#  Delocalization of dark and bright excitons in flat-band materials and the
optical properties of V2O5

Vitaly Gorelov  LSI, CNRS, CEA/DRF/IRAMIS, École Polytechnique, Institut
Polytechnique de Paris, F-91120 Palaiseau, France  European Theoretical
Spectroscopy Facility  corresponding author<EMAIL_ADDRESS>Lucia Reining  LSI, CNRS, CEA/DRF/IRAMIS, École Polytechnique, Institut
Polytechnique de Paris, F-91120 Palaiseau, France  European Theoretical
Spectroscopy Facility  Martin Feneberg  Institute of Physics, Otto von
Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
Rüdiger Goldhahn  Institute of Physics, Otto von Guericke University
Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany  André Schleife
Department of Materials Science and Engineering, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA  Materials Research Laboratory,
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
National Center for Supercomputing Applications, University of Illinois at
Urbana-Champaign, Urbana, Illinois 61801, USA  Walter R. L. Lambrecht
Department of Physics, Case Western Reserve University, Cleveland,
OH-441-6-7079, USA  Matteo Gatti  LSI, CNRS, CEA/DRF/IRAMIS, École
Polytechnique, Institut Polytechnique de Paris, F-91120 Palaiseau, France
European Theoretical Spectroscopy Facility  Synchrotron SOLEIL, L’Orme des
Merisiers Saint-Aubin, BP 48 F-91192 Gif-sur-Yvette, France

(August 27, 2024)

###### Abstract

The simplest picture of excitons in materials with atomic-like localization of
electrons is that of Frenkel excitons, where electrons and holes stay close
together, which is associated with a large binding energy. Here, using the
example of the layered oxide V2O5 , we show how localized charge-transfer
excitations combine to form excitons that also have a huge binding energy but,
at the same time, a large electron-hole distance, and we explain this
seemingly contradictory finding. The anisotropy of the exciton delocalization
is determined by the local anisotropy of the structure, whereas the exciton
extends orthogonally to the chains formed by the crystal structure. Moreover
we show that the bright exciton goes together with a dark exciton of even
larger binding energy and more pronounced anisotropy. These findings are
obtained by combining first principles many-body perturbation theory
calculations, ellipsometry experiments, and tight binding modelling, leading
to very good agreement and a consistent picture. Our explanation is general
and can be extended to other materials.

##  1 Introduction

It is common belief that the exciton binding energy directly correlates with
its spatial localisation [1]. The classical Wannier and Frenkel textbook
models[2, 3] describe, respectively, the limiting cases of weakly bound and
delocalised excitons, typically found in ordinary semiconductors, and tightly
bound and localised excitons, observed in molecular or noble gas solids.
Excitons that stand out from this conventional expectation have always
attracted considerable attention, as recently witnessed by the surge of
interest in two-dimensional (2D) materials [4]. In fact, exciton binding
energies in 2D materials are always large, irrespective of their spatial
extent [5], which can be rationalised as the consequence of reduced effective
screening in low dimensions [6]. Remarkably, there has been a long standing
controversy whether strongly bound charge transfer excitons, occurring for
example in alkali halides, can be considered as Frenkel excitons [2, 7, 8]:
the origin of their spatial delocalisation has remained largely unexplained.
In low-dimensional transition-metal oxides the interplay between charge
localisation, electron interactions, and reduced dimensionality gives rise to
a wide variety of intriguing physical properties while posing a great
challenge for their theoretical interpretation [9]. These complex materials
therefore offer an ideal playground for unconventional exciton physics.

Vanadium pentoxide (V2O5) is a layered oxide that can be a prototype example
of such a challenging material. It is a “ladder compound” with a unique
crystal structure: the layers consist of double zigzag oxygen and vanadium
chains along the $y$ direction connected by a bridge oxygen forming V-O-V
rungs along the $x$ direction, and vanadyl oxygen atoms located below and
above the vanadium atoms (see Fig.2(a)). The three-dimensional (3D) bulk V2O5
is thus expected to exhibit also properties of 2D or even quasi-1D
character[10, 11]. V2O5 is supposed to be a simple band insulator, with a
relatively wide gap opening between filled O $2p$ bands and empty V $3d$
bands. The peculiar crystal structure is manifested in the weakly dispersing
top-valence bands and, above the band gap, in two narrow bands separated from
higher energy conduction bands[12, 13, 14, 15] (see Fig.4 and Supplementary
Figure 1). These split-off bands are due to V $d_{xy}$ orbitals that have the
smallest overlap with O $2p$ and hence the smallest bonding-antibonding
splitting. As a result, they have a pronounced quasi-1D character, dispersing
mainly along the chain $y$ direction. Unlike other vanadium oxides[16], pure
bulk V2O5 remains non-magnetic and insulating at all temperatures. Due to
doping with alkali intercalation, or because of vanadyl oxygen vacancies, the
split-off bands can be partially occupied, leading to charge ordering and
interesting 1D magnetic properties such as a possible 1D spin-Peierls
transition at low temperature [17, 18, 19, 20, 21].

The band structure appears hence to be well understood, but an unusually large
discrepancy between the calculated band gap and the experimental one has
persisted for a long time. The experimental band gap is widely accepted to be
$2.35$ eV, as determined from Tauc plots based on absorption spectra, with a
slightly different onset depending on crystal direction[22]. This value is
larger, but only to a small extent, than the Kohn-Sham 1.7 eV band gap[13, 14,
15] calculated in the local density approximation (LDA) of density functional
theory (DFT). However, the state-of-the-art theoretical approach for band
structures, namely, the GW approximation[23] to the self-energy of many-body
perturbation theory, yielded a much larger gap[24, 25, 15], with the indirect
minimum gap ranging from 3.5 to 4.0 eV depending on whether a perturbative
G0W0 scheme [26] or quasiparticle (QP) self-consistent GW[27] (QSGW) was
adopted (see Supplementary Table 1).

