In Fig. (left) we show the population of the first diabatic state, V 00 , as function of the propagation time with 0, 1, 2, 6 and 23 bath modes. For the 2D, 2D + 1BM and 2D + 2BM models, the standard direct-product approach has been used with a 1 fs time step. With this short time step, we can observe that the population presents small oscillations up to 150 fs with a period of about 1-2 fs. For 2D + 6BM and 2D + 23BM, the Smolyak scheme has been used with a 5 fs time step and the small oscillations observed with the 1 fs time step are not visible anymore although they are present in the dynamics. It is worth noting, that this large time step does not prevent us from performing an exact propagation. However, it enables us to reduce the number of Hamiltonian actions required to propagate by a factor two for full propagation time. As it appears clearly from the results just discussed, the time-dependent quantum treatment with a sparse grid scheme is feasible for the studied vibronic model with a large number of degrees of freedom (up to 25). The computational time necessary to obtain the results in Fig. employing the present Smolyak FIG. : Left: Population of the first diabatic state, V00, using a quantum dynamics propagation approach for different model sizes, from the 2D system without the environment to the 2D + 23BM system+bath model which includes all modes listed in Section IV. Right: Population of the S1 adiabatic state using a quantum dynamics propagation approach for different model sizes, from the 2D to the 2D + 23BM (Ls = 6 has been used for 2D + 23BM model).