}|$ to the
spectrum, as shown by their representation on the LDA band structure in the
top panel of Fig. 10. The corresponding cumulative sum $S_{\lambda}(\omega)$,
bottom panel of Fig. 10, is indeed not always zero: it has even two distinct
peaks, before progressively decreasing to zero, giving rise to a dark exciton.
This suggests the occurrence of destructive interference of contributions
$\tilde{\rho}_{vck}A_{\lambda}$ of different sign, involving transitions over
a large range of energy. Moreover, it also shows that including not enough
conduction bands in the BSE hamiltonian (1) would produce a weak excitonic
peak in the spectrum. It is another indication that an independent-particle
picture is here inadequate, whereas the strong electron-hole interaction
manifest itself as the (positive or negative) interference of many electron-
hole pairs.

Figure 10:  Contributions of independent transitions to the dipole strength of
the lowest energy dark exciton in the XANES spectrum at the L1 edge. (Top
panel) The size of the circle is proportional to
$|\tilde{\rho}_{vck}A_{\lambda}|$. (Bottom panel) Corresponding cumulative sum
$S_{\lambda}(\omega)$. The zero of the energy axis has been set to the
smallest independent-particle transition energy and the intensity normalised
to the largest value.

Figure 11:  Exciton correlation function
$\Psi_{\lambda}(\mathbf{r}_{h},\mathbf{r}_{e})$ for the lowest bright exciton
in the optical spectrum and at the prepeak at the L1 edge. The position of the
hole $\mathbf{r}_{h}$ is fixed at $\mathbf{r}_{h}^{0}$ (see black ball). The
color plots show the corresponding electron density distribution
$|\Psi_{\lambda}(\mathbf{r}_{h}^{0},\mathbf{r}_{e})|^{2}$ in the $xy$ plane
perpendicular to the $z$ axis contaning the hole. In order to avoid nodes of
the orbitals, the hole position has been slightly displaced from an oxygen
atom for the optical exciton, and from an aluminum atom for the $L_{1}$ edge.
(This explains why the density distributions are not symmetric around
$\mathbf{r}_{h}^{0}$). The intensity follows a blue-cyan-green-yellow-orange-
red gradient, and goes from 0 (blue) to the maximum value of the square of the
excitonic wavefunctions (red).

Fig. 11 displays the electron density distribution
$|\Psi_{\lambda}(\mathbf{r}_{h}^{0},\mathbf{r}_{e})|^{2}$ for a fixed position
of the hole $\mathbf{r}_{h}^{0}$ for the wavefunction of the lowest bright
excitons in the spectra. In the color plots, we consider a cut of the three-
dimensional distribution in the $xy$ plane, perpendicular to the $z$ axis,
containing the hole. In all cases, the hole position (represented by the black
ball in Fig. 11) has been chosen slightly away from the atoms, in order to
avoid the nodes of the orbitals. This is the reason why the electron
distribution is not symmetrical around the hole. For an uncorrelated electron-
hole pair, the electron density would be delocalised all over the crystal,
corresponding to a Bloch wavefunction. The effect of the electron-hole
correlation is instead to localise the electron density around the hole.

For the optical spectrum (left panel), the hole has been placed near an O
atom, consistently with the main character of the valence band (see Sec.
III.1). Here we discover that the electron charge is also surprisingly located
at the O atoms, and quite delocalised in the $xy$ plane. This picture is
indeed in contrast with the naive expectation of a charge transfer O $\to$ Al
nature of the exciton, which is based on the largely ionic character of the
electronic properties of $\alpha$-Al2O3. However, the strong Al-O
hybridisation of the bottom conduction bands makes it possible for the exciton
to localise entirely on the O atoms. The nature of the exciton in
$\alpha$-Al2O3 therefore turns out to be similar to what foundRohlfing and
Louie (1998); Gatti and Sottile (2013) in other ionic materials like LiF,
where, analogously, for a hole fixed at a F atom, the electron charge is
located mainly on F atoms as well.

Finally, the right panel of Fig. 11 shows the wavefunction of the first bright
exciton in the prepeak of the L1 edge. The hole is localised close to an Al
atom. The resulting electron charge has partially the shape of a deformed $2p$
orbital pointing to the next neighbor O atom. In this case, the electron
charge is entirely localised around the same Al site, displaying the atomic
character of the core exciton.

