# Charge ordering in Ir dimers in the ground state of Ba5AlIr2O11

Vamshi M. Katukuri  Current addres: Max Planck Institute for Solid State
Physics, Heisneberstrasse 1, 70569, Stuttgart Germany<EMAIL_ADDRESS>Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015
Lausanne, Switzerland  Xingye Lu<EMAIL_ADDRESS>Swiss Light Source, Photon
Science Division, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
Center for Advanced Quantum Studies and Department of Physics, Beijing Normal
University, Beijing 100875, China  D. E. McNally  Swiss Light Source, Photon
Science Division, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
Marcus Dantz  Swiss Light Source, Photon Science Division, Paul Scherrer
Institut, CH-5232 Villigen PSI, Switzerland  Vladimir N. Strocov  Swiss Light
Source, Photon Science Division, Paul Scherrer Institut, CH-5232 Villigen PSI,
Switzerland  M. Moretti Sala  European Synchrotron Radiation Facility, BP 220,
F-38043 Grenoble Cedex, France  M. H. Upton  Advanced Photon Source, Argonne
National Laboratory, Argonne, Illinois 60439, USA  J. Terzic  G. Cao
Department of Physics and Astronomy, University of Kentucky, Lexington,
Kentucky 40506, USA  Department of Physics, University of Colorado at Boulder,
Boulder, CO 80309  Oleg V. Yazyev  Institute of Physics, École Polytechnique
Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland  Thorsten Schmitt
<EMAIL_ADDRESS>Swiss Light Source, Photon Science Division, Paul
Scherrer Institut, CH-5232 Villigen PSI, Switzerland

###### Abstract

It has been well established experimentally that the interplay of electronic
correlations and spin-orbit interactions in Ir4+ and Ir5+ oxides results in
insulating $J_{\rm eff}$=1/2 and $J_{\rm eff}$=0 ground states, respectively.
However, in compounds where the structural dimerization of iridum ions is
favourable, the direct Ir $d$–$d$ hybridisation can be significant and takes a
key role. Here, we investigate the effects of direct Ir $d$–$d$ hybridisation
in comparison with electronic correlations and spin-orbit coupling in
Ba5AlIr2O11, a compound with Ir dimers. Using a combination of ab initio many-
body wave function quantum chemistry calculations and resonant inelastic X-ray
scattering (RIXS) experiments, we elucidate the electronic structure of
Ba5AlIr2O11. We find excellent agreement between the calculated and the
measured spin-orbit excitations. Contrary to the expectations, the analysis of
the many-body wave function shows that the two Ir (Ir4+ and Ir5+) ions in the
Ir2O9 dimer unit in this compound preserve their local $J_{\rm eff}$ character
close to 1/2 and 0, respectively. The local point group symmetry at each of
the Ir sites assumes an important role, significantly limiting the direct
$d$–$d$ hybridisation. Our results emphasize that minute details in the local
crystal field (CF) environment can lead to dramatic differences in electronic
states in iridates and 5$d$ oxides in general.

Dimerization or clustering of TM atoms is observed in many TM compounds, e.g.
in vanadium oxides Imada  _et al._ (1998); Pen  _et al._ (1997) and titanates
Khomskii and Mizokawa (2005) where dimers of spin singlets akin to the Peierls
state in one dimension Peierls (1991) are stabilized when the $t_{2g}$
orbitals of the TM $d$-manifold are partially filled. In these systems, the TM
ions tend to have a strong direct (intra-dimer) $d$–$d$ overlap that result in
molecular-like orbitals with appreciable bonding-antibonding splitting.
Consequently, the local electronic structure depends on the intra-dimer
hopping integral ($t_{d}$), intra-atomic Hund’s coupling ($J_{H}$) and inter-
atomic ($U$) Coulomb interactions and electron filling of the orbitals
localized at TM clusters. Alternatively, dimerization of TM ions can also be
favourable from crystallographic considerations, particularly in compounds
with heavy TM ions, e.g. 5$d$ ions, where the $d$ orbitals are more spread
out. A number of dimerized or cluster 4$d$ and 5$d$ compounds Terasaki  _et
al._ (2016); Nag  _et al._ (2016); Dey  _et al._ (2017); Nag  _et al._ (2018)
with intriguing properties have been synthesized recently. Novel physical
phenomena have been observed in these compounds, e.g. the inelastic X-ray
scattering analogue of Young’s double slit experiment has been realized in
Ba3CeIr2O9 Revelli  _et al._ (2019), where the molecular orbital formation
within the Ir dimers is crucial. In lacunar spinels Ga$M_{4}$$X_{8}$ ($M$=Nb,
Mo, Ta and W and $X$=S, Se and Te), spin-orbit coupled molecular $J_{\rm eff}$
states Kim  _et al._ (2014a); Jeong  _et al._ (2017) and topological
superconductivity Park  _et al._ (2020) have been proposed where molecular
orbital formation within the tetrahedral cluster of $M$ ions is the key.

