ISTPC/2021/Lecture_2/ISTPC_Loos_2.tex

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% methods
\newcommand{\evGW}{ev$GW$}	
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\newcommand{\scGW}{sc$GW$}	
\newcommand{\GOWO}{$G_0W_0$}	
\newcommand{\GOW}{$G_0W$}	
\newcommand{\GWO}{$GW_0$}	
\newcommand{\GW}{$GW$}		
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\newcommand{\GWSOSEX}{{\GW}+SOSEX}		
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% operators
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% energies
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% orbital energies
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
\newcommand{\eGW}[1]{\epsilon^\text{\GW}_{#1}}
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\newcommand{\deKS}[1]{\Delta\epsilon^\text{KS}_{#1}}
\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
\newcommand{\eHOMO}{\epsilon_\text{HOMO}}
\newcommand{\eLUMO}{\epsilon_\text{LUMO}}

\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}}

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\newcommand{\OmsCIS}[1]{{}^{1}\Omega^\text{CIS}_{#1}}
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\newcommand{\OmsBSE}[1]{{}^{1}\Omega^\text{BSE}_{#1}}

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% Matrices
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%	*************
%	* HEAD DATA *
%	*************
	\title[$GW$/BSE methods in chemistry]{
		$GW$/BSE methods in chemistry:
		Computational aspects
	} 
        \author[PF 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),\\
                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 Charged excitations: 
			\begin{itemize}
				\item One-shot $GW$ (\GOWO)
				\item Partially self-consistent $GW$ (\evGW)
				\item Self-consistent $GW$ (\qsGW)
				\item $GW$ vs GF
			\end{itemize}
		\item Neutral excitations
			\begin{itemize}
				\item Configuration interaction with singles (CIS)
				\item Time-dependent Hartree-Fock (TDHF)
				\item Random-phase approximation (RPA)
				\item Time-dependent density-functional theory (TDDFT)
				\item Bethe-Salpeter equation (BSE) formalism
			\end{itemize}
		\item Total energies
			\begin{itemize}
				\item Plasmon formula
				\item Galitski-Migdal formulation 
				\item Adiabatic connection fluctuation-dissipation theorem (ACFDT)
			\end{itemize}
	\end{itemize}
\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 neutral 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}{Fundamental and optical gaps}
	\begin{center}
		\includegraphics[width=\textwidth]{fig/gaps}
	\end{center}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{The MBPT chain of actions}
	\begin{center}
		\includegraphics[width=0.7\textwidth]{fig/BSE-GW}
	\end{center}
\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}{Green's function and dynamical screening}
	\begin{block}{One-body Green's function}
	\begin{equation}
		\blue{G}(\br_1,\br_2;\yo) 
		= \sum_i \frac{\MO{i}(\br_1) \MO{i}(\br_2)}{\yo - \e{i}{} - i\eta}	
		+ \sum_a \frac{\MO{a}(\br_1) \MO{a}(\br_2)}{\yo - \e{a}{} + i\eta}	
	\end{equation}
	\end{block}
	\begin{block}{Non-interacting 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}
	\end{block}
	\begin{block}{Dielectric function}
		\begin{equation}
			\epsilon(\br_1,\br_2;\omega) = \delta(\br_1 - \br_2) - \int \frac{P(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
		\end{equation}
	\end{block}
	\begin{block}{Dynamically-screened Coulomb potential}
		\begin{equation}
			\highlight{W}(\br_1,\br_2;\omega) =  \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
		\end{equation}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Dynamical screening in a 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}_{pq,rs}(\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 RPA calculation (pseudo-hermitian linear problem)}
		\begin{equation}
			\begin{pmatrix}
				\bA{}{}		&	\bB{}{}	\\
				-\bB{}{}	&	-\bA{}{}	\\
			\end{pmatrix}
			\cdot
			\begin{pmatrix}
				\orange{\bX{m}{}}	\\
				\orange{\bY{m}{}}	\\
			\end{pmatrix}
			=
			\orange{\Om{m}{}}
			\begin{pmatrix}
				\orange{\bX{m}{}}	\\
				\orange{\bY{m}{}}	\\
			\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{}{} + \bY{}{})_m = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{}
			\\
			(\bX{}{} - \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 \bX{m}{} = \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}
		    \blue{\eGW{p}} = \e{p}^{\KS} + \green{Z_{p}} [\red{\Sig{pp}{\xc}}(\e{p}^{\KS}) - V_{p}^{\xc} ]
		\end{equation}
	\end{block}
	\begin{block}{Renormalization factor or spectral weight}
		\begin{equation}
			\green{Z_{p}} = \qty[ 1 - \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \e{p}^{\KS}} ]^{-1}
			\qq{with} 0 \le \green{Z_{p}} \le 1
		\end{equation}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Perturbative {\GW} with linearized solution}
	\begin{block}{Linearized {\GOWO}~subroutine}
		\begin{algorithmic}
			\Procedure{{\GOWO}lin}{}
				\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 $\orange{\bB{}{\RPA}}$
				\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ 
				\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}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Perturbative {\GW} with graphical solution}
	\begin{block}{Graphical {\GOWO}~subroutine}
		\begin{algorithmic}
			\Procedure{{\GOWO}graph}{}
				\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 $\orange{\bB{}{\RPA}}$
				\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ 
				\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}{Partially self-consistent eigenvalue $GW$}
	\begin{block}{{\evGW} subroutine}
		\begin{algorithmic}
			\Procedure{partially self-consistent {\evGW}}{}
				\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 $\orange{\bB{}{\RPA}}$
				\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ 
				\State Form screened ERIs $\violet{\ERI{pq}{m}}$
				\State Set $\blue{\beGnWn{-1}} = \beKS$  and $n = 0$
				\While{$\max{\abs{\bDelta}} < \tau$}
					\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}{Quasiparticle self-consistent {\GW} (\qsGW)}
	\begin{block}{{\qsGW} subroutine}
		\begin{algorithmic}
			\Procedure{partially self-consistent {\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: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow}  \ERI{pq}{rs}$
					\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$
					\State Compute RPA eigenvalues $\orange{\Om{m}{\RPA}}$ and eigenvectors $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ 
					\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 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}{TD-DFT and BSE in practice: Casida-like equations}
	\begin{block}{Linear response problem}
		\begin{equation*}
		    \boxed{\begin{pmatrix}
				\red{\bA{}{}} & \orange{\bB{}{}}  
				\\
				\orange{-\bA{}{}}  & \red{-\bB{}{}}
			\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( \varepsilon_a^\text{\violet{KS}} - \varepsilon_i^\text{\violet{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( \varepsilon_a^{\green{GW}} - \varepsilon_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}{TDHF and CIS: removing the correlation part}
	\begin{block}{Linear response problem}
		\begin{equation*}
		    \boxed{\begin{pmatrix}
				\red{\bA{}{}} & \orange{\bB{}{}}  
				\\
				\orange{-\bA{}{}}  & \red{-\bB{}{}}
			\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( \varepsilon_a^\text{\green{HF}} - \varepsilon_i^\text{\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( \varepsilon_a^\text{\green{HF}} - \varepsilon_i^\text{\green{HF}} ) \delta_{ij} \delta_{ab} 
		    + 2 \blue{(ia|bj)} - \yellow{(ij|ba)}
		\end{equation}
	\end{block}
	%
\end{frame}

