Furthermore, in spin-PLDM, the ZPE matrix U (0) = 1 (identity matrix at time 0) needs to be propagated according to U (∆t) = exp[-i Ĥ∆t]U (0), which is done in the time ∆t basis after the ZPE matrix has been rotated to the ∆t basis such that the Hamiltonian is diagonal [ (∆t)] and only applies a phase shift in addition to the time-overlap transformation. The step-wise ZPE matrix basis rotation and subsequent time-propagation can be compactly written as U (∆t) = e -i[ (∆t)]∆t × S T (0, ∆t) × U (0).