Excitonic effects may explain the difference between the quasiparticle and
optical gap, but have not been studied previously. Indeed, the Tauc plot
analysis[22, 28, 29] is not applicable when excitonic effects are present.
Direct and inverse photoemission measurements[30] on V2O5 films, which yield
the quasiparticle gap and should be free from excitonic effects, have obtained
the largest value for the gap, of 2.8 eV. However, there is a large
uncertainty on these data because of the limited resolution of these
spectroscopies and the uncertainties concerning the position of the Fermi
level. Moreover, it is still substantially smaller than the calculated
quasiparticle gap. The origin of this discrepancy has been a matter of debate
for a long time. One hypothesis, in particular, was an inadequate description
of screening in the GW calculations. First, since V2O5 is a strongly polar
material, a source of additional screening could be the lattice polarization.
In Ref. 15, on the basis of a model by Bechstedt et al.[31], the self-energy
was rescaled by a factor 0.38 improving the agreement of the onset energy with
ellipsometry data[32]. Later work showed that the effect was largely
overestimated and that it should rather reduce the gap by an amount of the
order of 0.2 eV, way too small to account for the gap overestimate [33, 34].
While the formation of self-trapped electron polarons has been found to be
important in V2O5 [35, 36, 37], they are not expected to occur at the time
scale of an optical absorption and can thus not explain the discrepancy. On
the purely electronic side, the accuracy of QSGW calculations is limited by
the fact that no electron-hole interaction effects are included in the
calculation of $W$. Inclusion of this effect is estimated to reduce the self-
energy by roughly 20% [38], which is, however, much too little in order to
restore agreement. Moreover, these modifications of the screening do not lead
to significant improvement of the spectral shape.

On the other hand, there is also some discrepancy between different optical
experiments. Reflectance and ellipsometry measurements[39, 32, 40, 41, 42]
have obtained spectra for the complex dielectric function
$\epsilon(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$ that agree on
the main peak positions but differ in the onset energy of the first peak (see
Supplementary Figure 2). The discrepancies between different optical
measurements may be due to the sample quality; indeed, it has been shown that
the growth conditions and synthesis methods can significantly shift the
absorption band edge[43, 44, 45, 35, 42]. Presence of strong excitonic effects
hence remains the most promising hypothesis. Indeed, it is well known that
exciton binding is stronger in lower dimensions. The topic of excitons in
strongly anisotropic, in particular layered, materials meets increasing
interest[46, 47, 48, 49, 50, 51, 52]. However, typical values for the exciton
binding energy in such materials are rather of the order of 100 meV, whereas,
here, a difference of more than 1 eV between experiment and theory has to be
explained. On the other hand, optical properties of V2O5 have, to the best of
our knowledge, always been calculated neglecting excitonic effects[53, 15,
54], and the material is too peculiar to rely on extrapolation from other
compounds. In particular, the above materials mostly show significant in-plane
dispersion of the electronic bands, which is not true for V2O5.

At first sight, on the basis of the Wannier model, one would not expect strong
exciton binding in V2O5, with its anisotropic dielectric constant of the order
of 5 [55, 51, 56]. Indeed, for example several transitions metal oxide
perovskites have similar dielectric constants and an exciton binding energy
that lies rather in the range of 100-200 meV[57]. However, the rich playground
of possible excitations in materials that are more complex than common
semiconductors or insulators has been explored only very partially. V2O5 may
be representative for a class of materials, where the electronic charge has a
pronounced atomic-like character, a band ordering suggesting charge transfer
excitations, and a non-trivial, anisotropic crystal structure. While
significant many-body effects can be expected, it is impossible to predict
their strength and nature without taking these various aspects into account.

In the present work we show that V2O5 is the prototype example for a material
whose optical properties are dominated by very strong electron-hole binding,
with excitons of noteworthy properties. We calculate optical spectra including
excitonic effects from first principles by solving the Bethe-Salpeter equation
(BSE)[58], and we validate the results with state-of-the-art ellipsometry
experiments on high-quality single crystals. The very good agreement allows us
to use the calculated data to highlight the intriguing nature of bright and
dark excitons. In particular, we propose an exciton model that explains the
counter-intuitive anisotropy and delocalisation of these charge transfer
excitons, whose relevance goes well beyond the specific case of V2O5 .

##  2 Results

Figure 1: Absorption spectra of V2O5 for $x$, $y$ and $z$ polarization (the
V2O5 layers are in the $xy$ plane). The experimental spectra obtained from
ellipsometry measurements are compared with calculated BSE spectra in the
three directions (note the change of intensity scale for the $z$ direction in
the right panel). The vertical arrows mark the position of the dark excitons
below the absorption onset.

###  2.1 Theoretical and experimental dielectric function

We performed state-of-the-art QSGW calculations to obtain an accurate QP band
structure (see Fig.4 and Supplementary Figure 1). Our results, with a direct
gap at $\Gamma$ of 4.4 eV and an indirect minimum gap of 3.8 eV, agree to
within 0.4 eV with the previous QSGW calculations[15], confirming the large
discrepancy with respect to the widely accepted experimental value for the gap
near 2.35 eV [22] (see Supplementary Table 1). The subsequent solution of the
electron-hole BSE yields the absorption spectrum in Fig.1. For the in-plane
polarizations, it shows a very pronounced peak at the onset, which is situated
at about 3.1 eV, more than 1 eV below the direct QP gap. This shows that there
is indeed a huge excitonic effect in this material. Our experimental results
in Fig.1 confirm the ab initio prediction: in the in-plane directions, they
also show a very strong peak at the onset around $\sim$ 3 eV, similar to, but
stronger than in the experimental results of Mokerov et al.[39] and Losurdo et
al.[40] (see Supplementary Figure 3). Most importantly, the broader peaks
found by Parker et al.[32] below 3 eV are not confirmed by the present data.
Our experiment and calculations also agree on the remaining features of the
spectrum, in particular, the agreement is very good both in $x$ and $y$
direction. Along the chains, in the $y$ direction, the spectral shape is very
well reproduced, and also smaller features such as the shoulder between 4 and
5 eV match. In $x$ direction, theory also correctly describes the 3-peak
structure and the shoulder on the high energy side of the first peak. Absolute
intensities are in qualitative agreement; note that on the experimental side,
peak heights may show some variation according to the sample, and in
particular to the surface roughness that enters the data analysis, and that
broadening due to phonons is not included in the calculations. For the real
part of the spectra see Supplementary Figure 4. The $z$ direction, i.e. out-
of-plane polarization, is not directly accessible by our experiments due to
the fact that the $z$ axis is aligned normal to the sample surface.
Consequently, no high-quality experimental data is available for the out-of-
plane polarization but the information that $\epsilon_{2}$ is of lower
amplitude is reliable. On this basis, the qualitative agreement of theory an
experiment for the $z$ direction is very satisfying.

The agreement between calculated and measured spectra resolves the long-
standing apparent discrepancy between theory and experiment on the band gap of
V2O5, indicating that the first peaks at $\sim$ 3.1 eV in optical absorption
are strongly bound excitons about 1.2 eV below the smallest (located at $Z$)
direct QP gap of 4.3 eV.

