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This is a summary of modern chemical reaction theories in the light of the reaction path concept of potential energy hypersurfaces (PES). It presents a simple mathematical analysis of stationary and critical points of the PES and aims to provide the tools necessary for studying chemical reactions.

The main subjects are:
- a comprehensive mathematical description of molecular systems,
- a new reaction path concept,
- an algorithm for following the reaction path.
The reaction path's tangent is determined by an excitation vector and the saddle points surrounding a minimizer can be localized without further information. A procedure appropriate to trace these reaction paths is presented.