##  IV  Conclusions

In summary, we have presented a norm-conserving pseudopotential approach that
permits one to evaluate optical and XANES spectra on the same footing, using
the same basis set for valence and shallow-core electrons. We have validated
the approach by comparison with full potential all-electron calculations, at
three different levels of theory, independent-particle approximation, RPA and
full excitonic calculation, within the BSE formalism. We have applied this
approach to study the optical and semi-core excitations of corundum
$\alpha$-Al2O3, a promising material for its optical and structural
properties. Both regimes, optical and XANES, present strong many-body effects
that require the highest level of theory for an accurate and quantitative
description. The BSE calculations show good agreement with experiments, when
available, but more importantly permit one to explain the physical origin of
the various excitations, thanks to a thorough analysis of the excitons. The
small anisotropy in the optical regime, for instance, reveals a different
order of excitons in the $z$ and perpendicular $xy$ directions: the first
exciton in bright along $z$, followed by dark excitons, while it is the
contrary in the perpendicular $xy$ direction. This splitting appears also for
the L2,3 edge. The dark/bright character of the excitons in the optical, L1
and L2,3 edges is analysed both by projecting the excitonic eigenvectors on
the LDA band structure, as well as by looking at the cumulative function, Eq.
(3). The first analysis tool is particularly useful to understand the origin
of each exciton, in terms of the single-particle transitions and of the atomic
characters of the single bands; the cumulative function can reveal purely
many-body effects, like the distructive interference that takes place at the
L1 edge, making the first exciton dark. In addition, the excitonic
wavefunction, by showing the localization of the different excitons, can
reveal counter-intuitive behaviour, like the electron localization on the
oxygen atom, for the bright exciton in the optical spectrum, in contrast to a
naive charge-transfer O$\to$Al character.

This work opens the way to the treatment of other shallow-core spectroscopies,
like electron energy loss near-edge structures (ELNES). Moreover, the unified
footing to tackle shallow core, valence, and conduction states, will be
particularly useful to describe Resonant Inelastic X-ray Scattering (RIXS) and
X-ray Raman Scattering (XRS).

###### Acknowledgements.

We thank the French Agence Nationale de la Recherche (ANR) for financial
support (Grant Agreements No. ANR-19-CE30-0011). Computational time was
granted by GENCI (Project No. 544).

