#  Electronic structure and optoelectronic properties of halide double
perovskites: Fundamental insights and design of a theoretical workflow

Mayank Gupta  Condensed Matter Theory and Computational Lab, Department of
Physics, Indian Institute of Technology Madras, Chennai 600036, India  Center
for Atomistic Modelling and Materials Design, Indian Institute of Technology
Madras, Chennai 600036, India  Susmita Jana  Condensed Matter Theory and
Computational Lab, Department of Physics, Indian Institute of Technology
Madras, Chennai 600036, India  Center for Atomistic Modelling and Materials
Design, Indian Institute of Technology Madras, Chennai 600036, India  B. R. K.
Nanda<EMAIL_ADDRESS>Condensed Matter Theory and Computational Lab,
Department of Physics, Indian Institute of Technology Madras, Chennai 600036,
India  Center for Atomistic Modelling and Materials Design, Indian Institute
of Technology Madras, Chennai 600036, India

###### Abstract

Like single perovskites, halide double perovskites (HDP) have truly emerged as
efficient optoelectronic materials since they display superior stability and
are free of toxicity. However, challenges still exist due to either wide and
indirect bandgaps or parity-forbidden transitions in many of them. The lack of
understanding in chemical bonding and the formation of parity-driven valence
and conduction band edge states have hindered the design of optoelectronically
efficient HDPs. In this study, we have developed a theoretical workflow using
a multi-integrated approach involving ab-initio density functional theory
(DFT) calculations, model Hamiltonian studies, and molecular orbital picture
leading to momentum matrix element (MME) estimation. This workflow gives us
detailed insight into chemical bonding and parity-driven optical transition
between edge states. In the process, we have developed a band-projected
molecular orbital picture (B-MOP) connecting free atomic orbital states
obtained at the Hartree-Fock level and orbital-resolved DFT bands. From the
B-MOP, we show that the nearest neighbor cation-anion interaction determines
the position of atom-resolved band states, while the second neighbor cation-
cation interactions determine the shape and width of band dispersion and,
thereby, MME. The latter is critical to quantify the optical absorption
coefficient. Considering both B-MOP and MME, we demonstrate a mechanism of
tailoring bandgap and optical absorptions through chemical doping at the
cation sites. Furthermore, the cause of bandgap bowing, a common occurrence in
doped HDPs, is explained by ascribing it to chemical effect and structural
distortion.

††preprint: APS

##  I Introduction:

In the last couple of decades, organic and inorganic halide single perovskites
(HSPs) of the formula ABX3 (e.g. CsPbI3) have gained enormous research
attention as they demonstrate promising optoelectronic properties Jena  _et
al._ (2019); Protesescu  _et al._ (2015), solar cell applications Kojima  _et
al._ (2009), non-trivial topological quantum phases which bring the dimension
of orbitronics Gupta and Nanda (2022); Kepenekian  _et al._ (2015) and
topotronics Jin  _et al._ (2012); Kashikar  _et al._ (2020); Kore  _et al._
(2020). At the same time, there appears to be a large number of disadvantages
associated with this class of compounds. The most significant one is the lack
of stability on prolonged exposure to light and heat. As most of the promising
HSPs are lead (Pb) based, toxicity remains another concern. The halide double
perovskites (HDPs) are emerging as an alternate class of compounds which to
some extent, overcome the aforementioned disadvantages. HDP has a general
formula of A2BB′X6 where A is a monovalent cation of Group-I, B and B′ are
metals with +1 (K, Na, Ag, Au, Cu, In, Tl) and +3 (Bi, Sb, In, Tl) oxidation
states, and X is a halide. Most commonly, in HDPs, A-site is Cs, and Cl and Br
are considered halogen sites.

Compared to the HSPs, HDPs are in general more stable Bibi  _et al._ (2021)
and environmental friendly. They create a large chemical configurational
space, and therefore this family is capable of exhibiting diverse electronic
structures and, in turn, are suitable for a wide range of applications. These
include photovoltaic solar cells Zhang  _et al._ (2022), photodetectors Lei
_et al._ (2018), photocatalysis Muscarella and Hutter (2022); Li  _et al._
(2022a), CO2 reduction Pi  _et al._ (2022), spintronics Ning  _et al._ (2020),
X-ray detectors Zhu  _et al._ (2020); Zhang  _et al._ (2020), water splitting
Sikarwar  _et al._ (2023), etc. The HDPs are also being actively examined as
solar cell absorbers. However, the issue of indirect bandgap in some of them
and parity forbidden transition in others McClure  _et al._ (2016); Zelewski
_et al._ (2019); Volonakis  _et al._ (2017); Liu  _et al._ (2021); Ravi  _et
al._ (2018) are the bottleneck which needs to be addressed. For example,
Cs2AgBiBr6 and Cs2AgSbCl6 possess indirect bandgap. On the other hand, though
Cs2AgInCl6 exhibits a direct band gap of 3.23 eV, parity forbidden transition
at $\Gamma$, leads to very weak optical absorption near the band gap. The next
optical transition in this system happens at $\sim$ 4.46 eV, which is much
higher for an ideal solar cell material Zhou  _et al._ (2017). In a recent
study, it has been revealed that, for B = In+, TI+ and B′ = Sb3+, Bi3+, HDPs
show favorable optical absorption suitable for thin-film solar cell
applicationsSlavney  _et al._ (2018). Unfortunately, Tl+ is toxic, and In+
tends to be unstable against oxidation and form mixed-valence compounds with
distorted and complex crystal structures Xiao  _et al._ (2017).

Despite a few disadvantages, HDPs have attracted considerable attention due to
their simple, robust, and easy synthesis process. Since the valence band and
conduction band edges are formed out of the covalent hybridization among the
orbitals of metal cations (B, B′) and halide anions (X), cationic and anionic
mixing naturally becomes an effective strategy to manipulate the electronic
properties and optical behavior. Taking into account these advantages, many
design principles have been proposed experimentally Prasanna  _et al._ (2017)
and theoretically Zhao  _et al._ (2017); Cai  _et al._ (2019); Zhao  _et al._
(2018) to modify the electronic structure so as to achieve better
optoelectronic performances.

Recent studies Appadurai  _et al._ (2021); Manna  _et al._ (2019) have shown
that, Cs2AgInxBi1-xCl6 and Cs2AgBixSb1-xCl6 produce high photoluminescence for
the range 0.8 $\leq$ x $\leq$ 0.9. The reasons are attributed to the
manipulation of bandgap and parity. In another study, Athrey and Sudakar
Dakshinamurthy and Sudakar (2022); C. Dakshinamurthy and Sudakar (2021) have
experimentally demonstrated distortion drive nonlinear bandgap variation and
self-trapped exciton (STE) emission on the cationic intermix systems Cs2(Ag,
Na)BiCl6. Interestingly, the anionic intermixing (Cs2AgBiBr6-xClx) results in
linear bandgap variation Dakshinamurthy  _et al._ (2023). In a combined
theoretical and experimental study, Slavney et al. Slavney  _et al._ (2017)
reported a change in the bandgap from 1.95 to 1.4 eV in MA2AgBiBr6 by Tl
doping at Ag and Bi site, which is close to the ideal bandgap for photovoltaic
applications. Many other studies are carried out to demonstrate the tuning of
optical properties in HDPs by a similar cation intermixing approach Du  _et
al._ (2017); Karmakar  _et al._ (2018); Tran  _et al._ (2017); Zhang  _et al._
(2022). However, these isolated investigations with limited scopes do not
reveal the universal mechanism that alters the electronic structure at the
vicinity of the Fermi level. Hence, there is a lack of guiding principles
which can be utilized to design HDPs for electronic applications through
controlled cationic and anionic intermixing. Developing materials design
workflow has become necessary as HDPs are now being intensely investigated in
search of stable and highly efficient solar cell materials.

In this study, by considering a set of prototype compounds Cs2BB′Cl6 (B = Ag,
Na, In, Tl and B′ = In, Tl, Bi, Sb), we develop a theoretical workflow to
establish the relationship between cationic and anionic intermixing and the
electronic structure, as well as, the optical absorption in the HDPs. The
theoretical workflow, schematically illustrated in Fig. 1, is based on density
functional theory (DFT) calculations, an elegant Slater-Koster formalized
tight-binding (SK-TB) model, and band projected molecular orbital picture
(B-MOP). The optical absorption study is carried out by calculating the
momentum matrix elements (MME), which are the outcomes of the solution to
model Hamiltonian. Through the workflow, we understand the chemical bonding
and parity-driven optical transition between the edge states. With the aid of
B-MOP, we infer that the nearest neighbor cation-anion interaction determines
the position of atom-resolved band states. On the other hand, second neighbor
cation-cation interactions determine shape and width of the band dispersion
and, hence, the MMEs. The imaginary part of the dielectric constant and in
turn the optical absorptions are calculated using the MMEs. With the aid of
both B-MOPs and MMEs, we demonstrate how chemical doping at the cation site
can tailor the bandgap and optical absorption. As a byproduct, we demonstrate
how the chemical effect and structural distortion together cause bandgap
bowing, a common occurrence in doped HDPs.

Figure 1: A schematic summarizing the design principle to calculate and
predict the efficient optoelectronic properties of HDPs.

##  II Designing Approach and Computational Details

We will first briefly discuss the crystal structure of HDPs. As shown in Fig.
2 (a), it has a single unit rhombohedral primitive unit cell (Fm$\bar{3}$m)
with two organic or inorganic monovalent A cations, one monovalent B, one
trivalent B′ cations, and six halogen anions. A conventional crystal structure
of HDP is a cubic unit cell and contains four formula units. The salient
feature of the crystal structure is the presence of BX6 and B′X6 octahedra
which are alternately arranged and connected with corner-sharing X-anions in
all three directions. The A+ cations occupy the cuboctahedral cavities
positions.

The approach to design the theoretical workflow is summarized in the flowchart
shown in Fig. 1. Hartree-Fock calculations on sample free atoms A, B, B′, and
X provide the free atomic orbital energy levels. This, in combination with the
DFT calculated band structure, establishes B-MOP describing the possible
chemical bondings of the prototype compounds. The B-MOPs enable us to design a
parametric tight-binding model Hamiltonian and construction of chemical
configuration space. The variation in the parameter and configuration can
contribute towards searching for desired electronic structure and optical
absorption deterministic momentum matrix so as to maximize optoelectronic
efficiencies. Each component of the flowchart is further described in detail
in the remaining part of the paper.

The DFT electronic structure calculations are performed on a set of HDPs (see
Table 1) using full-potential linearized augmented plane-wave (FP-LAPW) method
as implemented in the WIEN2k simulation tool Blaha  _et al._ (2019). For
structural relaxation, we have used pseudopotential-based Vienna ab-initio
Simulation Package (VASP)Kresse and Furthmüller (1996) within the framework of
the projector-augmented waves (PAW) method. Relaxations are performed via the
conjugate gradient algorithm until the residual force in each atom is $<$ 0.01
eV/Å. A $k$-mesh of size 6 $\times$ 6 $\times$ 6 is used for the Brillouin
zone (BZ) sampling, PBE generalized gradient approximation (GGA) Kresse and
Joubert (1999); Blöchl (1994) is employed for the exchange-correlation
functional with the energy cutoff 400 eV, and the convergence criterion for
total energy is set to 10-6 eV. The lattice constants of HDPs after relaxation
are provided in Table 1.

Figure 2: (a) The rhombohedral (one formula unit) unit cell (shown in shaded
gray color) inside a conventional (four formula unit) face-centered cubic cell
of the halide double perovskite A2BB′X6. The 2nd nearest-neighbor (NN)
electron-electron hopping interactions take place between B and B′ cations and
are mediated by halogen X anions. The 4th-NN interactions occur between the
same metal cations $i.e.$, B to B and B′ to B′. (b and c) The first Brillouin
zone for the rhombohedral and conventional unit cells, respectively. The
$k$-path used for band structure calculations is shown in green color.

The GGA-PBE functional underestimates the bandgap as compared to the
experimental bandgap, and hence the GGA-PBE along with the modified
Becke–Johnson (mBJ) Tran and Blaha (2009) potential is used to calculate the
electronic band structure of HDPs. The results are in good agreement with the
experimental bandgap (see Table 1). The number of plane waves is determined by
setting $R_{MT}K_{MAX}$ to 7.0 for all the compounds. The BZ integration is
carried out with a Monkhorst–Pack grid with the k-mesh size of 8 $\times$ 8
$\times$ 8 (yielding 35 irreducible points). The calculations include spin-
orbit coupling (SOC) effect. For the model Hamiltonian studies, we have
developed a few codes using MATLAB Inc. (2022), and the package is made
available online Gupta and Nanda (2023).

Table 1: DFT obtained GGA+mBJ bandgaps for various halide double perovskites along with the nature of bandgap, orbital compositions of valence and conduction bands, and valence electron counts (VEC). Crystal formula | Lattice Constant(Å) | Lattice Constant(Å) | Bandgap | Bandgap | Nature of | VBM | CBM | VEC
---|---|---|---|---|---|---|---|---
| (GGA relaxed) | (Experimental) | GGA-mBJ(eV) | exp (eV) | bandgap |  |  |
Cs2AgBiBr6 | 11.465 | 11.271 Zhang  _et al._ (2022); C. Dakshinamurthy and Sudakar (2022) | 1.90 | 2.19 Zhang  _et al._ (2022) | Indirect | Ag-$e_{g}$+Bi-$s$ | Bi-$p$ | 48
Cs2AgBiCl6 | 10.936 | 10.760 Dakshinamurthy and Sudakar (2022) | 2.98 | 2.65 Dakshinamurthy and Sudakar (2022) | Indirect | Ag-$e_{g}$+Bi-$s$ | Bi-$p$ | 48
Cs2AgSbCl6 | 10.809 | 10.699 Karmakar  _et al._ (2018) | 2.41 | 2.57 Karmakar  _et al._ (2018) | Indirect | Ag-$e_{g}$+Sb-$s$ | Sb-$p$ | 48
Cs2AgInCl6 | 10.560 | 10.480 Zhang  _et al._ (2022) | 3.28 | 3.57 Dahl  _et al._ (2019) | Direct | Ag-$e_{g}$ | In-$s$ | 46
Cs2AgTlCl6 | 10.784 | 10.559 Slavney  _et al._ (2018) | 0.64 | 1.96 Slavney  _et al._ (2018) | Direct | Ag-$e_{g}$ | Tl-$s$ | 46
Cs2AgAsCl6 | 9.14 | - | 2.30 | - | Indirect | Ag-$e_{g}$+As-$s$ | As-$p$ | 48
Cs2AgGaCl6 | 9.00 | - | 2.89 | - | Direct | Ag-$e_{g}$ | Ga-$s$ | 46
Cs2CuBiCl6 | 10.60 | - | 0.82 | 1.57 Neelu  _et al._ (2023) | Indirect | Cu-$e_{g}$+Bi-$s$ | Bi-$p$ | 48
Cs2CuInCl6 | 10.34 | - | 0.25 | - | Direct | Cu-$e_{g}$ | In-$s$ | 46
Cs2InBiCl6 | 11.345 | - | 0.53 | - | Direct | In-$s$ | Bi-$p$ | 40
Cs2InSbCl6 | 11.212 | - | 0.64 | - | Direct | In-$s$ | Sb-$p$ | 40
Cs2TlBiCl6 | 11.547 | - | 2.09 | - | Direct | Tl-$s$ | Bi-$p$ | 40
Cs2TlSbCl6 | 11.420 | - | 2.01 | - | Direct | Tl-$s$ | Sb-$p$ | 40
Cs2TlBiBr6 | 12.069 | - | 1.34 | - | Direct | Tl-$s$ | Bi-$p$ | 40
Cs2TlSbBr6 | 11.944 | - | 1.27 | - | Direct | Tl-$s$ | Sb-$p$ | 40
Cs2NaBiCl6 | 11.026 | 10.833 C. Dakshinamurthy and Sudakar (2021) | 4.15 | 3.0 Dakshinamurthy and Sudakar (2022) | Indirect | Bi-$s$ | Bi-$p$ | 38
Cs2NaSbCl6 | 10.930 | - | 3.99 | - | Indirect | Sb-$s$ | Sb-$p$ | 38
Cs2NaInCl6 | 10.730 | 10.533 Gray  _et al._ (2020) | 5.37 | 4.15 Li  _et al._ (2022b) | Direct | Cl-$p$ | In-$s$ | 36
Cs2KInCl6 | 11.156 | 10.770 Cong  _et al._ (2021) | 5.89 | - | Direct | Cl-$p$ | In-$s$ | 36
Cs2KBiCl6 | 11.498 | - | 4.32 | - | Direct | Bi-$s$ | Bi-$p$ | 38
Cs2NaBiBr6 | 11.615 | 11.357 Lamba  _et al._ (2021) | 3.29 | 3.10 Lamba  _et al._ (2021) | Indirect | Bi-$s$ | Bi-$p$ | 38

As the next step, we perform first-principles calculations to estimate the
electronic properties of a set of HDPs. For all HDPs discussed here, we have
used A = Cs and X = Cl as an example, and we believe that results and
description will not change by replacing A and X site atoms with their
equivalent unless the crystal symmetry is destroyed. Since not all HDPs are
experimentally synthesized in their pristine phase, we have conceived the
hypothetical cubic Fm$\bar{3}$m structure and performed the full relaxation
(both atomic positions and crystal lattice parameters) for all of them. The
relaxed lattice constant in comparison to the available experimental lattice
constant and bandgap calculated with GGA+mBJ+SOC functional is listed in Table
1. The obtained band structures (calculated along the path shown in Fig. 2
(b)) are further used to map the B-MOP as shown in Figs. 3 and 4. Our aim in
studying the pristine phases of HDPs is to understand the properties of the
end member crystals, which help to predict the properties of the cation mixed
phases.