Figure 2: Electron density distribution for dark and bright excitons. (a)
Crystal structure of V2O5 . V atoms are grey, while O atoms are red. Selected
atoms indicate the V-O-V rungs. (b)-(c) Electron density distribution
$|\Psi_{\lambda}(\mathbf{r}_{h}^{0},\mathbf{r}_{e})|^{2}$ for the first
exciton $\lambda$ that is bright in $x$ direction and (d)-(e) for the lowest-
energy dark exciton. In both cases, the position of the hole
$\mathbf{r}_{h}^{0}$ is fixed close to a bridge oxygen Ob atom (light blue
dot). Since it is displaced along a non-symmetric direction from the Ob atom
the resulting electronic distribution is also slightly asymmetric. (b) and (d)
views in the plane of V2O5 layers (i.e., $xy$ planes) ; (c) and (e) views
normal to V2O5 layers (i.e., $zy$ planes). Ellipses centered at the hole
position represent the calculated exciton mean radius
$\langle|r^{\alpha}_{\lambda}|\rangle$. The isosurface value for bright and
dark excitons is chosen to be 1% of the maximum electron density.

###  2.2 Exciton extent and anisotropy

We thus reveal a strong excitonic effect and strong anisotropy of the
absorption spectrum in this material, which requires explanation. The very
good agreement between the calculations and experiment allows us to use the
BSE results for the analysis. The BSE can be formulated as a two-body
Schrödinger equation in the basis of transitions between QSGW bands, with
eigenvalues $E_{\lambda}$ and eigenvectors $\bar{A}^{vc{\bf k}}_{\lambda}$,
where $v$ ($c$) stands for valence (conduction) bands and ${\bf k}$ is a
wavevector in the first Brillouin zone (BZ) (see Methods). In the final
spectrum, the coefficients $\bar{A}^{vc{\bf k}}_{\lambda}$ mix the QSGW
transitions $vc{\bf k}$, and the contribution to the spectral intensity for
each excitonic transition $\lambda$ is $|\sum_{vc{\bf k}}\bar{A}^{vc{\bf
k}}_{\lambda}\tilde{\rho}_{vc{\bf k}}|^{2}$, where $\tilde{\rho}_{vc{\bf k}}$
are the oscillator strengths of the QSGW transitions. This kind of calculation
can therefore reveal both bright and dark excitons, the latter corresponding
to $E_{\lambda}$ with vanishing $|\sum_{vc{\bf k}}\bar{A}^{vc{\bf
k}}_{\lambda}\tilde{\rho}_{vc{\bf k}}|^{2}$. We find that indeed, the first
transitions correspond to dark excitons, not visible in the spectrum in any of
the three polarization directions. They are found at an energy as low as
$\sim$2.6 eV (see arrows in Fig.1), i.e., they have a binding energy as large
as $E_{b}=1.6$ eV.

Inspection of the real-space electron-hole correlation functions
$|\Psi_{\lambda}(\mathbf{r}_{h},\mathbf{r}_{e})|^{2}$ (see Methods Eq. (5)),
reveals further surprises. Fig.2(b)-(e) depict $|\Psi_{\lambda}|^{2}$ for the
first dark exciton and the lowest energy exciton that is bright in the $x$
direction. With the hole fixed at position $\mathbf{r}_{h}^{0}$ next to a
bridge Ob atom (light blue dot), where the top-valence states contributing to
these excitons are mostly located (see the analysis below with explanation of
Fig.2), the corresponding electron distribution
$|\Psi_{\lambda}(\mathbf{r}_{h}^{0},\mathbf{r}_{e})|^{2}$ is represented as
function of ${\bf r}_{e}$. For both excitons we find that the electronic
charge is entirely centered on V atoms, illustrating the Ob $2p$ \- V $d_{xy}$
charge transfer nature of the excitons, while the envelope functions globally
have ellipsoidal shapes. While the dark exciton (Fig.2(d),(e)) is more
localised than the bright exciton (Fig.2(b),(c)), both excitons are
delocalised across different V2O5 layers (see Figs.2(c) and 2(e)) and, to an
even greater extent, within the V2O5 layers (see Figs. 2(b) and 2(d)), in
stark contrast to what one would expect for such huge binding energies. We
have calculated the exciton mean radius over the supercell volume $\Omega$ for
each direction $\alpha=\\{x,y,z\\}$ as
$\langle|r^{\alpha}_{\lambda}|\rangle=\frac{1}{\Omega}\int_{\Omega}d{\bf
r}_{e}|r^{\alpha}_{e}-r^{0\alpha}_{h}||\Psi_{\lambda}(\mathbf{r}^{0}_{h},\mathbf{r}_{e})|^{2}$,
finding an ellipsoid with axes (5.4, 2.3, 3.6)Å and (7.7, 4.2, 5.4)Å, which
span more than a half unit cell and more than an entire unit cell, for the
dark and bright exciton, respectively. This result contradicts an extreme
Frenkel picture of charge transfer excitons, where the exciton wavefunction
would be perfectly localised within the V-O-V unit. A most counter-intuitive
finding is the anisotropy, with especially the dark exciton extending in $x$
direction rather than along the chains, as the band structure would instead
suggest. The reason will become clear in the following analysis, which
highlights additional unusual facts.

###  2.3 Analysis of the spectra

Figure 3: Analysis of the absorption spectra of V2O5 for x, y, and z
polarization. The calculated JDOS and independent-particle (IPA) spectra from
the QSGW band structure are compared with the RPA spectra that include local
field effects (LFE) and the BSE spectra that also include excitonic effects.
The excitonic JDOS from the BSE, which is the density of excitonic states at
zero momentum as a function of energy, is also shown for comparison. Discrete
bound excitonic states inside the gap give rise to a JDOS with small
intensity, not visible at the scale of the plot. The vertical arrows mark the
position of the dark excitons below the absorption onset.