The interplay of inter-site electron hopping ($t$), $J_{H}$, $U$ and the
strong atomic spin-orbit coupling (SOC) in $5d$ and in some $4d$ compounds
result in the $J_{\rm eff}$ physics Kim  _et al._ (2008, 2009); Katukuri  _et
al._ (2012); Moretti Sala  _et al._ (2014); Witczak-Krempa  _et al._ (2014).
For instance, in compounds with Ir4+ ($d^{5}$) configuration in an octahedral
environment, e.g. in Sr2IrO4 Kim  _et al._ (2008, 2009), the strong SOC leads
to completely filled $J_{\rm eff}$ = 3/2 and half-filled $J_{\rm eff}$ = 1/2
levels. Similarly, in Ba2YIrO6 and NaIrO3, the Ir5+ ions realise a completely
filled $J_{\rm eff}$= 3/2 and empty $J_{\rm eff}$= 1/2 sub-manifolds Abragam
and Bleaney (1970); Khaliullin (2013), resulting in a non-magnetic $J_{\rm
eff}=0$ ground state Kusch  _et al._ (2018). In dimerized systems, $t_{d}$ can
be much larger and successively may play a dominant role compared to other
local interactions, which could result in the breakdown or a significant
modification of the $J_{\rm eff}$ physics. Thus, it is crucial to identify the
role of these multiple physical interactions in Ir dimer systems to gain a
better understanding of the electronic and magnetic properties of these
materials.

In this letter, we illustrate how subtle crystal structure details are
extremely important to precisely understand the electronic structure of 5$d$
compounds where structural dimerization or clustering is prevalent. Using
state-of-the-art ab initio many-body electronic structure methods in
combination with high-resolution resonant inelastic X-ray scattering (RIXS)
experiments, we present a detailed analysis of the electronic structure of
Ir2O9 dimers in Ba5AlIr2O11and unravel the nature of electronic ground and
excited states of Ba5AlIr2O11. While we find an excellent agreement between
the RIXS spectra and the calculated excitations, analysis of the many-body
wave functions reveal a nearly complete charge separation – Ir4+ and Ir5+–
within the dimers in the ground state, in contrary to an earlier report of
formation of molecular orbitals in Ba5AlIr2O11 Wang  _et al._ (2019). The
strong SOC of the Ir4+ and Ir5+ ions results in $J_{\rm eff}$= 1/2 and $J_{\rm
eff}$= 0 local configurations, respectively, and thus we conclude that a
localized $J_{\rm eff}$ picture is more appropriate in Ba5AlIr2O11.

Ba5AlIr2O11 contains dimers composed of crystallographically inequivalent Ir
cations encaged in face sharing O6 octahedra Müller-Buschbaum and Lang (1989);
Terzic  _et al._ (2015), see Fig. 3a and 3b, and Supplementary material (SM)
Fig. S1 sup . At 210 K, a lattice distortion is believed to lower the symmetry
of the crystal and enhance the charge disproportionation leading to charge
ordering that correspond to Ir4+ and Ir5+ valence configurations Terzic  _et
al._ (2015). However, analysis of RIXS spectrum of Ba5AlIr2O11 using density
functional theory and model Hamiltonian calculations Wang  _et al._ (2019) has
proposed the formation of hybridized dimer orbitals, debunking the charge
disproportionation phenomenon. Nevertheless, given the complex low-symmetry
crystal environment and the interplay of spin and orbital degrees of freedom
in Ba5AlIr2O11, it is unclear if the dimer orbitals are actually realised in
the ground state.

Figure 1: (a) Incident-energy dependence of elementary excitations below 1.2
eV for Ba5AlIr2O11 measured around Ir $L_{3}$ edge with $Q=(23.5,0,2.5)$ at 20
K. (b) RIXS spectra (below 1.2 eV) measured at $E_{i}=11.215$ keV (marked as
white dashed line in (a)). (c) High energy excitations (1-8 eV) measured with
the same setup as that for (b). Green line is a Multi-Gaussian fitting of the
raw data in red open circles (with error bars).

Results: The RIXS spectra shown in Fig. 1 was measured on single crystals
grown by flux method Terzic  _et al._ (2015) at the ID20 beam line of the
European Synchrotron Radiation Facility (ESRF) with $\sim$25 meV resolution
Moretti Sala  _et al._ (2018) and the 27-ID-B beamline with $\sim$30 meV
resolution at the Advanced Photon Source (APS), with $\pi$ polarization at a
scattering angle close to $2\theta=90^{\circ}$. The incident-energy dependence
of RIXS spectra across the Ir-$L_{3}$ edge ($E_{i}=11.215$keV) at the zone
center Q=(23.5, 0, 2.5) is shown in Fig. 1(a). While the same $E_{i}$ as
determined from previous measurements on iridates such as Sr2IrO4 and Ba2YIrO6
Kim  _et al._ (2012); Kusch  _et al._ (2018) was chosen, we find that the
maximum of the resonance is not at $E_{i}$ in Ba5AlIr2O11 as the precise CF
environment around Ir ions and the mixing of the valence states influences the
resonance energy. However, we see that the energies of the modes remain
unchanged in a broad range around $E_{i}$.

The features marked by A-K in Fig. 1(b) are incident-energy independent Raman
modes as shown in Fig. 1(a). These modes correspond to intrinsic electronic
transitions between various occupied and unoccupied states, and therefore
provide direct information about low energy electronic structure. To resolve
all the Raman modes and determine the low energy electronic structure, we show
in Figs. 1(b) and 1(c) high statistic energy spectra collected at $E_{i}$
(white dashed line in Fig. 1(a)). In Fig. 1(b), several sharp Raman modes
below 1 eV and a broad peak at 1.2 eV, named $A$ to $K$ are determined by
fitting the spectra using multiple gaussians. The sum of the fitting curves is
shown as a green solid curve. In Fig. 1(c), higher energy excitations up to 8
eV are shown. This spectrum is decomposed into several peaks and
interestingly, these modes show very little momentum dependence (see SM Fig.
S6 sup ), indicating that all of them correspond to local spin-orbital ($d-d$)
excitations and reflecting the low energy electronic structure.