## References

  * van Bokhoven and Lamberti (2016) J. van Bokhoven and C. Lamberti, eds.,  _X-Ray Absorption and X-Ray Emission Spectroscopy: Theory and Applications_ (Wiley, 2016).
  * de Groot  _et al._ (2021) F. M. de Groot, H. Elnaggar, F. Frati, R. pan Wang, M. U. Delgado-Jaime, M. van Veenendaal, J. Fernandez-Rodriguez, M. W. Haverkort, R. J. Green, G. van der Laan, Y. Kvashnin, A. Hariki, H. Ikeno, H. Ramanantoanina, C. Daul, B. Delley, M. Odelius, M. Lundberg, O. Kuhn, S. I. Bokarev, E. Shirley, J. Vinson, K. Gilmore, M. Stener, G. Fronzoni, P. Decleva, P. Kruger, M. Retegan, Y. Joly, C. Vorwerk, C. Draxl, J. Rehr, and A. Tanaka, Journal of Electron Spectroscopy and Related Phenomena 249, 147061 (2021).
  * Fujikawa (1983) T. Fujikawa, Journal of the Physical Society of Japan 52, 4001 (1983).
  * Tyson  _et al._ (1992) T. A. Tyson, K. O. Hodgson, C. R. Natoli, and M. Benfatto, Phys. Rev. B 46, 5997 (1992).
  * Ahlers  _et al._ (1998) D. Ahlers, G. Schütz, V. Popescu, and H. Ebert, Journal of Applied Physics 83, 7082 (1998).
  * Rehr and Albers (2000) J. J. Rehr and R. C. Albers, Rev. Mod. Phys. 72, 621 (2000).
  * Rehr  _et al._ (2009) J. J. Rehr, J. J. Kas, M. P. Prange, A. P. Sorini, Y. Takimoto, and F. Vila, Comptes Rendus Physique 10, 548 (2009).
  * Rehr  _et al._ (2010) J. J. Rehr, J. J. Kas, F. D. Vila, M. P. Prange, and K. Jorissen, Phys. Chem. Chem. Phys. 12, 5503 (2010).
  * de Groot (2005) F. de Groot, Coordination Chemistry Reviews 249, 31 (2005), synchrotron Radiation in Inorganic and Bioinorganic Chemistry.
  * De Groot and Kotani (2008) F. De Groot and A. Kotani,  _Core level spectroscopy of solids_ (CRC press, 2008).
  * Haverkort  _et al._ (2012) M. W. Haverkort, M. Zwierzycki, and O. K. Andersen, Phys. Rev. B 85, 165113 (2012).
  * Mo and Ching (2000) S.-D. Mo and W. Y. Ching, Phys. Rev. B 62, 7901 (2000).
  * Gougoussis  _et al._ (2009) C. Gougoussis, M. Calandra, A. P. Seitsonen, and F. Mauri, Phys. Rev. B 80, 075102 (2009).
  * Taillefumier  _et al._ (2002) M. Taillefumier, D. Cabaret, A.-M. Flank, and F. Mauri, Phys. Rev. B 66, 195107 (2002).
  * Bunău and Calandra (2013) O. Bunău and M. Calandra, Phys. Rev. B 87, 205105 (2013).
  * Mazevet  _et al._ (2010) S. Mazevet, M. Torrent, V. Recoules, and F. Jollet, High Energy Density Physics 6, 84 (2010).
  * Hetényi  _et al._ (2004) B. Hetényi, F. De Angelis, P. Giannozzi, and R. Car, The Journal of Chemical Physics 120, 8632 (2004).
  * Prendergast and Galli (2006) D. Prendergast and G. Galli, Phys. Rev. Lett. 96, 215502 (2006).
  * Gao  _et al._ (2009) S.-P. Gao, C. J. Pickard, A. Perlov, and V. Milman, Journal of Physics: Condensed Matter 21, 104203 (2009).
  * Prentice  _et al._ (2020) J. C. A. Prentice, J. Aarons, J. C. Womack, A. E. A. Allen, L. Andrinopoulos, L. Anton, R. A. Bell, A. Bhandari, G. A. Bramley, R. J. Charlton, R. J. Clements, D. J. Cole, G. Constantinescu, F. Corsetti, S. M.-M. Dubois, K. K. B. Duff, J. M. Escartín, A. Greco, Q. Hill, L. P. Lee, E. Linscott, D. D. O’Regan, M. J. S. Phipps, L. E. Ratcliff, A. R. Serrano, E. W. Tait, G. Teobaldi, V. Vitale, N. Yeung, T. J. Zuehlsdorff, J. Dziedzic, P. D. Haynes, N. D. M. Hine, A. A. Mostofi, M. C. Payne, and C.-K. Skylaris, The Journal of Chemical Physics 152, 174111 (2020), https://doi.org/10.1063/5.0004445 .
  * Hjalmarson  _et al._ (1981) H. P. Hjalmarson, H. Büttner, and J. D. Dow, Phys. Rev. B 24, 6010 (1981).
  * Lie  _et al._ (1999) K. Lie, R. Brydson, and H. Davock, Phys. Rev. B 59, 5361 (1999).
  * Triguero  _et al._ (1998) L. Triguero, L. G. M. Pettersson, and H. Ågren, Phys. Rev. B 58, 8097 (1998).
  * Klein  _et al._ (2021) B. P. Klein, S. J. Hall, and R. J. Maurer, Journal of Physics: Condensed Matter 33, 154005 (2021).
  * Rehr  _et al._ (2005) J. J. Rehr, J. A. Soininen, and E. L. Shirley, Physica Scripta 2005, 207 (2005).
  * Liang  _et al._ (2017) Y. Liang, J. Vinson, S. Pemmaraju, W. S. Drisdell, E. L. Shirley, and D. Prendergast, Phys. Rev. Lett. 118, 096402 (2017).
  * Onida  _et al._ (2002) G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (2002).