The calculated spectra can be analysed as the combination of the joint density
of states (JDOS), which reflects the density of electron-hole transition
energies, and the dipole matrix elements weighting the transition intensities.
Let us start with the anisotropy of the spectra when comparing $z$
polarization to directions in the $xy$ plane. This anisotropy is extremely
pronounced, although the bands have weak dispersion in all directions. Below 6
eV it is due to single-particle matrix elements, as one can see by comparing
in Fig.3 the independent-particle spectrum to the JDOS stemming from the GW
band structure. The matrix elements in the spectra enhance the first peak in
the two in-plane directions $x$ and $y$ and strongly reduce it perpendicular
to the planes, thus highlighting the layered character of the material. The
spectrum is even more suppressed in $z$ direction, with a dramatic effect at
higher energy, when, beyond the independent-particle approximation (IPA),
crystal Local Field Effects (LFE) are taken into account in the Random Phase
Approximation (RPA). LFE reflect the response on length scales of and below
the crystal unit cell, and make the response function very sensitive to
crystal anisotropies. The huge LFE perpendicular to the planes are reminiscent
of the strong depolarization effects observed in finite systems, which
indicates strong confinement in the planes [59]. Within the BSE, the effective
unscreened electron-hole exchange interaction is responsible for LFE in the
RPA, while the direct screened electron-hole attraction $-W$ describes
excitonic effects beyond the RPA. The LFE within the RPA can modify in
principle the transition energies in the JDOS and the electron-hole
wavefunctions in the matrix elements. Our RPA calculations have shown that,
although transition energies are affected in principle, there is no visible
change in the JDOS in practice. The suppression of the spectral weight is
instead entirely due to changes in the electron-hole wavefunctions, which
changes the matrix elements. Interestingly, we can now also clearly
distinguish the behavior in $x$ and $y$ direction: in $y$ direction, along the
chains, LFE are negligible, whereas perpendicular to the chains there is a
sizable blueshift of oscillator strength and some suppression of intensity.
This additional anisotropy, with electrons delocalized along the chains, gives
the material a character between a 2D and a quasi 1D system.

Changes of the JDOS between the IPA and the RPA are not analyzed usually.
Sometimes comparisons of JDOS are done between IPA and full BSE calculations,
where it is commonly found that, besides the eventual appearance of bound
states, modifications of the JDOS are small, even when spectra change
dramatically[7, 60]. This is true even in monolayers, where, besides the
appearance of bound excitons, only minor (less than 0.1 eV) shifts of the JDOS
are observed [61]. This may be understood by considering the effective
electron-hole interaction as a perturbation (see Methods). In an extended
system, first-order perturbation theory cannot change transition energies,
which explains the stability of the JDOS, both in the RPA and full BSE
calculations [58]. Intensities, instead, may change already at first order,
and the size of the changes is inversely proportional to energy differences
between the unperturbed transitions. The spectral range shown in Fig.3 is
dominated by transitions from the top-valence bands to the split-off
conduction bands, which are dispersing very little. Flat bands yield small
energy differences between the transitions and can therefore explain huge
changes of intensities, as observed here already in the RPA, even in first-
order perturbation theory.

V2O5, however, behaves differently from other materials when the direct
screened electron-hole interaction $-W$ is included in the BSE. Contrary to
common findings, now also the JDOS undergoes significant changes all over the
spectral range, as Fig.3 shows. In particular, there is an overall redshift of
the order of half an eV. Note that the exciton binding energy is defined as
$E_{b}=E_{c}-E_{v}-E_{\lambda}$, where $E_{c}-E_{v}$ is the energy of the
lowest direct transition in QSGW, and $E_{\lambda}$ is the transition energy
in the BSE. The substantial shift of the JDOS may require care for the
definition of a binding energy extracted from experiment. This means that the
excitonic effects are so strong that also a large spectral range above the
continuum onset is dominated by significantly bound excitons. Instead, at
higher energy, transitions that are more dispersing contribute to the spectra,
resulting in a 0.15eV red shift of the excitonic JDOS, which is in fact closer
to other layered materials [61]. Finally, with respect to the BSE-JDOS, the
BSE spectrum is governed by matrix element effects that are dramatic. They
completely suppress the intensity of the two dark excitons at $\sim$2.6 eV,
and create the huge oscillator strength of the bright excitons close to 3.1
eV. Such drastic matrix element effects are typically observed in low-
dimensional systems [61].

Figure 4:  Contributions to the optical absorption intensity. (a) For first
bright excitons contributing to the main peaks in $x$ and $y$ directions (at
3.06 and 3.05 eV respectively). Size of the circles indicates $|\sum_{vc{\bf
k}}\bar{A}^{vc{\bf k}}_{\lambda}\tilde{\rho}_{vc{\bf k}}|$ summed over
equivalent ${\bf k}$ values. (b) For bright and dark exciton in $x$ direction.
Size of the circles indicates $|\bar{A}^{vc{\bf
k}}_{\lambda}\tilde{\rho}_{vc{\bf k}}|$ In both (a) and (b), transitions are
vertical, so each transition links two circles of equal size, one in a valence
and one in a conduction band, at the same ${\bf k}$ point.

###  2.4 Nature of the excitons

To get a substantiated, though simple, explanation for our observations, the
origin of the excitons must be analyzed. Fig.4(a) shows partial contributions
to the intensity $|\sum_{{\bf k}_{equiv}}\bar{A}^{vc{\bf
k}}_{\lambda}\tilde{\rho}_{vc{\bf k}}|$ projected onto the band structure,
where the sum is performed over ${\bf k}$ points that are equivalent by
symmetry, for a bright exciton in $x$ direction and a bright exciton in $y$
direction at the absorption onset. The absolute value is indicated by the size
of the circles. The circles always appear in pairs of equal size, one in a
conduction and one in a valence band, at the same ${\bf k}$ point. The
analysis shows that the character of excitons that are bright in $x$ and $y$
directions, respectively, is different. The excitons at the absorption onset
that are bright in $x$ direction are dark along $y$, and vice versa. The
darkness of the excitons in certain directions is due to the negligibly small
oscillator strengths of the single particle transitions that are mixed to
yield the excitonic transitions. In $x$ direction, all peaks are coming
predominantly from transitions around the S-Y-$\Gamma$ and Z-T regions in the
Brillouin zone. In the Z-T region the transitions are from the bridge Ob
$p_{z}$ to the split-off V $d_{xy}$ states [15]. Excitons that are bright in
$y$ directions are made mainly of transitions around $\Gamma$-X and
$\Gamma$-Z-T between the top valence and the split-off conduction bands. The
higher energy excitons, e.g. in $x$ direction at 4.55 eV, instead also mix
transitions from lower lying valence and higher conduction bands (see
Supplementary Figure 5).

The importance of the split-off conduction bands for the spectrum in the low
energy range explains many of the observations, including the fact that the
calculations are quite delicate. In particular it is interesting to note that
approximating the QSGW corrections to the LDA band structure by a rigid shift,
as it is often done with the “scissor operator approximation”, would not be
adequate here. Using a scissor, the highest valence and lowest conduction
bands are not matching the QSGW band structure sufficiently well, leading to a
splitting of the first two excitonic peaks into three peaks of equal
intensity. This is shown in the Supplementary Figure 6. Indeed, although the
discrepancies between QSGW and the scissor appear to be small in the band
structure, the fact that excitonic effects in this material are so much due to
mixing of transitions close in energy makes the result exceptionally sensitive
to these changes.

