where ω ν is the frequency of the ν-th nuclear normal mode, E k is the vertical transition energy of state k, and the electronic-nuclear coupling is λ kl ν . Here, we investigate two widely tested LVC models: the pyrazine model, 104 and the benzene cation radical model. The pyrazine model is a standard conical intersection model used to study non-adiabatic dynamics. It is a three-modes two-states model, with two tuning coordinates, R1 and R2 , and one coupling coordinate, R3 . The parameters are given in Tab. II of Appendix G. The initial electronic state is prepared on state |2 . The benzene radical cation model is a five-modes three-states model with three tuning coordinates, R1 , R2 and R3 , and two coupling coordinates, R4 and R5 . The parameters are given in Tab. III of Appendix G. The initial electronic state 51 is prepared on state |3 .