deletions | additions       diff --git a/Complex DFT.tex b/Complex DFT.tex  index 804bb11..cdf6fb8 100644  --- a/Complex DFT.tex  +++ b/Complex DFT.tex 
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and lifetime.  Mathematically, resonances can be defined as poles of the scattering  matrix or cross-section at complex energies.\cite{PhysRev.56.750,Hatano01022008} 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 close to that energy. 
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define the energy $\epsilon_0$ and width $\Gamma$ of the  resonance. More details can be found in the many reviews of complex scaling~\cite{simon1973resonances,Reinhardt_1982,Ho19831}.  \begin{figure}  \centering  \includegraphics{fig-cs-spectrum}  \includegraphics{fig-cs-potential}  \caption{Spectrum (left) of 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 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}. 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 
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for independent particles or  in combination with DFT and  selected  XC functionals\cite{Larsen:2013cw}. 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 with the kinetic operator and various potentials. 
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for a stationary point, with the idea that this minimizes the energy.  In 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}. argued~\cite{WM07}.  The complex-scaled versions  of the Kohn--Sham equations thereby become similar to the usual ones:  \begin{align} 
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\right] \psi_{\theta n}(\vec r) = \psi_{\theta n}(\vec r) \epsilon_{\theta n}.  \end{align}  The effective potential $v_\theta(\vec r)$ is the functional derivative  of the energy functional with respect to the density $n_\theta(\ve $n_\theta(\vec  r)$, and therefore consists of the terms  \begin{align}  v_\theta(\vec r) \equiv \fdiff{E}{n_\theta(\vec r)} =  
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stationary points~\cite{WM07} of the  functional~\cite{Whitenack_2010,WW11}.  \begin{figure}   \includegraphics{ionization-He}  \caption{Ionization rate of He atom in an electric field as a function of 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