  * Besley  _et al._ (2009) N. A. Besley, M. J. G. Peach, and D. J. Tozer, Phys. Chem. Chem. Phys. 11, 10350 (2009).
  * Bunău and Joly (2012) O. Bunău and Y. Joly, Phys. Rev. B 85, 155121 (2012).
  * Bunău and Joly (2012) O. Bunău and Y. Joly, Journal of Physics: Condensed Matter 24, 215502 (2012).
  * Strinati (1982) G. Strinati, Phys. Rev. Lett. 49, 1519 (1982).
  * Strinati (1984) G. Strinati, Phys. Rev. B 29, 5718 (1984).
  * Shirley (1998) E. L. Shirley, Phys. Rev. Lett. 80, 794 (1998).
  * Carlisle  _et al._ (1999) J. A. Carlisle, E. L. Shirley, L. J. Terminello, J. J. Jia, T. A. Callcott, D. L. Ederer, R. C. C. Perera, and F. J. Himpsel, Phys. Rev. B 59, 7433 (1999).
  * Shirley (2000) E. Shirley, Journal of Physics and Chemistry of Solids 61, 445 (2000).
  * Martin  _et al._ (2016) R. M. Martin, L. Reining, and D. M. Ceperley,  _Interacting Electrons: Theory and Computational Approaches_ (Cambridge University Press, 2016).
  * Bechstedt (2014) F. Bechstedt,  _Many-Body Approach to Electronic Excitations: Concepts and Applications_ , Springer Series in Solid-State Sciences (Springer Berlin Heidelberg, 2014).
  * Botti  _et al._ (2007) S. Botti, A. Schindlmayr, R. D. Sole, and L. Reining, Reports on Progress in Physics 70, 357 (2007).
  * Wills  _et al._ (2010) J. Wills, M. Alouani, P. Andersson, A. Delin, O. Eriksson, and O. Grechnyev,  _Full-Potential Electronic Structure Method: Energy and Force Calculations with Density Functional and Dynamical Mean Field Theory_ , Springer Series in Solid-State Sciences (Springer Berlin Heidelberg, 2010).
  * Andersen (1975) O. K. Andersen, Phys. Rev. B 12, 3060 (1975).
  * Sjöstedt  _et al._ (2000) E. Sjöstedt, L. Nordström, and D. Singh, Solid State Communications 114, 15 (2000).
  * Madsen  _et al._ (2001) G. K. H. Madsen, P. Blaha, K. Schwarz, E. Sjöstedt, and L. Nordström, Phys. Rev. B 64, 195134 (2001).
  * Payne  _et al._ (1992) M. C. Payne, M. P. Teter, D. C. Allan, T. A. Arias, and J. D. Joannopoulos, Rev. Mod. Phys. 64, 1045 (1992).
  * Ku and Eguiluz (2002) W. Ku and A. G. Eguiluz, Phys. Rev. Lett. 89, 126401 (2002).
  * Delaney  _et al._ (2004) K. Delaney, P. García-González, A. Rubio, P. Rinke, and R. W. Godby, Phys. Rev. Lett. 93, 249701 (2004).
  * Tiago  _et al._ (2004) M. L. Tiago, S. Ismail-Beigi, and S. G. Louie, Phys. Rev. B 69, 125212 (2004).
  * van Schilfgaarde  _et al._ (2006a) M. van Schilfgaarde, T. Kotani, and S. V. Faleev, Phys. Rev. B 74, 245125 (2006a).
  * Friedrich  _et al._ (2006) C. Friedrich, A. Schindlmayr, S. Blügel, and T. Kotani, Phys. Rev. B 74, 045104 (2006).
  * Gómez-Abal  _et al._ (2008) R. Gómez-Abal, X. Li, M. Scheffler, and C. Ambrosch-Draxl, Phys. Rev. Lett. 101, 106404 (2008).
  * Luppi  _et al._ (2008) E. Luppi, H.-C. Weissker, S. Bottaro, F. Sottile, V. Veniard, L. Reining, and G. Onida, Phys. Rev. B 78, 245124 (2008).
  * Klimeš  _et al._ (2014) J. c. v. Klimeš, M. Kaltak, and G. Kresse, Phys. Rev. B 90, 075125 (2014).
  * Hamann (2013) D. R. Hamann, Phys. Rev. B 88, 085117 (2013).
  * Note (1) The same hypothesis is made when the core orbitals are obtained from a calculation of the isolated atomShirley (2004); Blöchl (1994); Unzog  _et al._ (2022).
  * French (1990) R. H. French, Journal of the American Ceramic Society 73, 477 (1990).
  * French  _et al._ (1994) R. H. French, D. J. Jones, and S. Loughin, Journal of the American Ceramic Society 77, 412 (1994).
  * Tanaka and Adachi (1996) I. Tanaka and H. Adachi, Phys. Rev. B 54, 4604 (1996).
  * Cabaret  _et al._ (1996) D. Cabaret, P. Sainctavit, P. Ildefonse, and A.-M. Flank, Journal of Physics: Condensed Matter 8, 3691 (1996).
  * Ildefonse  _et al._ (1998) P. Ildefonse, D. Cabaret, P. Sainctavit, G. Calas, A.-M. Flank, and P. Lagarde, Physics and Chemistry of Minerals 25, 112 (1998).
  * van Bokhoven  _et al._ (2001) J. A. van Bokhoven, T. Nabi, H. Sambe, D. E. Ramaker, and D. C. Koningsberger, Journal of Physics: Condensed Matter 13, 10247 (2001).
  * Strinati (1988) G. Strinati, Rivista del Nuovo Cimento 11, 1 (1988).
  * Hedin (1965) L. Hedin, Phys. Rev. 139, A796 (1965).
