From structure mining to unsupervised exploration of atomic octahedral
networks

R. Patrick Xian1,2,∗ Ryan J. Morelock3 Ido Hadar4
Charles B. Musgrave3 Christopher Sutton2,∗

1Department of Engineering, University of Cambridge, Cambridge, United
Kingdom.

2Department of Chemistry and Biochemistry,
University of South Carolina, Columbia, SC, USA.

3Department of Chemical and Biological Engineering,
University of Colorado Boulder, Boulder, CO, USA.

4The Institute of Chemistry and the Center for Nanoscience and Nanotechnology,
The Hebrew University of Jerusalem, Jerusalem, Israel.

∗Corresponding authors: R.P.X. (xrpatrick AT gmail.com), C.S. (cs113 AT
mailbox.sc.edu)

> Networks of atom-centered coordination octahedra commonly occur in inorganic
> and hybrid solid-state materials. Characterizing their spatial arrangements
> and characteristics is crucial for relating structures to properties for
> many materials families. The traditional method using case-by-case
> inspection becomes prohibitive for discovering trends and similarities in
> large datasets. Here, we operationalize chemical intuition to automate the
> geometric parsing, quantification, and classification of coordination
> octahedral networks. We find axis-resolved tilting trends in ABO3 perovskite
> polymorphs, which assist in detecting oxidation state changes. Moreover, we
> develop a scale-invariant encoding scheme to represent these networks,
> which, combined with human-assisted unsupervised machine learning, allows us
> to taxonomize the inorganic framework polytypes in hybrid iodoplumbates
> (AxPbyIz). Consequently, we uncover a violation of Pauling’s third rule and
> the design principles underpinning their topological diversity. Our results
> offer a glimpse into the vast design space of atomic octahedral networks and
> inform high-throughput, targeted screening of specific structure types.

## Introduction

The coordination polyhedron is a recurring theme in chemistry from molecules
to solids [1, 2, 3, 4, 5, 6]. Among inorganic compounds, coordination
octahedra (COs) are one of the most common local atomic environments of metal-
centered ions [7]. Networks of COs, or simply octahedral networks (ONs),
display elaborate spatial patterns based on their orientation (i.e. tilting),
distortion, and connectivity, which underlie the stability and functionalities
of materials. The most prominent representatives of materials containing ONs
are the perovskite structure family [8]. Besides the corner-sharing ONs in
classic perovskites, their variants are prevalent in binary and multinary
compounds [1, 2, 9, 10], metal-rich clusters [11] (with anion-centered COs),
numerous hydroxide-based crystalline minerals [5, 6] and functional materials
[12, 13, 14, 15], etc. Besides their ubiquitous presence, these networks are a
convenient abstraction of the underlying atomistic identity and are enumerable
algebraically [5], providing an intuitive surrogate to first-principles
approaches for explaining the structure and dynamics of materials [16, 17].
Therefore, it is not an understatement that cataloguing and documenting the
architecture and distribution of ONs greatly facilitates our understanding of
matter [18, 3].

Designing materials that contain ONs may be viewed through the lens of spatial
patterning or tiling. Although the ON design space extends in multiple
directions [19], in this work, we computationally examined two dimensions:
chemical diversity and network connectivity, as illustrated in Fig. 1a, taking
the perovskite family of materials as examples. Along the chemical diversity
axis, alloying multiple ions can create new properties. Single perovskites
have the formula unit ABX3, where A, B are cations and X is an anion. The
oxide subclass of single perovskites (ABO3), which are well-studied, has a
total member count approaching $5\times 10^{3}$. This number quickly explodes
to $10^{7}$ for double oxide perovskites (AA′BB′O6) [20, 21] and much more for
higher-order multinary perovskites. The network connectivity of the inorganic
framework may be modified by pairing with selected organic cations in these
structures, which in turn alter the structural stability and electronic
properties without tuning the chemical diversity of the ON. This results in
the distinctive connectivity patterns that characterize the hybrid (i.e.
containing both organic and inorganic parts) perovskites and their extended
families of materials, called perovskitoid or hybrid metalates [13, 22, 23,
24].

Figure 1: Spatial complexity of octahedral networks. a, Illustration of the
spatial complexity of octahedral networks (ONs) along two directions: chemical
diversity and network connectivity. The difference in chemical composition is
represented by color. The types of connectivity between octahedra are
categorized as corner-, edge-, and face-sharing. b, A progression of exemplary
materials containing ONs are shown from left to right. The structures are
taken from materials databases (single perovskite CaTiO3) and publications
(double perovskite Cs2AgBiBr6, layered nonstoichiometric oxide Li4+xNi1-xWO6
[25], hybrid multinary perovskite (C3H10N)4AgInCl8 [14], and molecular
perovskite [DAB]Mn2(H2POO)6 [26]). The legend at the bottom indicates the
types of coordination octahedra present in the corresponding crystal
structures directly above each octahedron.

Existing methods for the quantification of ONs in chemical structures date
back at least half a century ago: The Glazer tilt system proposed in the 1970s
[27] classifies the spatial orientation of corner-sharing COs in perovskites
and is widely adopted by the community. The periodic nets introduced by Wells
from the 1950s to the 1970s [18, 2] and the structural hierarchy hypothesis
introduced by Hawthorne shortly afterwards [5] provide an intuitive
simplification of the ONs for understanding their similarity and complexity.
In recent decades, advances in chemical synthesis have greatly expanded the
diversity of compounds containing ONs (see Fig. 1b), which exhibit seemingly
unlimited possible arrangements. Although the classification nomenclature for
these modular structures is still under active debate, the materials families
containing ONs are increasingly viewed as a continuum with shared chemical
provenance and connectivity patterns [28, 29]. Moreover, modern atomic-
resolution electron microscopy is capable of characterizing local disordering
at heterogeneous interfaces [30], creating opportunities to broadly sample the
configurational space and cross-examine experimental results with theoretical
predictions. Driven by the growing quantity of structural data and the
community efforts towards developing compact and efficient representations of
molecular and materials systems [31, 32], we present here a computational
workflow for the structural analysis and discovery of atomic ONs to obtain
summary statistics and occurrence ranking. These are enabled by geometry
processing [33], unsupervised machine learning (manifold learning and
clustering), and human label refinement with the use of a scale-invariant
coordination network encoding (CNE) scheme. The collection of these methods
reveals the octahedral tilting trends in oxide perovskite polymorphs and
guides the identification of inorganic framework polytypes in hybrid
iodoplumbates (i.e. lead iodide salts), which led us to discover the
preference of connectivity in contrast with Pauling’s third rule [1, 34] in
these materials and the power-law distribution of polytypes.

## Results

Coordination network parsing. Central to the computational workflow in Fig. 2a
is the parsing of atomic structures, first into coordination environments, and
then into a coordination network. The coordination environment here refers
specifically to the first neighbors of a central atom with a specific symmetry
[1, 7], while the long-range connectivity between the environments is
described by the coordination network. At the intermediate range, structural
motifs such as network motif—the smallest recurrent subnetwork that captures
the predominant connectivity patterns of the whole [35]—are formed. Each level
of the structural organization may be evaluated for its specific properties.
For example, the COs are viewed as superatomic single units in an ON that form
a specific network motif, which summarizes their connectivity pattern. The
spatial arrangement of the network motifs, thanks to symmetry, produces the
ionic frameworks. We refer to structures with chemically identical frameworks
but distinguishable network motifs as framework polytypes. This hierarchical
organization is chemically interpretable [3, 5] and facilitates an efficient
programmatic implementation for structural parsing. We highlight here two
aspects that make the approach effective here and leave the details in
Methods.

  1. (1)

Generalized definition of the axial tilting angle: An important CO-level
property is the tilting angle. Glazer’s original work on octahedral tilting
[27] considers only small octahedral distortions in orthogonal lattices with
corner-sharing motif. Here, we retain the Cartesian basis as the reference
axes and use their angles formed with the corresponding octahedral diagonals
as the tilting angles to accommodate other structure types involving COs. This
procedure is automated by computationally assigning a right-hand spiral
ordering of the vertices, as shown in Fig. 2b, contingent upon a screw axis.
It also accommodates oblique lattices and more varieties of materials families
where the discussion of axial tilting of COs is still meaningful [15, 19].

  2. (2)

Bottom-up network construction: Because geometric complexes [36] formed by
adjoining coordination polyhedra are not guaranteed to be convex, searching
for connected geometric components makes their identification challenging. Our
approach builds the coordination network data structure from the bottom up as
a set of connected polyhedra, therefore the spatial relationships between the
building units can be queried. We specify the motif of an ON by the properties
of two graphs: (i) the mesh graph, constructed using atoms as the vertices and
the unique edges of the coordination polyhedra as edges. The mesh specifies
the boundary of the nearest-neighbor coordination environment and may contain
non-manifold geometry at the adjoining vertices and edges [33]; (ii) the
inter-unit graph, which uses the building units (i.e. the COs) as the vertices
and the connectivity type as edges (see Methods), is a part-based
representation of the atomic structure [5, 37]. The combination of (i) and
(ii) leads to fine-grained discernibility and bypasses (bond) distances [31]
to enable scale invariance for distinguishing between materials structures, as
will be benchmarked in the Supplementary Information.

Figure 2: Computational analysis of octahedral networks. a, Schematic of the
computational workflow for parsing ONs, which extracts geometric parameters
and connectivity from crystallographic or computational materials science
data. b, Spatial assignment of atoms in a BX6 octahedral coordination (B atom
in blue, X atoms in red, connected by chemical bonds in grey). The vertex
indices are ordered according to the right-hand spiral convention, with atoms
1 and 6 being the apical atoms and atoms 2-5 the basal atoms. The inter-unit
connectivity is computed to reconstruct the ON within the structure. The
relationships between neighboring COs are used to construct the inter-unit
graph and the mesh graph representations of the octahedral network, as shown
with an example in b for a corner-sharing 2-mer of edge-sharing 2-mers.

Next, we use our approach to analyze large materials structure datasets to
gain insights into the design of complex (e.g. multivalent) perovskites and
hybrid materials.