Table 1: Relative energies (eV) of the excitation levels calculated at CASSCF+NEVPT2 level of theory. The first column contains non-relativistic multiplet structure, the multiplet symbols on the left correspond to the octahedral ($O_{h}$) symmetry. The degeneracy of the states is split in Ba5AlIr2O11 due to the lowered symmetry in the two octahedra due to the anisotropic crystal fields, see text. Spin-orbit coupled multiplet structure is shown in the second column. Note that each state is doubly degenerate (Kramers doublet). The corresponding peaks in the RIXS data in Fig. 1a are shown in column 3.  CASSCF+NEVPT2 | \+ SOC (x 2) | Ir L-edge RIXS
---|---|---
${}^{4}A_{1}$ – | 0.00 | 0.00 | 0.00
${}^{2}T_{1}$ – | 0.03, 0.08, 0.10 | 0.18 | 0.18 (A)
${}^{2}A_{1}$ – | 0.14 | 0.24 , 0.27 | 0.28 (B) 0.33 (C)
${}^{4}T_{1}$ – | 0.16, 0.17, 0.17 |  | 0.44 (D)
${}^{2}E_{1}$ – | 0.18, 0.23 | 0.48, 0.62 | 0.50 (E)
${}^{4}E_{1}$ – | 0.25, 0.28 | 0.55, 0.56, 0.58 | 0.56 (F)
${}^{2}T_{2}$ – | 0.77, 0.80, 0.94 | 0.60, 0.78 | 0.75 (G)
${}^{4}T_{2}$ – | 0.84, 0.86, 0.95 | 0.84, 0.90 | 0.82 (H)
${}^{2}T_{3}$ – | 0.86, 0.86,0.90 | 1.14 – 1.20 (4) | 0.98 (I), 1.00 (J)
${}^{2}A_{2}$ – | 0.91 | 1.21, 1.24 | 1.20 (K)
${}^{2}E_{2}$ – | 1.00, 1.01 | 1.37 – 1.43 (4) | 1.4 (L)
${}^{2}A_{3}$ – | 1.03 | 1.47 |
${}^{2}T_{4}$ – | 1.10,1.10,1.12 | 1.48, 1.53, 1.56 |
${}^{2}E_{3}$ – | 1.60, 1.63 | 1.73 – 1.80 (5) | 1.77 (M)
${}^{2}T_{5}$ – | 1.68, 1.78, 1.81 | 2.02, 2.04 |
|  | 2.11, 2.17, 2.20 | 2.17 (N)
|  | 2.59 |
|  | 2.63 – 2.77 (4) | 2.70 (O)

We now turn to the RIXS results measured using O-$K$ edge (Fig. 2 and Fig. S2
in SM sup ) carried out at the ADRESS beamline of the Swiss Light Source at
the Paul Scherrer Institut, with $\sim$70 meV energy resolution for both
$\sigma$ and $\pi$ polarizations at a scattering angle of 2$\theta$=130 deg.
Strocov  _et al._ (2010); Ghiringhelli  _et al._ (2006), see SM, Fig. S1. With
the presence of strong hybridization between O $2p$ orbitals and Ir $5d$
orbitals, O-$K$ RIXS is sensitive to various elementary excitations of
iridates Lu  _et al._ (2018). Figs. 2(a) and 2(b) are RIXS maps collected at
O-$K$ edge with $\pi$ and $\sigma$ polarizations at $25^{\circ}$ grazing
incidence. Besides the sharp spin-orbital excitations ($E\approx 0.26,0.57$
eV) below $1$ eV consistent with those measured with Ir-$L_{3}$ edge, two high
energy excitations at $E\approx 2.26$ and $2.71$ eV have also been observed.
Note the RIXS maps in Fig. 2 contains substantial fluorescence which is absent
in the results collected at Ir $L_{3}$ edge (Fig. 1), indicating complex
energy levels/bands of oxygen ions. In addition, significant polarization
dependence of the excitations have also been observed, which we attribute to
the overlap between light polarization (electric field ${\bf E}$) and the
different O $2p$ orbitals hybridized with different Ir $5d$ orbitals (for
details, see sup ).

Figure 2: RIXS results of Ba5AlIr2O11 as measured at O-$K$ edge. (a, b) Energy
dependence of RIXS spectra for Ba5AlIr2O11 taken around O-$K$ edge with
$\delta=-40^{\circ}$ ($Q=(0.93,0,0.65)$).

To decipher the nature of the rich excitation spectrum observed in RIXS
spectra and to examine the formation of dimer orbitals in Ba5AlIr2O11, we
performed many-body ab initio cluster-in-embedding quantum chemistry (QC)
calculations, starting from the crystal structure reported in Ref. Müller-
Buschbaum and Lang (1989); Terzic  _et al._ (2015). These are based on the
construction of the exact wave function for the atoms in the cluster using
configuration interaction wave function theory – complete active space self-
consistent field (CASSCF) and multireference perturbation methods Helgaker
_et al._ (2000). The calculations were performed on a cluster containing one
Ir2O9 dimer unit, two neighboring AlO4 tetrahedra and the surrounding 15 Ba2+
ions. All the calculations were performed using ORCA quantum chemistry program
Neese (2012), see SM sup  for all the computational details.

