deletions | additions       diff --git a/Complex DFT.tex b/Complex DFT.tex  index c85e16b..d9b7a14 100644  --- a/Complex DFT.tex  +++ b/Complex DFT.tex 
...
equation with the boundary condition that the wave must asymptotically  have the form  \begin{align}  \phi(r) \sim \frac{\mathrm{e}^{\ii k}}{r}/r\quad\textrm{as}\quad k}}{r}\quad\textrm{as}\quad  r\rightarrow\infty\ , \end{align}  where the momentum $k$ is complex and has a negative imaginary part.  This causes the state to diverge exponentially in space as 
...
states and operators are represented  by means of the transformation  \begin{align}  \hat R_\theta 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 
...
stationary points~\cite{WM07} of the  functional~\cite{Whitenack_2010,WW11}.  Figure~\ref{fig:cs-ionization-He} Fig.~\ref{fig:cs-ionization-He}  shows calculated ionization rates of for  the He 1s state 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 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  ionization potential for weak fields. As the LDA local density approximation  is known to produce inaccurate ionization potentials due to its wrong asymptotic form at large distances, the LDA it  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 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