where e L L is the decay dynamics propagation according to the stochastic Lindblad approach, and e L Ĥ is the PLDM propagation in Eq. 61, coupled to the nuclear update with the force described in Eq. 62. In the current work, the PLDM update e L Ĥ ∆t is performed using the Chebyshev Series Expansion outlined in Sec. III B, and the nuclear update is performed using velocity verlet. The stochastic Lindblad update e L L is performed using as outlined in Eq. A3 and Eqs. A5-A6 for ∆t/2 before and after the PLDM propagation. We refer to Eq. 69 as the L-PLDM approach. Note that although we formally express the propagation as the reduced density matrix in Liouville space, all of the dynamical propagation has been performed in Hilbert space for the forward and backward coefficients.