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ISTPC/2022/GFQC/ISTPC_Loos_QFQC.tex

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Lecture 2 2022
2022-06-21 11:29:41 +02:00
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\newcommand{\mr}{\multirow}
\newcommand{\br}{\bm{r}}
\newcommand{\ree}{r_{12}}
\newcommand{\T}[1]{#1^{\intercal}}
% methods
\newcommand{\evGW}{ev$GW$}
\newcommand{\qsGW}{qs$GW$}
\newcommand{\scGW}{sc$GW$}
\newcommand{\GOWO}{$G_0W_0$}
\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]{$G_{#1}W_{#1}$}
\newcommand{\GOF}{$G_0F2$}
\newcommand{\GF}{$GF2$}
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\newcommand{\RPA}{\text{RPA}}
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% operators
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% energies
\newcommand{\Ec}{E_\text{c}}
\newcommand{\EHF}{E_\text{HF}}
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\newcommand{\EcGMGW}{E_\text{c}^\text{GM@GW}}
\newcommand{\EcGMGF}{E_\text{c}^\text{GM@GF2}}
\newcommand{\EcGMGWSOSEX}{E_\text{c}^\text{GM@GW+SOSEX}}
\newcommand{\EcMP}{E_c^\text{MP2}}
\newcommand{\EcGF}{E_c^\text{\GF}}
\newcommand{\EcGOF}{E_c^\text{\GOF}}
\newcommand{\Eg}[1]{E_\text{g}^{#1}}
\newcommand{\IP}{\text{IP}}
\newcommand{\EA}{\text{EA}}
% orbital energies
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\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
\newcommand{\eGF}[1]{\epsilon^\text{\GF}_{#1}}
\newcommand{\eGOF}[1]{\epsilon^\text{\GOF}_{#1}}
\newcommand{\de}[1]{\Delta\epsilon_{#1}}
\newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
\newcommand{\deKS}[1]{\Delta\epsilon^\text{KS}_{#1}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
\newcommand{\eHOMO}[1]{\epsilon_\text{HOMO}^{#1}}
\newcommand{\eLUMO}[1]{\epsilon_\text{LUMO}^{#1}}
\newcommand{\cHF}[1]{c^\text{HF}_{#1}}
\newcommand{\cKS}[1]{c^\text{KS}_{#1}}
% Matrix elements
\newcommand{\A}[2]{A_{#1}^{#2}}
\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}}
\newcommand{\B}[2]{B_{#1}^{#2}}
\newcommand{\tB}[2]{\Tilde{B}_{#1}^{#2}}
\renewcommand{\S}[1]{S_{#1}}
\newcommand{\ABSE}[1]{A^\text{BSE}_{#1}}
\newcommand{\BBSE}[1]{B^\text{BSE}_{#1}}
\newcommand{\ARPA}[1]{A^\text{RPA}_{#1}}
\newcommand{\BRPA}[1]{B^\text{RPA}_{#1}}
\newcommand{\dABSE}[1]{\delta A^\text{BSE}_{#1}}
\newcommand{\dBBSE}[1]{\delta B^\text{BSE}_{#1}}
\newcommand{\G}[1]{G_{#1}}
\newcommand{\Po}[1]{P_{#1}}
\newcommand{\W}[1]{W_{#1}}
\newcommand{\Wc}[1]{W^\text{c}_{#1}}
\newcommand{\vc}[1]{v_{#1}}
\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
\newcommand{\SigGW}[1]{\Sigma^\text{\GW}_{#1}}
\newcommand{\SigGWSOSEX}[1]{\Sigma^\text{\GWSOSEX}_{#1}}
\newcommand{\SigGF}[1]{\Sigma^\text{\GF}_{#1}}
\newcommand{\Z}[1]{Z_{#1}}
% excitation energies
\newcommand{\OmRPA}[1]{\Omega^\text{RPA}_{#1}}
\newcommand{\OmCIS}[1]{\Omega^\text{CIS}_{#1}}
\newcommand{\OmTDHF}[1]{\Omega^\text{TDHF}_{#1}}
\newcommand{\OmBSE}[1]{\Omega^\text{BSE}_{#1}}
\newcommand{\spinup}{\downarrow}
\newcommand{\spindw}{\uparrow}
\newcommand{\singlet}{\uparrow\downarrow}
\newcommand{\triplet}{\uparrow\uparrow}
\newcommand{\Oms}[1]{{}^{1}\Omega_{#1}}
\newcommand{\OmsRPA}[1]{{}^{1}\Omega^\text{RPA}_{#1}}
\newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}}
\newcommand{\OmsTDHF}[1]{{}^{1}\Omega^\text{TDHF}_{#1}}
\newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}}
\newcommand{\Omt}[1]{{}^{3}\Omega_{#1}}
\newcommand{\OmtRPA}[1]{{}^{3}\Omega^\text{RPA}_{#1}}
\newcommand{\OmtCIS}[1]{{}^{3}\Omega^\text{CIS}_{#1}}
\newcommand{\OmtTDHF}[1]{{}^{3}\Omega^\text{TDHF}_{#1}}
\newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}}
\newcommand{\MO}[1]{\phi_{#1}}
\newcommand{\ERI}[2]{(#1|#2)}
\newcommand{\rbra}[1]{(#1|}
\newcommand{\rket}[1]{|#1)}
\newcommand{\sERI}[2]{[#1|#2]}
\newcommand{\sig}{\sigma}
\newcommand{\sigp}{\sigma'}
% Matrices
\newcommand{\bF}{\bm{F}}
\newcommand{\bFHF}{\bm{F}^\text{HF}}
\newcommand{\bH}{\bm{H}}
\newcommand{\bh}{\bm{h}}
\newcommand{\bvc}{\bm{v}}
\newcommand{\bSig}{\bm{\Sigma}}
\newcommand{\bSigX}{\bm{\Sigma}^\text{x}}
\newcommand{\bSigC}{\bm{\Sigma}^\text{c}}
\newcommand{\bSigGW}{\bm{\Sigma}^\text{\GW}}
\newcommand{\bSigGWSOSEX}{\bm{\Sigma}^\text{\GWSOSEX}}
\newcommand{\bSigGF}{\bm{\Sigma}^\text{\GF}}
\newcommand{\be}{\bm{\epsilon}}
\newcommand{\bDelta}{\bm{\Delta}}
\newcommand{\beHF}{\bm{\epsilon}^\text{HF}}
\newcommand{\beKS}{\bm{\epsilon}^\text{KS}}
\newcommand{\bcHF}{\bm{c}^\text{HF}}
\newcommand{\bcKS}{\bm{c}^\text{KS}}
\newcommand{\beGW}{\bm{\epsilon}^\text{\GW}}
\newcommand{\beGnWn}[1]{\bm{\epsilon}^\text{\GnWn{#1}}}
\newcommand{\bcGnWn}[1]{\bm{c}^\text{\GnWn{#1}}}
\newcommand{\beGF}{\bm{\epsilon}^\text{\GF}}
\newcommand{\bde}{\bm{\Delta\epsilon}}
\newcommand{\bdeHF}{\bm{\Delta\epsilon}^\text{HF}}
\newcommand{\bdeGW}{\bm{\Delta\epsilon}^\text{GW}}
\newcommand{\bdeGF}{\bm{\Delta\epsilon}^\text{GF2}}
\newcommand{\bO}{\bm{0}}
\newcommand{\bI}{\bm{1}}
\newcommand{\bOm}[2]{\bm{\Omega}_{#1}^{#2}}
\newcommand{\bA}[2]{\bm{A}_{#1}^{#2}}
\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}}
\newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}}
\newcommand{\bK}[2]{\blue{\bm{K}}_{#1}^{#2}}
\newcommand{\bP}[2]{\red{\bm{P}}_{#1}^{#2}}
\newcommand{\yo}{\yellow{\omega}}
\newcommand{\la}{\yellow{\lambda}}
\newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}}
\usepackage{tikz}
\usetikzlibrary{arrows,positioning,shapes.geometric}
\usetikzlibrary{decorations.pathmorphing}
\tikzset{snake it/.style={
decoration={snake,
amplitude = .4mm,
segment length = 2mm},decorate}}
% *************
% * HEAD DATA *
% *************
\title[Green's function-based methods in chemistry]{
Green's function-based methods in chemistry
}
\author[PF Loos (\url{https://pfloos.github.io/WEB_LOOS})]{Pierre-Fran\c{c}ois LOOS}
\date{ISTPC 2022 --- June 24th, 2022}
\institute[CNRS@LCPQ]{
Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\
Universit\'e de Toulouse, CNRS, UPS, Toulouse, France.
