diff --git a/2021/Lecture_2/ISTPC_Loos_2.tex b/2021/Lecture_2/ISTPC_Loos_2.tex
index 66bd371..6f30430 100644
--- a/2021/Lecture_2/ISTPC_Loos_2.tex
+++ b/2021/Lecture_2/ISTPC_Loos_2.tex
@@ -62,6 +62,7 @@
 \newcommand{\GOF}{$G_0F2$}			
 \newcommand{\GF}{$GF2$}
 \newcommand{\KS}{\text{KS}}
+\renewcommand{\HF}{\text{HF}}
 \newcommand{\RPA}{\text{RPA}}
 \newcommand{\RPAx}{\text{RPAx}}
 \newcommand{\BSE}{\text{BSE}}
@@ -86,14 +87,14 @@
 \newcommand{\EcMP}{E_c^\text{MP2}}
 \newcommand{\EcGF}{E_c^\text{\GF}}
 \newcommand{\EcGOF}{E_c^\text{\GOF}}
-\newcommand{\Egap}{E_\text{gap}}
+\newcommand{\Eg}[1]{E_\text{g}^{#1}}
 \newcommand{\IP}{\text{IP}}
 \newcommand{\EA}{\text{EA}}
 
 % orbital energies
 \newcommand{\nSat}[1]{N_{#1}^\text{sat}}
 \newcommand{\eSat}[2]{\epsilon_{#1,#2}}
-\newcommand{\e}[1]{\epsilon_{#1}}
+\newcommand{\e}[2]{\epsilon_{#1}^{#2}}
 \newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
 \newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
 \newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
@@ -106,8 +107,8 @@
 \newcommand{\deHF}[1]{\Delta\epsilon^\text{HF}_{#1}}
 \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{\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}}
@@ -173,6 +174,7 @@
 \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}}
@@ -201,6 +203,7 @@
 \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}{\bm{c}}
 \newcommand{\bX}[2]{\bm{X}_{#1}^{#2}}
 \newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}}
 \newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}}
@@ -258,14 +261,14 @@ decoration={snake,
 %-----------------------------------------------------
 \begin{frame}{Today's program}
 	\begin{itemize}
-		\item Charged excitations: 
+		\item \textbf{Charged excitations}
 			\begin{itemize}
 				\item One-shot $GW$ (\GOWO)
-				\item Partially self-consistent $GW$ (\evGW)
-				\item Self-consistent $GW$ (\qsGW)
-				\item $GW$ vs GF
+				\item Partially self-consistent eigenvalue $GW$ (\evGW)
+				\item Quasiparticle self-consistent $GW$ (\qsGW)
 			\end{itemize}
-		\item Neutral excitations
+		\bigskip
+		\item \textbf{Neutral excitations}
 			\begin{itemize}
 				\item Configuration interaction with singles (CIS)
 				\item Time-dependent Hartree-Fock (TDHF)
@@ -273,7 +276,8 @@ decoration={snake,
 				\item Time-dependent density-functional theory (TDDFT)
 				\item Bethe-Salpeter equation (BSE) formalism
 			\end{itemize}
-		\item Total energies
+		\bigskip
+		\item \textbf{Total energies}
 			\begin{itemize}
 				\item Plasmon formula
 				\item Galitski-Migdal formulation 
@@ -283,6 +287,67 @@ decoration={snake,
 \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 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}
+			\bigskip
+			\item $m$ indexes \purple{the $OV$ single excitations} ($i \to a$)
+		\end{itemize}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Useful papers}
+	\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. 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{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665
+	\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}
@@ -313,37 +378,125 @@ decoration={snake,
 \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}
+		\\
+		\bigskip
+		\pub{Blase et al. JPCL 11 (2020) 7371}
 	\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}
+\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}
 %-----------------------------------------------------
 
@@ -352,8 +505,8 @@ decoration={snake,
 	\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}	
+		= \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}{Non-interacting polarizability}
