# The Polytope Formalism: application to molecular constitution and the prospect of a complete description of Chemical Space

## TL;DR

The Polytope Formalism offers a unified mathematical framework to describe chemical structures, isomerism, and reactivity, enabling a complete and rigorous exploration of Chemical Space.

## Contribution

Extending the Polytope Formalism to molecular constitution and isomerism, enabling a unified and mathematically rigorous description of Chemical Space.

## Key findings

- The formalism connects molecular constitution and stereoisomerism through atom connectivity.
- It includes traditional isomers and their interconversion intermediates in complete species sets.
- The framework supports a discretised encoding of potential-energy surfaces and isomerisation mechanisms.

## Abstract

The principles of the Polytope Formalism – first developed for stereoisomerism – are here extended to molecular constitution (including constitutional isomerism), highlighting a deep connection between these two aspects of structure and opening the way toward a unified description of all isomerism. A key feature of this development is that it is based solely on atom connectivity, with explicit inclusion of subvalent and hypervalent species. The resulting complete sets of possible species include traditional isomers and their interconversion intermediates (transition states, higher-order saddle points, etc.), providing a powerful tool for elucidating isomerisation mechanisms. This is demonstrated through increasingly complex examples of H-tautomerism. The corresponding networks of species and interconversion pathways map directly onto their associated potential-energy surfaces and thus function as a “discretised encoding” of them. Because this framework accommodates a multidimensional implementation of transition-state theory, it describes and organises the behaviour of the chemical system. Beyond stereoisomerism and molecular constitution, the same mathematical and conceptual principles may be applied to the quantum chemical aspects (nuclear, electronic, and rovibrational states), yielding a fully discretised and physically grounded representation of molecular systems. In doing so, the Polytope Formalism provides a universal framework for the automated exploration of chemical behaviour and Chemical Space – integrating rigorous theory, digital representation, and data-driven discovery within a single coherent framework.

The Polytope Formalism establishes a single unifying framework to represent chemical structure, isomerism, and reactivity giving a mathematically rigorous and complete description of Chemical Space.