  The application of topology, a branch of mathematics, to the study of
electronic states in crystalline materials has had a revolutionary impact on
the field of condensed matter physics. For example, the development of
topological band theory has delivered new approaches and tools to characterize
the electronic structure of materials, resulting in the discovery of new phases
of matter with exotic properties. In the framework of topological band theory,
the crossings between energy levels of electrons are characterized by
topological invariants, which predict the presence of topological boundary
states. Given the frequency of energy level crossings on the potential energy
surface in molecules, the applicability of these concepts to molecular systems
could be of great interest for our understanding of reaction dynamics. However,
challenges arise due to differing quantum mechanical descriptions of solids and
molecules. Out work aims to bridge the gap between topological band theory and
molecular chemistry. We propose that the Euler Class, a topological invariant,
can be used to categorize and analyse the distribution of nonadiabatic
couplings on the potential energy surface. To exemplify this connection, we
introduce a model system with two distinct regimes that are characterized by
different values of the Euler Class, yet identical potential energy surfaces.
Contrary to expectations set by the Born-Oppenheimer approximation, we propose
that these two regimes don't exhibit identical dynamics, due to a qualitatively
distinct distribution of nonadiabatic couplings.