@@ -361,12 +514,10 @@ decoration={snake,
 			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{block}{Dielectric function and dynamically-screened Coulomb potential}
 		\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}
@@ -375,7 +526,7 @@ decoration={snake,
 %-----------------------------------------------------
 
 %-----------------------------------------------------
-\begin{frame}{Dynamical screening in a basis}
+\begin{frame}{Dynamical screening in the orbital basis}
 	\begin{block}{Spectral representation of $W$}
 		\begin{equation}
 		\begin{split}
@@ -406,35 +557,35 @@ decoration={snake,
 	\begin{block}{Direct RPA calculation (pseudo-hermitian linear problem)}
 		\begin{equation}
 			\begin{pmatrix}
-				\bA{}{}		&	\bB{}{}	\\
-				-\bB{}{}	&	-\bA{}{}	\\
+				\bA{}{\RPA}		&	\bB{}{\RPA}	\\
+				-\bB{}{\RPA}	&	-\bA{}{\RPA}	\\
 			\end{pmatrix}
 			\cdot
 			\begin{pmatrix}
-				\orange{\bX{m}{}}	\\
-				\orange{\bY{m}{}}	\\
+				\orange{\bX{m}{\RPA}}	\\
+				\orange{\bY{m}{\RPA}}	\\
 			\end{pmatrix}
 			=
 			\orange{\Om{m}{}}
 			\begin{pmatrix}
-				\orange{\bX{m}{}}	\\
-				\orange{\bY{m}{}}	\\
+				\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}
+			\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}{}
+			(\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{m}{} + \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}{}
+			(\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)}
@@ -463,8 +614,8 @@ decoration={snake,
 	\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}			 
+			= 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}
@@ -488,26 +639,67 @@ decoration={snake,
 	\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} ]
+		    \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}
-	\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}
+			\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 \& Loos, 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}{Linearized {\GOWO}~subroutine}
+	\begin{block}{}
 		\begin{algorithmic}
-			\Procedure{{\GOWO}lin}{}
+			\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 $\orange{\bB{}{\RPA}}$
+				\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 $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ 
 				\Comment{\alert{This is a $\order*{N^6}$ step!}}
 				\State Form screened ERIs $\violet{\ERI{pq}{m}}$
@@ -519,17 +711,31 @@ decoration={snake,
 			\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}{Graphical {\GOWO}~subroutine}
+	\begin{block}{}
 		\begin{algorithmic}
-			\Procedure{{\GOWO}graph}{}
+			\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 $\orange{\bB{}{\RPA}}$
+				\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 $\orange{\bX{m}{\RPA}+\bY{m}{\RPA}}$ 
 				\Comment{\alert{This is a $\order*{N^6}$ step!}}
 				\State Form screened ERIs $\violet{\ERI{pq}{m}}$
@@ -544,17 +750,24 @@ decoration={snake,
 %-----------------------------------------------------
 
 %-----------------------------------------------------
-\begin{frame}{Partially self-consistent eigenvalue $GW$}
-	\begin{block}{{\evGW} subroutine}
+\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{partially self-consistent {\evGW}}{}
+			\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 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{$\max{\abs{\bDelta}} < \tau$}
+					\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 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 $\blue{\eGnWn{p}{n}}$
@@ -568,16 +781,28 @@ decoration={snake,
 \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}{{\qsGW} subroutine}