The analysis allows us to set up a minimal tight-binding (TB) model (for
details, see Supplementary Note 7) that should describe the exciton that is
bright in $\hat{\bf x}$ direction: with the hole localized mainly on the
bridge oxygen $p_{z}$, and the main contribution to the electron stemming from
neighbouring V $d_{xy}$, neglecting the slight buckling of V-O-V rungs in
$\hat{\bf z}$ direction, the TB wavefunctions of the corresponding valence and
conduction band read

$\displaystyle\varphi_{v\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum_{\bf
R}e^{i\mathbf{k}{\bf R}}\Phi_{v}(\mathbf{r}-{\bf R})$
$\displaystyle\varphi_{c\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum_{\bf
R}\Big{(}e^{i\mathbf{k}({\bf R}-{\bf d})}\Phi_{c}({\bf r}-{\bf R}+{\bf
d})+e^{i\mathbf{k}({\bf R}+{\bf d})}\Phi_{c}(\mathbf{r}-{\bf R}+\sigma_{x}{\bf
d})\Big{)}\,,$ (1)

where $N$ is the number of the unit cells that are identified by lattice
vectors ${\bf R}$, $\Phi$ are atomic orbitals, and ${\bf d}$ is the distance
between the Ob, located at the origin, and the next neighbour V atoms that are
related by a mirror plane $\sigma_{x}$ perpendicular to $\hat{\bf x}$. The odd
character of the matrix elements of $\nabla$,
$\tilde{\rho}_{vck_{x}}=-\tilde{\rho}_{vck_{x}}$, becomes immediately clear.
Note that the symmetry ${\bf k}\to-{\bf k}$ is restored if the second V-O-V
rung in the unit cell is taken into account. The matrix elements of the bare
exchange and direct screened electron-hole interactions are evaluated as
prescribed by Eqs. (3) and (4) in Methods. Taking into account a single pair
of bands and neglecting the overlap between orbitals on different sites, this
yields
$v_{\mathbf{k}\mathbf{k}^{\prime}}\approx\frac{2v}{N}\cos[(k_{x}-k_{x}^{\prime})d_{x}]e^{i(k_{z}-k_{z}^{\prime})d_{z}}$
with $v\equiv 2\sum_{{\bf R}}\int d{\bf r}d{\bf r}^{\prime}\,\Phi_{v}({\bf
r})\Phi^{*}_{v}({\bf r}^{\prime})\bar{v}_{c}({\bf r}-{\bf r}^{\prime}+{\bf
R})\Phi_{c}({\bf r}^{\prime}+{\bf d})\Phi^{*}_{c}({\bf r}+{\bf d})$ for the
electron-hole exchange, and, supposing the screened Coulomb interaction
$W(\mathbf{r},\mathbf{r}^{\prime})$ to be short ranged and only dependent on
the distance $|\mathbf{r}-\mathbf{r}^{\prime}|$, one obtains for the direct
electron-hole interaction
$-W_{\mathbf{k}\mathbf{k}^{\prime}}\approx-\frac{w}{N}\,{\rm
cos}([k_{x}-k_{x}^{\prime}]d_{x})e^{i(k_{z}-k_{z}^{\prime})d_{z}}$ with
$w\equiv 2\int d{\bf r}d{\bf r}^{\prime}\,|\Phi_{c}({\bf r}+{\bf
d})|^{2}W({\bf r}-{\bf r}^{\prime})|\Phi_{v}({\bf r}^{\prime})|^{2}$
determined by the next-neighbour screened interaction. The resulting BSE can
be solved analytically. Besides the continuum solutions, there is a solution
$A^{\mathbf{k}}\propto{\rm cos}(k_{x}d_{x})e^{ik_{z}d_{z}}$ that is even in
$k_{x}$, with a binding energy of the order of $w-2v$, and an odd solution
$A^{\mathbf{k}}\propto{\rm sin}(k_{x}d)e^{ik_{z}d_{z}}$ with reduced binding
energy. Since the matrix elements are odd, the solution with smaller binding
energy must be the bright exciton. The very strongly bound dark exciton,
instead, is predicted to stem from the same single-particle transitions but
with even $A^{\mathbf{k}}$. This is confirmed by the analysis of the ab initio
results in Fig.4(b): as for the bright excitons, the single-particle
transitions $\tilde{\rho}_{vc{\bf k}}$ that dominate the dark exciton are not
dipole forbidden, so each contribution to the oscillator strength separately
is not zero, but of the same order as for the bright excitons, in contrast to
other materials with interesting strongly bound dark excitons, such as, e.g.,
hBN, CrI3, and transition metal dichalcogenides, where dark excitons are made
of dipole forbidden matrix elements [62, 63, 64, 65]. As we have verified in
the ab initio calculations, in V2O5 the oscillator strengths are odd functions
of ${\bf k}$, while the coefficients $\bar{A}^{vc{\bf k}}_{\lambda}$ are even
functions in $\mathbf{k}$ . Therefore, when the contributions from ${\bf k}$
and $-{\bf k}$ points are summed together, they cancel each other and make the
exciton dark, as predicted by the model. Such a robust dark exciton can play a
key role for many disexcitation processes[66, 67, 68, 69], such as phonon-
assisted luminescence and light emission.

The model further allows us to understand the electron-hole correlation, using
the $A^{\mathbf{k}}$ and tight-binding wavefunctions to yield the squared
electron-hole wavefunction

$\displaystyle|\Psi(0,\mathbf{r}_{e})|^{2}$ $\displaystyle=$
$\displaystyle|\sum_{\mathbf{k}}A^{\mathbf{k}}\varphi^{*}_{v\mathbf{k}}(0)\varphi_{c\mathbf{k}}(\mathbf{r}_{e})|^{2}$
$\displaystyle\propto$ $\displaystyle\Big{(}|\Phi_{c}(\mathbf{r}_{e}+{\bf
d})|^{2}+|\Phi_{c}(\mathbf{r}_{e}+\sigma_{x}{\bf
d})|^{2}\Big{)}\Big{(}1\pm\frac{\sin(2\pi d_{x}/a)}{4\pi d_{x}/a}\Big{)}^{2}$
$\displaystyle+\delta_{R_{y},0}\delta_{R_{z},0}\frac{\sin^{2}(2\pi
d_{x}/a)}{4\pi^{2}/a^{2}}\sum_{R_{x}\neq
0}\Big{(}\frac{|\Phi_{c}(\mathbf{r}_{e}-\hat{\bf x}R_{x}+{\bf
d})|^{2}}{(R_{x}-2d_{x})^{2}}+\frac{|\Phi_{c}(\mathbf{r}_{e}-\hat{\bf
x}R_{x}+\sigma_{x}{\bf d})|^{2}}{(R_{x}+2d_{x})^{2}}\Big{)}\Bigg{]},$

