Lastly, when ĤQC + ĤC = 0 (no nuclear DOFs present in the system Hamiltonian), the L-MFE approach in Eq. 72 (or equivalently Eq. 21) provides identical results as those obtained by solving Eq. 12, regardless of the choice of dt (as long as it is small enough to provide a stable integration of -i ℏ [ ĤQ , ρ]). This is because in the L-MFE approach, the decay dynamics are designed to exactly match the analytical time evolution of the reduced density matrix elements. Consequentially, if the Hamiltonian is 0 and only jump operator dynamics are present, the choice of dt for L-MFE could be arbitrarily large and still give the correct dynamics. On the other hand, under the same condition when ĤQC + ĤC = 0, the Ehrenfest+R approach is only accurate up to first order in dt (see Eq. A28 and Eq. B7).