Hierarchical topological analysis of crystal structures: the skeletal net concept

Topological analysis of crystal structures faces the problem of the `correct' or the `best' assignment of bonds to atoms, which is often ambiguous. A hierarchical scheme is used where any crystal structure is described as a set of topological representations, each of which corresponds to a particular assignment of bonds encoded by a periodic net. In this set, two limiting nets are distinguished, complete and skeletal, which contain, respectively, all possible bonds and the minimal number of bonds required to keep the structure periodicity. Special attention is paid to the skeletal net since it describes the connectivity of a crystal structure in the simplest way, thus enabling one to find unobvious relations between crystalline substances of different composition and architecture. The tools for the automated hierarchical topological analysis have been implemented in the program package ToposPro. Examples, which illustrate the advantages of such analysis, are considered for a number of classes of crystalline substances: elements, intermetallics, ionic and coordination compounds, and molecular crystals. General provisions of the application of the skeletal net concept are also discussed.

1. Introduction

At present, a huge amount of crystallographic data on about two million crystal structure determinations are accumulated in worldwide electronic databases such as the Cambridge Structural Database (CSD) (Groom et al., 2016), the Inorganic Crystal Structure Database (ICSD) (Hellenbrandt, 2004), the Crystallography Open Database (COD) (Gražulis et al., 2009) or the Powder Diffraction File (PDF) (Villars & Cenzual, 2021). To arrange this information and to establish structural relations between different groups of chemical substances, one needs rigorous methods of structure description which can be algorithmized and implemented in user-friendly software. The corresponding mathematical models represent the crystal structure as a set of lattices which obey space-group theory. Since the very beginning of the X-ray analysis era, crystal chemistry used the model of close packing (Coxeter et al., 1989) and the structure type concept (Lima-de-Faria et al., 1990), in which the arrangement of atoms in the crystal space was the main criterion of the classification. Initially, these models were applied to rather simple inorganic structures (metals or ionic compounds) but then they were expanded to molecular crystals in Kitaigorodskii's theory (Kitaigorodsky, 1961) and in the concept of structural class (Belsky & Zorkii, 1977). Selected sublattices were considered instead of the whole structure in the models of cation arrays (O'Keeffe & Hyde, 2007). Although the initial descriptors of these models were naturally geometrical (space group, occupied Wyckoff positions, number of atoms in the unit cell etc.), the local connectivity was explicit or implicit like coordination numbers (CNs) of atoms or molecules in close packings.

Another route to the description and comparative analysis of crystal structures was proposed by Wells (1954, 1977) in a series of papers and books. Wells introduced the concept of periodic net by establishing contacts within the crystallographic lattices. It was a purely topological approach where attention was mainly paid to connectivity, not to geometric positions of atoms. Such an approach is naturally crystallo-chemical since it accounts for interatomic or intermolecular bonding, which is most important in understanding the properties and structure of crystalline chemical substances. Importantly, the connectivity of the whole crystal, not of the closest coordination environment, was treated in the periodic net model. Pearson (1972) also used topological models for cataloging intermetallic crystals, but he separated only 2-periodic nets, thus describing not the whole structure. The periodic net concept was developed by other authors (Delgado-Friedrichs et al., 2005; Klee, 2004; Eon, 2005; Thimm, 2009) where graph theory was expanded to periodic systems of points. Specialized software such as Systre (Delgado-Friedrichs & O'Keeffe, 2003) or ToposPro (Blatov et al., 2014), and electronic databases RCSR (Reticular Chemistry Structure Resource) (O'Keeffe et al., 2008) and TopCryst (Shevchenko et al., 2022) were then created and used for the large-scale exploration and classification of all available crystal structures (Alexandrov et al., 2011).

However, the topological description faces an essential problem: in many cases, the interatomic bonds cannot be unambiguously assigned because of their different strength. As a result, the same structure can be treated in different ways and it is often impossible to choose the correct or the `best' description. A possible solution could be in assigning several topological representations to the same crystal structure (Blatov, 2006). Each representation is described by a periodic net where one of several chemically reasonable ways of choosing interatomic contacts is realized. Blatov (2006) proposed a hierarchical scheme for the generation of a sequence of topological representations by subsequent removal of the net edges in accordance with the strength of the corresponding interatomic contacts. This scheme was then used for the hierarchical analysis of molecular packings (Aman et al., 2014) and an important concept of critical net was introduced (Banaru & Banaru, 2020), which is the net with a minimal number of the strongest intermolecular contacts required for connecting all molecules in the crystal. In this work, we extend this idea to inorganic crystals as the concept of skeletal net and show that this extended concept can be used for comprehensive crystallo-chemical systematics and establishing unobvious relations between crystalline substances of different composition and architecture.