where $a$ is the lattice parameter in $\hat{\bf x}$ direction, the upper
(lower) sign in $\pm$ refers to the dark (bright) exciton, respectively, and
the hole has been placed on the bridge oxygen at $\mathbf{r}_{h}=0$. The model
predicts that the excitons are localized in $\hat{\bf y}$ and $\hat{\bf z}$
directions, but decay only quadratically with the distance in $\hat{\bf x}$
direction due to a contribution that depends on the O-V distance $d_{x}$. For
$d_{x}\to 0$ this longer-range contribution would approach zero, and the
picture would reduce to the Frenkel exciton. For $d_{x}\neq 0$, instead, the
first term in (2.4) yields the delocalization over the charge transfer unit
that one would expect from the Frenkel picture, whereas the second term
switches on the much larger $1/R_{x}^{2}$ extension. Indeed, although the
model takes only the next neighbour transition into account, it correctly
predicts that the exciton is significantly more extended than the next
neighbour distance. In other words, the delocalization in $\hat{\bf x}$
direction is due to the charge transfer nature of the main underlying single-
particle transition, and not to the long range of the Coulomb interaction.
Interestingly, it is also independent of the interaction strength , whereas
the binding energy depends linearly on the interaction strength. This limited
role of the interaction strength is a general characteristic of excitons
stemming from isolated groups of weakly dispersing transitions (see
Supplementary Note 8 for a proof), independently of the present tight binding
model. The tight binding analysis, instead, shows that the local environment
of the hole, with the V-O-V bridge, determines the shape of the exciton,
rather than the atomic chains, which explains the counter-intuitive anisotropy
of the wavefunction. Of course, the model is simple, but it is validated by
predicting further interesting details that are indeed observed in the ab
initio results, in particular, the weaker intensity of the density on the
next-neighbour V atom in the bright, as compared to the dark exciton, or the
ratio between the dark and bright electron-hole binding energy (see
Supplementary Note 7). Because of this difference in density, looking at
Figs.2(b) and 2(d) one would expect higher intensity for the lowest bound
state, which instead, as predicted by the model and found in the ab initio
calculations, is dark exclusively because of the symmetry of the mixing
coefficients.

##  3 Discussion

In conclusion, we have shown that excitons in flat band materials may have
intriguing properties which contradict textbook expectations. Flat bands
correspond to localized electronic states, and as one would expect, the
exciton binding energy is in general very large[34]. However, when the single-
particle excitations that are mixed to form an exciton are charge transfer
excitations with a mirror symmetry, the exciton wavefunction delocalizes. This
delocalization is neither isotropic nor does it follow the overall anisotropy
of the crystal structure, but it shows an anisotropy governed by the local
motif of the charge transfer unit. Our work predicts in particular that
materials with such a local geometry, where excitations take place from a
central atom to two equivalent, symmetrically placed, neighbouring atoms, will
show a bound exciton that may be more or less bright, according to the
character and the overlap of the involved orbitals. On top of this, one will
also find an even more strongly bound exciton that will always be dark because
of destructive interference. While it may not be detected in optical
measurements, its presence can be inferred from that of a strongly bound
bright exciton and the appropriate local geometry. With the hole on the
central atom, the electron distribution of both the dark and bright excitons
has a large extension perpendicular to the mirror plane.

A very good material to illustrate these effects is V2O5. As we have shown
using first principles calculations and ellipsometry measurements, its optical
properties are dominated by huge anisotropy and excitonic effects. Strongly
bound dark and bright excitons are found in the gap, consistently with our
prediction, and even the excitation energies in the continuum are shifted
significantly, contrary to usual observations. In the lowest dark and bright
excitons, the electron density for a hole on a bridge oxygen extends
perpendicular to the atomic chains along which electrons disperse. Our
explanation of the absorption spectra, based on a consistent picture emerging
from state-of-the-art ab initio calculations within many-body perturbation
theory, a simple tight-binding model, and our ellipsometry measurements,
solves the long-standing puzzle of the optical properties of V2O5, and may
serve as a guideline for the understanding of electronic excitations in other
charge transfer insulators. Moreover, our work illustrates that one cannot
simply extrapolate knowledge gained from textbook semiconductors and
insulators, such as the link between the dielectric constant, the electron-
hole distance and the binding energy, to the advanced materials of current
interest. This gives new freedom to engineering properties of materials; for
example, we have shown that for materials where excitons are dominated by
isolated groups of weakly dispersing excitations, the screening of the
electron-hole interaction has a direct impact on the exciton binding energy
but not on the electron-hole wavefunction. Therefore, by modifying the
screening[70] one can tune the absorption onset while keeping intact the
spectral shape with its very strong intensity at the onset. Altogether, the
present work may give guidelines to identify or combine materials where
excitons exhibit properties tailored for specific needs.

##  4 Methods

###  4.1 Optical properties from the Bethe-Salpeter equation

We solved the electron-hole Bethe-Salpeter equation (BSE) using the QSGW QP
eigenvalues and wavefunctions, and the screened direct electron-hole
interaction given by the statically screened $W$ evaluated in the Random Phase
Approximation (RPA) using QSGW ingredients. The BSE is an in principle exact
equation for the two-particle correlation function[58]. In the GW
approximation[23], the BSE with a statically screened Coulomb interaction $W$
can be cast as an effective two-particle Schrödinger equation for the two-
particle correlation function of the electron-hole (e-h) pair[71, 72, 7, 73]:
$H_{\rm
exc}\Psi_{\lambda}(\mathbf{r}_{h},\mathbf{r}_{e})=E_{\lambda}\Psi_{\lambda}(\mathbf{r}_{h},\mathbf{r}_{e})$.
This equation is usually expressed in a basis of pairs of orbitals. In gapped
systems at zero temperature, only pairs of an occupied $v{\bf k}$ and an
unoccupied $c{\bf k}$ orbitals contribute to an absorption spectrum, so the
pair corresponds to a direct transition $|vc{\bf k}\rangle$. In this basis the
resonant part of the Hamiltonian reads $\langle vc{\bf k}|H_{\rm
exc}|v^{\prime}c^{\prime}{\bf k}^{\prime}\rangle=E_{vc{\bf k}}+\langle vc{\bf
k}|\bar{v}_{c}-W|v^{\prime}c^{\prime}{\bf k}^{\prime}\rangle$. Here the energy
$E_{vc{\bf k}}$ is the difference between an unoccupied and an occupied
quasiparticle (QP) state, calculated in the QSGW. The effective electron-hole
exchange interaction $\bar{v}_{c}$ is given by the microscopic components of
the bare Coulomb interaction and is responsible for LFE. The screened Coulomb
interaction $-W$ represents the direct electron-hole interaction and accounts
for excitonic effects. If it is neglected, the BSE yields the RPA. The matrix
elements in terms of QP orbitals $\varphi$ read

