deletions | additions
diff --git a/photoemission2.tex b/photoemission2.tex
index bf58b4d..982a688 100644
--- a/photoemission2.tex
+++ b/photoemission2.tex
...
Starting from a set of orbitals localized in $A$ at $t=0$ it is possible to derive a
time propagation time-propagation scheme with time step $\Delta t$ by recursively applying
the discrete
time evolution time-evolution operator $\hat{U}(\Delta t)\equiv \hat{U}(t+\Delta t,t)$ and splitting
the components with Eq.~\eqref{eq:mask_split}.
The result can be written in a closed form for $\phi^A_i(\vec{r},t)$, represented in real space, and
$\phi^B_i(\vec{k},t)$, in momentum space, with the following structure
...
\end{array}
\right. .
\end{align}
The
momentum resolved momentum-resolved photoelectron probability is then obtained directly from
the momentum components as~\cite{DeGiovannini_2012}
\begin{equation}
P(\vec{k})=\lim_{t\rightarrow \infty}\sum_i^N |\phi^B_i(\vec{k},t)|^2\,,
\end{equation}
while the
energy resolved energy-resolved probability follows by direct integration
$P(E)=\int_{E=|\vec{k}|^2/2}{\rm d}\vec{k}P(\vec{k})$.
In Eq.~\eqref{eq:FMM_prop_aux} we introduced the Volkov propagator $\hat{U}_{\rm v}(\Delta t)$ for the wavefunctions
in $B$.
It is the
time evolution time-evolution operator associated with the Hamiltonian $\hat{H}_{\rm v}$ describing free electrons in
an oscillating field.
Given a time dependent vector field $\vec{A}(t)$, the Hamiltonian
$\hat{H}_{\rm v}=\frac{1}{2}(-i\vec{\nabla}-\frac{\vec{A}(t)}{c})^2$ expressed in the velocity gauge is diagonal in
momentum and
can be naturally
applies applied to $\phi^B_i(\vec{k},t)$.
For all
the systems that can be described by
an a Hamiltonian such that $\hat{H}(\vec{r},t)=\hat{H}_{\rm v}(\vec{r},t)$
for $\vec{r} \in B$ and all times
$t$ the equations $t$, Eqs.
\eqref{eq:FMM_prop} and \eqref{eq:FMM_prop_aux} are equivalent to a time propagation in the entire space
$A\cup B$.
In
particular particular, it exactly describes situations
when where the electrons follow trajectories crossing the boundary separating $A$ and $B$ as illustrated in
Fig.~\ref{fig:pes_sheme}~(b). Fig.~\ref{fig:pes_sheme}(b).
In Octopus we discretize Eq.~\eqref{eq:FMM_prop_aux} in real and momentum space and co-propagate
the complete set of orbitals $\phi^A_i(\vec{r},t)$ and $\phi^B_i(\vec{k},t)$.
...
instability.
In fact, substituting the Fourier integrals in \eqref{eq:FMM_prop_aux} with Fourier sums (usually evaluated with FFTs)
imposes periodic boundary conditions that spuriously reintroduces charge that was supposed to disappear.
This is illustrated with a
one dimensional one-dimensional example in
Fig.~\ref{fig:pes_nfft}~(a) Fig.~\ref{fig:pes_nfft}(a) where a wavepacket
launched towards the left edge of the simulation box reappears from the
other. other edge.
An alternative discretization strategy is
that of using zero padding.
This is done by embedding the
systems 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). 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 distance of $2(\alpha -1)L$ before appearing from the other side.
In doing 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). 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 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.
...
obtain accurate results.
We conclude briefly summarizing some of the most important features and applications of our approach.
The method allows
us to retrieve
$P(\vec{k})$, the most resolved quantity
$P(\vec{k})$ available in experiments nowadays.
In addition, it is very flexible with respect to the definition of the external
field and can operate in a wide range of situations.
In the
strong field regime strong-field regime, it can handle interesting situations, for instance,
when the electrons follow trajectories extending beyond the simulation box, or when the target system
is a large molecule.
This
constitute constitutes a step forward compared to the standard theoretical tools employed in the field which, in the
large majority of cases, invoke the
single active electron single-active-electron approximation.
In Ref.~\cite{DeGiovannini_2012} the code was successfully employed to study the photoelectron angular distributions
of nitrogen dimers under a strong infrared laser field.
The method can efficiently describe situations where more
that than one laser pulse
are is involved.
This includes, for instance,
time resolved 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 the effect moving ions at the classical level
in Ref.~\cite{CrawfordUranga_2014}.
Finally, we point out that our method is by no means restricted to the study of
light induced light-induced
ionization but can be applied to characterize ionization induced by other processes, for example, ionization taking place after a proton collision.
