Application of Catastrophe Theory to Molecules

Abstract

There are many applications of catastrophe theory in different fields of science. We will apply this theory to molecules. The theory deals with the mathematical methods of investigation of analytic behavior of a smooth function and it was developed by Arnold and Thorn [1–4]. The theory of singularity of the function( the other name of it is the catastrophe theory) is the generalization of the investigation of the minimum and maximum of the function. The application of this theory to molecules is based on finding of the extremal points of the function p(r,X), which is the total density of the electron charge of a molecule. Basing on such an investigation, which was done by Srebnik and Bader [5], Collard and Hall have suggested to use the catastrophe theory for the investigation of structural instabilities in a molecule [6]. This suggestion was developed in brilliant works of Bader with collaborators[7–11]. They have shown, that in a three-atom molecule and in some cases, in more complicated molecules the structural instabilities can be described by a catastrophe, which was called by Thorn as the elliptic ombilic.