2 - Detecting Chirality

Summary

For complex molecules, topology can be an important tool to detect chirality. Nonetheless, there is no mathematically precise definition of chemical chirality that will work for every molecule, because different molecules exhibit different levels of flexibility. Most bonds flex and stretch a little. For large molecules these little bits can add up to quite a lot of flexibility. If we wish to discuss the concept of chirality from a topological point of view, then we need to commit ourselves to a mathematical definition, even if that definition does not correspond precisely to the chemical concept of chirality for every molecule. We would like our definition of topological chirality to have the property that, for any molecule, if the molecular graph is topologically chiral then the molecule will be chemically chiral even if the converse is not true. With this goal in mind, we start with the following definition.

Definition. A graph embedded in three-dimensional space is topologically achiral if it can be deformed to its mirror image. Otherwise it is topologically chiral.

If a molecule can convert itself to its mirror image, then the transformation that the molecule goes through to get to the mirror image corresponds to a deformation from the graph to its mirror image. In particular, though individual bonds of the molecule may rotate or flex, bonds do not pass through one another at room temperature. Analogously, we do not permit one edge of a graph to pass through another edge during a deformation of a graph.