Here, we briefly outline the QD scheme used in this work to connect the spin-mapping approaches, originally formulated in the diabatic quantum representation, with the adiabatic electronic structure calculations. For more details on the QD scheme in general, we refer the reader to Refs. 57,62,70,71. Upon performing the electronic structure calculation at the initial nuclear configuration R(0) to obtain the necessary adiabatic quantities: diagonal energies µ (0) and nuclear gradients Eq. 4). After calculating the initial force F(0) in the mapping representation (Eq. 19) and propagating the nuclear DOFs via a velocity-Verlet scheme, a second electronic structure calculation is performed at the updated nuclear coordinated R(∆t) to obtain the necessary adiabatic quantities: diagonal energies µ (∆t), nuclear gradients