saving work: OK for GW
This commit is contained in:
parent
5a5a288dd1
commit
8c79468776
|
|
@ -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<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{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}}
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user