  * Albrecht  _et al._ (1998) S. Albrecht, L. Reining, R. Del Sole, and G. Onida, Phys. Rev. Lett. 80, 4510 (1998).
  * Benedict  _et al._ (1998) L. X. Benedict, E. L. Shirley, and R. B. Bohn, Phys. Rev. Lett. 80, 4514 (1998).
  * Rohlfing and Louie (2000) M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927 (2000).
  * Vinson  _et al._ (2011) J. Vinson, J. J. Rehr, J. J. Kas, and E. L. Shirley, Phys. Rev. B 83, 115106 (2011).
  * Vinson and Rehr (2012) J. Vinson and J. J. Rehr, Phys. Rev. B 86, 195135 (2012).
  * Gilmore  _et al._ (2015) K. Gilmore, J. Vinson, E. Shirley, D. Prendergast, C. Pemmaraju, J. Kas, F. Vila, and J. Rehr, Computer Physics Communications 197, 109 (2015).
  * Gilmore  _et al._ (2021) K. Gilmore, J. Pelliciari, Y. Huang, J. J. Kas, M. Dantz, V. N. Strocov, S. Kasahara, Y. Matsuda, T. Das, T. Shibauchi, and T. Schmitt, Phys. Rev. X 11, 031013 (2021).
  * Geondzhian and Gilmore (2018) A. Geondzhian and K. Gilmore, Phys. Rev. B 98, 214305 (2018).
  * Dashwood  _et al._ (2021) C. D. Dashwood, A. Geondzhian, J. G. Vale, A. C. Pakpour-Tabrizi, C. A. Howard, Q. Faure, L. S. I. Veiga, D. Meyers, S. G. Chiuzbăian, A. Nicolaou, N. Jaouen, R. B. Jackman, A. Nag, M. García-Fernández, K.-J. Zhou, A. C. Walters, K. Gilmore, D. F. McMorrow, and M. P. M. Dean, Phys. Rev. X 11, 041052 (2021).
  * Vinson (2022) J. Vinson, Phys. Chem. Chem. Phys. 24, 12787 (2022).
  * Olovsson  _et al._ (2009a) W. Olovsson, I. Tanaka, P. Puschnig, and C. Ambrosch-Draxl, Journal of Physics: Condensed Matter 21, 104205 (2009a).
  * Olovsson  _et al._ (2009b) W. Olovsson, I. Tanaka, T. Mizoguchi, P. Puschnig, and C. Ambrosch-Draxl, Phys. Rev. B 79, 041102 (2009b).
  * Olovsson  _et al._ (2011) W. Olovsson, I. Tanaka, T. Mizoguchi, G. Radtke, P. Puschnig, and C. Ambrosch-Draxl, Phys. Rev. B 83, 195206 (2011).
  * Vorwerk  _et al._ (2017) C. Vorwerk, C. Cocchi, and C. Draxl, Phys. Rev. B 95, 155121 (2017).
  * Vorwerk  _et al._ (2019) C. Vorwerk, B. Aurich, C. Cocchi, and C. Draxl, Electronic Structure 1, 037001 (2019).
  * Vorwerk  _et al._ (2020) C. Vorwerk, F. Sottile, and C. Draxl, Phys. Rev. Research 2, 042003 (2020).
  * Laskowski and Blaha (2010) R. Laskowski and P. Blaha, Phys. Rev. B 82, 205104 (2010).
  * Yao  _et al._ (2022) Y. Yao, D. Golze, P. Rinke, V. Blum, and Y. Kanai, Journal of Chemical Theory and Computation 18, 1569 (2022).
  * Vorwerk  _et al._ (2022) C. Vorwerk, F. Sottile, and C. Draxl, Phys. Chem. Chem. Phys. 24, 17439 (2022).
  * Unzog  _et al._ (2022) M. Unzog, A. Tal, and G. Kresse, Phys. Rev. B 106, 155133 (2022).
  * Note (2) There is also the possibility to include $-W$ and exclude $\bar{v}_{c}$, which corresponds to the description of spin-triplet excitations.
  * Gonze  _et al._ (2016) X. Gonze, F. Jollet, F. Abreu Araujo, D. Adams, B. Amadon, T. Applencourt, C. Audouze, J.-M. Beuken, J. Bieder, A. Bokhanchuk, E. Bousquet, F. Bruneval, D. Caliste, M. Côté, F. Dahm, F. Da Pieve, M. Delaveau, M. Di Gennaro, B. Dorado, C. Espejo, G. Geneste, L. Genovese, A. Gerossier, M. Giantomassi, Y. Gillet, D. Hamann, L. He, G. Jomard, J. Laflamme Janssen, S. Le Roux, A. Levitt, A. Lherbier, F. Liu, I. Lukačević, A. Martin, C. Martins, M. Oliveira, S. Poncé, Y. Pouillon, T. Rangel, G.-M. Rignanese, A. Romero, B. Rousseau, O. Rubel, A. Shukri, M. Stankovski, M. Torrent, M. Van Setten, B. Van Troeye, M. Verstraete, D. Waroquiers, J. Wiktor, B. Xu, A. Zhou, and J. Zwanziger, Comput. Phys. Commun. 205, 106 (2016).
  * (84) L. Reining, V. Olevano, F. Sottile, S. Albrecht, and G. Onida, “The exc code,” https://etsf.polytechnique.fr/software/Ab_Initio/, unpublished.
  * Gulans  _et al._ (2014) A. Gulans, S. Kontur, C. Meisenbichler, D. Nabok, P. Pavone, S. Rigamonti, S. Sagmeister, U. Werner, and C. Draxl, Journal of Physics: Condensed Matter 26, 363202 (2014).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
  * Troullier and Martins (1991) N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991).
  * van Setten  _et al._ (2018) M. van Setten, M. Giantomassi, E. Bousquet, M. Verstraete, D. Hamann, X. Gonze, and G.-M. Rignanese, Computer Physics Communications 226, 39 (2018).
