Uninodal nets

A thousand uninodal nets were described over the years by Fischer, Koch and Sowa in their enumeration of homogeneous sphere packings, exploring systematically all crystal systems [18]. 

Figure 1.3.4 In the 3-connected uninodal net 4.8.10 in fact, the angle be is included in the rhr-a the vertex 1 with the three edges a,b,c has circuit with 10 edges [1,2,3,4,5,6,7,8,9,10] that Vertex Symbol 4.8.12 (angles ac 4-ring, ab is not a ring because it is the sum of two...

There are a number of uninodal nets that are also self-penetrating, notably the (8,4) net (Figure 19a) and the (12,3) net (Figure 19b). A compound containing two interpenetrating self-penetrating (12,3) nets has even been reported. ... 

The Fischer symbol was created for the description of different packings of identical spheres and is consequently only applicable to uninodal nets. It has the format p/n/JU. 

Figure 6.10 Two views each of the uninodal nets otk (IO/3/t3), utj 10/3/t2) and atm < 10/3/15) Note that the iitk net is chiral.

The overwhelming richness of structural types can be seen from the fact that there are over 140 recognized zeolite networks (thermally stable and microporous), this number growing by about six every year [2,3]. On the theoretical side there are estimates that the number of plausible regular tetrahedral frameworks exceeds 100 000, f2j and the uninodal nets alone number more than 150 [4]. The lUPAC approved Structure Commission of the International Zeolite Association are in charge of the assignment of three letter codes to each unique zeolite structure and publish them in a web-based database [5J. 

There are a number of other tetrahedral nets with quoted examples in Ockwig et al. [6] but since there are only one or two examples of each net, we will not treat them in any detail. A summary of the uninodal nets is shown in Table 7.1 and in Figure 7.21. 

Table 7.1 Summary of very rare four-oonneeted uninodal nets [6]. These nets are shown and described in Figure 7.21.

Uninodal nets with square planar nodes... 

M. O Keeffe and N.E. Brese, Uninodal 4-Connected 3d Nets. 1. Nets without 3-Rings or 4-Rings. Acta Crystallogn, Sect. A Found. Crystallogr., 1992, 48, 663-669. 

A fundamental contribution is represented by a famous series of articles and books on crystal chemistry published many years ago by A. F. Wells [4], who analysed and classified a great number of nets. Fie emphasized the importance of describing a crystal structure in terms of its basic topology such a description not only provides a simple and elegant way of representing the structures but also evidences relations between structures that are not always apparent from the conventional descriptions. Wells introduced a method for the systematic generation of 3D arrays from 2D nets and also described many hypothetical motifs that were successively discovered within the realm of coordination polymers or of other extended systems. Flis results included a list of many simple nets described with only one kind of node (uninodal) or with two nodes of different connectivity (mainly binodal 3,4-connected). 

An empirical search by O Keeffe [17] for uninodal four-connected nets resulted in the recognition of 168 distinct types (many of which were unseen by Wells). 

Within uninodal 3-connected nets this is special in that it contains relatively large shortest rings (12-gons), the largest n-gons considered by Wells. 

Noll W (1968) Chemistry and technology of sihcones. Academic Press, Inc London O Keeffe M, Brese NE (1992) Uninodal 4-cormected 3D nets. Nets without 3-rings or 4-rings. Acta Cryst A 48 663-669... 

Figure 19 The uninodal self-penetrating (a) (12,3) and (b) (8,4) nets. Catenane-like motifs are highlighted in each.

The diamond net (dia) is uninodal, four connected with tetrahedral nodes and contains six-membered rings. Figure 4.15 shows one node and the four links ABCD. 

The nodes may still be symmetry related so that the nets based on three-connected nodes and Z, = 4 may still be uninodal, see Figure 5.2. 

It is also important to realise that the symmetry of the net very likely is different from the actual crystallographic symmetry of the compound in question. For example, we can build diamond nets from two different types of building blocks, so that in the crystal it is impossible to have a symmetry relation between the nodes, but the 3D-net will still be uninodal since this is a property of the dia net. 

There are a number of other uninodal (10,3) nets derived by Wells [21] and by Fischer and Koch [22]. We give a summary of these nets in Table 6.1 and show the utk, utj and utm nets in Figure 6.10. 

Table 6.1 Other known uninodal (10,3)-nets [21-23]. Measurements refer to the most symmetric case and may deviate substantially if the net is distorted.

The next size is 9, but although there are numerous nets of this type, none of them are uninodal so for the moment we will move on to the nets based on... 

The (8,3)-a and (8,3)-b nets (8 -cta and 8 -ctb) are the two uninodal (8,3) nets and they can both be constructed with perfect symmetry around the vertex, that is angles of 120 and equal lengths of the connectors [211. Wells labelled the first (8,3)-a and the second (8,3) b. The corresponding three letter codes are eta and etb (Fischer symbol 3/8/h2 and 3/8/hl). Unfortunately, they have both the same vertex symbol 8-8 8j, and genus 4, which could make it difficult to distinguish between them. However, eta is chiral, and etb not this should make the identification easier. The nets are shown in Figure 6.11 and Figure 6.12. 

Figure 7.21 Some rare uninodal four-connected nets, see Table 7.1. Some characteristics of these nets are mmt interconnected sheets of six-rings in boat configuration, zni columns of squares linked by two corners and interlinked by the remaining two, cag parallel interlinked crankshafts, gsi four-fold helices of opposing chirality, each helix links to six other helices, neb parallel zigzag bands, unh, chiral, five-rings.

A large number of other five-connected nets have been described (the RCSR lists 184 uninodal five-connected nets) but only a handful of other molecule based nets are known [9]. We will just pick one of these 184 and show in the next section. 

The RCSR currently contains some 75 uninodal seven-connected nets, but only two contain four-membered rings, svn and wfq, the remaining having three-membered rings as shortest circuits. In general, we get closer and closer to close packing as we draw more and more lines, so that the links between the nodes need to be long if we should consider the structure as 3D-net. 

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