The topological analysis of $ELF$: an introduction.

The topological analysis of the gradient field of the electron localization function $ELF$ is a powerful interpretative tool which enables a position space partition of the electron density of a molecule or of a solid. It is based on the mathematical theory of the dynamical systems which is particularly suitable to extract qualitative information from data. In many respects this approach can be considered as a good starting point to build a mathematical theory of the chemical bonding. However, as many other tools its use is not as straightforward as it looks at first glance. There are sometimes traps, cul de sac and mirages which are tempting for caution less people. This page intend to briefly address the mathematical background of the topological analysis and to provide an HOW TO to use it as safely as possible in chemical applications.