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Merge pull request #1 from pfloos/overleaf-2020-02-03-1521 

Updates from Overleaf
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Pierre-Francois Loos 2020-02-03 16:27:20 +01:00 committed by GitHub
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BSE-PES.tex
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@ -337,6 +337,7 @@ with $\epsilon_{\lambda}$ the dielectric function at coupling constant $\lambda$
\\
\times \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} - \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta}),
\end{multline}
where the spectral weights at coupling strength $\lambda$ read
\begin{equation}
\sERI{pq}{m} = \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}.
@ -344,6 +345,7 @@ where the spectral weights at coupling strength $\lambda$ read
In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}{\IS}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed by solving the linear eigenvalue problem \eqref{eq:LR} with the following matrix elements
\begin{subequations}
\label{eq:LR_RPA}
\begin{align}
\label{eq:LR_RPA}
\ARPA{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2 \IS \ERI{ia}{jb},
@ -357,6 +359,7 @@ The relation between the BSE formalism and the well-known RPAx approach can be o
%namely setting $\epsilon_{\lambda}({\bf r},{\bf r}'; \omega) = \delta({\bf r}-{\bf r}')$
so that $W^{\lambda}$ reduces to the bare Coulomb potential. In that limit, the $GW$ quasiparticle energies reduce to the Hartree-Fock eigenvalues, and Eqs.~\ref{eq:LR_BSE} to the RPAx equations:
\begin{subequations}
\label{eq:LR_RPAx}
\begin{align}
\label{eq:LR_RPAx}
\ARPAx{ia,jb}{\IS} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \IS \qty[ 2 \ERI{ia}{jb} - \ERI{ij}{ab} ],