  * Zhou  _et al._ (2020) J. S. Zhou, L. Reining, A. Nicolaou, A. Bendounan, K. Ruotsalainen, M. Vanzini, J. J. Kas, J. J. Rehr, M. Muntwiler, V. N. Strocov, F. Sirotti, and M. Gatti, Proceedings of the National Academy of Sciences 117, 28596 (2020).
  * Sturm  _et al._ (1990) K. Sturm, E. Zaremba, and K. Nuroh, Phys. Rev. B 42, 6973 (1990).
  * Quong and Eguiluz (1993) A. A. Quong and A. G. Eguiluz, Phys. Rev. Lett. 70, 3955 (1993).
  * Zhou  _et al._ (2018) J. S. Zhou, M. Gatti, J. J. Kas, J. J. Rehr, and L. Reining, Phys. Rev. B 97, 035137 (2018).
  * Marinopoulos and Grüning (2011) A. G. Marinopoulos and M. Grüning, Phys. Rev. B 83, 195129 (2011).
  * Lorin  _et al._ (2021) A. Lorin, M. Gatti, L. Reining, and F. Sottile, Phys. Rev. B 104, 235149 (2021).
  * Newnham and Haan (1962) E. E. Newnham and Y. M. Haan, Zeitschrift fur Kristallographie - Crystalline Materials 117, 235 (1962).
  * Mackrodt  _et al._ (2019) W. C. Mackrodt, M. Rérat, F. S. Gentile, and R. Dovesi, Journal of Physics: Condensed Matter 32, 085901 (2019).
  * Ahuja  _et al._ (2004) R. Ahuja, J. M. Osorio-Guillen, J. S. de Almeida, B. Holm, W. Y. Ching, and B. Johansson, Journal of Physics: Condensed Matter 16, 2891 (2004).
  * Santos  _et al._ (2015) R. Santos, E. Longhinotti, V. Freire, R. Reimberg, and E. Caetano, Chemical Physics Letters 637, 172 (2015).
  * Will  _et al._ (1992) F. G. Will, H. G. DeLorenzi, and K. H. Janora, Journal of the American Ceramic Society 75, 295 (1992).
  * Crist (2004) B. Crist,  _Handbooks of Monochromatic XPS Spectra: Volume 2 : Commercially Pure Binary Oxides_ (XPS International LLC, 2004).
  * Willand  _et al._ (2013) A. Willand, Y. O. Kvashnin, L. Genovese, A. Vázquez-Mayagoitia, A. K. Deb, A. Sadeghi, T. Deutsch, and S. Goedecker, The Journal of Chemical Physics 138, 104109 (2013).
  * Lejaeghere  _et al._ (2014) K. Lejaeghere, V. V. Speybroeck, G. V. Oost, and S. Cottenier, Critical Reviews in Solid State and Materials Sciences 39, 1 (2014).
  * Prandini  _et al._ (2018) G. Prandini, A. Marrazzo, I. E. Castelli, N. Mounet, and N. Marzari, npj Computational Materials 4, 2057 (2018).
  * Lejaeghere  _et al._ (2016) K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. D. Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. D. Marco, C. Draxl, M. Dułak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Grånäs, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iuşan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. Küçükbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordström, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunström, A. Tkatchenko, M. Torrent, D. Vanderbilt, M. J. van Setten, V. V. Speybroeck, J. M. Wills, J. R. Yates, G.-X. Zhang, and S. Cottenier, Science 351, aad3000 (2016), https://www.science.org/doi/pdf/10.1126/science.aad3000 .
  * Gonze  _et al._ (1991) X. Gonze, R. Stumpf, and M. Scheffler, Phys. Rev. B 44, 8503 (1991).
  * Puschnig and Ambrosch-Draxl (2002) P. Puschnig and C. Ambrosch-Draxl, Phys. Rev. B 66, 165105 (2002).
  * Rangel  _et al._ (2020) T. Rangel, M. D. Ben, D. Varsano, G. Antonius, F. Bruneval, F. H. da Jornada, M. J. van Setten, O. K. Orhan, D. D. O’Regan, A. Canning, A. Ferretti, A. Marini, G.-M. Rignanese, J. Deslippe, S. G. Louie, and J. B. Neaton, Computer Physics Communications 255, 107242 (2020).
  * Albrecht (1999) S. Albrecht,  _Optical Absorption Spectra of Semiconductors and Insulators: ab initio calculations of many-body effects_ , Ph.D. thesis, Ecole Polytechnique, Palaiseau (1999).
  * Puschnig (2002) P. Puschnig,  _Excitonic Effects in Organic Semi-Conductors - An Ab-initio Study within the LAPW Method_ , Ph.D. thesis (2002).
  * Freysoldt  _et al._ (2008) C. Freysoldt, P. Eggert, P. Rinke, A. Schindlmayr, and M. Scheffler, Phys. Rev. B 77, 235428 (2008).
  * Fuchs  _et al._ (2008) F. Fuchs, C. Rödl, A. Schleife, and F. Bechstedt, Phys. Rev. B 78, 085103 (2008).
  * Goedecker (1993) S. Goedecker, Phys. Rev. B 47, 9881 (1993).
  * Singh (1991) D. Singh, Phys. Rev. B 43, 6388 (1991).
