Graph-Based Internal Coordinate Analysis for Transition State Characterization

We present graphRC, a method for rapid transition state (TS) mode analysis using internal coordinates derived from molecular graphs. The imaginary mode of a TS describes the direction of atomic motion at the saddle point, providing a local approximation to the reaction coordinate, while Intrinsic Reaction Coordinate (IRC) and Quick Reaction Coordinate (QRC) calculations trace the full pathway to adjacent minima. In all cases, displacements are expressed in Cartesian coordinates and do not directly describe changes in bonding. By translating these into internal coordinate changes (bonds, angles, and dihedrals), graphRC provides chemical insight into the TS mode and reaction coordinate trajectories without prior knowledge of reactant and product structures. Molecular connectivity is determined using xyzgraph, a flexible graph builder validated across 4846 structures spanning 61 elements and 490 element-pair bond types, with close agreement to DFT-derived bonding. Initial validation on 16 diverse TS achieved 100% identification of bond changes, rotations, and inversions, with zero false positives compared to IRC and QRC connectivity. Across 395 TS covering organic, organometallic, and catalytic transformations, normal-mode analysis alone detects the primary bond change in every case, with high agreement to IRC-derived connectivity. This enables programmatic TS verification at a fraction of the cost of formal reaction coordinate calculations, complementing more rigorous methods with rapid, interpretable analysis.