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\newcommand{\fdiff}[2]{\frac{\delta #1}{\delta #2}} % functional derivative
In this section we discuss the calculation of resonant electronic states by
means of the
complex scaling method complex-scaling method, as implemented in Octopus. By
resonant states ``resonant states," we mean metastable electronic states
of finite systems, such as atoms or molecules, with a characteristic energy
and lifetime.
Mathematically, resonances can be defined as poles of the scattering
matrix or cross-section at complex energies~\cite{PhysRev.56.750,Hatano01022008}.
If a pole is close to the
real energy
axis axis, it will produce a large, narrow peak in the
cross-section of scattered continuum
states close to that energy.
Another states.
One way
to create a resonance resonances can arise is
to apply from application of an
electric field strong enough to ionize the system through tunnelling.
Resonant states may temporarily capture incoming electrons or electrons
excited from bound states, making them important intermediate states
...
outgoing component but not an incoming one.
They can be determined by solving the
time-independent Schrödinger
equation with the boundary condition that the
wave wavefunction must asymptotically
have the form
\begin{align}
\phi(r) \sim \frac{\mathrm{e}^{\ii k}}{r}\quad\textrm{as}\quad r\rightarrow\infty\ ,
...
from DFT. However it turns out that a suitably chosen analytic
continuation of a Siegert state is localized, and this form can be used
to derive information from the state.
This is the idea
beyond behind the
\emph{complex scaling}\cite{aguilarcombes,Balslev:1971ez} \emph{complex-scaling} \cite{aguilarcombes,Balslev:1971ez} method where
states and operators are represented
by means of the transformation
\begin{align}
\hat R_\theta\, \psi(\vec r) = \ee^{\ii N \theta / 2} \psi(\vec r \ee^{\ii\theta})\ ,
\end{align}
where $N$ is the number of spatial dimensions to which the scaling operation
is applied, and $\theta$ is a fixed \emph{scaling angle} which
deteremines determines
the path in the complex plane along which the analytic continuation is taken.
The transformation maps the Hamiltonian to a non-Hermitian operator
$\hat H_\theta = \hat R_\theta \hat H \hat R_{-\theta}$.
...
resonance~\cite{simon1973resonances,Reinhardt_1982,Ho19831}.
A typical example of a spectrum of the transformed Hamiltonian $\hat H_\theta$
is shown
on in Figure~\ref{fig:cs-spectrum}, and the corresponding potential
and lowest bound and resonant states
on in Fig.~\ref{fig:cs-potential-wfs}.
The bound-state energies are unchanged while the continuum rotates by
$-2 \theta$ around the origin. Finally resonances appear as isolated
eigenvalues in the fourth quadrant once $\theta$ is sufficiently large
to ``uncover'' them from the continuum.
Importantly, matrix elements (and in
particular energies) of states are independent of $\theta$ as long as
the states are localized and well
presented represented numerically --- this
ensures that all physical
bound-state characteristics of the untransformed Hamiltonian are retained.
...
for independent particles or
in combination with DFT and
selected
xc XC functionals~\cite{Larsen:2013cw}.
The energy functional in KS-DFT consists of several
terms that are all expressible as integrals of the density or the
wavefunctions with the kinetic operator and various potentials.
...
\varphi_{\theta n}(\vec r)\\
+ \ee^{-\ii\theta} \frac12
\iint \dee \vec r \, \dee \rprime \,
\frac{n_\theta(\vec
r)n_\theta(\rprime)}{\Vert\vec r)n_\theta(\rprime)}{\left| \vec r -
\rprime\Vert}\\ \rprime \left|}\\
\quad+ E_\xc^\theta[n_\theta]
+ \int\dee \vec r\, v_{\mathrm{ext}}(\vec r \ee^{\ii \theta}) n_\theta(\vec r)\ ,
\end{multline}
%
with
the, now complex, the now-complex electron density
\begin{align}
n_\theta(\vec r) = \sum_n f_n \varphi_{\theta n}^2(\vec r)\ ,
\end{align}
with occupation numbers $f_n$, and complex-scaled KS states $\varphi_{\theta n}(\vec r)$.
Note that no complex conjugation is performed
of on the left component in
matrix elements such as the density or kinetic energy.
In order to define the complex-scaled
xc XC potential, it is necessary to perform an analytic continuation procedure~\cite{Larsen:2013cw}.
In standard
DFT DFT, the KS equations
are obtained by taking the derivative of the energy functional with respect
to the wavefunctions. Solving the equations corresponds to searching
for a stationary point, with the idea that this minimizes the energy.
...
continued pseudopotentials, and where the Hartree potential
\begin{align}
v_\Ha^\theta(\vec r) &=
\ee^{-\ii\theta}\int\dee \rprime\, \frac{n_\theta(\vec
r')}{\Vert r')}{\left| \vec r'-\vec
r\Vert} r\right|}
\end{align}
is determined by solving the Poisson equation defined by the complex density.
Together with the xc potential
...
stationary points~\cite{WM07} of the
functional~\cite{Whitenack_2010,WW11}.
Fig.~\ref{fig:cs-ionization-He} shows calculated ionization rates for the
He~1s~state He~1$s$~state in a
uniform Stark-type electric field as a function of field strength.
In the limit of weak electric fields, the simple approximation
by Ammosov, Delone and Krainov~\cite{adk}, which
is depends
only on the ionization potential, approaches the accurate reference calculation
by Scrinzi and co-workers~\cite{PhysRevLett.83.706}.
This demonstrates that the ionization rate is determined largely by the
...
Meanwhile exact exchange, which is known to produce
accurate ionization energies, predicts ionization rates
much closer to the reference calculation. The key property
of the
xc XC functional that allows accurate determination of decay rates
from complex-scaled DFT therefore appears to be that it must yield
accurate ionization potentials, which is linked to its ability
to reproduce the correct asymptotic form of the potential at large distances