Octahedral tilting distribution in oxide perovskite polymorphs. The
discussions on octahedral tilting trends are normally based on the examination
of specific subclasses of single perovskite structures that have been
synthesized [16, 38]. Proxies for the tilting angle [39, 40, 41], such as the
B-O-B angle, have often been used to estimate the octahedra orientation, and
typically only small datasets of experimental data [42] have been examined in
attempts to uncover trends between these proxies and materials properties.
However, a large proportion of the abundant oxide perovskites found on earth
persist in kinetically trapped metastable states, either as polymorphs with
free energies above the convex hull vs. composition phase diagram or as
precursors or intermediates that form during chemical synthesis [43].
Consequently, understanding the large design space of oxide perovskites
requires evaluating many configurations, including those of metastable
polymorphs.

Using the workflow described in Fig. 2, we provide here a more complete
picture enabled by the availability of a large dataset of oxide perovskite
polymorphs from high-throughput computation [44] (see Methods), which includes
a total of over 2000 structures divided into ten different Glazer tilt
classes.

Figure 3: Octahedral tilting trends in orthorhombic ABO3 polymorphs. a,
Dependence on the Goldschmidt tolerance factor [45] for the axis-resolved
tilting angles of single oxide perovskite polymorphs within the tilt class
a+b-b-. The dashed lines highlight the linear trend. The density estimation is
shown in b with corresponding colors. c, An example series of lanthanide oxide
perovskite structures exhibiting tilting microtrends. The three structures
shown here are orthorhombic LaTiO3, GdTiO3, and LuTiO3 with the TiO6
coordination octahedra colored in light blue. The structures are displayed
along the crystallographic a axis as indicated in the compass. d, Map of the
Goldschmidt tolerance factor through the computationally accessed chemical
space. Representative tilting microtrends found within a through e, $A$-site
and f, $B$-site cation substitution.

We illustrate our results in Fig. 3 using orthorhombic polymorphs in the
a+b-b- (#13) tilt classes, while additional results are provided in
Supplementary Information (Section 1.1). Two distinctive trends stand out in
Fig. 3: (1) There exists a clear dependence (the global trends) of tilting
angles on the Goldschmidt tolerance factor [45] for almost all non-cubic
polymorphs; (2) When ranked by the periodicity of the elements, we observe
repeating patterns (the microtrends) of the tilting angles. The observation
indicates that, in certain classes of polymorphs, the atomic structures
obtained through isovalent substitution of $A$-site (or $B$-site) cations are
highly predictable, which gives rise to a periodic table of oxide perovskite
polymorphs. The microtrends reflect the dependence of chemical substitution on
the structure (or property) in a substitution series. Fig. 3e illustrates that
the $A$-site substitution series, such as lanthanide metalates (LnBO3), has a
monotonic change in all tilting angles as the ionic radius decreases (the
lanthanide contraction). The microtrends are similar in orthorhombic
lanthanide titanates (LnTiO3), manganites (LnMnO3), and gallates (LnGaO3). The
structural changes also correlate with the changes in their experimental
properties as previously reported [46, 47]. In $B$-site substitution series
(see Fig. 3f), a similar pattern emerges, albeit across a broader range of
substituted cations.

These tilting microtrends within a large dataset can identify outlier
materials directly based on their structural deviations. In the case of oxide
perovskites, we find that the europium and ytterbium perovskites break the
tilting trends (see Fig. 3e). Subsequent charge analysis of all cations (see
Supplementary Section 1.2) in the orthorhombic LnTiO3 and LnMnO3 series show
that europium and ytterbium ions are in the +2 oxidation state, forming
divalent-tetravalent perovskites (A2+B4+O3), whereas the remaining compounds
in the series are trivalent perovskites (A3+B3+O3). Typically, assigning
oxidation states in large computational screening datasets is a complicated
task, but our analysis shows that the structural trend obtained from COs is a
convenient indicator for changes in the oxidation state of cations.

Classification of octahedral connectivity in hybrid iodoplumbates. Small
organic molecules have been traditionally used to template the ON [48],
resulting in a wide range of modular (or recombination) structures [49, 4]
with a multimeric ON as the inorganic framework. A multimeric ON contains more
than a single CO in its motif. Besides complex oxides [50], hybrid metalates
[24] such as iodoplumbates are prominent examples. Existing terminologies used
to describe the ON connectivity pattern tend to be unstructured. For example,
purely corner-sharing network motifs are classified by (i) the cleaving plane
index [48, 13], (ii) homologous series [51, 52], which contain stacked
octahedral layers of varying thickness, and (iii) corrugated network [53],
which contains W-shaped octahedral sequences with mixed cis and trans neighbor
configurations. These classification conventions are mostly derived from the
crystallographic shear structures [17].

Figure 4: Landscape of iodoplumbate octahedral networks. a, Existing
stoichiometries of hybrid iodoplumbates (AxPbyIz) mined from the Cambridge
Structural Database. b, Relationship between the number of coordination
octahedra ($N_{\mathrm{CO}}$) and the Euler characteristic per octahedron
($\chi^{\mathrm{ON}}/N_{\mathrm{CO}}$). Each point represent an example from
hybrid iodoplumbates, which is colored according to the framework
dimensionality [54] of the material. c, Distribution of octahedral networks in
AxPbyIz classified by their connectivity in a pie chart. Composite
connectivity types are represented in parentheses.

Moreover, the umbrella term perovskitoid from the halide perovskite community
[22] was originally used to denote 1D structures with face-sharing octahedra,
but now generally refers to the extended family of hybrid metalates without
purely corner-sharing ONs. In general, these attempts lack specificity in
capturing the variety of low-dimensional structures and are incompatible with
high-throughput computational screening.

Previous attempts at linking stoichiometry to dimensionality [55, 56] showed
limited success in small case studies [57, 58], but became less effective with
the increasing complexity of the materials design space. In our view, the
limitation is due to the restricted feature space and problem formulation
(i.e. predicting dimensionality using stoichiometry). A taxonomy should also
distinguish the topological aspect of the structural components [59], which is
equivalent to polymorph (or polytype) identification. Our procedure builds on
these requirements for motif assignment in hybrid metalates, which aims to
distinguish the inorganic framework polytypes, regardless of the organic
constituent. To this end, we analyzed $\sim$ 970 compounds from the Cambridge
Structural Database that are hybrid iodoplumbates (see Methods), AxPbyIz,
where A is an organic cation, and the [PbyIz]n- polynuclear anion induces
polytypism. The hybrid iodoplumbates are the most studied members of the
hybrid metalate materials family that also includes hybrid perovskites [55,
56, 24].

We first explored the stoichiometric and topological diversity of AxPbyIz
compounds using our computational workflow. In Fig. 4a, the stoichiometric
ratio between lead (Pb) and iodine (I) in the empirical formula is on the
order of 3:1, yielding an average formula of (APbI3)y. As the number of Pb
atoms in the formula unit increases so does the scatter along the magenta
dashed line, indicating stoichiometries that more frequently deviate from this
3:1 ratio due to the growing complexity of network connectivity. When
representing the polynuclear anions ([PbyIz]n-, with $n$ being the charge it
bears) by 3D meshes, we can calculate the relationship between the number of
COs and the Euler characteristic [60, 61] of the mesh graph representing the
ONs. The results in Fig. 4b lie within a region bounded by the non-sharing
limit, where isolated octahedra (i.e. 0D hybrid perovskites [62]) exist. As
the number of COs increase, the Euler characteristic generally decreases in
value, indicating a departure from a convex shape. Details of the counting
procedure are provided in Methods.

The connectivity analysis reveals that a significant proportion of existing
materials contain only face-sharing octahedra, while some also contain corner-
sharing octahedra co-existing with face- or edge-sharing octahedra in the same
structure. A detailed statistical summary of our results following polytype
identification (see Fig. 4c and Extended Data Fig. 1) shows that in hybrid
iodoplumbates, the nearest-neighbor connectivity pattern of COs occurs with
the trend,

$\text{corner-sharing PbI}_{6}>\text{face-sharing PbI}_{6}\gg\text{edge-
sharing PbI}_{6}.$ (1)

In terms of probability of connectivity ($P$), we can equivalently write
$P_{\mathrm{PbI}_{6}}^{\mathrm{CS}}>P_{\mathrm{PbI}_{6}}^{\mathrm{FS}}\gg
P_{\mathrm{PbI}_{6}}^{\mathrm{ES}}$. This observation violates Pauling’s third
rule [1] for structure prediction of ionic compounds, which orders corner-
sharing (CS) as the most likely connectivity type between neighboring
coordination environments, followed by edge-sharing (ES), and then face-
sharing (FS). The violation leads to a modified preference in connectivity,
which have been observed in layered oxides [63], oxides under non-ambient
conditions [64], and oxides with large anion size and high coordination number
[34]. However, for halides, no systematic study is yet present. From a
structural prediction perspective, Pauling’s third rule still provides a good
estimate for the dominant connectivity type between coordination environments,
which is CS in our case for a homogeneously connected coordination network
(i.e. containing only a single connectivity type). For those with two distinct
connectivity types, our summary shows that the polytype with coexisting CS and
FS PbI6 octahedra, the two main connectivity types in Eq. (1), is the most
common, followed by the other two possibilities,

$\displaystyle\text{corner- \& face-sharing PbI}_{6}$
$\displaystyle\gg\text{corner- \& edge-sharing PbI}_{6}$
$\displaystyle\sim\text{face- \& edge-sharing PbI}_{6}.$ (2)

This may also be written more compactly in the probability of connectivity as
$P_{\mathrm{PbI}_{6}}^{\mathrm{CS,FS}}\gg
P_{\mathrm{PbI}_{6}}^{\mathrm{CS,ES}}\approx
P_{\mathrm{PbI}_{6}}^{\mathrm{FS,ES}}$, and
$P_{\mathrm{PbI}_{6}}^{\mathrm{CS,FS}}\propto
P_{\mathrm{PbI}_{6}}^{\mathrm{CS}}P_{\mathrm{PbI}_{6}}^{\mathrm{FS}}$,
similarly for the other two probabilities for mixed connectivity. Eq. (2) may
be regarded as the second-order extension of Pauling’s third rule [1] for the
specific type of coordination environment encountered here.

Using the stoichiometric and topological information of the PbyIz motif, we
construct the CNE scheme, a vector representation [65] of the ON (see
Methods). We then perform dimensionality reduction and clustering
interactively to identify the distribution of polytypes, as shown in Fig. 5a.
Manifold learning (see Methods) is used here for dimensionality reduction
because it provides an interpretable (due to well-separated clusters)
visualization [66] of the underlying structure of the dataset to facilitate
the subsequent clustering. We validate the clusters derived from the
algorithms using both external and internal metrics along with polytype labels
generated by selective visualization and human refinement. The approach
uncovers the underlying clusters within the dataset [67], which in our case
correspond to structures with a shared ON motif. Details of the procedure are
provided in Methods and Supplementary Section 2. We rank the framework
polytypes according to their occurrence in the dataset, resulting in a power-
law distribution (see Extended Data Fig. 1a), and the top twenty entries (out
of $\sim$ 70) are listed in Fig. 5b. Some of the polytypes are visualized
using a selected example in Fig. 5c.