2. Method

We use the model of a periodic net to describe any crystal structure irrespective of its composition or the kind of chemical bonding. The nodes and edges of the net correspond to atoms and interatomic contacts, respectively. Initially, all contacts are considered which correspond to the faces of the Voronoi polyhedra of the atoms. The faces are arranged in the order of their solid angles, which roughly estimate the strength of interatomic contacts for each kind of A–B atomic pair (Blatov, 2004) (Fig. 1). The resulting net is called complete (Blatov, 2006) as it includes all possible chemical bonds which are provided by exchanging electrons and hence require a direct contact of the atoms in the space. Any net derived from a particular crystal structure characterizes a topological representation of this structure; the complete net corresponds to the complete representation. All other representations are partial since they can be generated from the complete representation by a subsequent removal of weak interatomic contacts and/or by other simplification procedures (Shevchenko & Blatov, 2021). Additional chemical criteria can be applied to select the contacts that a crystal chemist usually considers when describing the crystal structure. In particular, the universal Domains algorithm (Blatov, 2016) can discriminate any kind of bonding (valence or hydrogen bonds, specific or van der Waals interactions) in crystal structures of any nature and uses Slater radii of atoms (Slater, 1964) as well as geometrical parameters of hydrogen and halogen bonds (Blatov et al., 2021). Usually the contacts with solid angles less than 1.5% of the total solid angle 4π steradian are discarded since this value was estimated as a typical error of the solid-angle determination in an X-ray experiment (Blatov, 2004).

Figure 1 The Voronoi polyhedron of an Au atom in the crystal structure of β′-AuCd. 14 Au–Au and Cu–Cu contacts are arranged according to the solid angles (Ω) of the corresponding faces of the polyhedron expressed as a percentage of the total solid angle 4π steradian: four Au–Cd contacts shown by gray cylinders (Ω = 9.3%); four Au–Cd contacts with Ω = 9.1% (dashed blue lines); two Au–Au contacts with Ω = 7.1% (dashed red lines); two Au–Au contacts with Ω = 5.6% (dashed black lines); and two Au–Au contacts with Ω = 0.5% (bold green lines). The last two contacts with Ω < 1.5% are too weak (Blatov et al., 2021) and should be discarded from the complete net. The Au and Cd atoms are shown as yellow and magenta balls, respectively.

This hierarchical scheme, in which all nets are in `supernet–subnet' relationships (Blatov et al., 2019), enables one to treat a particular crystal structure at different levels of its organization and to find structural relations between crystalline substances of different architecture (Blatov, 2006). For example, the crystal structure of β′-AuCd (Chang & Read, 1951) can be considered as a hexagonal close packing (hcp) with 12-coordinated atoms, although it is low-symmetrical with space group Pmma and contains additional weak contacts with solid angles less than 1.5% of the total solid angle 4π steradian (Fig. 1), which should be discarded according to Blatov (2004). Hereafter, to designate net topologies we use the RCSR bold three-letter symbols (O'Keeffe et al., 2008) or ToposPro NDk symbols (Blatov et al., 2021). This representation is complete as all 12 robust contacts in the nearest environment of both Au and Cd atoms are taken into account. However, due to the non-equivalence of the contacts, three partial representations can also be generated with 10-coordinated cco, 8-coordinated bcu and 4-coordinated sql topologies (Fig. 2). According to the TopCryst database (Shevchenko et al., 2022), which accumulates the occurrences of nets in crystal structures, the cco topology is rare, while the two other partial nets are abundant and hence deserve special consideration since they represent typical atomic motifs. The square lattice sql net is 2-periodic and can be a subject for the representation of intermetallic structures as packings of layers (Pearson, 1972). However, the most fruitful is the body-centered cubic bcu partial representation as it relates the low-temperature β′-AuCd phase with the high-temperature β₁-AuCd phase, which belongs to the CsCl structure type (Chang & Read, 1951) with the bcu topology. This structural relation was used by Chang & Read (1951) to interpret in detail the experimentally observed martensitic transition β₁-AuCd → β′-AuCd. This hcp–bcu topological transformation also indicates the relationship between body-centered cubic packing and close packings, which is well known in crystal chemistry.