+	\begin{block}{}
 		\begin{algorithmic}
-			\Procedure{partially self-consistent {\qsGW}}{}
+			\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: $\ERI{\mu\nu}{\lambda\sigma} \stackrel{\blue{\bcGnWn{n-1}}}{\rightarrow}  \ERI{pq}{rs}$
-					\State Construct RPA matrices $\orange{\bA{}{\RPA}}$ and $\orange{\bB{}{\RPA}}$
+					\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 Form screened ERIs $\violet{\ERI{pq}{m}}$
 					\State Evaluate $\red{\bSigC}(\blue{\beGnWn{n-1}})$ and form 
@@ -593,6 +818,104 @@ decoration={snake,
 \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}{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{center}
+			\includegraphics[width=0.4\textwidth]{fig/dyn}
+			\\
+			\bigskip
+			\pub{Authier \& Loos, JCP 153 (2020) 184105} [see also \pub{Romaniello et al, JCP 130 (2009) 044108}]
+		\end{center}
+	\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{equation}
+			\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
+		\end{equation}
+		\begin{equation}
+			\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{equation}
+		\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*}
@@ -619,7 +942,7 @@ decoration={snake,
 	%
 	\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}
+		    \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}
 			\\
@@ -629,17 +952,67 @@ decoration={snake,
 	%		
 	\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}
+		    \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}{TDHF and CIS: removing the correlation part}
+
+%-----------------------------------------------------
+\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 BSE 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}
@@ -665,7 +1038,7 @@ decoration={snake,
 	%
 	\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)}
+		    \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}
@@ -680,27 +1053,57 @@ decoration={snake,
 	\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} 
+		    = \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 $\bA{}{}$ matrix at a given level of theory
+				\State Compute $\red{\bA{}{}}$ matrix at a given level of theory
 				\If{$\TDA$} 
-					\State Diagonalize $\bA{}{}$ to get $\Om{m}{\TDA}$ and $\bX{m}{\TDA}$
+					\State Diagonalize $\red{\bA{}{}}$ to get $\highlight{\Om{m}{\TDA}}$ 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 $(\bX{}{} + \bY{}{})_m = \Om{m}{-1/2} (\bA{}{} - \bB{}{})^{+1/2} \cdot \bZ{m}{}$
+					\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}
@@ -708,10 +1111,86 @@ decoration={snake,
 \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}{Correlation energy}
+\begin{frame}{Properties}
+	\begin{block}{Oscillator strength (length gauge)}
+		\begin{equation}
+			\boxed{f_m = \frac{2}{3} \Om{m}{} \qty[ (\mu_m^x)^2 + (\mu_m^y)^2 + (\mu_m^z)^2 ]}
+		\end{equation}
+	\end{block}
+	\begin{block}{Transition dipole}
+		\begin{equation}
+			\boxed{\mu_m^x = \sum_{ia} (i|x|a) (\bX{m}{} + \bY{m}{})_{ia}}
+			\qquad
+			(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{\expval{\hat{S}^2}_m = \expval{\hat{S}^2}_0 + \underbrace{\Delta \expval{\hat{S}^2}_m}_{\text{\pub{JCP 134101 (2011) 134}}}}
+			\qquad
+			\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}
+	\begin{block}{Spin-flip formalism}
+		\begin{center}
+			\includegraphics[width=0.5\textwidth]{fig/SFBSE}
+			\\
+			\bigskip
+			\pub{Monino \& Loos, JCTC 17 (2021) 2852}
+		\end{center}
+	\end{block}
+\end{frame}
+%-----------------------------------------------------
+
+%-----------------------------------------------------
