OK with Lecture 2
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@ -237,7 +237,7 @@ decoration={snake,
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Computational aspects
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}
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\author[PF Loos (\url{https://www.irsamc.ups-tlse.fr/loos/})]{Pierre-Fran\c{c}ois LOOS}
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\date{Online ISTPC 2021 school --- April 27th, 2021}
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\date{Online ISTPC 2021 school --- June 8th, 2021}
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\institute[CNRS@LCPQ]{
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Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\
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Universit\'e de Toulouse, CNRS, UPS, Toulouse, France.
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@ -263,6 +263,7 @@ decoration={snake,
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Today's program}
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\begin{itemize}
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@ -271,20 +272,21 @@ decoration={snake,
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\item One-shot $GW$ (\GOWO)
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\item Partially self-consistent eigenvalue $GW$ (\evGW)
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\item Quasiparticle self-consistent $GW$ (\qsGW)
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\item Other self-energies (GF2, SOSEX, T-matrix, etc)
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\end{itemize}
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\bigskip
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\item \textbf{Neutral excitations}
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\begin{itemize}
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\item Configuration interaction with singles (CIS)
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\item Time-dependent Hartree-Fock (TDHF)
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\item Random-phase approximation (RPA)
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\item Configuration interaction with singles (CIS)
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\item Time-dependent Hartree-Fock (TDHF) or RPA with exchange (RPAx)
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\item Time-dependent density-functional theory (TDDFT)
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\item Bethe-Salpeter equation (BSE) formalism
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\end{itemize}
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\bigskip
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\item \textbf{Total energies}
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\begin{itemize}
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\item Plasmon formula
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\item Plasmon (or trace) formula
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\item Galitski-Migdal formulation
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\item Adiabatic connection fluctuation-dissipation theorem (ACFDT)
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\end{itemize}
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@ -292,6 +294,11 @@ decoration={snake,
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\section{Context}
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\begin{frame}
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\tableofcontents[currentsection]
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\end{frame}
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%-----------------------------------------------------
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\begin{frame}{Assumptions \& Notations}
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\begin{block}{Let's talk about notations}
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@ -506,6 +513,11 @@ decoration={snake,
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\section{Charged excitations}
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\begin{frame}
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\tableofcontents[currentsection]
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\end{frame}
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%-----------------------------------------------------
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\begin{frame}{Green's function and dynamical screening}
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\begin{block}{One-body Green's function}
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@ -892,6 +904,11 @@ decoration={snake,
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\end{frame}
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%-----------------------------------------------------
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%-----------------------------------------------------
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\section{Neutral excitations}
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\begin{frame}
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\tableofcontents[currentsection]
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\end{frame}
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%-----------------------------------------------------
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\begin{frame}{Dynamical vs static kernels}
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\begin{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}}
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@ -934,6 +951,7 @@ decoration={snake,
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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\begin{frame}{L\"owdin partitioning technique}
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\begin{block}{Folding or dressing process}
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\begin{equation}
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@ -1257,26 +1275,31 @@ decoration={snake,
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Correlation energy at the $GW$ level}
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\begin{block}{RPA correlation energy: plasmon (or trace) formula}
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\section{Total energies}
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\begin{frame}
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\tableofcontents[currentsection]
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\end{frame}
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%-----------------------------------------------------
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\begin{frame}{Correlation energy at the $GW$ or BSE level}
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\begin{block}{RPA@$GW$ correlation energy: plasmon (or trace) formula}
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\begin{equation*}
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\label{eq:Ec-RPA}
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\EcRPA
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= \frac{1}{2} \qty[ \sum_{p} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ]
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= \frac{1}{2} \sum_{m} \qty( \Om{m}{\RPA} - \Om{m}{\TDA} )
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\green{\EcRPA}
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= \frac{1}{2} \qty[ \sum_{p} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
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= \frac{1}{2} \sum_{m} \qty( \orange{\Om{m}{\RPA}} - \orange{\Om{m}{\TDA}} )
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\end{equation*}
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\end{block}
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\begin{block}{Galitskii-Migdal functional}
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\begin{equation*}
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\label{eq:GM}
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\EcGM
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\green{\EcGM}
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= \frac{-i}{2}\sum_{pq}^{\infty}\int \frac{d\omega}{2\pi} \red{\SigC{pq}}(\omega) \blue{\G{pq}}(\omega) e^{i\omega\eta}
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= 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\e{a}{} - \e{i}{} + \orange{\Om{m}{\RPA}}}
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= 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\eGW{a} - \eGW{i} + \orange{\Om{m}{\RPA}}}
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\end{equation*}
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\end{block}
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\begin{block}{Adiabatic connection}
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\begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection}
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\begin{equation}
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\Ec^\text{ACFDT} = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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\green{\Ec^\text{ACFDT}} = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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\end{equation}
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\end{block}
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@ -1290,9 +1313,14 @@ decoration={snake,
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\boxed{
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\green{\Ec^\text{ACFDT}}
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})
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\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_{k=1}^{K} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})
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}
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\end{equation}
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$\la$ is the \textbf{strength} of the electron-electron interaction:
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\begin{itemize}
