An Analytical Semiclassical Propagation Method for Multi-State Nonadiabatic Scattering Matrices

An analytical semiclassical propagation method is developed for multi-state nonadiabatic scattering matrices in coupled-channel molecular scattering dynamics. The formalism describes the global scattering process through sequential propagation of local transition matrices (called \(I\)-matrix) associated with crossing and avoided-crossing regions along reactive coordinates. By combining WKB propagation with analytical nonadiabatic transition formulas, the method provides an efficient treatment of multichannel scattering while preserving essential quantum interference effects. The theory is generalized to arbitrary \(N\)-state systems and naturally incorporates multiple transition regions and repeated couplings among electronic states (or reactive ro-vibrational states). Numerical results show good agreement with exact quantum calculations over a wide collision-energy range for three-state scattering and hundred-state scattering as numerical examples. The present approach provides both computational efficiency and transparent physical interpretation for complex nonadiabatic scattering phenomena.