Persistent homology is a relatively new mathematical field. Given an increasing sequence of spaces (using a filtration parameter), persistent homology measures the range over which certain features persist, including connected components, one-dimensional holes, two-dimensional voids, etc. Persistent homology already has several interesting chemistry applications. In the analysis of point cloud data (e.g., Cartesian coordinates of atoms) it has helped characterize different conformations of molecules, solution phase organization , and to identify chemical reactivity using filtration values based upon Euclidean distance. In the case of manifolds, sublevelset persistent homology has been applied to the probability surfaces of activated processes and in the characterization of energy landscapes. In the latter, the sublevelset filtration is obtained by taking thresholds of the energy function V . The topology changes precisely when V obtains the value of a critical point (a minima, a critical point of index one, or a critical point of higher index). Let V : X → R be an energy function defined on a configurational space X of all nuclei within the system. The sublevelset of X at energy E, denoted V -1 ((-∞, E]) := {x ∈ X | V (x) ≤ E}, contains all configurations x ∈ X with energy V (x) ≤ E. Note that when comparing two energy values E ≤ E , we have an inclusion V -1 ((-∞, E]) ⊆ V -1 ((-∞, E ]), and thus we obtain an increasing filtration as the energy value E increases. The persistent homology of this sublevelset filtration counts the number of connected components (0-dimensional homology), loops (1-dimensional homology), voids (2-dimensional homology), and higher order features of a given real-valued function. Furthermore, persistent homology tracks the energy value at which each topological feature first appears and ends. We use birth to refer to the first appearance of a topological feature and death to denote the end of that topological feature. For example, Figure presents the potential energy landscape of 2-methylpentane along with 9 of its sublevelsets at different energy values. In the sublevelset persistence barcode, each topological feature is represented by an interval whose left endpoint gives the birth energy value, and whose right endpoint gives the death energy value. Red bars are for 0-dimensional features (connected components), blue bars are for 1-dimensional features (loops), and green bars are for 2-dimensional features (voids).