deletions | additions       diff --git a/figures/fig-cs-potential/caption.tex b/figures/fig-cs-potential/caption.tex  index a8a1b0c..35fdd17 100644  --- a/figures/fig-cs-potential/caption.tex  +++ b/figures/fig-cs-potential/caption.tex 
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Replace The parameter $\theta$ is a fixed number that one must choose, with  the original theorems proven for $0<\theta<\pi/4$. The spatial coordinates $\ve  r$ are rotated to $\ve r \ee^{\ii\theta}$ in the complex plane, and one  hence works in terms of the analytic continuations of both states and  operators. The idea of complex scaling is that for suitable $\theta$,  the analytic continuation of the Siegert states are \emph{localized}  instead of diverging,  and are eigenstates of the non-Hermitian  operator $\hat H_\theta$. 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 all physical  bound-state characteristics of the untransformed Hamiltonian are retained.  A typical example of a spectrum 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.  Octopus 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.  The energy functional in Kohn--Sham DFT consists of several  terms that are all expressible as integrals of the density or the  wavefunctions.  The functional is complex-scaled by rotating the integration contour  of every term by $\theta$ in the complex plane.  The energy functional thereby expressible as  % Problem: with \ve r', the arrow overlaps with the apostrophe.  % 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'}  \begin{align}  E_\theta &= \ee^{-\ii2\theta}  \sum_n \int\dee\ve r\, \psi_{\theta n}(\ve r) \left(-\frac12 \nabla^2\right)  \psi_{\theta n}(\ve r)  + \ee^{-\ii\theta} \frac12  \iint \dee \ve r \, \dee \rprime \,  \frac{n_\theta(\ve r)n_\theta(\rprime)}{\Vert\ve r - \rprime\Vert}\nonumber\\  &\quad+ E_\xc^\theta[n_\theta]  + \int\dee \ve r\, v_{\mathrm{ext}}(\ve r \ee^{\ii \theta}) n_\theta(\ve r),  \end{align}  with the electron density  \begin{align}  n_\theta(\ve r) = \sum_n f_n \psi_{n \theta}^2(\ve r),  \end{align}  occupation numbers $f_n$, and complex-scaled Kohn--Sham states $\psi_{n \theta}(\ve r)$.  We also need to define the complex-scaled XC potential $E_\xc^\theta[n_\theta]$.  Note how 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 such as the density or kinetic energy as is the correct procedure  when calculating matrix elements of states that would have been purely real  in the unscaled case.  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 text 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}.  The complex-scaled versions  of the Kohn--Sham thereby become similar to the usual ones:  \begin{align}  \left[-\frac12 \ee^{-\ii2\theta}\nabla^2 + v_{\mathrm{eff}}^\theta(\ve r)  \right] \psi_\theta(\ve r) = \psi_\theta(\ve r) \epsilon_\theta.  \end{align}  The effective potential $v_{\mathrm{eff}}^\theta(\ve r)$ 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 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 your caption \begin{align}  v_\Ha^\theta(\ve r) &=  \ee^{-\ii\theta}\int\dee \ve r\, \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)}.  \end{align}  blahblah......  The complex-scaled Hartree potential is determined by solving the  Poisson equation  \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}  \end{align}  The lowest-energy resonance of a system is then found by searching for  stationary points\cite{WM07} of the functional.  \cite{doi:10.1021/jz9001778,WW11} % daniel JPhysChemLett, PRL  ....