##  III Construction of Band Projected Molecular-Orbit Picture (B-MOP)

A MOP examines the possible chemical bondings and provides us with a broader
picture of the electronic structure of a material and its universality in a
given family. Therefore, without carrying out comprehensive electronic
structure calculations, it is possible through MOP to develop an insight into
how the modulations in the electronic structure across a family due to
chemical substitution and doping. Here, we construct the MOP in three steps:
First, the free atomic energy levels of the valence orbitals are estimated
using the Hartree-Fock theory, and their relative positions in the energy
scale are obtained. In the second step, orbital projected band structures are
carried out using DFT. In the third step, the band centers of the projected
bands are linked to the free atomic orbital energy levels so as to obtain
probable chemical bonding and their strengths and finally draw the MOP. Such
an attempt of linking the schematical MOP with eigenstates and eigenvalues in
the momentum space has never been done before. Therefore, to distinguish it
from the conventional not-to-scale MOPs, we name it as band projected MOP
(B-MOP).

Figure 3 shows B-MOPs for Ag-based HDPs. It infers the formation of bonding
and antibonding spectrum arising from {Ag-($s$, $d$); Bi/Sb/In/Tl-($s$, $p$)}
– X-$p$ covalent hybridizations. The bonding spectrum consists of $e_{g}-p$,
$t_{2g}-p$, $\sigma_{s-p}$, $\sigma_{p-p}$, $\pi_{p-p}$ interactions, and the
antibonding spectrum consists of $e_{g}-p^{*}$, $t_{2g}-p^{*}$,
$\sigma_{s-p}^{*}$, $\sigma_{p-p}^{*}$, $\pi_{p-p}^{*}$ interactions. The
conservation of basis leaves behind eight non-bonding states combinedly formed
by the X-$p$ orbitals. The strength of covalent hybridization is measured by
the energy difference between the two corresponding bonding and antibonding
pair. The B-MOPs suggest the strength of hybridization in the increasing order
as $s-p$ $>$ $p-p$ $>$ $d-p$. Now the valence and conduction band edge states
(VBE and CBE) will be determined from the valence electron count (VEC), which
is defined as the sum of the valence electrons in the constituent member
element. In the case of Cs2AgBiCl6 (Cs2AgSbCl6), the VEC is found to be 48
(see Table 1 for other HDPs), and therefore the electron filling implies that
VBE is formed by Ag-$e_{g}$ – X-$p$ hybridized orbitals and CBE is formed by
Bi-$p$ – X-$p$ (Sb-$p$ – X-$p$) hybridized orbitals. Similarly, for Cs2AgInCl6
(Cs2AgTlCl6), with VEC 46, the VBE is formed by Ag-$e_{g}$ – X-$p$ and CBE is
formed by In-$s$ – X-$p$ (Tl-$s$ – X-$p$) hybridized orbitals, respectively.

Figure 3: The band projected molecular orbital picture (B-MOP) of halide
double perovskites as envisaged from the following molecular hybridizations:
B(Ag)-{$s$, $d$}–Cl-$p$, B(In, Tl)-{$s$, $p$}–Cl-$p$ and B′(Bi, Sb, In,
Tl)-{$s$, $p$}–Cl-$p$ atomic orbitals produces the bonding and antibonding
orbitals along with the nonbonding Cl-$p$ orbitals. The free atomic energy
levels are estimated from Hartree-Fock’s theory. The interactions among the B
and B′ states are not represented explicitly in these MOPs.

From B-MOPs (Fig. 3 (a, b)), we describe some of the main features that play
an important role in establishing the electronic structure in HDPs. For the
Ag-based compounds, these are: (I) The $t_{2g}$ and $e_{g}$ orbitals dominated
bands are narrow, indicating weaker interactions with the Cl-$p$ states,
whereas B/B′-$s$ based orbitals dominated bands are wider due to their
stronger interactions with Cl-$p$ states. (II) Depending on the VEC, Ag-HDPs
demonstrate two kinds of bandgaps. For Cs2AgBiCl6 and Cs2AgSbCl6, the anti-
bonding spectrum $\sigma^{*}_{s-p}$ is occupied to form the VBE with VBM at X
while $\sigma^{*}_{p-p}$ constitute the CBE with CBM at $\Gamma$. For
Cs2AgInCl6 and Cs2AgTlCl6, which have two less VEC compared to Cs2AgBiCl6 and
Cs2AgSbCl6, the $\sigma^{*}_{s-p}$ is unoccupied to form the CBE with CBM at
$\Gamma$ and $e^{*}_{g-p}$ is occupied to from the VBE with VBM lies on the
$\Gamma-X$ flat band. (III) Bandgap variation: The free atomic energies of B
and B′-atoms are turning out to be the deterministic factor for the magnitude
of the bandgap as they influence the position of the bands. For example, the
bandgap difference of $\sim$2.64 eV between Cs2AgInCl6 and Cs2AgTlCl6 can be
attributed to the free atomic energy difference between In-$s$ and Tl-s
orbitals (see Fig. 3 (c, d)). Since the VBM is formed by $e^{*}_{g-p}$ bonding
for both cases, the bandgap is determined by the position of the
$\sigma^{*}_{s-p}$ led CBM. With In-$s$ energy 3 eV higher than that of
Tl-$s$, the latter forms a narrow bandgap of 0.64 eV, and the former forms a
wide bandgap of 3.3 eV. In the case of Cs2AgBiCl6 and Cs2AgSbCl6, where Bi/Sb-
Cl $\sigma^{*}_{s-p}$ and $\sigma^{*}_{p-p}$ form the VBE and CBE
respectively, the bandgaps are of a nearly similar order. This is due to the
fact that both Bi-$s$ and Bi-$p$ energies are lower by a similar magnitude
with respect to that of Sb-$s$ and Sb-$p$ energies. Hence, the relative
positioning of VBM and CBM are similar for both compounds.

Similarly, the salient features that we obtain from B-MOP for the HDPs
(Cs2InBiCl6, Cs2InSbCl6, Cs2TiBiBr6, Cs2TlSbBr6 without Ag) are as follows
(Fig. 3 (e-h)). (I) All are direct bandgap systems with B-Cl-
$\sigma^{*}_{s-p}$ and B′-Cl-$\sigma^{*}_{p-p}$ forming the VBE and CBE,
respectively. (II) When B is In, the system exhibits a narrower bandgap; when
it is Tl, it exhibits a wider bandgap. It is largely attributed to the fact
that In-$s$ free atomic energy levels are higher than that of Tl-$s$ by 3.0
eV. Hence, in the case of the latter, the VBE goes up to increase the
separation between CBE and VBE. Furthermore, if we compare the case of B′ as
Bi and Sb, the former show a smaller bandgap as Bi-$p_{1/2}$ free atomic
energy level is lower than that of Sb-$p_{1/2}$ to lower the position of CBE.

When the group-1A atoms (Na, K, etc.) occupy the B-site of HDPs, the B-MOP for
such compounds is illustrated in Fig. 4. The B site acts as an electron donor
and does not participate in the band formation. The interaction of four
B′-{$s$, $p$} orbitals with 18 Cl-$p$ orbitals give rise to four bonding
states ($\sigma_{s-p},\sigma_{p-p}$, and $\pi_{p-p}$), four corresponding
anti-bonding states ($\sigma_{s-p}^{*},\sigma_{p-p}^{*}$, and $\pi_{p-p}^{*}$)
and fourteen flat bands (shown in yellow). When B′ is Bi, the VEC is 38, and
therefore the anti-bonding bands $\sigma_{s-p}^{*}$ and $\sigma_{p-p}^{*}$
form the VBE and CBE respectively. It results in a wide and indirect bandgap
system (4.0 - 4.5 eV). There is a narrow separation between $\sigma_{s-p}^{*}$
and flat bands. When B′ is In, the VEC is reduced by two, and the
$\sigma_{s-p}^{*}$ forms the CBE while the Cl-$p$ flat bands form the VBE.
These systems exhibit a wide bandgap to the tune of 5.5 - 6.0 eV
approximately. B-MOP for four more compounds, namely Cs2AgAsCl6, Cs2CuBiCl6,
Cs2AgGaCl6, and Cs2CuInCl6 have been studied. The first two compounds belong
to the group of Cs${}_{2}A$gBiCl6 and Cs2AgSbCl6, and the remaining two belong
to the group of Cs2AgInCl6 and Cs2AgTlCl6. The detailed analysis of these four
compounds is provided in Section XV of SI.

The replacement of Cl by Br nearly replicates the B-MOP Dakshinamurthy  _et
al._ (2023). However, since Br-$p$ free atomic energy level is $\sim$ 0.5 eV
higher than that of Cl-$p$, we notice a couple of changes in the electronic
structure, and the most important of them is the reduction of the bandgap. The
cause of it can be explained by taking the example of Cs2AgBiBr6 and
Cs2AgBiCl6. Here, the prominent interactions that define the VBE and CBE are
Ag-$d$ – Br/Cl-$p$ and Bi-$p$ – Br/Cl-$p$, respectively. We find that the
Br-$p$ energy levels are comparable to that of Ag-$d$ energy levels while the
Cl-$p$ energy levels lie $\sim$ 0.5 eV below. Therefore, the Ag-$d$ — Br-$p$
interaction is stronger to push the antibonding $e_{g}^{*}$ band above for the
Br compound as compared to that of the Cl compound. Since there is a large
onsite energy difference between Bi-$p$ and Br/Cl-$p$ orbitals
($\epsilon_{(Bi-p)}-\epsilon_{(Br/Cl-p)}$), the positioning of the CBE
($\sigma_{(p-p)}^{*}$) is less affected by ($\epsilon_{(BrCl-p)}$). Therefore,
the CBE does not see a similar shift as the VBE does, and hence, the Br-based
HDPs generally see a lower bandgap. Further discussions are made in the
supplementary information (SI) (Section XIX).

Figure 4: The B-MOP of HDPs as envisaged from the molecular hybridization of
(Na, K)-{$s$, $p$}–X-$p$ and (In, Bi)-{$s$, $p$}–X-$p$ atomic orbitals
produces the bonding and antibonding orbitals along with the nonbonding X-$p$
orbitals.

##  IV Band gap improvement using HSE06+G0W0 analysis

DFT is considered as a reliable tool for calculating the fundamental
properties of materials in their ground states. However, the common exchange-
correlation approximations tend to underestimate quasiparticle (QP) bandgaps.
To obtain accurate QP energies, Green’s function-based ab-initio many-body
perturbation theory (MBPT) can be employed. In the framework of MBPT, the QP
eigensystem is determined through the solution of one-particle equations,
incorporating a non-local, energy-dependent self-energy operator referred to
as $\Sigma$. In practice, $\Sigma$ is frequently approximated as iGW, where W
denotes the screened Coulomb interaction and G represents the one-particle
Green’s function. Instead of iterative evaluations of $\Sigma$ at each step,
G0W0, a computationally more efficient one-shot approach, is frequently
employed. In the context of modeling halide perovskites, it has become evident
that achieving an accurate representation of the electronic structure hinges
on the calculation of many-body QP energies. This approach effectively
corrects localized electronic states, reducing the mixing of orbitals between
B-eg or B′-s and X-$p$ near the valence band maximum (VBM) and conduction band
minimum (CBM). Consequently, the method enhances accuracy by precisely
pinpointing the positions of the VBM and CBM, ultimately leading to more
reliable bandgap values.

Methodology: In the case of HDPs, G0W0 calculations have been performed on top
of the orbitals obtained from hybrid exchange functional (HSE06@G0W0) with
0.25 mixing in Hartree-Fock exchange which is included in the VASP Kresse and
Furthmüller (1996). In addition to the SOC parameter, we have taken the number
of virtual bands to be almost three times the number of occupied bands. The
HSE06@G0W0 band spectra are obtained using VASP interfaced with Wannier90
software Mostofi  _et al._ (2008).

QP band gap correction: While it is computationally expensive to carry out
HSE06@G0W0 calculations for all the compounds, for demonstration, we have
chosen four compounds, namely Cs2AgBiCl6, Cs2AgInCl6, Cs2InBiCl6, and
Cs2NaInCl6, one each from the categories listed in Table S10. The resulting
band gap values from the HSE06@G0W0 method, considering SOC effect, have been
listed in Table 2, and they exhibit a strong agreement with the experimental
values. The HSE0@G0W0 obtained band structures are shown in Fig. 5. For
materials with indirect semiconducting characteristics like Cs2AgBiCl6,
HSE06@G0W0 yields an exact band gap of 2.65 eV. For the direct bandgap
semiconductors, Cs2AgInCl6, Cs2NaInCl6, and Cs2InBiCl6 the QP direct gap
improves with values 2.86 eV, 4.31 eV and 0.72 eV respectively. Despite this
shift in the band gap value, the overall character of the band spectra closely
resembles that generated by the GGA-mBJ approach, as shown in Fig. 5.

Table 2: Band gap comparison in HDPs calculated using different exchange-correlation functionals.  Structure | Exp. | GGA-mBJ | HSE@G0W0
---|---|---|---
| (eV) | (eV) | (with SOC) (eV)
Cs2AgBiCl6 | 2.65 Dakshinamurthy and Sudakar (2022) | 2.98 | 2.65
Cs2AgInCl6 | 3.02 Dahl  _et al._ (2019) | 3.28 | 2.86
Cs2InBiCl6 | - | 0.53 | 0.72
Cs2NaInCl6 | 4.15 Li  _et al._ (2022b) | 5.37 | 4.31
Figure 5: DFT band structure of Cs2AgBiCl6, Cs2AgInCl6, Cs2InBiCl6, and
Cs2NaInCl6 calculated from HSE06@G0W0+SOC method.

In Figs. 3 and 4, we have constructed the B-MOPs by considering the band
structures obtained from DFT-GGA+mBJ calculations. Similar B-MOPs can be
constructed for the DFT-HSE06@G0W0 calculations by adding a constant energy
shift to the conduction band spectrum. This is due to the fact that the band
dispersion does not change with HSE06@G0W0.

##  V Construction of Tight-Binding Model Hamiltonian

While the B/B′-X interactions primarily build the MOP and hence provide a
broad overview of electronic structure, the next-neighbor B′-B interactions
influence the band dispersion considerably. As we will see later, the optical
properties are driven by these second-neighbor interactions. To provide a
deeper insight to the band dispersion, in the following section, we develop a
tight-binding (TB) model Hamiltonian involving B-B′, B′-B′, and B-B
interactions. The model is constructed based on linear combination of atomic
orbitals (LCAO) and within the framework of Slater-Koster (SK) formalism. In
the second quantization notation, the spin-orbit coupled SK-TB Hamiltonian is
given as:

$H=\sum_{i,\alpha}\epsilon_{i\alpha}c^{\dagger}_{i\alpha}c_{i\alpha}+\sum_{{\braket{ij}};\alpha,\beta}t_{i\alpha
j\beta}(c^{\dagger}_{i\alpha}c_{j\beta}+h.c.)+\lambda\vec{L}\cdot\vec{S},$ (1)

where $i$, $j$ and $\alpha$, $\beta$ are indices for the sites and orbitals,
respectively. The first term represents the onsite energy ($\epsilon$), while
the second term is for the hopping integrals, with $t$ being the hopping
strength. The effective tight-binding Hamiltonian matrix includes second
nearest-neighbor metal-metal (B-B′) interactions, as well as fourth-neighbor
B-B and B′-B′ interactions. The third term in the Hamiltonian represents the
atomic spin-orbit coupling (SOC) effect, with the strength of the SOC
$\lambda$.