Figure 2 Representations of the crystal structure of β′-AuCd: (left) the complete representation of the 12-coordinated hcp topology; (middle) a partial representation of the 10-coordinated cco topology; (right) a partial representation of the 8-coordinated bcu topology. A partial representation as a stacking of 4-coordinated sql layers is shown by balls and cylinders; one of the layers is depicted by green cylinders. The coordination polyhedron of an Au atom in the form of a distorted twinned cuboctahedron, a distorted two-capped cube or a distorted cube is shown in each picture. The dashed edges that subsequently transform the sql stacking into the bcu, cco and hcp nets are colored in blue, red and black, respectively. The Au and Cd atoms are shown as yellow and magenta balls, respectively.

However, this approach does not directly answer the question of which representation is the `best' for the classification of a particular crystal structure and for establishing relations with other crystal structures. As a result, even similar crystal structures can be assigned to different topological types if the bonding in them was treated in different ways. For example, almost all coordination environments of Cu<sup>II</sup> are distorted due to the Jahn–Teller effect, with four short Cu–X contacts and one or two longer Cu–X contacts. Analysis of 49 934 Cu–O contacts in 6158 inorganic copper compounds deposited in the ICSD shows that the distribution of their distances has two maxima at 1.95 (7) and 2.35 (8) Å (Fig. 3). The distances vary in a wide range and this often leads to ambiguous assignment of CNs to copper atoms. If only the distances around the first maximum are considered as bonds then CN = 4; if all contacts are taken into account then CN = 5 or 6. In structural descriptions, the authors often give just a list of the distances without assigning them to bonds, but for the topological description such an assignment is crucial. A possible argument for accepting a particular topology is the abundance of this topology in other crystal structures, as was mentioned above for β′-AuCd. For example, in 52 different structure determinations of La<sub>2</sub>CuO<sub>4</sub> deposited in the ICSD, 106 symmetry-independent Cu–O distances range within 1.78–2.47 Å with two distinct groups of contacts at average distances 1.91 (4) and 2.41 (3) Å. Although the structure determinations were performed in five space groups (Cm, Cmce, Cmmm, Fmmm and I4/mmm) and hence belong to different structure types, the overall topologies of their complete nets are the same in all cases. This means that all La<sub>2</sub>CuO<sub>4</sub> structures belong to the same topological type. Such a conclusion shows the advantages of the topological approach in the search for structural relations. According to the TopCryst database, accounting for the former group of the Cu–O distances results in CN(Cu) = 4 and the overall topology 4,5,6,9T2, which occurs only in La<sub>2</sub>CuO<sub>4</sub>, while assigning all six Cu–O contacts to bonds leads to the 6<sup>3</sup>,9T2 topology which occurs in 278 crystal structures of other compounds, many of which belong to the K<sub>2</sub>NiF<sub>4</sub> structure type (Balz & Plieth, 1955) with a regular octahedral environment of Ni atoms and unambiguous assignment of bonds. However, such an `abundance' argument cannot be used if the topologies of all possible representations are unique or rare. Moreover, it cannot replace chemical arguments and should be used only if chemical reasons do not enable one to make an unambiguous choice between several possible assignments of bonding.

Figure 3 Distribution of 49 934 Cu–O distances in 6158 inorganic copper compounds.

To gain unambiguity in the topological classification we propose the concept of skeletal net, which is the net where all weak contacts are subsequently removed until the net preserves its periodicity. Thus the complete and skeletal nets can be considered as maximal and minimal nets in the sense that the former contains all possible bonds in the structure, while the latter loses as many weak bonds as possible, although keeping the periodicity of the complete net. For example, the bcu net is skeletal for the crystal structure of β′-AuCd while the sql net is not skeletal because it has a lower periodicity than the initial complete net (Fig. 2). We do not use further the term minimal net as it has another meaning in graph theory (Beukemann & Klee, 1992): it contains the minimum set of edges that preserves the net periodicity irrespective of the weight of the edges, thus having no chemical background. In contrast, skeletal net can contain additional edges, which are not responsible for the net periodicity, but correspond to strong bonds forming stable structural groups. The skeletal net concept extends the notion of critical net (Banaru & Banaru, 2020) while resting upon the same ideas. Unlike critical net, skeletal net can be determined for any kind of crystal structures, not only molecular crystals, and not only for the whole structure, but for any independent structural unit. The complete and skeletal nets establish limits for other possible representations of the crystal structure where some weak bonds can be taken into account for some reasons.

