deletions | additions       diff --git a/Complex DFT.tex b/Complex DFT.tex  index d7dc313..4c0dbb5 100644  --- a/Complex DFT.tex  +++ b/Complex DFT.tex 
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matrix elements such as the density or kinetic energy.  In order to define the complex-scaled xc potential, it is necessary to perform an analytic continuation procedure~\cite{Larsen:2013cw}.  In standard DFT the Kohn--Sham 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.  In our case we cannot minimize the energy functional as it is  complex-valued~\cite{WM07}.  The complex-scaled versions  of the Kohn--Sham KS  equations thereby become similar to the usual ones: \begin{align}  \left[-\frac12 \ee^{-\ii2\theta}\nabla^2 + v_\theta(\vec r)  \right] \varphi_{\theta n}(\vec r) = \varphi_{\theta n}(\vec r) \epsilon_{\theta n}\ . 
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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  Kohn--Sham 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