deletions | additions       diff --git a/photoemission2.tex b/photoemission2.tex  index 996b9be..baea2f2 100644  --- a/photoemission2.tex  +++ b/photoemission2.tex 
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This is done by embedding the system into a simulation box enlarged by a factor $\alpha>1$, extending   the orbitals with zeros in the outer region as shown in Fig.~\ref{fig:pes_nfft}(b).  In this way, the periodic boundaries are pushed away from the simulation box and the wavepackets have to travel   an additional distanceof  $2(\alpha -1)L$ before appearing reappearing  from the other side. In doing so so,  the computational cost is increased by adding $(\alpha -1)n$ points for each orbital. This cost can be greatly reduced using a special grid with only two additional points placed at $\pm \alpha L$   as shown in Fig.~\ref{fig:pes_nfft}(c). 
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In this case we can drop the equation for $\varphi^B_i$ responsible for the ingoing flow  and obtain the set  \begin{align}\label{eq:MM_prop_aux}  \left\{  \begin{array}{l}  \varphi^A_i(\vec{r},t+\Delta t) = M \hat{U}(\Delta t) \phi^A_i(\vec{r},t)\\ \phi^A_i(\vec{r},t)\ ,\\  \varphi^B_i(\vec{r},t+\Delta t) = 0 \\ \ ,\\  \vartheta^A_i(\vec{k},t+\Delta t) = \frac1{  (2\pi)^{\frac{3}{2}}} \int {\rm d}\vec{r} e^{-\mathrm{i}\vec{k}\cdot\vec{r}} (1-M) \hat{U}(\Delta t)  \phi^A_i(\vec{r},t) \\ \ ,\\  \vartheta^B_i(\vec{k},t+\Delta t) = \hat{U}_{\rm v}(\Delta t) \phi^B_i(\vec{k},t) \phi^B_i(\vec{k},t)\ .  \end{array}  \right. .  \end{align}  This new set of equations together with \eqref{eq:FMM_prop} lifts the periodic conditions at the   boundaries and secures numerical stability for arbitrary long time propagations.