deletions | additions       diff --git a/Sternheimer.tex b/Sternheimer.tex  index 0a8b007..16cd5e6 100644  --- a/Sternheimer.tex  +++ b/Sternheimer.tex 
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where $\varphi_k(\vec r)$ are the wave-functions of the static  Kohn-Sham Hamiltonian $H$ obtained by taking $\lambda=0$  \begin{equation}  H\varphi_k(\vec r) =\epsilon_\io\varphi_\io(\vec =\epsilon_k\varphi_k(\vec  r) \,,  \end{equation}  and $\delta\varphi_{\io}(\vec r, \omega)$ are the first order variations of 
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%\begin{multline}  \begin{equation}  \label{eq:sternheimer}  \left\{\op{\HH} \left\{H  - \epsilon_\io\pm\omega \epsilon_k\pm\omega  + \imi\eta\right\}\delta\varphi_{\io}(\vec \mathrm{i}\eta\right\}\delta\varphi_{k}(\vec  r, \pm\omega) = \\ -\mathrm{P}_c\,\delta{\op{\HH}}(\pm\omega) \varphi_\io(\vec -\mathrm{P}_c\,\delta{H}(\pm\omega) \varphi_k(\vec  r) \,,  \end{equation}  %\end{multline} 
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%\begin{multline}  \begin{equation}  \label{eq:h1}  \delta{\op{\HH}}(\omega)=  \delta{\op{\vv}}(\vec{r}) \delta{H}(\omega)=  \delta{V}(\vec{r})  +\mint{r'} \frac{\delta{n}(\vec{r}',\omega)}{|\vec{r}-\vec{r}'|}  +\mint{r'} f_{\rm xc}(\vec r, \vec r', \omega)\,\delta{n}(\vec{r'}, \omega)  \ . 
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finite \(\eta\) allows us to solve numerically the Sternheimer  equation close to re\-so\-nan\-ces, as it removes the divergences of  this equation.} The projector \(\mathrm{P}_c\) effectively removes  the components of \(\delta\varphi_{\io}(\vec \(\delta\varphi_{k}(\vec  r, \pm\omega)\) in the subspace of the occupied ground-state wave-functions. In linear  response, these components do not contribute to the variation of the  density\footnote{This is straightforward to prove by expanding the  variation of the wave-functions in terms of the ground-state  wave-functions, using standard perturbation theory, and then  replacing the resulting expression in the variation of the density,Eq.~(\ref{eq:varrho}).},  and therefore we can safely ignore the projector for first-order response calculation. This is important for  large systems as the cost of the calculation of the projections scales  quadratically with the number of orbitals.  The first term of \(\delta{\op{\HH}}(\omega)\) \(\delta{H}}(\omega)\)  comes from the external perturbative field, while the next two represent the variation of the  Hartree and exchange-correlation potentials. The exchange-correlation  kernel is a functional of the ground-state density $n$, and is