Between the approximate FSSH and the numerically exact MCTDH, we can find approaches based on the multiple spawning framework, which express the molecular wavefunction in an adaptively sized basis of traveling multidimensional Gaussians with frozen width, also commonly referred to as trajectory basis functions (TBFs). The phase space centers of the TBFs evolve according to classical equations of motion, whereas their amplitudes are coupled through the TDSE. To describe the transfer of amplitude between different electronic states, methods of the multiple spawning varieties allow their basis set to grow whenever a TBF reaches a region of high nonadiabaticity. In the complete basis set limit, multiple spawning approaches have been shown to converge to the exact quantum dynamics. However, to arrive at a computationally tractable implementation of the multiple spawning idea that is geared towards on-the-fly dynamics -AIMS one has to introduce a set of controlled approximations to the Hamiltonian matrix elements coupling the TBF amplitudes.