  Real-time time-dependent density functional theory (RT-TDDFT) is known to be
hindered by the very small time step (attosecond or smaller) needed in the
numerical simulation due to the fast oscillation of electron wavefunctions,
which significantly limits its range of applicability for the study of
ultrafast dynamics. In this paper, we demonstrate that such oscillation can be
considerably reduced by optimizing the gauge choice using the parallel
transport formalism. RT-TDDFT calculations can thus be significantly
accelerated using a combination of the parallel transport gauge and implicit
integrators, and the resulting scheme can be used to accelerate any electronic
structure software that uses a Schr\"odinger representation. Using absorption
spectrum, ultrashort laser pulse, and Ehrenfest dynamics calculations for
example, we show that the new method can utilize a time step that is on the
order of $10\sim 100$ attoseconds in a planewave basis set, and is no less than
$5\sim 10$ times faster when compared to the standard explicit 4th order
Runge-Kutta time integrator. Thanks to the significant increase of the size of
the time step, we also demonstrate that the new method is more than 10 times
faster in terms of the wall clock time when compared to the standard explicit
4th order Runge-Kutta time integrator for silicon systems ranging from 32 to
1024 atoms
