The first scheme is the full restart scheme, where we simply restart the dynamics from the beginning each time an update is carried out and in each restart we use the newest PES. This scheme is the safest option as a constant PES is used in the propagation of the dynamics meaning that the dynamics are solely dependent on the most accurate PES and we have no issues with the conservation of energy. However, this scheme will also be costly as a lot of time will be used in the wave function propagation. We therefore also propose a partial restart scheme. In the partial restart scheme we restart the calculation at a previous time before the PES was updated, t bef . The PES is then continuously updated from the old PES at t bef to the new PES at t upd . The procedure for this will be described below. This partial restart scheme will be more efficient than the full restart as much less time will be spent in the wave function propagation and if t bef is sufficiently long back in time the simulation of the dynamics is expected to be largely unaffected. We note that the partial restart scheme formally does not solve the problem of losing exact energy conservation but is expected to strongly reduce the numerical consequence.