and finally by the second half time-step of the mapping variables {ϕ n , θ n } with a similar Verlet scheme as outlined in Eq. 128. Thus, in principle, the non-adiabatic mapping dynamics in the SU (N ) representation does not need the MMST mapping variables. On the other hand, an alternative but numerically simpler way (compared to propagating {ϕ n , θ n }) to propagate dynamics is to obtain the initial values of the angles for {ϕ n , θ n } through either the sampling or focusing approach described in Sec. IV, then transform them into the MMST Cartesian mapping variables {q n , p n } through Eq. 121, and directly propagate the EOMs with these MMST variables through Eq. 117 and Eq. 119. This is an easier approach to implement into computer code, because these equations are much simpler than the corresponding EOMs in Eq. 113. In addition, there are several previously developed symplectic integrators to propagate these EOMs, which one can take advantage of. Our numerical tests suggest that identical numerical accuracy of the results are generated from this approach and the approach in Eq. 128.