  * Tomiki  _et al._ (1993) T. Tomiki, Y. Ganaha, T. Shikenbaru, T. Futemma, M. Yuri, Y. Aiura, S. Sato, H. Fukutani, H. Kato, T. Miyahara, A. Yonesu, and J. Tamashiro, Journal of the Physical Society of Japan 62, 573 (1993).
  * French  _et al._ (1998) R. H. French, H. Müllejans, and D. J. Jones, Journal of the American Ceramic Society 81, 2549 (1998).
  * Weigel  _et al._ (2008) C. Weigel, G. Calas, L. Cormier, L. Galoisy, and G. S. Henderson, Journal of Physics: Condensed Matter 20, 135219 (2008).
  * van Bokhoven  _et al._ (1999) J. A. van Bokhoven, H. Sambe, D. E. Ramaker, and D. C. Koningsberger, The Journal of Physical Chemistry B 103, 7557 (1999).
  * Mizoguchi  _et al._ (2009) T. Mizoguchi, I. Tanaka, S.-P. Gao, and C. J. Pickard, Journal of Physics: Condensed Matter 21, 104204 (2009).
  * van Schilfgaarde  _et al._ (2006b) M. van Schilfgaarde, T. Kotani, and S. Faleev, Phys. Rev. Lett. 96, 226402 (2006b).
  * O’Brien  _et al._ (1991) W. L. O’Brien, J. Jia, Q.-Y. Dong, T. A. Callcott, J.-E. Rubensson, D. L. Mueller, and D. L. Ederer, Phys. Rev. B 44, 1013 (1991).
  * O’Brien  _et al._ (1993) W. L. O’Brien, J. Jia, Q.-Y. Dong, T. A. Callcott, D. R. Mueller, D. L. Ederer, and C.-C. Kao, Phys. Rev. B 47, 15482 (1993).
  * Cabaret  _et al._ (2005) D. Cabaret, E. Gaudry, M. Taillefumier, P. Sainctavit, and F. Mauri, Physica Scripta 2005, 131 (2005).
  * Cabaret and Brouder (2009) D. Cabaret and C. Brouder, Journal of Physics: Conference Series 190, 012003 (2009).
  * Brouder  _et al._ (2010) C. Brouder, D. Cabaret, A. Juhin, and P. Sainctavit, Phys. Rev. B 81, 115125 (2010).
  * Manuel  _et al._ (2012) D. Manuel, D. Cabaret, C. Brouder, P. Sainctavit, A. Bordage, and N. Trcera, Phys. Rev. B 85, 224108 (2012).
  * Nemausat  _et al._ (2016) R. Nemausat, C. Brouder, C. Gervais, and D. Cabaret, Journal of Physics: Conference Series 712, 012006 (2016).
  * Delhommaye  _et al._ (2021) S. Delhommaye, G. Radtke, C. Brouder, S. P. Collins, S. Huotari, C. Sahle, M. Lazzeri, L. Paulatto, and D. Cabaret, Phys. Rev. B 104, 024302 (2021).
  * Marinopoulos  _et al._ (2002) A. G. Marinopoulos, L. Reining, V. Olevano, A. Rubio, T. Pichler, X. Liu, M. Knupfer, and J. Fink, Phys. Rev. Lett. 89, 076402 (2002).
  * Vast  _et al._ (2002) N. Vast, L. Reining, V. Olevano, P. Schattschneider, and B. Jouffrey, Phys. Rev. Lett. 88, 037601 (2002).
  * Dash  _et al._ (2007) L. Dash, F. Bruneval, V. Trinité, N. Vast, and L. Reining, Computational Materials Science 38, 482 (2007), selected papers from the International Conference on Computational Methods in Sciences and Engineering 2004.
  * Huotari  _et al._ (2010) S. Huotari, J. A. Soininen, G. Vankó, G. Monaco, and V. Olevano, Phys. Rev. B 82, 064514 (2010).
  * Cudazzo  _et al._ (2014) P. Cudazzo, K. O. Ruotsalainen, C. J. Sahle, A. Al-Zein, H. Berger, E. Navarro-Moratalla, S. Huotari, M. Gatti, and A. Rubio, Phys. Rev. B 90, 125125 (2014).
  * Ruotsalainen  _et al._ (2021) K. Ruotsalainen, A. Nicolaou, C. J. Sahle, A. Efimenko, J. M. Ablett, J.-P. Rueff, D. Prabhakaran, and M. Gatti, Phys. Rev. B 103, 235136 (2021).
  * Note (3) It is well known that local field effects, expressed as electron-hole exchange interaction in the BSE framework, are essential to get the correct branching ratios between L2 and L3 components, see e.g. Gilmore  _et al._ (2015); Vinson and Rehr (2012); Ankudinov  _et al._ (2005). However, in the present case the neglect of spin-orbit coupling does not allow us to resolve the two components. For $\alpha$-Al2O3 an electron–hole exchange energy of 0.3 eV has been estimated Weigel  _et al._ (2008); O’Brien  _et al._ (1991).
  * Rohlfing and Louie (1998) M. Rohlfing and S. G. Louie, Phys. Rev. Lett. 81, 2312 (1998).
  * Gatti and Sottile (2013) M. Gatti and F. Sottile, Phys. Rev. B 88, 155113 (2013).
  * Shirley (2004) E. L. Shirley, Journal of Electron Spectroscopy and Related Phenomena 136, 77 (2004), progress in Core-Level Spectroscopy of Condensed Systems.
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Ankudinov  _et al._ (2005) A. L. Ankudinov, Y. Takimoto, and J. J. Rehr, Phys. Rev. B 71, 165110 (2005).


