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saving work: OK for GW

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2021/Lecture_2/ISTPC_Loos_2.tex
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@ -7,6 +7,7 @@
\usetheme{Warsaw} \usetheme{Warsaw}
%\usecolortheme{seahorse} %\usecolortheme{seahorse}
\usepackage{mathpazo,libertine} \usepackage{mathpazo,libertine}
\usepackage[compat=1.1.0]{tikz-feynman}
\usepackage{algorithmicx,algorithm,algpseudocode} \usepackage{algorithmicx,algorithm,algpseudocode}
\algnewcommand\algorithmicassert{\texttt{assert}} \algnewcommand\algorithmicassert{\texttt{assert}}
@ -21,9 +22,10 @@
colorlinks=true, colorlinks=true,
linkcolor=cyan, linkcolor=cyan,
filecolor=magenta, filecolor=magenta,
urlcolor=blue, urlcolor=cyan,
citecolor=purple citecolor=purple
} }
\urlstyle{same}
\definecolor{darkgreen}{RGB}{0, 180, 0} \definecolor{darkgreen}{RGB}{0, 180, 0}
\definecolor{fooblue}{RGB}{0,153,255} \definecolor{fooblue}{RGB}{0,153,255}
@ -56,6 +58,7 @@
\newcommand{\GOW}{$G_0W$} \newcommand{\GOW}{$G_0W$}
\newcommand{\GWO}{$GW_0$} \newcommand{\GWO}{$GW_0$}
\newcommand{\GW}{$GW$} \newcommand{\GW}{$GW$}
\newcommand{\GT}{$GT$}
\newcommand{\GOWOSOSEX}{{\GOWO}+SOSEX} \newcommand{\GOWOSOSEX}{{\GOWO}+SOSEX}
\newcommand{\GWSOSEX}{{\GW}+SOSEX} \newcommand{\GWSOSEX}{{\GW}+SOSEX}
\newcommand{\GnWn}[1]{$G_{#1}W_{#1}$} \newcommand{\GnWn}[1]{$G_{#1}W_{#1}$}
@ -203,6 +206,7 @@
\newcommand{\btA}[2]{\bm{\Tilde{A}}_{#1}^{#2}} \newcommand{\btA}[2]{\bm{\Tilde{A}}_{#1}^{#2}}
\newcommand{\btB}[2]{\bm{\Tilde{B}}_{#1}^{#2}} \newcommand{\btB}[2]{\bm{\Tilde{B}}_{#1}^{#2}}
\newcommand{\bB}[2]{\bm{B}_{#1}^{#2}} \newcommand{\bB}[2]{\bm{B}_{#1}^{#2}}
\newcommand{\bC}[2]{\bm{C}_{#1}^{#2}}
\newcommand{\bc}{\bm{c}} \newcommand{\bc}{\bm{c}}
\newcommand{\bX}[2]{\bm{X}_{#1}^{#2}} \newcommand{\bX}[2]{\bm{X}_{#1}^{#2}}
\newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}} \newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}}
@ -231,7 +235,7 @@ decoration={snake,
$GW$/BSE methods in chemistry: $GW$/BSE methods in chemistry:
Computational aspects Computational aspects
} }
\author[PF Loos]{Pierre-Fran\c{c}ois LOOS} \author[PF Loos (\url{https://www.irsamc.ups-tlse.fr/loos/})]{Pierre-Fran\c{c}ois LOOS}
\date{Online ISTPC 2021 school --- April 27th, 2021} \date{Online ISTPC 2021 school --- April 27th, 2021}
\institute[CNRS@LCPQ]{ \institute[CNRS@LCPQ]{
Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\
@ -292,7 +296,7 @@ decoration={snake,
\begin{block}{Let's talk about notations} \begin{block}{Let's talk about notations}
\begin{itemize} \begin{itemize}
\item We consider \blue{closed-shell systems} (2 opposite-spin electrons per orbital) \item We consider \blue{closed-shell systems} (2 opposite-spin electrons per orbital)
\item We only deal with \blue{singlet excited states} but triplets can also be obtained \item We only deal with \blue{singlet excited states} but \purple{triplets} can also be obtained
\bigskip \bigskip
\item Number of \green{occupied orbitals} $O$ \item Number of \green{occupied orbitals} $O$
\item Number of \alert{vacant orbitals} $V$ \item Number of \alert{vacant orbitals} $V$