Relative energies of the multiplet structure of the Ir2O9 dimer unit obtained
from CASSCF + NEVPT2 (N-electron valence perturbation theory) Angeli  _et al._
(2001) calculations are shown in Table 1. An active space of nine electrons in
six orbitals (three $t_{2g}$ orbitals on each iridium) was considered in the
CASSCF calculation which sufficiently captures the important static
correlations (i.e. near degeneracies) in Ba5AlIr2O11. In the NEVPT2
calculation, the correlations involving all the neighboring occupied oxygen
$2p$ and iridium $5s$, $5p$ orbitals as well as all the unoccupied orbitals
are accounted for, accurately describing the O-2p to Ir-d charge transfer
effects and other dynamic correlation effects. It is important to note that
the intra- (Hund’s coupling $J_{H}$) and inter-site ($U$) Coulomb interactions
and the hybridization between different orbitals are included in the
calculation, accurate within the basis-set limit.

The lowest nine quartet ($s=\frac{3}{2}$) and 24 doublet ($s=\frac{1}{2}$)
scalar relativistic states (first column in Table 1) are first computed and
then are allowed to admix via the SOC, resulting in 84 states, see the second
column of Table 1. It can be seen that the excitation energies obtained from
CASSCF+NEVPT2+SOC calculations are in excellent agreement with the peaks
observed in RIXS experiments, except for peak $D$. This peak is related to the
electron-hole exciton which is also observed in other iridate materials such
as Sr2IrO4 Kim  _et al._ (2014b); Lu  _et al._ (2018) and Na2IrO3 Gretarsson
_et al._ (2013). Such excitations are not considered in the current QC
calculations111To simulate electron-hole exciton peaks one would need to
calculate N-1 (removal of an electron) state calculations.. Further, our
calculations reveal excitations from the $t_{2g}$ to $e_{g}$ manifold starting
at 3.4 eV which correspond to RIXS peaks P and Q.

To elucidate the origin of these excitations, we first analyze the scalar-
relativistic multiplet structure. When the two iridium ions in the dimer unit
are in cubic environment ($O_{h}$ symmetry), the low energy multiplet
structure is a result of the interaction of the ground state ${}^{2}T_{1g}$
multiplet of the Ir4+ ion Katukuri  _et al._ (2014) and the ${}^{3}T_{1g}$
ground state term of the Ir5+ ion. In addition, the lowest ${}^{1}T_{1g}$ and
${}^{1}E_{1g}$ Kusch  _et al._ (2018) singlet states contribute significantly
to the low energy spin-orbit excitations Kusch  _et al._ (2018). The resulting
spectrum contains ${}^{4}T_{1g},^{4}T_{2g},^{4}A_{1g},^{4}E_{1g},$ quartets
and 11 doublet terms – ${}^{2}T_{1g,2g,3g,4g,5g}$, ${}^{2}A_{1g,2g,3g}$,
${}^{2}E_{1g,2g,3g}$ 222See direct product tables for $O_{h}$ point group
representations, e.g. in Ref. Atkins  _et al._ (1993). However, in Ba5AlIr2O11
the Ir ions are enclosed in distorted octahedra resulting in low symmetry CFs
and splitting of the $t_{2g}$ levels at each Ir ion 333For example, for pure
trigonal distortions, the $t_{2g}$ orbitals are split into $a_{1g}$ and
$e_{g}$.. Further, the small Ir-Ir intra-dimer distance of 2.73 Å in
Ba5AlIr2O11 (2.698 Å in elemental iridium) may result in direct overlap of the
Ir $d$ orbitals and the formation of bonding and antibonding states Streltsov
and Khomskii (2016). Consequentially, the multiplet degeneracies in the
spectrum are split.

To understand the formation of bonding and antibonding dimer orbitals in
Ba5AlIr2O11, we plot the evolution of orbital energies as a function of
Ir1-Ir2 intra-dimer distance ($d$) in Fig. 3(c). The six levels for each $d$
correspond to the CASSCF canonical orbital 444The canonical orbitals are
eigenfunctions of the so-called (effective single particle) Fock operator.
energies of the six $t_{2g}$-like orbitals in the Ir2O9 dimer unit. The colour
variations of the energy levels represent the orbital compositions 555The
orbital compositions are obtained from the Löwdin natural orbital
decomposition. from Ir1, Ir2 and O ions. Interestingly, for $d\geq 2.73$ Å, we
find 20% and 13% hybridization for Ir1 $5d$ – O $2p$ and Ir2 $5d$ – O $2p$,
respectively, while there is negligible direct Ir1-Ir2 $d$-orbital
hybridization. The $a_{1g}$ orbital of Ir1 contains 4.5% contribution from
$a_{1g}$ orbital of Ir2 and vice versa. For $d=2.65$ Å, a significant direct
Ir1-Ir2 $d$-orbital hybridization is observed. We find this hybridization
increasing up to 25% for $d$ =$2.45$ Å, resulting in large bonding –
antibonding energy separation, as seen in the corresponding orbital plots in
Fig. 3(d) . Note that for $d=2.73$ Å orbitals with predominantly Ir1 character
are at higher energies than those of Ir2 character, reflecting different on-
site orbital energies. This is a direct consequence of the difference in the
valence configurations of Ir1 and Ir2 ions and the Ir1,2 $5d$ – O $2p$
hybridization.

Figure 3: Crystal structure of Ba5AlIr2O11 : (a) unit cell, (b) Ir2O9 dimer
units connected along $b$. $d$ is the intra-dimer Ir1-Ir2 distance. (c)
Orbital (relative) energy-level diagram of the six $t_{2g}$ orbitals, shown in
(d), for different $d$, the lowest energy orbital is set to zero. The length
of red, black and blue colours of each level are proportional to the
percentage contributions from Ir1, Ir2 and O ions, respectively.

