Fig. presents the dynamics of three different spinboson models, where LSC-IVR and Ehrenfest dynamics are known to provide less accurate results. This includes the electronic asymmetry in models presented in panels (a) and (b), as well as the strong system-bath coupling in the model presented in panel (c). The details of the parameters are provided in Tab. I of Appendix G. The Spin Mapping results (black solid lines) are compared to LSC-IVR (blue dashed lines), Ehrenfest (green dashed lines) and exact results (red dots). We used the same time-step and number of trajectories that has been used in the previous model. The s = W choice of the spin mapping method slightly underestimates the oscillations in the population dynamics at low temperature, but gives more accurate results compared to the LSC-IVR method (blue dashed line) or Ehrenfest Dynamics (green dashed line). Both the sampled and the focused initial conditions provide almost identical results. Fig. presents the population dynamics of state |2 in the pyrazine model. The results are obtained from the current Spin Mapping approach (red solid) with both focused and sampled initial conditions, and compared to Ehrenfest dynamics (green dash), as well as to the recently developed Generalized Discrete Truncated Wigner Approximation (GDTWA) 51 approach (blue dashed line). The Spin Mapping approach generates very accurate population dynamics compared to the exact results, regardless of the initial conditions of the mapping variables, although the early dynamics is slightly more accurate when focusing the initial conditions, while the long-time dynamics seems closer to the exact result when using the sampled initial conditions. Interestingly, the current Spin Mapping approach almost coincides with the GDTWA approach. It was claimed that in the GDTWA approach, the discrete sampling of the phase space is the key to provide more accurate results compared to the spin mapping formalism, when comparing the population dynamics generated from GDTWA and spin-PLDM. 105,106 However, it seems that this is because the spin-PLDM is less accurate than the current spin mapping approach or spin-LSC for the LVC model, even though the former is more accurate than the latter for spin-boson type problems. 105,106 Future work is needed to further explore the numerical performance of these approaches. Fig. presents the numerical results of a three-state model for the benzene radical cation. This is a particularly challenging model due to the multi-state dynamics and the presence of conical intersection, especially for traditional mixed quantum-classical methods such as Ehrenfest (panel a) and Fewest Switches Surface Hopping 3 (panel c), both of which generate less accurate population dynamics. The spin mapping formalism, on the other hand, gives almost quantitatively accurate dynamics for this challenging system. For the Spin Mapping results presented here, we used 10 5 trajectories for the sampled initial condition and 10 4 for the focused initial condition, and a nuclear time-step of dt = 0.5 a.u. as well as a mapping time-step of dt map = dt nuc /16. While the focused initial conditions provide slightly more accurate short-time dynamics up to 150 fs (panel b), the sampled initial conditions seem to generate population closer to the exact result at a longer time.