A coming together of mathematics and chemistry

The definition of structure and structural stability for a molecular system as presented above is an example of the application of a general mathematical theory of structural stability. This work has evolved, under the general headings of differential topology and qualitative dynamics, a theory of 

An equivalence relation for molecular graphs is defined as follows two molecular graphs are equivalent if and only if they are associated with two points of the same structural region. An equivalence class of molecular graphs is called a molecular structure. It is then seen that a unique molecular 

As discussed above and illustrated in Fig. 3.4, the partitioning of nuclear configuration space obtained as a result of the definition of molecular structure leads to the concept of a structure diagram. The space R is partitioned into a finite number of structural regions with their boundaries, as defined by the catastrophe set, denoting the configurations of unstable structures. This information constitutes a system s structure diagram, a 

The examples previously discussed with reference to the structure diagram demonstrated the existence of two kinds of catastrophe points, called bifurcation and conflict points. Both types of instabilities were illustrated in terms of the behaviour observed for molecular charge distributions. What we now show is that the existence of these two kinds of catastrophes and just these two, is a direct consequence of a theorem of structural stability stated by Palis and Smale in 1970. This theorem predicts what are the two basic mechanisms for structural change in a chemical system. 

In order to state the conditions under which one will observe a catastrophe of the conflict type, it is necessary to introduce a number of definitions and the notion of a transversal or stable interseetion of two submanifolds of three-dimensional space. 

---