Figure 6: The DFT (red) and tight-binding bands (blue) for the HDPs. The
squared momentum matrix elements (P2) corresponding to the valence band edge
(VBE) to conduction band edge (CBE) transition are also shown for each of the
compounds. The parity of the VBE and CBE are mentioned using Koster notations.

A full Hamiltonian involves the interactions among both B and B′ valence
orbitals as well as X-$p$ orbitals. The present model Hamiltonian with minimal
basis is constructed by excluding the X-X interactions while the B-X and B′-X
interactions are mapped into effective B-B′ interactions Kashikar  _et al._
(2021). Such a mapping can be validated through Loẅdin downfold technique
Löwdin (1951); Gupta and Nanda (2022). The choice of the basis set is crucial
and depends on the atomic orbitals contributing to the bands near the Fermi
level. For example, in Ag-based double perovskites, the bands near the Fermi
level are contributed from B(Ag)-{$s$, $d$} and B′-{$s$, $p$} orbitals, while
in Cs2(In, Tl)(Bi, Sb)Cl6, B-{$s$, $p$} and B′-{$s$, $p$} orbitals contribute
to the bands near the Fermi level. The size of the Hamiltonian matrix is
dependent on the chosen basis set. The Hamiltonian matrix can be expressed as

$H=\left[\begin{array}[]{cc}H_{BB}&H_{BB^{\prime}}\\\
H_{BB^{\prime}}^{\dagger}&H_{B^{\prime}B^{\prime}}\\\ \end{array}\right]$ (2)

Here, $H_{BB}$ and $H_{B^{\prime}B^{\prime}}$ are the interaction submatrices
between the same cations $i.e.$, B-B and B′-B′ which correspond to the 4th
neighbor interactions and $H^{BB^{\prime}}$/$H^{B^{\prime}B}$ are interaction
matrices between two different cations B-B′ and B′-B which correspond to the
2nd neighbor interactions (among $d$ and $p$ orbitals for Ag-based HDPs and
$p$ \- $p$ orbitals interactions for Cs2(In/Tl)(Bi/Sb)Cl6) as shown in Fig. 2
(a) and Fig. S1 in SI. By considering the SOC effect, the basis size is
doubled, and sub-matrices take shape:

$H_{BB/B^{\prime}B^{\prime}}=\left[\begin{array}[]{cc}H^{\uparrow\uparrow}&H^{\uparrow\downarrow}\\\
H^{\dagger\downarrow\uparrow}&H^{\downarrow\downarrow}\\\
\end{array}\right],H_{BB^{\prime}/B^{\prime}B}=\left[\begin{array}[]{cc}H^{\uparrow\uparrow}&0\\\
0&H_{\downarrow\downarrow}.\\\ \end{array}\right]$ (3)

Here, $H^{\uparrow\uparrow}_{BB/B^{\prime}B^{\prime}}$ and
$H^{\downarrow\downarrow}_{BB/B^{\prime}B^{\prime}}$ are the Hamiltonian sub-
matrices corresponding to up and down spin components and are connected
through the time-reversal symmetry. The non-vanishing $H^{\uparrow\downarrow}$
and $H^{\downarrow\uparrow}$ elements of $H_{BB/B^{\prime}B^{\prime}}$ are due
to the SOC effect.

The components of the Hamiltonian matrix, describing the interaction between
any two atomic orbitals, say $\alpha$ at site position $\vec{R}_{i}$ and
$\beta$ at site at position $\vec{R}_{j}$, is calculated using SK integrals
($f_{\alpha\beta}$) Slater and Koster (1954)

$\displaystyle h_{\alpha\beta}^{ij}(k)$ $\displaystyle=$ $\displaystyle
f_{\alpha\beta}(t;l,m,n)e^{i\vec{k}\cdot(\vec{R}_{j}-\vec{R}_{i})},$ (4)
$\displaystyle h_{\alpha\beta\sigma\sigma^{\prime}}(k)$ $\displaystyle=$
$\displaystyle\sum_{\braket{j}}h_{\alpha\beta}^{ij}(k)\delta_{\sigma\sigma^{\prime}},$
(5) $\displaystyle H_{\alpha\beta\sigma\sigma^{\prime}}$ $\displaystyle=$
$\displaystyle
h_{\alpha\beta\sigma\sigma^{\prime}}+h_{\alpha\beta\sigma\sigma^{\prime}}^{SOC}$
(6)

The $f_{\alpha\beta}(t;l,m,n)$ depend on the tight-binding hopping parameters
$t$ and direction cosines ($l$, $m$, $n$) connecting the site $j$ to $i$. The
$h^{SOC}_{\alpha\beta\sigma\sigma^{\prime}}$ is driven by the atomistic SOC
$\lambda\bm{L\cdot S}$. The required $h_{\alpha\beta\sigma\sigma^{\prime}}$ as
well as $h_{\alpha\beta\sigma\sigma^{\prime}}^{SOC}$ depend on the basis
chosen, which varies from compound to compound, see Section XIII of SI for
details. As shown in Fig. 6, the bands obtained from the model Hamiltonian are
fitted with that of the DFT, and the resulted parameters are listed in Table
S10, S11, and S12 of Section XIV of SI.

Some of the critical inferences obtained by analyzing the TB parameters are as
follows: (I) The 4th nearest-neighbor (B-B/B′-B′) hopping interactions are
very weak (0 - 50 meV) suggesting that the dispersion is mainly driven through
the 2nd nearest-neighbor B-B′ interactions which are 1-2 order higher in
strength. (II) Only when Ag and Bi/Sb occupy B and B′ sites, the B-$s$ –
B′-$s$ interactions become negligible due to large onsite energy differences.
In the rest of the members, this interaction is significant enough to
influence the dispersion of VBE and CBE (See the B-MOP in Fig. 3). (III) For
HDPs where the ionic Na+ and K+ occupy the B-site, the dispersions are due to
B′-B′ and B-X interactions. (IV) The SOC strength is estimated for both B and
B′ for each of the compounds considered (see Table S10, S11, and S12 in SI).
For Bi and Tl, it is $\sim$0.5 eV, and for other B and B′ elements, it is
$\sim$0.2 eV. Interestingly, these numbers are comparable to the hopping
interactions $t_{\alpha\beta}$. Tight-binding bands for Cs2AgAsCl6,
Cs2CuBiCl6, Cs2AgGaCl6, and Cs2CuInCl6 are provided in the SI, Section XV.

Furthermore, our findings indicate that while the SOC of a compound has a
deterministic effect on its bandgap, it does not influence the parity
eigenvalues of its VBE and CBE.

##  VI Optical Properties Calculation

The optical absorption coefficient $\alpha(\omega)$ of a material is
determined by its frequency ($\omega$) dependent dielectric constant
($\epsilon(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$):

$\alpha(\omega)=\omega\sqrt{\frac{-\epsilon_{1}(\omega)+\sqrt{\epsilon_{1}^{2}(\omega)+\epsilon_{2}^{2}(\omega)}}{2}},$
(7)
$\begin{split}\epsilon_{1}(\omega)=1+\frac{2}{\pi}C\int_{0}^{\inf}\frac{\omega^{\prime}\epsilon_{2}(\omega^{\prime})}{\omega^{\prime
2}-\omega^{2}}d\omega^{\prime},\\\ \\\
\epsilon_{2}(\omega)=\frac{e^{2}\hbar^{2}}{\pi
m_{e}^{2}\omega^{2}}\sum_{v,c}\int_{BZ}d^{3}k|P_{v,c}|^{2}\times\\\
\delta(E_{c}(\vec{k})-E_{v}(\vec{k})-\hbar\omega).\end{split}$

Here, C is the Cauchy principal value of the integral; $e$ and $m_{e}$,
respectively, are charge and mass of an electron. Pv,c in the expression of
$\epsilon_{2}(\omega)$ are the MME corresponding to a transition from valence
band at energy $E_{v}$ to conduction band at energy $E_{c}$. The Dirac-delta
function switches on the MME contribution when a transition occurs from one
state to another. The MME for periodic Block functions $u_{\vec{k}\beta}$
$e^{i\vec{k}\cdot\vec{R}}$ are obtained as follows Lee  _et al._ (2018):

$\bra{\vec{k},\beta}\vec{P}\ket{\vec{k},\beta^{\prime}}=\frac{m_{e}}{\hbar}\bra{u_{\vec{k}\beta}}\frac{\partial
H(\vec{k})}{\partial\vec{k}}\ket{u_{\vec{k}\beta^{\prime}}}.$ (8)

In the case of optical transition, the transitions from the top valence to the
bottom conduction band are generally considered. Therefore, the component of
relevant MME can be expressed as follows,

$(P_{v,c})_{x,y,z}=\frac{m_{e}}{\hbar}\sum_{\beta,\beta^{\prime}}u_{\vec{k}\beta,c}\frac{\partial
H_{\beta\beta^{\prime}}}{\partial k_{x,y,z}}u_{\vec{k}\beta^{\prime},v}.$ (9)

Here, $u_{\vec{k}\beta,c}$ and $u_{\vec{k}\beta^{\prime},v}$ respectively
represent the eigenvectors associated with the energy eigenvalues $E_{v}$ and
$E_{c}$. We have calculated the squared MME along the same high symmetric
$k$-path as that of the band structure. A detailed derivation to calculate the
optical properties of materials using the SK-TB model is given in Section XVI
of the SI.

In Fig. 6, we have shown the band structures for a series of HDPs, and for
each of them, the parity eigenvalues (indicated through Koster notations) of
VBE and CBE are estimated at the high symmetry points, as shown in the Figure.
Corresponding to each of the band structures, the P2 (VBE $\rightarrow$ CBE)
is also plotted. According to Laporte’s rule Laporte and Meggers (1925), which
is applied only for the centrosymmetric systems as is the case here, the
optical transition between the VBE and CBE at any given k-point is allowed
only when they have opposite parities (odd-even). The validation of it comes
through the P2 plot. oing beyond the simplistic parity analysis, a thorough
group theoretical symmetry analysis is also carried out in Sections XI and XII
in the SI to elaborate the selection rules governing the optical transition.

For example, in the case of Cs2AgBiCl6 and Cs2AgSbCl6, the transition between
VBE and CBE at the high symmetry points W, $\Gamma$, and X is allowed due to
opposite parities while it is forbidden at L due to same parity (odd-odd).
Agreeing with it, the P2 value vanishes at L and is finite elsewhere. In the
case of Cs2AgInCl6 and Cs2AgTlCl6, the P2 is zero along the path $\Gamma$-X as
both the CBE and VBE have even parity along this path. Therefore, even though
these two systems have a direct bandgap, the lowest energy transitions are not
allowed, and hence the optical bandgap (defined through transition at L)
differs from the electronic bandgap. The compounds Cs2(In/Tl)(Bi/Sb)Cl6 are
direct bandgap systems with VBM and CBM lying at $\Gamma$. The finite value of
P2 implies the lowest energy allowed transition, which is further verified by
the parity analysis. Hence, these compounds are much more promising for
optoelectronic applications. In the case of Cs2(Na/K)BiCl6, where the VBE and
CBE are formed by Bi-$s$ and Bi-$p$ respectively (see Fig. 4), while the
opposite parity allows the optical transition between VBE and CBE, the P2 is
found to be negligible (of the order $\sim$ 10-2). On the other hand, the same
parities for VBE and CBE of Cs2(Na/K)InCl6 forbid the optical transition, and
expectedly, P2 is found to be zero. We may note that, since these systems are
ionic with large charge transfer from Na/K to Cl, the transition dipole moment
for this VBE to CBE transition is naturally weak Locardi  _et al._ (2019).

The optical transition analysis through Fig. 6 is based on a given $k$-path
(W-L-$\Gamma$-X-L), which does not necessarily provide the complete picture to
understand the transition due to the whole BZ, we have calculated the joint
densities of states (JDOS). The JDOS provides a measure of the number of all
possible optical transitions between the occupied valence band and the
unoccupied conduction band separated by photon energy $\hbar\omega$.

$JDOS=\frac{e^{2}\hbar^{2}}{\pi
m_{e}^{2}\omega^{2}}\sum_{v,c}\int_{BZ}d^{3}k\hskip
2.84544pt\delta(E_{c}(\vec{k})-E_{v}(\vec{k})-\hbar\omega).$ (10)

The JDOS, obtained for the lowest electronic transition, is plotted in the
upper panel of Fig. 7. Also, $\epsilon_{2}(\omega)$, which can be best
described as the MME modulated JDOS, is shown in the lower panel of Fig. 7.

As discussed in the above paragraph, the optical transition in Na/K based
compound is negligible (see Fig. 7 (f)) even though the JDOS (Fig. 7 (c))
shows some transition probabilities between 4 to 6.5 eV. This is due to the
fact that in these ionic systems, the dipole-dipole transition is very weak.
The $\epsilon_{2}(\omega)$ gives a measure of optical bandgap and how it
differs from electronic bandgap (as inferred from the JDOS alone). For the
case of Cs2InBiCl6, Cs2InSbCl6, Cs2TlBiCl6, and Cs2TlSbCl6, peaks of JDOS and
$\epsilon_{2}$ suggest the direct and strong optical transitions (see Fig. 7
(b and e)). In Cs2AgTlCl6, the first small peak in JDOS below 1 eV is
suppressed in $\epsilon_{2}(\omega)$, implying the optical bandgap is $\sim$ 1
eV (see Fig. 7 (a, d)). Because of a similar reason, in the case of Cs2AgInCl6
the optical bandgap is estimated to be $\sim$ 3.6 eV. Cs2AgBiCl6 and
Cs2AgSbCl6 have large JDOS values however their $\epsilon_{2}$ curves show
rather different features. The optical transition for Cs2AgSbCl6 is quite weak
compared to Cs2AgBiCl6.

Figure 7: (a-c) Model Hamiltonian obtained joint densities of states (JDOS) of
HDPs. The JDOS is calculated for the transition between VBE and CBE. (d-f)
Calculated $\epsilon_{2}(\omega)$ showing the effect of parity forbidden
transition and effective optical bandgap of these compounds.

##  VII Tailoring of the optoelectronic properties: Doping on cationic sites

Figure 8: (Upper panel) DFT obtained orbital projected band structures of
Cs2AgInxBi1-xCl6. The calculations are performed with a structurally relaxed
four-formula unit supercell. (Lower panel) The P2 plot, which is obtained from
the TB model Hamiltonian designed for the supercell (see SI). Figure 9: Same
as Figure 8 but for the compound Cs2Na1-xAgxInCl6.

With a good understanding of the electronic structure and optical behavior of
the pristine HDPs, in this section, we will show how the optoelectronic
properties can be tailored by cation intermixing. For demonstration, we have
considered two cases (a) Cs2AgInxBi1-xCl6 and (b) Cs2Na1-xAgxInCl6. A detailed
description of the TB model for cation-intermixed HDPs supercells and
corresponding TB bands fitted DFT bands are provided in Sections XVII, XVIII,
and Fig. S4 of the SI, respectively. In the first case, the end member
Cs2AgBiCl6 has an indirect bandgap (unfavorable for optical transition), and
Cs2AgInCl6 has a direct bandgap, however, with parity forbidden transitions.
Furthermore, the CBE and VBE of the former are made up of Bi-$p$ and
Ag-$e_{g}$/Bi-$s$ orbitals, respectively, while for the latter, these are made
up of In-$s$ and Ag-$e_{g}$ orbitals, respectively. A careful look at the
B-MOP suggests that with Bi and In intermixing, the CBE will be dominated by
Bi-$p$ while VBE will be by Ag-eg states as well as dopant Bi-$s$ states. Upon
dilution, the Bi-states are expected to be localized and alter the shape of
the corresponding band. Thereby, the optical absorption spectra are also
expected to change. To verify, in Fig. 8, we have plotted the DFT band
structure of Cs2AgInxBi1-xCl6 ($x$ = 0.25, 0.5, and 0.75) and P2, which are
estimated from the model Hamiltonian. The salient features of the band
structures are as follows: (I) With increasing In, the Bi-$s$ state dominates
the VBE while the Ag-eg dominated bands are pushed below. (II) The CBE is
always formed by the Bi-$p$ characters. However, its shape changes with doping
concentration, which implies new interactions, Bi-$p$ and In-$s$ states. (III)
For the case of $x$ = 0.75, both VBE and CBE are narrower across the BZ path
and also have a direct bandgap with the VBM and CBM lying at Z. Interestingly,
at every high symmetry point (except at R), VBE and CBE have opposite parities
to allow optical transitions across the path. The P2 plot further
substantiates it.