+\begin{frame}{Correlation energy at the $GW$ level}
 	\begin{block}{RPA correlation energy: plasmon formula}
 		\begin{equation*}
 			\label{eq:Ec-RPA}
@@ -725,7 +1204,7 @@ decoration={snake,
 			\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}}}
+			= 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}
@@ -824,10 +1303,17 @@ decoration={snake,
 
 	\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}}
+			\A{ia,jb}{\lambda,\RPAx} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \lambda \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]
 			\qquad 
 			\B{ia,jb}{\lambda,\RPAx} = \lambda \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
 		\end{equation}
+		\begin{equation}
+			\boxed{
+				\Ec^\RPAx 
+				= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
+				\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPAx} - \Tr(\bA{}{\RPAx}) ] 
+			}
+		\end{equation}
 	\end{block}
 	
 \end{frame}
@@ -838,10 +1324,18 @@ decoration={snake,
 \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}}
+			\A{ia,jb}{\lambda,\BSE} = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \lambda \qty[2 \ERI{ia}{bj} - W_{ij,ab}^{\lambda}(\omega = 0) ]
 			\qquad 
 			\B{ia,jb}{\lambda,\BSE} = \lambda \qty[2 \ERI{ia}{jb} - W_{ib,ja}^{\lambda}(\omega = 0)]
 		\end{equation}
+		\begin{equation}
+			\boxed{
+				\Ec^\BSE 
+				= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
+				\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \Om{m}{\BSE} - \Tr(\bA{}{\BSE}) ] 
+			}
+		\end{equation}
+
 	\end{block}
 	\begin{block}{$\lambda$-dependent screening}
 		\begin{equation}
@@ -857,66 +1351,31 @@ decoration={snake,
 \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}	
+\begin{frame}{ACFDT in a computer}
+	\begin{block}{ACFDT correlation energy from BSE}
+		\begin{algorithmic}
+			\Procedure{ACFDT for BSE}{}
+				\State Compute $GW$ quasiparticle energies $\beGW$ and interaction kernel $\bK{}{}$
+				\State Get Gauss-Legendre weights and roots $\{w_k,\lambda_k\}_{1\le k \le N_\text{grid}}$
+				\State $\Ec \gets 0$
+				\For{$k=1,\ldots,N_\text{grid}$}
+					\State Compute $W^{\lambda_k}$ 
+					\State Perform BSE calculation at $\lambda = \lambda_k$ to get $\bX{}{\lambda_k}$ and $\bY{}{\lambda_k}$
+					\State Form two-particle density matrix $\bP{}{\lambda_k}$
+					\State $\Ec \gets \Ec + w_k \Tr( \bK{}{} \bP{}{\lambda_k})$
+				\EndFor
+			\EndProcedure
+		\end{algorithmic}
+	\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}
+\begin{frame}
+	\begin{center}
+		\includegraphics[width=0.8\textwidth]{fig/TOC_BSE}
+	\end{center}
 \end{frame}
 %-----------------------------------------------------
 
diff --git a/2021/Lecture_2/fig/0-0.pdf b/2021/Lecture_2/fig/0-0.pdf
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index 0000000..1e03dd0
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index 0000000..5833f25
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diff --git a/2021/Lecture_2/fig/Jablonski.pdf b/2021/Lecture_2/fig/Jablonski.pdf
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diff --git a/2021/Lecture_2/fig/Jablonski.tex b/2021/Lecture_2/fig/Jablonski.tex
new file mode 100644
index 0000000..9b5805a
--- /dev/null
+++ b/2021/Lecture_2/fig/Jablonski.tex
@@ -0,0 +1,112 @@
+\documentclass{standalone}
+%\usepackage[top=3cm,left=0cm,right=0cm,bottom=3cm]{geometry}
+\usepackage{mathtools,physics,bm,xcolor}
+
+\usepackage{tikz}
+% shadows only for title
+\usetikzlibrary{decorations.pathmorphing,shadows} 
+
+
+\pagestyle{empty}
+
+\begin{document}
+
+% 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}
+
