From 724adf96c4a2c5c142f55ea09cc1fabe0a47db2b Mon Sep 17 00:00:00 2001 From: pfloos Date: Wed, 9 Nov 2022 10:27:37 +0100 Subject: [PATCH 1/2] modifs Sec IV --- Notes/Notes.rty | 10 ++++----- Notes/Notes.tex | 55 ++++++++++++++++++++++++++++--------------------- 2 files changed, 37 insertions(+), 28 deletions(-) diff --git a/Notes/Notes.rty b/Notes/Notes.rty index 7af1bea..54e8606 100755 --- a/Notes/Notes.rty +++ b/Notes/Notes.rty @@ -122,12 +122,12 @@ \newcommand{\xc}{\text{xc}} \newcommand{\x}{\text{x}} -\newcommand{\GW}{\text{GW}} +\newcommand{\GW}{GW} \newcommand{\GF}{\text{GF(2)}} -\newcommand{\GT}{\text{$GT$}} -\newcommand{\evGW}{ev$GW$} -\newcommand{\qsGW}{qs$GW$} -\newcommand{\GOWO}{$G_0W_0$} +\newcommand{\GT}{GT} +\newcommand{\evGW}{\text{ev}$GW$} +\newcommand{\qsGW}{\text{qs}GW} +\newcommand{\GOWO}{G_0W_0} %%% Notations %%% diff --git a/Notes/Notes.tex b/Notes/Notes.tex index ae0959e..5dcd9a7 100644 --- a/Notes/Notes.tex +++ b/Notes/Notes.tex @@ -21,6 +21,9 @@ \usepackage[normalem]{ulem} \newcommand{\titou}[1]{\textcolor{red}{#1}} +\newcommand{\trashPFL}[1]{\textcolor{\red}{\sout{#1}}} +\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}} + \newcommand{\ant}[1]{\textcolor{green}{#1}} % addresses @@ -276,36 +279,38 @@ The central equation of MBPT in practice is the following \label{eq:quasipart_eq} \bF{}{} + \bSig(\omega) = \omega \mathbb{1}. \end{equation} -However, in order to use it we need to rely on approximations of the self-energy $\bSig(\omega)$. +\PFL{Not quite. You're missing the eigenvectors to make it a non-linear eigenvalue problem.} +However, in order to use it we need to rely on approximations of the dynamical self-energy $\bSig(\omega)$. %%%%%%%%%%%%%%%%%%%%%% \subsection{Self-energies and quasiparticle equations} \label{sec:folded} %%%%%%%%%%%%%%%%%%%%%% -In the following, we will focus on the GF(2), GW and GT approximations. +In the following, we will focus on the GF(2), $GW$ and $GT$ approximations. The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory. \begin{align} \label{eq:GF2_selfenergy} - \Sigma_{pq}^{GF(2)}(\omega) &= \sum_{ija} \frac{W_{pa,ij}W_{qa,ij}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\ - &+ \sum_{iab} \frac{W_{pi,ab}W_{qi,ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag + \Sigma_{pq}^{\text{GF(2)}}(\omega) + & = \sum_{ija} \frac{W_{pa,ij}^{\text{GF(2)}}W_{qa,ij}^{\text{GF(2)}}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\ + & + \sum_{iab} \frac{W_{pi,ab}^{\text{GF(2)}}W_{qi,ab}^{\text{GF(2)}}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag \end{align} with \begin{equation} \label{eq:GF2_sERI} W^{\GF}_{pq,rs}= \frac{1}{\sqrt{2}}\aeri{pq}{rs} \end{equation} -On the other hand, the GW self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy. +On the other hand, the $GW$ self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy. \begin{equation} \label{eq:GW_selfenergy} - \Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{W_{pi,v} W_{qi,v}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{av} \frac{W_{pa,v}W_{qa,v}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag + \Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{W_{pi,v}^{\GW} W_{qi,v}^{\GW}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{av} \frac{W_{pa,v}^{\GW}W_{qa,v}^{\GW}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag \end{equation} with \begin{equation} \label{eq:GW_sERI} - W_{pq,v}^\GW = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v}^{\dRPA} + \bY_{v}^{\dRPA} )_{ia} + W_{pq,v}^{\GW} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{v}^{\dRPA} + \bY_{v}^{\dRPA} )_{ia} \end{equation} -Finally, the GT approximation corresponds to another approximation to the polarizability than in GW, namely the one coming from pp-hh-RPA +Finally, the $GT$ approximation corresponds to another approximation to the polarizability than in $GW$, namely the one coming from pp-hh-RPA The corresponding self-energies read as \begin{equation} \label{eq:GT_selfenergy} @@ -317,14 +322,14 @@ with \eri{pq}{\chi^{N+2}_m} &= \sum_{c Date: Wed, 9 Nov 2022 10:33:11 +0100 Subject: [PATCH 2/2] more modifs Sec IV --- Notes/Notes.tex | 20 ++++++++++++-------- 1 file changed, 12 insertions(+), 8 deletions(-) diff --git a/Notes/Notes.tex b/Notes/Notes.tex index 5dcd9a7..06f1860 100644 --- a/Notes/Notes.tex +++ b/Notes/Notes.tex @@ -289,12 +289,12 @@ However, in order to use it we need to rely on approximations of the dynamical s In the following, we will focus on the GF(2), $GW$ and $GT$ approximations. The GF($n$) formalism is defined such that the self-energy includes every diagram up to $n$-th order of M\"oller-Plesset perturbation theory. -\begin{align} +\begin{equation} \label{eq:GF2_selfenergy} \Sigma_{pq}^{\text{GF(2)}}(\omega) - & = \sum_{ija} \frac{W_{pa,ij}^{\text{GF(2)}}W_{qa,ij}^{\text{GF(2)}}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} \notag \\ - & + \sum_{iab} \frac{W_{pi,ab}^{\text{GF(2)}}W_{qi,ab}^{\text{GF(2)}}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag -\end{align} + = \sum_{ija} \frac{W_{pa,ij}^{\text{GF(2)}}W_{qa,ij}^{\text{GF(2)}}}{\omega + \epsilon _a -\epsilon_i -\epsilon_j - \ii \eta} + + \sum_{iab} \frac{W_{pi,ab}^{\text{GF(2)}}W_{qi,ab}^{\text{GF(2)}}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} +\end{equation} with \begin{equation} \label{eq:GF2_sERI} @@ -303,7 +303,9 @@ with On the other hand, the $GW$ self-energy is obtained by taking the RPA polarizability and removing the vertex correction in the exact definition of the self-energy. \begin{equation} \label{eq:GW_selfenergy} - \Sigma_{pq}^{\GW}(\omega) = \sum_{iv} \frac{W_{pi,v}^{\GW} W_{qi,v}^{\GW}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + \sum_{av} \frac{W_{pa,v}^{\GW}W_{qa,v}^{\GW}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag + \Sigma_{pq}^{\GW}(\omega) + = \sum_{iv} \frac{W_{pi,v}^{\GW} W_{qi,v}^{\GW}}{\omega - \epsilon_i + \Omega_{v}^{\dRPA} - \ii \eta} + + \sum_{av} \frac{W_{pa,v}^{\GW}W_{qa,v}^{\GW}}{\omega - \epsilon_a - \Omega_{v}^{\dRPA} + \ii \eta} \notag \end{equation} with \begin{equation} @@ -314,13 +316,15 @@ Finally, the $GT$ approximation corresponds to another approximation to the pola The corresponding self-energies read as \begin{equation} \label{eq:GT_selfenergy} - \Sigma_{pq}^{\GT}(\omega) = \sum_{im} \frac{\eri{pi}{\chi^{N+2}_m}\eri{qi}{\chi^{N+2}_m}}{\omega + \epsilon_i - \Omega_{m}^{N+2} + \ii \eta} + \sum_{am} \frac{\eri{pa}{\chi^{N-2}_m}\eri{qa}{\chi^{N-2}_m}}{\omega + \epsilon_a - \Omega_{m}^{N-2} - \ii \eta} \notag + \Sigma_{pq}^{\GT}(\omega) + = \sum_{iv} \frac{W_{pi,v}^{N+2} W_{qi,v}^{N+2}}{\omega + \epsilon_i - \Omega_{v}^{N+2} + \ii \eta} + + \sum_{av} \frac{W_{pa,v}^{N-2} W_{qa,v}^{N-2}}{\omega + \epsilon_a - \Omega_{v}^{N-2} - \ii \eta} \notag \end{equation} with \begin{align} \label{eq:GT_sERI} - \eri{pq}{\chi^{N+2}_m} &= \sum_{c