From 8c79468776ae82f650661cfb1166173afcf20954 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 4 Jun 2021 15:53:52 +0200 Subject: [PATCH] saving work: OK for GW --- 2021/Lecture_2/ISTPC_Loos_2.tex | 92 +++++++++++++++++++++++++++------ 1 file changed, 76 insertions(+), 16 deletions(-) diff --git a/2021/Lecture_2/ISTPC_Loos_2.tex b/2021/Lecture_2/ISTPC_Loos_2.tex index 6f30430..589ce6b 100644 --- a/2021/Lecture_2/ISTPC_Loos_2.tex +++ b/2021/Lecture_2/ISTPC_Loos_2.tex @@ -7,6 +7,7 @@ \usetheme{Warsaw} %\usecolortheme{seahorse} \usepackage{mathpazo,libertine} +\usepackage[compat=1.1.0]{tikz-feynman} \usepackage{algorithmicx,algorithm,algpseudocode} \algnewcommand\algorithmicassert{\texttt{assert}} @@ -21,9 +22,10 @@ colorlinks=true, linkcolor=cyan, filecolor=magenta, - urlcolor=blue, + urlcolor=cyan, citecolor=purple } +\urlstyle{same} \definecolor{darkgreen}{RGB}{0, 180, 0} \definecolor{fooblue}{RGB}{0,153,255} @@ -56,6 +58,7 @@ \newcommand{\GOW}{$G_0W$} \newcommand{\GWO}{$GW_0$} \newcommand{\GW}{$GW$} +\newcommand{\GT}{$GT$} \newcommand{\GOWOSOSEX}{{\GOWO}+SOSEX} \newcommand{\GWSOSEX}{{\GW}+SOSEX} \newcommand{\GnWn}[1]{$G_{#1}W_{#1}$} @@ -203,6 +206,7 @@ \newcommand{\btA}[2]{\bm{\Tilde{A}}_{#1}^{#2}} \newcommand{\btB}[2]{\bm{\Tilde{B}}_{#1}^{#2}} \newcommand{\bB}[2]{\bm{B}_{#1}^{#2}} +\newcommand{\bC}[2]{\bm{C}_{#1}^{#2}} \newcommand{\bc}{\bm{c}} \newcommand{\bX}[2]{\bm{X}_{#1}^{#2}} \newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}} @@ -231,7 +235,7 @@ decoration={snake, $GW$/BSE methods in chemistry: 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} \institute[CNRS@LCPQ]{ Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ @@ -292,7 +296,7 @@ decoration={snake, \begin{block}{Let's talk about notations} \begin{itemize} \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 \item Number of \green{occupied orbitals} $O$ \item Number of \alert{vacant orbitals} $V$ @@ -310,7 +314,7 @@ decoration={snake, %----------------------------------------------------- %----------------------------------------------------- -\begin{frame}{Useful papers} +\begin{frame}{Useful papers/programs} \begin{itemize} \item \red{mol$GW$:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149 \bigskip @@ -318,7 +322,7 @@ decoration={snake, \bigskip \item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022 \bigskip - \item \purple{FHI-AIMS:} Caruso et al. 86 (2012) 081102 + \item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102 \bigskip \item \orange{Review:} \begin{itemize} @@ -329,7 +333,8 @@ decoration={snake, \item Blase et al. JPCL 11 (2020) 7371 \end{itemize} \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{frame} %----------------------------------------------------- @@ -368,7 +373,7 @@ decoration={snake, \end{block} \begin{block}{What can you calculate with BSE?} \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 Correlation and total energies \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}}} \end{equation} \end{block} - \begin{block}{Non-interacting polarizability} + \begin{block}{Polarizability} \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' \end{equation} @@ -659,7 +664,7 @@ decoration={snake, \includegraphics[width=0.7\textwidth]{fig/QP} \\ \bigskip - \pub{V\'eril \& Loos, JCTC 14 (2018) 5220} + \pub{V\'eril et al, JCTC 14 (2018) 5220} \end{center} \end{column} \begin{column}{0.5\textwidth} @@ -700,7 +705,7 @@ decoration={snake, \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 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!}} \State Form screened ERIs $\violet{\ERI{pq}{m}}$ \For{$p=1,\ldots,N$} @@ -736,7 +741,7 @@ decoration={snake, \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 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!}} \State Form screened ERIs $\violet{\ERI{pq}{m}}$ \For{$p=1,\ldots,N$} @@ -764,9 +769,10 @@ decoration={snake, \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 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 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 $\violet{\ERI{pq}{m}}$ \For{$p=1,\ldots,N$} \State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$ @@ -800,14 +806,16 @@ decoration={snake, \Procedure{{\qsGW}}{} \State Perform HF calculation to get $\beHF$ and $\bcHF$ \green{(optional)} \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: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow} \ERI{pq}{rs}$ + \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 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 Evaluate $\red{\bSigC}(\blue{\beGnWn{n-1}})$ and form $\red{\Tilde{\Sigma}^{\co}} \leftarrow \qty[ \red{\bSigC}(\blue{\beGnWn{n-1}})^\dag + \red{\bSigC}(\blue{\beGnWn{n-1}}) ]/2$ - \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 $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$ \State $n \leftarrow n + 1$ @@ -831,6 +839,58 @@ decoration={snake, \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