A Riemannian Framework for Chemical Reaction Dynamics: Geodesics, Curvature, and Transition-State Stability

Abstract We present a differential-geometric formulation of chemical reaction dynamics in which the nuclear configuration space is treated as a curved manifold whose metric is induced by the covariant Hessian of the potential energy surface (PES). In this Riemannian setting, reaction pathways emerge as geodesics of a curvature-weighted action, providing an intrinsic, coordinate-independent alternative to steepest-descent and chain-of-states methods. The second variation of the action δ2S furnishes a rigorous stability index that identifies transition states as regions of negative sectional curvature and links vibrational structure directly to curvature eigenvalues through the Jacobi operator. The transition state (TS) corresponds intrinsically to a region where the manifold’s signature becomes pseudo-Riemannian, characterized by local instability where δ2S < 0. Furthermore, the linearization of the geodesic equation yields a Jacobi operator, whose eigenvalues reproduce harmonic and anharmonic vibrational frequencies, directly connecting spectroscopic data to curvature invariants. Applied to the dissociation channels of water (H2O → H + OH) and (H2O → H2 + O), this framework reproduces fundamental vibrational frequencies, rotational constants, and anharmonicities with 1–5% accuracy and generates detailed stability maps that resolve hyperbolic, parabolic, and elliptic regions along the minimal path. Beyond reproducing known mechanistic information, the formalism reveals geometric features─channel anisotropy, curvature-driven bifurcations, and instability corridors─that remain hidden in traditional approaches. By reframing reaction dynamics through intrinsic geometry rather than extrinsic coordinates, this work lays the foundation for a unified geometric description of reaction pathways, stability, and spectroscopy.