Additive energy functions have predictable landscape topologies

Recent work (J. Chem. Phys. 154, 114114) has demonstrated that sublevelset persistent homology provides a compact representation of the complex features of an energy landscape in $3N$-dimensions. This includes information about all transition paths between local minima (connected by critical points of index > 1), and allows for differentiation of energy landscapes that may appear similar when considering only the lowest energy pathways (as tracked by other representations like disconnectivity graphs using index 1 critical points). Using the additive nature of the conformational potential energy landscape of n-alkanes, it became apparent that some topological features --- such as the number of sublevelset persistence bars --- could be proven. This work expands the notion of predictable energy landscape topology to any additive intramolecular energy function, including the number of sublevelset persistent bars as well as the birth and death times of these topological features. This amounts to a rigorous methodology to predict the relative energies of all topological features of the conformational energy landscape in 3N dimensions (without the need for dimensionality reduction). This approach is demonstrated for branched alkanes of varying complexity and connectivity patterns. More generally, this result explains how the sublevelset persistent homology of an additive energy landscape can be computed from the individual terms comprising that landscape.