Similarly, in the second case, we are doping Ag at the Na site of Cs2NaInCl6.
As already been discussed, the Cs2NaInCl6 does not exhibit any optical
transition as the P2 vanishes, thanks to the identical parity of CBE and VBE
(see Fig. 6). Fig. 9 shows the orbital projected band structure of
Cs2Na1-xAgxCl6 ( $x$ = 0.25, 0.5, and 0.75) and the corresponding P2 plot. The
salient features of the band structures are as follows: (I) The Ag-$d$
characters tend to dominate the VBE, and for diluted Ag concentration, the VBE
becomes flatter like the impurity bands. (II) The shape of CBE remains largely
unchanged though it becomes wider with increasing Ag concentration. (III) The
parities of CBE and VBE have altered at certain $k$ points, and therefore, P2
is no longer vanishing. Recent studies by Luo $et.al.$ Luo  _et al._ (2018),
report the existence of self-trapped excitons (STEs) and, thereby, broadband
and white light emission in the Cs2Na1-xAgxCl6. The present study suggests
that the STEs are formed by the flat bands, and the localized carrier at the
Ag-site becomes the source of the white light emission.

Figure 10: JDOS (shown in red) and imaginary part of the dielectric constant
(shown in blue) for cation intermixed Cs2AgInxBi1-xCl6 (upper panel) and
Cs2Na1-xAgxInCl6 (lower panel).  Figure 11: (a) The schematic illustration of
the orbital resolved band structure in the vicinity of Fermi level and
demonstration of bandgap bowing in Cs2AgInxBi1-xCl6 and Cs2Na1-xAgxInCl6. (b)
The bandgap (Eg) as a function of doping concentration $x$ in Cs2AgInxBi1-xCl6
(left) and Cs2Na1-xAgxInCl6 (right). The DFT obtained bandgap values are shown
in black circular dots, and the polynomial fitted curves are shown in blue.
The variation of the lattice constant ($a$) as a function of $x$ is also
shown.

The above analysis on cation intermixed HDPs is made by examining the
eigenspectrum along certain k-paths. To substantiate the conclusions made, we
now consider the full BZ and estimate JDOS and $\epsilon_{2}$ for
Cs2AgIn1-xBixCl6 and Cs2Na1-xAgxInCl6 in Fig. 10. The red curve represents the
JDOS for the transition from VBE to CBE, and the blue curve represents
$\epsilon_{2}$. In the former case, we indeed observed increasing JDOS and
$\epsilon_{2}$ with increasing In concentration. as shown in Fig. 10 (a-c).
The direct and strong optical transition makes the system highly
photoluminescent, which agrees with the recent experimental studies by
Appadurai $et.al.$ Appadurai  _et al._ (2021). In the case of
Cs2Na1-xAgxInCl6, any amount of Ag doping induces optical transition.

##  VIII Bandgap bowing effect

The band positioning due to cation doping, as discussed in the previous
section, can be schematically summarized through Figs. 11 (a, c). It shows
that with doping, not only there is a reconstitution of VBE and CBE, but also
there is a shift of these edge bands. Either the VBE goes up, or CBE comes
down, or both happen simultaneously to reduce the bandgap with respect to the
parent pristine compounds. This effect, in general, is called bandgap bowing
which occurs less often than the linear change in the bandgap as defined by
Vegard’s law. To quantify the bandgap bowing, in Fig. 11 (c), we have
estimated the bandgap as a function of doping concentration in
Cs2AgInxBi1-xCl6 and Cs2Na1-xAgxInCl6. The cause of bandgap bowing in the HDPs
is a matter of debate in experimental and density functional studies
Dakshinamurthy and Sudakar (2022); C. Dakshinamurthy and Sudakar (2021); Manna
_et al._ (2019); Li  _et al._ (2020); Han  _et al._ (2020); Li  _et al._
(2020); Goyal  _et al._ (2018). These studies collectively proposed three
probable factors for bandgap bowing: These are (I) change in lattice constant,
(II) octahedral distortion, and (III) chemical effect. Through Figs. 8, 9 and
11 (a, b), and related discussion, we have already discussed how the chemical
effect plays a role in band repositioning. Our B-MOP indeed has shown that the
free atomic orbital energies play a major role in determining the position of
VBE and CBE. A similar observation was made by D. Han $et.al.$ Han  _et al._
(2020).

Figure 12: (a) Eg for Cs2AgInxBi1-xCl6 as a function of $x$, which are
estimated by using the lattice parameter of Cs2AgInCl6 (10.56 Å; solid cyan
line) and Cs2AgBiCl6 (10.936 Å; solid magenta line) and keeping the structures
unrelaxed. (b) Eg as a function of $x$ for the volume optimized
Cs2AgInxBi1-xCl6, and with and without atomic position relaxation. The
relaxation distorts the octahedra.

To understand the role of the lattice on the bandgap bowing, we have carried
out three hypothetical experiments on Cs2AgInxBi1-xCl6. (I) Across the
concentration range, the lattice parameter of the doped system is taken as
that of Cs2AgInCl6 (10.560 Å), and band structure is calculated without
relaxing the system. The resulting bandgap as a function of concentration $x$
is shown in Fig. 12 (a) (cyan solid line). This is a case of increasing
compression with decreasing $x$. We find that, in this case, there is minimal
variation in the bandgap. (II) The bandgap as a function of $x$ is now
calculated by taking the lattice parameter as that of Cs2AgBiCl6 (10.936 Å).
This is a case of increasing expansion with increasing $x$. The bandgap falls
sharply as $x$ reaches 0.5 and then remains almost unchanged. Together these
two experiments imply that (see Fig. 11 (a)), (i) the presence of In-$s$
orbital pushes down the Bi-$p$ states in the Bi-rich system, and (ii) in the
In-rich system, the expansion significantly reduces the bandgap. (iii) In the
third experiment, we carried out volume optimization and calculated the
bandgap with and without structural relaxation. The relaxation primarily
distorts the octahedra in this system. The results are plotted in Fig. 12 (b).
We find the bowing is larger when the octahedra symmetry is maintained, while
it reduces when the octahedra are distorted.

For further analyzing the role of octahedral distortion on the bandgap, we
have calculated the orbital resolved DFT band structures of Cs2AgIn0.5Bi0.5Cl6
with and without octahedra distortion. The distortion of the octahedra
includes the compression of InCl6 and expansion of AgCl6, as shown in Fig. 13.
The compression strengthens the hybridization between the In-{$s$, $p$} and
Cl-$p$ orbitals, and therefore corresponding antibonding states
($\sigma_{s-p}^{*}$, $\sigma_{p-p}^{*}$, and $\pi_{p-p}^{*}$ ) go higher in
energy. Since the BiCl6 octahedra remain largely unaffected, the position of
Bi-$p$ dominated antibonding states (blue curves) is less perturbed. This
leads to a swap in the CBE character and an increase in the bandgap from 1.67
eV to 2.32 eV with distortion (as can be seen in Fig. 13), and as a
consequence, the bandgap bowing is weakened, which is in agreement with Fig.
12 (b).

Figure 13: Volume optimized Cs2AgIn0.5Bi0.5Cl6 crystal structure (a) without
and (b) with the relaxation of the atomic position. The relaxation distorts
the octahedra. The orbital resolved band structures of (a) and (b) are shown
in (c) and (d), respectively.

Our overall analysis implies that the chemical effect and lattice expansion
tend to increase the bandgap bowing while octahedral distortion and lattice
compression reduce it. In the case of Cs2Na1-xAgxInCl6 the bandgap bowing is
primarily driven by the chemical effect as the lattice parameter variation is
minimal. Here, the reduction of the bandgap is driven by the appearance of the
Ag-$d$ states as much above the Cl-$p$ states as shown schematically in Fig.
11 (b).

##  IX Conclusion and outlook

In conclusion, we carried out a comprehensive electronic structure study by
employing density functional calculations, model Hamiltonian formulation, and
optoelectronic study by estimating the momentum matrix elements (MME). From
our results and analysis, we developed a theoretical workflow to study the
electronic and optoelectronic properties of halide double perovskites
(pristine and doped) for photovoltaic applications. In this work, we devise
band projected molecular orbital picture (B-MOP) as an efficient tool to
analyze the electronic structure of covalent hybridized systems in general and
halide double perovskites (HDPs) in particular. Based on our understanding of
the electronic structure, we could successfully categorize the HDPs into five
categories which are based on the valence electron configuration of B and B′,
characters of the valence and conduction band edge states, and bandgap as well
as optical transition behavior. The list is summarized in Table S10.

Table 3: Summary of different categories (valence electron configurations) in HDPs and their electronic and optical behaviors.  Category | B |  B′ | VB | CB | Nature
---|---|---|---|---|---
$d^{10}s^{0}$(B) | Group 11 | Group 15 |  Ag-eg |  Bi/Sb-$p$ | indirect,
\- $d^{10}s^{2}$(B′) | (eg. Ag) | (eg. Bi, Sb) |  |  | allowed
$d^{10}s^{0}$ | Group 11 | Group 13 |  Ag-eg |  In/Tl-$s$ | direct,
\- $d^{10}s^{0}$ | (eg. Ag) | (eg. In, Tl) |  |  | forbidden
$d^{10}s^{2}$ | Group 13 | Group 15 |  In/Tl-$s$ | Bi/Sb-p | direct,
\- $d^{10}s^{2}$ | (eg. In, Tl) | (eg. Bi, Sb) |  |  | allowed
$s^{0}$ | Group 1 | Group 13 |  X-$p$ |  In/Tl-$s$ | direct,
\- $d^{10}s^{0}$ | (eg. Na, K) | (eg. In, Tl) |  |  | forbidden
$s^{0}$ \- $s^{2}$ | Group 1 | Group 15 |  Bi/Sb-$s$ |  Bi/Sb-$p$ | indirect,
| (eg. Na, K) | (eg. Bi, Sb) |  |  | allowed
|  |  |  |  | (weak)

The B-MOP obtained from nearest-neighbor cation-anion interactions determines
the position and orbital character of the bands. The tight-binding model,
which is based on second-neighbor cation-cation interactions, provides insight
to determine shape and width of band dispersion. Our study suggests that
second-neighbor cation-cation interactions turn out to be deterministic
factors for MME and, thereby, optoelectronic properties of HDPs.

Through our design principle, we show the possibilities of tuning the bandgap
and optical absorption by doping. To demonstrate, we took two prototype
examples of doping at the cationic site: Cs2AgInxBi1-xCl6 and Cs2Na1-xAgxInCl6
for $x$ = 0.25, 0.5, and 0.75. We obtain the maximum transition dipole moment
at $x$ = 0.75 for Cs2AgInxBi1-xCl6 and at $x$ = 0.5 for Cs2Na1-xAgxInCl6 which
are found to be in good agreement with the previous experimental finding. Our
analysis also provides an interesting insight into the bandgap gap bowing,
which seems to be a common occurrence in HDPs. We show that the chemical
effect plays an enhancement factor in bandgap bowing, while the octahedra
distortion tends to minimize it.

The findings of the present study provide guiding principles to design
efficient optoelectronic materials out of HDPs. We hope that these findings
will stimulate theoretical and experimental research on alloying of HDPs to
realize their applications in the area of photovoltaic and optoelectronics.
The tight-binding model developed in this work (for pristine and cation
intermixed) is generic and can be employed to study the electronic structure
and optical behavior of 3D HDPs and their 2D and 1D counterparts Ji  _et al._
(2021), including vacancy-induced perovskites.

##  X Acknowledgment

This work is funded by the Department of Science and Technology (DST), India,
through the Department of Science and Technology (DST) mission innovation for
funding through grant number DST/TMD/IC-MAP/2K20/03 (C). We acknowledge HPCE
IIT Madras for providing the computational facility.

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Supplementary Information for “Electronic structure and optoelectronic
properties of halide double perovskites: Fundamental insights and design of a
theoretical workflow”

##  XI Symmetry analysis

As mentioned in the manuscript, all of the halide double perovskites (HDPs)
adopt a cubic crystal structure featuring double space group symmetry of
Fm$\bar{3}$m (no. 225). Within this symmetry group, optical transitions occur
along the high-symmetry points of the Brillouin zone (BZ), namely $\Gamma$, X,
L, and W (see Fig. 2(b) in the main manuscript). These high-symmetry points
correspond to specific point groups such as $O_{h}$, $D_{3d}$, $D_{2d}$,
$D_{4h}$ respectively. The character tables for these point groups are
provided in the following tables (Tables S1 to S4). Analyzing these tables
will facilitate our discussion of allowed and forbidden optical transitions
from a group theory perspective.