@ -310,7 +314,7 @@ decoration={snake,
%----------------------------------------------------- %-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Useful papers} \begin{frame}{Useful papers/programs}
\begin{itemize} \begin{itemize}
\item \red{mol$GW$:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149 \item \red{mol$GW$:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
\bigskip \bigskip
@ -318,7 +322,7 @@ decoration={snake,
\bigskip \bigskip
\item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022 \item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
\bigskip \bigskip
\item \purple{FHI-AIMS:} Caruso et al. 86 (2012) 081102 \item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
\bigskip \bigskip
\item \orange{Review:} \item \orange{Review:}
\begin{itemize} \begin{itemize}
@ -329,7 +333,8 @@ decoration={snake,
\item Blase et al. JPCL 11 (2020) 7371 \item Blase et al. JPCL 11 (2020) 7371
\end{itemize} \end{itemize}
\bigskip \bigskip
\item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 \item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com})
\end{itemize} \end{itemize}
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
@ -368,7 +373,7 @@ decoration={snake,
\end{block} \end{block}
\begin{block}{What can you calculate with BSE?} \begin{block}{What can you calculate with BSE?}
\begin{itemize} \begin{itemize}
\item Singlet and triplet neutral excitations (vertical absorption energies) \item Singlet and triplet optical excitations (vertical absorption energies)
\item Oscillator strengths (absorption intensities) \item Oscillator strengths (absorption intensities)
\item Correlation and total energies \item Correlation and total energies
\end{itemize} \end{itemize}
@ -509,7 +514,7 @@ decoration={snake,
+ \underbrace{\sum_a \frac{\MO{a}(\br_1) \MO{a}(\br_2)}{\yo - \e{a}{} + i\eta}}_{\text{\red{addition part = EAs}}} + \underbrace{\sum_a \frac{\MO{a}(\br_1) \MO{a}(\br_2)}{\yo - \e{a}{} + i\eta}}_{\text{\red{addition part = EAs}}}
\end{equation} \end{equation}
\end{block} \end{block}
\begin{block}{Non-interacting polarizability} \begin{block}{Polarizability}
\begin{equation} \begin{equation}
P(\br_1,\br_2;\omega) = - \frac{i}{\pi} \int \blue{G}(\br_1,\br_2;\omega+\omega') \blue{G}(\br_1,\br_2;\omega') d\omega' P(\br_1,\br_2;\omega) = - \frac{i}{\pi} \int \blue{G}(\br_1,\br_2;\omega+\omega') \blue{G}(\br_1,\br_2;\omega') d\omega'
\end{equation} \end{equation}
@ -659,7 +664,7 @@ decoration={snake,
\includegraphics[width=0.7\textwidth]{fig/QP} \includegraphics[width=0.7\textwidth]{fig/QP}
\\ \\
\bigskip \bigskip
\pub{V\'eril \& Loos, JCTC 14 (2018) 5220} \pub{V\'eril et al, JCTC 14 (2018) 5220}
\end{center} \end{center}
\end{column} \end{column}
\begin{column}{0.5\textwidth} \begin{column}{0.5\textwidth}
@ -700,7 +705,7 @@ decoration={snake,
\State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$ \State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$ \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\beKS$ and $\ERI{pq}{rs}$ \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\beKS$ and $\ERI{pq}{rs}$