}
\titlegraphic{
\includegraphics[width=0.3\textwidth]{fig/jarvis}
\\
\vspace{0.05\textheight}
\includegraphics[height=0.05\textwidth]{fig/UPS}
\hspace{0.2\textwidth}
\includegraphics[height=0.05\textwidth]{fig/ERC}
\hspace{0.2\textwidth}
\includegraphics[height=0.05\textwidth]{fig/LCPQ}
\hspace{0.2\textwidth}
\includegraphics[height=0.05\textwidth]{fig/CNRS}
}
\begin{document}
%-----------------------------------------------------
\begin{frame}
\titlepage
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Today's program}
\begin{itemize}
\item \textbf{Charged excitations}
\begin{itemize}
\item One-shot $GW$ (\GOWO)
\item Partially self-consistent eigenvalue $GW$ (\evGW)
\item Quasiparticle self-consistent $GW$ (\qsGW)
\item Other self-energies (GF2, SOSEX, T-matrix, etc)
\end{itemize}
\bigskip
\item \textbf{Neutral excitations}
\begin{itemize}
\item Random-phase approximation (RPA)
\item Configuration interaction with singles (CIS)
\item Time-dependent Hartree-Fock (TDHF) or RPA with exchange (RPAx)
\item Time-dependent density-functional theory (TDDFT)
\item Bethe-Salpeter equation (BSE) formalism
\end{itemize}
\bigskip
\item \textbf{Correlation energy}
\begin{itemize}
\item Plasmon (or trace) formula
\item Galitski-Migdal formulation
\item Adiabatic connection fluctuation-dissipation theorem (ACFDT)
\end{itemize}
\end{itemize}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section{Context}
\begin{frame}
\tableofcontents[currentsection]
\end{frame}
%-----------------------------------------------------
\begin{frame}{Assumptions \& Notations}
\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 \purple{triplets} can also be obtained
\bigskip
\item Number of \green{occupied orbitals} $O$
\item Number of \alert{vacant orbitals} $V$
\item \violet{Total number of orbitals} $N = O + V$
\bigskip
\item $\MO{p}(\br)$ is a (real) \blue{spatial orbital}
\item $i,j,k,l$ are \green{occupied orbitals}
\item $a,b,c,d$ are \alert{vacant orbitals}
\item $p,q,r,s$ are \violet{arbitrary (occupied or vacant) orbitals}
\item $\mu,\nu,\lambda,\sigma$ are \purple{basis function indexes}
\bigskip
\item $m$ indexes \purple{the $OV$ single excitations} ($i \to a$)
\end{itemize}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Useful papers/programs}
\begin{itemize}
\item \red{mol$GW$:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
\bigskip
\item \green{Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528
\bigskip
\item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
\bigskip
\item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
\bigskip
\item \orange{Reviews \& Books:}
\begin{itemize}
\item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344
\item Onida et al. Rev. Mod. Phys. 74 (2002) 601
\item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022
\item Golze et al. Front. Chem. 7 (2019) 377
\item Blase et al. JPCL 11 (2020) 7371
\item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press)
\end{itemize}
\bigskip
\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}
%-----------------------------------------------------
%-----------------------------------------------------
\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{columns}
\begin{column}{0.4\textwidth}
\centering
\includegraphics[width=0.8\linewidth]{fig/pentagon}
\\
\pub{Hedin, Phys Rev 139 (1965) A796}
\end{column}
\begin{column}{0.6\textwidth}
\begin{block}{What can you calculate with $GW$?}
\begin{itemize}
\item Ionization potentials (IPs) given by occupied MO energies
\item Electron affinities (EAs) given by virtual MO energies
\item Fundamental (HOMO-LUMO) gap (or band gap in solids)
\item Correlation and total energies
\end{itemize}
\end{block}
\begin{block}{What can you calculate with BSE?}
\begin{itemize}
\item Singlet and triplet optical excitations (vertical absorption energies)
\item Oscillator strengths (absorption intensities)
\item Correlation and total energies
\end{itemize}
\end{block}
\end{column}
\end{columns}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{The MBPT chain of actions}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig/BSE-GW}
\\
\bigskip
\pub{Blase et al. JPCL 11 (2020) 7371}
\end{center}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Photochemistry: Jablonski diagram}
% colors
\definecolor{turquoise}{rgb}{0 0.41 0.41}
\definecolor{rouge}{rgb}{0.79 0.0 0.1}
\definecolor{vert}{rgb}{0.15 0.4 0.1}
\definecolor{mauve}{rgb}{0.6 0.4 0.8}
\definecolor{violet}{rgb}{0.58 0. 0.41}
\definecolor{orange}{rgb}{0.8 0.4 0.2}
\definecolor{bleu}{rgb}{0.39, 0.58, 0.93}
\begin{center}
\begin{tikzpicture}[scale=0.7]
% styles
\tikzstyle{elec} = [line width=2pt,draw=black!80]
\tikzstyle{vib} = [thick,draw=black!30]
\tikzstyle{trans} = [line width=2pt,->]
\tikzstyle{transCI} = [trans,dashed,draw=vert]
\tikzstyle{transCS} = [trans,dashed,draw=violet]
\tikzstyle{relax} = [draw=orange,ultra thick,decorate,decoration=snake]
\tikzstyle{rv} = [rotate=90,text=orange,pos=0.5,yshift=3mm]
% fondamental
\path[elec] (0,0) -- ++ (14,0)
node[below,pos=0.5,yshift=-1mm] {Ground state $S_0$};
\path[vib] (0,0.2) -- ++ (14,0);
\path[vib] (0,0.4) -- ++ (13,0);
\foreach \i in {1,2,...,30} {
