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diff --git a/photoemission2.tex b/photoemission2.tex
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\begin{align}\label{eq:FMM_prop_aux}
\varphi^A_i(\vec{r},t+\Delta t)& = M \hat{U}(\Delta t) \phi^A_i(\vec{r},t)\ ,\\
\varphi^B_i(\vec{r},t+\Delta t)& =
\frac{M}{(2\pi)^{3/2}}\int {\rm d}\vec{k}
e^{\mathrm{i}\vec{k}\cdot\vec{r}} \mathrm{e}^{\mathrm{i}\vec{k}\cdot\vec{r}} \hat{U}_{\rm v}(\Delta t)
\phi^B_i(\vec{k},t) \ ,\\
\vartheta^A_i(\vec{k},t+\Delta t)& =
\frac1{(2\pi)^{3/2}} \int {\rm d}\vec{r}
e^{-\mathrm{i}\vec{k}\cdot\vec{r}} \mathrm{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) \\
&- \frac1{(2\pi)^{3/2}} \int {\rm d}\vec{r}
e^{-\mathrm{i}\vec{k}\cdot\vec{r}} \mathrm{e}^{-\mathrm{i}\vec{k}\cdot\vec{r}}
\varphi^B_i(\vec{r},t+\Delta t)\ .
\end{align}
The momentum-resolved photoelectron probability is then obtained directly from
...
as shown in Fig.~\ref{fig:pes_nfft}(c).
Since the new grid has non uniform spacing a non-equispaced FFT (NFFT) is used~\cite{Kunis_2006,Keiner_2009}. With this strategy, a price is paid in momentum space where the maximum momentum $k_{\rm max}$ is reduced
by a factor $\alpha$ compared to ordinary FFT.
In Octopus we implemented all
the three strategies: bare FFT, zero padding with FFT and zero padding with NFFT.
All these discretization strategies are numerically stable for a propagation time
approximately equivalent to the time that it takes
to for a wavepacket with the highest momentum considered to
be reintroduced in the simulation box.
For longer times we can employ a modified set of equations.
It can be derived from \eqref{eq:MM_prop_aux} under the assumption that the electron flow is only outgoing.
...
\varphi^A_i(\vec{r},t+\Delta t) = M \hat{U}(\Delta 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}}} (2\pi)^{{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)\ .
\end{array}
\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.
A consequence of this approximation is the fact that the removal of charge is performed only in the equation for $\varphi^A_i$ by means of a multiplication
for by $M(\vec{r})$.
This is equivalent to the use of a mask function boundary absorber that is known to present reflections
in an energy range that depends on $M(\vec{r})$~\cite{DeGiovannini:2014wo}.
Carefully choosing the most appropriate mask function thus becomes of key importance in order to
...
This includes, for instance, time-resolved measurements where pump and probe setups are employed.
In Ref.~\cite{DeGiovannini_2013} Octopus was used to monitor the time evolution of the $\pi\rightarrow\pi^*$
transition in ethylene molecules with photoelectrons.
The study was later extended
including to include the effect
of moving ions at the classical
level
in Ref.~\cite{CrawfordUranga_2014}. level~\cite{CrawfordUranga_2014}.
Finally, we point out that our method is by no means restricted to the study of light-induced
ionization but can be applied to characterize ionization induced by other processes, for example, ionization taking place after a proton collision.