%-----------------------------------------------------
\begin{frame}{Linear response}
	\begin{block}{General linear response problem}
		\begin{algorithmic}
			\Procedure{Linear response}{}
				\State Compute $\bA{}{}$ matrix at a given level of theory
				\If{$\TDA$} 
					\State Diagonalize $\bA{}{}$ to get $\Om{m}{\TDA}$ and $\bX{m}{\TDA}$
				\Else
					\State Compute $\bB{}{}$ matrix at a given level of theory
					\State Diagonalize $\bA{}{} - \bB{}{}$ to form $(\bA{}{} - \bB{}{})^{1/2}$
					\State Form and diagonalize $(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2}$ 
					to get $\Om{m}{2}$ and $\bZ{m}{}$
					\State Compute $(\bX{}{} + \bY{}{})_m = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{}$
				\EndIf
			\EndProcedure
		\end{algorithmic}
	\end{block}
\end{frame}
%-----------------------------------------------------



%-----------------------------------------------------
\begin{frame}{Correlation energy}
	\begin{block}{RPA correlation energy: plasmon formula}
		\begin{equation*}
			\label{eq:Ec-RPA}
			\EcRPA 
			= \frac{1}{2} \qty[ \sum_{p} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ]
			= \frac{1}{2} \sum_{m} \qty( \Om{m}{\RPA} - \Om{m}{\TDA} )
		\end{equation*}
	\end{block}
	\begin{block}{Galitskii-Migdal functional}
		\begin{equation*}
			\label{eq:GM}
			\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}{\e{a} - \e{i} + \orange{\Om{m}{\RPA}}}
		\end{equation*}
	\end{block}
\end{frame}
%-----------------------------------------------------