\path[vib] (0,0.4 + \i*0.2) -- ++ ({2 + 10*exp(-0.2*\i)},0);
}
% T1
\path[elec] (11,4) -- ++ (3,0) node[anchor=south west] {$T_1$};
\foreach \i in {1,2,...,6} {
\path[vib] (11,4 + \i*0.2) -- ++ (3,0);
}
% S1
\path[elec] (4,5) node[anchor=south east] {$S_1$} -- ++ (5,0);
\foreach \i in {1,2,...,6} {
\path[vib] (4,5 + \i*0.2) -- ++ (5,0);
}
\foreach \i in {1,2,...,12} {
\path[vib] ({7.5 - 1*exp(-0.3*\i)},6.2+\i*0.2) -- (9,6.2+\i*0.2);
}
% S2
\path[elec] (4,8) node[anchor=south east] {$S_2$} -- ++ (2,0);
\foreach \i in {1,2,...,6} {
\path[vib] (4,8 + \i*0.2) -- ++ (2,0);
}
% absorption
\path[trans,draw=turquoise] (4.5,0) -- ++(0,9)
node[rotate=90,pos=0.35,text=turquoise,yshift=-3mm] {\small Absorption};
% fluo
\path[trans,draw=rouge](7,5) -- ++(0,-4.4)
node[rotate=90,pos=0.5,text=rouge,yshift=-3mm] {\small Fluorescence};
% phosphorescence
\path[trans,draw=mauve] (13,4) -- ++(0,-3.4)
node[rotate=90,pos=0.5,text=mauve,yshift=-3mm] {\small Phosphorescence};
% Conversion interne
\path[transCI] (4,5) -- ++(-1.9,0) node[below,pos=0.5,text=vert] {\small IC};
\path[transCI] (6,8) -- ++(1.3,0) node[above,pos=0.5,text=vert] {\small IC};
% Croisement intersysteme
\path[transCS] (9,5) -- ++(2,0) node[below,pos=0.5,text=violet] {\small ISC};
\path[transCS] (11,4) -- ++(-2.5,0) node[below,pos=0.5,text=violet] {\small ISC};
% relaxation vib
\path[relax] (5.5,8.8) -- ++(0,-0.8) node[rv] {\small \textbf{VR}};
\path[relax] (8,8) -- ++(0,-3) node[rv] {\small \textbf{VR}};
\path[relax] (1,5) -- ++(0,-5) node[rv] {\small \textbf{VR}};
\path[relax] (11.5,5) -- ++(0,-1) node[rv] {\small \textbf{VR}};
\end{tikzpicture}
\end{center}
%\tiny
%\begin{itemize}
% \item[] \tikz {\path[line width=2pt,->,dashed,draw=vert]
% (0,0) -- (1,0) node[above,pos=0.5,text=vert] {IC};} Internal Conversion,
% $S_i\,\longrightarrow\,S_j$ non radiative transition.
%
% \item[] \tikz {\path[line width=2pt,->,dashed,draw=violet]
% (0,0) -- (1,0) node[above,pos=0.5,text=violet] {ISC};} InterSystem Crossing,
% $S_i\,\longrightarrow\,T_j$ non radiative transition.
%
% \item[] \tikz {\path[line width=2pt,draw=orange,ultra thick,
% decorate,decoration=snake] (0,0) -- (1,0) node[above,pos=0.5,text=orange] {RV};}
% Vibrationnal Relaxation.
%\end{itemize}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Photochemistry: absorption, emission, and 0-0}
\begin{center}
\includegraphics[width=0.5\textwidth]{fig/0-0}
\\
\textbf{\alert{Vertical excitation energies cannot be computed experimentally!!!}}
\end{center}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section{Charged excitations}
\begin{frame}
\tableofcontents[currentsection]
\end{frame}
%-----------------------------------------------------
\begin{frame}{Green's function and dynamical screening}
\begin{block}{One-body Green's function}
\begin{equation}
\blue{G}(\br_1,\br_2;\yo)
= \underbrace{\sum_i \frac{\MO{i}(\br_1) \MO{i}(\br_2)}{\yo - \e{i}{} - i\eta}}_{\text{\green{removal part = IPs}}}
+ \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}{Polarizability}
\begin{equation}
P(\br_1,\br_2;\yo) = - \frac{i}{\pi} \int \blue{G}(\br_1,\br_2;\yo+\omega') \blue{G}(\br_1,\br_2;\omega') d\omega'
\end{equation}
\end{block}
\begin{block}{Dielectric function and dynamically-screened Coulomb potential}
\begin{equation}
\epsilon(\br_1,\br_2;\yo) = \delta(\br_1 - \br_2) - \int \frac{P(\br_1,\br_3;\yo) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation}
\begin{equation}
\highlight{W}(\br_1,\br_2;\yo) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\yo) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Dynamical screening in the orbital basis}
\begin{block}{Spectral representation of $W$}
\begin{equation}
\begin{split}
\highlight{W}_{pq,rs}(\yo)
& = \iint \MO{p}(\br_1) \MO{q}(\br_1) \highlight{W}(\br_1,\br_2;\yo) \MO{r}(\br_2) \MO{s}(\br_2) d\br_1 d\br_2
\\
& = \underbrace{\ERI{pq}{rs}}_{\text{(static) exchange part}}
+ \underbrace{2 \sum_m \violet{\ERI{pq}{m}} \violet{\ERI{rs}{m}}
\qty[ \frac{1}{\yo - \orange{\Om{m}{\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\RPA}} - i \eta} ]}_{\text{(dynamical) correlation part } \highlight{W}^{\co}_{pq,rs}(\yo)}
\end{split}
\end{equation}
\end{block}
\begin{block}{Electron repulsion integrals (ERIs)}
\begin{equation}
\ERI{pq}{rs} = \iint \frac{\MO{p}(\br_1) \MO{q}(\br_1) \MO{r}(\br_2) \MO{s}(\br_2)}{\abs{\br_1 - \br_2}} d\br_1 d\br_2
\end{equation}
\end{block}
\begin{block}{Screened ERIs (or spectral weights)}
\begin{equation}
\violet{\ERI{pq}{m}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\RPA}+\bY{m}{\RPA}})_{ia}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Computation of the dynamical screening}
\begin{block}{Direct (ph-)RPA calculation (pseudo-hermitian linear problem)}
\begin{equation}
\begin{pmatrix}
\bA{}{\RPA} & \bB{}{\RPA} \\
-\bB{}{\RPA} & -\bA{}{\RPA} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\orange{\bX{m}{\RPA}} \\
\orange{\bY{m}{\RPA}} \\
\end{pmatrix}
=
\orange{\Om{m}{\RPA}}
\begin{pmatrix}
\orange{\bX{m}{\RPA}} \\
\orange{\bY{m}{\RPA}} \\
\end{pmatrix}
\end{equation}
\begin{equation}
\qq*{For singlet states:} \A{ia,jb}{\RPA} = \delta_{ij} \delta_{ab} (\e{a}{} - \e{i}{}) + 2\ERI{ia}{bj}
