  The goal of this paper is to provide an analysis of the "toolkit" method used
in the numerical approximation of the time-dependent Schr\"odinger equation.
The "toolkit" method is based on precomputation of elementary propagators and
was seen to be very efficient in the optimal control framework. Our analysis
shows that this method provides better results than the second order Strang
operator splitting. In addition, we present two improvements of the method in
the limit of low and large intensity control fields.