The effect of the low crystal filed symmetry at two different Ir ions can be
estimated by computing the $t_{2g}$ splittings, $\delta$, at each of the Ir
ions from restricted active space (RAS) Malmqvist  _et al._ (1990)
calculations where the $d$ orbital occupation at the other Ir ion is
constrained. We find considerably large $t_{2g}$ splittings of
$\delta_{1}=0.58$ and $\delta_{2}=0.60$ eV for Ir1 and Ir2, respectively. Such
large values compete with the SOC strength of $\sim 0.5$ eV of the Ir ions to
considerably reduce the effect of SOC, thus resulting in the modification of
the local spin-orbit multiplet structure.

The scalar-relativistic ground state realized in Ba5AlIr2O11 is the double
exchange ${}^{4}A_{1g}$ multiplet Streltsov  _et al._ (2017), an orbitally
non-degenerate high-spin quartet, with wave function
${}^{4}\psi_{0}=\alpha|d_{1}^{4},d_{2}^{5}\rangle+\beta|d_{1}^{3},d_{2}^{6}\rangle+\gamma|d_{1}^{5},d_{2}^{4}\rangle$
with $\alpha^{2}=0.89,\beta^{2}=0.09$ and $\gamma^{2}=0.02$, where $d_{i}^{n}$
corresponds to $n$ electrons in Iri $d$ orbitals. The lowest ${}^{2}T_{1}$
doublet state is 40 meV higher with wave function weights
$\alpha^{2}=0.85,\beta^{2}=0.11$ and $\gamma^{2}=0.02$. We further find that
the weight of $|d_{1}^{4},d_{2}^{5}\rangle$ configuration in all the excited
multiplet wave functions is greater than 95%. It is interesting to note that
excluding all the configurations involving hopping of electrons from Ir1 to
Ir2 and vice versa in the wave function preserve the spin-orbit spectrum
except for an overall shift $\leq$ 50 meV. The double exchange ground state as
well as the dominant contribution of $|d_{1}^{4},d_{2}^{5}\rangle$
configuration imply charge separation within the dimer units. Further, the
natural orbital occupations obtained from the CASSCF calculations are close to
4 and 5 for Ir1 and Ir2 ions, respectively. Thus, we conclude that the two Ir
ions in Ba5AlIr2O11 host different ionic states – Ir${}_{1}^{5+}$ and
Ir${}_{2}^{4+}$ – which results in charge ordering within the dimers and the
low energy excitations are strictly local to individual Ir ions and not among
dimer orbitals.

The SOC results in the admixture of all the 15 non-relativistic multiplet
terms shown in Table 1. Addition of angular momenta of two $l_{\rm eff}$=1
($l_{\rm eff,1}$ = $l_{\rm eff,2}$ = 1) sites with spins $s_{1}=1/2$ and
$s_{2}=1$ gives rise to 84 effective total angular momentum ($J_{\rm eff}$)
states. In Ba5AlIr2O11, due to the non-cubic CFs, all degeneracies are removed
except for the Kramers doublet degeneracy. From the analysis of the wave
functions, we assign the peaks A-C to excitations from the Ir1 $J_{\rm eff}=0$
to $J_{\rm eff}=1$ states. The peak F and satellite feature H consists of
excitations from Ir1 $J_{\rm eff}=0$ to $J_{\rm eff}=2$ and Ir2 $J_{\rm
eff}=1/2$ to $J_{\rm eff}=3/2$ states that are split due to non-cubic CFs. The
peak G originates from excitations involving Ir1 $J_{\rm eff}=0$ and $J_{\rm
eff}=2$ states as well. The peaks I-K are the result of simultaneous on-site
excitations at Ir1 and Ir2 ions, see Fig. S5 in SM sup .

At the first sight, an insignificant Ir-Ir intra-dimer $d$-orbital
hybridization in Ba5AlIr2O11 might be surprising, even though the distance
between Ir sites is close to that of Ir metal. However, due to the
crystallographic in-equivalence of the two Ir ions in the dimer unit and the
different O6 arrangement, the symmetry of split $t_{2g}$ orbitals at each of
the Ir ions is very different and subsequently a little direct overlap is
realized. In fact, for dimer systems with structurally equivalent ions such as
Ba3InIr2O9 Dey  _et al._ (2017), we find a considerable hybridization
resulting in delocalized dimer orbitals Katukuri and Yazyev . It would be
interesting to characterize the local electronic structure in other face-
sharing Ir-dimer compounds such as Ba3ZnIr2O9 and Ba3ZrIr2O9 Sakamoto  _et
al._ (2006) where the Ir dimer unit occupancy is eight and ten respectively.

In conclusion, we have measured both Ir $L_{3}$ and O $K$-edge RIXS spectra
and observed multiple spin-orbital excitations. Our ab initio quantum
chemistry calculations reproduce very well the excitation spectrum up to 3.5
eV observed in the RIXS measurements. We find charge ordering within the Ir-
dimers with Ir${}_{1}^{5+}{\rm(}d^{4}$) and Ir${}_{2}^{4+}{\rm(}d^{5}$)
configurations. We have established a direct connection between the
excitations in Ir2O9 dimer unit and those at individual Ir ions. The
appearance of multiple peaks is a direct consequence of strong non-cubic CFs
originating from the distorted octahedral environment around the Ir ions. In
spite of small intra-dimer Ir-Ir distance, the direct $d$–$d$ hybridization is
relatively weak and the bonding–antibonding splitting is negligible compared
to the non-cubic CF splittings. Alternatively, we find increased intra-dimer
configuration mixing due to strong electron-electron interactions,
particularly the $|d_{1}^{3},d_{2}^{6}\rangle$ configuration stabilizing the
ground state. This strongly supports nearly complete charge ordering within
the Ir dimers in Ba5AlIr2O11 and refutes the suggested formation of dimer
orbitals Wang  _et al._ (2019). Our results highlight the importance of minute
details of the crystal structure to understand the electronic and magnetic
properties of clustered Iridates and TM magnets in general and calls for re-
investigating several already studied materials with accurate ab initio many-
body calculations. Finally, we emphasize that the combination of RIXS and
quantum chemistry calculations is an excellent tool to unambiguously decipher
complicated electronic structures.

