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Pierre-Francois Loos 2022-11-09 10:27:37 +01:00
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10
Notes/Notes.rty
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@ -122,12 +122,12 @@
\newcommand{\xc}{\text{xc}} \newcommand{\xc}{\text{xc}}
\newcommand{\x}{\text{x}} \newcommand{\x}{\text{x}}
\newcommand{\GW}{\text{GW}} \newcommand{\GW}{GW}
\newcommand{\GF}{\text{GF(2)}} \newcommand{\GF}{\text{GF(2)}}
\newcommand{\GT}{\text{$GT$}} \newcommand{\GT}{GT}
\newcommand{\evGW}{ev$GW$} \newcommand{\evGW}{\text{ev}$GW$}
\newcommand{\qsGW}{qs$GW$} \newcommand{\qsGW}{\text{qs}GW}
\newcommand{\GOWO}{$G_0W_0$} \newcommand{\GOWO}{G_0W_0}
%%% Notations %%% %%% Notations %%%

55
Notes/Notes.tex
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@ -21,6 +21,9 @@
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}} \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}} \newcommand{\ant}[1]{\textcolor{green}{#1}}
% addresses % addresses
@ -276,36 +279,38 @@ The central equation of MBPT in practice is the following
\label{eq:quasipart_eq} \label{eq:quasipart_eq}
\bF{}{} + \bSig(\omega) = \omega \mathbb{1}. \bF{}{} + \bSig(\omega) = \omega \mathbb{1}.
\end{equation} \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} \subsection{Self-energies and quasiparticle equations}
\label{sec:folded} \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. 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{align}
\label{eq:GF2_selfenergy} \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 \\ \Sigma_{pq}^{\text{GF(2)}}(\omega)
&+ \sum_{iab} \frac{W_{pi,ab}W_{qi,ab}}{\omega + \epsilon _i -\epsilon_a -\epsilon_b + \ii \eta} \notag & = \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} \end{align}
with with
\begin{equation} \begin{equation}
\label{eq:GF2_sERI} \label{eq:GF2_sERI}
W^{\GF}_{pq,rs}= \frac{1}{\sqrt{2}}\aeri{pq}{rs} W^{\GF}_{pq,rs}= \frac{1}{\sqrt{2}}\aeri{pq}{rs}
\end{equation} \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} \begin{equation}
\label{eq:GW_selfenergy} \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} \end{equation}
with with
\begin{equation} \begin{equation}
\label{eq:GW_sERI} \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} \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 The corresponding self-energies read as
\begin{equation} \begin{equation}
\label{eq:GT_selfenergy} \label{eq:GT_selfenergy}
@ -317,14 +322,14 @@ with
\eri{pq}{\chi^{N+2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N+2} \notag \\ \eri{pq}{\chi^{N+2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N+2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N+2} \notag \\
\eri{pq}{\chi^{N-2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N-2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N-2} \notag \eri{pq}{\chi^{N-2}_m} &= \sum_{c<d} \aeri{pq}{cd} \bX_{cd,m}^{N-2} + \sum_{k<l} \aeri{pq}{kl} \bY_{kl,m}^{N-2} \notag
\end{align} \end{align}
The two RPA problems giving the eigenvectors needed to build the GW and GT self-energies are given in Appendix~\ref{sec:rpa}. The two RPA problems giving the eigenvectors needed to build the $GW$ and $GT$ self-energies are given in Appendix \ref{sec:rpa}.
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
\subsection{The unfolded equations} \subsection{The unfolded equations}
\label{sec:unfolded} \label{sec:unfolded}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
Following Schirmer for the GF(2) case or Bintrim \etal, the non-linear quasi-particle equations for each approximations can be unfolded in larger linear problems Following Schirmer for the GF(2) case or Bintrim \etal, the non-linear quasi-particle equations for each approximation can be unfolded in larger linear problems
\begin{equation} \begin{equation}
\label{eq:unfolded_equation} \label{eq:unfolded_equation}
\bH \bc_{(s)} = \epsilon_s \bc_{(s)} \bH \bc_{(s)} = \epsilon_s \bc_{(s)}
@ -346,33 +351,37 @@ The expression of the coupling blocks $\bV{}{}$ and the diagonal blocks $\bC{}{}
\begin{align} \begin{align}
\label{eq:GF2_unfolded} \label{eq:GF2_unfolded}
V^\text{2h1p}_{p,klc} & = \frac{1}{\sqrt{2}}\aeri{pc}{kl} V^\text{2h1p}_{p,klc} & = \frac{1}{\sqrt{2}}\aeri{pc}{kl}
& \\
V^\text{2p1h}_{p,kcd} & = \frac{1}{\sqrt{2}}\aeri{pk}{dc} \\ V^\text{2p1h}_{p,kcd} & = \frac{1}{\sqrt{2}}\aeri{pk}{dc}
\\
C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik} C^\text{2h1p}_{ija,klc} & = \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} \delta_{ik}
& \\
C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd} \notag C^\text{2p1h}_{iab,kcd} & = \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} \delta_{bd}
\end{align} \end{align}
\item \textbf{GW} \item \textbf{GW}
\begin{align} \begin{align}
\label{eq:GW_unfolded} \label{eq:GW_unfolded}
V^\text{2h1p}_{p,klc} & = \eri{pc}{kl} V^\text{2h1p}_{p,klc} & = \eri{pc}{kl}
&
V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc} \notag \\
C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} & &
\\ \\
C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd} \notag & & V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}
\\
C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik}
\\
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd}
\end{align} \end{align}
\item \textbf{GT} \item \textbf{GT}
\begin{align} \begin{align}
\label{eq:GT_unfolded} \label{eq:GT_unfolded}
V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl} V^\text{2h1p}_{p,klc} & = \aeri{pc}{kl}
& \\
V^\text{2p1h}_{p,kcd}&= \aeri{pk}{cd} \notag \\ V^\text{2p1h}_{p,kcd} & = \aeri{pk}{cd}
C^\text{2h1p}_{ija,klc} &= \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ac} & & \\ \\
C^\text{2p1h}_{iab,kcd} &= \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{ik} & & \notag C^\text{2h1p}_{ija,klc} & = \qty[ \qty( \epsilon_i + \epsilon_j - \epsilon_a) \delta_{jl} \delta_{ac} - \aeri{ij}{kl} ] \delta_{ac}
\\
C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \epsilon_a + \epsilon_b - \epsilon_i) \delta_{ik} \delta_{ac} + \aeri{ab}{cd} ] \delta_{ik}
\end{align} \end{align}
\end{itemize} \end{itemize}
The downfolding procedure to obtain the GW self-energy is derived in details in Appendix~\ref{sec:downfolding}. The downfolding procedure to obtain the $GW$ self-energy is derived in details in Appendix~\ref{sec:downfolding}.
\textbf{\textcolor{red}{That would be nice to add electron-hole T matrix to see if it also correspond to one term that can be found in the CI below.}} \textbf{\textcolor{red}{That would be nice to add electron-hole T matrix to see if it also correspond to one term that can be found in the CI below.}}