$\langle t|\bar{v}_{c}|t^{\prime}\rangle\equiv 2\int d{\bf r}d{\bf
r}^{\prime}\,\varphi^{*}_{c{\bf k}}({\bf r})\varphi_{v{\bf k}}({\bf
r})\bar{v}_{c}({\bf r},{\bf r}^{\prime})\varphi_{c^{\prime}{\bf
k}^{\prime}}({\bf r}^{\prime})\varphi_{v^{\prime}{\bf k}^{\prime}}^{*}({\bf
r}^{\prime})$ (3)

and

$\langle t|W|t^{\prime}\rangle\equiv\int d{\bf r}d{\bf
r}^{\prime}\,\varphi_{c{\bf k}}^{*}({\bf r})\varphi_{c^{\prime}{\bf
k}^{\prime}}({\bf r})W({\bf r},{\bf r}^{\prime})\varphi_{v{\bf k}}({\bf
r}^{\prime})\varphi^{*}_{v^{\prime}{\bf k}^{\prime}}({\bf r}^{\prime})\,,$ (4)

where $t\equiv v,c,{\bf k}$. The two-particle correlation function is

$\Psi_{\lambda}(\mathbf{r}_{h},\mathbf{r}_{e})=\sum_{vc{\bf
k}}\bar{A}_{\lambda}^{vc{\bf k}}\varphi^{*}_{v{\bf k}}({\bf
r}_{h})\varphi_{c{\bf k}}({\bf r}_{e}).$ (5)

In the Tamm-Dancoff approximation, which neglects the coupling between
resonant and antiresonant transitions and is usually a good approximation for
absorption spectra of solids (we have verified that this is the case also for
V2O5 ), the macroscopic dielectric function is obtained from the BSE as:

$\epsilon_{M}(\omega)=1-\lim_{\mathbf{q}\to
0}\frac{8\pi}{N_{k}\Omega_{0}q^{2}}\sum_{\lambda}\frac{\left|\sum_{t}\bar{A}_{\lambda}^{t}\tilde{\rho}_{t}(\mathbf{q})\right|^{2}}{\omega-
E_{\lambda}+i\eta},$ (6)

with $\Omega_{0}$ the unit cell volume and $N_{\bf k}$ the number of ${\bf k}$
points, and where the oscillator strengths are
$\tilde{\rho}_{t}(\mathbf{q})=\int\varphi^{*}_{v\mathbf{k}-\mathbf{q}}(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}\varphi_{c\mathbf{k}}(\mathbf{r})d\mathbf{r}$.
Each exciton $\lambda$ contributes with strength
$\left|\sum_{t}\bar{A}_{\lambda}^{t}\tilde{\rho}_{t}\right|^{2}$ to the
absorption spectrum
$\epsilon_{2}(\omega)\equiv\textrm{Im}\epsilon_{M}(\omega)$. If it is
negligibly small, the exciton is said to be dark. Setting all exciton
strengths to 1 gives the excitonic JDOS:
$\frac{2}{N_{k}\Omega_{0}\omega^{2}}\sum_{\lambda}\delta(\omega-E_{\lambda})$.
The RPA or the independent-particle JDOS are obtained analogously by
neglecting either only $-W$ or also $\bar{v}_{c}$.

Analysis in terms of perturbation theory: Perturbation theory may help to
understand the impact of flat bands on excitonic effects. To zero order, a
transition $\lambda$ corresponds to a band-to band transition $v_{\lambda}{\bf
k}_{\lambda}\to c_{\lambda}{\bf k}_{\lambda}$, so
$E^{(0)}_{\lambda}=E_{c_{\lambda}{\bf k}_{\lambda}}-E_{v_{\lambda}{\bf
k}_{\lambda}}$ and $\bar{A}_{\lambda}^{vc{\bf
k},(0)}=\delta_{vv_{\lambda}}\delta_{cc_{\lambda}}\delta_{{\bf k}{\bf
k}_{\lambda}}$. First-order changes of the transition energies are given by
the diagonal elements $\langle t_{\lambda}|\bar{v}_{c}|t_{\lambda}\rangle$ and
$\langle t_{\lambda}|W|t_{\lambda}\rangle$, which are vanishing in an infinite
system because of the normalization of the wavefunctions over the crystal
volume [58], as one can also see directly in the tight binding matrix elements
in the Supplementary Note 7.

To first order, the corrections to intensities
$\sum_{t}\bar{A}_{\lambda}^{t}\tilde{\rho}_{t}$ are

$\sum_{t}\bar{A}_{\lambda}^{t,(1)}\tilde{\rho}_{t}=\sum_{t\neq
t_{\lambda}}\frac{\langle t|\bar{v}_{c}|t_{\lambda}\rangle-\langle
t|W|t_{\lambda}\rangle}{E^{(0)}_{t_{\lambda}}-E^{(0)}_{t}}\tilde{\rho}_{t}\,.$
(7)

This expression shows that the sum over $t$ makes the result converge to a
non-vanishing correction even for an infinite system with dense Brillouin zone
sampling. It also shows that changes in intensities are large when band-to-
band transitions lie close in energy, because the differences appear in the
denominator. Finally, for flat bands, where the differences can be very small
even at distant ${\bf k}$-points, small changes in band energies can
significantly alter the results.

###  4.2 Computational details

We adopted the $Pmmn$ orthorhombic crystal structure of V2O5 with 14 atoms per
unit cell and the experimental lattice parameters[74] $a=11.512$ Å , $b=3.564$
Å and $c=4.368$ Å. We used norm-conserving Troullier-Martins[75]
pseudopotentials, including $3s$ and $3p$ semicore states for vanadium (total
of 112 electrons), which have been already validated in previous studies on
vanadates[76, 77]. The LDA ground-state calculation converged with an energy
cutoff of 100 Hartree and $4\times 4\times 4$ $\mathbf{k}$-point grid. In the
GW calculations, done using the Godby-Needs plasmon pole model[78]
($\omega_{p}$ = 26 eV) and validated with the contour deformation integration
technique[79], the dielectric function was computed using $6\times 6\times 6$
$k$-point grid and 350 bands and had a size of 4.9 Hartree, while the self-
energy required 700 bands and 52 Hartree cutoff energy (see Supplementary
Figure 2). Within the QSGW scheme[27] all O $2p$ bands (i.e., 30 topmost
occupied bands) and 22 empty bands were calculated self-consistently. The BSE
Hamiltonian was built using QSGW QP energies, wavefunctions and statically
screened $W$ with a $6\times 6\times 6$ $k$-point grid, and 15 valence and 16
conduction bands (see Supplementary Note 6 for more details). A 0.1eV Gaussian
broadening was applied to the resulting spectra. LDA and GW calculations were
carried out with ABINIT [80], while the EXC code[81] was used for BSE
calculations.

