From 5a5a288dd105fdac27b1f766e298d7aef189d4f2 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Wed, 2 Jun 2021 11:03:57 +0200 Subject: [PATCH] saving work --- 2021/Lecture_2/ISTPC_Loos_2.tex | 741 +++++++++++++++++++++----- 2021/Lecture_2/fig/0-0.pdf | Bin 0 -> 23158 bytes 2021/Lecture_2/fig/BSE-GW.pdf | Bin 0 -> 94258 bytes 2021/Lecture_2/fig/BSE1.png | Bin 0 -> 223553 bytes 2021/Lecture_2/fig/BSE3.png | Bin 0 -> 245009 bytes 2021/Lecture_2/fig/G0W0.png | Bin 0 -> 469511 bytes 2021/Lecture_2/fig/GWSph.pdf | Bin 0 -> 64888 bytes 2021/Lecture_2/fig/Jablonski.pdf | Bin 0 -> 78243 bytes 2021/Lecture_2/fig/Jablonski.tex | 112 ++++ 2021/Lecture_2/fig/QP.pdf | Bin 0 -> 38718 bytes 2021/Lecture_2/fig/SFBSE.pdf | Bin 0 -> 20967 bytes 2021/Lecture_2/fig/TOC_BSE.pdf | Bin 0 -> 59198 bytes 2021/Lecture_2/fig/dyn.pdf | Bin 0 -> 48894 bytes 2021/Lecture_2/fig/evGW.png | Bin 0 -> 473904 bytes 2021/Lecture_2/fig/photochemistry.pdf | Bin 0 -> 36113 bytes 2021/Lecture_2/fig/photochemistry.tex | 120 +++++ 2021/Lecture_2/fig/qsGW1.png | Bin 0 -> 462919 bytes 2021/Lecture_2/fig/qsGW2.png | Bin 0 -> 156122 bytes 18 files changed, 832 insertions(+), 141 deletions(-) create mode 100755 2021/Lecture_2/fig/0-0.pdf create mode 100644 2021/Lecture_2/fig/BSE-GW.pdf create mode 100644 2021/Lecture_2/fig/BSE1.png create mode 100644 2021/Lecture_2/fig/BSE3.png create mode 100644 2021/Lecture_2/fig/G0W0.png create mode 100644 2021/Lecture_2/fig/GWSph.pdf create mode 100644 2021/Lecture_2/fig/Jablonski.pdf create mode 100644 2021/Lecture_2/fig/Jablonski.tex create mode 100644 2021/Lecture_2/fig/QP.pdf create mode 100644 2021/Lecture_2/fig/SFBSE.pdf create mode 100644 2021/Lecture_2/fig/TOC_BSE.pdf create mode 100644 2021/Lecture_2/fig/dyn.pdf create mode 100644 2021/Lecture_2/fig/evGW.png create mode 100644 2021/Lecture_2/fig/photochemistry.pdf create mode 100644 2021/Lecture_2/fig/photochemistry.tex create mode 100644 2021/Lecture_2/fig/qsGW1.png create mode 100644 2021/Lecture_2/fig/qsGW2.png 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 new file mode 100755 index 0000000000000000000000000000000000000000..6d83727056dc3d83339e909e5b0451bca36ae498 GIT binary patch literal 23158 zcmbrk19W9ivo{=bV%xTD+nLz5&6(J?Z6_1kwryu(Ctv12&-32ru6y70eQSMd?