All net topologies reported in this paper were determined using the ToposPro program package (Blatov et al., 2014) and the TopCryst database containing the reference topological types (Shevchenko et al., 2022). All examples of crystal structures presented below were taken from the ICSD. The skeletal nets were constructed with ToposPro version 5.5.2.1; a detailed tutorial of the construction procedure is given in the supporting information.

3. Skeletal nets: examples

3.1. Elements

In many cases, all chemical bonds in an elementary substance are of the same nature which results in the equivalence of the complete and skeletal nets. For example, face-centered cubic metals or diamond-type non-metals have the fcu or dia topology, respectively, for both kinds of nets. All bonds in these structures are equivalent and breaking them leads to an unconnected set of atoms. However, if even two symmetry-independent sets of bonds exist, there can be differences in the complete and skeletal nets. For example, the connectivity in body-centered cubic metals is usually considered in two different ways: with CNs of atoms 8 or 14 (8 + 6) and the topologies bcu or bcu-x, which correspond to skeletal and complete nets, respectively. More complicated cases can be found in low-symmetrical phases. For example, a high-pressure phase of strontium, Sr-III, was solved in two space groups, Imma (Winzenick & Holzapfel, 1996) and I4₁/amd (Allan et al., 1998), and in both studies it was related to the β-Sn structure, which also has the tetragonal I4₁/amd symmetry. However, if the β-Sn structure is usually treated as a 6-coordinated bsn net, the Sr atoms in the Sr-III phase should have CN = 10 since such a number of Sr—Sr bonds have rather short (less than 3.7 Å) distances and all of them correspond to the faces of the Sr Voronoi polyhedron with solid angles larger than 4.5%. However, the skeletal nets of both Sr-III and β-Sn are 4-coordinated diamondoid (dia) with the four strongest contacts, and their complete nets have also the same deh-d topology. The bsn net can also be found in both structures as a partial net in the sequence: deh-d (10-coordinated complete net) → bsn (6-coordinated partial net) → dia (4-coordinated skeletal net) (Fig. 4). The topological approach enables us to make such a conclusion irrespective of different reported space symmetries of the Sr-III structures. Thus, the comparison of the skeletal nets enables one to find a topological similarity and the matching of other possible representations provides the final conclusion about the full topological equivalence of Sr-III and β-Sn, despite different space symmetry (in the case of the Imma phase) and interatomic distances.

Figure 4 Topological representations of the Sr-III (I41/amd) crystal structure: the solid green lines depict the skeletal diamondoid (dia) net, the dashed blue and red lines show additional Sr–Sr contacts that subsequently transform the skeletal net into a partial 6-coordinated bsn (β-Sn) net and to the complete 10-coordinated deh-d net. The coordination polyhedron in the form of a distorted octahedron is shown for a Sr atom in the bsn net.

3.2. Intermetallics

Topological classification of intermetallic structures is a great challenge because of essential ambiguity in assigning interatomic contacts in many cases. Metal atoms can combine in different ratios and the nature of the metallic bond allows them to have many neighbors in a wide range of distances. Moreover, some compounds of metals and metalloids (B, Si, As, Te) or even non-metals (C, N, P) are often considered intermetallic due to their metallic properties. In such structures, the range of strengths of interatomic bonds can even be wider than in compounds of metals. The skeletal net concept could become a powerful tool for establishing structural relations in this group of substances. Let us consider the crystal structure of Sc₂CoIn (Gulay et al., 2018) where the authors reported CNs of Sc (13), Co (8) and In (12). The Co–Co contacts with distances of 3.29 Å were ignored while the Sc–Sc and In–In contacts of the same length were counted. This point is debatable and shows the problem of assigning interatomic contacts mentioned above. In any case, high and different CNs indicate quite complicated and unusual topology of the structure. With the Voronoi approach we obtain CN = 13, 14 and 14 for Sc, Co and In, respectively. However, in the skeletal net, all atoms have CN = 8 and the topology of the net is body-centered cubic (bcu). The authors mentioned that the unit cell of Sc₂CoIn could be obtained by twinning the CsCl-type unit cell, and the structure could be derived from the ScCo crystal structure, which belongs to the CsCl structure type. This finding confirms the results of our topological analysis since CsCl has the bcu topology, and hence, the bcu skeletal net indeed underlies the Sc₂CoIn architecture (Fig. 5). This conclusion is also supported by the fact that the 13,14,14-coordinated complete net is topologically close to the bcu-x net, which describes an extended (8 + 6)-coordinated body-centered net: adding a Sc–Sc contact of length 4.17 Å results in the transformation of the 13,14,14-coordinated net into a 14-coordinated bcu-x net (Fig. 5).