\qquad
\B{ia,jb}{\RPA} = 2\ERI{ia}{jb}
\end{equation}
\end{block}
\begin{block}{Non-hermitian to hermitian}
\begin{equation}
(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{m}{} = \Om{m}{2} \, \bZ{m}{}
\end{equation}
\begin{gather}
(\bX{m}{} + \bY{m}{}) = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{}
\\
(\bX{m}{} - \bY{m}{}) = \Om{m}{+1/2} (\bA{}{} - \bB{}{})^{-1/2} \cdot \bZ{m}{}
\end{gather}
\end{block}
\begin{block}{Tamm-Dancoff approximation (TDA)}
\begin{equation}
\bB{}{} = \bO \quad \Rightarrow \quad \bA{}{} \cdot \orange{\bX{m}{}} = \orange{\Om{m}{\TDA} \bX{m}{}}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{The self-energy}
\begin{block}{$GW$ Self-energy}
\begin{equation}
\underbrace{\red{\Sig{}{\xc}}(\br_1,\br_2;\yo)}_{\text{$GW$ self-energy}}
= \underbrace{\purple{\Sig{}{\x}}(\br_1,\br_2)}_{\text{\purple{exchange}}}
+ \underbrace{\red{\Sig{}{\co}}(\br_1,\br_2;\yo)}_{\text{\red{correlation}}}
= \frac{i}{2\pi} \int \blue{G}(\br_1,\br_2;\yo+\omega') \highlight{W}(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega'
\end{equation}
\end{block}
\begin{block}{Exchange part of the (static) self-energy}
\begin{equation}
\purple{\Sig{pq}{\x}} = - \sum_{i} \ERI{pi}{iq}
\end{equation}
\end{block}
\begin{block}{Correlation part of the (dynamical) self-energy}
\begin{equation}
\red{\Sig{pq}{\co}}(\yo)
= 2 \sum_{im} \frac{\violet{\ERI{pi}{m}} \violet{\ERI{qi}{m}}}{\yo - \e{i}{} + \orange{\Om{m}{\RPA}} - i \eta}
+ 2 \sum_{am} \frac{\violet{\ERI{pa}{m}} \violet{\ERI{qa}{m}}}{\yo - \e{a}{} - \orange{\Om{m}{\RPA}} + i \eta}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Quasiparticle equation}
\begin{block}{Dyson equation}
\begin{equation}
\qty[ \blue{G}(\br_1,\br_2;\yo) ]^{-1}
= \underbrace{\qty[ G_{\KS}(\br_1,\br_2;\yo) ]^{-1}}_{\text{KS Green's function}}
+ \red{\Sig{}{\xc}}(\br_1,\br_2;\yo) - \underbrace{\upsilon^{\xc}(\br_1)}_{\text{KS potential}} \delta(\br_1 - \br_2)
\end{equation}
\end{block}
\begin{block}{Non-linear quasiparticle (QP) equation}
\begin{equation}
\yo = \eKS{p} + \red{\Sig{pp}{\xc}}(\yo) - V_{p}^{\xc}
\qq{with}
V_{p}^{\xc} = \int \MO{p}(\br) \upsilon^{\xc}(\br) \MO{p}(\br) d\br
\end{equation}
\end{block}
\begin{block}{Linearized QP equation}
\begin{equation}
\red{\Sig{pp}{\xc}}(\yo) \approx \red{\Sig{pp}{\xc}}(\eKS{p}) + (\yo - \eKS{p}) \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \eKS{p}}
\qq{$\Rightarrow$}
\blue{\eGW{p}} = \eKS{p} + \green{Z_{p}} [\red{\Sig{pp}{\xc}}(\eKS{p}) - V_{p}^{\xc} ]
\end{equation}
\begin{equation}
\underbrace{\green{Z_{p}}}_{\text{renormalization factor}} = \qty[ 1 - \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \eKS{p}} ]^{-1}
\qq{with} 0 \le \green{Z_{p}} \le 1
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Solutions of the non-linear QP equation: {\evGW}@HF/6-31G for \ce{H2} at $R = 1$ bohr}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig/QP}
\\
\bigskip
\pub{V\'eril et al, JCTC 14 (2018) 5220}
\end{center}
\end{column}
\begin{column}{0.5\textwidth}
\begin{center}
\includegraphics[width=\textwidth]{fig/GWSph}
\\
\bigskip
\pub{Loos et al, JCTC 14 (2018) 3071}
\end{center}
\end{column}
\end{columns}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{$GW$ flavours}
\begin{block}{Acronyms}
\begin{itemize}
\bigskip
\item perturbative $GW$, one-shot $GW$, or \green{\GOWO}
\bigskip
\item \orange{\evGW} or eigenvalue-only (partially) self-consistent $GW$
\bigskip
\item \red{\qsGW} or quasiparticle (partially) self-consistent $GW$
\bigskip
\item \violet{\scGW} or (fully) self-consistent $GW$
\bigskip
\end{itemize}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Perturbative {\GW} with linearized solution}
\begin{block}{}
\begin{algorithmic}
\Procedure{{\GOWO}lin@KS}{}
\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 $\orange{\bOm{}{\RPA}}$ 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{$p=1,\ldots,N$}
\State Compute diagonal of the self-energy $\red{\SigC{pp}}(\yo)$ at $\yo = \eKS{p}$
\State Compute renornalization factors \green{$\Z{p}$}
\State Evaluate $\blue{\eGOWO{p}} = \eKS{p} + \green{\Z{p}} \qty{ \Re[\red{\SigC{pp}}(\eKS{p})] - V_{p}^{\xc} }$
\EndFor
\EndProcedure
\end{algorithmic}
\end{block}
\bigskip
For contour deformation technique, see, for example, \pub{Duchemin \& Blase, JCTC 16 (2020) 1742}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Example from \texttt{QuAcK} (\ce{Ne}/cc-pVDZ)}
\begin{center}
\includegraphics[width=0.55\textwidth]{fig/G0W0}
\\
\bigskip
\pub{https://github.com/pfloos/QuAcK}
\end{center}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Perturbative {\GW} with graphical solution}
\begin{block}{}
\begin{algorithmic}
\Procedure{{\GOWO}graph@KS}{}
\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 $\orange{\Om{}{\RPA}}$ 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 $\red{\SigC{pp}}(\yo)$
\State Solve $\yo = \eKS{p} + \Re[\red{\SigC{pp}}(\yo)] - V_{p}^{\xc}$ to get $\blue{\eGOWO{p}}$ via Newton's method
\EndFor
\EndProcedure
\end{algorithmic}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Newton's method}
\centering
\url{https://en.wikipedia.org/wiki/Newton\%27s_method}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Partially self-consistent eigenvalue \GW}
