. Increasing the σ i,ν widths to account for the broadening of the trajectories distribution as the dynamics proceeds makes the Gaussians very flat: as a consequence the quantum momentum becomes smaller, because the gradient of the nuclear density almost vanishes. In such conditions, the CT-TSH method tends to TSH. The complementary drawback of the frozen Gaussian option, i.e., the large fraction of lonely trajectories, is effectively fixed by the application of ODC to such trajectories. Note that, before the fraction of lonely trajectories starts to be relevant, i.e., up to 500-600 fs (see Fig. ), the CT-TSH-FGAC and CT-TSH-FGLA algorithms are able to maintain a good agreement between P m (t) and F m (t). The same is true for the ππ * excitation, up to 200-300 fs (see Fig. ). consistency throughout the dynamics. Decoherence corrections, such as the ODC algorithm, are designed to restore the internal consistency of the TSH procedure in an ad hoc manner, resulting in F m (t) ≃ P m (t), as seen on the right panel of Fig. . As shown in Table , for the trajectories starting in the S 1 (nπ * ) state, the excited state population decay obtained with TSH is the slowest; the decoherence correction (ODC) changes TSH results, perhaps too much, accelerating the decay significantly; finally, the CT-TSH decay is faster than in TSH, but not as fast as TSH-ODC decay. We can conclude that the decoherence correction applied on TSH, and the coupling among the trajectories, correct TSH in the same direction, accelerating the decay rate.