Vertex symbol

The first term in (13), also called the diagonal term (Berry 1985), originates from periodic orbit pairs (p,p ) related through cyclic permutations of the vertex symbol code. There are typically n orbits of that kind and all these orbits have the same amplitude A and phase L. The corresponding periodic orbit pair contributions is (in general) g n - times degenerate where n is the length of the orbit and g is a symmetry factor (g = 2 for time reversal symmetry). 

The coordination sequences and the vertex symbol are unique for a particular framework topology, i.e. they can be used to distinguish between different zeolite framework types unambiguously. In this way, frameworks with the same topologies can be easily identified. Currently, it is easier to calculate the coordination sequences and vertex symbol using computer program based on crystallographic data. 

In this database, there are lots of query tools. Users could search specified frameworks by choosing space group, number of T-atoms, framework density, unit-cell volume, TD10, framework energy, and cell parameters, etc. Furthermore, other topological parameters, such as vertex symbol and coordination sequences, could also be used to search a framework. 

It should be noted that all the structures in this database have been fully refined assuming a SiC>2 composition. Only structures with low energies comparable with known frameworks are stored in this database. To make sure that all the structures in this database are unique, coordination sequences and vertex symbols are used to get rid of duplicated frameworks. 

Nomenclature for Single Nets Schlafli and Vertex Symbols... 

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...

Given this inadequacy other descriptors are needed. To differentiate nets with the same Schlafli Symbol a modified version of it, the Vertex Symbol (VS, or long symbol introduced by O Keeffe [1]) appears more useful. In a VS the size of the shortest rings at each angle is given with a subscript to denote the number of such rings. Solid state chemists define a ring as a circuit that has the property that there is no shorter path ( short cut ) between any two vertices on the circuit than the shortest one that is part of the circuit (see Fig. 1.3.4) [28]. 

Not all the angles of a 4-connected net necessarily possess rings. When there is not a ring an asterisk is inserted in the VS. Thus for the 4-connected CdS04 net (the net of the Cd and S atoms, with the -O- links considered as edges, cds, Schlafli Symbol 658) the Vertex Symbol is 6.6.6.6.62.. ... 

Vertex Symbols can be computed for higher connectivities, but since the number of angles increases as the square of the coordination number, they soon become cumbersome the Vertex Symbol for the 6-coordinated net of the primitive cubic lattice (pcu) is 4.4.4.4.4.4.4.4.4.4.4.4.. . , and a 12-coordinated net has (12x11)/2 = 66 angles. 

The Vertex Symbols, however, are also unable to fully characterize nets. For example diamond and lonsdaleite (hexagonal diamond, Ion) have the same VS = 62.62.62.62.62.62 and, therefore, other descriptors must be considered. 

Ion (right). The two nets have the same Schlafli and Vertex Symbol VS (top), but different Coordination Sequence (CS). The rings multi-... 

Zeolite frameworks can be classified according to various schemes (e.g. by pore opening, by structural subunit, by channel system, by framework density, by loop configurations, by vertex symbols, and/or by coordination sequences). Most of these features are defined in the introductory pages of the Atlas of Zeolite Framework Types and then given... 

While this is a good starting point, it is by no means an adequate method to name individual nets. It is better suited to classify nets in categories such as the (10,3)-nets (see Figure 6). Two more advanced designators, automatically calculated by OLEX and TOPOS, are now in common use, the point symbol and the vertex symbol. 

Figure 15 The bnn net has a short symbol 4 6" and a long symbol 4-4-4-4-4-4 6 6 6. The three different types of cycles in this structure are color coded, showing one type of 4-gon and two types of hexagons. Note that the black cycle is short circuited by the green link and thus is not a ring in the mathematical sense, giving the of the vertex symbol.

D Schlafli symbol Vertex symbol (VS) Vertex type Regular tilings Polyhedra and Uses shortest rings sql 4,4, hcb 6,3 rco 3,4, ... 

