The adiabatic Born-Oppenheimer approximation is prevalent in electronic-structure simulations and molecular dynamics studies, since it significantly reduces computational cost, however, within this approximation ultrafast electron dynamics is inaccessible. Achieving a computationally affordable, accurate description of real-time electron dynamics through time-dependent quantum-mechanical theory arguably is one of the greatest challenges in computational materials physics and chemistry today.

Several groups are currently exploring real-time time-dependent density functional theory as a possible route and we recently implemented this technique into the highly parallel Qbox/Qb@ll codes. The numerical integration of the time-dependent Kohn-Sham equations is highly non-trivial: Using a plane-wave basis set leads to large Hamiltonians which constrains what integrators can be used without losing computational efficiency. Here, we studied various integrators for propagating the single-particle wave functions explicitly in time, while achieving high parallel scalability of the plane-wave pseudopotential implementation. We compare a fourth-order Runge-Kutta scheme that we found to be conditionally stable and accurate to an enforced time reversal symmetry algorithm. Both are well-suited for highly parallelized supercomputers as proven by excellent performance on a large number of nodes on Blue Gene based ``Sequoia'' at LLNL and Cray XE6 based "Blue Waters" at NCSA. This allows us to apply our scheme to materials science simulations involving hundreds of atoms and thousands
of electrons.