Potential energy surfaces (PESs) are central to understanding chemical structure, stability, and reactivity, yet they are typically explored through local sampling rather than represented as complete, structured objects. In this presentation, the Polytope Formalism is shown to provide a discrete, mathematically complete encoding of PES topology through its application to molecular constitution.
By representing atom connectivity as configurations within an abstract-polytopal space, the formalism systematically generates all constitutional isomers and their associated interconversion intermediates, including transition states and higher-order saddle points. The adjacency relations between configurations define elementary reaction graphs whose structure corresponds directly to the network of the associated PES.
When energetic information is assigned to these graph vertices and edges, the reaction graph functions as a compressed, navigable representation of the PES — capturing both its topology (connectivity of configurations) and its topography (relative energies and barriers). This facilitates a multidimensional extension of transition-state theory and provides a rigorous framework for analysing mechanistic pathways.
Through worked examples involving tautomerism and macrocyclic systems, it is shown how this approach transforms PES analysis from local exploration into a globally organised representation. The Polytope Formalism thus establishes a foundation for automated reaction discovery, mechanistic classification, and a mathematically grounded mapping of Chemical Space.