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

Exponential propagators for the Schr\"odinger equation with a time-dependent potential