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and lifetime.
Mathematically, resonances can be defined as poles of the scattering
matrix or cross-section at complex
energies. energies.\cite{PhysRev.56.750,Hatano01022008}
If a pole is close to the
real energy axis it will produce a large, narrow peak in the
cross-section of scattered continuum states
around close to that energy.
Another way to create a resonance is to apply 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
in many processes.
Another way to create a resonance is to apply an
electric field strong enough to ionize the system through tunnelling.
%Resonances are typically seen in scattering experiments
%as a peak of given energy and width, with the width being proportional
%to the lifetime. For example a resonance will correspond a peak in
%the electron scattering cross section. A long-lived resonant state can be thought
%of as an ``almost bound state'', but the properties of resonant states are
%nevertheless very different, and different methods must be used to calculate them.
The defining characteristic of a resonant state is that it has an
outgoing component but not an incoming one.
Such states are often called Siegert
states\cite{PhysRev.56.750}.
They can be determined by solving the
time-independent Schrödinger
equation with the boundary condition that the wave must asymptotically
have the form% $r\phi(r) \sim \exp(+\ii k r)$ as $r \rightarrow \infty$.
\begin{align}
\phi(r) \sim \exp(\ii k r)/r\quad\textrm{as
$r\rightarrow\infty$}. $r\rightarrow\infty$},
\end{align}
The where the momentum $k$ is complex
with and has a negative imaginary part.
This causes the state to diverge exponentially in space as
$r\rightarrow\infty$.
Its energy also has The state can further be ascribed a complex
energy, likewise with a negative imaginary part, causing it to decay
exponentially
over time
uniformly at every point in
space.
Since they have complex energies they are not eigenstates of any Hermitian operator,
and indeed do not belong to the Hilbert space.
This precludes the direct calculation of resonances with
the standard computational methods known from DFT. space uniformly.
The method Resonant states are not eigenstates of
\emph{complex scaling}\cite{aguilarcombes,Balslev:1971ez}
circumvents this problem by a transformation
which maps any Hermitian operator and
in particular do not reside within the
Hamiltonian to Hilbert space. This precludes their direct
calculation with the standard computational methods
from DFT. However it turns out that a
non-Hermitian operator
$\hat H_\theta = \hat R_\theta \hat H \hat R_{-\theta}$, where suitably chosen analytic
continuation of a Siegert state is localized, and this form can be used
to derive information from the
operator $\hat R_\theta$ state.
This is
the idea behond the
\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(\ve r) = \ee^{\ii N \theta / 2} \psi(\ve r \ee^{\ii\theta}),
\end{align}
and where $N$ is the number of
spatial dimensions
in to which the scaling operation
is
applied.
The parameter applied, and $\theta$ is a fixed
number that one must choose, with \emph{scaling angle} which deteremines
the
original theorems proven for $0<\theta<\pi/4$. The spatial coordinates $\ve
r$ are rotated to $\ve r \ee^{\ii\theta}$ path in the complex
plane, and one
hence works in terms of plane along which the analytic
continuations of both states and
operators. continuation is taken.
The transformation maps the Hamiltonian to a non-Hermitian operator
$\hat H_\theta = \hat R_\theta \hat H \hat R_{-\theta}$.
The
idea of complex scaling is that for a suitable $\theta$,
the analytic continuation of the Siegert states
are \emph{localized}
instead $\psi(\ve r)$ of
diverging,
and the original Hamiltonian are
square-integrable
eigenstates
$\psi_\theta(\ve r)$
of
the non-Hermitian
operator $\hat
H_\theta$. Importantly, H_\theta$, and their eigenvalues $\epsilon_0 - \ii\Gamma/2$
define the
matrix elements energy $\epsilon_0$ and width $\Gamma$ of
localized states, the
resonance. More details can be found in
particular the
energies, are independent many reviews of
$\theta$. This ensures all physical
bound-state characteristics complex scaling~\cite{simon1973resonances,doi:10.1146/annurev.pc.33.100182.001255,Ho19831}.
\begin{figure}
\centering
\includegraphics{fig-cs-spectrum}
\includegraphics{fig-cs-potential}
\caption{Spectrum (left) of
the untransformed one-dimensional complex-scaled single-particle
Hamiltonian
with potential
$v(x) = 3(x^2 - 2) \ee^{-x^2 / 4}$ and $\theta=0.5$.
The lowest-energy resonance, here located close to the origin,
does not lie exactly on the real axis but has an
imaginary part of about $-10^{-5}$. Right: The potential (blue)
and the real (fully drawn) and imaginary (dotted) parts of the
two bound and three lowest resonant wavefunctions. For improved visualization,
the wavefunctions
are
retained. vertically displaced by the real parts of their energies.}
\label{fig:cs-spectrum}
\end{figure}
A typical example of a spectrum
of the transformed Hamiltonian $\hat H_\theta$
is shown in Figure \ref{fig:cs-spectrum}.
The bound-state energies are unchanged, the continuum rotates by
$-2 \theta$ around the origin, and 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 numerically --- this
ensures that all physical
bound-state characteristics of the untransformed Hamiltonian are retained.
Our implementation supports calculations with complex scaling
for independent particles or
in combination with DFT and
selected
XC
functionals~\cite{Larsen:2013cw}.
This combination relies on complex-scaling the DFT energy functional. functionals\cite{Larsen:2013cw}.
The energy functional in Kohn--Sham DFT consists of several
terms that are all expressible as integrals of the density or the
wavefunctions. wavefunctions with the kinetic operator and various potentials.
