Topological Characterization of Molecular Energy Landscapes Using Sublevel-Set Persistent Homology

The study of conformational spaces and potential energy surface (PES) functions is fundamental for understanding the structural and dynamical properties of molecules with one or more rotational degrees of freedom. In this work, the topological characteristics of conformational spaces and PES functions are investigated for a set of molecules including ethane, butane, and butadiene, which possess one rotational degree of freedom, as well as n-pentane with two rotational degrees of freedom. Sublevel-set persistent homology was applied to the potential energy functions in order to characterize the topology of the associated energy landscapes. This approach allows for the identification of topological changes during the sublevel filtration process, which can be associated with the presence of critical points in the energy landscape, including minima (index 0), transition states (index-1), and maxima (index-2). Furthermore, the method provides information about the global connectivity and structural organization of the conformational landscape. The results show that sublevel-set persistent homology successfully reproduces the energy hierarchy and connectivity between molecular conformers, providing a coherent topological description of the molecular energy landscape. These findings demonstrate that persistent homology constitutes a useful framework for studying the topology of conformational spaces and potential energy surfaces in molecular systems.