deletions | additions       diff --git a/Complex DFT.tex b/Complex DFT.tex  index ecf1f01..47a87f4 100644  --- a/Complex DFT.tex  +++ b/Complex DFT.tex 
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are obtained by taking the functional derivative of the energy functional with respect  to the density. Solving the equations corresponds to searching  for a stationary point, with the idea that this minimizes the energy.  In our case case,  we cannot minimize the energy functional functional,  as it is complex-valued~\cite{WM07}. complex-valued~\cite{WM07}, but we can still search for stationary points to find the resonances~\cite{Whitenack_2010,WW11}.  The complex-scaled version  of the KS equations thereby become similar to the usual ones:  \begin{align} 
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\ee^{-\ii\theta}\int\dee \rprime\, \frac{n_\theta(\vec r')}{\left| \vec r'-\vec r\right|}  \end{align}  is determined by solving the Poisson equation defined by the complex density.  Together with the xc XC  potential, \begin{align}  v_\xc^\theta(\vec r) &= \fdiff{E_\xc^\theta[n_\theta]}{n_\theta(\vec r)},  \end{align}  this defines a self-consistency cycle very similar to ordinary  KS 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}.  Fig.~\ref{fig:cs-ionization-He} shows calculated ionization rates for the He~1$s$~state in a  uniform Stark-type electric field as a function of field strength.