Computational Method. Using the EOMs expressed in Eq. 113 or Eq. 104 and the out-of-equilibrium TCF expressions that we derived in Eq. 73, we use the spin mapping approach under the linearized approximation to study the non-adiabatic dynamics of model systems. Here, we briefly summarize the details of the propagation algorithm. In this paper, we present numerical results with both sampled and focused initial conditions. The sampling of {θ n } or {Θ n } and {ϕ n } is done over the phase space volume element dΩ defined in Eqs. 21-22, through a Metropolis-Hasting algorithm. The results are presented using r s = r s = r W . The focused initial condition are described in Eq. 81 and the procedure in Sec. IV. To propagate the dynamics, we use the simple Verlet algorithm because of the conjugate relation between Θ n and ϕ n (see Eq. 109), as well as the relation between Θn and θn in Eq. 109a (to use the EOMs in Eq. 113).