###### Acknowledgements.

V. M. K. and O. V. Y. would like to acknowledge funding from Swiss NSF NCCR
MARVEL and the Sinergia grant NanoSkyrmionics CR- SII5 171003. The work at PSI
is supported by the Swiss NSF through the NCCR MARVEL and the Sinergia network
Mott Physics Beyond the Heisenberg Model (MPBH). X. L. acknowledges financial
support from the European Community’s Seventh Framework Programme
(FP7/20072013) under Grant agreement No. 290605 (Cofund; PSI-Fellow). The work
at BNU is supported by the National Natural Science Foundation of China under
Grant No. 11734002 and 11922402. M. D. was partially funded by the Swiss
National Science Foundation within the D-A-CH programme (SNSF Research Grant
200021L 141325). This research used resources of the Advanced Photon Source, a
U.S. Department of Energy (DOE) Office of Science User Facility operated for
the DOE Office of Science by Argonne National Laboratory under Contract No.
DE-AC02-06CH11357.

## References

  * Imada  _et al._ (1998) Masatoshi Imada, Atsushi Fujimori, and Yoshinori Tokura, “Metal-insulator transitions,” Rev. Mod. Phys. 70, 1039–1263 (1998).
  * Pen  _et al._ (1997) H. F. Pen, J. van den Brink, D. I. Khomskii, and G. A. Sawatzky, “Orbital Ordering in a Two-Dimensional Triangular Lattice,” Phys. Rev. Lett. 78, 1323–1326 (1997).
  * Khomskii and Mizokawa (2005) D. I. Khomskii and T. Mizokawa, “Orbitally Induced Peierls State in Spinels,” Phys. Rev. Lett. 94, 156402 (2005).
  * Peierls (1991) R. Peierls,  _More Surprises in Theoretical Physics_ (Princeton University Press, 1991).
  * Terasaki  _et al._ (2016) Ichiro Terasaki, Shun Ito, Taichi Igarashi, Shinichiro Asai, Hiroki Taniguchi, Ryuji Okazaki, Yukio Yasui, Kensuke Kobayashi, Reiji Kumai, Hironori Nakao, and Youichi Murakami, “Novel Charge Ordering in the Trimer Iridium Oxide BaIrO3,” Crystals 6 (3), 27 (2016).
  * Nag  _et al._ (2016) Abhishek Nag, S. Middey, Sayantika Bhowal, S. K. Panda, Roland Mathieu, J. C. Orain, F. Bert, P. Mendels, P. G. Freeman, M. Mansson, H. M. Ronnow, M. Telling, P. K. Biswas, D. Sheptyakov, S. D. Kaushik, Vasudeva Siruguri, Carlo Meneghini, D. D. Sarma, Indra Dasgupta, and Sugata Ray, “Origin of the Spin-Orbital Liquid State in a Nearly $J=0$ Iridate ${\mathrm{Ba}}_{3}{\mathrm{ZnIr}}_{2}{\mathrm{O}}_{9}$,” Phys. Rev. Lett. 116, 097205 (2016).
  * Dey  _et al._ (2017) Tusharkanti Dey, M. Majumder, J. C. Orain, A. Senyshyn, M. Prinz-Zwick, S. Bachus, Y. Tokiwa, F. Bert, P. Khuntia, N. Büttgen, A. A. Tsirlin, and P. Gegenwart, “Persistent low-temperature spin dynamics in the mixed-valence iridate ${\mathrm{Ba}}_{3}{\mathrm{InIr}}_{2}{\mathrm{O}}_{9}$,” Phys. Rev. B 96, 174411 (2017).
  * Nag  _et al._ (2018) Abhishek Nag, Sayantika Bhowal, F. Bert, A. D. Hillier, M. Itoh, Ilaria Carlomagno, C. Meneghini, T. Sarkar, R. Mathieu, I. Dasgupta, and Sugata Ray, “${\mathrm{Ba}}_{3}M{\mathrm{Ir}}_{2}{\mathrm{O}}_{9}$ hexagonal perovskites in the light of spin-orbit coupling and local structural distortions,” Phys. Rev. B 97, 064408 (2018).
  * Revelli  _et al._ (2019) A. Revelli, M. Moretti Sala, G. Monaco, P. Becker, L. Bohatý, M. Hermanns, T. C. Koethe, T. Fröhlich, P. Warzanowski, T. Lorenz, S. V. Streltsov, P. H. M. van Loosdrecht, D. I. Khomskii, J. van den Brink, and M. Grüninger, “Resonant inelastic x-ray incarnation of Young’s double-slit experiment,” Sci. Adv. 5, eaav4020 (2019).
  * Kim  _et al._ (2014a) Heung-Sik Kim, Jino Im, Myung Joon Han, and Hosub Jin, “Spin-orbital entangled molecular $j_{\rm eff}$ states in lacunar spinel compounds,” Nat Commun 5, 3988 (2014a).
  * Jeong  _et al._ (2017) Min Yong Jeong, Seo Hyoung Chang, Beom Hyun Kim, Jae-Hoon Sim, Ayman Said, Diego Casa, Thomas Gog, Etienne Janod, Laurent Cario, Seiji Yunoki, Myung Joon Han, and Jungho Kim, “Direct experimental observation of the molecular $J_{\rm eff}$ = 3/2 ground state in the lacunar spinel GaTa4Se8,” Nat Commun 8, 782 (2017).
