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Pierre-Francois Loos 2024-06-20 17:32:08 +02:00
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2024/GFQC.md
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# Green's Functions in Quantum Chemistry # Green's Functions in Quantum Chemistry
- Physical interpretation of the one- and two-body self-energy
- Lowdin partitioning technique - Lowdin partitioning technique
- Definition of the Green's function - Definition of the Green's function
- Physical interpretation of the one- and two-body self-energy
- Dyson equation - Dyson equation
- Matrix representation of G - Matrix representation of G
- Example for the HF Green's function - Example for the HF Green's function

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2024/GFQC/ISTPC_Loos_QFQC.tex
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@ -292,6 +292,20 @@ decoration={snake,
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Fundamental and optical gaps}
\begin{center}
\includegraphics[width=\textwidth]{fig/gaps}
\end{center}
\begin{equation}
\underbrace{\Eg{\KS}}_{\text{KS gap}} = \eLUMO{\KS} - \eHOMO{\KS} \ll \underbrace{\green{\Eg{GW}}}_{\text{\green{{\GW} gap}}} = \eLUMO{GW} - \eHOMO{GW}
\end{equation}
\begin{equation}
\underbrace{\blue{\Eg{\text{opt}}}}_{\text{\blue{optical gap}}} = E_1^N - E_0^N = \underbrace{\red{\Eg{\text{fund}}}}_{\text{\red{fundamental gap}}} + \underbrace{\purple{E_\text{B}}}_{\text{\purple{excitonic effect}}}
\end{equation}
\end{frame}
%-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\section{Motivations} \section{Motivations}
\begin{frame} \begin{frame}
@ -411,11 +425,11 @@ decoration={snake,
\qq{$\Rightarrow$} \qq{$\Rightarrow$}
\bG_0^{-1}(\yo) - \bSig(\yo) = \bO \bG_0^{-1}(\yo) - \bSig(\yo) = \bO
\qq{$\Rightarrow$} \qq{$\Rightarrow$}
\yo \bI - \be - \bSig(\yo) = \bO \det[\yo \bI - \be - \bSig(\yo)] = 0
\end{equation} \end{equation}
\begin{block}{Diagonal approximation} \begin{block}{Diagonal approximation}
\begin{equation} \begin{equation}
\yo \bI - \be - \bSig(\yo) = \bO \det[\yo \bI - \be - \bSig(\yo)] = 0
\qq{$\Rightarrow$} \qq{$\Rightarrow$}
\yo - \e{p}{\HF} - \Sig{pp}{}(\yo) = 0 \yo - \e{p}{\HF} - \Sig{pp}{}(\yo) = 0
\end{equation} \end{equation}
@ -434,13 +448,18 @@ decoration={snake,
\begin{frame}{Spectral Function} \begin{frame}{Spectral Function}
The following decomposition of the self-energy
\begin{equation} \begin{equation}
\bSig(\yo) = \Re \bSig(\yo) + i \Im \bSig(\yo) \bSig(\yo) = \Re \bSig(\yo) + i \Im \bSig(\yo)
\end{equation} \end{equation}
leads to the following expression for the spectral function (related to photoemission spectra)
\begin{equation} \begin{equation}
\begin{split}
\bA{}{}(\yo) \bA{}{}(\yo)
= - \frac{1}{\pi} \Im \abs{\bG(\yo)} & = - \frac{1}{\pi} \Im \abs{\bG(\yo)}
= - \frac{1}{\pi} \frac{\abs{\Im \bSig(\yo)}}{\qty[\yo \bI - \be - \Re \bSig(\yo)]^2 + \qty[ \Im \bSig(\yo)]^2} \\
& = - \frac{1}{\pi} \frac{\abs{\Im \bSig(\yo)}}{\qty[\yo \bI - \be - \Re \bSig(\yo)]^2 + \qty[ \Im \bSig(\yo)]^2}
\end{split}
\end{equation} \end{equation}
\end{frame} \end{frame}
@ -499,19 +518,6 @@ decoration={snake,
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Fundamental and optical gaps (\copyright~Bruno Senjean)}
\begin{center}
\includegraphics[width=\textwidth]{fig/gaps}
\end{center}
\begin{equation}
\underbrace{\Eg{\KS}}_{\text{KS gap}} = \eLUMO{\KS} - \eHOMO{\KS} \ll \underbrace{\green{\Eg{GW}}}_{\text{\green{{\GW} gap}}} = \eLUMO{GW} - \eHOMO{GW}
\end{equation}
\begin{equation}
\underbrace{\blue{\Eg{\text{opt}}}}_{\text{\blue{optical gap}}} = E_1^N - E_0^N = \underbrace{\red{\Eg{\text{fund}}}}_{\text{\red{fundamental gap}}} + \underbrace{\purple{E_\text{B}}}_{\text{\purple{excitonic effect}}}
\end{equation}
\end{frame}
%-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Hedin's pentagon} \begin{frame}{Hedin's pentagon}