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\label{fig:pes_nfft}  Scheme illustratingthe  different discretization strategies for Eq.~\eqref{eq:FMM_prop_aux Eq.~\eqref{eq:FMM_prop_aux} in one dimension.  In all the cases an initial wavepacket (green) is launched towards the left side of a simulation box of   length $L$ and discretized in $n$ sampling points spaced by $\Delta x$. $A$ and $B$ indicate the   space partitions corresponding to Fig.~\ref{fig:pes_sheme}.  Owing to the discretization of the Fourier integrals periodic conditions are imposed at the boundaries and  the wavepacket wraps around the edges of the simulation box (red).   The time evolution is portrayed together with a momentum space representation (yellow), with spacing $\Delta k$  and maximum momentum $k_{\rm max}$, in three situations differing in the strategy used to map real and  momentum spaces:(a) Fast Fourier transform (FFT), (b) FFT extended with zeros (zero padding) in a box enlarged   by a factor $\alpha$, and (c) zero padding with NFFT.