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ \State Compute RPA eigenvalues $\orange{\bOm{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$
\Comment{\alert{This is a $\order*{N^6}$ step!}} \Comment{\alert{This is a $\order*{N^6}$ step!}}
\State Form screened ERIs $\violet{\ERI{pq}{m}}$ \State Form screened ERIs $\violet{\ERI{pq}{m}}$
\For{$p=1,\ldots,N$} \For{$p=1,\ldots,N$}
@ -736,7 +741,7 @@ decoration={snake,
\State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$ \State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$ \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\beKS$ and $\ERI{pq}{rs}$ \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\beKS$ and $\ERI{pq}{rs}$
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ \State Compute RPA eigenvalues $\orange{\Om{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$
\Comment{\alert{This is a $\order*{N^6}$ step!}} \Comment{\alert{This is a $\order*{N^6}$ step!}}
\State Form screened ERIs $\violet{\ERI{pq}{m}}$ \State Form screened ERIs $\violet{\ERI{pq}{m}}$
\For{$p=1,\ldots,N$} \For{$p=1,\ldots,N$}
@ -764,9 +769,10 @@ decoration={snake,
\State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$ \State Perform KS calculation to get $\beKS$, $\bcKS$, and $\bm{V}^{\xc}$
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$ \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\bcKS}{\rightarrow} \ERI{pq}{rs}$
\State Set $\blue{\beGnWn{-1}} = \beKS$ and $n = 0$ \State Set $\blue{\beGnWn{-1}} = \beKS$ and $n = 0$
\While{$\max{\abs{\bDelta}} < \tau$} \While{$\max{\abs{\bDelta}} > \tau$}
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\blue{\beGnWn{n-1}}$ and $\ERI{pq}{rs}$ \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\blue{\beGnWn{n-1}}$ and $\ERI{pq}{rs}$
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ \State Compute RPA eigenvalues $\orange{\Om{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$
\Comment{\alert{This is a $\order*{N^6}$ step!}}
\State Form screened ERIs $\violet{\ERI{pq}{m}}$ \State Form screened ERIs $\violet{\ERI{pq}{m}}$
\For{$p=1,\ldots,N$} \For{$p=1,\ldots,N$}
\State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$ \State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$
@ -800,14 +806,16 @@ decoration={snake,
\Procedure{{\qsGW}}{} \Procedure{{\qsGW}}{}
\State Perform HF calculation to get $\beHF$ and $\bcHF$ \green{(optional)} \State Perform HF calculation to get $\beHF$ and $\bcHF$ \green{(optional)}
\State Set $\blue{\beGnWn{-1}} = \beHF$, $\blue{\bcGnWn{-1}} = \bcHF$ and $n = 0$ \State Set $\blue{\beGnWn{-1}} = \beHF$, $\blue{\bcGnWn{-1}} = \bcHF$ and $n = 0$
\While{$\max{\abs{\bDelta}} < \tau$} \While{$\max{\abs{\bDelta}} > \tau$}
\State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow} \ERI{pq}{rs}$ \State AO to MO transformation for ERIs: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow} \ERI{pq}{rs}$
\Comment{\alert{This is a $\order*{N^5}$ step!}}
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\blue{\beGnWn{n-1}}$ and $\ERI{pq}{rs}$ \State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\blue{\beGnWn{n-1}}$ and $\ERI{pq}{rs}$
