On the Curvature and Phase Structure of Molecular Vibrations: An Ω-Model Perspective

We present a unified geometric model of molecular vibration based on the Ω-Model, in which dual-frequency torsional curvature replaces classical potential energy surfaces. By treating longitudinal and poloidal oscillations as curvature-based dynamics and incorporating resonance, bond order, and complex phase interactions, we predict vibrational energies from first principles. Crucially, we show that vibrational energy in π-bonded systems can manifest as pure torsional phase curvature—an imaginary quantity in the Ω-formalism—which corresponds to non-dissipative, IR-inactive energy storage. We demonstrate accurate predictions for H2 and CH, partial predictions for CO, and uncover a new curvature matrix structure that bridges torsional and longitudinal modes. This framework offers a natural geometric explanation for molecular resonance, zero-point energy, and vibrational anharmonicity, and may serve as a foundational bridge between curvature geometry and quantum field theory in molecules.