\begin{block}{}
\begin{algorithmic}
\Procedure{{\evGW}@KS}{}
\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{$\max{\abs{\bDelta}} > \tau$}
\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$ from $\blue{\beGnWn{n-1}}$ and $\ERI{pq}{rs}$
\State Compute RPA eigenvalues $\orange{\Om{}{\RPA}}$ 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 $\red{\SigC{pp}}(\yo)$
\State Solve $\yo = \eKS{p} + \Re[\red{\SigC{pp}}(\yo)] - V_{p}^{\xc}$ to get $\blue{\eGnWn{p}{n}}$
\EndFor
\State $\bDelta = \blue{\beGnWn{n}} - \blue{\beGnWn{n-1}}$
\State $n \leftarrow n + 1$
\EndWhile
\EndProcedure
\end{algorithmic}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Example from \texttt{QuAcK} (\ce{Ne}/cc-pVDZ)}
\begin{center}
\includegraphics[width=0.5\textwidth]{fig/evGW}
\\
\bigskip
\pub{https://github.com/pfloos/QuAcK}
\end{center}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Quasiparticle self-consistent {\GW} (\qsGW)}
\begin{block}{}
\begin{algorithmic}
\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{$\max{\abs{\bDelta}} > \tau$}
\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 $\orange{\Om{}{\RPA}}$ 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 $\bFHF$ 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 $n \leftarrow n + 1$
\EndWhile
\EndProcedure
\end{algorithmic}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Example from \texttt{QuAcK} (\ce{Ne}/cc-pVDZ)}
\begin{center}
\includegraphics[width=0.45\textwidth]{fig/qsGW1}
\hspace{0.1\textwidth}
\includegraphics[width=0.4\textwidth]{fig/qsGW2}
\\
\bigskip
\pub{https://github.com/pfloos/QuAcK}
\end{center}
\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}
\\
\bigskip
\includegraphics[width=\textwidth]{fig/Tmatrix}
\\
\pub{Martin, Reining \& Ceperley, Interacting Electrons (Cambridge University Press)}
\end{column}
\end{columns}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section{Neutral excitations}
\begin{frame}
\tableofcontents[currentsection]
\end{frame}
%-----------------------------------------------------
\begin{frame}{Dynamical vs static kernels}
\begin{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}}
\begin{equation}
\begin{pmatrix}
\bA{}{}(\yo) & \bB{}{}(\yo)
\\
-\bB{}{}(-\yo) & -\bA{}{}(-\yo)
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{}{}
\\
\bY{}{}
\end{pmatrix}
=
\yo
\begin{pmatrix}
\bX{}{}
\\
\bY{}{}
\end{pmatrix}
\qq{\alert{\bf Hard to solve!}}
\end{equation}
\end{block}
\begin{block}{Static BSE vs dynamic BSE for \ce{HeH+}/STO-3G}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig/dyn}
\end{center}
\end{column}
\begin{column}{0.5\textwidth}
Dynamical kernels can give you more than static kernels... Sometimes, too much...
\end{column}
\end{columns}
\bigskip
\center
\pub{Authier \& Loos, JCP 153 (2020) 184105} [see also \pub{Romaniello et al, JCP 130 (2009) 044108}]
\end{block}
\end{frame}
%-----------------------------------------------------
\begin{frame}{L\"owdin partitioning technique}
\begin{block}{Folding or dressing process}
\begin{equation}
\underbrace{\bH{}{} \cdot \bc = \yo \, \bc}_{\text{A large linear system with $N$ solutions\ldots}}
\qq{$\Rightarrow$}
\begin{pmatrix}
\overbrace{\bH_1}^{N_1 \times N_1} & \T{\bh} \\
\bh & \underbrace{\bH_2}_{N_2 \times N_2} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bc_1 \\
\bc_2 \\
\end{pmatrix}
= \yo
\begin{pmatrix}
\bc_1 \\
\bc_2 \\
\end{pmatrix}
\qquad N = N_1 + N_2
\end{equation}
\begin{align}
\qq*{\bf Row \#2:}
& \bh \cdot \bc_1 + \bH_2 \cdot \bc_2 = \yo \, \bc_2
& \qq{$\Rightarrow$}
& \bc_2 = (\yo \, \bI - \bH_2)^{-1} \cdot \bh \cdot \bc_1
\\
\qq*{\bf Row \#1:}
& \bH_1 \cdot \bc_1 + \T{\bh} \cdot \bc_2 = \yo \, \bc_1
& \qq{$\Rightarrow$}
& \underbrace{\Tilde{\bH}_1(\yo) \cdot \bc_1 = \yo \, \bc_1}_{\text{A smaller non-linear system with $N$ solutions\ldots}}
\end{align}
\begin{equation}
\boxed{
\underbrace{\Tilde{\bH}_1(\yo)}_{\text{Effective Hamitonian}}
= \bH_1 + \T{\bh} \cdot (\yo \, \bI - \bH_2)^{-1} \cdot \bh
}
\end{equation}
\begin{equation}
\qq*{Static approx. (e.g.~$\yo = 0$):}
\underbrace{\Tilde{\bH}_1(\yo = 0)}_{\text{A smaller linear system with $N_1$ solutions\ldots}}
= \bH_1 - \underbrace{\T{\bh} \cdot \bH_2^{-1} \cdot \bh}_{\text{approximations possible...}}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{TD-DFT and BSE in practice: Casida-like equations}
\begin{block}{Linear response problem}
\begin{equation*}
\boxed{\begin{pmatrix}
\red{\bA{}{}} & \orange{\bB{}{}}
\\
\orange{-\bB{}{}} & \red{-\bA{}{}}
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{m}{}
\\
\bY{m}{}
\end{pmatrix}
=
\highlight{\Om{m}{}}
\begin{pmatrix}
\bX{m}{}
\\
\bY{m}{}
\end{pmatrix}}
\end{equation*}
\end{block}
%
\begin{block}{Blue pill: TD-DFT within the \alert{adiabatic} approximation}
\begin{gather}
\red{A}_{ia,jb} = \qty( \e{a}{\green{\KS}}- \e{i}{\green{\KS}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} + \yellow{f}^{\yellow{xc}}_{ia,bj}
\qquad
\orange{B}_{ia,jb} = 2 \blue{(ia|jb)} + \yellow{f}^{\yellow{xc}}_{ia,jb}
\\
\yellow{f}^{\yellow{xc}}_{ia,bj} = \iint \phi_{i}(\br{})\phi_{a}(\br{}) \frac{\delta^2 E^{xc} }{\delta\rho(\br{}) \delta\rho(\br{}')} \phi_{b}(\br{})\phi_{j}(\br{}) d\br{} d\br{}'
