Appendix 2. The concept of “underlying net” and crystal structure representation. Methods of simplification. 

When we explore the topological properties of Coordination Polymers / Metal Organic Frameworks (CP/MOFs) the key notion is the concept of underlying net. While the whole structure topology is reflected by an atomic net including all atoms as nodes and all interatomic bonds as edges, the nodes and edges of the underlying net correspond to structural groups and connections between them. In fact, any structure representation that separates building blocks (for instance, molecules in molecular packings or ligands in coordination compounds) implicitly applies the underlying net concept. The resulting simpler representation allows one to find structural resemblances between chemically quite different compounds.

In ToposPro where a number of structure representations can be built in an automated mode; each representation can be transformed to an underlying net by simplification procedures (see Module 4). Any structure representation in ToposPro corresponds to a particular way to choose structural units; this way depends on the task under consideration. For any class of chemical compounds some standard representation can be proposed that conforms to a conventional crystallochemical description (see Table below). The standard representation of valence-bonded CP/MOFs considers metal atoms and organic ligands as structure units, ignoring guest ions and molecules. Then the procedure of contracting multiatomic groups to their centroids is applied to all ligands to obtain a primary simplified net. This net can be directly transformed to the underlying net by the procedure of secondary simplification that consists in successive removing 0-coordinated nodes (intraframework molecular moieties), 1-coordinated nodes (terminal/dangling ligands) and transforming 2-coordinated nodes (bridge µ2 ligands or metal atoms connected to only two bridge ligands) into edges. If the bridge µn ligands are connected with more than two metal atoms (n>2) they form additional nodes of the underlying net.

Standard representations for some types of crystal structures

Structure type Structure units = simplified net nodes Type of bond within the structure units Type of bond between the structure units = simplified net edges
valence-bonded CP/MOFs metal atoms, ligands valence valence
H-bonded CP/MOFs complex groups valence H-bond
organic crystals, supramolecular complexes organic molecules valence intermolecular: H-bonds, specific (e.g. Halogen Bond), van der Waals

The standard representation is not the only way to obtain the primary simplified net for a CP/MOF. If the metal atoms form a cluster or a polynuclear coordination group this group can be considered as a structure unit and a node of a simplified net. To recognize these groups automatically, ToposPro uses the following criterion for distinguishing intra- and intercluster bonds. Let us determine for each kth bond the size Nk of the minimal ring of bonds to which the kth bond belongs (Nk is the number of bonds in the ring) and arrange all bonds in ascending order of Nk: N1 ≤ N2 ≤ N3 If in this sequence there is such Nm that

Nm+1 – Nm > 2 (1)

then all bonds with k ≤ m are referred to as intracluster, the other bonds are considered intercluster. Condition (1) means that intracluster bonds belong to small rings, i.e. they are allocated rather compactly forming clusters, while intercluster bonds connect the clusters together and become edges of the simplified net. If several pairs of Nk obey (1), there are several ways to cluster the bonds and several cluster representations of the CP/MOF. In most cases only one representation corresponds to finite complex groups, and the resulting underlying net can be unambiguously obtained.

A modification of cluster representation is the skeleton representation. In this case the user should specify the maximal size (N) of rings that will be accounted to determine the atoms belonging to the structure skeleton. If the atom belongs to at least one ring of size more than N, this atom is included into the skeleton, otherwise it is contracted to the skeleton atoms. In practice, this means that atoms belonging only to small rings (e.g. benzene rings) are removed from the structure, and the small rings are replaced by edges of the simplified skeleton net. The larger N the simpler is the skeleton. This representation is especially useful when analyzing entanglements since in this case small rings being impossible to be catenated just hinder the topological analysis.

Special structure representations are edge nets and ring nets. Edge net can be constructed by placing new nodes in the middle of the edges of the initial net, connecting new nodes by new edges and removing old nodes and edges. The edge net is complete if all the edges in the initial net are centered by new nodes and all old nodes and edges are removed; otherwise the edge net is partial. Similarly to edge net, one can construct a ring net by putting nodes in the centers of rings of the initial net, removing the nodes of the initial net belonging to the centered rings, and adding new edges between the centers of adjacent rings. Depending on whether all rings are centered or not, the ring net can be complete or partial. A special kind of ring net is Hopf ring net. That is the net whose nodes and edges correspond to rings and Hopf links (interlocking of 2 rings) between them. By default, ToposPro marks Hopf links as “H bonds” between the ring centers.

Hopf, multiple crossing, and the three simplest three-component links. The corresponding edges of the ring nets that connect the ring net nodes are shown by arrows. For the Borromean link, the ring net fragment contains an additional node in the center of the link. The program Knotplot (R. G. Scharein; was used to draw the link pictures.

Hopf ring net directly characterizes the catenation pattern, i.e. the method of catenation of the rings for the whole system of entangled nets, if the kind of network and the degree of interpenetration are fixed. For example, if a set of structures containing two interpenetrating diamondoid networks is under consideration, it is sufficient to compare their Hopf ring nets to find the differences in their catenation patterns.


Pay attention, that 0, 1, 2-coordinated nodes of simplified net should be deleted. You will obtain the underlying net with nodes of coordination > 2, which can then be classified by means of the ADS program. Otherwise, you would not properly classify the net topology. The 2-c nodes may be kept in the underlying net only in the rare cases where the entanglement is formed between 2-member rings (called also loops), see: Yang J., Ma J.-F, Batten S. Polyrotaxane metal-organic frameworks (PMOFs) Chem. Comm. 2012, 48, 7899-7912.

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