where now {φ} and {φ } are independently and randomly sampled between 0 and 2π. The time-evolution of the mapping variables is identical to Eq. 18 for the forward and backward DOFs, while the state-dependent nuclear force F e (R) used to propagate the nuclei P/M = F e (R, Z, Z ) is instead computed identically to the standard PLDM prescription as the average over the forward and backward paths,