\end{gather}
\end{block}
%
\begin{block}{Red pill: BSE within the \alert{static} approximation}
\begin{gather}
\red{A}_{ia,jb} = \qty( \e{a}{\green{GW}} - \e{i}{\green{GW}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} - \purple{W}^\text{stat}_{ij,ba}
\qquad
\orange{B}_{ia,jb} = 2 \blue{(ia|jb)} - \purple{W}^\text{stat}_{ib,ja}
\\
\purple{W}^\text{stat}_{ij,ab} \equiv \purple{W}_{ij,ab} (\omega = 0) = (ij|ab) - W^{c}_{ij,ab}(\omega = 0)
\end{gather}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{The bridge between TD-DFT and BSE}
\begin{block}{}
\begin{center}
\begin{tabular}{lcr}
\hline
\bf \red{TD-DFT} & \bf \purple{Connection} & \bf \violet{BSE}
\\
\hline
\\
\red{One-point density} & & \violet{Two-point Green's function}
\\
$\rho(1)$ & $\rho(1) = -iG(11^{+})$ & $G(12)$
\\
\\
\red{Two-point susceptibility} & & \violet{Four-point susceptibility}
\\
$\chi(12) = \pdv{\rho(1)}{U(2)}$ & $\chi(12) = -i L(12;1^+2^+)$ & $L(12;34) = \pdv{G(13)}{U(42)}$
\\
\\
\red{Two-point kernel} & & \violet{Four-point kernel}
\\
$K(12) = v(12) + \pdv{V^{xc}(1)}{\rho(2)}$ & & $i \Xi(1234) = v(13) \delta(12) \delta(34) - \pdv{\Sigma^{xc}(12)}{G(34)}$ \\
\hline
\end{tabular}
\end{center}
\end{block}
\bigskip
For dynamical correction within BSE, see, for example, \pub{Loos \& Blase, JCP 153 (2020) 114120}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{BSE in a computer}
\begin{block}{Vertical excitation energies from BSE}
\begin{algorithmic}
\Procedure{BSE@GW}{}
\State Compute $GW$ quasiparticle energies \blue{$\eGW{p}$} at the {\GOWO}, {\evGW}, or {\qsGW} level
\State Compute static screening $\highlight{W^\text{stat}_{pq,rs}}$
\State Construct BSE matrices $\orange{\bA{}{\BSE}}$ and $\orange{\bB{}{\BSE}}$ from \blue{$\eGW{p}$}, $\ERI{pq}{rs}$, and $\highlight{W^\text{stat}_{pq,rs}}$
\State Compute lowest eigenvalues $\orange{\Om{m}{\BSE}}$ and eigenvectors $\orange{\bX{m}{\BSE}+\bY{m}{\BSE}}$ \green{(optional)}
\Comment{\alert{This is a $\order*{N^4}$ step!}}
\EndProcedure
\end{algorithmic}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Removing the correlation part: TDHF and CIS}
\begin{block}{Linear response problem}
\begin{equation*}
\boxed{\begin{pmatrix}
\red{\bA{}{}} & \orange{\bB{}{}}
\\
\orange{-\bB{}{}} & \red{-\bA{}{}}
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{m}{}
\\
\bY{m}{}
\end{pmatrix}
=
\highlight{\Om{m}{}}
\begin{pmatrix}
\bX{m}{}
\\
\bY{m}{}
\end{pmatrix}}
\end{equation*}
\end{block}
%
\begin{block}{TDHF = RPA with exchange (RPAx)}
\begin{align}
\red{A}_{ia,jb} & = \qty( \e{a}{\green{\HF}} - \e{i}{\green{\HF}} ) \delta_{ij} \delta_{ab} + 2 \blue{(ia|bj)} - \yellow{(ij|ba)}
&
\orange{B}_{ia,jb} & = 2 \blue{(ia|jb)} - \yellow{(ib|ja)}
\end{align}
\end{block}
%
\begin{block}{Linear response problem within the Tamm-Dancoff approximation}
\begin{equation}
\boxed{\red{\bA{}{}} \cdot \bX{m}{} = \highlight{\Om{m}{}} \, \bX{m}{} }
\end{equation}
\end{block}
%
\begin{block}{TDHF within TDA = CIS}
\begin{equation}
\red{A}_{ia,jb}
= \qty( \e{a}{\green{\HF}} - \e{i}{\green{\HF}} ) \delta_{ij} \delta_{ab}
+ 2 \blue{(ia|bj)} - \yellow{(ij|ba)}
\end{equation}
\end{block}
%
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Relationship between CIS, TDHF, DFT and TDDFT}
\center
\begin{tikzpicture}
\usetikzlibrary{shapes.misc}
\begin{scope}[very thick,
node distance=3cm,on grid,>=stealth',
box/.style={rectangle,draw,fill=green!40}],
\node [box, align=center] (CIS) {\textbf{CIS}};
\node [box, align=center] (HF) [left=of CIS, yshift=1cm] {\textbf{HF}};
\node [box, align=center] (TDHF) [right=of CIS, yshift=1cm] {\textbf{TDHF}};
\node [box, align=center] (DFT) [below=of HF] {\textbf{DFT}};
\node [box, align=center] (TDDFT) [below=of TDHF] {\textbf{TDDFT}};
\node [box, align=center] (TDA) [below=of CIS] {\textbf{TDA}};
\path
(CIS) edge [<-] node[below,sloped]{CI} (HF)
(CIS) edge [<-] node[below,sloped]{$\bB{}{}=\bO$} (TDHF)
(HF) edge [->] node[above]{linear response} (TDHF)
(HF) edge [<->] node[left]{$\upsilon_\text{x}^\text{HF}$ vs $\upsilon_\text{xc}$} (DFT)
(TDHF) edge [<->] node[right]{$\upsilon_\text{x}^\text{HF}$ vs $\upsilon_\text{xc}$} (TDDFT)
(DFT) edge [->] node[above]{linear response} (TDDFT)
(DFT) edge [->] node[below,sloped]{CI} node[strike out,sloped]{\alert{$\cross$}} (TDA)
(TDDFT) edge [->] node[below,sloped]{$\bB{}{}=\bO{}{}$} (TDA)
;
\end{scope}
\end{tikzpicture}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Linear response}
\begin{block}{General linear response problem}
\begin{algorithmic}
\Procedure{Linear response}{}
\State Compute $\red{\bA{}{}}$ matrix at a given level of theory (RPA, RPAx, TD-DFT, BSE, etc)
\If{$\TDA$}
\State Diagonalize $\red{\bA{}{}}$ to get $\highlight{\Om{m}{\TDA}}$ and $\bX{m}{\TDA}$
\Else
\State Compute \orange{$\bB{}{}$} matrix at a given level of theory
\State Diagonalize $\red{\bA{}{}} - \orange{\bB{}{}}$ to form $(\red{\bA{}{}} - \orange{\bB{}{}})^{1/2}$
\State Form and diagonalize $(\red{\bA{}{}} - \orange{\bB{}{}})^{1/2} \cdot (\red{\bA{}{}} + \orange{\bB{}{}}) \cdot (\red{\bA{}{}} - \orange{\bB{}{}})^{1/2}$
