Using EF equations as starting point, the coupledtrajectory mixed quantum-classical (CTMQC) algorithm has been introduced by approximating classically the nuclear TDSE (5) , yielding a representation of the nuclear wavefunction in terms of coupled trajectories . Specifically, the need to reconstruct at each time the nuclear wavefunction/density appearing in the electronnuclear coupling operator Û [χ, Φ](R, t) from the distribution of trajectories requires that they are propagated in parallel. In CTMQC, the trajectories at time t are generated by Newton's equation with forces determined from the TDVP and the TDPES at time t 68 , which in turn can be calculated from the electronic conditional amplitude at time t. Therefore, Φ(r, t; R(t)), associated to the trajectory R(t), is expressed using the BH-like expansion similarly to TSH and MTE, namely Φ(r, t; R(t)) = l C l (R(t), t)ϕ l (r; R(t)). However, we note that in CTMQC we retain the dependence of the expansion coefficients on R(t), which is not the case in TSH and MTE. Such a dependence is critical to obtain the additional coupled-trajectory (CT) contributions in the evolution equations of the electronic coefficients ĊIT l (t) + ĊCT l (t), which are obtained simply by inserting the BH-like expression of Φ(r, t; R(t)) into Eq. ( ). Here, the term ĊCT l (t) can be evaluated only if the positions of all trajectories in the ensemble are known. Using the representation of the electronic conditional amplitude in terms of the adiabatic states to calculate the TDVP and the TDPES, the coupled trajectories of the ensemble are generated with forces that can be cast as F MF (t) + F CT (t), where the first term is the same as in MTE while the second term can only be determined at time t if the positions of all trajectories at that time are known, similarly to ĊCT l (t). While CTMQC can be viewed as derived from EF equations by making controlled approximations, it suffers from some drawbacks that can be related to such approximations, e.g., it is not guaranteed that the classi-cal energy of the ensemble of trajectories is conserved and the calculation of the forces is computationally expensive as it requires the full NAC vectors (3N n -dimensional vectors). To circumvent both issues simultaneously, the coupled-trajectory Tully surface hopping (CTTSH) method has been introduced . In CTTSH, the trajectories are generated by purely adiabatic forces F AD (t), as in TSH, and this dynamics is combined with the idea of stochastic hops from one adiabatic PES to another; energy conservation after the hops is imposed at the level of a single trajectory. Nonetheless, in CTTSH the coupling among the trajectories is retained since the electronic evolution is the same as in CTMQC, ĊIT l (t) + ĊCT l (t), as it has been demonstrated that the coupled-trajectory contribution is crucial to encode information about the nuclear dynamics and the delocalization of the nuclear trajectories in the electronic dynamics to achieve decoherence . CTTSH inherits some issues from TSH, e.g., the presence of frustrated hops and the dependence on the procedure for velocity rescaling . Therefore, the combined coupled-trajectory Tully surface hopping (CCTTSH) method has been recently introduced that exploits the coupled-trajectory nature of the algorithm to circumvent the issue of frustrated hops by imposing energy conservation over the ensemble of trajectories rather than at the level of a single trajectory. With the number of frustrated hops largely reduced thanks to the "energy sharing" idea of CCTTSH, our observations seem to suggest a limited dependence of CCTTSH results on the velocity rescaling procedure. O α (t), (7) with O α (t) the value of the property O for the trajectory α. In Section IV, O α (t) will be the population of the electronic states, the nuclear positions and momenta or the nuclear kinetic energy. Note that in the case of the TSH-based methods illustrated above, the electronic populations can be estimated either as the trajectory-averaged populations