We formulate time-dependent quantum dynamics with a basis set in which the classical limit arises in a natural fashion. The basis set is time-dependent and can be used either for all degrees of freedom or together with time-independent basis functions, grids, etc. The basis-set is driven by classical mechanical equations of motion governed by an effective potential derived from the Dirac-Frenkel variational principle. We furthermore formulate an operator version of the theory. Here the coupling between the basis set functions located around the classical trajectories is obtained by solving the quantum matrix problem in a second quantization frame. In this approximation the quantum theory scales as (3N)(2), where N is the number of particles.