  Ultrafast dynamics in chemical systems provide a unique access to fundamental
processes at the molecular scale. A proper description of such systems is often
very challenging because of the quantum nature of the problem. The concept of
matrix product states (MPS), however, has proven its performance in describing
such correlated quantum system in recent years for a wide range of
applications. In this work, we continue the development of the MPS approach to
study ultrafast electron dynamics in quantum chemical systems. The method
combines time evolution schemes, such as fourth-order Runge-Kutta and Krylov
space time evolution, with MPS, in order to solve the time-dependent
Schr\"odinger equation efficiently. This allows for describing electron
dynamics in molecules on a full configurational interaction (CI) level for a
few femtoseconds after excitation. As a benchmark, we compare MPS based
calculations to full CI calculations for a chain of hydrogen atoms and for the
water molecule. Krylov space time evolution is in particular suited for the MPS
approach, as it provides a wide range of opportunities to be adjusted to the
reduced MPS dimension case. Finally, we apply the MPS approach to describe
charge migration effects in iodoacetylene and find direct agreement between our
results and experimental observations.