TABLE S1: Character table for the double point group D2d at symmetry point W ((0.25, 0.5, 0.75): w.r.t. primitive vector; (0.0, 0.5, 1.0): w.r.t. conventional vectors). $D_{2d}$ | E | $C_{2}$ | $2C^{\prime}_{2}$ | $2S_{4}$ | $2\sigma_{d}$ | Functions
---|---|---|---|---|---|---
$W_{1}$ | 1 | 1 | 1 | 1 | 1 | $x^{2}+y^{2},z^{2},xyz$
$W_{2}$ | 1 | 1 | -1 | 1 | -1 | $R_{z},z(x^{2}-y^{2})$
$W_{3}$ | 1 | 1 | 1 | -1 | -1 | $x^{2}-y^{2}$
$W_{4}$ | 1 | 1 | -1 | -1 | 1 | $z,xy,(x^{2}+y^{2})z,z^{3}$
$W_{5}$ | 2 | -2 | 0 | 0 | 0 |  | $(x,y),(R_{x},R_{y})$
---
$(zx,yz),{x(x^{2}+y^{2}),y(x^{2}+y^{2})},(xz^{2},yz^{2})$,
${x(x^{2}-3y^{2}),y(3x^{2}-y^{2})}$
$W_{6}$ | 2 | 0 | 0 | $\sqrt{2}$ | 0 |
$W_{7}$ | 2 | 0 | 0 | -$\sqrt{2}$ | 0 |
TABLE S2: Character table for the double point group D3d at symmetry point L ((0.0, 0.0, 0.0): w.r.t primitive vector; (0.5, 0.5, 0.5): w.r.t. conventional vectors.) $D_{3d}$ | E | $2C_{3}$ | $3C^{\prime}_{2}$ | $i$ | $2S_{6}$ | 3$\sigma_{d}$ | Functions
---|---|---|---|---|---|---|---
$L_{1}^{+}$ | 1 | 1 | 1 | 1 | 1 | 1 | $x^{2}+y^{2},z^{2}$
$L_{2}^{+}$ | 1 | 1 | -1 | 1 | 1 | -1 | $R_{z}$
$L_{3}^{+}$ | 2 | -1 | 0 | 2 | -1 | 0 | $(R_{x},R_{y})(xy,x^{2}-y^{2}),(zx,yz)$
$L_{1}^{-}$ | 1 | 1 | 1 | -1 | -1 | -1 | $x(x^{2}-3y^{2})$
$L_{2}^{-}$ | 1 | 1 | -1 | -1 | -1 | 1 | $z$
$L_{3}^{-}$ | 2 | -1 | 0 | -2 | 1 | 0 | $(x,y)$
$L_{4}^{+}$ | 2 | 1 | 0 | 2 | 1 | 0 |
$L_{5}^{+}$ | 1 | -1 | $i$ | 1 | -1 | -$i$ |
$L_{6}^{+}$ | 1 | -1 | -$i$ | 1 | -1 | -$i$ |
$L_{4}^{-}$ | 2 | 1 | 0 | -2 | 1 | 0 |
$L_{5}^{-}$ | 1 | -1 | $i$ | -1 | 1 | -$i$ |
$L_{6}^{-}$ | 1 | -1 | -$i$ | -1 | 1 | $i$ |
TABLE S3: Character table for the double point Group Oh at point $\Gamma$ ((0.0, 0.0, 0.0): w.r.t primitive and conventional vectors.) $O_{h}$ | $E$ | $3C_{2}$ | $8C_{3}$ | $6C_{4}$ | $6C_{2}^{\prime}$ | $i$ | $3\sigma$ | $8S_{6}$ | $6S_{4}$ | $6\sigma_{d}$ | Functions
---|---|---|---|---|---|---|---|---|---|---|---
$\Gamma_{1}^{+}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $x^{2}+y^{2}+z^{2}$
$\Gamma_{2}^{+}$ | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 |
$\Gamma_{3}^{+}$ | 2 | 2 | -1 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | ($x^{2}-y^{2},2z^{2}-x^{2}-y^{2}$)
$\Gamma_{4}^{+}$ | 3 | -1 | 0 | 1 | -1 | 3 | -1 | 0 | 1 | -1 | ($R_{x},R_{y},R_{z}$)
$\Gamma_{5}^{+}$ | 3 | -1 | 0 | -1 | 1 | 3 | -1 | 0 | -1 | 1 | $(xy,yz,zx)$
$\Gamma_{1}^{-}$ | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 |
$\Gamma_{2}^{-}$ | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | $xyz$
$\Gamma_{3}^{-}$ | 2 | 2 | -1 | 0 | 0 | -2 | -2 | 1 | 0 | 0 |
$\Gamma_{4}^{-}$ | 3 | -1 | 0 | 1 | -1 | -3 | 1 | 0 | -1 | 1 |  | $(x,y,z)$
---
$(x^{3},y^{3},z^{3})$,
${x(y^{2}+z^{2}),y(z^{2}+x^{2}),z(x^{2}+y^{2})}$
$\Gamma_{5}^{-}$ | 3 | -1 | 0 | -1 | 1 | -3 | 1 | 0 | 1 | -1 | ${x(z^{2}-y^{2}),y(x^{2}-z^{2}),z(y^{2}-x^{2})}$
$\Gamma_{6}^{+}$ | 2 | 0 | 1 | $\sqrt{2}$ | 0 | 2 | 0 | 1 | $\sqrt{2}$ | 0 |
$\Gamma_{7}^{+}$ | 2 | 0 | 1 | -$\sqrt{2}$ | 0 | 2 | 0 | 1 | -$\sqrt{2}$ | 0 |
$\Gamma_{8}^{+}$ | 4 | 0 | -1 | 0 | 0 | 4 | 0 | -1 | 0 | 0 |
$\Gamma_{6}^{-}$ | 2 | 0 | 1 | $\sqrt{2}$ | 0 | -2 | 0 | -1 | -$\sqrt{2}$ | 0 |
$\Gamma_{7}^{-}$ | 2 | 0 | 1 | -$\sqrt{2}$ | 0 | -2 | 0 | -1 | $\sqrt{2}$ | 0 |
$\Gamma_{8}^{-}$ | 4 | 0 | -1 | 0 | 0 | -4 | 0 | 1 | 0 | 0 |
TABLE S4: Character Table for the double point group $D_{4h}$ at point X ((0.0, 0.5, 5.0): w.r.t primitive vector; (0.0, 0.0, 1.0): w.r.t. conventional vectors.) $D_{4h}$ | $E$ | $2C_{4}$ | $C_{2}$ | $2C_{2}^{\prime}$ | $2C_{2}^{\prime\prime}$ | $i$ | $2S_{4}$ | $\sigma_{h}$ | 2$\sigma_{v}$ | $2\sigma_{d}$ | Functions
---|---|---|---|---|---|---|---|---|---|---|---
$X_{1}^{+}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $x^{2}+y^{2},z^{2}$
$X_{2}^{+}$ | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | $R_{z}$
$X_{3}^{+}$ | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | $x^{2}-y^{2}$
$X_{4}^{+}$ | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | $xy$
$X_{5}^{+}$ | 2 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | 0 | 0 | $(R_{x},R_{y}),(zx,yz)$
$X_{1}^{-}$ | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 |
$X_{2}^{-}$ | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | $z,(x^{2}+y^{2})z,z^{3}$
$X_{3}^{-}$ | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | $xyz$
$X_{4}^{-}$ | 3 | -1 | 0 | 1 | -1 | -3 | 1 | 0 | -1 | 1 | $z(x^{2}-y^{2})$
$X_{5}^{-}$ | 2 | 0 | -2 | 0 | 0 | -2 | 0 | 2 | 0 | 0 |  | $(x,y)$
---
${x(x^{2}+y^{2}),y(x^{2}+y^{2})},(xz^{2},yz^{2}),$
${x(x^{2}-3y^{2}),y(3x^{2}-y^{2})}$
$X_{6}^{+}$ | 2 | $\sqrt{2}$ | 0 | 0 | 0 | 2 | $\sqrt{2}$ | 0 | 0 | 0 |
$X_{7}^{+}$ | 2 | -$\sqrt{2}$ | 0 | 0 | 0 | 2 | $\sqrt{2}$ | 0 | 0 | 0 |
$X_{6}^{-}$ | 2 | $\sqrt{2}$ | 0 | 0 | 0 | -2 | -$\sqrt{2}$ | 0 | 0 | 0 |
$X_{7}^{-}$ | 2 | -$\sqrt{2}$ | 0 | 0 | 0 | -2 | $\sqrt{2}$ | 0 | 0 | 0 |

###  XI.1 Band-diagram analysis

The irreducible representations (IR) at the band edges can be explained via
group theoretical analysis. As an example, for Cs2AgBiCl6 the conduction band
and the valence band primarily arise from the contributions from Ag$-e_{g}$,
Bi$-p,s$ and Cl$-p$ states. Consequently, these bands can be expressed as a
linear combination of $|j,m\rangle$ states like -
$|\frac{1}{2},\frac{1}{2}\rangle$, $|\frac{3}{2},\frac{1}{2}\rangle$,
$|\frac{5}{2},\frac{3}{2}\rangle$. By closely analyzing the band structure,
for the symmetry point W and the corresponding D2d point group, we can deduce
that only $W_{6}$ and $W_{7}$ represent these kind of orbital combinations
such as -
$\phi(|\frac{1}{2},\frac{1}{2}\rangle,|\frac{1}{2},\bar{\frac{1}{2}}\rangle)$;
$\phi(|\frac{3}{2},\frac{1}{2}\rangle,-|\frac{3}{2},\bar{\frac{3}{2}}\rangle)$;
$\phi(|\frac{3}{2},\bar{\frac{3}{2}}\rangle,-|\frac{3}{2},\frac{3}{2}\rangle)$;
$\phi(|\frac{5}{2},\bar{\frac{3}{2}}\rangle$,
$|\frac{5}{2},\frac{3}{2}\rangle)$ and many more. Likewise, the IRs at other
symmetry points L ($L_{5}^{-}$, $L_{4}^{-}$), $\Gamma$ ($\Gamma_{8}^{+}$,
$\Gamma_{6}^{-}$), X ($X_{6}^{+}$, $X_{6}^{-}$) also represent similar orbital
combinations which are expected from our calculations. Therefore, the IRs of
the band diagrams obtained from DFT agree well with our group theoretical
analysis.

##  XII Explanation of Optical Transitions from Group Theory

Here, we are going to discuss the optical transitions primarily driven by the
direct products of IRs as well as the parity of these representations. For a
comprehensive discussion of these transitions, we have written the direct
product tables (Table S5 \- S8) for the aforementioned point groups at their
corresponding high symmetry points.

TABLE S5: Direct product table for group $D_{2d}$. $D_{2d}$ | $W_{1}$ | $W_{2}$ | $W_{3}$ | $W_{4}$ | $W_{5}$ | $W_{6}$ | $W_{7}$
---|---|---|---|---|---|---|---
$W_{1}$ | $W_{1}$ | $W_{2}$ | $W_{3}$ | $W_{4}$ | $W_{5}$ | $W_{6}$ | $W_{7}$
$W_{2}$ |  | $W_{2}$ | $W_{3}$ | $W_{4}$ | $W_{5}$ | $W_{6}$ | $W_{7}$
$W_{3}$ |  |  | $W_{1}$ | $W_{2}$ | $W_{5}$ | $W_{7}$ | $W_{6}$
$W_{4}$ |  |  |  | $W_{1}$ | $W_{5}$ | $W_{7}$ | $W_{6}$
$W_{5}$ |  |  |  |  |  $W_{1}$+$W_{2}$+$W_{3}$+$W_{4}$ |  $W_{6}$+$W_{7}$ |  $W_{6}$+$W_{7}$
$W_{6}$ |  |  |  |  |  |  $W_{1}$+$W_{2}$+$W_{5}$ |  $W_{3}$+$W_{4}$+$W_{5}$
$W_{7}$ |  |  |  |  |  |  |  $W_{1}$+$W_{2}$+$W_{5}$
TABLE S6: Direct product table for group $D_{3d}$. $D_{3d}$ | $L_{4}^{+}$ | $L_{5}^{+}$ | $L_{6}^{+}$ | $L_{4}^{-}$ | $L_{5}^{-}$ | $L_{6}^{-}$
---|---|---|---|---|---|---
$L_{1}^{+}$ | $L_{4}^{+}$ | $L_{5}^{+}$ | $L_{6}^{+}$ | $L_{4}^{-}$ | $L_{5}^{-}$ | $L_{6}^{-}$
$L_{2}^{+}$ | $L_{4}^{+}$ | $L_{6}^{+}$ | $L_{5}^{+}$ | $L_{4}^{-}$ | $L_{6}^{-}$ | $L_{5}^{-}$
$L_{3}^{+}$ |  $L_{4}^{+}$+$L_{5}^{+}$+$L_{6}^{+}$ | $L_{4}^{+}$ | $L_{4}^{+}$ |  $L_{4}^{-}$+$L_{5}^{-}$+$L_{6}^{-}$ | $L_{4}^{-}$ | $L_{4}^{-}$
$L_{1}^{-}$ | $L_{4}^{-}$ | $L_{5}^{-}$ | $L_{6}^{-}$ | $L_{4}^{+}$ | $L_{5}^{+}$ | $L_{6}^{+}$
$L_{2}^{-}$ | $L_{4}^{-}$ | $L_{6}^{-}$ | $L_{5}^{-}$ | $L_{4}^{+}$ | $L_{6}^{+}$ | $L_{5}^{+}$
$L_{3}^{-}$ |  $L_{4}^{-}$+$L_{5}^{-}$+$L_{6}^{-}$ | $L_{4}^{-}$ | $L_{4}^{-}$ |  $L_{4}^{+}$+$L_{5}^{+}$+$L_{6}^{+}$ | $L_{4}^{+}$ | $L_{4}^{+}$
$L_{5}^{+}$ |  $L_{1}^{+}$+$L_{2}^{+}$+$L_{3}^{+}$ | $L_{3}^{+}$ | $L_{3}^{+}$ |  $L_{1}^{-}$+$L_{2}^{-}$+$L_{3}^{-}$ | $L_{3}^{-}$ | $L_{3}^{-}$
$L_{6}^{+}$ |  | $L_{2}^{+}$ | $L_{1}^{+}$ | $L_{3}^{-}$ | $L_{2}^{-}$ | $L_{1}^{-}$
$L_{7}^{+}$ |  |  | $L_{2}^{+}$ | $L_{3}^{-}$ | $L_{1}^{-}$ | $L_{2}^{-}$
$L_{5}^{-}$ |  |  |  |  $L_{1}^{+}$+$L_{2}^{+}$+$L_{3}^{+}$ | $L_{3}^{+}$ | $L_{3}^{+}$
$L_{6}^{-}$ |  |  |  |  | $L_{2}^{+}$ | $L_{1}^{+}$
$L_{7}^{-}$ |  |  |  |  |  | $L_{2}^{+}$
TABLE S7: Direct product for group $D_{4h}$ $D_{4h}$ | $X_{6}^{+}$ | $X_{7}^{+}$ | $X_{6}^{-}$ | $X_{7}^{-}$
---|---|---|---|---
$X_{1}^{+}$ | $X_{6}^{+}$ | $X_{7}^{+}$ | $X_{6}^{-}$ | $X_{7}^{-}$
$X_{2}^{+}$ | $X_{6}^{+}$ | $X_{7}^{+}$ | $X_{6}^{-}$ | $X_{7}^{-}$
$X_{3}^{+}$ | $X_{7}^{+}$ | $X_{6}^{+}$ | $X_{7}^{-}$ | $X_{6}^{-}$
$X_{4}^{+}$ | $X_{7}^{+}$ | $X_{6}^{+}$ | $X_{7}^{-}$ | $X_{6}^{-}$
$X_{5}^{+}$ |  $X_{6}^{+}$+$X_{7}^{+}$ |  $X_{6}^{+}$+$X_{7}^{+}$ |  $X_{6}^{-}$+$X_{7}^{-}$ |  $X_{6}^{-}$+$X_{7}^{-}$
$X_{1}^{-}$ | $X_{6}^{-}$ | $X_{7}^{-}$ | $X_{6}^{+}$ | $X_{7}^{+}$
$X_{2}^{-}$ | $X_{6}^{-}$ | $X_{7}^{-}$ | $X_{6}^{+}$ | $X_{7}^{+}$
$X_{3}^{-}$ | $X_{7}^{-}$ | $X_{6}^{-}$ | $X_{7}^{+}$ | $X_{6}^{+}$
$X_{4}^{-}$ | $X_{7}^{-}$ | $X_{6}^{-}$ | $X_{7}^{+}$ | $X_{6}^{+}$
$X_{5}^{-}$ |  $X_{6}^{-}$+$X_{7}^{-}$ |  $X_{6}^{-}$+$X_{7}^{-}$ |  $X_{6}^{+}$+$X_{7}^{+}$ |  $X_{6}^{+}$+$X_{7}^{+}$
$X_{6}^{+}$ |  $X_{1}^{+}$+$X_{2}^{+}$+$X_{5}^{+}$ |  $X_{3}^{+}$+$X_{4}^{+}$+$X_{5}^{+}$ |  $X_{1}^{-}$+$X_{2}^{-}$+$X_{5}^{-}$ |  $X_{3}^{-}+$+$X_{4}^{-}$+$X_{5}^{-}$
$X_{7}^{+}$ |  |  $X_{1}^{+}$+$X_{2}^{+}$+$X_{5}^{+}$ |  $X_{1}^{-}$+$X_{2}^{-}$+$X_{5}^{-}$ |  $X_{3}^{-}+$+$X_{4}^{-}$+$X_{5}^{-}$
$X_{6}^{-}$ |  |  |  $X_{1}^{+}$+$X_{2}^{+}$+$X_{5}^{+}$ |  $X_{3}^{+}$+$X_{4}^{+}$+$X_{5}^{+}$
$X_{7}^{-}$ |  |  |  |  $X_{1}^{+}$+$X_{2}^{+}$+$X_{5}^{+}$
TABLE S8: Direct product for group $O_{h}$ $O_{h}$ | $\Gamma_{6}^{+}$ | $\Gamma_{7}^{+}$ | $\Gamma_{8}^{+}$ | $\Gamma_{6}^{-}$ | $\Gamma_{7}^{-}$ | $\Gamma_{8}^{-}$
---|---|---|---|---|---|---
$\Gamma_{6}^{+}$ |  $\Gamma_{1}^{+}$\+ $\Gamma_{4}^{+}$ |  $\Gamma_{2}^{+}$+$\Gamma_{5}^{+}$ |  $\Gamma_{3}^{+}$+$\Gamma_{4}^{+}$+$\Gamma_{5}^{+}$ |  $\Gamma_{1}^{-}$\+ $\Gamma_{4}^{-}$ |  $\Gamma_{2}^{-}$+$\Gamma_{5}^{-}$ |  $\Gamma_{3}^{-}$+$\Gamma_{4}^{-}$+$\Gamma_{5}^{-}$
$\Gamma_{7}^{+}$ |  |  $\Gamma_{1}^{+}$\+ $\Gamma_{4}^{+}$ |  $\Gamma_{3}^{+}$+$\Gamma_{4}^{+}$+$\Gamma_{5}^{+}$ |  $\Gamma_{2}^{-}$+$\Gamma_{5}^{-}$ |  $\Gamma_{1}^{-}$\+ $\Gamma_{4}^{-}$ |  $\Gamma_{3}^{-}$+$\Gamma_{4}^{-}$+$\Gamma_{5}^{-}$
$\Gamma_{8}^{+}$ |  |  |  $\Gamma_{1}^{+}$\+ $\Gamma_{2}^{+}$+$\Gamma_{3}^{+}$+2$\Gamma_{4}^{+}$+2$\Gamma_{5}^{+}$ |  $\Gamma_{3}^{-}$+$\Gamma_{4}^{-}$+$\Gamma_{5}^{-}$ |  $\Gamma_{3}^{-}$+$\Gamma_{4}^{-}$+$\Gamma_{5}^{-}$ |  $\Gamma_{1}^{-}$\+ $\Gamma_{2}^{-}$+$\Gamma_{3}^{-}$+2$\Gamma_{4}^{-}$+2$\Gamma_{5}^{-}$
$\Gamma_{6}^{-}$ |  |  |  |  $\Gamma_{1}^{+}$\+ $\Gamma_{4}^{+}$ |  $\Gamma_{2}^{+}$\+ $\Gamma_{5}^{+}$ |  $\Gamma_{3}^{-}$+$\Gamma_{4}^{-}$+$\Gamma_{5}^{+}$
$\Gamma_{7}^{-}$ |  |  |  |  |  $\Gamma_{1}^{+}$\+ $\Gamma_{4}^{+}$ |  $\Gamma_{3}^{-}$+$\Gamma_{4}^{-}$+$\Gamma_{5}^{+}$
$\Gamma_{7}^{-}$ |  |  |  |  |  |  $\Gamma_{1}^{+}$\+ $\Gamma_{2}^{+}$+$\Gamma_{3}^{+}$+$2\Gamma_{4}^{+}$+$2\Gamma_{5}^{+}$