We find that a number of clusters already show high purity in Fig. 5a, where
homogeneous clusters tend to have a more globular shape than heterogeneous
ones. The shorthand notations in 5a correspond to the indices in Fig. 5c and
its potential variants. For example, $\text{c}_{4}(1)$ indicates CS 1-layer
polytype, or single-layer 2D hybrid perovskite, $\text{c}_{11}(2)$ indicates
having the FS chain polytype with two independent chains per unit cell, 3Dp
represents 3D CS hybrid perovskites. Clusters with two major polytypes are
indicated with a slash, such as $\text{c}_{2}/\text{c}_{5}$. A complete
description of the naming rules for the polytypes is provided in the
Supplementary Information and more examples are given in Supplementary Table
5. Our observation is that the major clusters represent common network motifs:
CS $n$-layers (of different layer thickness $n$), FS chains (with different
numbers of polymeric chains per unit cell), 0D and 3D hybrid perovskite
structures. Some of the lesser-known polytypes are also aggregated into high-
purity clusters, including those with mixed connectivity such as ES chain of
FS 3-mers (Fig. 5c10), CS layer of FS 3-mers (Fig. 5c9), and CS chain of ES
2-mer of FS 2-mers (Fig. 5c7), which has a 1D arrangement of COs similar to
the columnar substructure of the V2O3 corundum (see Extended Data Fig. 2).
Although the mixed connectivity in this polytype due to the [Pb2I6]2- anion is
in seeming violation of Pauling’s third rule, it’s surprisingly more common
than many other framework polytypes found in the AxPbyIz dataset. A caveat to
mention is that the clusters in Fig. 5c with the same label generally differ
in the packing of the organic cation and its charge ($n/x$ for
[A${}_{x}]^{n+}$), which is not considered explicitly in polytype
identification but emerges due to the correlation between the characteristics
of the organic cation and the inorganic framework it supports. Chemically
speaking, the organic packing is a part of the structural motifs of these
materials but are not the network motif defined by the PbI6 COs.

Moreover, we compare the effectiveness of the CNE with an established atom-
centered representation, the smooth overlap of atomic positions (SOAP) [68], a
kernel-based symmetry-invariant representation that prioritizes on distance
information, which has been used to visualize materials design spaces [69].
The results from our controlled numerical experiments (see Supplementary Fig.
5 and Supplementary Table 4) show that, because of the scale dependence of the
kernel, clusters from the SOAP representation are generally less pure than
those obtained from the CNE scheme, and exhibit more elongated shapes, which
are less beneficial for the subsequent polytype identification.

Figure 5: Computational taxonomy of octahedral networks in AxPbyIz. a,
Embedding space visualization of PbyIz octahedral networks. Some of the major
clusters are labeled with shorthand notations in connection to c (see Text).
The black circles and the colors distinguish the identified clusters from
unsupervised machine learning. b, The top twenty polytypes in the occurrence
ranking derived from the clustering in a and human label refinement. The
colors in the bar chart represent the grouping by dimensionality. The
representative structures are shown in c1-c11, from monomeric to polymeric
octahedral networks with diverse motifs. The names of the motifs are c1, non-
sharing 1-mer (0D perovskite), c2, straight CS 2-layer; c3, corrugated CS
1-layer; c4, straight CS chain; c5, corrugated CS chain; c6, ES chain of ES
2-mers; c7, CS chain of ES 2-mer of FS 2-mers; c8, isolated FS 3-mers; c9, CS
layer of FS 3-mers; c10, ES chain of FS 3-mers; c11, FS chain. In terms of
framework dimensionality, polytype c1 and c8 are 0D, polytype c2, c5, c6, c7,
c10, c11 are 1D, polytype c3, c4, c9 are 2D. The solid blue box oulines the
unit cell of the respective structure.

The polytype occurrence ranking in in Fig. 5b reveals more information
supplementary to the connectivity trends discussed in Eqs. (1)-(2). By cross-
examination, we find a scarcity of structures with 2D layers entirely composed
of edge-sharing PbI6 octahedra (only 3 out of $\sim$ 970). In contrast, the 2D
edge-sharing motif is the majority polytype of PbI2 [70], a parent compound of
AxPbyIz, as well as the structure of the “analogous perovskite” compound BiI3
(in the sense that ReO3 is structurally analogous to ABO3) [71] that has been
reported [72]. This is likely due to the balance of charges since infinite
edge-sharing PbI6 octahedral layers are charge neutral and would require
vacancies to bear charges [13] and incorporate $A$-site cations. In fact, the
more common types of ONs in AxPbyIz with edge-sharing connectivity are
overwhelmingly 1D, featuring a short secondary unit, such as an edge-sharing
2-mer or 3-mer, stacked up through corner-sharing or edge-sharing (e.g. Fig.
5c6). Impressively, the 1D multimeric polytypes with a single connectivity
type (i.e. only containing CS, ES, or FS octahedra), traditionally found in
materials with distinct oxidation states [73], are all present within this
single family of materials via pairing with different organic cations,
indicating the tremendous framework variability of hybrid materials. These
observations suggest that polyatomic organic cations can stabilize more
diverse ONs than those typically found in the (parent) inorganic materials,
where only monatomic cations exist.

In addition to rationalizing the observed trends, looking across polytypes
with similar building units help identify design principles relating to (i)
dimensionality tuning by changing connectivity, and (ii) building unit
augmentation. For principle (i), we find that the dimensionality may be tuned
by the octahedral connectivity within the inorganic framework. This enables
the use of ES or FS multimers to construct, in principle, ONs of any dimension
(0D-3D), expanding the perovskite motif with single CS octahedra. Although
dimensional tuning has been discussed in the context of purely inorganic
materials [74], for hybrid metalates such as iodoplumbates, the vast space of
admissible organic cations suggests the possibility of fine-tuning the
material properties by changing the inorganic framework. For principle (ii),
we find that accommodation of the unit network to different organic cations
may also be achieved by augmenting the secondary unit, such as from 1-mers or
2-mers to 6-mers, which are all found in the hybrid iodoplumbate dataset.
These principles are further discussed and illustrated with examples in
Supplementary Figs. 7-8.

## Discussion

From a computational perspective, our approach marries chemical intuition with
geometry processing and the landscape of machine learning methods using
object-centric representations [65]. The objects here refer to the atom-
centered COs present in the source data, rather than the atoms reported from
simulation or experiment. Because of this object focus, the outcome is highly
interpretable. The present work motivates further developments in
preprocessing methods that are compatible with specific, object-valued data.
For example, existing approaches often require a symmetry-checking stage [40,
75] that becomes computationally expensive for large datasets of complex
structures. This necessitates simpler procedures followed by data cleaning
using chemical intuition, which produces noisier data (e.g. with missed or
inaccurately identified coordination environments). As such, the scalability
of our approach is heavily dependent upon the efficiency and accuracy of
coordination environment detection techniques.

From a materials science perspective, our oxide and iodoplumbate results
deserve further contemplation with regard to their published energetics and
related material properties. Dimensional confinement effects [76], which stem
from changes in octahedral connectivity, significantly modify the electronic
bandgap and optoelectronic properties in hybrid iodoplumbates and related
ternary halides [77, 22, 78]. In halide compositions, for instance, purely ES
connectivity increases the bandgap by $\sim$ 0.5 eV over purely CS
connectivity, and FS increases the bandgap by another $\sim$ 0.5 eV [77]. This
is comparable to the bandgap broadening observed from compositional tuning,
i.e. by switching from iodide to bromide anions ($\sim$ 0.5 eV) and bromide to
chloride anions ($\sim$ 0.6 eV) [79]. Both effects are related to the reduced
overlap of the atomic orbitals due to the lower dimensionality or the smaller
anions. Moreover, the electronic dimensionality affects the photoluminescence
properties of iodoplumbates. ES and FS motifs can lead to charge localization
which can enhance the photoluminescence quantum yield up to 100% [24]. On the
contrary, CS and mixed-connectivity motifs can be beneficial for charge
transport and modified emissions (e.g. broad emission [53]). For multinary
hybrid metalates, the dimensionality and connectivity of their octahedral
networks can also be altered by their synthesis conditions [80]. The diversity
of ON motif catalogued in the present work motivates a thorough investigation
into the relationship between connectivity patterns and materials properties
[81, 82] with band structures computed with density functional theory (DFT).
We envision a computational study whereby the functionalities could be tuned
for a fixed ON, with pairing moieties that are optimized in an approach that
resembles the docking mechanism in protein chemistry. Insights from such a
study would buttress targeted screenings for specific network connectivity
patterns, and likely reduce the resource cost and attrition rate from random
search-based screening methods [83].

Overall, our results reaffirm the necessity to design task-dependent
representations for machine learning-assisted exploration of the materials
design space [31] and the need to explicitly incorporate connectivity
information in existing distance-based representations [68, 37]. The
representation from CNE may be merged with those for organic molecules to
build interpretable predictors and generative models of hybrid functional
material properties. The modified Pauling’s third rule in the materials
examined here and by others [63, 64, 34] motivates its reformulation using a
fully probabilistic approach that reflects chemical specificity and serves
growing data. Our approach may also be adapted to extract insights from other
analogous, yet less investigated materials families featuring octahedral
networks, such as inorganic and hybrid metalates with nonmetal or metalloid
ligands, high-entropy materials [84], coordination polymers, and molecular
perovskites [15], which have even more complex design spaces and highly
tunable structure-property landscapes.

### Methods

Materials datasets and their curation. The oxide perovskite (ABO3) polymorph
dataset was obtained from high-throughput computational screening of
compositionally valid compounds. Although details of the data generation have
been reported elsewhere [44], we provide here a brief summary of the
computational method: The structures of the oxide perovskites were generated
using the software SPuDS [85], which accounts for electrostatic structural
distortions using bond-valence theory. The SPuDS-generated structures were
further optimized using density functional theory within the Vienna Ab-initio
Simulation Program (VASP 5.4.1) and under periodic boundary conditions
utilizing projector augmented wave (PAW) pseudopotentials and the Perdew-
Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) exchange-
correlation functional [86]. Electronic self-consistent field energies were
converged to within $10^{-6}$ eV and forces were converged to within 0.01
eV/Å. The dataset is represented in the pymatgen-style json format, and a
custom parser code (https://github.com/Sutton-Research-Lab/parsetta) was
developed for querying subsets and filtering.

