A. Different flavors of independent-and coupled-trajectory schemes Tully surface hopping (TSH) is probably the most famous and widely used algorithm to simulate nonadiabatic dynamics . It can be deduced intuitively from the BH representation. In TSH, the nuclear wavepackets are replaced by swarms of independent trajectories and each trajectory either evolves classically on an adiabatic PES, thus identifying the active/force state, or hops from one PES to another in the region of strong nonadiabaticity according to a stochastic procedure, as for instance the fewest-switches scheme. TSH is a purely quantum-classical framework, thus the quantum character of the molecule is limited to the electronic wavefunction, such that the electrons are described by Φ(r, R(t), t) for a given nuclear trajectory R(t) of the swarm. The electronic wavefunction is then expressed using a BHlike expansion Φ(r, R(t), t) = l C l (t)ϕ l (r; R(t)) and an electronic TDSE driven by Ĥel (r, R(t)) is introduced to determine the evolution equation for the expansion coefficients ĊIT l (t) (the subscript IT stands here for "independent trajectory"). Each nuclear trajectory of the ensemble is generated by Newton's equation with adiabatic (AD) forces F AD (t) calculated from the gradient of E l (R(t)) and the hopping probability from state l to state k at time t depends on various parameters, such as the nuclear velocity Ṙ(t) and the NAC vectors d lk (R(t)). While being very powerful and intuitive, TSH may suffer from several drawbacks caused by overcoherence , frustrated hops and velocity rescaling . Overcoherence represents the situation of a trajectory evolving in state k, i.e., for which the active/force state is k, but being associated to a time-dependent electronic state that is a linear combination of adiabatic states. The term "overcoherence" thus exemplifies the disconnect between the electronic and the nuclear dynamics, and is cured by decoherence corrections aiming to restore the connection between the two sets of evolution equations. Frustrated hops and velocity rescaling are related to the hopping procedure, since in TSH the classical energy is conserved for each trajectory, and this condition is enforced after the hops. Therefore, a change in potential energy after a hop is compensated by a change in kinetic energy applied by rescaling the velocity of the trajectory. When a hop occurs towards an electronic state of higher potential energy than the active/force state, the hop can be frustrated if the trajectory does not have enough kinetic energy to compensate for the potential energy change. If a hop is successful, the kinetic energy of the trajectory is modified to ensure energy conversation by rescaling the velocity either isotropically in all directions or only along the direction of the NAC vector between the two electronic states involved in the hop. The presence and treatment of frustrated hops as well as the method for the rescaling the velocity can strongly alter the outcome of a TSH simulation.