Fig. presents the population dynamics of Model 6 (Eq. 76f), which is a more challenging 3-state Hamiltonian that involves multiple electronic couplings to test the validity of the L-MFE method. The diagonal populations generated from L-MFE (dotted) again match Lindblad dynamics (solid lines). In Fig. , all of the coherences, Im[ρ 10 ] (orange), Im[ρ 12 ] (yellow), and Im[ρ 20 ] (red) match perfectly with Lindblad dynamics. Note that due to the electronic couplings between states |0⟩ and |1⟩, as well as between states |0⟩ and |2⟩, all of the coherences in general will be non-zero at a given point in time. Using the random phase (governed by Eq. 66) for state |0⟩, all of these detailed features are captured, further demonstrating the exact equivalence between the L-MFE method and Lindblad dynamics when no nuclear DOFs are present. Fig. presents the population dynamics of Model 7 (Eq. 76g) to assess the importance of which state the random phase is applied to. Fig. ,c present the population and coherence dynamics, respectively, of the L-MFE method versus exact Lindblad dynamics. The L-MFE method applies the random phase only to state |0⟩ and its dynamics match the exact results. Fig. present the incorrect population and coherence dynamics, respectively, of a modified L-MFE method when the random phase (θ ξ ) is only applied to state |1⟩ instead of state |0⟩, such that c 1,ξ (t + dt) = e iθ ξ e -Γdt/2 c 1,ξ (t). The populations dynamics of this modified L-MFE method show incorrect oscillation magnitudes, most notably that the oscillations between states |1⟩ and |2⟩ are considerably smaller than the oscillations of the exact dynamics. This corresponds to the coherence dynamics of Im[ρ 12 ] that are smaller than the corresponding exact coherence dynamics. This coherence inaccuracy is caused by the over-decoherence of state |1⟩ due to the application of the random phase to state |1⟩. This result highlights the importance of applying the random phase to the correct state.