%-----------------------------------------------------
\begin{frame}{Adiabatic connection fluctuation dissipation theorem (ACFDT)}
	\begin{block}{Adiabatic connection}
		\begin{equation}
			\boxed{
				\Ec^\text{ACFDT} 
				= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
				\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} w_k \Tr( \bK{}{} \bP{}{\lambda_k}) 
			}
		\end{equation}
	\end{block}
	\begin{block}{Interaction kernel}
		\begin{equation}
			\bK{}{} = 
			\begin{pmatrix}
				\btA{}{} & \btB{}{}
				\\
				\btB{}{}  & \btA{}{}
			\end{pmatrix}
			\qquad
			\tA{ia,jb}{} =  2\lambda\ERI{ia}{bj}
			\qquad
			\tB{ia,jb}{} =  2\lambda\ERI{ia}{jb}
		\end{equation}
	\end{block}
	\begin{block}{Correlation part of the two-particle density matrix}
		\begin{equation}
			\bP{}{\lambda} = 
			\begin{pmatrix}
				\bY{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bY{}{\lambda} \cdot \T{(\bX{}{\lambda})}
				\\
				\bX{}{\lambda} \cdot \T{(\bY{}{\lambda})}  & \bX{}{\lambda} \cdot \T{(\bX{}{\lambda})}
			\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}
		\begin{equation}
			\boxed{\int_a^b f(x) w(x) dx \approx \sum_k \underbrace{w_k}_{\text{weights}} f(\underbrace{x_k}_{\text{roots}})}
		\end{equation}
	\end{block}
	\begin{block}{Quadrature rules}
		\begin{center}
			\begin{tabular}{llll}
				\hline
				Interval $[a,b]$	&	Weight function $w(x)$								&	Orthogonal polynomials					&	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}
			 \\
			 \bigskip
			 \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}
			\A{ia,jb}{\lambda,\RPA} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\lambda\ERI{ia}{bj}
			\qquad 
			\B{ia,jb}{\lambda,\RPA} = 2\lambda\ERI{ia}{jb}
		\end{equation}
		\begin{equation}
			\boxed{
				\Ec^\RPA 
				= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
				= \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ] 
			}
		\end{equation}
	\end{block}

	\begin{block}{RPAx matrix elements}
		\begin{equation}
			\A{ia,jb}{\lambda,\RPAx} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \underbrace{\lambda \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]}_{\tA{ia,jb}{\lambda,\RPAx}}
			\qquad 
			\B{ia,jb}{\lambda,\RPAx} = \lambda \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
		\end{equation}
	\end{block}
	
\end{frame}
%-----------------------------------------------------
	
	
%-----------------------------------------------------
\begin{frame}{ACFDT at the BSE level}
	\begin{block}{BSE matrix elements}
		\begin{equation}
			\A{ia,jb}{\lambda,\BSE} = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \underbrace{\lambda \qty[2 \ERI{ia}{bj} - W_{ij,ab}^{\lambda}(\omega = 0) ]}_{\tA{ia,jb}{\lambda,\BSE}}
			\qquad 
			\B{ia,jb}{\lambda,\BSE} = \lambda \qty[2 \ERI{ia}{jb} - W_{ib,ja}^{\lambda}(\omega = 0)]
		\end{equation}
	\end{block}
	\begin{block}{$\lambda$-dependent screening}
		\begin{equation}
			\highlight{W}_{pq,rs}^{\lambda}(\yo) 
			= \ERI{pq}{rs}
			+ 2 \sum_m \violet{\ERI{pq}{m}^{\lambda}} \violet{\ERI{rs}{m}^{\lambda}}
			\qty[ \frac{1}{\yo - \orange{\Om{m}{\lambda,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\lambda,\RPA}} - i \eta} ]
		\end{equation}
		\begin{equation}
			\violet{\ERI{pq}{m}^{\lambda}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\lambda,\RPA}+\bY{m}{\lambda,\RPA}})_{ia}
		\end{equation}
	\end{block}
\end{frame}
%-----------------------------------------------------



%-----------------------------------------------------
\begin{frame}{The bridge between TD-DFT and BSE}
	\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{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}{Useful papers}
	\begin{itemize}
		\item \red{molGW:} 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. 86 (2012) 081102
		\bigskip
		\item \orange{Review:} 
			\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
			\end{itemize}
		\bigskip
		\item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665
	\end{itemize}
\end{frame}
%-----------------------------------------------------

\end{document}