Point (ii) above concerns the theoretical representation adopted to describe the electronic dynamics. When aiming to simulate the ultrafast relaxation process of a photo-excited system via conical intersections or avoided crossings between electronic states of the same spin multiplicity, i.e., internal conversions, the Hamiltonian given in Eq. ( ) with x = r and r = R is able to fully capture this dynamics. Therefore, usually, the adiabatic representation is employed, formed by the eigenstates of the electronic Hamiltonian, i.e., ĤBO (r, R) in Eq. (1). In this representation, a peculiar parametric dependence on the nuclear configuration R appears, as consequence of the dependence of ĤBO on R, and stimulates steadily interests in various communities due to its potential relation with geometric phases (see Section IV A). The form of the electronic Hamiltonian ĤBO is diagonal in this representation and the coupling between pairs of electronic states is encoded in the nonadiabatic coupling vectors -connecting two electronic states via the nuclear displacement operator. Due to the very localized nature of these nonadiabatic coupling vectors, local diabatization schemes are often employed to solve the electronic evolution equation locally in the diabatic basis, designed to capture even with a finite integration time step the localized degeneracy region between two electronic states along the nuclear evolution. When describing processes as intersystem crossings involving electronic states with different spin multiplicity, the spin-orbit coupling ĤSOC (x, R), with x = r, σ, has to be included in the molecular Hamiltonian to open spin-forbidden relaxation pathways upon photoexcitation. In such situations, the question presents itself as to whether the spin-diabatic or the spin-adiabatic representation is most suited for the simulations . The former is composed by the eigenstates of ĤBO , and the pairs of electronic states are coupled either via the nonadiabatic coupling vectors or via the spin-orbit couplings, depending on the spin multiplicity; the latter is formed by the eigenstates of ĤBO + ĤSOC , and the couplings between pairs of states becomes fully of nonadiabatic character. Also in this case, the literature presents various options depending on the algorithm and on the code employed for the simulations. Dynamics in the presence of spin-orbit coupling was analyzed based on CTMQC and pointed out that, while at the exactfactorization level there is no difference between the two representations, as expected, the approximations underlying the CTMQC algorithm perform better in the spin-adiabatic basis than in the spin-diabatic basis. A very similar issue is encountered when an external laser field is added to the molecular Hamiltonian, i.e., ĤBO (r, R) + Ĥext (r, R,t), since now the ideas of field-adiabatic or field-diabatic bases can be introduced . Also in this case, authors discussed this point in relation to various trajectory-based algorithms for nonadiabatic dynamics, and in the context of the exact factorization only the particular case of a periodic drive was considered, and CT-MQC was combined with the Floquet formalism 147 , designed to solve a TDSE with a Hamiltonian that is periodic in time.