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I'm following Matteo Gatti's slides to repeat the derivation of macroscopic dielectric function $\epsilon_M$:

$$\epsilon_M=\dfrac{1}{\epsilon^{-1}_{\vec{G}=0,\vec{G}'=0}(\vec{q},\omega)}.$$

On page 41 one needs to perform the following derivations:

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Here are my derivations:

enter image description here

Note:

  • $V_{tot}$ and $V_{ext}$ is a periodic potential in real space.

One can see to obtain the final result I have introduced a delta function, is it reasonable? Or do I miss something to arrive at the final result?

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Perhaps it's missing a prefactor? If you split the sum as $$ \sum_{\textbf{q}'} \sum_{\textbf{G}'} V_{ext}(\textbf{q}' + \textbf{G}', \omega) \int d\textbf{r}_1 d\textbf{r}_2 e^{-i(\textbf{q}+\textbf{G})\cdot \textbf{r}_1} \epsilon^{-1}(\textbf{r}_1, \textbf{r}_2, \omega) e^{i(\textbf{q}'+\textbf{G}')\cdot \textbf{r}_2}, $$ then you can shift $\textbf{G'} \rightarrow \textbf{G}' - \textbf{q}' + \textbf{q}$, which yields $$ \sum_{\textbf{q}'} \sum_{\textbf{G}'} V_{ext}(\textbf{q} + \textbf{G}', \omega) \int d\textbf{r}_1 d\textbf{r}_2 e^{-i(\textbf{q}+\textbf{G})\cdot \textbf{r}_1} \epsilon^{-1}(\textbf{r}_1, \textbf{r}_2, \omega) e^{i(\textbf{q}+\textbf{G}')\cdot \textbf{r}_2} \\ = \sum_{\textbf{q}'} \sum_{\textbf{G}'} V_{ext}(\textbf{q} + \textbf{G}', \omega) \epsilon^{-1}_{\textbf{G}\textbf{G}'}(\textbf{q}, \omega). $$

The sum over $\textbf{q}'$ is simply the number of $\textbf{q}$-points, that's why I think it's missing a prefactor.

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