where the nuclear DOFs obeys Eq. 98a-Eq. 98b. To solve Eq. 113b, one can use the expressions of Ωαn+1,n and Ωβn+1,n in Eq. D5 and Eq. D6, respectively, which are only functions of Ω k (Eq. B2-Eq. B4) that depends on {θ n , ϕ n }. For a two-level system, it is straightforward to show that Eqs. 112-107 reduce back to Eqs. E9a-E9b, which are the EOMs for the SU (2) mapping formalism derived in the previous work of spin mapping non-adiabatic ring polymer molecular dynamics. One can evolve each Θ n , or equivalently θ n by using the chain rule in Eq. 110, and ϕ n using a velocity Verlet algorithm ({Θ n } being the generalized conjugate momenta of {ϕ n }), which does not require using the derivative of the potential.