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Julien Toulouse 2019-10-01 18:14:02 +02:00
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Manuscript/GW-srDFT.tex
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@ -235,7 +235,7 @@ $(G_\text{0})^{-1}(\b{r},\b{r}',\omega)= (\omega + (1/2) \nabla_\b{r}^2 + v_\tex
\begin{equation}
\bar{\Sigma}^{\cal B}[n](\b{r},\b{r}') = \bar{v}^{\cal B}[n](\b{r}) \delta(\b{r}-\b{r}'),
\end{equation}
with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$. The solution of the Dyson equation~(\ref{Dyson}) gives the Green function $G^{\cal B}(\b{r},\b{r}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bar{\Sigma}^{\cal B}[n]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bar{\Sigma}^{\cal B}$. Of course, in the complete-basis-set (CBS) limit, the basis-set correction vanishes, $\bar{\Sigma}^{{\cal B}\to \text{CBS}}=0$, and the Green function becomes exact, $G^{{\cal B}\to \text{CBS}}=G$.
%From Julien:
%\begin{equation}
@ -265,6 +265,14 @@ with $\bar{v}^{\cal B}[n](\b{r}) = \delta \bar{E}^{\cal B}[n] / \delta n(\b{r})$
\subsection{The GW Approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The Dyson equation can be written with an arbitrary reference
\begin{equation}
(G^{\cal B})^{-1} = (G_\text{ref}^{\cal B})^{-1}- \left( \Sigma_\text{Hxc}^{\cal B}[G^{\cal B}]- \Sigma_\text{ref}^{\cal B} \right) - \bar{\Sigma}^{\cal B}[n_{G^{\cal B}}],
\end{equation}
where $(G_\text{ref}^{\cal B})^{-1} = (G_\text{0}^{\cal B})^{-1} - \Sigma_\text{ref}^{\cal B}$. For example, if the reference is Hartree-Fock (HF), $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = \Sigma_\text{Hx,HF}^{\cal B}(\b{r},\b{r}')$ is the HF nonlocal self-energy, and if the reference is Kohn-Sham, $\Sigma_\text{ref}^{\cal B}(\b{r},\b{r}') = v_\text{Hxc}^{\cal B}(\b{r}) \delta(\b{r}-\b{r}')$ is the local Hxc potential.
Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.