distribution of $\mathcal{N}(0,\sigma^{2})$ where $\sigma$ is generated from
the posterior distribution of $\sigma$.

Based on the previous argument, a principled framework we adopt is the
following: we draw a sample $(\sigma_{1},\dots,\sigma_{n})$ from the posterior
distribution $\pi(\sigma|\mathbf{x})$. For each of the individual
$\sigma_{i},i\in\\{1,\dots,n\\}$, we generate a random value from
$\mathcal{N}(0,\sigma_{i}^{2})$. From this sample of a mixture of normal
distributions, we compute the empirical $p$-th quantile, $Q(p)$. This is then
the actual $p$-th quantile of the posterior marginal distribution
$\mathcal{U}(s)|\mathbf{x}$.

##  Appendix C Detection Results for Pointings with Detected UDGs

###  C.1 Pointings with Contaminating Bright Galaxies

Figure 11: Posterior results for V8-WFC3 pointing. (a) Posterior mean
intensity $\lambda(s)$; (b) Posterior mean spatial random effect
$\exp(\mathcal{U}(s))$; (c) Excursion function $F_{C}(s)$ with $C=Q(0.95)$.
Black points are locations of GC candidates. Black crosses are the locations
of previously known UDG candidates. Figure 12: Posterior results for V10-ACS
pointing. (a) Posterior mean intensity $\lambda(s)$; (b) Posterior mean
spatial random effect $\exp(\mathcal{U}(s))$; (c) Excursion function
$F_{C}(s)$ with $C=Q(0.95)$. Black points are locations of GC candidates.
Black cross is the location of previously known UDG candidate. Figure 13:
Posterior results for V15-ACS pointing. (a) Posterior mean intensity
$\lambda(s)$; (b) Posterior mean spatial random effect $\exp(\mathcal{U}(s))$;
(c) Excursion function $F_{C}(s)$ with $C=Q(0.95)$. Black points are locations
of GC candidates. Black crosses are the locations of previously known UDG
candidates.

###  C.2 Pointings without Contaminating Bright Galaxies

Figure 14: Posterior results for V7-ACS pointing. (a) Posterior mean spatial
random effect $\exp(\mathcal{U}(s))$; (b) Excursion function $F_{C}(s)$ with
$C=Q(0.95)$. Black points are locations of GC candidates. Black cross is the
location of previously known UDG candidate. Figure 15: Posterior results for
V7-WFC3 pointing. (a) Posterior mean spatial random effect
$\exp(\mathcal{U}(s))$; (b) Excursion function $F_{C}(s)$ with $C=Q(0.95)$.
Black points are locations of GC candidates. Black cross is the location of
previously known UDG candidate. Figure 16: Posterior results for V9-ACS
pointing. (a) Posterior mean spatial random effect $\exp(\mathcal{U}(s))$; (b)
Excursion function $F_{C}(s)$ with $C=Q(0.95)$. Black points are locations of
GC candidates. Black cross is the location of previously known UDG candidate.
Figure 17: Posterior results for V10-WFC3 pointing. (a) Posterior mean spatial
random effect $\exp(\mathcal{U}(s))$; (b) Excursion function $F_{C}(s)$ with
$C=Q(0.95)$. Black points are locations of GC candidates. Black crosses are
the locations of previously known UDG candidates.

##  Appendix D Prior Distribution Parameter for $R_{k}$

Table 4: Galaxy ID, coordinates, visit pointings, half-light radius $R_{e}$, and mean of prior distribution for $R_{k}$. Galaxies listed here are all contaminating galaxies in pointings with detected UDGs.  ID | R.A. | Dec. | Pointing | $R_{e}$ | Mean $R_{k}(\times R_{e})$
---|---|---|---|---|---
| (J2000) | (J2000) |  | (kpc) | (kpc)
IC 312 | 03 18 08.41 | +41 45 15.75 | V8-ACS | 2.0 | 10.0 ($5\times$)
SDSS J031813.09+414809.0 | 03 18 13.20 | +41 48 09.17 | V8-ACS | 1.3 | 3.9 ($3\times$)
LEDA 12319 aaThis galaxy is a dE but a simple inspection by eye shows its GC system is much more extended than its light profile. Hence, a five-times ratio is adopted. | 03 18 38.91 | +41 40 08.19 | V8-WFC3 | 2.2 | 11.0 ($5\times$)
LEDA 12315 | 03 18 35.39 | +41 40 01.03 | V8-WFC3 | 1.0 | 3.0 ($3\times$)
$\left[\rm BM99\right]$-317 bb[BM99]-NNN catalog was obtained by Brunzendorf & Meusinger (1999). | 03 20 19.11 | +41 42 42.90 | V10-ACS | 3.4 | 10.2 ($3\times$)
PGC 012437 | 03 19 52.97 | +41 18 08.52 | V11-ACS | 2.6 | 13.0 ($5\times$)
UGC 02673 | 03 20 01.64 | +41 15 04.15 | V11-ACS | 3.3 | 16.5 ($5\times$)
PCC 3164 | 03 18 52.25 | +41 16 02.94 | V15-ACS | 3.5 | 10.5 ($3\times$)
LEDA 12356 | 03 19 04.30 | +41 13 59.00 | V15-ACS | 2.0 | 10.0 ($5\times$)