The structure data for hybrid iodoplumbates (AxPbyIz) were obtained from the
Cambridge Structural Database (CSD) [87], accumulated until early 2022. We
directly queried the CSD via a chemical search of lead and iodine compounds,
which yielded over 1020 entries. We next removed entries that were not hybrid
iodoplumbates, including (i) structures that did not contain organic
components (removed by checking the chemical species beyond Pb and I in the
compound) and (ii) structures with lead coordination environments other than
octahedral (removed during the initial coordination environment detection). We
also kept only the compounds with a Pb stoichiometry of up to 6 in the
chemical formula. The final number of stoichiometrically-ordered data entries
was 971, which includes fewer than 6 entries with duplicated IDs in the CSD.
These duplicates were kept because of their different measurement conditions.

The majority of processing steps were carried out using crystmorph, which
implements the workflow in Fig. 2 for the analysis of coordination networks.
We adopted a chemical environment parser [75] to analyze the structures by
iteratively identifying the space groups of possible coordination environments
centered locally on every atom. For the examples used in this work, the
analysis was performed for the metal-centered coordination environments, i.e.
on lead for the iodoplumbates, to speed up the process. These groupings are
coordination octahedra and we represent them as base objects in crystmorph.
Subsequent algebraic operations between COs described in the following two
subsections are implemented for either the atomic level (mesh graph) or the
CO-level (inter-unit graph). The characteristics and diversity of the dataset
are illustrated in Extended Data Fig. 1.

Octahedral axial tilt calculation. A convex octahedron ($O$) comprises a
convex set of points in $\mathbb{R}^{3}$ as the vertices, while the pairs and
trios of adjacent vertices form the edges and faces, respectively. The
vertices, edges and faces are also known as the elements of the polyhedron.

$O=\\{s_{i}\\}_{i=6}=\\{s_{1},...,s_{6}\\}$ (3)

Each point $s_{i}$ is determined by its spatial coordinates
$(x_{i},y_{i},z_{i})$. In computational geometry, convex octahedra are convex
hulls of their vertices. Existing fast algorithms for convex hull construction
can determine the edges and faces of the octahedron 3D point set [88] and can
calculate the octahedral volume without specifically ordering the vertices.

$V(O)=V(\mathrm{Conv}(O))$ (4)

To order the vertices according to the right-hand spiral convention, we first
define the spiral axis unit vector $\mathbf{e}_{j}$ $(j=x,y,z)$ and order the
atoms according to their projection onto the spiral axis. This separates the
coordination atoms into apical and basal atoms. Then, we order the basal atoms
with respect to the basal plane origin in right-handed (or left-handed)
fashion accordingly. The ordering of the vertices of a graph is also called
graph canonization [89]. The general steps of the tilting angle calculation
are provided as follows, starting with coordinates of the unordered vertices
and a set of axis vectors,

  1. (i)

Calculate the projection of the coordinates of each vertex onto a given axis
vector.

  2. (ii)

Rank the projected coordinates of the vertices along the axis vector.

  3. (iii)

Order the vertices by their ranking: the topmost and the lowermost vertices
are the apical atoms, the remaining become the basal atoms.

  4. (iv)

Check the vertex ordering by comparing with octahedral diagonals.

  5. (v)

Calculate the apical vector, which connects the apical atoms.

  6. (vi)

Calculate the tilting angle using the dot product between the apical vector
and the given axis vector.

  7. (vii)

Iterate over steps (i) - (vi) for the other axis vectors.

Going from unordered to ordered vertices requires to determine a permutation
matrix for the vertex ordering vector. Although simple to implement and fast,
the projection-based method described above is subject to errors in structured
with oblique lattices or with large tilting angles (e.g. > 30∘). We introduces
an ordering check as step (iv) based on the assumption that the three longest
line segments connecting any two vertices are the diagonals (such as the
vertex pairs $\\{1,6\\}$, $\\{2,4\\}$, $\\{3,5\\}$ in Fig. 2b) of the
octahedron. This assumption is true if the octahedron is not extremely
distorted, which is satisfied for the materials family studied here. Note that
for a given orientation (defined by an axis vector), only one of the
octahedral diagonals overlaps with the apical vector, unless two of diagonals
are symmetrically equivalent with respect to the given orientation. The cases
that fail the check are subject to one more permutation for correction.
Another caveat to mention is that the ordering of the basal atoms contains
rotational ambiguity (i.e. can be shifted clockwise or counterclockwise by the
same index, without violating the spiral ordering), but this does not affect
the ordering check because the vertices that form the octahedral diagonals in
the basal plane always differ by 2 in their ordering.

Octahedral network connectivity analysis. Construction of infinite polyhedra
was first systematically discussed by Coxeter [90, 91], which requires all
constituents to be connected in all dimensions. This would only apply to the
3D networks for those formed by PbI6 COs or other compositions. The octahedral
network we refer to in this work may also be finite in one to three
dimensions, which is a special case of a polyhedral complex [92]. For
connectivity analysis, the center atom is ignored. The network analysis
proceeds through three steps: (i) Individual coordination octahedra (COs);
(ii) Adjacency matrix of the octahedral network inter-unit graph; (iii)
Properties of the mesh graph and the inter-unit graph. We define the inter-
unit graph $G_{I}(V_{I},E_{I})$ as the graph where each coordination
environment is represented by a node and their the number of shared atoms
between environments as edge weight [93]. The mesh graph $G_{M}(V_{M},E_{M})$
forms the collective boundary of the octahedral network. It has a similar
meaning in computer graphics when discussing the triangulated mesh of an
object [94].

Geometric combinatorics. The analysis starts from the COs detected by a 3D
Voronoi partitioning algorithm [75]. In crystmorph, we also provide a faster
version using only the nearest neighbors. We consider the network formed by
bringing together individual octahedra ($O_{i}$, $i=1,2,...,N_{\mathrm{CO}}$)
with various connectivity types, a process called gluing in topology [36]. The
ON (in the quotient space formed by gluing) may be expressed as,

$S^{\mathrm{ON}}=\bigcup_{i=1}^{N_{\mathrm{CO}}}O_{i}.$ (5)

Although elaborate gluing schemes exist for polyhedra [95], here we formulate
gluing as the real-space intersection(s) of two or more octahedra. Its
chemical validity is grounded on the layer-stacking characteristics of crystal
growth [8].

Reconstruction of the network is achieved through combinatorial query of the
spatial relationship between the COs, which recovers the adjacency matrix of
the inter-unit graph ($A^{\mathrm{IG}}$). We define the connectivity of glued
COs using the intersection of the individual octahedra. The intersection
$\Delta_{12}$ between octahedra $O_{1}$ and $O_{2}$ is

$\Delta_{12}=O_{1}\cap O_{2}=\\{s_{i}|s_{i}\in O_{1},s_{i}\in
O_{2},i=1,2,...,N_{\mathrm{CO}}\\}.$ (6)

Two octahedra are corner-sharing if they share only one vertex and that their
convex hulls have no spatial overlap (non-overlapping condition) except at the
shared vertex (the cardinality of $\Delta_{12}$ is Card($\Delta_{12}$) = 1).
Similarly, octahedra $O_{1}$ and $O_{2}$ are edge-sharing if and only if they
share only one edge (or two adjacent vertices, Card($\Delta_{12}$) = 2) and
their convex hulls have no other overlap. The face-sharing criterion requires
the existence of three adjacent, non-collinear vertices (Card($\Delta_{12}$) =
3) mutual to the two adjoining octahedra. For scenarios where octahedral metal
centers have fractional site occupancies of two or more atoms, only the atom
with the highest fractional occupancy is used to avoid overcounting and/or
miscounting polyhedrons. These definitions are summarized into the following
equation,

$A_{i,j}^{\mathrm{IG}}=\begin{cases}0,&\text{if $O_{i}$ and $O_{j}$ are non-
sharing},\\\ 1,&\text{if $O_{i}$ and $O_{j}$ are corner-sharing},\\\
2,&\text{if $O_{i}$ and $O_{j}$ are edge-sharing},\\\ 3,&\text{if $O_{i}$ and
$O_{j}$ are face-sharing}.\end{cases}$ (7)

The matrix $A^{\mathrm{IG}}$ is symmetric
($A_{i,j}^{\mathrm{IG}}=A_{j,i}^{\mathrm{IG}}$) and generally exhibits a block
structure. The limiting cases are the 0D structures [62], where all entries
are zero, and the 3D structures, where the blocks are highly overlapping.

To calculate the mesh graph (MG) characteristics of the ON, we used the
information gathered from the entries of $A^{\mathrm{IG}}$. In our problem,
the MG contains only closed surface mesh. A general treatment of the counting
problem may be solved with a simple heuristic: the sharing of each adjacent
pair of iodine (for iodoplumbates) atoms amounts to an edge-deletion
operation. Similarly, a face-deletion operation amounts to sharing of an
adjacent iodine trio, etc. Then, we used the following formulas to calculate
the numbers of unique vertices ($N_{V}^{\mathrm{MG}}$), edges
($N_{E}^{\mathrm{MG}}$) and faces ($N_{F}^{\mathrm{MG}}$), respectively.

$\displaystyle N_{V}^{\mathrm{MG}}$
$\displaystyle=6N_{\mathrm{CO}}-3N_{3}^{\mathrm{IG}}-2N_{2}^{\mathrm{IG}}-N_{1}^{\mathrm{IG}}$
(8) $\displaystyle N_{E}^{\mathrm{MG}}$
$\displaystyle=12N_{\mathrm{CO}}-3N_{3}^{\mathrm{IG}}-N_{2}^{\mathrm{IG}}$ (9)
$\displaystyle N_{F}^{\mathrm{MG}}$
$\displaystyle=8N_{\mathrm{CO}}-2N_{3}^{\mathrm{IG}}$ (10)

The terms $N_{3}^{\mathrm{IG}}$, $N_{2}^{\mathrm{IG}}$, and
$N_{1}^{\mathrm{IG}}$ are, respectively, the number of 3-connected (face-
sharing), 2-connected (edge-sharing), and 1-connected (vertex-sharing) entries
in the triangular submatrix of $A^{\mathrm{IG}}$ due to its symmetry. Eqs
(8)-(10) may be derived using the intuitive definition of gluing in Eq. (5).
The initially isolated $N_{\mathrm{CO}}$ convex octahedra, each having 6
vertices, 12 edges, and 8 faces, are glued together. Each instance of face-,
edge- and corner-sharing reduces the number of vertices by 3, 2, and 1,
respectively, as in Eq. (8). Each instance of face- and edge-sharing reduces
the number of edges by 3 and 1, respectively, as in Eq. (9). On the contrary,
the total number of faces is only affected by the number face-sharing of
octahedra, as in Eq. (10).