+    % 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] {\large 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] {\large $T_1$};
+    \foreach \i in {1,2,...,6} {
+        \path[vib] (11,4 + \i*0.2) -- ++ (3,0);
+    }
+
+    % S1
+    \path[elec] (4,5) node[anchor=south east] {\large $S_1$} -- ++ (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] {\large $S_2$} -- ++ (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] {\large Absorption};
+
+    % fluo
+    \path[trans,draw=rouge](7,5) -- ++(0,-4.4)
+        node[rotate=90,pos=0.5,text=rouge,yshift=-3mm] {\large Fluorescence};
+
+    % phosphorescence
+    \path[trans,draw=mauve] (13,4) -- ++(0,-3.4)
+        node[rotate=90,pos=0.5,text=mauve,yshift=-3mm] {\large Phosphorescence};
+
+    % Conversion interne
+    \path[transCI] (4,5) -- ++(-1.9,0) node[below,pos=0.5,text=vert] {\large IC};
+    \path[transCI] (6,8) -- ++(1.3,0)  node[above,pos=0.5,text=vert]  {\large IC};
+
+    % Croisement intersysteme
+    \path[transCS] (9,5)  -- ++(2,0)    node[below,pos=0.5,text=violet] {\large ISC};
+    \path[transCS] (11,4) -- ++(-2.5,0) node[below,pos=0.5,text=violet] {\large ISC};
+
+    % relaxation vib
+    \path[relax] (5.5,8.8) -- ++(0,-0.8) node[rv] {\textbf{VR}};
+    \path[relax] (8,8)     -- ++(0,-3)   node[rv] {\textbf{VR}};
+    \path[relax] (1,5)     -- ++(0,-5)   node[rv] {\textbf{VR}};
+    \path[relax] (11.5,5)  -- ++(0,-1)   node[rv] {\textbf{VR}};
+
+\end{tikzpicture}
+
+\end{center}
+
+
+\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{document}
\ No newline at end of file
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diff --git a/2021/Lecture_2/fig/TOC_BSE.pdf b/2021/Lecture_2/fig/TOC_BSE.pdf
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diff --git a/2021/Lecture_2/fig/evGW.png b/2021/Lecture_2/fig/evGW.png
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diff --git a/2021/Lecture_2/fig/photochemistry.pdf b/2021/Lecture_2/fig/photochemistry.pdf
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diff --git a/2021/Lecture_2/fig/photochemistry.tex b/2021/Lecture_2/fig/photochemistry.tex
new file mode 100644
index 0000000..bd40ba2
--- /dev/null
+++ b/2021/Lecture_2/fig/photochemistry.tex
@@ -0,0 +1,120 @@
+\documentclass{standalone}
+\usepackage{tikz}
+\usepackage{siunitx}
+\usepackage{mathtools,physics,bm,xcolor}
+%\usetikzlibrary{arrows.meta}
+%\tikzset{myarr/.style={
+%        {Triangle[width=4pt, length=4pt]}-{Triangle[width=4pt, length=4pt]},
+%}}
+\usepackage{tikz}
+    \usetikzlibrary{intersections}
+\usepackage{pgfplots}
+    \usepgfplotslibrary{fillbetween}
+  \definecolor{darkgreen}{RGB}{0, 180, 0}
+  
+\begin{document}  
+
+\begin{tikzpicture}[scale=2]
+
+	% x axis
+	\draw [->] (0,0) -- (0,4);
+	\node [left] at (0,4) {Energy};
+	% x axis
+	\draw [->] (0,0) -- (4,0);
+	\node [below] at (4,0) {Nuclear coordinates};
+	% absorption 
+	\draw [thick, blue, <->]  (1,1) -- (1,3);
+	% emission 
+	\draw [thick, red, <->]  (2,1.5) node[along]{$E^{fluo}$} -- (2,2.5);
+	% adiabatic 
+	\draw [thick, darkgreen, <->]  (2.5,1) node[right]{$E^\text{adia}$} -- (2.5,2.5);
+	% 0-0 
+	\draw [thick, blue, <->]  (3,1.1) node[along]{$E^{0-0}$} -- (3,2.7);
+
+%  \node [right] at (2.65,-2) {$\theta$};
+%  \draw [thick] (-2,-2.05) node[below]{\SI{70}{\degree}} -- (-2,-1.95);
+%  \draw [thick] (0,-2.05) node[below]{\SI{90}{\degree}} -- (0,-1.95);
+%  \draw [thick] (2,-2.05) node[below]{\SI{110}{\degree}} -- (2,-1.95);
+
+  % Theta = 75 rectangular
+%  \node[draw,circle,minimum size=85pt,opacity=0.3,thick] (a) at (-2,-1) {};
+%  \node [above] at (-2,-0.2) {$D_{2h}$};
