which gives the probability for the trajectory α to hop from the BO PES m to the BO PES l from time t to time t+dt. TSH results presented in Section VI are obtained including the so-called energy decoherence (ED) correction , we will thus referred to TSH-ED henceforth. In this case, the electronic coefficients along the trajectory α associated to the non-active states, say m, are exponentially damped over a typical decoherence time