###  XII.1 Direct transitions

The direct optical transitions between VBE and CBE states are guided by two
key rules:
(i) For a non-trivial transition, the direct product of the involved states
must exhibit electric dipole characteristics consistent with the symmetries of
their respective point groups.
(ii) Transitions between the two states are permissible only if they possess
opposite parities, in accordance with Laporte’s rule Laporte and Meggers
(1925).
Table S9 lists the distinct types of direct transitions that are pertinent to
various HDP structures along the high-symmetry points. As an example, we can
consider the direct transitions within the Cs2AgBiCl6 structure. At point W,
there is only one possible transition from $W_{6}$ to $W_{7}$ between VBE and
CBE. From the Table S9 we note that the direct product between $W_{6}$ and
$W_{7}$ is expressed as the summation of three representations $W_{3}$,
$W_{4}$ and $W_{5}$. Among these three representations, $W_{4}$ has the
character of electric dipole operator (A2) for D2d point group. Hence, this
transition is allowed. At L, the direct product between $L_{5}^{-}+L_{6}^{-}$
and $L_{4}^{-}$ can be expressed as $L_{1}^{+}$+$L_{2}^{+}$+$L_{3}^{+}$, which
does not include the electric dipole operator $L_{2}^{-}$
(A1u)+$L_{3}^{-}$(Eu) for the point group D3d. Hence, This transition is
forbidden which is also supported by the Laporte rule as well as by the P2
(see Fig. 6 of the main manuscript). Again, the transitions between VBE and
CBE along $\Gamma$ and X point are also allowed ((Table S9)) following the
direct product rule as demonstrated in Table S8 and S7 respectively.

TABLE S9: Direct products between two IRs for all types of allowed transitions at high symmetry points in the HDP systems.  W: |  |  $W_{6}\otimes W_{7}$ = $W_{2}\oplus W_{3}\oplus W_{5}$
---
$W_{6}\otimes W_{6}$ = $W_{1}\oplus W_{4}\oplus W_{5}$
$W_{7}\otimes W_{7}$ = $W_{1}\oplus W_{4}\oplus W_{5}$
$\bf{\Gamma:}$ |  |  $\Gamma_{8}^{+}\otimes\Gamma_{6}^{-}$ = $\Gamma_{3}^{-}\oplus\Gamma_{4}^{-}\oplus\Gamma_{5}^{-}$
---
$\Gamma_{6}^{+}\otimes\Gamma_{6}^{-}$ = $\Gamma_{1}^{-}\oplus\Gamma_{4}^{-}$
L: |  $L_{4}^{+}\otimes L_{4}^{-}$ = $L_{1}^{-}\oplus L_{2}^{-}\oplus L_{3}^{-}$
X: |  $X_{6}^{+}\otimes X_{6}^{-}$ = $X_{1}^{-}\oplus X_{2}^{-}\oplus X_{5}^{-}$

###  XII.2 Indirect Transitions

As we can observe from Fig. 6 of the main manuscript there are only two
indirect bandgap semiconductors Cs2AgBiCl6 and Cs2AgSbCl6 which represent
finite optical transition along their band spectra. Given that indirect
semiconductors primarily involve transitions between VBM and CBM via indirect
processes, our discussion on selection rules will specifically pertain to
these two cases only. As evident from the dispersion spectrum (Fig. 6 in the
main manuscript), there are only two specific types of indirect transitions,
namely $W_{6}\rightarrow L_{4}^{-}$ and $X_{6}^{+}\rightarrow L_{4}^{-}$.
Since indirect transitions are mainly phonon-mediated interactions, we need to
analyze the active phonon modes for their respective point groups. As
determined from the character table of D2d and D3d for W and X point
respectively, it can be written that $W_{6}\otimes L_{4}^{-}$ = $W_{1}\oplus
W_{2}\oplus W_{3}\oplus W_{4}$. Now, $W_{1}$ and $W_{3}$ have the optical
phonon, and $W_{4}$ has the acoustic phonon characteristics. Therefore, the
transition $W_{6}\rightarrow L_{4}^{-}$ is a phonon-mediated allowed
transition. Similarly, $X_{6}^{+}\otimes L_{4}^{-}$ = $X_{1}^{-}\oplus
X_{2}^{-}\oplus X_{5}^{-}$. This transition is also allowed only through the
acoustic phonon modes $X_{2}^{-}$ and $X_{5}^{-}$. Likewise, indirect
transitions in other HDPs can be successfully demonstrated through an
elementary group theoretical approach.

##  XIII Tight-Binding model Hamiltonian

Fig. S1: Schematic illustration of various hopping interactions up to the
fourth nearest neighbor (NN) in halide double perovskites. The Hamiltonian
presented below involves effective interactions among the B and B′ orbitals to
describe the band structure in the vicinity of Fermi level. The contribution
of NN B-X and B′-X interactions is incorporated in these effective
interactions. It can be understood by examining the downfold technique such as
Loẅdin downfold Löwdin (1951); Gupta and Nanda (2022).

The generic tight-binding (TB) Hamiltonian incorporating covalent
interactions, as illustrated in Fig. S1, and spin-orbit coupling (SOC) effect
for HDP can be written as

$H=\sum_{i,\alpha}\epsilon_{i\alpha}c^{\dagger}_{i\alpha}c_{i\alpha}+\sum_{{\braket{ij}};\alpha,\beta}t_{i\alpha
j\beta}(c^{\dagger}_{i\alpha}c_{j\beta}+h.c.)+\lambda\vec{L}\cdot\vec{S},$
(11)

The details of the Hamiltonian are provided in Section V of the main text.
Here, we have explained the components of the Hamiltonian matrix. The complete
Hamiltonian matrix containing interactions between B and B′ is expressed as

$H=\left[\begin{array}[]{cc}H_{BB}&H_{BB^{\prime}}\\\
H_{BB^{\prime}}^{\dagger}&H_{B^{\prime}B^{\prime}}\\\ \end{array}\right]$ (12)

Where $H_{BB/B^{\prime}B^{\prime}}$ are the Hamiltonian sub-matrices
containing interactions among the same cations (i.e., the onsite energy and
fourth-neighbor $BB$ and $B^{\prime}B^{\prime}$ interactions) and the
Hamiltonian sub-matrices $H_{BB^{\prime}}$ and $H_{BB^{\prime}}^{\dagger}$
contain the second neighbor $BB^{\prime}$ interactions between two different
cations (see Fig. S1).

$H_{BB/B^{\prime}B^{\prime}}=\left[\begin{array}[]{cc}H^{\uparrow\uparrow}&0\\\
0&H^{\downarrow\downarrow}\\\
\end{array}\right],H_{BB^{\prime}/B^{\prime}B}=\left[\begin{array}[]{cc}H^{\uparrow\uparrow}&0\\\
0&H^{\downarrow\downarrow}\\\ \end{array}\right]$ (13)

The Hamiltonian H${}_{BB/B^{\prime}B^{\prime}}$ for 4th NN interactions, when
$s$ and $p$ orbitals of B and B′ are contributing to bands around the Fermi
are in basis set order $\ket{s^{B/B^{\prime},\uparrow/\downarrow}}$,
$\ket{p_{x}^{B/B^{\prime},\uparrow/\downarrow}}$,
$\ket{p_{y}^{B/B^{\prime},\uparrow/\downarrow}}$, and
$\ket{p_{z}^{B/B^{\prime},\uparrow/\downarrow}}$ is given as

$H^{\uparrow\uparrow/\downarrow\downarrow}_{BB^{\prime}/B^{\prime}B/BB/B^{\prime}B^{\prime}}=\left[\begin{array}[]{ccccc}\epsilon_{s}+h_{ss}&h_{sp_{x}}&h_{sp_{y}}&h_{sp_{z}}\\\
h_{sp_{x}}^{*}&\epsilon_{p}+h_{p_{x}p_{x}}&h_{p_{x}p_{y}}&h_{p_{x}p_{z}}\\\
h_{sp_{y}}^{*}&h_{p_{x}p_{y}}^{*}&\epsilon_{p}+h_{p_{y}p_{y}}&h_{p_{y}p_{z}}\\\
h_{sp_{z}}^{*}&h_{p_{x}p_{z}}^{*}&h_{p_{y}p_{z}}^{*}&\epsilon_{p}+h_{p_{z}p_{z}}\\\
\end{array}\right]$ (14)

This is also the Hamiltonian matrix for 2nd neighbor B-B′ interactions when
$s$ and $p$ orbitals of B and B′ are contributing, however, in that case,
hopping parameters will be different.

Furthermore, the Hamiltonian matrix for 4th neighbor B-B interactions, when an
Ag atom is sitting at B-site, in basis set order
$\ket{s^{B,\uparrow/\downarrow}}$, $\ket{d_{z^{2}}^{B,\uparrow/\downarrow}}$,
$\ket{d_{x^{2}-y^{2}}^{B,\uparrow/\downarrow}}$,
$\ket{d_{xy}^{B,\uparrow/\downarrow}}$,
$\ket{d_{xz}^{B,\uparrow/\downarrow}}$, and
$\ket{d_{yz}^{B,\uparrow/\downarrow}}$, can be expressed as

$H^{\uparrow\uparrow/\downarrow\downarrow}_{BB}=\left[\begin{array}[]{ccccccc}\epsilon_{s}+h_{ss}&h_{sd_{z^{2}}}&h_{sd_{x^{2}-y^{2}}}&h_{sd_{xy}}&h_{sd_{xz}}&h_{sd_{yz}}\\\
h_{sd_{z^{2}}}^{*}&\epsilon_{de_{g}}+h_{d_{z^{2}}d_{z^{2}}}&h_{d_{z^{2}}d_{x^{2}-y^{2}}}&h_{d_{z^{2}}d_{xy}}&h_{d_{z^{2}}d_{xz}}&h_{d_{z^{2}}d_{yz}}\\\
h_{sd_{x^{2}-y^{2}}}^{*}&h_{d_{x^{2}-y^{2}}d_{z^{2}}}&\epsilon_{d_{x^{2}-y^{2}}}+h_{d_{x^{2}-y^{2}}d_{x^{2}-y^{2}}}&h_{d_{x^{2}-y^{2}}d_{xy}}&h_{d_{x^{2}-y^{2}}d_{xz}}&h_{d_{x^{2}-y^{2}}d_{yz}}\\\
h_{sd_{xy}}^{*}&h_{d_{xy}d_{z^{2}}}&h_{d_{x^{2}-y^{2}}d_{x^{2}-y^{2}}}&\epsilon_{dxy}+h_{d_{xy}d_{xy}}&h_{d_{xy}d_{xz}}&h_{d_{xy}d_{yz}}\\\
h_{sd_{xz}}^{*}&h_{d_{xz}d_{z^{2}}}&h_{d_{xz}d_{x^{2}-y^{2}}}&h_{d_{xz}d_{xy}}&\epsilon_{dxz}+h_{d_{xz}d_{xz}}&h_{d_{xz}d_{yz}}\\\
h_{sd_{yz}}^{*}&h_{d_{yz}d_{z^{2}}}&h_{d_{yz}d_{x^{2}-y^{2}}}&h_{d_{yz}d_{xy}}&h_{d_{yz}d_{xz}}&\epsilon_{dyz}+h_{d_{yz}d_{yz}}\\\
\end{array}\right]$ (15)

and the Hamiltonian sub-matrix $H_{BB^{\prime}/B^{\prime}B}$, containing
interactions between $d$ and $p$ orbitals from two different NN cations can be
expressed as

$H^{\uparrow\uparrow/\downarrow\downarrow}_{BB^{\prime}}=\left[\begin{array}[]{cccc}h_{ss}&h_{sp_{x}}&h_{sp_{y}}&h_{sp_{z}}\\\
h_{d_{z^{2}}s}&h_{d_{z^{2}}p_{x}}&h_{d_{z^{2}}p_{y}}&h_{d_{z^{2}}p_{z}}\\\
h_{d_{x^{2}-y^{2}}s}&h_{d_{x^{2}-y^{2}}p_{x}}&h_{d_{x^{2}-y^{2}}p_{y}}&h_{d_{x^{2}-y^{2}}p_{z}}\\\
h_{d_{xy}s}&h_{d_{xy}p_{x}}&h_{d_{xy}p_{y}}&h_{d_{xy}p_{z}}\\\
h_{d_{xz}s}&h_{d_{xz}p_{x}}&h_{d_{xz}p_{y}}&h_{d_{xz}p_{z}}\\\
h_{d_{yz}s}&h_{d_{yz}p_{x}}&h_{d_{yz}p_{y}}&h_{d_{yz}p_{z}}\\\
\end{array}\right].$ (16)

Each of the above elements of the Hamiltonian matrix $h_{ij}$ are expressed in
terms of Slater-Koster Slater and Koster (1954) form. The SOC matrix for
$d$-orbital basis set in basis order $\ket{s^{B,\uparrow}}$,
$\ket{d_{z^{2}}^{B,\uparrow}}$, $\ket{d_{x^{2}-y^{2}}^{B,\uparrow}}$,
$\ket{d_{xy}^{B,\uparrow}}$, $\ket{d_{xz}^{B,\uparrow}}$,
$\ket{d_{yz}^{B,\uparrow}}$, $\ket{s^{B,\downarrow}}$,
$\ket{d_{z^{2}}^{B,\downarrow}}$, $\ket{d_{x^{2}-y^{2}}^{B,\downarrow}}$,
$\ket{d_{xy}^{B,\downarrow}}$, $\ket{d_{xz}^{B,\downarrow}}$,
$\ket{d_{yz}^{B,\downarrow}}$ and $p$-orbital basis set in the basis order
$\ket{s^{B/B^{\prime},\uparrow}}$, $\ket{p_{x}^{B/B^{\prime},\uparrow}}$,
$\ket{p_{y}^{B/B^{\prime},\uparrow}}$, $\ket{p_{z}^{B/B^{\prime},\uparrow}}$,
$\ket{s^{B/B^{\prime},\downarrow}}$, $\ket{p_{x}^{B/B^{\prime},\downarrow}}$,
$\ket{p_{y}^{B/B^{\prime},\downarrow}}$,
$\ket{p_{z}^{B/B^{\prime},\downarrow}}$ are

$H_{d}^{SOC}=\lambda\left[\begin{array}[]{cccccccccccc}0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&-\sqrt{3}&i\sqrt{3}\\\ 0&0&0&-2i&0&0&0&0&0&0&1&i\\\
0&0&2i&0&0&0&0&0&0&0&-i&1\\\ 0&0&0&0&0&-i&0&\sqrt{3}&-1&i&0&0\\\
0&0&0&0&i&0&0&-i\sqrt{3}&-i&-1&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&\sqrt{3}&i\sqrt{3}&0&0&0&0&0&0\\\ 0&0&0&0&-1&i&0&0&0&2i&0&0\\\
0&0&0&0&-i&-1&0&0&-2i&0&0&0\\\ 0&-\sqrt{3}&1&i&0&0&0&0&0&0&0&i\\\
0&-i\sqrt{3}&-i&1&0&0&0&0&0&0&-i&0\\\ \end{array}\right],$ (17)
$H_{p}^{SOC}=\left[\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\
0&0&-i\lambda&0&0&0&0&\lambda\\\ 0&i\lambda&0&0&0&0&0&-i\lambda\\\
0&0&0&0&0&-\lambda&i\lambda&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&-\lambda&0&0&i\lambda&0\\\ 0&0&0&-i\lambda&0&-i\lambda&0&0\\\
0&\lambda&i\lambda&0&0&0&0&0\\\ \end{array}\right]$ (18)

##  XIV Estimation of TB hopping parameters

TB hoppings for various HDPs obtained from fitting with DFT band structures
are provided in Tables S10, S11, and S12.