+%  \draw [thick] (a.35) -- (a.145) -- (a.-145) -- (a.-35) -- (a.35);
+%  \node[fill,draw,circle, label=above right:H, minimum size=3pt,inner sep=0pt] at (a.35) {};
+%  \node[fill,draw,circle, label=below right:H, minimum size=3pt,inner sep=0pt] at (a.-35) {};
+%  \node[fill,draw,circle, label=below left:H, minimum size=3pt,inner sep=0pt] at (a.-145) {};
+%  \node[fill,draw,circle, label=above left:H, minimum size=3pt,inner sep=0pt] at (a.145) {};
+%  \draw [dashed,thick] (-2,-1) -- (a.35);
+%  \draw [dashed,thick] (-2,-1) -- (a.-35);
+%  \draw [thick] (-1.8,-1) arc [start angle=0,end angle=35,radius=0.2];
+%  \draw [thick] (-1.8,-1) arc [start angle=0,end angle=-35,radius=0.2];
+%  \node [right] at (-1.8,-1) {$\theta$};
+  
+   % Theta = 90 square
+%  \node[draw,circle,minimum size=85pt,opacity=0.3,thick] (b) at (0,-1) {};
+%  \node [above] at (0,-0.2) {$D_{4h}$};
+%  \draw [thick] (b.45) -- (b.135) -- (b.-135) -- (b.-45) -- (b.45);
+%  \node[fill,draw,circle, label=above right:H, minimum size=3pt,inner sep=0pt] at (b.45) {};
+%  \node[fill,draw,circle, label=below right:H, minimum size=3pt,inner sep=0pt] at (b.-45) {};
+%  \node[fill,draw,circle, label=below left:H, minimum size=3pt,inner sep=0pt] at (b.-135) {};
+%  \node[fill,draw,circle, label=above left:H, minimum size=3pt,inner sep=0pt] at (b.135) {};
+%  \draw [thick,myarr,dashed] (b.-135) -- (b.45);
+%  \node [above left] at (0,-1) {$d$};
+
+   % Theta = 105 rectangular
+%  \node[draw,circle,minimum size=85pt,opacity=0.3,thick] (c) at (2,-1) {};
+%  \node [above] at (2,-0.2) {$D_{2h}$};
+%  \draw [thick] (c.55) -- (c.125) -- (c.-125) -- (c.-55) -- (c.55);
+%  \node[fill,draw,circle, label=above right:H, minimum size=3pt,inner sep=0pt] at (c.55) {};
+%  \node[fill,draw,circle, label=below right:H, minimum size=3pt,inner sep=0pt] at (c.-55) {};
+%  \node[fill,draw,circle, label=below left:H, minimum size=3pt,inner sep=0pt] at (c.-125) {};
+%  \node[fill,draw,circle, label=above left:H, minimum size=3pt,inner sep=0pt] at (c.125) {};
+%  \draw [dashed,thick] (2,-1) -- (c.55);
+%  \draw [dashed,thick] (2,-1) -- (c.-55);
+%  \draw [thick] (2.2,-1) arc [start angle=0,end angle=55,radius=0.2];
+%  \draw [thick] (2.2,-1) arc [start angle=0,end angle=-55,radius=0.2];
+%  \node [right] at (2.2,-1) {$\theta$};
+  %%%%%%% Define Potential Function %%%%%%%
+%    \pgfmathsetmacro{\DeGS}{1}
+%    \pgfmathsetmacro{\RoGS}{1}
+%    \pgfmathsetmacro{\alphaGS}{1}
+%    \pgfmathsetmacro{\DeES}{1.2}
+%    \pgfmathsetmacro{\RoES}{1.2}
+%    \pgfmathsetmacro{\alphaES}{1.2}
+%    \pgfmathdeclarefunction{GS}{1}{%
+%        \pgfmathparse{%
+%            \DeGS*((1-exp(-\alphaGS*(#1-\RoGS)))^2-1)%
+%        }%
+%    }%
+%    \pgfmathdeclarefunction{ES}{1}{%
+%        \pgfmathparse{%
+%            \DeES*((1-exp(-\alphaES*(#1-\RoES)))^2-1)%
+%        }%
+%    }%
+%%%%%%%% Energy Levels %%%%%%%
+%    \pgfmathdeclarefunction{energyGS}{1}{%
+%        \pgfmathparse{%
+%            -\DeGS+(#1+.5) - (#1+.5)^2/(1*\DeGS)
+%        }%
+%    }%
+%    \pgfmathdeclarefunction{energyES}{1}{%
+%        \pgfmathparse{%
+%            -\DeES+(#1+.5) - (#1+.5)^2/(1*\DeES)
+%        }%
+%    }%
+%
+%    \begin{axis}[
+%            axis lines=none,
+%            smooth,
+%            no markers,
+%            domain=0:4,
+%            xmax=10,
+%            ymax=10,
+%            scale=1
+%        ]
+%        \addplot [black, samples=50, name path global=GSCurve] {GS(x)};
+%        \addplot [black, samples=50, name path global=ESCurve] {ES(x)};
+%    \end{axis}
+\end{tikzpicture}
+
+\end{document}
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