Figure 5 Topological representations of the Sc2CoIn crystal structure: the solid green lines depict the skeletal body-centered cubic (bcu) net, the dashed blue lines show additional contacts that transform the skeletal net into a 13,14,14-coordinated complete net. The dashed red line shows an additional Sc–Sc contact that transforms the complete net into a bcu-x net. One of the coordination polyhedra of atoms in the bcu net is shaded.

If an intermetallic compound contains metalloids, which can connect each other by strong covalent bonds, the skeletal net directly represents the substructure of the metalloid atoms. For example, the ThSi₂-type silicides incorporate a 3-periodic silicon substructure, which is described by a ths net (Delgado-Friedrichs et al., 2005). It is the net that can be found as the skeletal net if one discards all contacts between metal and silicon atoms, which are weaker than the Si—Si bonds. Note that this kind of skeletal net admits unconnected parts of the initial structure (isolated metal atoms) while keeping the overall 3-periodic architecture of the crystal. In fact, such topological representation corresponds to the treatment of the structure as a silicon framework with the interstitial metal atoms occupying the cages of the framework.

3.3. Ionic compounds

In simple binary ionic compounds like NaCl, the skeletal net coincides with the complete net since all bonds are of the same kind and strength. However, when the number of different cations and/or anions is higher, several representations can be derived and the skeletal net enables one to find more structural relations than the complete topological description. For example, analysis of the crystal structures of simple sulfates reveals six compounds MSO₄ (M = Sr, Ba, Pb, Eu, VO, UO₂) with the same skeletal net of the sra (SrAl₂) topology. To obtain this topology, the skeletal nets generated by the algorithm described above should be additionally simplified by transforming the 2-coordinated bridging atoms to the net edges and removing terminal atoms (Shevchenko & Blatov, 2021) (Fig. 6). In the skeletal net, the sulfur atoms keep their connectivity while the M atoms lose all but the four strongest bonds. The number of bonds to be lost is different: one (M = VO, UO₂) or four (M = Sr, Ba, Pb, Eu) and, as a result, different complete nets produce the same skeletal topology. We have found that the sra topology underlies many other inorganic salts, not only simple ones like β-VOPO₄, Li₂CrO₄·2H₂O or β-CuAlCl₄, but also double salts M1M2XO₄ where M1 is usually an alkali atom, which loses all its bonds in the skeletal net, and M2 is a d metal. In fact, the skeletal net represents such compounds as a M2XO₄ framework, whose cages are occupied by the M1 atoms. Such consideration is natural since the sra topology describes the zeolite ABW framework and some of the M1M2XO₄ salts are referred to as ABW zeolites (Diego Gatta et al., 2012).

Figure 6 A fragment of the crystal structure of barite (BaSO4): thin green lines depict the skeletal net; blue dashed lines are added to form the complete net; thick black cylinders replace the bridging oxygen atoms and form a simplified skeletal net. Each oxygen atom replaced with a black cylinder forms a triangle together with this cylinder; one of these triangles is shaded pink.

3.4. Coordination compounds

Coordination compounds contain at least two types of bonds: within ligands and between donor atoms of ligands and coordination centers (metal atoms). Since ligands are molecules or ions that are stable in the reaction media, the bonds of the former type are stronger than the bonds of the latter type. Therefore the ligands are usually kept in the skeletal net as they were in the complete net, while the metal–ligand bonds can be partially broken. In many cases, such transformation of the complete net does not change the underlying topology, which can be determined in the standard consideration of the coordination network (Shevchenko & Blatov, 2021). However, sometimes one can obtain a simpler topology, which underlies the structure, thus getting a better insight into the structure organization. For example, Tronic et al. (2007) described the crystal structure of (CuCN)₇(Pym)₂ (Pym = pyrimidine) as self-penetrating, i.e. containing catenated rings of connected atoms, by considering only strong coordination bonds Cu—C or Cu—N at distances of no more than 2.1 Å, but accounting for the Cu⋯Cu interactions of length 2.71–2.93 Å. The complete net of this structure contains additional Cu–C contacts of lengths 2.39 and 2.49 Å. The skeletal net suggests ignoring the Cu⋯Cu interactions and the weakest Cu–C contacts (2.49 Å), thus resulting, after simplification, in two disconnected interpenetrating 3-periodic nets of the ths topology, while the discarded Cu–C contacts unite these nets into a single self-penetrating 3-periodic network (Fig. 7). These two alternative representations fully correspond to the topological description of this structure given in the TopCryst system.

