Lecture 2 2022

2022/GFQC/fig/BSE-GW.tex

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+\documentclass{standalone}
+\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,multimedia,hyperref,tabularx,amscd,pgfgantt,mhchem,physics}
+\usetikzlibrary{shapes.gates.logic.US,trees,positioning,arrows}
+
+\usepackage{tgchorus}
+\usepackage[T1]{fontenc}
+
+\begin{document}
+
+\begin{tikzpicture}
+	\begin{scope}[very thick,
+		node distance=5cm,on grid,>=stealth',
+		theo1/.style={rectangle,draw,fill=red!20},
+		theo2/.style={rectangle,draw,fill=orange!20},
+		theo3/.style={rectangle,draw,fill=green!40},
+		exp1/.style={rectangle,draw,fill=cyan!40},
+		exp2/.style={rectangle,draw,fill=violet!40}]
+		
+		\node [theo1, text width=7cm, align=center]	(KS)		
+		{\textbf{\LARGE Kohn-Sham DFT}
+			$$
+				\qty[ -\frac{\nabla^2}{2} + v_\text{ext} + V^{\text{Hxc}} ] \phi_p^{\text{KS}} = \varepsilon^{\text{KS}}_p \phi_p^{\text{KS}}
+			$$	
+		};									
+
+		\node [theo2, text width=7cm, align=center]	(GW)	[below=of KS, yshift=2cm]				
+		{\textbf{\LARGE $GW$ approximation} 
+			$$   
+				\varepsilon_p^{GW} = \varepsilon_p^{\text{KS}} +
+				\mel{\phi_p^{\text{KS}}}{\Sigma^{GW}(\varepsilon_p^{GW}) - V^{\text{xc}}}{\phi_p^{\text{KS}}}
+			$$
+
+		};												
+
+		\node [theo3, text width=7cm, align=center]	(BSE)	[below=of GW, yshift=2cm]									
+		{\textbf{\LARGE Bethe-Salpeter equation}
+			$$
+				\begin{pmatrix}
+					\bm{A} & \bm{B}  \\
+					-\bm{B}^*  & -\bm{A}^{*}
+				\end{pmatrix}
+				\begin{pmatrix}
+					\bm{X}_m  \\
+					\bm{Y}_m
+				\end{pmatrix}
+				=
+				\Omega_{m}
+				\begin{pmatrix}
+					\bm{X}_m  \\
+					\bm{Y}_m
+				\end{pmatrix}
+			$$
+		};
+								
+
+		\node [exp1, align=center] 					(photo)	[right=of GW, xshift=3cm]										
+		{\LARGE (Inverse) \\ \LARGE  photoemission \\ \LARGE  spectroscopy};
+
+		\node [exp2, align=center] 					(abs)	[right=of BSE, xshift=3cm]			
+		{\LARGE  Optical \\ \LARGE spectroscopy};
+		
+		
+		\path 
+		(KS) 			edge [->,color=black] 			node [right,black] {\LARGE Fundamental gap}		(GW)
+		(GW) 			edge [->,color=black] 			node [right,black] {\LARGE Excitonic effect}	(BSE)
+		(photo) 		edge [<->,color=black] 			node [above,black] {Ionization potentials}	node [below,black] {Electron affinities}	(GW)
+		(abs) 			edge [<->,color=black] 			node [above,black] {Optical excitations}	(BSE)
+		;
+
+		
+	\end{scope}
+\end{tikzpicture}
+
+\end{document}  

2022/GFQC/fig/Jablonski.tex

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+\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}

2022/GFQC/fig/photochemistry.tex

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+\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}