TABLE S10: TB hopping parameters (in eV) for halide perovskites family. | hopping | $Cs_{2}AgBiCl_{6}$ | $Cs_{2}AgSbCl_{6}$ | $Cs_{2}AgAsCl_{6}$ | $Cs_{2}AgInCl_{6}$ | $Cs_{2}AgTlCl_{6}$ | $Cs_{2}AgGaCl_{6}$ | $Cs_{2}CuBiCl_{6}$ | $Cs_{2}CuInCl_{6}$
---|---|---|---|---|---|---|---|---|---
| parameters |  B = Ag,B′ = Bi |  B = Ag,B′ = Sb |  B = Ag,B′ = As |  B = Ag,B′ = In |  B = Ag,B′ = Tl |  B = Ag,B′ = Ga |  B = Cu,B′ = Bi |  B = Cu,B′ = In
| $\epsilon_{s}^{BB}$ | 7.76 | 5.86 | 6.16 | 7.15 | 7.76 | 7.15 | 5.4 | 5.2
| $\epsilon_{e_{g}}^{BB}$ | -0.40 | -1.1 | -0.92 | -0.08 | 0.04 | -0.05 | -0.17 | -0.08
| $\epsilon_{t_{2g}}^{BB}$ | -0.91 | -1.81 | -1.61 | -0.82 | -1.16 | -0.76 | -0.70 | -0.55
B-B | $t_{dd\sigma}^{BB}$ | -0.002 | 0.0 | 0.0 | 0.01 | 0.016 | 0.0 | 0.0 | 0.0
| $t_{dd\pi}^{BB}$ | -0.008 | 0.0 | 0.0 | 0.007 | -0.017 | 0.0 | 0.0 | 0.0
| $t_{dd\delta}^{BB}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| $t_{ss\sigma}^{BB}$ | 0.0 | 0.03 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| $t_{sd\sigma}^{BB}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| $\epsilon_{s}^{B^{\prime}B^{\prime}}$ | -1.85 | -1.16 | -1.75 | 4.6 | 1.46 | 4.15 | -3.2 | 1.44
| $\epsilon_{p}^{B^{\prime}B^{\prime}}$ | 4.18 | 3.41 | 3.30 | 7.98 | 6.81 | 7.40 | 2.3 | 5.8
| $t_{ss}^{B^{\prime}B^{\prime}}$ | -0.03 | -0.065 | -0.01 | 0.0 | 0.048 | 0.0 | -0.03 | -0.06
B′-B′ | $t_{sp}^{B^{\prime}B^{\prime}}$ | 0.03 | 0.02 | 0.03 | 0.018 | 0.126 | 0.018 | 0.025 | 0.0
| $t_{pp\sigma}^{B^{\prime}B^{\prime}}$ | -0.028 | -0.076 | -0.06 | -0.092 | -0.078 | -0.092 | -0.055 | -0.25
| $t_{pp\pi}^{B^{\prime}B^{\prime}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | -0.014 | 0.0
| $t_{ss}^{BB^{\prime}}$ | 0.0 | 0.0 | 0.24 | 0.375 | 0.532 | 0.384 | 0.22 | 0.25
| $t_{sp}^{BB^{\prime}}$ | 0.475 | 0.57 | 0.60 | 0.48 | 0.54 | 0.48 | 0.57 | 0.54
B-B′ | $t_{sd\sigma}^{BB^{\prime}}$ | 0.193 | 0.25 | 0.32 | 0.42 | 0.39 | 0.44 | 0.12 | 0.27
| $t_{pd\sigma}^{BB^{\prime}}$ | 0.54 | 0.63 | 0.67 | 0.63 | 0.84 | 0.64 | 0.38 | 0.22
| $t_{pd\pi}^{BB^{\prime}}$ | 0.0 | 0.14 | 0.0 | 0.24 | 0.2 | 0.24 | 0.02 | 0.12
SOC | $\lambda^{B}$ | 0.09 | 0.09 | 0.09 | 0.09 | 0.09 | 0.09 | 0.0 | 0.06
| $\lambda^{B^{\prime}}$ | 0.54 | 0.17 | 0.1 | 0.16 | 0.24 | 0.05 | 0.16 | 0.16

TABLE S11: TB hopping parameters (in eV) for halide perovskites family. Cation- | TB parameters | $CsPbBr_{3}$ | $Cs_{2}InBiCl_{6}$ | $Cs_{2}InSbCl_{6}$ | $Cs_{2}TlBiCl_{6}$ | $Cs_{2}TlSbCl_{6}$
---|---|---|---|---|---|---
cation |  |  | B = In, B′ = Bi | B = In, B′ = Sb | B = Tl, B′ = Bi | B = Tl, B′ = Sb
| $\epsilon_{s}^{BB}$ | -1.36 | -0.98 | -1.16 | -1.0 | -1.08
| $\epsilon_{p}^{BB}$ | 4.15 | 4.5 | 4.07 | 5.24 | 4.6
B-B | $t_{ss\sigma}^{BB}$ | 0.0 | -0.053 | -0.008 | -0.026 | 0.0
| $t_{sp\sigma}^{BB}$ | 0.0 | -0.016 | 0.036 | 0.044 | 0.028
| $t_{pp\sigma}^{BB}$ | 0.0 | 0.0 | 0.0 | 0.04 | 0.0
| $t_{pp\pi}^{BB}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| $\epsilon_{s}^{B^{\prime}B^{\prime}}$ | -1.36 | -3.156 | -2.676 | -1.22 | -1.62
| $\epsilon_{p}^{B^{\prime}B^{\prime}}$ | 4.15 | 2.42 | 1.9 | 3.24 | 2.8
| $t_{ss}^{B^{\prime}B^{\prime}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
B′-B′ | $t_{sp\sigma}^{B^{\prime}B^{\prime}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| $t_{pp\sigma}^{B^{\prime}B^{\prime}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| $t_{pp\pi}^{B^{\prime}B^{\prime}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0
| $t_{ss\sigma}^{BB^{\prime}}$ | -0.23 | -0.377 | -0.303 | -0.21 | -0.22
| $t_{sp\sigma}^{BB^{\prime}}$ | 0.547 | 0.44 | 0.51 | 0.50 | 0.47
B-B′ | $t_{pp\sigma}^{BB^{\prime}}$ | 0.805 | 0.665 | 0.665 | 0.675 | 0.72
| $t_{pp\pi}^{BB^{\prime}}$ | 0.105 | 0.09 | 0.09 | 0.089 | 0.078
SOC | $\lambda^{B}$ | 0.52 | 0.16 | 0.16 | 0.46 | 0.37
| $\lambda^{B^{\prime}}$ | 0.52 | 0.56 | 0.17 | 0.49 | 0.17
TABLE S12: TB hopping parameters (in eV) for halide perovskites family. Cation- | TB parameters | $Cs_{2}NaInCl_{6}$ | $Cs_{2}NaBiCl_{6}$ | $Cs_{2}KInCl_{6}$ | $Cs_{2}KBiCl_{6}$
---|---|---|---|---|---
cation |  | B = Na, B′ = In | B = Na, B′ = Bi | B = K, B′ = In | B = K, B′ = Bi
| $\epsilon_{s}^{B}$ | 12.4 | 9 | 15 | 15
| $\epsilon_{p}^{B}$ | 13.5 | 10 | 15 | 15
B-B | $t_{ss\sigma}^{B}$ | – | – | – | –
| $t_{sp\sigma}^{B}$ | – | – | – | –
| $t_{pp\sigma}^{B}$ | – | – | – | –
| $t_{pp\pi}^{B}$ | – | – | – | –
| $\epsilon_{s}^{B^{\prime}B^{\prime}}$ | 5.8 | -0.08 | 6.21 | 0.0
| $\epsilon_{p}^{B^{\prime}B^{\prime}}$ | 9.64 | 5.20 | 9.35 | 5.12
| $t_{ss\sigma}^{B^{\prime}B^{\prime}}$ | -0.036 | -0.034 | 0.010 | 0.01
B′-B′ | $t_{sp\sigma}^{B^{\prime}B^{\prime}}$ | 0.018 | 0.05 | 0.05 | 0.04
| $t_{pp\sigma}^{B^{\prime}B^{\prime}}$ | -0.152 | -0.08 | -0.15 | -0.01
| $t_{pp\pi}^{B^{\prime}B^{\prime}}$ | 0 | 0.02 | 0 | 0
| $t_{ss\sigma}^{BB^{\prime}}$ | 0.0 | 0.0 | -0.33 | -0.40
| $t_{sp\sigma}^{BB^{\prime}}$ | 0.0 | 0.0 0 | 0.0 | 0.0
B-B′ | $t_{pp\sigma}^{BB^{\prime}}$ | 0.0 | 0.0 | 0.0 | 0.0
| $t_{pp\pi}^{BB^{\prime}}$ | 0.0 | 0.0 | 0.0 | 0.0
SOC | $\lambda^{B}$ | 0.0 | 0.0 | 0.0 | 0.0
| $\lambda^{B^{\prime}}$ | 0.16 | 0.52 | 0.16 | 0.52

##  XV B-MOP and tight-binding model for $\bf{Cs_{2}AgAsCl_{6}}$,
$\bf{Cs_{2}AgGaCl_{6}}$, $\bf{Cs_{2}CuBiCl_{6}}$, and $\bf{Cs_{2}CuInCl_{6}}$

We have added four more compounds to our list of HDPs and performed the model
Hamiltonian calculations for the same. These compounds are Cs2AgAsCl6,
Cs2AgGaCl6, Cs2CuBiCl6, and Cs2CuInCl6. We have obtained the B-MOP for all
these compounds and performed the tight-binding calculations. Upon careful
examination, we deduce a consistent pattern wherein electronic characteristics
remain largely unaltered, apart from the bandgap, when newly substituted
cations belong to the same group within the periodic table. This phenomenon is
exemplified by Cs2AgB′Cl6, where B′ corresponds to group 15 elements (As, Sb,
Bi) within the periodic table. In each of these three compounds, the band
structure consistently exhibits the qualities of an indirect bandgap, with the
VBM situated at point X, and the CBM located at the high-symmetry L. Notably,
as we turn B′ from As to Bi within this framework, a remarkable trend unfolds.
The mBJ bandgap experiences a progressive increase, escalating from 2.26 to
2.98 eV, while concurrently, the dispersion of the edge bands diminishes.
Similar observations have been made for Cs2AgB′Cl6, where B′ corresponds to
group 13 elements (Ga, In, Tl). The nature of bandgap remains direct with both
VBM and CBM situated at $\Gamma$. The calculated MME also shows the same
behavior with the parity-forbidden transition from $\Gamma$ to X for all three
cases (see Fig. S3) however, the bandgap is maximum for B′ = In. The reason
behind it can be explained through the free atomic energy of B′-$s$ orbitals.
The TB fitting parameters for all four new compounds are given in Table S10.

In another case, we altered B from Ag to Cu and conducted electronic structure
calculations using various B′ elements. The calculated band structures, B-MOP
results, tight-binding models, MME, and parity values from symmetry analysis
are illustrated in Fig. S3.

Fig. S2: The band projected molecular orbital picture (B-MOP) of halide double
perovskites as envisaged from the following molecular hybridizations: B(Ag,
Cu)-$s$, $d$-Cl-$p$ and B′(Ga, As, Bi, In)-$s$, $p$-Cl-$p$ atomic orbitals
produces the bonding and antibonding orbitals along with the nonbonding Cl-$p$
orbitals. The free atomic energy levels are estimated from Hartree-Fock’s
theory.  Fig. S3: DFT (red) fitted tight-binding bands (blue) of Cs2AgAsCl6,
Cs2AgGaCl6, Cs2CuBiCl6, and Cs2CuInCl6. The squared momentum matrix elements
(P2) corresponding to the valence band edge (VBE) to conduction band edge
(CBE) transition are also shown for each of the compounds. The parity of the
VBE and CBE are mentioned using Koster notations.

##  XVI Mathematical formulation to calculate optical properties

We will start from the Fermi Golden rule and then derive the expressions for
the momentum matrix element (MME).

Fermi Golden rule: The Fermi Golden rule states that transition probability
per unit time from initial state $\ket{i}$ to a set of final states $\ket{f}$
is given to first-order approximation:

$W_{i\rightarrow
f}=\frac{2\pi}{\hbar}|\bra{\textbf{k},\beta}\vec{P}\ket{\textbf{k},\beta^{\prime}}|^{2}\rho(E_{F})$
(19)

where $\bra{\textbf{k},\beta}\vec{P}\ket{\textbf{k},\beta^{\prime}}$ is matrix
element of perturbation H′ between final and initial states, and $\rho(E_{f})$
is density of states at energy EF. The density of states for transition
between states $E_{v}$ and $E_{c}$ with incident photon energy $\hbar\omega$,
can be approximated to $\delta(E_{c}-E_{v}-\hbar\omega)$.

As a starting point to derive the momentum matrix element (MME), let us
consider $n^{th}$ tight-binding eigenstate formed with a linear combination of
atomic orbitals,

$\ket{n,\textbf{k}}=\sum_{\alpha}c_{n\alpha}(\textbf{k})\ket{\alpha,\textbf{k}}$
(20)

where $\ket{\alpha,\textbf{k}}$ is an atomic orbital eigenstate in $k$ space,
$\alpha$ is the atomic orbital, and $k$ is momentum. In periodic systems, the
state $\ket{\alpha,\textbf{k}}$ can be defined as the Bloch sum of localized
orbitals

$\ket{\alpha,\textbf{k}}=\frac{1}{\sqrt{N}}\sum_{\textbf{R}}e^{i\textbf{k}\cdot\textbf{R}}\ket{\alpha,\textbf{R}}.$
(21)

Where R is lattice vector and N is number of lattice sites. The system
Hamiltonian $H$ is then diagonalized within eigenstate to obtain the energy
eigenvalue $E_{n\textbf{k}}$ of the system.

