First of all, it is important to rationalize the use of the LZ transition probabilities at the low level in the MTS-TSH method presented in section II B. For that purpose, we have run a reference trajectory for about 100 femtoseconds (fs) on the second excited state of system I. This trajectory was performed with the PBE0 functional and a standard velocity Verlet algorithm. The transitions probabilities where calculated at each time step using the FS method (see section II A 1) but no electronic transitions were allowed such that the system 1: Initialize positions, velocities, and running electronic state: x, v, J 2: Initialize high and low level forces: F high , F low 3: Get effective force: