which can be easily obtained via finite difference wavefunction overlaps of the polaritonic states throughout the trajectory. This procedure is drastically cheaper than the direct computation of the non-adiabatic coupling vectors themselves, as one only needs to compute the non-adiabatic coupling vectors to rescale the nuclei at the moment of a hop. Further, one can ignore the asymmetric nuclear velocity rescaling altogether and perform uniform energy-based rescaling, which is known to provide slightly worse dynamics but alleviates the computation of the vector non-adiabatic couplings altogether, providing a substantial speed-up in ab initio simulations. It is also worth mentioning that the first-order non-adiabatic couplings of the electronic subsystem d αβ (R(t ) ) can be approximated using the scalar nonadiabatic coupling and the diagonal gradients. 163;164