This counting method is generalizable to higher-order adjacency tensors [96]
when more than two coordination octahedra are adjoined at a single vertex.
This results in a general counting rule for the elements of the complex (i.e.
mesh graph representation of the octahedral network) which may be written as a
telescopic sum as follows,

$N_{X}^{\mathrm{MG}}=a_{X}N_{\mathrm{CO}}+\sum_{i=1}^{3}\left(-b_{X,i}N_{i}^{(2)}+c_{X,i}N_{i}^{(3)}-...\right)\,(X=V,E,F).$
(11)

The parenthesized numbers in the superscript of $N$ indicate the body order of
connectivity considered, with ordinary graphs (or “2-graphs”, corresponding to
the IG in this example, or $N_{i}^{(2)}\coloneqq N_{i}^{\mathrm{IG}}$) having
two-body connectivity, first-order hypergraphs (or “3-graphs”) having three-
body connectivity, etc. The summation index $i$ in Eq. (11) has the same
meaning as in Eqs. (8)-(10). Deriving Eq. (11) requires measure theory and the
general form of the inclusion-exclusion principle [60], which we leave for
future work. We focus here on the truncated version used in this work up to
$N^{(2)}$.

Euler characteristic. We calculate the Euler characteristic [61] for the MG
according to,

$\chi^{\mathrm{MG}}=N_{V}^{\mathrm{MG}}-N_{E}^{\mathrm{MG}}+N_{F}^{\mathrm{MG}}$
(12)

where $N_{V}$, $N_{E}$, and $N_{F}$ are the counts of unique vertices, edges
and faces of the mesh graph, respectively. Assuming only two-body connectivity
and substituting Eqs. (8)-(10) into Eq. (12), we obtain the Euler
characteristic,

$\chi^{\mathrm{MG}}=2N_{\mathrm{CO}}-2N_{3}^{\mathrm{IG}}-N_{2}^{\mathrm{IG}}-N_{1}^{\mathrm{IG}}.$
(13)

Since the $N_{i}^{\mathrm{IG}}$ $(i=1,2,3)$ terms are nonnegative integers,
the right-hand side of Eq. (13) may be used to establish bounds and relations
for $\chi^{\mathrm{MG}}$, so we write
$\chi^{\mathrm{MG}}(N_{i}^{\mathrm{IG}})$.

Firstly, we consider the cases when the ON can be partitioned into $n$
unconnected convex objects. This requires that $\chi^{\mathrm{MG}}=2n$ for
$n\in\mathbb{N},0<n\leqslant N_{\mathrm{CO}}$. One limiting case is when
$n=N_{\mathrm{CO}}$, where the ON may be partitioned into convex objects only
when $N_{3}^{\mathrm{IG}}=N_{2}^{\mathrm{IG}}=N_{1}^{\mathrm{IG}}=0$, and
$\chi^{\mathrm{MG}}$ is reduced to $2N_{\mathrm{CO}}$. This leads to the upper
bound,

$\sup_{\begin{subarray}{c}N_{i}^{\mathrm{IG}}\geqslant 0,\\\
N_{i}^{\mathrm{IG}}\in\mathbb{N}\end{subarray}}\frac{\chi^{\mathrm{MG}}(N_{i}^{\mathrm{IG}})}{N_{\mathrm{CO}}}=2,$
(14)

which is the non-sharing limit specified in Fig. 5b. This is the scenario with
completely isolated octahedra (also known as zero-dimensional structures
[62]). Technically speaking, in this limiting case, the quotient space of ON
in Eq. (5) becomes a disjoint union of the subspaces containing individual
octahedra.

Secondly, the space-filling 3D framework completely made of corner-sharing
octahedra is another special case. In this case, the term
$N_{3}^{\mathrm{IG}}=N_{2}^{\mathrm{IG}}=0$. If we ignore the unit-cell
boundary, each octahedron is connected to 6 others, resulting in
$N_{3}^{\mathrm{IG}}\sim 3N_{\mathrm{CO}}$, which yields
$\chi^{\mathrm{MG}}/N_{\mathrm{CO}}\sim-1$. This limiting case is not attained
because of the existence of unit-cell boundaries, but it represents one of the
most connected ONs. Moreover, the limiting cases of octahedral networks with
any other dimensionality and modularity may be related to $\chi^{\mathrm{MG}}$
using a similar type of analysis.

Unsupervised polytype identification. The human-assisted unsupervised machine
learning approach combines the outcome of dimensionality reduction and
clustering with human label refinement. For modular materials composed of
connected coordination polyhedra [4], the “natural groupings” from chemical
intuition should possess similar types of connectivity, which was our initial
assumption. Several major polytypes exist for hybrid iodoplumbates [56, 24],
while some rare polytypes exist with very few or even just one example.
Identifying the characteristic structural groupings of the hybrid
iodoplumbates is carried out using dimensionality reduction and clustering.
Here, we combined Uniform Manifold Approximation and Projection (UMAP) [97]
with common-nearest-neighbor (CommonNN) clustering [98] to identify the
characteristic structural groupings of the hybrid iodoplumbates.

Coordination network encoding. The representation obtained from the CNE scheme
includes the stoichiometries and the octahedral network parameters described
previously. For each material, we construct the feature vector for the
coordination network as a six-tuple,

$\mathcal{M}=(y,z,N_{\mathrm{CO}},N_{V}^{\mathrm{MG}},N_{E}^{\mathrm{MG}},N_{F}^{\mathrm{MG}}),$
(15)

where $y$ and $z$ are the stoichiometries of Pb and I in the chemical formula
AxPbyIz, $N_{\mathrm{CO}}$ is the number of COs detected in the structure. The
parameters of the mesh graph in Eq. (15) are calculated using the truncated
expressions as in Eqs. (8)-(10). The feature variability comes from the
different definitions of the crystal structure, including (i) the space group
of the unit cell, (ii) the number of formula units within the unit cell, and
(iii) the boundary of the unit cell. These factors influence all terms in the
feature vector. A full list of features for the AxPbyIz compounds is provided
in Supplementary Table 6.

Dimensionality reduction. UMAP is a nonlinear dimensionality reduction
algorithm, which minimizes the fuzzy set cross-entropy ($L_{\mathrm{UMAP}}$)
between the input data in the high-dimensional manifold (e.g. the $p$ values)
and its counterpart in the low-dimensional target embedding (e.g. the $q$
values).

$L_{\mathrm{UMAP}}=\sum_{i\neq
j}\left(p_{ij}\log\left(\frac{p_{ij}}{q_{ij}}\right)+\left(1-p_{ij}\right)\log\left(\frac{1-p_{ij}}{1-q_{ij}}\right)\right)$
(16)

The $i$ and $j$ are the data indices, which in our case refer to the indices
of distinct materials. The $L_{\mathrm{UMAP}}$ term is minimized via
stochastic gradient descent to obtain the optimal embedding [97] given a
hyperparameter setting. We employed the implementation in umap
(https://github.com/lmcinnes/umap) with a Euclidean distance between data
points in the input feature space specified by Eq. (15).

Interactive clustering. CommonNN is a density-based clustering algorithm that
can effectively recognize clusters with a more irregular shape (i.e. deviating
from globular) [98]. We used its implementation in scikit-learn-extra
(https://github.com/scikit-learn-contrib/scikit-learn-extra), and tuned its
clustering hyperparameters via grid search by maximizing the silhouette score
[99]. Our procedure began with a mix of homogeneous and heterogeneous clusters
that were progressively refined and separated based upon their distinguishable
features. (see Supplementary Table 3). Representative members of each cluster
were then visually inspected by a human expert with a crystallographic
software (Vesta 3 [100]) to produce the polytype labels according to the
naming rules described in the Supplementary Section 2.3. We assessed the
quality of the initial clustering using a set of internal and external cluster
validation metrics [101] and report the results in Supplementary Table 4. A
list of procedures is also provided in Supplementary Table 3.

### Acknowledgments

We thank G. Csanyi (University of Cambridge) and H.-C. zur Loye (University of
South Carolina) for helpful discussions. The work was supported by the Air
Force Research Laboratory through the research grant (FA8655-21-1-7010). CBM
and RJM were supported by the U.S. Department of Energy (DOE), Office of
Energy Efficiency and Renewable Energy (EERE), Hydrogen and Fuel Cell
Technologies Office (HFTO), and specifically the HydroGEN Advanced Water
Splitting Materials Consortium, established as part of the Energy Materials
Network under this same office (award DE-EE0008088). CBM and RJM also
acknowledge support from the National Science Foundation (awards NSF
CHEM-1800592 and CBET-2016225). C.S. acknowledges start-up support from the
University of South Carolina.

### Data Availability

The single oxide perovskite polymorph dataset is available on GitHub
(https://github.com/rymo1354/gii_minimization/tree/main/structures_and_energies).
The hybrid iodoplumbate dataset, including the calculated features and labels,
is provided as supplementary files.

### Code Availability

The code for the analysis of octahedral networks is accessible on GitHub
(https://github.com/RealPolitiX/crystmorph). The final version will be made
public upon acceptance of the manuscript.

### Author contributions

R.P.X. conceived and coordinated the project, developed the theory and code
for the quantification of octahedral networks and carried out the analysis.
R.J.M. provided the oxide perovskite dataset and conducted the charge analysis
under the supervision of C.B.M.. I.H. provided support on the crystal
chemistry and nomenclature of hybrid iodoplumbates. C.S. and C.B.M. provided
support on the problem formulation and interpretation of results. R.P.X. wrote
the initial draft with inputs from R.J.M. and I.H.. All authors contributed to
discussion and revision of the manuscript to its final form.

### Competing interests

The authors declare no competing interests in the content of the article.

