# Graph-Based Internal Coordinate Analysis for Transition State Characterization

## TL;DR

This paper introduces graphRC, a new method that uses molecular graphs to analyze chemical reaction pathways and transition states by translating atomic movements into chemical insights.

## Contribution

The novel contribution is graphRC, a graph-based approach for rapid and interpretable transition state analysis using internal coordinates.

## Key findings

- graphRC accurately identifies bond changes, rotations, and inversions in transition states with 100% accuracy compared to IRC and QRC.
- Across 395 transition states, normal-mode analysis with graphRC detects primary bond changes with high agreement to IRC-derived results.
- The method enables programmatic verification of transition states at a lower computational cost than traditional reaction coordinate calculations.

## Abstract

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.