###  4.3 Experimental details

Two V2O5 samples grown by floating zone melting[82] with the surface normal
parallel to $z$ ($\parallel c$) were investigated by spectroscopic
ellipsometry in two different geometries, once with the $x$ direction
($\parallel a$) in the plane of incidence and once perpendicular to the plane
of incidence. Both samples were measured under several angles of incidence
($\Phi=50^{\circ}$, $60^{\circ}$, and $70^{\circ}$). The ellipsometric
parameters $\Psi$ and $\Delta$ were determined for photon energies between 0.5
and 6 eV using generalized ellipsometry. Spectral resolution was set to 1.3nm
corresponding to about 10meV at 3eV. More details about the experimental
procedure can be found e.g. in Ref. 83

Experimental data of both samples were modeled in an anisotropic two layer
model consisting of the bulk V2O5 and an anisotropic Bruggeman[84] effective
medium approximated surface roughness consisting of 50$\%$ oxide and 50$\%$
void. A model independent point-by-point fit to the experimental data yielded
the complex elements of the dielectric tensor
$\varepsilon=\varepsilon_{1}+i\varepsilon_{2}$. It is important to take the
uniaxial structure of the material correctly into account to describe the peak
amplitudes correctly. Due to the surface orientation of the samples,
components $\varepsilon_{x}$ and $\varepsilon_{y}$ are determined with high
accuracy while $\varepsilon_{z}$ has higher error bars. Nevertheless, the
amplitude of $\varepsilon_{2z}$ does not exceed 3 in the investigated spectral
range.

## Data availability

The data that support this work are available in the article and Supplementary
information file.

## Acknowledgements

This work benefited from the support of EDF in the framework of the research
and teaching Chair “Sustainable energies” at Ecole Polytechnique.
Computational time was granted by GENCI (Project No. 544). WRLL was supported
by the U.S. Department of Energy - Basic Energy Sciences (DOE-BES) grant no.
DE-SC0008933. This material is in part based upon work supported by the
National Science Foundation under grant no. DMR-1555153. This research is part
of the Blue Waters sustained-petascale computing project, which is supported
by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and
the state of Illinois. Blue Waters is a joint effort of the University of
Illinois at Urbana-Champaign and its National Center for Supercomputing
Applications. The experimental part of this work was performed in the
framework of GraFOx, a Leibniz-Science Campus partially funded by the Leibniz
association. We thank Carsten Hucho and Arno Wirsig (Paul-Drude-Institut für
Festkörperelektronik, Leibniz-Institut im Forschungsverbund Berlin e.V.,
Germany) for the generous access to their samples.

## Author contributions

V.G., L.R., W.R.L.L. and M.G. designed research; A.S. did the first
preliminary BSE calculations ; V.G. performed all the fully first principles
QSGW and BSE calculations; M.F. conducted the experiments; V.G., L.R., and M.G
analysed the excitons and proposed the model; V.G., L.R., W.R.L.L. and M.G.
wrote the manuscript with contributions from all authors. All authors reviewed
the manuscript.

## Competing interests

The authors declare no competing interests.

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## Figure legends

Figure 1. Absorption spectra of V2O5 for $x$, $y$ and $z$ polarization (the
V2O5 layers are in the $xy$ plane). The experimental spectra obtained from
ellipsometry measurements are compared with calculated BSE spectra in the
three directions (note the change of intensity scale for the $z$ direction in
the right panel). The vertical arrows mark the position of the dark excitons
below the absorption onset.

Figure 2. Electron density distribution for dark and bright excitons. (a)
Crystal structure of V2O5 . V atoms are grey, while O atoms are red. Selected
atoms indicate the V-O-V rungs. (b)-(c) Electron density distribution
$|\Psi_{\lambda}(\mathbf{r}_{h}^{0},\mathbf{r}_{e})|^{2}$ for the first
exciton $\lambda$ that is bright in $x$ direction and (d)-(e) for the lowest-
energy dark exciton. In both cases, the position of the hole
$\mathbf{r}_{h}^{0}$ is fixed close to a bridge oxygen Ob atom (light blue
dot). Since it is displaced along a non-symmetric direction from the Ob atom
the resulting electronic distribution is also slightly asymmetric. (b) and (d)
views in the plane of V2O5 layers (i.e., $xy$ planes) ; (c) and (e) views
normal to V2O5 layers (i.e., $zy$ planes). Ellipses centered at the hole
position represent the calculated exciton mean radius
$\langle|r^{\alpha}_{\lambda}|\rangle$. The isosurface value for bright and
dark excitons is chosen to be 1% of the maximum electron density.

Figure 3. Analysis of the absorption spectra of V2O5 for x, y, and z
polarization. The calculated JDOS and independent-particle (IPA) spectra from
the QSGW band structure are compared with the RPA spectra that include local
field effects (LFE) and the BSE spectra that also include excitonic effects.
The excitonic JDOS from the BSE, which is the density of excitonic states at
zero momentum as a function of energy, is also shown for comparison. Discrete
bound excitonic states inside the gap give rise to a JDOS with small
intensity, not visible at the scale of the plot. The vertical arrows mark the
position of the dark excitons below the absorption onset.

Figure 4. Contributions to the optical absorption intensity. (a) For first
bright excitons contributing to the main peaks in $x$ and $y$ directions (at
3.06 and 3.05 eV respectively). Size of the circles indicates $|\sum_{vc{\bf
k}}\bar{A}^{vc{\bf k}}_{\lambda}\tilde{\rho}_{vc{\bf k}}|$ summed over
equivalent ${\bf k}$ values. (b) For bright and dark exciton in $x$ direction.
Size of the circles indicates $|\bar{A}^{vc{\bf
k}}_{\lambda}\tilde{\rho}_{vc{\bf k}}|$ In both (a) and (b), transitions are
vertical, so each transition links two circles of equal size, one in a valence
and one in a conduction band, at the same ${\bf k}$ point.