In Fig. , we compare the accuracy of several propagators for one step propagation during ∆t, as a function of the uncertainty parameter z (defined in Eq. 33). The red and orange curves present the standard symplectic integrator (using velocity-valet, Vel-Ver) and RK-4 propagators, respectively. For velocity verlet propagation, we further divided the integration into 1000 steps, with δt = ∆t/1000. For RK-4, we use δt = ∆t/100 and 100 steps for the ∆t propagation. The light-cyan curve is the accuracy of the normal Chebyshev (Cheb) expansion without the separation of manifolds and the shift of the energy axis. We considered the first six terms of the Chebyshev expansion in Eq. 32 for the ∆t propagation. For low z values up to z ≈ 0.25, the quantum propagation via normal Chebyshev expansion for a single time step ∆t is much more accurate compared to RK-4 or the velocity verlet. As we increase z, the accuracy of Chebyshev reduces and, around z ≈ 0.6, it becomes worse than the velocity verlet propagation, due to the limited number of expansion terms (only six). Further increasing the number of terms will reduce the error. The dark-gray curve represents the Chebyshev propagator with the separate manifold and the shifting of the energy axis which we will refer to as the shifted Chebyshev (Shift-Cheb). As we separate the manifolds and shift the energy of the manifold, the accuracy of Chebyshev increases by orders of magnitude and it consistently performs much better than velocity verlet and RK-4 propagation even at a high value of z, with only six items used in the expansion.