Representations of Energy Landscapes by Sublevelset Persistent Homology: An Example With n-Alkanes

TL;DR

This paper demonstrates how sublevelset persistent homology can effectively characterize the topology of energy landscapes in n-alkanes, providing a comprehensive topological analysis beyond traditional methods.

Contribution

It introduces the use of sublevelset persistent homology for energy landscapes and provides a complete characterization for n-alkanes, supported by analytical and sampled data comparisons.

Findings

Complete topological characterization of n-alkane energy landscapes.

Validation of persistent homology as a tool for analyzing high-dimensional energy landscapes.

Support for using topological metrics to assess sampling fidelity in molecular simulations.

Abstract

Encoding the complex features of an energy landscape is a challenging task, and often chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e. 2- or 3-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is more information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane $C_m H_{2m+2}$ potential energy landscapes, for all $m$, and in all homological dimensions. We further compare both the analytical configurational potential energy landscapes and sampled data from molecular dynamics simulation, using the united and all-atom descriptions of the intramolecular interactions. In turn, this supports the application of distance metrics to quantify sampling fidelity and lays the foundation for future work regarding new metrics that quantify differences between the topological features of high-dimensional energy landscapes.