$H(k)\ket{n,\textbf{k}}=E_{nk}\ket{n,\textbf{k}}\\\ $ (22)

The connection between position and momentum operator is derived through
commutator relation:

$\begin{split}[H,\textbf{r}]&=\left[\frac{p^{2}}{2m_{e}}+V_{0}(\textbf{r}),\textbf{r}\right]\\\
[H,x]&=\frac{1}{2m_{e}}([p_{x}[p_{x},x]+[p_{x},x]p_{x}])\\\
[H,x]&=\frac{-i\hbar}{m_{e}}p_{x}\\\ \end{split}$ (23)

Similarly, one can obtain $[H,y]=(-i\hbar/m_{e})p_{y}$ and
$[H,z]=(-i\hbar/m_{e})p_{z}$. Hence,

$[H,\textbf{r}]=\frac{-i\hbar}{m_{e}}\textbf{p}$ (24)

or,

$\textbf{p}=\frac{im_{e}}{\hbar}[H,\textbf{r}]$ (25)

$\begin{split}&\bra{n,k}\textbf{p}\ket{m,k}=\bra{n,k}\frac{im_{e}}{\hbar}[H,\textbf{r}]\ket{m,k}\\\
&=\frac{im_{e}}{\hbar}\bra{n,k}H\textbf{r}-\textbf{r}H\ket{m,k}\\\
&=\frac{im_{e}}{\hbar}\sum_{\beta}c_{n\beta}^{*}(k)\frac{1}{\sqrt{N}}\sum_{R^{\prime}}\bra{\beta,R^{\prime}}e^{-i\vec{k}\cdot\vec{R^{\prime}}}(H\textbf{r}-\textbf{r}H)\sum_{\alpha}c_{m\alpha}(k)\sum_{R}\frac{1}{\sqrt{N}}\ket{\alpha,R}e^{i\vec{k}\cdot\vec{R}}\\\
&=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{RR^{\prime}}e^{1k(R-R^{\prime})}\bra{\beta
R^{\prime}}H\textbf{r}-\textbf{r}H\ket{\alpha R}\\\ &=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{RR^{\prime}}e^{ik(R-R^{\prime})}(\bra{\beta
R^{\prime}}H\textbf{r}\ket{\alpha R}-\bra{\beta
R^{\prime}}\textbf{r}H\ket{\alpha R})\\\ &=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{RR^{\prime}}e^{ik(R-R^{\prime})}(\bra{\beta
R^{\prime}}H\sum_{\gamma}\ket{\gamma R}\bra{\gamma R}\textbf{r}\ket{\alpha
R}\\\ &\hskip 170.71652pt-\bra{\beta
R^{\prime}}\textbf{r}\sum_{\gamma}\ket{\gamma R^{\prime}}\bra{\gamma
R^{\prime}}H\ket{\alpha R})\\\ &=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{RR^{\prime}}e^{ik(R-R^{\prime})}(\sum_{\gamma}\bra{\beta
R^{\prime}}H\ket{\gamma R}\ (R\delta_{\gamma\alpha}+d_{\alpha\gamma})-\\\
&\hskip 170.71652pt\sum_{\gamma}\bra{\gamma R^{\prime}}H\ket{\alpha
R}(R^{\prime}\delta_{\gamma\beta}+d_{\gamma\beta}))\\\ &=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{RR^{\prime}}e^{ik(R-R^{\prime})}\sum_{\gamma}(\bra{\beta
R^{\prime}}H\ket{\gamma R}R\delta_{\gamma\alpha}-\bra{\gamma
R^{\prime}}H\ket{\alpha R}R^{\prime}\delta_{\gamma\beta}\\\ &\hskip
170.71652pt+d_{\alpha\gamma}\bra{\beta R^{\prime}}H\ket{\gamma
R}-d_{\gamma\beta}\bra{\gamma R^{\prime}}H\ket{\alpha R})\\\
&=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{RR^{\prime}}e^{ik(R-R^{\prime})}(\bra{\beta
R^{\prime}}H\ket{\alpha R}(R-R^{\prime})+\\\ &\hskip
170.71652pt\sum_{\gamma}(\bra{\beta R^{\prime}}H\ket{\gamma
R}d_{\alpha\gamma}-\bra{\gamma R^{\prime}}H\ket{\alpha R}d_{\gamma\beta}))\\\
&=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{RR^{\prime}}e^{ik(R-R^{\prime})}(R-R^{\prime})\bra{\beta
R^{\prime}}H\ket{\alpha R}+\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\\\ &\hskip
170.71652pt\sum_{RR^{\prime}}e^{ik(R-R^{\prime})}\sum_{\gamma}(\bra{\beta
R^{\prime}}H\ket{\gamma R}d_{\alpha\gamma}-\bra{\gamma R^{\prime}}H\ket{\alpha
R}d_{\gamma\beta})\\\ &=\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)(-i\nabla_{k})\sum_{R}\sum_{R^{\prime}}e^{ikR}e^{-ikR^{\prime}}\bra{\beta
R^{\prime}}H\ket{\alpha R}+\frac{im_{e}}{\hbar
N}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\\\ &\hskip
170.71652pt\sum_{R}\sum_{R^{\prime}}e^{ikR}e^{-ikR^{\prime}}\sum_{\gamma}(\bra{\beta
R^{\prime}}H\ket{\gamma R}d_{\alpha\gamma}-\bra{\gamma R^{\prime}}H\ket{\alpha
R}d_{\gamma\beta})\\\
&=\frac{m_{e}}{\hbar}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\nabla_{k}\bra{\beta
k}H\ket{\alpha
k}+\frac{im_{e}}{\hbar}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\sum_{\gamma}(\bra{\beta
k}H\ket{\gamma k}d_{\alpha\gamma}-\\\ &\hskip 170.71652pt\bra{\gamma
k}H\ket{\alpha k}d_{\gamma\beta})\\\
&=\frac{m_{e}}{\hbar}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\nabla_{k}\bra{\beta
k}H\ket{\alpha
k}+\frac{im_{e}}{\hbar}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)(E_{\beta
k}-E_{\alpha k})d_{\alpha\beta}\\\
&=\frac{m_{e}}{\hbar}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)\nabla_{k}\bra{\beta
k}H\ket{\alpha
k}+(E_{n,k}-E_{m,k})\frac{im_{e}}{\hbar}\sum_{\alpha\beta}c_{n\beta}^{*}(k)c_{m\alpha}(k)d_{\alpha\beta}\\\
\end{split}$ (26)

Recall that the energy eigenstate $\ket{n,k}$ can be written in the Bloch
form:

$\ket{n,\textbf{k}}=e^{i\textbf{k}\cdot\textbf{r}}\ket{u_{n\textbf{k}}},$ (27)

where the cell-periodic wave function $\ket{u_{n\textbf{k}}}$ satisfies

$\hat{H}_{u}(\textbf{k})\ket{u_{n\textbf{k}}}=\epsilon_{n\textbf{k}}\ket{u_{n\textbf{k}}}$
(28)

Thanks to $\ket{u_{n\textbf{k}}}$, the MME for periodic Block functions
$u_{\textbf{k}\beta}$ $e^{i\textbf{k}\cdot\textbf{R}}$ are obtained as
follows:

$\bra{\textbf{k},\beta}\vec{P}\ket{\textbf{k},\beta^{\prime}}=\frac{m_{e}}{\hbar}\bra{u_{\textbf{k}\beta}}\frac{\partial
H(\textbf{k})}{\partial\textbf{k}}\ket{u_{\textbf{k}\beta^{\prime}}}.$ (29)

In the case of optical transition, the transition from the top valence to the
bottom conduction band is considered. Therefore, the component of relevant MME
can be expressed as follows,

$(P_{v,c})_{x,y,z}=\frac{m_{e}}{\hbar}\sum_{\beta,\beta^{\prime}}u_{\textbf{k}\beta,c}\frac{\partial
H_{\beta\beta^{\prime}}}{\partial k_{x,y,z}}u_{\textbf{k}\beta^{\prime},v}.$
(30)

Here, $u_{\textbf{k}\beta,c}$ and $u_{\textbf{k}\beta^{\prime},v}$
respectively represent the eigenvectors associated with the energy eigenvalues
$E_{v}$ and $E_{c}$. The real and imaginary parts of the dielectric constant
can be calculated as:

$\begin{split}\epsilon_{1}(\omega)&=1+\frac{2}{\pi}C\int_{0}^{\inf}\frac{\omega^{\prime}\epsilon_{2}(\omega^{\prime})}{\omega^{\prime
2}-\omega^{2}}d\omega^{\prime},\\\ \\\
\epsilon_{2}(\omega)&=\frac{e^{2}\hbar^{2}}{\pi
m_{e}^{2}\omega^{2}}\sum_{v,c}\int_{BZ}d^{3}k|P_{v,c}|^{2}\times\delta(E_{c}(\textbf{k})-E_{v}(\textbf{k})-\hbar\omega).\end{split}$

and

$\epsilon(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$ (31)

Here, C is the Cauchy principal value of integral; $e$ and $m_{e}$,
respectively, are charge and mass of an electron. Pv,c in
$\epsilon_{2}(\omega)$ is MME corresponding to a transition from the valence
band at energy $E_{v}$ to the conduction band at energy $E_{c}$. The Dirac-
delta function switches on MME contribution when a transition occurs from one
state to another. Ec, Ev and eigenvectors are taken from our developed TB
model. The optical absorption coefficient $\alpha(\omega)$ of a material is
determined by its frequency-dependent dielectric constant:

$\alpha(\omega)=\omega\sqrt{\frac{-\epsilon_{1}(\omega)+\sqrt{\epsilon_{1}^{2}(\omega)+\epsilon_{2}^{2}(\omega)}}{2}}$
(32)

##  XVII TB band structure of cation doped halide double perovskites

From the hopping interaction strength, we observed that electron’s hoppings
are confined to 2nd neighbor B-B′ interactions. Therefore TB fitting
parameters obtained for the pristine unit cell can be further used to
calculate the electronic properties of cation-doped double perovskites. Here,
we have designed the model for Cs2AgIn1-xBixCl6 and Cs2Na1-xAgxInCl6 at $x$ =
0.25, 0.50, and 0.75. To obtain TB band structure, we have designed four
formula supercell, from rhombohedral unit cell to cubic supercell as shown in
Fig. 2 of the main text, which contains eight metal cation sites in it. In the
supercell crystal structure, four sites are filled with Ag cations and rest
sites have been distributed between In and Bi cations depending on the value
of $x$ in Cs2AgIn1-xBixCl6. Similarly, the Hamiltonian for Cs2Na1-xAgxInCl6 is
also designed. The DFT band structure for both Cs2AgIn1-xBixCl6 and
Cs2Na1-xAgxInCl6 compounds, calculated along high symmetry k-path shown in
Fig. 2 (c) in the main text, are shown in Fig. S4.

Fig. S4: DFT (red) fitted TB (red) bands of cation doped halide perovskites
Cs2AgIn1-xBixCl6 and Cs2Ag1-xNaxInCl6.

##  XVIII TB fitted onsite energy of different orbitals for cation doped HDP

The TB hopping parameters, while cation doping, are fixed (do not vary with
$x$) and taken from previously fitted pristine compounds. However, the onsite
energies vary as a function of $x$ and are plotted in Fig. S5. The polynomial
expressions of onsite energies of Cs2AgIn1-xBixCl6 and Cs2Ag1-xNaxInCl6 are
provided in Eq. 23 and 24. Since X-{$p$} orbitals are omitted from the model
Hamiltonian, and M-X interactions are downfolded to effective onsite
energiesRaviJCP. Only onsite energies of Ag-{$s$, $d$}, In-{$s$, $p$}, and
Bi-{$s$, $p$} are varying as a function of $x$.

In Fig. S5, we have shown the variation of different onsite interaction
parameters for these two cation doped HDPs as a function of $x$. The
polynomial expressions for onsite energy vs. doping concentration $x$ for
Cs2AgIn1-xBixCl6 are;

$\displaystyle\epsilon_{s}^{Ag}(x)$ $\displaystyle=$
$\displaystyle-5.01x^{4}+19.25x^{3}-16.25x^{2}+2.41x+7.15,$ (33)
$\displaystyle\epsilon_{eg}^{Ag}(x)$ $\displaystyle=$
$\displaystyle-8.70x^{4}+18.46x^{3}-10.34x^{2}+0.28x-0.08,$
$\displaystyle\epsilon_{t2g}^{Ag}(x)$ $\displaystyle=$
$\displaystyle-6.08x^{4}+15.34x^{3}-9.46x^{2}+0.14x-0.82,$
$\displaystyle\epsilon_{s}^{In}(x)$ $\displaystyle=$ $\displaystyle
4.60x^{3}-3.58x^{2}-3.32x+5.58,$ $\displaystyle\epsilon_{p}^{In}(x)$
$\displaystyle=$ $\displaystyle 5.97x^{3}-8.48x^{2}+1.90x+7.94,$
$\displaystyle\epsilon_{s}^{Bi}(x)$ $\displaystyle=$
$\displaystyle-5.97x^{3}+6.64x^{2}-3.20x-0.66,$
$\displaystyle\epsilon_{p}^{Bi}(x)$ $\displaystyle=$ $\displaystyle
0.53x^{3}+2.64x^{2}-2.09x+4.04.$

Similarly, for Cs2Ag1-xNaxInCl6, the expressions are:

$\displaystyle\epsilon_{s}^{Ag}(x)$ $\displaystyle=$ $\displaystyle
0.1783x^{3}-1.125x^{2}+1.907x+6.19,$ (34) $\displaystyle\epsilon_{eg}^{Ag}(x)$
$\displaystyle=$ $\displaystyle 0.045x^{3}-0.285x^{2}+0.554x-0.395,$
$\displaystyle\epsilon_{t2g}^{Ag}(x)$ $\displaystyle=$ $\displaystyle
0.063x^{3}-0.410x^{2}+0.806x-1.280,$ $\displaystyle\epsilon_{s}^{In}(x)$
$\displaystyle=$ $\displaystyle 0.075x^{3}-0.378x^{2}+0.267x+4.632,$
$\displaystyle\epsilon_{p}^{In}(x)$ $\displaystyle=$ $\displaystyle
0.135x^{3}-0.948x^{2}+1.917x+6.872,$

Fig. S5: Various onsite energies of molecular orbitals as a function of $x$ in
Cs2AgInxBi1-xCl6 (left) and Cs2Na1-xAgxInCl6 (right). The black circular dots
represent the exact values obtained from curve-fitting, and the blue curve
corresponds to the polynomial fitting of the black dots. Fig. S6: The orbital
resolved band structured of Cs2AgBiCl6-xBrx as a function of $x$. The change
in the bandgap from Br to Cl is approximately to the free atomic energy
difference between Cl and Br ($\sim$0.5 eV).

##  XIX Discussion on Bromine doping in Chlorine based HDP

As stated in our B-MOP, the positions of the bands in the energy spectrum are
obtained from the molecular hybridization of (B, B′)-{$s$,$p$/$s$,$d$} – X-$p$
atomic orbitals. For example, in the case of Cs2AgBiBr6 and Cs2AgBiCl6, the
prominent interactions that define the valence band spectrum and conduction
band spectrum are Ag-$d$ – Br/Cl-$p$ and Bi-$p$ – Br/Cl-$p$, respectively. We
find that the Br-$p$ energy levels are comparable to that of Ag-$d$ energy
levels while the Cl-$p$ energy levels lie $\sim$ 0.5 eV below (9.96 eV for
Cl-$p$ and 9.46 eV for Br-$p$ orbitals). Therefore, the Ag-$d$ — Br-$p$
interaction is stronger to push the antibonding $e_{g}^{*}$ band above for the
Br compound as compared to that of the Cl compound. Since there is a large
onsite energy difference between Bi-$p$ and Br/Cl-$p$ orbitals
($\epsilon_{(Bi-p)}-\epsilon_{(Br/Cl-p)}$), the positioning of the conduction
band minimum ($\sigma_{(p-p)}^{*}$) is less affected by
($\epsilon_{(BrCl-p)}$). Thus, we believe that change in the bandgap is mainly
due to the shift in the position of the valence band while changing Cl to Br.

To illustrate the bandgap variation as a function of halide anion, in Fig. S6,
we have plotted the orbital resolved band structures of Cs2AgBiCl6-xBrx and
bandgap as a function of $x$. The band structures are calculated considering
the computationally less expensive GGA functional, as the objective is to
examine the trend. The replacement of the Br anion from Cl anion gradually
reduces the GGA bandgap of the compound from 1.34 eV for Cs2AgBiCl6 to 0.94 eV
Cs2AgBiBr6. As depicted in Fig. S6, we have noted that the orbital
contribution of the edge bands remains consistent between Br and Cl.
Additionally, as expected, we note that the contributions of Cl-$p$ orbitals
(red) and Br-$p$ orbitals (blue) are more in the valence band spectrum in
comparison to the conduction band spectrum. The net reduction in the bandgap
is $\approx$ 0.45 eV, which is roughly the same order as the onsite energy
difference between Cl-$p$ and Br-$p$ orbitals.