deletions | additions       diff --git a/Complex DFT.tex b/Complex DFT.tex  index 3f04044..e9fb10f 100644  --- a/Complex DFT.tex  +++ b/Complex DFT.tex 
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\section{Complex density % Commands to unify notation (vectors, constants, etc.) in this section.  \newcommand{\ve}[1]{\mathbf{#1}} % Vectors with bold. Change to \vec for   \newcommand{\ee}{\mathrm e} % 2.718281828...  \newcommand{\ii}{\mathrm i} % sqrt -1  \newcommand{\xc}{\mathrm{xc}} % exchange--correlation  \newcommand{\Ha}{\mathrm H} % hartree  \newcommand{\dee}{\mathrm d} % dx, dr  \newcommand{\fdiff}[2]{\frac{\delta #1}{\delta #2}} %  functional theory}  Ask, Umberto derivative  \section{Complex scaling and resonances}  We shall here discuss the calculation of resonant electronic states by  means of the complex scaling method as implemented in Octopus. By  resonant states we mean metastable electronic states  on finite systems, such as atoms or molecules, with a characteristic energy  and lifetime.  Mathematically, resonances can be defined as poles of the scattering  matrix or cross-section at complex energies. 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 around that energy.  Resonant states may temporarily capture incoming electrons or electrons  excited from bound states, making them important intermediate states  in many processes. Another way to create a resonance is to apply an  electric field strong enough to ionize the system through tunnelling.  %Resonances are typically seen in scattering experiments  %as a peak of given energy and width, with the width being proportional  %to the lifetime. For example a resonance will correspond a peak in  %the electron scattering cross section. A long-lived resonant state can be thought  %of as an ``almost bound state'', but the properties of resonant states are  %nevertheless very different, and different methods must be used to calculate them.  The defining characteristic of a resonant state is that it has an  outgoing component but not an incoming one.  Such states are often called Siegert  states\cite{PhysRev.56.750}. They can be determined by solving the  time-independent Schrödinger  equation with the boundary condition that the wave must asymptotically  have the form% $r\phi(r) \sim \exp(+\ii k r)$ as $r \rightarrow \infty$.  \begin{align}  \phi(r) \sim \exp(\ii k r)/r\quad\textrm{as $r\rightarrow\infty$}.  \end{align}  The momentum $k$ is complex with negative imaginary part.  This causes the state to diverge exponentially in space as $r\rightarrow\infty$.  Its energy also has negative imaginary part,  causing it to decay exponentially over time uniformly at every point in space.  Since they have complex energies they are not eigenstates of any Hermitian operator,  and indeed do not belong to the Hilbert space.  This precludes the direct calculation of resonances with  the standard computational methods known from DFT.  The method of \emph{complex scaling}\cite{aguilarcombes,Balslev:1971ez}  circumvents this problem by a transformation  which maps the Hamiltonian to a non-Hermitian operator  $\hat H_\theta = \hat R_\theta \hat H \hat R_{-\theta}$, where the operator $\hat R_\theta$ is  \begin{align}  \hat R_\theta \psi(\ve r) = \ee^{\ii N \theta / 2} \psi(\ve r \ee^{\ii\theta}),  \end{align}  and $N$ is the number of dimensions in which the scaling operation is applied.