We adopt the local diabatization (LD) algorithm for the electronic wavefunction propagation, which is numerically stable in strong coupling regions and exempt from the so-called "trivial crossing" problem . Therefore, Eq. 12 is reformulated in the LD formalism as shown in Appendix A. The fourth order Runge-Kutta (RK4) integrator was employed for the propagation of the electronic coefficients, since the presence of the term Ċα m,ct in Eq. 11 prevents us to use the propagator employed in the LD scheme (see Appendix A).