Formation of a cage structure

We have demonstrated that Thom s theory of elementary catastrophes finds a direct application in the analysis of structural instabilities which correspond to the making and/or opening of a ring structure. The usefulness of Thom s classification theorem is a consequence of the fact that all the changes in Vp that are involved in such a process occur on a. two-dimensional submanifold of the behaviour space of the electronic coordinates. Clearly, more complex cases of structural changes are to be expected, cases whose complete description will necessitate the use of the full three-dimensional behaviour space. Such a case is illustrated by the formation of a cage structure. 

We have previously noted that the bifurcation set predicted by eqn (4.3) partitions a given control plane w 0 into two regions. At any point (u, v) contained in the region which is bounded by the hypocycloid-shaped cross-section of the bifurcation set, the function /(x, y ft) of eqn (4.3) exhibits two saddle points and another critical point, which is a local maximum if w 0, 

The preceding discussions illustrate the simple way in which the unfolding of the elliptic umbilic singularity (eqn (4.3)) accounts for structural changes which accompany special symmetry-preserving deformations. It is to be realized that the model afforded by this equation works only because the portion of the behaviour space in which these structural changes take place is of dimension two. This is a consequence of the preservation of the symmetry plane in all the distortions studied so far. 

Difficulties are encountered as more general deformations are considered, deformations of the [l.l.l]propellane molecule which require the use of the full three-dimensional behaviour space. At the present time, Thom s classification theorem does not cover situations which involve more than two 

Cross-sections of the structure diagram for CjHg for that portion of its control or nuclear configuration space in which the plane of the apical carbon atoms is a symmetry plane. The structure at the origin of the control space is that given in Fig. 4.5(b) for the bifurction catastrophe point. An increase or decrease in w corresponds to an increase or decrease, respectively, in the separation between the bridgehead carbon atoms. 

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