Spin-Orbit Induced Non-Adiabatic Dynamics: An Exact $\Omega$-Representation

Transforming rovibronic Hamiltonians of molecular systems from the $\Lambda S$ (Hund's case a) basis to the adiabatic $\Omega$ representation is widely used to "remove" spin-orbit coupling (SOC) and enable single-state treatments of spectra and dynamics. We show that this simplification is only apparent: the SOC elimination necessarily generates sizeable non-adiabatic couplings (NACs) from the nuclear kinetic energy operator. Neglecting these spin-orbit-induced NACs causes severe errors in rovibronic energies and transition properties. Using an analytically tractable two electronic state model and high-accuracy variational benchmarks, we derive the exact conditions for numerical equivalence between $\Omega$ and $\Lambda S$ formulations and quantify how missing NAC terms and bond-length-dependent spin factors degrade predictions. We implement a complete $\Omega$-representation workflow in Duo for diatomics, fully transforming all Hamiltonian terms and enabling side-by-side $\Omega$ vs $\Lambda S$ calculations. For common single-state pipelines (e.g., LEVEL), we provide diagnostics that flag unsafe regimes and practical remedies to restore accuracy. The results deliver actionable guidance for spectroscopy, photophysics, and kinetics: $\Omega$-based single-state approximations are reliable only when interacting states are well separated in the Franck-Condon region; otherwise, explicit non-adiabatic terms are required - even for "forbidden" transitions.