\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ \State Compute RPA eigenvalues $\orange{\Om{}{\RPA}}$ and eigenvectors $\orange{\bX{}{\RPA}+\bY{}{\RPA}}$
\Comment{\alert{This is a $\order*{N^6}$ step!}}
\State Form screened ERIs $\violet{\ERI{pq}{m}}$ \State Form screened ERIs $\violet{\ERI{pq}{m}}$
\State Evaluate $\red{\bSigC}(\blue{\beGnWn{n-1}})$ and form \State Evaluate $\red{\bSigC}(\blue{\beGnWn{n-1}})$ and form
$\red{\Tilde{\Sigma}^{\co}} \leftarrow \qty[ \red{\bSigC}(\blue{\beGnWn{n-1}})^\dag + \red{\bSigC}(\blue{\beGnWn{n-1}}) ]/2$ $\red{\Tilde{\Sigma}^{\co}} \leftarrow \qty[ \red{\bSigC}(\blue{\beGnWn{n-1}})^\dag + \red{\bSigC}(\blue{\beGnWn{n-1}}) ]/2$
\State Form $\purple{\Tilde{\bF}} = \bFHF + \red{\Tilde{\Sigma}^{\co}}$ \State Form $\bFHF$ from $\blue{\bcGnWn{n-1}}$ and then $\purple{\Tilde{\bF}} = \bFHF + \red{\Tilde{\Sigma}^{\co}}$
\State Diagonalize $\purple{\Tilde{\bF}}$ to get $\blue{\beGnWn{n}}$ and $\blue{\bcGnWn{n}}$ \State Diagonalize $\purple{\Tilde{\bF}}$ to get $\blue{\beGnWn{n}}$ and $\blue{\bcGnWn{n}}$
\State $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$ \State $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$
\State $n \leftarrow n + 1$ \State $n \leftarrow n + 1$
@ -831,6 +839,58 @@ decoration={snake,
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Other self-energies}
\begin{columns}
\begin{column}{0.7\textwidth}
\begin{block}{Second-order Green's function (GF2) \pub{[Hirata et al. JCP 147 (2017) 044108]}}
\begin{equation}
\Sig{pq}{\text{GF2}}(\yo)
= \frac{1}{2} \sum_{iab} \frac{\mel{iq}{}{ab}\mel{ab}{}{ip}}{\yo + \e{i}{} - \e{a}{} - \e{b}{}}
+ \frac{1}{2} \sum_{ija} \frac{\mel{aq}{}{ij}\mel{ij}{}{ap}}{\yo + \e{a}{} - \e{i}{} - \e{j}{}}
\end{equation}
\end{block}
\begin{block}{T-matrix \pub{[Romaniello et al. PRB 85 (2012) 155131; Zhang et al. JPCL 8 (2017) 3223]}}
\begin{equation}
\Sig{pq}{GT}(\omega)
= \sum_{im} \frac{\braket*{pi}{\green{\chi_m^{N+2}}} \braket*{qi}{\green{\chi_m^{N+2}}}}{\yo + \e{i}{} - \green{\Om{m}{N+2}}}
+ \sum_{am} \frac{\braket*{pa}{\blue{\chi_m^{N-2}}} \braket*{qa}{\blue{\chi_m^{N-2}}}}{\yo + \e{i}{} - \blue{\Om{m}{N-2}}}
\end{equation}
\begin{gather}
\braket*{pi}{\green{\chi_m^{N+2}}} = \sum_{c<d} \mel{pi}{}{cd} \green{X_{cd}^{N+2,m}} + \sum_{k<l} \mel{pi}{}{kl} \green{Y_{kl}^{N+2,m}}
\\
\braket*{pa}{\blue{\chi_m^{N-2}}} = \sum_{c<d} \mel{pa}{}{cd} \blue{X_{cd}^{N-2,m}} + \sum_{k<l} \mel{pa}{}{kl} \blue{Y_{kl}^{N-2,m}}
\end{gather}
\begin{equation}
\qq*{\purple{pp-RPA problem:}}
\begin{pmatrix}
\bA{}{} & \bB{}{}
\\
-\bB{}{\intercal} & -\bC{}{}
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{m}{N\pm2}
\\
\bY{m}{N\pm2}
\end{pmatrix}
=
\Om{m}{N\pm2}
\begin{pmatrix}
\bX{m}{N\pm2}
\\
\bY{m}{N\pm2}
\end{pmatrix}
\end{equation}
\end{block}
\end{column}
\begin{column}{0.35\textwidth}
\includegraphics[width=\textwidth]{fig/Sigma}
\end{column}
\end{columns}
\end{frame}
%-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Dynamical vs static kernels} \begin{frame}{Dynamical vs static kernels}
\begin{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}} \begin{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}}