to get $\highlight{\Om{m}{2}}$ and $\bZ{m}{}$
\State Compute $\sqrt{\highlight{\Om{m}{2}}}$ and $(\bX{m}{} + \bY{m}{}) = \highlight{\Om{m}{-1/2}} (\red{\bA{}{}} - \orange{\bB{}{}})^{1/2} \cdot \bZ{m}{}$
\EndIf
\EndProcedure
\end{algorithmic}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Form linear response matrices}
\begin{block}{Linear-response matrices for BSE}
\begin{algorithmic}
\Procedure{Form $\red{\bA{}{}}$ for singlet states}{}
\State Set $\red{\bA{}{}} = \bO$
\State $ia \gets 0$
\For{$i=1, \ldots, O$}
\For{$a=1, \ldots, V$}
\State $ia \gets ia + 1$
\State $jb \gets 0$
\For{$j=1, \ldots, O$}
\For{$b=1, \ldots, V$}
\State $jb \gets jb + 1$
\State $\red{A_{ia,jb}} = \delta_{ij} \delta_{ab} (\e{a}{\green{GW}} - \e{i}{\green{GW}})
+ 2\blue{(ia|bj)} - \yellow{(ij|ba)} + \purple{W^{\co}_{ij,ba}}(\omega = 0)$
\EndFor
\EndFor
\EndFor
\EndFor
\EndProcedure
\end{algorithmic}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Properties}
\begin{block}{Oscillator strength (length gauge)}
\begin{equation}
\boxed{\green{f_m} = \frac{2}{3} \orange{\Om{m}{}} \qty[ (\blue{\mu_m^x})^2 + (\blue{\mu_m^y})^2 + (\blue{\mu_m^z})^2 ]}
\end{equation}
\end{block}
\begin{block}{Transition dipole}
\begin{equation}
\boxed{\blue{\mu_m^x} = \sum_{ia} \red{(i|x|a)} \orange{(\bX{m}{} + \bY{m}{})_{ia}}}
\qquad
\red{(p|x|q)} = \int \MO{p}(\br) \,x\, \MO{q}(\br) d\br
\end{equation}
\end{block}
\begin{block}{Monitoring possible spin contamination \pub{[Monino \& Loos, JCTC 17 (2021) 2852]}}
\begin{equation}
\boxed{\purple{\expval{\hat{S}^2}_m} = \violet{\expval{\hat{S}^2}_0} + \underbrace{\Delta \expval{\hat{S}^2}_m}_{\text{\pub{JCP 134101 (2011) 134}}}}
\qquad
\violet{\expval{\hat{S}^2}_0} = \frac{n_\alpha - n_\beta}{2} \qty( \frac{n_\alpha - n_\beta}{2} + 1 ) + n_\beta + \sum_p (p_\alpha|p_\beta)
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Example from \texttt{QuAcK} (\ce{H2O}/cc-pVDZ)}
\begin{center}
\includegraphics[height=0.45\textwidth]{fig/BSE1}
\hspace{0.05\textwidth}
\includegraphics[height=0.45\textwidth]{fig/BSE3}
\\
\bigskip
\pub{https://github.com/pfloos/QuAcK}
\end{center}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Open-shell systems and double excitations}
\begin{block}{Spin-flip formalism (H2/cc-pVQZ)}
\begin{center}
\includegraphics[width=0.28\textwidth]{fig/SFBSE}
\includegraphics[width=0.4\textwidth]{fig/H2}
\includegraphics[width=0.3\textwidth]{fig/H2_QuAcK}
\\
\bigskip
\pub{Monino \& Loos, JCTC 17 (2021) 2852}
\end{center}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section{Correlation energy}
\begin{frame}
\tableofcontents[currentsection]
\end{frame}
%-----------------------------------------------------
\begin{frame}{Correlation energy at the $GW$ or BSE level}
\begin{block}{RPA@$GW$ correlation energy: plasmon (or trace) formula}
\begin{equation*}
\label{eq:Ec-RPA}
\green{\EcRPA}
= \frac{1}{2} \qty[ \sum_{p} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
= \frac{1}{2} \sum_{m} \qty( \orange{\Om{m}{\RPA}} - \orange{\Om{m}{\TDA}} )
\end{equation*}
\end{block}
\begin{block}{Galitskii-Migdal functional}
\begin{equation*}
\label{eq:GM}
\green{\EcGM}
= \frac{-i}{2}\sum_{pq}^{\infty}\int \frac{d\omega}{2\pi} \red{\SigC{pq}}(\omega) \blue{\G{pq}}(\omega) e^{i\omega\eta}
= 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\blue{\eGW{a}} - \blue{\eGW{i}} + \orange{\Om{m}{\RPA}}}
\end{equation*}
\end{block}
\begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection}
\begin{equation}
\green{\Ec^\text{ACFDT}} = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Adiabatic connection fluctuation dissipation theorem (ACFDT)}
\begin{block}{Adiabatic connection}
\begin{equation}
\boxed{
\green{\Ec^\text{ACFDT}}
= \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la
\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_{k=1}^{K} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})
}
\end{equation}
$\la$ is the \textbf{strength} of the electron-electron interaction:
\begin{itemize}
\item $\la = 0$ for the \green{non-interacting system}
\item $\la = 1$ for the \alert{physical system}
\end{itemize}
\end{block}
\begin{block}{Interaction kernel}
\begin{equation}
\bK{}{} =
\begin{pmatrix}
\btA{}{} & \btB{}{}
\\
\btB{}{} & \btA{}{}
\end{pmatrix}
\qquad
\tA{ia,jb}{} = 2\ERI{ia}{bj}
\qquad
\tB{ia,jb}{} = 2\ERI{ia}{jb}
\end{equation}
\end{block}
\begin{block}{Correlation part of the two-particle density matrix}
\begin{equation}
\bP{}{\la} =
\begin{pmatrix}
\bY{}{\la} \cdot \T{(\bY{}{\la})} & \bY{}{\la} \cdot \T{(\bX{}{\la})}
\\
\bX{}{\la} \cdot \T{(\bY{}{\la})} & \bX{}{\la} \cdot \T{(\bX{}{\la})}
\end{pmatrix}
-
\begin{pmatrix}
\bO & \bO
\\
\bO & \bI
\end{pmatrix}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Gaussian quadrature}
\begin{block}{Numerical integration by quadrature}
\textit{``A $K$-point \orange{Gaussian quadrature} rule is a quadrature rule constructed to yield an exact result for polynomials up to degree $2K-1$ by a suitable choice of the \violet{roots $x_k$} and \purple{weights $w_k$} for $k = 1, \ldots, K$.''}
\begin{equation}
\boxed{\int_{\red{a}}^{\red{b}} f(x) \purple{w(x)} dx \approx \sum_k^{K} \underbrace{\purple{w_k}}_{\text{\purple{weights}}} f(\underbrace{\violet{x_k}}_{\text{\violet{roots}}})}