Delaney symbol or D-symbol 3D Schlafli symbol Vertex symbol (VS) Extended Schlafli symbol Extended Schlafli symbol 2- periodic nets All tilings Regular tilings 3- Periodic nets Uses shortest rings sql 4 , hcb 6, cem 3, 4 pcu 4,3,4 pts (4.4.82.82.83.881(4.4.87.87.87.87)... 

Face symbol Delaney symbol of D-symbol Circuit symbol Vertex symbol Extended Schlafli symbol Polyhedra and cages (tiles) All tilings pcu 4. 6 rco [3. 4 ], pts [42.82]+ 8 ], pcu [4 ... 

Unfortunately there are no lUPAC or lUCr recommendations, or even a consensus among the scientists in the field, about the nomenclature of 3D-nets, and several naming systems are currently in use. As noted by O Keeffe et al. some have many names and symbols...other structures have no names at all and they give the example of the srs net that is also know as (10,3)-a , Laves net , Y , 3/10/cl , SrSi2 and labyrinth graph of the gyroid surface [1,2]. It can also be described by various sets of numbers, the most complete being lOs lOs lOs, referred to as the vertex symbol. Unfortunately, there is no present nomenclature that creates a set of numbers for each net that can be proven to be unique. 

This Babylonian naming situation is of course a state of some concern, but there may be an acceptable solution at hand by the use of the shorthand notation, lower case three letter codes, proposed by O Keeffe et al. [1,3,4] and, when appropriate, the vertex symbol [5] also advocated by O Keeffe. 

Figure 4.7 For the four-connected square grid net we get Schiafli or short symbol 4 . 6 using circuits, but we count only four 4-rings since the last four pairs of links ac and bd) all contain shortcuts and there are no alternative longer paths without shortcuts giving vertex symbol 4.4.4.4.1. (see section 4.2.4).

The extended Schlafli notation has been put forward by O Keeffe [5,9] and is, as the name indicates, an expansion of the Schlafli notation. This notation is also called vertex symbol or long notation. Now we will not only note the size of the smallest rings of a link pair, but we will also count how many of these there are. 

For link pairs without smallest rings, that is all connections contain a short cut back to the node, we write or oo ) and thus for the square-grid net in Figure 4.7 the vertex symbol will be 4-4-44- -. ... 

Although there seems to be a theoretical possibility of having nets with the same set of vertex symbol and CIO value, for practical purposes this it is an... 

Using the vertex symbol notation the srs net is a lOj-lOs-IOs net, where we count all five 10-membered circuits for each link pair. Two of the five circuits at a node can be found in Figure 4.8. The CIO value for this net is 529. 

O Keeffe and co-workers have recently published a free-access web-based searchable database with both hypothetical and synthesised 3D-nets [3]. All nets are listed with their long (vertex) notation, as well as all the types of nodes and links in each net and a number of other parameters. This is a valuable help for the identification of nets once the ring analysis giving the vertex symbols is finished. 

Note that tbe CIO values may differ with one unit depending on if the first nude is counted or not. We arc not aware of any observed nets having different topology and the same CIO value but recently it has been shown tliat there exists two topological distinctive tetrahedral based nets with the same vertex symbols and identical coordination sequences up to the 16 shell [10]. 

In this section we will use two nets to show how to count the rings using the Schlafli and the vertex symbol (extended Schlafli) notation described earlier. 

If we want to use the vertex symbol, the link pairs have to be arranged with opposing pairs AB,CD,AC,BD,AD,BC, In this particular case, wc wiU sec that there is only one type of ring and the sequence we have chosen here is arbitrary. The resulting vertex symbol for the dia net is 62-62 62 62 6 6 and the rings are shown in Figure 4.17. 

Figure 4.17 All shortest rings in the dia net. The resulting vertex symbol is 6i-6i-62-62 62 62-...

---