Extended Data Fig. 1: Statistical summary of octahedral networks in AxPbyIz.
a, The ranking distribution of the inorganic framework polytypes in the
AxPbyIz dataset in a log-log plot. b, Distribution of the structures
differentiated by their PbI6 octahedral connectivity types. Here, CS (corner-
sharing) only, ES (edge-sharing) only, and FS (face-sharing) only includes the
structures with the respective unary connectivity type. The structures
containing mixed connectivity are divided into binary and ternary connectivity
types and labeled with the constituent types in parenthesis. c, Framework
dimensionality of the structures. f, Distribution of the layer thickness in 2D
layered perovskites (polytype name CS n-layer, n = 1, 2, …), along with an
example structure in e showing a CS 2-layer polytype. f, Distribution of 1D FS
chains per unit cell, along with an example structure in g showing four
symmetry-independent FS chains per unit cell, which is indicated by the solid
blue box.

Extended Data Fig. 2: Corundum-like columns in AxPbyIz. a, Illustration of the
V2O3 in corundum with a single column highlighted, which contains all of
corner-, edge-, and face-sharing octahedra. The cyan octahedra has the
composition VO6 and the oxygen atoms are shown in red. Examples of similar
columnar structure with mixed connectivity in the octahedral network of hybrid
iodoplumbates are compared in b, (C10H26N2O)Pb2I6 and c, C24H41N4Pb2I6, both
shown in their respective side (left) and top (right) views. The blue box
indicates the unit cell in all figures.

> Supplementary Information
>  From structure mining to unsupervised exploration of atomic octahedral
> networks

###### Contents

  1. 1 Octahedral tilting trend and outlier discovery
    1. 1.1 Tilting distributions in oxide perovskite polymorphs
    2. 1.2 Oxidation state analysis
  2. 2 Unsupervised learning of octahedral network motifs
    1. 2.1 Comparison of representation and machine learning methods
    2. 2.2 Human label refinement and quality assessment
    3. 2.3 Queryable naming rules for inorganic framework polytypes
    4. 2.4 Diversity and surprises in iodoplumbate octahedral networks

##  1 Octahedral tilting trend and outlier discovery

###  1.1 Tilting distributions in oxide perovskite polymorphs

Using the same procedure as demonstrated in the main text Fig. 3, we analyzed
the other perovskite tilt classes, the results are shown in Supplementary Fig.
1 for tilt classes a+a+a+ (#3), a+a+c- (#5) and a-b-b- (#13), in Supplementary
Fig. 2 for tilt classes a-a-a- (#14), a0b+b+ (#16) and a0b+c- (#17), and in
Supplementary Fig. 3 for tilt classes a0b-b- (#20), a0a0c+ (#21) and a0a0c-
(#22). The index number of the tilt class [27, 102] is in parenthesis.

Supplementary Figure 1: Tilting angle distributions (part I). For the Glazer
tilt classes of a+a+a+ (#3), a+a+c- (#5) and a-b-b- (#13) for single oxide
perovskites (ABO3). The empirical marginal distribution of the tilting angles
along different axes are shown as histograms on the right, along with the
respective estimated distributions (black lines).

Supplementary Figure 2: Tilting angle distributions (part II). For the Glazer
tilt classes of a-a-a- (#14), a0b+b+ (#16) and a0b+c- (#17) for single oxide
perovskites (ABO3). The marginal distribution of the tilting angles along
different axes are shown as histograms on the right, along with the respective
estimated distributions (black lines).

Supplementary Figure 3: Tilting angle distributions (part III). For the Glazer
tilt classes of a0b-b- (#20), a0a0c+ (#21) and a0a0c- (#22) for single oxide
perovskites (ABO3). The marginal distribution of the tilting angles along
different axes are shown as histograms on the right, along with the respective
estimated distributions (black lines). The $\theta_{z}$ angles for tilt
classes #21 and #22 are strictly zero, therefore no distribution is shown.

Most of the computationally optimized atomic structures of ABO3 in the dataset
show a unique set of tilting angles ($\theta_{x}$, $\theta_{y}$,
$\theta_{z}$), such as tilt classes #3, #5, #10, #16, #17, #21-#23. Others
show two or more sets, largely due to the oblique lattice system and the
imperfect symmetry constraints. For presentation, the calculated tilting
angles for each material are averaged along each axis. Some of the structures
in the geometry optimization end up in a different phase are also reflected in
the axis-resolved tilting angles, such as the those with angles less than 20∘
and larger than 0∘ in Supplementary Fig. 1b.

###  1.2 Oxidation state analysis

We assigned oxidation states using the DFT+U onsite magnetic moment(s) rather
than through Bader charge analysis [103] due to charge self-regulation that
arises in the 3d transition metals [104], whereby switching oxidation states
negligibly changes the net charge(s) contained within the zero flux surfaces
of the electron charge density. The onsite magnetization is therefore a better
indicator of 3d transition metal (e.g., Mn and Ti) oxidation states in ABO3
perovskites than the Bader charge.

The oxidation states of the orthorhombic lanthanide manganite (LnMnO3)
perovskites and lanthanide titanate (LnTiO3) perovskites with space group
$Pnma$ were deduced from the DFT-computed $d$-orbital magnetic moments of Mn
and Ti sites, which occupy the B-sites of these perovskites. The formal
oxidation states of the lanthanide A-sites (Ln), Mn or Ti B-sites, and oxygen
(O) X-sites of each composition were constrained to be charge-balanced, such
that they summed to zero. Tables S1 and S2 order the perovskite compositions
by Ln atomic number, and tabulate the $d$-orbital magnetic moments of Mn and
Ti, as well as the $f$-orbital magnetic moments of the A-site lanthanides.

Supplementary Table 1: Oxidation states of lanthanide manganite (LnMnO3). Assignment of formal oxidation states to the orthorhombic LnMnO3 perovskites using DFT-computed, site-specific magnetic moments. The compounds are ordered by the A-site atomic number. Compound | A1 | A2 | A3 | A4 | B1 | B2 | B3 | B4 | Aoxi | Boxi
---|---|---|---|---|---|---|---|---|---|---
LaMnO3 | 0.008 | 0.008 | 0.008 | 0.008 | 3.881 | 3.881 | 3.881 | 3.881 | +3 | +3
PrMnO3 | 0 | 0 | 0 | 0 | 3.919 | 3.919 | 3.919 | 3.919 | +3 | +3
NdMnO3 | 0 | 0 | 0 | 0 | 3.832 | 3.832 | 3.832 | 3.832 | +3 | +3
SmMnO3 | 0 | 0 | 0 | 0 | 3.888 | 3.888 | 3.888 | 3.888 | +3 | +3
EuMnO3 | 6.392 | 6.392 | 6.394 | 6.394 | 3.677 | 3.677 | 3.677 | 3.677 | +2, +3 | +4, +3
GdMnO3 | 6.878 | 6.878 | 6.878 | 6.878 | 3.881 | 3.881 | 3.881 | 3.881 | +3 | +3
TbMnO3 | 0 | 0 | 0 | 0 | 3.883 | 3.883 | 3.883 | 3.883 | +3 | +3
DyMnO3 | 0 | 0 | 0 | 0 | 3.82 | 3.82 | 3.82 | 3.82 | +3 | +3
HoMnO3 | 0 | 0 | 0 | 0 | 3.819 | 3.819 | 3.819 | 3.819 | +3 | +3
ErMnO3 | 0 | 0 | 0 | 0 | 3.818 | 3.818 | 3.818 | 3.818 | +3 | +3
YbMnO3 | 0 | 0 | 0 | 0 | 3.065 | 3.065 | 3.065 | 3.065 | +2 | +4
LuMnO3 | 0 | 0 | 0 | 0 | 3.879 | 3.879 | 3.879 | 3.879 | +3 | +3

In the manganite perovksite series, Mn3+ has the electronic configuration
[Ar]3d4 and Mn4+ has the electronic configuration [Ar]3d3. Mn with DFT-
computed magnetic moments that approach 4 are most likely in the +3 oxidation
state (high-spin), whereas Mn with magnetic moments that approach 3 are most
likely in the +4 oxidation state (high-spin). For compositions with Ln = La,
Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, and Lu, the magnitudes of Mn magnetic moments
are greater than or equal to 3.818, suggesting that Mn is in the +3 oxidation
state; to ensure charge balance, these lanthanides must also adopt the +3
oxidation state. The magnetic moment of Mn in YbMnO3 is 3.065, suggesting that
Mn adopts the +4 oxidation state in this composition, and that Yb must
therefore adopt the +2 oxidation state to ensure charge balance. The magnetic
moment of Mn in the EuMnO3 composition is 3.677, suggesting it is between the
+3 and +4 oxidation states. Accordingly, Eu (which has electronic
configurations of [Xe]4f66s2 and [Xe]4f76s2 in the +3 and +2 oxidation states,
respectively) has a magnetic moment of 6.392, or between the expected
magnetizations of 6 for Eu3+ (high-spin) and 7 for Eu2+ (high-spin).

