General Information

The mathematical (periodic) graph \(G(V,E)\) is the foundation of a topology or net. It is defined by its vertices \(v\in V\) and edges \(e\in E\) that describe the connections between vertex \(i\in V\) and \(j \in V\). The vertices of the graph itself are not positioned in space. In order to construct a crystallographic unit cell, the labeled quotient graph can be used for the embedding of the net: Embedding means positioning the graphs vertices in a unit cell in euclidean (3-dimensional) space. Without any constraints, the number of embeddings of a topology is infinite, but there is one particular embedding called faithful embedding, maximum symmetry embedding or barycentric embedding, for which the vertices are positioned in such a way as to constrain the average edge length to one, to ensure that vertices do not collide (are not at the same position in space) and that edges do not intersect each other. This embedding is possible for almost all chemically relevant topologies, exceptions are called instable nets.

Common Symbols

nets

The coordination of a net is the set of coordination numbers of the included vertices. e.g. \((3,4)\)
The connectivity of a net describes the set of connections between vertices identified by their coordination number. e.g. \((3-3,3-4)\)
An attempt to name topologies is by using RCSR symbols, which are a three letter code given to a particular topology. Whenever possible, the three letters are an abbreviation for some characteristic of the net (like dia=diamond net, qtz= quartz net or pts = the net of the Pt and S atoms in PtS), most names however are given arbitrarily.
The transitivity of a net, given in terms of an integer quadruple \(pqrs\), denotes the number of distinct vertices (\(p\)), edges (\(q\)), faces (\(r\)) and tiles (\(s\)). The smaller the numbers, the simpler is the respective net. According to the empirically grounded minimum transitivity principle, nature prefers topologies that have small transitivity.
A regular net is one of the five nets with transitivity \(1111\), which are dia. bcu, nbo, srs and pcu
quasiregular nets feature transitivity \(1112\), as e.g. fcu
semiregular nets are allowed to also have larger numbers of distinct faces \(r\) (\(11rs\)).
Nets with only one kind of vertex are called vertex transitive, nets with only one kind of edge are denoted as edge transitive.
Uninodal nets are nets with just one distinct vertex, hence its transitivity is \(1qrs\)
A topology belongs to one of the 230 spacegroups, which contain the translational invariances (symmetries) of the unit cell.

vertices

Vertices have a coordination of \(c \ge 3\). By definition, a vertex with a coordination of two is not a vertex, but an edge between two vertices.
The Symmetry of a vertex is usually given as Hermann-Maugin, or Schoenflies label which describes the symmetry operations possible to the coordination figure of the vertex, it exists only in a particular embedding. The corresponding order denotes the count of the symmetry equivalent copies of the unique point inside the full unit cell.
In addition there exist a lot more symbols like the Vertex Symbol, the Point Symbols, the Extended Point Symbols, Face Symbols, [...], which exist to help uniquely characterize a net.

Deconstruction of Crystal Structures to Topologies

Crystal structures of MOFs are relatively easy to deconstruct due to the structural integrity of the organic linker and evident partitioning into metal-node and organic linker. HKUST-1 for example can be easily deconstructed by assigning one vertex to each linker and one vertex to each paddlewheel metal-node, resulting in the tbo topology (see also: RTA).

For larger linkers (with e.g. multiple aromatic rings) however, there exist multiple ways: Apart from the single vertex representation, every sub fragment can be assigned a vertex :

Both representations have their advantages and disadvantages: Whereas the single-node representation features vertices with more irregular geometries, the multi-node representations are 'cleaner' with respect to the individual vertices' deviation from regular geometries. The latter also features further degrees of freedom of characterization, which is the angles between the planes spanned by the respective three-coordinated vertex connectors. In contrast to that, one has to bare in mind that the multi-node representation alchemically cuts fragments that chemically belong together into topologically distinct entities. By way of not considering the single-node representation during construction one misses isoreticular isomers as introduced in Isoreticular Isomers. This would have been the case in Ref. [5], where the multi-node representation would not have yielded alternative insertion orientations for the tetracarboxylate linkers. Hence, only in the single-node representation distinct orientations of the linker resulting in isoreticular isomerism can be detected. For many topologies, the RCSR includes both nets, as for example the fof, fog and tfb which are all nbo derived nets where a 4-c vertex has been replaced by two 3-c vertices. For structure prediction purposes it is important to consider both, hence by searching for only for one of them, one may miss important topologies.