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\item $\la = 0$ for the \green{non-interacting system}
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\item $\la = 1$ for the \alert{physical system}
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\end{itemize}
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\end{block}
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\begin{block}{Interaction kernel}
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\begin{equation}
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@ -1330,12 +1358,14 @@ decoration={snake,
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%-----------------------------------------------------
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\begin{frame}{Gaussian quadrature}
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\begin{block}{Numerical integration by quadrature}
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\textit{``A $K$-point \orange{Gaussian quadrature} rule is a quadrature rule constructed to yield an exact result for polynomials up to degree $2K-1$ by a suitable choice of the \violet{roots $x_k$} and \purple{weights $w_k$} for $k = 1, \ldots, K$.''}
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\begin{equation}
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\boxed{\int_{\red{a}}^{\red{b}} f(x) \purple{w(x)} dx \approx \sum_k \underbrace{\purple{w_k}}_{\text{\purple{weights}}} f(\underbrace{\violet{x_k}}_{\text{\violet{roots}}})}
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\boxed{\int_{\red{a}}^{\red{b}} f(x) \purple{w(x)} dx \approx \sum_k^{K} \underbrace{\purple{w_k}}_{\text{\purple{weights}}} f(\underbrace{\violet{x_k}}_{\text{\violet{roots}}})}
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\end{equation}
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\end{block}
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\begin{block}{Quadrature rules}
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\begin{center}
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\small
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\begin{tabular}{llll}
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\hline
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\red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{Name} \\
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@ -1350,7 +1380,6 @@ decoration={snake,
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\hline
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\end{tabular}
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\\
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\bigskip
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\url{https://en.wikipedia.org/wiki/Gaussian_quadrature}
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\end{center}
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\end{block}
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@ -1361,7 +1390,7 @@ decoration={snake,
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\begin{frame}{ACFDT at the RPA/RPAx level}
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\begin{block}{RPA matrix elements}
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\begin{equation}
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\orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\la\ERI{ia}{bj}
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\orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\violet{\eHF{a}} - \violet{\eHF{i}}) + 2\la\ERI{ia}{bj}
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\qquad
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\orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb}
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\end{equation}
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@ -1376,7 +1405,7 @@ decoration={snake,
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\begin{block}{RPAx matrix elements}
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\begin{equation}
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\orange{\A{ia,jb}{\la,\RPAx}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \la \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]
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\orange{\A{ia,jb}{\la,\RPAx}} = \delta_{ij} \delta_{ab} (\violet{\eHF{a}} - \violet{\eHF{i}}) + \la \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]
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\qquad
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\orange{\B{ia,jb}{\la,\RPAx}} = \la \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
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\end{equation}
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\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
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}
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\end{equation}
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If exchange added to kernel, i.e., $\bK{}{} = \bK{}{\x}$, then \pub{[Angyan et al. JCTC 7 (2011) 3116]}
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\begin{equation}
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\green{\Ec^\RPAx}
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= \frac{1}{4} \int_0^1 \Tr( \bK{}{\x} \bP{}{\la}) d\la
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\alert{=} \frac{1}{4} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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@ -1397,7 +1431,7 @@ decoration={snake,
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\begin{frame}{ACFDT at the BSE level}
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\begin{block}{BSE matrix elements}
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\begin{equation}
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\orange{\A{ia,jb}{\la,\BSE}} = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \la \qty[2 \ERI{ia}{bj} - \highlight{W}_{ij,ab}^{\la}(\omega = 0) ]
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\orange{\A{ia,jb}{\la,\BSE}} = \delta_{ij} \delta_{ab} (\violet{\eGW{a}} - \violet{\eGW{i}}) + \la \qty[2 \ERI{ia}{bj} - \highlight{W}_{ij,ab}^{\la}(\omega = 0) ]
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\qquad
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\orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)]
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\end{equation}
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@ -1412,10 +1446,10 @@ decoration={snake,
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\end{block}
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\begin{block}{$\la$-dependent screening}
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\begin{equation}
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\highlight{W}_{pq,rs}^{\la}(\yo)
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\highlight{W}_{pq,rs}^{\la}(\omega)
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= \ERI{pq}{rs}
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+ 2 \sum_m \violet{\ERI{pq}{m}^{\la}} \violet{\ERI{rs}{m}^{\la}}
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\qty[ \frac{1}{\yo - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\la,\RPA}} - i \eta} ]
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\qty[ \frac{1}{\omega - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\omega + \orange{\Om{m}{\la,\RPA}} - i \eta} ]
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\end{equation}
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\begin{equation}
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\violet{\ERI{pq}{m}^{\la}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\la,\RPA}+\bY{m}{\la,\RPA}})_{ia}
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@ -1430,10 +1464,10 @@ decoration={snake,
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\begin{algorithmic}
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\Procedure{ACFDT for BSE}{}
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\State Compute $GW$ quasiparticle energies $\blue{\beGW}$ and interaction kernel $\bK{}{}$
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\State Get Gauss-Legendre weights and roots $\{\purple{w_k},\violet{\lambda_k}\}_{1\le k \le N_\text{grid}}$
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\State Get Gauss-Legendre weights and roots $\{\purple{w_k},\violet{\lambda_k}\}_{1\le k \le K}$
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\State $\green{\Ec} \gets 0$
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\For{$k=1,\ldots,N_\text{grid}$}
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\State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}$
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\For{$k=1,\ldots,K$}
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\State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}(\omega = 0)$
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\State Perform BSE calculation at $\la = \violet{\lambda_k}$ to get $\bX{}{\violet{\lambda_k}}$ and $\bY{}{\violet{\lambda_k}}$
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\State Form two-particle density matrix $\bP{}{\violet{\lambda_k}}$
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\State $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$
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@ -1447,7 +1481,9 @@ decoration={snake,
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%-----------------------------------------------------
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\begin{frame}
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\begin{center}
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\includegraphics[width=0.8\textwidth]{fig/TOC_BSE}
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\includegraphics[width=0.7\textwidth]{fig/TOC_BSE}
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\\
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\pub{Loos et al. JPCL 11 (2020) 3536}
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\end{center}
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\end{frame}
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%-----------------------------------------------------
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2021/Lecture_2/fig/Sigma.png
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