Supplementary Table 2: Oxidation states of lanthanide titanate (LnTiO3). Assignment of formal oxidation states to the orthorhombic LnTiO3 perovskites using DFT-computed, site-specific magnetic moments. The compounds are ordered by the A-site atomic number. Compound | A1 | A2 | A3 | A4 | B1 | B2 | B3 | B4 | Aoxi | Boxi
---|---|---|---|---|---|---|---|---|---|---
LaTiO3 | 0.056 | 0.056 | 0.056 | 0.056 | 0.783 | 0.783 | 0.784 | 0.781 | +3 | +3
CeTiO3 | 0.931 | 0.951 | 0.946 | 0.939 | 0.821 | 0.822 | 0.821 | 0.822 | +3 | +3
PrTiO3 | 0 | 0 | 0 | 0 | 0.816 | 0.816 | 0.816 | 0.816 | +3 | +3
NdTiO3 | 0 | 0 | 0 | 0 | 0.817 | 0.817 | 0.817 | 0.817 | +3 | +3
SmTiO3 | 0 | 0 | 0 | 0 | 0.817 | 0.817 | 0.817 | 0.817 | +3 | +3
EuTiO3 | 6.638 | 6.638 | -6.638 | -6.638 | 0 | 0 | 0 | 0 | +2 | +4
GdTiO3 | 6.755 | 6.755 | 6.755 | 6.755 | -0.726 | -0.727 | -0.727 | -0.727 | +3 | +3
TbTiO3 | 0 | 0 | 0 | 0 | 0.82 | 0.82 | 0.82 | 0.82 | +3 | +3
DyTiO3 | 0 | 0 | 0 | 0 | 0.821 | 0.821 | 0.821 | 0.821 | +3 | +3
HoTiO3 | 0 | 0 | 0 | 0 | 0.823 | 0.823 | 0.823 | 0.823 | +3 | +3
ErTiO3 | 0 | 0 | 0 | 0 | 0.824 | 0.824 | 0.824 | 0.824 | +3 | +3
TmTiO3 | 0 | 0 | 0 | 0 | 0.826 | 0.826 | 0.826 | 0.826 | +3 | +3
YbTiO3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +2 | +4
LuTiO3 | 0 | 0 | 0 | 0 | 0.829 | 0.829 | 0.829 | 0.829 | +3 | +3

In the titanate perovksite series, Ti3+ has the electronic configuration
[Ar]3d1 and Ti4+ has the electronic configuration [Ar]. Ti with DFT-computed
magnetic moments that approach 1 are most likely in the +3 oxidation state,
whereas Mn with magnetic moments that approach 0 are most likely in the +4
oxidation state. For compositions with Ln = La, Ce, Pr, Nd, Sm, Gd, Tb, Dy,
Ho, Er, Tm and Lu, the magnitudes of Ti magnetic moments are greater than or
equal to 0.727, suggesting that Ti is in the +3 oxidation state; to ensure
charge balance, these lanthanides must also adopt the +3 oxidation state. The
magnetic moment of Mn in YbMnO3 is 3.065, suggesting that Mn adopts the +4
oxidation state in this composition, and that Yb must therefore adopt the +2
oxidation state to ensure charge balance. The magnetic moments of Ti in the
EuTiO3 and YbTiO3 compositions are 0, which suggests that Ti is in the +4
state in these compositions. Yb and Eu must therefore adopt the +2 oxidation
states for these compositions to ensure charge balance.

##  2 Unsupervised learning of octahedral network motifs

###  2.1 Comparison of representation and machine learning methods

The entire data processing pipeline for polytype identification discussed in
the Main Text is summarized in Supplementary Fig. 4. The data featurization
step uses the coordination network encoding (CNE) scheme, which yields a
scale-invariant representation of the connectivity pattern among coordination
environments. To identify hybrid materials by their inorganic framework
polytypes, a dimensionality reduction step is required to improve the
clustering outcome. We find that, compared with linear dimensionality
reduction methods, nonlinear methods result in more separation between
compartments of data, such as t-distributed stochastic neighbor embedding
(tSNE) [105] and uniform manifold approximation and projection (UMAP) [97]
result in more separable embeddings to enable downstream clustering and
discover subpopulations. To check the effectiveness of the CNE in
distinguishing between inorganic framework polytypes, we compared its outcome
against that of the smooth overlap of atomic positions (SOAP) representation
[68], as implemented in dscribe [106].

The results, shown in Supplementary Fig. 5, include three sets of 2D
embeddings from the two representations and principal component analysis
(PCA), tSNE, and UMAP, respectively. For the numerical experiment, we used the
scikit-learn implementation [107] of PCA, the parallelized implementation of
tSNE in openTSNE [108] (https://github.com/pavlin-policar/openTSNE), which
includes the exaggeration technique in optimization [105, 109, 110] to
increase the distances between groupings of similar data points and thereby
improves the clusterability of the low-dimensional embedding. The data
clustering used the feature matrix obtained from the corresponding encoding
method (CNE and SOAP) and subsequent feature scaling. The original
implementation of UMAP in [97] was used for the numerical experiment.

Supplementary Figure 4: Polytype identification workflow. The workflow starts
with crystal structure data, which are converted into scale-free features of
the octahedral network after identification of the coordination environment.
Then, dimensionality reduction is carried out, followed by interactive
clustering and human labeling. The labels are assigned inorganic framework
polytype names.

The low-dimensional embedding obtained from linear dimensionality reduction,
such as PCA, has been used previously for visualizing chemical space [69]. Yet
for grouping and labeling the subpopulations of structures (here being the
polytypes of AxPbyIz), its use is limited because of the clutteredness as a
result of the linear mapping. The outcome is also sensitive to extreme values,
which in our case corresponds to materials with large unit cells. As a result,
in Supplementary Fig. 5a-b, we only show the central parts of these
embeddings. For nonlinear dimensionality reduction methods like tSNE and UMAP,
the subpopulations are clearly revealed and significantly easier to be
separated by clustering algorithms.

Between the two representations compared here, we find that the clusters
derived from the CNE are predominantly globular, which requires only simple
(e.g. convex, linear) decision boundaries to isolate. Comparatively, the scale
dependence and higher dimensionality of the SOAP representation produces a
more complex low-dimensional embedding, with elongated cluster shapes,
resulting in extended and highly nonlinear boundaries between clusters, which
makes the subsequent clustering more challenging. The elongated shapes also
indicate the heterogeneity of the clusters, which has been confirmed upon
human inspection. The general appearances are similar between the outcomes
from tSNE and UMAP starting from the same representation, as shown in
Supplementary Fig. 5c-f. Quantitative comparisons between the low-dimensional
embeddings are presented in the next subsection.

###  2.2 Human label refinement and quality assessment

The dimensionality reduction and clustering provides a starting point to
facilitate human validation and annotation of the octahedral network
connectivity, because, to our knowledge, no tools exist yet for this type of
work. For clusters that exhibit apparent substructures in the low-dimensional
embedding, we conducted re-clustering to extract the underlying structure. We
summarize the steps in three steps, as listed in Supplementary Table 3. The
quality of unsupervised machine learning (dimensionality reduction and
clustering) is essential for the subsequent clustering and label refinement.
Human judgement of the clustering outcome requires the preconceived notion of
similarity between crystal structures. Here, the cluster members (i.e.
materials) generally possess the same kind of connectivity within their
inorganic frameworks, regardless of the organic component incorporated into
the structure.

Supplementary Table 3: Human label refinement procedure. The detail of human operations to corresponding algorithm output depends on the cluster size and purity. The example cluster sizes provided here are typical values for the hybrid iodoplumbate dataset only. Algorithm output | Cluster size | Cluster purity | Human operation
---|---|---|---
Large clusters |  10 - 30+ |  Very high | Label one member,
from initial clustering | check 1-3 others
Medium clusters |  10 - 15 |  Medium to high | Label one member, check 1-3
from re-clustering | others and potential outliers
Small clusters left |  $\leqslant$ 10 |  Very low to low |  Label every member
behind or unique cases

Although the occurrence of the octahedral network motifs exhibits a power-law
distribution when ranked by their occurence in the dataset (see Supplementary
Fig. 1), a considerable amount of clusters (before refinement) already have
high purity. We compare here the clustering quality from different
representations (CNE and SOAP) and dimensionality reduction algorithms (PCA,
tSNE, and UMAP), using human-refined labels to quantitatively assess the
result of the overall computational workflow. The assessment uses different
metrics [101] as listed in Supplementary Table 4. Among the metrics, both the
silhouette score [99] and the Davies-Bouldin index [111] are used for internal
validation of the clustering. They are calculated using the features and
algorithm-generated cluster labels. They quantify the spatial separation of
clusters using intracluster and intercluster distances. For the silhouette
score, which is bounded within [-1, 1], the larger the value, the better the
clustering quality, whereas for the Davies-Bouldin index, which has a lower
bound of 0 and no upper bound, a better clustering results in a smaller value
[112]. The adjusted Rand index [113] is an external validation metric that
compares the clustering outcome with a known standard, which in this case is
the human-refined polytype labels after refinement. The metric is adjusted for
random guessing and is generally bounded within [0, 1], with 0 indicating that
the clustering is indistinguishable from a random assignment and 1 meaning
that it is identical to the standard, i.e. human expert labels, up to label
permutation. A negative value may also be obtained sometimes, which indicates
less accuracy than random assignment [114]. We verified the 0 baseline with
randomly generated cluster labels.

Supplementary Figure 5: Visual comparison of unsupervised machine learning
outcomes. The dependence of the clustering outcome on the material
representation and dimensionality reduction algorithm used (written on the
upper right corner of each subfigure). We use distinct colors and the grey
disks drawn in the background of the scatter plot to distinguish clusters
identified by the common-nearest-neighbor algorithm [98]. The radius of the
disk is correlated with the size of the cluster. Those identified as “noise
samples” by the clustering algorithm are not shown with a disk. The scatter
plot in the subfigure e is identical to Fig. 5a in the main text. Quantitative
metrics derived from these embeddings are presented in Supplementary Table 4.
Supplementary Table 4: Benchmarks for AxPbyIz polytype identification. The
quality of outcome from different choices of representations and
dimensionality reduction algorithms are examined using the silhouette score
(in [-1, 1]) [99], Davies-Bouldin index (in [0, $\infty$]) [111], and adjusted
Rand index (generally in [0, 1]) [113]. The better score for the clustering in
each pair of pipelines in comparison (differing by the representation) is
highlighted in bold. The table is connected with Supplementary Fig. 5.

Pipeline1 | Representation | Algorithm | Silhouette | Davies-Bouldin2 | Adjusted Rand
---|---|---|---|---|---
1 | CNE | PCA | 0.473 | 0.415 | 0.146
2 | SOAP | PCA | -3.35e-2 | 0.793 | -7.74e-4
3 | CNE | tSNE | 0.870 | 0.209 | 0.206
4 | SOAP | tSNE | 0.659 | 0.343 | 4.95e-3
5 | CNE | UMAP | 0.743 | 0.354 | 0.208
6 | SOAP | UMAP | 0.519 | 0.449 | 1.50e-3

  * 1

Without loss of generality, the clustering in every pipeline is carried out
using the common-nearest-neighbor algorithm described in [98].

  * 2

Calculated without the outliers identified by the clustering algorithm.

The algorithm benchmark results can be compared visually in Supplementary Fig.
5, where the first two dimensions of the embeddings are shown. The calculated
cluster validation metrics in Supplementary Table 4 refers to these two
dimensions that account for most of the variance in the data. We identify the
following characteristics from the results:

  1. (1)

Starting with CNE generally achieves higher score than the SOAP representation
given the same dimensionality reduction and subsequent clustering algorithm.
In every case, a positive value for the adjusted Rand index is obtained,
indicating that the cluster assignment using unsupervised machine learning
algorithms is better than random guessing.