{ju{ 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+\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 diff --git a/2021/Lecture_2/fig/QP.pdf b/2021/Lecture_2/fig/QP.pdf new file mode 100644 index 0000000000000000000000000000000000000000..57c9da7d4132a46b614874fd9c0f36f62dd04904 GIT binary patch literal 38718 zcmV)7K*zr&P((&8F)lR4?5av(28Y+-a|L}g=dWMv9IJ_>Vma%Ev{3V58Xy-TYmO?DnS&tI{J_JJ;C+#^9~ zU@!)O2X;4jAT$u$mK#xJuszVtcD$L9aaplqUDLl z_y4p!K0be3|MR-PwU7AkKmUi1zxk*80QpA3qi{zSa5h{p0u7-~3a;gUbEwsn18s{o9@T+pwnhCuJjO z`=37k=i3m_N4nETO3%mt`fnfq<>P<6{g}`D|LO-ibl>kE-+r|E{_9b1GhULurI!Cp 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(4,0); + \node [below] at (4,0) {Nuclear coordinates}; + % absorption + \draw [thick, blue, <->] (1,1) -- (1,3); + % emission + \draw [thick, red, <->] (2,1.5) node[along]{$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$}; + %%%%%%% Define Potential Function %%%%%%% +% \pgfmathsetmacro{\DeGS}{1} +% \pgfmathsetmacro{\RoGS}{1} +% \pgfmathsetmacro{\alphaGS}{1} +% \pgfmathsetmacro{\DeES}{1.2} +% \pgfmathsetmacro{\RoES}{1.2} +% \pgfmathsetmacro{\alphaES}{1.2} +% \pgfmathdeclarefunction{GS}{1}{% +% \pgfmathparse{% +% \DeGS*((1-exp(-\alphaGS*(#1-\RoGS)))^2-1)% +% }% +% }% +% \pgfmathdeclarefunction{ES}{1}{% +% \pgfmathparse{% +% \DeES*((1-exp(-\alphaES*(#1-\RoES)))^2-1)% +% }% +% }% +%%%%%%%% Energy Levels %%%%%%% +% \pgfmathdeclarefunction{energyGS}{1}{% +% \pgfmathparse{% +% -\DeGS+(#1+.5) - (#1+.5)^2/(1*\DeGS) +% }% +% }% +% \pgfmathdeclarefunction{energyES}{1}{% +% \pgfmathparse{% +% -\DeES+(#1+.5) - (#1+.5)^2/(1*\DeES) +% }% +% }% +% +% \begin{axis}[ +% axis lines=none, +% smooth, +% no markers, +% domain=0:4, +% xmax=10, +% ymax=10, +% scale=1 +% ] +% \addplot [black, samples=50, name path global=GSCurve] {GS(x)}; +% \addplot [black, samples=50, name path global=ESCurve] {ES(x)}; +% \end{axis} +\end{tikzpicture} + +\end{document} diff --git a/2021/Lecture_2/fig/qsGW1.png b/2021/Lecture_2/fig/qsGW1.png new file mode 100644 index 0000000000000000000000000000000000000000..ae72804610c6a2d8f715c92c29ba869e7a72e9ce GIT binary patch literal 462919 zcmb@s1yo$kvM4-QaCav-!Gi^NcXtWy1h*l$ySoP0;O-hUxJz(n@PQzYeBU|up8MXp z|GNKrZ?DzUJ>6YhU9xL;Radx@f+R8m0RjL3K$eygQvm>=fp3KY?%i9@&b!DZ007C@ zN>o%yT2z!w$;rXo%GL}3kP1&uh1E)w#`L?VFQaS_f-Vf&5Icg7!w`uiBU499j)TUK zL4p?^j3Fu6S{BEm_xez9cmzM9jaKVT>o(=2rNqsRipZ(FfF8ne%)%ae8_> zZrWW3ICKG-)$ys}`T?!-`2@&=I0QJN#gC!kSYShX@I9_z5b$Z3m^bRzDG3}hLB4bjS7Sgg13RDoE>fo}q zLGiX6hXdhv>8Jw{duTtoo$I}2fMv4?e}z;W^0TF2!088UnL@xJo)8LF*n`MOc%0dE9!tE-z@7RnU3WMYwX&kl zL4BXvk^RLXLcBxxwI*^foGz{mU^GWA?Sy1%H}WoGEr_-e*<77re+X%1B>A!&Y)4iv zJ`;x4=*z$_OF~jpo4QB;?s`yW5W^rUgdTA)JpTC6s2%#r9~@1#7+IKAC@e@m2tMr9 z3o7`@;(D%?DV0(91Gk-q9w~fr{49QN(CXB&j+R-Zs5!v4fL7+`_@E?i-$dD6tmGKN z4#vFWHa_Mg$(PvA4ZcL3(PJ+TGw5a>#2wY9u`OjyDS8)$DYxPPX|Sz 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