Bonds graph

Figure 10 O-H radial distribution function as a function of density at 2000 K. At 34 GPa, we find a fluid state. At 75 GPa, we show a covalent solid phase. At 115 GPa, we find a network phase with symmetric hydrogen bonding. Graphs are offset by 0.5 for clarity.

So, the number of connections can be determined either from the "bond graph" [7], or even better, from the number of "propanes" that can be derived from the skeleton of the molecule under consideration. 

To build up a "bond graph", the bonds of a given molecule are enumerated and represented by a point, and then each pair of points is connected with a line whenever the corresponding bonds are adjacent. Therefore, the number of the resulting lines gives the number of "adjacent pairs" or connections (t)). For example,... 

With fused polycyclic systems and cycles bearing double bonds, in order to obtain simple "bond graphs", a careful distribution of points must be carried out. 

Bond graphs are familiar in organic chemistry where they are called molecular diagrams. The network of an organic molecule contains a finite number of atoms and bonds, and, because bonded atoms are always neighbours in three-dimensional space, such a bond graph can easily be drawn as a two-dimensional projection of the three-dimensional molecular structure. 

Fig. 2.4. The bond graph of NaCl (18189). The heavy line represents the spanning tree (Appendix 3). Compare this graph with the infinite three-dimensional network shown in Fig. 1.1.

Equations (3.3) and (3.4) have become known respectively as the valence sum rule and the loop, or equal valence, rule, and are known collectively as the network equations. Equation (3.4) represents the condition that each atom distributes its valence equally among its bonds subject to the constraints of eqn (3.3) as shown in the appendix to Brown (1992a). The two network equations provide sufficient constraints to determine all the bond valences, given a knowledge of the bond graph and the valences of the atoms. The solutions of the network equations are called the theoretical bond valences and are designated by the lower case letter 5. Methods for solving the network equations are described in Appendix 3. ... 

In NaCl (18189), this principle would require all atoms to be identical. Clearly this symmetry is already broken by the constraint imposed by the chemical formula which requires half the atoms to be Na" " and half CP. However, all the Na" " ions are indistinguishable from each other, and the same is true for the CP ions. The bonds likewise, six for each formula unit, are also equivalent in the bond graph (Fig. 2.4). The crystal structure (Fig. 1.1) is then determined by applying the principle of maximum symmetry to the constraints imposed by three-dimensional space as described in Section 11.2.2.4. The crystal structure is thus uniquely determined by the principle of maximum symmetry and the chemical and spatial constraints. 

In more complex compounds, particularly ternary and quaternary compounds, it is often not possible to maintain the equivalence of all atoms of the same element because they are required to form different numbers of bonds. For example, in the bond graph of CaCrFs (10286) shown in Fig. 2.5, Cr " " can only have its expected coordination number of six if one of the five F ions forms two bonds to Cr +. The equivalence of the remaining four F ions is broken by the spatial constraints (Section 12.3.5). 

Hitherto it has been assumed that the bond graph is bipartite, i.e. bonds only occur between a cation and an anion with no cation-cation or anion-anion bonds present. While the majority of inorganic compounds have bipartite bond graphs, there are a few, such as mercurous and peroxy compounds, that contain homoionic bonds. It is easy to see that there can be no electric flux linking two cations or two anions, so the ionic model predicts that no bond will exist between them. 

With a knowledge of the ideal coordination numbers expected for each of the ions, one can explore the crystal chemistry of a compound without prior knowledge of its structure or even its bond graph. It is not even necessary that the compound exist in order to explore its chemistry and to discover whether it is likely to be stable, and if so, what its properties might be. 

Fig. 7.6. The bond graph of oxalic acid dihydrate (OXACDH04) showing the observed bond valences. This graph shows the correct connectivity, but is not a three-dimensional molecular diagram.

Fig. 7.8. Bond graph of Ni(H20)6S04 (69127) showing the bond valences.

Fig. 8.9. Bond graphs of (a) ZnSb206 (30409) and (b) ZnV206 (30880) showing the theoretical bond valences.

Fig. 8.9(b), ZnV20g is able to crystallize with the brannerite structure whose theoretical bond valences, calculated from the network equations ((3.3) and (3.4)) and shown in Fig. 8.9(b), already predict an out-of-centre distortion for the ion. ZnV20g thus adopts a bond graph that supports the electronically induced distortion. In this case the adoption of a lower symmetry bond graph is favoured because it is able to reduce the bond strain. 

As pointed out above, the bond flux depends on the connectivity of the compound, that is, on the bond graph. This means that the length of a bond depends not only on its immediate environment, but also on the structure of the whole crystal or molecule of which the bond is part. Thus anions such as PO, which ideally are perfect tetrahedra, will often be distorted when they appear in crystals. However, this distortion can normally be predicted via the network equations provided the graph of the bond network is known. 

The principle of maximum symmetry requires that the crystal structure adopted by a given compound be the most symmetric that can satisfy the chemical constraints. We therefore expect to find high-symmetry environments around atoms wherever possible, but such environments are subject to constraints such as the relationship between site symmetry and multiplicity (eqn (10.2)) and the constraint that each atom will inherit certain symmetries from its bonded neighbours. The problems that arise when we try to match the symmetry that is inherent in the bond graph with the symmetry allowed by the different space groups are discussed in Section 11.2.2.4. 

The second principle is a rule that is derived from the properties of the bond graph and is known as the Coordination number rule (Rule 6.1). An alternative statement of the rule from that given in Section 6.3 is ... 

Rule 10.3 (Shubnikov s fundamental law). Atoms will occupy Wyckoff positions in the crystal that are compatible in both multiplicity and symmetry with the bond graph. 

The first step in any chemical approach to crystalline structure is to determine the short-range order, i.e. which atoms are bonded. The most convenient way of doing this is by means of the bond graph described in Section 2.5. In many cases all or most of the bond graph can be determined from first principles, since, except for the weakest bonds created in the post-crystallization stage, the bond graph is determined by the rules of chemistry, particularly the hierarchical principle (Rule 11.5), the valence matching principle (Rule 4.2), and the principle of maximum symmetry (Rule 3.1). 

Fig. 11.2. Bond graphs of (a) NaCl (18189), (b) CsCl (22173), and (c) ZnO (67454). In this and other bond graphs in the chapter, the spectrum and the highest possible crystallographic site symmetry allowed by the graph is shown but these site symmetries may not be mutually compatible.

Fig. 11.3. Possible bond graphs of ABO3 with 6-coordinate B, (a) 6-coordinate A, (b) 8-coordinate A, (c) a second graph for 8-coordinate A, (d) a third graph for 8-coordinate A, (e) 9-coordinate A, (f) 12-coordinate A.

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