用子水平集持久同调表示能量景观：以正构烷烃为例
Representations of Energy Landscapes by Sublevelset Persistent Homology: An Example With n-Alkanes
摘要：
对能量景观的复杂特征进行编码是一项具有挑战性的任务，化学家通常会沿着高度简化的空间（即二维或三维）追求最显著的特征（最小值和障碍）。尽管不连通图或合并树通过最低势垒路径总结了能量景观的局部最小值的连通性，但通过考虑不同能量阈值（或子水平集）下每个连通分量的拓扑结构，可以获得更多信息。我们建议sublevelset持续同源性作为一个适当的工具，用于此目的。我们的计算正构烷烃从丁烷到辛烷的配置相空间，让我们推测，然后证明，一个完整的表征sublevelset持久的同源性的烷烃$C_m H_{2m+2}$势能景观，所有$m$，并在所有的同调尺寸。我们进一步比较了解析构型势能景观和分子动力学模拟的采样数据，使用统一的和全原子的分子内相互作用的描述。反过来，这支持应用距离度量来量化采样保真度，并为未来关于量化高维能量景观拓扑特征之间差异的新度量的工作奠定了基础。
英文摘要：
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.
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