Metric Deformation and Topological Persistence of Molecular Configuration Spaces

Conformational analysis is commonly centered on potential energy surfaces, whereas changes in the intrinsic metric and topological organization of molecular configuration spaces have received less attention. Here, we introduce a correspondence-preserving framework that directly compares ideal geometric samples with their constrained MMFF94-relaxed realizations. Unlike energy-based conformational analysis, the proposed approach separates changes in pairwise structural geometry from changes in global topological organization. Configuration spaces were constructed for ethane, butane, butadiene, biphenyl, and n-pentane using one- and two-dimensional torsional domains. Pairwise root-mean-square deviation (RMSD) matrices after optimal rigid-body superposition were used to quantify metric deformation, while Vietoris–Rips persistent homology was used to compare the corresponding topological signatures. MMFF94 relaxation produced structured, conformation-dependent patterns of metric expansion and contraction rather than a uniform rescaling of the configuration spaces. Nevertheless, the dominant homological organization was preserved: the one-torsional systems retained (β0,β1)=(1,1), compatible with S1, whereas n-pentane retained (β0,β1,β2)=(1,2,1), compatible with T2. Persistence-diagram distances further showed that preservation of these dominant classes does not imply equality of the complete persistence representations. For the molecular systems, sampling schemes, and force-field model considered, the results demonstrate that molecular relaxation can systematically reorganize RMSD geometry without altering the dominant global topological features. The framework therefore provides a quantitative means of distinguishing metric deformation from topological change in corresponding molecular configuration spaces.