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

Abstract

Transforming rovibronic Hamiltonians of molecular systems from the $ΛS$ (Hund's case a) basis to the adiabatic $Ω$ 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 $Ω$ and $ΛS$ formulations and quantify how missing NAC terms and bond-length-dependent spin factors degrade predictions. We implement a complete $Ω$-representation workflow in Duo for diatomics, fully transforming all Hamiltonian terms and enabling side-by-side $Ω$ vs $ΛS$ 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: $Ω$-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.

Figures (4)

• Figure 1: Illustration of the potentials (top panels) and associated couplings (bottom panels) of the two-state coupled system in the $\Omega$-representation (left panels) and $\Lambda S$ representation (right panels). The DBOC-like corrections have been added to the $\Omega$ potentials.
• Figure 2: Illustrations of the spin eigenvalues (top) and TDMCs (bottom) as a function of bond length in the $\Omega$-representation. SOC-induced mixing swaps spin multiplicities between the $a$ and $B$ states, highlighting the emergence of a transition dipole moment. The corresponding constant spins in the $\Lambda S$ representation are also shown.
• Figure 3: Comparison of our computed $a\,^3\Sigma^-\leftarrow X\,^1\Sigma^+$$v=5\leftarrow0$ forbidden band intensities. The upper panel compares $\Lambda S$ (red) and $\Omega$ (blue) representations, showing their equivalence when all coupling terms are included. The lower panel compares our approximate Duo$\Omega$-calculation (orange) with a single-state approximation computed with Level (green dots).
• Figure 4: Computed band intensities for the studied system using Duo. The green lines show the forbidden $a\,^3\Sigma^-\leftarrow X\,^1\Sigma^+$ band, and the other colours (blue, red, and orange) show the full spectrum for the system of study.