  2. (2)

For both types of representations, the use of nonlinear dimensionality
reduction (e.g. tSNE, UMAP) improves the outcome from that of PCA, which
constrains to a linear embedding. The improvement is more significant for the
representation from CNE.

  3. (3)

The difference between choosing tSNE or UMAP for the pipeline (given
appropriate parameter settings) is less significant.

What’s also worth noting is that the CNE + PCA approach already produces
diamond-shaped patches in the embedding space, except that the clusters are
hard to delineate due to insufficient separation or clutteredness, this is
drastically improved using nonlinear dimensionality reduction, which contains
tunable parameters that can increase intercluster separation and reduce
intracluster distance. The linear embedding obtained from PCA tends to be
affected by extreme outliers (some are not shown due to limited space). In
this case, these outliers correspond to materials with a large unit cell
(therefore a large number of PbI6 octahedra).

###  2.3 Queryable naming rules for inorganic framework polytypes

There exists a few frameworks for the naming of modular (or recombination)
materials, largely based on the fragments [49]. For 3D frameworks, the
Ramsdell notation for polytypes [54] are also used. The types of frameworks
include isolated units (0D), chains (1D), sheets or layers (2D), and nets or
frameworks (3D) [2, 115, 116] and these conventions have been developed in the
context of purely inorganic binary compounds [115, 116]. For hybrid
iodoplumbates (and similarly for other halometalates), we can treat the
[PbyI${}_{z}]^{n-}$ part as an ionic binary compound, which is the inorganic
framework. We propose here a simple set of rules for generating searchable
framework names, which may also be queried to gather summary statistics. The
naming rules are based on the constituents of the framework and are
constructed in a multiscale fashion to reflect the levels of connectivity
present in the material structures. We list the rules below with the nickname
for each in bold letters.

  1. 1.

Degree of connectivity. Define the degree of connectivity as the number of
shared vertices between two octahedra, which could be 0, 1, 2, or 3, for non-
sharing (NS), corner-sharing (CS), edge-sharing (ES), or face-sharing (FS),
respectively. The naming starts from the octahedral units with the highest
degree of connectivity and proceeds in a decreasing order.

  2. 2.

Building units. In atomic ONs, the primary (building) units of the structure
are the coordination octahedra with atomic ligands, the secondary (building)
units are constructed from primary units through gluing operations such as CS,
ES, and FS, which forms multimers. Similarly, a tertiary (building) unit can
be constructed from gluing secodary units through CS, ES, FS, or a combination
of them.

  3. 3.

Multiplicity of building units. Find the smallest and most connected secondary
unit (a multimer) within the octahedral network. In graph-theoretical terms,
this amounts to the maximally connected subgraph of the inter-unit graph of
the ON. Then, identify the multiplicity of the building units. For a
multimeric secondary unit, the multiplicity is the number of octahedra. A
multimer (or $n$-mer) corresponds to a secondary unit with a multiplicity of
$n$ ($n$ = 1, 2, …, $\infty$). An octahedral polymer or polymeric chain of
octahedra is equivalent to an $\infty$-mer (infinite-mer). Naturally, a dimer,
trimer, tetramer, and pentamer, may be referred to as 2-mer, 3-mer, 4-mer, and
5-mer, respectively.

  4. 4.

Elementary and complex multimers. A multimer can be CS, ES, FS, or a mixture
of them, which are called, respectively, CS multimer, ES multimer, FS
multimer, or mixed-connectivity multimer. An elementary multimer has only one
type of connectivity, while a complex multimer has at least two types of
connectivity.

Supplementary Figure 6: Examples for naming inorganic framework polytypes. a,
Construction of the polytype “CS layer of FS 3-mers” starts from the primary
building unit, FS 3-mers (1), which are connected via CS (2) in two dimensions
to form a layer (3). The layers are spaced out to form the inorganic
framework. The inorganic framework is mixed with the organic sublattice to
form the hybrid iodoplumbate (4). b, Construction of the polytype “CS
framework of CS layer of ES 2-mers” starts from the primary building unit, ES
2-mers.

  5. 5.

Decorated multimers. A multimer can be linear (1D) or, equivalently, flat
(2D), zigzag (1D) or, equivalently, corrugated (2D), curved, or circular,
which are decorative terms to illustrate the fine features at larger spatial
scales (typically requires at least a 3-mer unit to represent). A straight
multimer is the default type, so the adjective may be dropped in its name, but
for other types, the adjective is necessary. They are the corrugated $n$-mer,
curved $n$-mer, or circular $n$-mer, respectively. In a corrugated multimer,
the octahedral units are connected consecutively on alternating sides of the
previous one. In a curved multimer, the octahedral units are getting
progressively closer to the initial one. In a circular multimer, the termial
octahedral unit overlaps completely with the initial one.

  6. 6.

Framework dimensionality. An octahedral framework can be polymeric (1D chain
or polymer), layered (2D layer), or scaffolded (3D framework). This level of
specification naturally yields the dimensionality of the framework [49, 117].

  7. 7.

Naming sequence. The name of the inorganic framwork is constructed
sequentially from the primary unit to the secondary unit and above. The name
is written in the possessive form “Un of Un-1 of … of U2 of U1”, with
increasing hierarchy from U1 to Un.

We note here that these above naming rules will not be able to cover all
possible combinations of coordination octahedra or ON architectures, as their
variability grows exponentially, neither can any of the conventions that have
been proposed so far in the crystallography community [54, 49, 10].
Nevertheless, these rules are able to handle the vast majority of cases for
halometalates (and likely for related metalates) involving ONs and contains a
level of generality within its scope.

Practically speaking, the rules we have proposed here produce an explicit
description in a composable and queryable name for generating a statistical
summary of the dataset as well as for machine learning tasks, such as
predicting the connectivity types, the framework dimensionality, the
multiplicity of building units, etc conditioned on the pairing moieties. These
characteristics of the inorganic framework are made explicit in the polytype
names. To illustrate the rationale behind these naming rules, we provide two
examples in Supplementary Fig. 6. More examples are given in Supplementary
Table 5.

###  2.4 Diversity and surprises in iodoplumbate octahedral networks

The frequency of occurrence of the ONs follows a power-law distribution, as
shown in Extended Data Fig. 1a, similar to many natural phenomena [118]. From
interactive clustering of hybrid iodoplumbate dataset described in this work,
we obtain the essential statistical summary of the inorganic framework
polytypes, as shown in Extended Data Fig. 1b-c. These quantitative information
and the associated examples allow us to arrive at the following three main
observations about the distribution of the inorganic framework.

  1. 1.

Pauling’s third rule is violated in hybrid iodoplumbates. Pauling’s third rule
says that the connected coordination polyhedra follows a stability trend where
corner-sharing (CS) $>$ edge-sharing (ES) $>$ face-sharing (FS) configurations
[1]. The statistical summary in Extended Data Fig. 1 shows that the polytypes
of hybrid iodoplumbates with only CS octahedra are the most common, occupying
52.7% of the surveyed structures. However, those with only FS PbI6 octahedra
are significantly more common than those with only ES PbI6 octahedra, which
occupy 28.8% and 7.1% of the surveyed structures, respectively. Moreover, the
co-existence of CS and FS PbI6 octahedra is observed in many structures. Both
of these observations are in stark contrast to Pauling’s third rule [1].

Supplementary Figure 7: Changing framework dimensionality by varying
connectivity. Examples with edge-sharing secondary units in hybrid
iodoplumbatesare selected for illustrating 0D-3D systems. a, 0D system: ES
2-mers of ES 3-mers. b, 1D system: CS chains of ES 2-mers. c, 1D system: ES
chains of ES 2-mers. d, 2D system: CS layers of ES chains. e, 2D system: CS
layers of ES chains of ES 2-mers. f, 3D system: CS frameworks of CS layers of
ES 2-mers. Supplementary Figure 8: Examples of modular design in hybrid
iodoplumbates. a, Augmentation of corner-sharing PbI6 octahedra in 1D
construct. The examples in (1)-(6) contain corner-sharing PbI6 octahedra
cleaved at lengths 1-5. b, Augmentation of face-sharing PbI6 octahedra in 0D
construct. The secondary units for the series in (1), (2), and (3) are
isolated face-sharing 2-mers, 3-mers, and 4-mers of PbI6 octahedra,
respectively. All structures have distinct pairing organic cations to
accommodate the change in the secondary unit.

  2. 2.

A continuum exists between 0D and 3D structures by varying network
connectivity. For hybrid iodoplumbates and its related families of materials,
the dimesionality of the system is determined by the spatial arrangements
between the organic and inorganic components (framework dimensionality).
Although representative members such as 3D perovskite or 1D face-sharing
hexagonal perovskite are common examples, our analysis of the hybrid
iodoplumbate dataset indicates that the network connectivity plays an
important role in dimensionality determination. An example with FS 3-meric
units is found in the Main Text Fig. 5. The 3-mer can form 0D structures
surrounded by an organc cation. When the 3-mers are connected via a shared I–I
edges, the structure can become 1D. When the 3-mers connect via shared iodine
vertices, the structure can become 2D with layers of CS FS 3-mers. Another
example, illustrated in Supplementary Fig. 7, starts with 0D structure with
isolated ES 3-mers, then connected through edge-sharing as polymer. The
polymers are connected to form layers in 2D and finally, the layers are
connected to form 3D structures with the organic filling in the gaps.

  3. 3.

Modular design from building unit augmentation. One design strategy for
building complex structures is by augmenting the secondary units. In
octahedral networks, this amounts to extending the number of connected
components. For layered hybrid perovskites, structural augmentation from
stacking an increasing number of corner-sharing layers along a defined
cleaving plane is well-known [14, 17]. This strategy, in principle, doesn’t
change the dimensionality of the system, but allows band structure tuning [52,
79]. We point out here from the large hybrid iodoplumbate dataset, analogous
strategies to extend the structural complexity in lower dimensional systems by
augmenting the secondary unit. They correspond to the accretional homologous
series first defined in inorganic crystallography [4], where the secondary
unit grows in size. The difference is that in hybrid materials, a sublattice
(here referring to the [PbyIz]n- framework) forms the homologous series, while
the organic cation changes to adapt to the size of the composite anion. We
illustrate this point with two example series in Supplementary Fig. 8: In the
first series, the corner-sharing motif is extended from 1 to 5 while keeping
the framework in 1D. In the second series, the face-sharing motif is extended
from 2 to 4 while keeping the framework in 0D.

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