It has been recommended that the best representation to use for trajectory surface hopping (TSH) calculations is the fully adiabatic basis in which the Hamiltonian is diagonal. Simulations of intersystem crossing processes with conventional TSH methods require an explicit computation of nonadiabatic coupling vectors (NACs) in the molecular-Coulomb-Hamiltonian (MCH) basis, also called the spin-orbit-free basis, in order to compute the gradient in the fully adiabatic basis (also called the diagonal representation). This explicit requirement destroys some of the advantages of the overlap-based algorithms and curvature-driven algorithms that can be used for the most efficient TSH calculations. Therefore, although these algorithms allow one to perform NAC-free simulations for internal conversion processes, one still requires NACs for intersystem crossing. Here, we show that how the NAC requirement is circumvented by a new computation scheme called the time-derivative-matrix scheme.