  * Park  _et al._ (2020) Moon Jip Park, GiBaik Sim, Min Yong Jeong, Archana Mishra, Myung Joon Han, and SungBin Lee, “Pressure-induced topological superconductivity in the spin–orbit Mott insulator GaTa4Se8,” npj Quantum Mater. 5, 41 (2020).
  * Kim  _et al._ (2008) B. J. Kim, Hosub Jin, S. J. Moon, J. Y. Kim, B. G. Park, C. S. Leem, Jaejun Yu, T. W. Noh, C. Kim, S. J. Oh, J. H. Park, V. Durairaj, G. Cao, and E. Rotenberg, “Novel $J_{\rm eff}=1/2$ Mott State Induced by Relativistic Spin-Orbit Coupling in Sr2IrO4,” Phys. Rev. Lett. 101, 076402–076402 (2008).
  * Kim  _et al._ (2009) B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi, and T. Arima, “Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr2IrO4,” Science 323, 1329–1332 (2009).
  * Katukuri  _et al._ (2012) Vamshi M. Katukuri, Hermann Stoll, Jeroen van den Brink, and Liviu Hozoi, “Ab initio determination of excitation energies and magnetic couplings in correlated quasi-two-dimensional iridates,” Phys. Rev. B 85, 220402 (2012).
  * Moretti Sala  _et al._ (2014) M. Moretti Sala, S. Boseggia, D. F. McMorrow, and G. Monaco, “Resonant X-Ray Scattering and the jeff = 1/2 Electronic Ground State in Iridate Perovskites,” Phys. Rev. Lett. 112, 026403 (2014).
  * Witczak-Krempa  _et al._ (2014) William Witczak-Krempa, Gang Chen, Yong Baek Kim, and Leon Balents, “Correlated Quantum Phenomena in the Strong Spin-Orbit Regime,” Annu. Rev. Condens. Matter Phys. 5, 57–82 (2014).
  * Abragam and Bleaney (1970) A. Abragam and B. Bleaney,  _Electron Paramagnetic Resonance of Transition Ions_ (Clarendon Press, Oxford, 1970).
  * Khaliullin (2013) Giniyat Khaliullin, “Excitonic magnetism in van vleck type ${d}^{4}$ mott insulators,” Phys. Rev. Lett. 111, 197201 (2013).
  * Kusch  _et al._ (2018) M. Kusch, V. M. Katukuri, N. A. Bogdanov, B. Büchner, T. Dey, D. V. Efremov, J. E. Hamann-Borrero, B. H. Kim, M. Krisch, A. Maljuk, M. Moretti Sala, S. Wurmehl, G. Aslan-Cansever, M. Sturza, L. Hozoi, J. van den Brink, and J. Geck, “Observation of heavy spin-orbit excitons propagating in a nonmagnetic background: The case of ${(\mathrm{Ba},\mathrm{Sr})}_{2}{\mathrm{YIrO}}_{6}$,” Phys. Rev. B 97, 064421 (2018).
  * Wang  _et al._ (2019) Y. Wang, Ruitang Wang, Jungho Kim, M. H. Upton, D. Casa, T. Gog, G. Cao, G. Kotliar, M. P. M. Dean, and X. Liu, “Direct Detection of Dimer Orbitals in ${\mathrm{Ba}}_{5}{\mathrm{AlIr}}_{2}{\mathrm{O}}_{11}$,” Phys. Rev. Lett. 122, 106401 (2019).
  * Müller-Buschbaum and Lang (1989) Hk. Müller-Buschbaum and Ch. Lang, “Ba5AlIr2O11: Eine neue Verbindung mit Iridium(IV, V),” Z. Anorg. Allg. Chem. 568, 29–34 (1989).
  * Terzic  _et al._ (2015) J. Terzic, J. C. Wang, Feng Ye, W. H. Song, S. J. Yuan, S. Aswartham, L. E. DeLong, S. V. Streltsov, D. I. Khomskii, and G. Cao, “Coexisting charge and magnetic orders in the dimer-chain iridate $\mathrm{B}{\mathrm{a}}_{5}\mathrm{AlI}{\mathrm{r}}_{2}{\mathrm{O}}_{11}$,” Phys. Rev. B 91, 235147 (2015).
  * (24) See Supplemental Material at [URL].
  * Moretti Sala  _et al._ (2018) M. Moretti Sala, K. Martel, C. Henriquet, A. Al Zein, L. Simonelli, Ch. J. Sahle, H. Gonzalez, M.-C. Lagier, C. Ponchut, S. Huotari, R. Verbeni, M. Krisch, and G. Monaco, “A high-energy-resolution resonant inelastic X-ray scattering spectrometer at ID20 of the European Synchrotron Radiation Facility,” J. Synchrotron Rad. 25, 580–591 (2018).
  * Kim  _et al._ (2012) Jungho Kim, D. Casa, M. H. Upton, T. Gog, Young-June Kim, J. F. Mitchell, M. van Veenendaal, M. Daghofer, J. van den Brink, G. Khaliullin, and B. J. Kim, “Magnetic Excitation Spectra of ${\mathrm{Sr}}_{2}{\mathrm{IrO}}_{4}$ Probed by Resonant Inelastic X-Ray Scattering: Establishing Links to Cuprate Superconductors,” Phys. Rev. Lett. 108, 177003 (2012).
  * Strocov  _et al._ (2010) V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Rad. 17, 631–643 (2010).
