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We also have implemented the Sternheimer formalism when non-integer occupations are used, as appropriate for metallic systems. In this case weighted projectors are added to both sides of the eq.~(\ref{eq:sternheimer}).\cite{DeGironcoli} We have generalized the equations to the dynamic case \cite{Strubbethesis}. The modified Sternheimer equation is  \begin{equation}  \left\{\hat H - \epsilon_n\pm\omega +  \mathrm{i}\eta + \sum_n \alpha_m \left| \varphi_m \right> \left| \varphi_m \right> \right\}\delta\varphi_{n}(\vec r, \pm\omega) = \\ \nonumber  -\left[ \tilde{\theta}_{{\rm F},n} - \sum_m \beta_{n,m} \left| \varphi_m \right> \left| \varphi_m \right> \right] \delta{\hat H}(\pm\omega) \varphi_n(\vec r) \,,  \end{equation}  where  \begin{equation}  \alpha_n = {\rm max} \left( \epsilon_{\rm F} - 3 \sigma - \epsilon_n, 0 \right) \\  \beta_{n,m} = \tilde{\theta}_{{\rm F},n} \tilde{\theta}_{n,m} + \tilde{\theta}_{{\rm F},m} \tilde{\theta}_{m,n} + \alpha_m \frac{\tilde{\theta}_{{\rm F},n} - \tilde{\theta}_{n,m}}{\epsilon_n - \epsilon_m \mp \omega} \tilde{\theta}_{m,n}  \end{equation}  $\sigma$ is the broadening energy and $\tilde{\theta}$ is the smearing scheme's approximation to the Heaviside function.