We have to calculate the hopping probabilities and assess if the system should hop to another state for both of these two steps. If a hopping event happens from current state |E I ⟩ to new |E K ⟩ due to quantum subsystem evolution itself governed by Eq. 27a, the velocities of the nuclei are re-scaled along the direction of the NAC d IK (R) in order to conserve the total energy. In particular, the NAC between two polaritonic states can be expressed as 46