  * Ghiringhelli  _et al._ (2006) G. Ghiringhelli, A. Piazzalunga, C. Dallera, G. Trezzi, L. Braicovich, T. Schmitt, V. N. Strocov, R. Betemps, L. Patthey, X. Wang, and M. Grioni, “SAXES, a high resolution spectrometer for resonant x-ray emission in the 400 - 1600 eV energy range,” Rev. Sci. Instrum. 77, 113108 (2006).
  * Lu  _et al._ (2018) X. Lu, P. Olalde-Velasco, Y. Huang, V. Bisogni, J. Pelliciari, S. Fatale, M. Dantz, J. G. Vale, E. C. Hunter, J. Chang, V. N. Strocov, R. S. Perry, M. Grioni, D. F. McMorrow, H. M. Rønnow, and T. Schmitt, “Dispersive magnetic and electronic excitations in iridate perovskites probed by oxygen $K$-edge resonant inelastic x-ray scattering,” Phys. Rev. B 97, 041102 (2018).
  * Helgaker  _et al._ (2000) T. Helgaker, P. Jørgensen, and J. Olsen,  _Molecular Electronic-Structure Theory_ (Wiley, Chichester, 2000).
  * Neese (2012) Frank Neese, “The ORCA program system,” WIREs Comput Mol Sci 2, 73–78 (2012).
  * Angeli  _et al._ (2001) Celestino Angeli, Renzo Cimiraglia, and Jean-Paul Malrieu, “N-electron valence state perturbation theory: a fast implementation of the strongly contracted variant,” Chem. Phys. Lett. 350, 297 – 305 (2001).
  * Kim  _et al._ (2014b) Jungho Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Khaliullin, and B. J. Kim, “Excitonic quasiparticles in a spin-orbit Mott insulator,” Nature Communications 5, 4453 EP – (2014b), article.
  * Gretarsson  _et al._ (2013) H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, Emil Bozin, Yogesh Singh, S. Manni, P. Gegenwart, Jungho Kim, A. H. Said, D. Casa, T. Gog, M. H. Upton, Heung-Sik Kim, J. Yu, Vamshi M. Katukuri, L. Hozoi, J. van den Brink, and Young-June Kim, “Crystal-Field Splitting and Correlation Effect on the Electronic Structure of ${A}_{2}{\mathrm{IrO}}_{3}$,” Phys. Rev. Lett. 110, 076402 (2013).
  * Note (1) To simulate electron-hole exciton peaks one would need to calculate N-1 (removal of an electron) state calculations.
  * Katukuri  _et al._ (2014) Vamshi M. Katukuri, K. Roszeitis, V. Yushankhai, A. Mitrushchenkov, H. Stoll, M. van Veenendaal, P. Fulde, J. van den Brink, and L. Hozoi, “Electronic structure of low-dimensional 4$d^{5}$ oxides: interplay of ligand distortions, overall lattice anisotropy, and spin-orbit interactions,” Inorg. Chem. 53, 4833 (2014).
  * Note (2) See direct product tables for $O_{h}$ point group representations, e.g. in Ref. Atkins  _et al._ (1993).
  * Note (3) For example, for pure trigonal distortions, the $t_{2g}$ orbitals are split into $a_{1g}$ and $e_{g}$.
  * Streltsov and Khomskii (2016) Sergey V. Streltsov and Daniel I. Khomskii, “Covalent bonds against magnetism in transition metal compounds,” Proc. Natl. Acad. Sci. USA 113, 10491–10496 (2016).
  * Note (4) The canonical orbitals are eigenfunctions of the so-called (effective single particle) Fock operator.
  * Note (5) The orbital compositions are obtained from the Löwdin natural orbital decomposition.
  * Malmqvist  _et al._ (1990) Per Aake. Malmqvist, Alistair. Rendell, and Bjoern O. Roos, “The restricted active space self-consistent-field method, implemented with a split graph unitary group approach,” The Journal of Physical Chemistry 94, 5477–5482 (1990).
  * Streltsov  _et al._ (2017) Sergey V. Streltsov, Gang Cao, and Daniel I. Khomskii, “Suppression of magnetism in ${\mathrm{Ba}}_{5}{\mathrm{AlIr}}_{2}{\mathrm{O}}_{11}$: Interplay of Hund’s coupling, molecular orbitals, and spin-orbit interaction,” Phys. Rev. B 96, 014434 (2017).
  * (44) Vamshi M. Katukuri and O. V. Yazyev, “Delocalized dimer orbitals in Ba3InIr2O9,” to be published .
  * Sakamoto  _et al._ (2006) Takeshi Sakamoto, Yoshihiro Doi, and Yukio Hinatsu, “Crystal structures and magnetic properties of 6H-perovskite-type oxides Ba3MIr2O9 (M=Mg, Ca, Sc, Ti, Zn, Sr, Zr, Cd and In),” J. Solid State Chem. 179, 2595 – 2601 (2006).
  * Atkins  _et al._ (1993) P. W. Atkins, M. S. Child, and C. S. G. Phillips,  _Tables for group theory_ (Oxford Univ. Press, London, 1993).