The goal of this paper is to provide an analysis of the ``toolkit'' method used in the numerical approximation of the time-dependent Schrödinger 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.