  We consider the numerical integration of the Schr\"odinger equation with a
time-dependent Hamiltonian given as the sum of the kinetic energy and a
time-dependent potential. Commutator-free (CF) propagators are exponential
propagators that have shown to be highly efficient for general time-dependent
Hamiltonians. We propose new CF propagators that are tailored for Hamiltonians
of said structure, showing a considerably improved performance. We obtain new
fourth- and sixth-order CF propagators as well as a novel sixth-order
propagator that incorporates a double commutator that only depends on
coordinates, so this term can be considered as cost-free. The algorithms
require the computation of the action of exponentials on a vector similarly to
the well known exponential midpoint propagator, and this is carried out using
the Lanczos method. We illustrate the performance of the new methods on several
numerical examples.