\end{equation}
\end{block}
\begin{block}{Quadrature rules}
\begin{center}
\small
\begin{tabular}{llll}
\hline
\red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{Name} \\
\hline
$[-1,1]$ & $1$ & Legendre $P_n(x)$ & Gauss-Legendre \\
$(-1,1)$ & $(1-x)^\alpha(1+x)^\beta, \quad \alpha,\beta > -1$ & Jacobi $P_n^{\alpha,\beta}(x)$ & Gauss-Jacobi \\
$(-1,1)$ & $1/\sqrt{1-x^2}$ & Chebyshev (1st kind) $T_n(x)$ & Gauss-Chebyshev \\
$[-1,1]$ & $\sqrt{1-x^2}$ & Chebyshev (2nd kind) $U_n(x)$ & Gauss-Chebyshev \\
$[0,\infty)$ & $\exp(-x)$ & Laguerre $L_n(x)$ & Gauss-Laguerre \\
$[0,\infty)$ & $x^\alpha \exp(-x), \quad \alpha > -1$ & Generalized Laguerre $L_n^\alpha(x)$ & Gauss-Laguerre \\
$(-\infty,\infty)$ & $\exp(-x^2)$ & Hermite $H_n(x)$ & Gauss-Hermite \\
\hline
\end{tabular}
\\
\url{https://en.wikipedia.org/wiki/Gaussian_quadrature}
\end{center}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{ACFDT at the RPA/RPAx level}
\begin{block}{RPA matrix elements}
\begin{equation}
\orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\violet{\eHF{a}} - \violet{\eHF{i}}) + 2\la\ERI{ia}{bj}
\qquad
\orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb}
\end{equation}
\begin{equation}
\boxed{
\green{\Ec^\RPA}
= \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la
= \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
}
\end{equation}
\end{block}
\begin{block}{RPAx matrix elements}
\begin{equation}
\orange{\A{ia,jb}{\la,\RPAx}} = \delta_{ij} \delta_{ab} (\violet{\eHF{a}} - \violet{\eHF{i}}) + \la \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]
\qquad
\orange{\B{ia,jb}{\la,\RPAx}} = \la \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
\end{equation}
\begin{equation}
\boxed{
\green{\Ec^\RPAx}
= \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la
\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
}
\end{equation}
If exchange added to kernel, i.e., $\bK{}{} = \bK{}{\x}$, then \pub{[Angyan et al. JCTC 7 (2011) 3116]}
\begin{equation}
\green{\Ec^\RPAx}
= \frac{1}{4} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{\x} \bP{}{\la}) d\la
\alert{=} \frac{1}{4} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{ACFDT at the BSE level}
\begin{block}{BSE matrix elements}
\begin{equation}
\orange{\A{ia,jb}{\la,\BSE}} = \delta_{ij} \delta_{ab} (\violet{\eGW{a}} - \violet{\eGW{i}}) + \la \qty[2 \ERI{ia}{bj} - \highlight{W}_{ij,ab}^{\la}(\omega = 0) ]
\qquad
\orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)]
\end{equation}
\begin{equation}
\boxed{
\green{\Ec^\BSE}
= \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la
\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\BSE}} - \Tr(\orange{\bA{}{\BSE}}) ]
}
\end{equation}
\end{block}
\begin{block}{$\la$-dependent screening}
\begin{equation}
\highlight{W}_{pq,rs}^{\la}(\omega)
= \ERI{pq}{rs}
+ 2 \sum_m \violet{\ERI{pq}{m}^{\la}} \violet{\ERI{rs}{m}^{\la}}
\qty[ \frac{1}{\omega - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\omega + \orange{\Om{m}{\la,\RPA}} - i \eta} ]
\end{equation}
\begin{equation}
\violet{\ERI{pq}{m}^{\la}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\la,\RPA}+\bY{m}{\la,\RPA}})_{ia}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{ACFDT in a computer}
\begin{block}{ACFDT correlation energy from BSE}
\begin{algorithmic}
\Procedure{ACFDT for BSE}{}
\State Compute $GW$ quasiparticle energies $\blue{\beGW}$ and interaction kernel $\bK{}{}$
\State Get Gauss-Legendre weights and roots $\{\purple{w_k},\violet{\lambda_k}\}_{1\le k \le K}$
\State $\green{\Ec} \gets 0$
\For{$k=1,\ldots,K$}
\State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}(\omega = 0)$
\State Perform BSE calculation at $\la = \violet{\lambda_k}$ to get $\bX{}{\violet{\lambda_k}}$ and $\bY{}{\violet{\lambda_k}}$
\Comment{\alert{This is a $\order*{N^6}$ step done many times!}}
\State Form two-particle density matrix $\bP{}{\violet{\lambda_k}}$
\State $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$
\EndFor
\EndProcedure
\end{algorithmic}
\end{block}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig/TOC_BSE}
\\
\pub{Loos et al. JPCL 11 (2020) 3536}
\end{center}
\end{frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin{frame}{Useful papers/programs}
\begin{itemize}
\item \red{mol$GW$:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
\bigskip
\item \green{Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528
\bigskip
\item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
\bigskip
\item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
\bigskip
\item \orange{Reviews \& Books:}
\begin{itemize}
\item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344
\item Onida et al. Rev. Mod. Phys. 74 (2002) 601
\item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022
\item Golze et al. Front. Chem. 7 (2019) 377
\item Blase et al. JPCL 11 (2020) 7371
\item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press)
\end{itemize}
\bigskip
\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}
%-----------------------------------------------------
\end{document}
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