Figure 7 Representations of the crystal structure of (CuCN)7(Pym)2: (top) skeletal net consisting of two interpenetrating nets; the red dashed lines show additional Cu–C contacts of length 2.49 Å that unite the networks into a single self-penetrating 3-periodic network; (bottom) the corresponding interpenetrating simplified skeletal nets of the ths topology. One of the interpenetrating nets is highlighted in yellow in both arrays.

3.5. Molecular crystals

Similar to coordination compounds, one can select two types of bonds in molecular crystals: intra- and intermolecular. Obviously, intermolecular contacts (hydrogen bonds, specific bonds like halogen or π–π interactions, van der Waals contacts) are much weaker than covalent intramolecular bonds, and intermolecular contacts can be partially broken in the skeletal net. If after this transformation one simplifies the skeletal net by squeezing molecules into their centers of mass, thus performing the standard simplification of the molecular network (Shevchenko & Blatov, 2021), the resulting simplified skeletal net in most cases coincides with the critical net proposed by Banaru & Banaru (2020). Many good examples of the description of molecular crystals by critical nets were provided by Banaru & Banaru (2020) and other works of Banaru and co-workers. A difference can occur if the skeletal net splits the complete net into an array of interpenetrating networks like in the example given above for coordination compounds.

4. Concluding remarks

The concept of skeletal net formalizes the topological description of crystal structures establishing the minimal set of chemical interactions to be considered in crystallo-chemical analysis. This does not mean that the skeletal net cannot be transformed further, it just points out what interactions should not be ignored without losing the overall structure connectivity. The strongest bonds are always preserved in the skeletal net, thus providing its difference from the minimal net. The skeletal net does not require all atoms of the structure to be connected in the same array; there should remain at least one group that keeps the initial periodicity. Such an approach enables one to find the main framework that supports the whole structure; all other low-periodic structural groups including isolated atoms are allocated in this framework. The subsequent analysis of the skeletal net can require additional topological transformations and the examples presented above show some of the transformations that also result in reasonable structure representations. The following general provisions of the application of the skeletal net concept should be taken into account.

(i) All 0-coordinated (isolated) and 1-coordinated (terminal) nodes of the skeletal net should be removed, and 2-coordinated (bridging) nodes should be transformed to edges for reducing the skeletal net to a canonical form that can then match a reference net from an electronic database. This procedure corresponds to the so-called secondary simplification of a periodic net (Shevchenko & Blatov, 2021).

(ii) The groups of nodes of the skeletal net, which contain the strongest bonds, represent the structural units that can be considered as a whole. This means that such units can be squeezed into their centers of mass, and the resulting simplified skeletal net will describe the connectivity of these units. Such a transformation corresponds to the cluster simplification procedure (Shevchenko & Blatov, 2021) and it is especially important for molecular crystals or coordination compounds that contain distinct complex units.

(iii) If the skeletal net contains isolated molecular units or atoms, the structure can be treated as porous and the porous framework is described by the periodic part of the skeletal net. This is a way to recognize porous structures using only topological criteria.

(iv) If the skeletal net contains an array of structural units of the same periodicity, for example, several interpenetrating 3-periodic nets, one could be interested in the links that unite the units into the whole framework. Such links can be easily found by the analysis of the sequence of nets between the skeletal and complete net. The obtained information could be useful for the design of interpenetrated arrays or vice versa for avoiding interpenetration when planning the synthesis of porous frameworks (Yaghi et al., 2019).

(v) The complete and skeletal nets establish the ranges within which other topological representations can be generated. This means that the crystal structure can be analyzed at different levels of topological complexity, and use of the skeletal net allows deeper structural relations to be found since many weak bonds are ignored that can be specific for a particular compound.

Although all examples in this paper are given for three-dimensional structures, the skeletal net concept is applicable to the low-periodic structures in the same way. It is implemented in crystallographic software, enabling one to perform a multi-level topological analysis of crystal structures of any chemical nature and complexity.