The functional is complex-scaled
as per the prescribed method
by rotating the
real-space integration contour
of every term by $\theta$ in the complex plane.
This makes the functional expressible
The energy
functional
thereby then becomes expressible as
% Problem: with \ve r', the arrow overlaps with the apostrophe.
% At least when vectors are denoted by arrows.
% This command fixes it, but maybe it isn't relevant if they change font anyway.
%\newcommand{\rprime}[0]{\ve r^{\,}{}'}
\newcommand{\rprime}[0]{\ve r'}
...
\end{align}
with the electron density
\begin{align}
n_\theta(\ve r) = \sum_n f_n
\psi_{n \theta}^2(\ve \psi_{\theta n}^2(\ve r),
\end{align}
occupation numbers $f_n$, and complex-scaled Kohn--Sham states
$\psi_{n \theta}(\ve $\psi_{\theta n}(\ve r)$.
We also need to define the complex-scaled XC potential $E_\xc^\theta[n_\theta]$.
Note
how that the wavefunctions $\psi_{\theta n}(\ve r)$ and density $n_\theta(\ve r)$
are complex, but no complex conjugation is performed of the left component in
matrix
elements. This the the correct procedure
when calculating matrix elements
of states that would such as the density or kinetic energy
(the ``unscaled'' wavefunctions are assumed to be purely
real
in since the system is finite, and the complex scaling transformation
subsequently affects the coordinates of left and right states equally, i.e.,
without complex conjugation).
The XC functional is complex-scaled by taking the analytic continuation
similarly to any other functional, and its exact form of course depends
on the
unscaled case. chosen approximation; see Ref.~\cite{Larsen:2013cw}.
In standard DFT the Kohn--Sham 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.
In
the present our case we cannot minimize the energy functional as it is
complex-valued, but we can search for a stationary point in exactly the same
way as has previously been argued\cite{WM07}.
The complex-scaled versions
of the Kohn--Sham
equations thereby become similar to the usual ones:
\begin{align}
\left[-\frac12 \ee^{-\ii2\theta}\nabla^2 +
v_{\mathrm{eff}}^\theta(\ve v_\theta(\ve r)
\right]
\psi_\theta(\ve \psi_{\theta n}(\ve r) =
\psi_\theta(\ve \psi_{\theta n}(\ve r)
\epsilon_\theta. \epsilon_{\theta n}.
\end{align}
The effective potential
$v_{\mathrm{eff}}^\theta(\ve $v_\theta(\ve r)$ is the functional derivative
of the energy functional with respect to the density $n_\theta(\ve r)$, and therefore
consists of the terms
\begin{align}
v_{\mathrm{eff}}^\theta(\ve v_\theta(\ve r) \equiv \fdiff{E}{n_\theta(\ve r)} =
v_\Ha^\theta(\ve r) + v_\xc^\theta(\ve r)
+ v_{\mathrm{ext}}(\ve r \ee^{\ii\theta})
\end{align}
with where $v_{\mathrm{ext}}(\ve r \ee^{\ii\theta})$ may represent atomic
potentials as analytically
continued pseudopotentials, and where the Hartree potential
\begin{align}
v_\Ha^\theta(\ve r) &=
\ee^{-\ii\theta}\int\dee
\ve r\, \rprime\, \frac{n_\theta(\ve r')}{\Vert \ve r'-\ve
r\Vert},\\
v_\xc^\theta(\ve r) &= \fdiff{E_\xc^\theta[n_\theta]}{n_\theta(\ve r)}. r\Vert}
\end{align}
blahblah......
The complex-scaled Hartree potential is determined by solving the Poisson equation
defined by the complex density.
Together with the XC potential
\begin{align}
E_\Ha^\theta = \ee^{-\ii\theta} \frac12
\iint \dee \ve r\, \dee \ve r'\,
\frac{\rho(\ve r)\rho(\ve r')}{\Vert\ve r - \ve r'\Vert} v_\xc^\theta(\ve r) &= \fdiff{E_\xc^\theta[n_\theta]}{n_\theta(\ve r)},
\end{align}
this defines a self-consistency cycle very similar to ordinary
Kohn--Sham DFT although more care must be taken to occupy the correct states,
as they are no longer simply ordered by energy.
The lowest-energy resonance of a system is then found by searching for
stationary points\cite{WM07} of the
functional.
\cite{Whitenack_2010,WW11} % daniel JPhysChemLett, PRL functional\cite{doi:10.1021/jz9001778,WW11}.
In practice one can obtain the correct complex-scaled form \begin{figure}
\includegraphics{ionization-He}
\caption{Ionization rate of
every
term by taking the derivative He atom in an electric field as a function of
the energy functional after imposing
the
... field strength. From Ref.~\cite{Larsen:2013cw}}
\label{fig:cs-ionization-He}
\end{figure}
Figure \ref{fig:cs-ionization-He} shows calculated ionization rates of the He 1s state in a
uniform Stark-type electric field as a function of field strength.
In the limit of weak electric fields, the simple perturbative approximation
by Ammosov, Delone and Krainov,\cite{adk} which is depends
only on the ionization potential, approaches the accurate correlated reference
calculation by Scrinzi.\cite{PhysRevLett.83.706}
This demonstrates that the ionization rate is determined largely by the
ionization potential for weak fields. As the LDA is known to produce
inaccurate ionization potentials
due to its wrong asymptotic form at large distances, the LDA necessarily yields
inaccurate rates at low fields.
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 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
from the system\cite{PhysRevA.49.2421}.
