A study of the geometrical phase effect on scattering processes: Validity of the extended-Longuet-Higgins formalism for a four-fold Jahn-Teller type model system

► Generalized Born-Oppenheimer equations are derived including the geometrical phase effect. ► The extended Longuet-Higgins theorem is applied on the formalism. ► A scattering model for fourfold electronic manifolds in Jahn-Teller systems is proposed. ► The geometrical phase effect is studied on scattering processes with the above model. ► A rationalization is provided in terms of symmetry-allowed reactive probabilities.