deletions | additions       diff --git a/Sternheimer.tex b/Sternheimer.tex  index 04ae8f0..3c99681 100644  --- a/Sternheimer.tex  +++ b/Sternheimer.tex 
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perturbative technique, it avoids the use of empty states, and has a  favorable scaling with the number of atoms.  Based on Octopus, we developed Octopus implements  a modified generalized  version of the Sternheimer equation that is able to cope with both static and dynamic response in and out of  resonance~\cite{Andrade_2007}. The method is suited for linear and  non-linear response; higher-order Sternheimer equations can be  obtained for higher-order variations. For second-order response,  however, wecan  apply the \(2N\,+\,1\) theorem (also known as Wigner's \(2N\,+\,1\) rule)~\cite{Gonze_1989,Corso_1996} to get the coefficients directly from first-order response variations.  In the Sternheimer formalism we consider the response to   monochromatic perturbative field \(\lambda  \delta{V}(\vec{r})\cos\left(\omega{t}\right)\). \delta{\hat v}(\vec{r})\cos\left(\omega{t}\right)\).  This perturbation induces a variation in the time-dependent Kohn-Sham (KS)  orbitals, that we denote $\delta\varphi_{n}(\vec r, \omega)$. These variations allows allow us  to calculate response quantities like observables, for example,  the frequency-dependent polarization. In order to calculate the variations of the orbitals we need to solve a linear equation that only depends on the occupied orbitals (atomic units are used throughout)  %\begin{multline}  \begin{equation}  \label{eq:sternheimer}  \left\{H \left\{\hat H  - \epsilon_n\pm\omega + \mathrm{i}\eta\right\}\delta\varphi_{n}(\vec r, \pm\omega) = -\mathrm{P}_c\,\delta{H}(\pm\omega) -\mathrm{\hat P}_c\,\delta{\hat H}(\pm\omega)  \varphi_n(\vec r) \,,  \end{equation}  %