In this section, we refer to the linearized method in the SU (N ) mapping formalism as the spin mapping (SM) approach, with the EOMs described in Eq. 98, or equivalently, in Eq. 113 or in Eq. 117. Here, we compare the numerical results obtained from the spin mapping formalism with other methods, including the LSC-IVR as well as the simple trajectory Ehrenfest method. The connection of these other method to the current formalism in the SU (N ) mapping formalism is discussed in Sec. F. Note that the current Spin Mapping approach is derived entirely based on the SU (N ) formalism, without the necessity to convert back to the Cartesian mapping variables of the MMST formalism that spin-LSC uses. Nevertheless, we found that the current approach generates numerically similar results from spin-LSC. As we have discussed, the underlying EOMs for the generalized spin mapping approach (within the linearization approximation) are identical to Ehrenfest dynamics and the MMST-based approach under a specific choice of r s . Nuclear initial conditions for all of these methods are also obtained from the Wigner transform of the initial nuclear density operator. In Appendix G, we connect several previously developed methods with the current method in the language of the SU (N ) mapping formalism.