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2021/Lecture_2/ISTPC_Loos_2.tex
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@ -237,7 +237,7 @@ decoration={snake,
Computational aspects Computational aspects
} }
\author[PF Loos (\url{https://www.irsamc.ups-tlse.fr/loos/})]{Pierre-Fran\c{c}ois LOOS} \author[PF Loos (\url{https://www.irsamc.ups-tlse.fr/loos/})]{Pierre-Fran\c{c}ois LOOS}
\date{Online ISTPC 2021 school --- April 27th, 2021} \date{Online ISTPC 2021 school --- June 8th, 2021}
\institute[CNRS@LCPQ]{ \institute[CNRS@LCPQ]{
Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\
Universit\'e de Toulouse, CNRS, UPS, Toulouse, France. Universit\'e de Toulouse, CNRS, UPS, Toulouse, France.
@ -263,6 +263,7 @@ decoration={snake,
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Today's program} \begin{frame}{Today's program}
\begin{itemize} \begin{itemize}
@ -271,20 +272,21 @@ decoration={snake,
\item One-shot $GW$ (\GOWO) \item One-shot $GW$ (\GOWO)
\item Partially self-consistent eigenvalue $GW$ (\evGW) \item Partially self-consistent eigenvalue $GW$ (\evGW)
\item Quasiparticle self-consistent $GW$ (\qsGW) \item Quasiparticle self-consistent $GW$ (\qsGW)
\item Other self-energies (GF2, SOSEX, T-matrix, etc)
\end{itemize} \end{itemize}
\bigskip \bigskip
\item \textbf{Neutral excitations} \item \textbf{Neutral excitations}
\begin{itemize} \begin{itemize}
\item Configuration interaction with singles (CIS)
\item Time-dependent Hartree-Fock (TDHF)
\item Random-phase approximation (RPA) \item Random-phase approximation (RPA)
\item Configuration interaction with singles (CIS)
\item Time-dependent Hartree-Fock (TDHF) or RPA with exchange (RPAx)
\item Time-dependent density-functional theory (TDDFT) \item Time-dependent density-functional theory (TDDFT)
\item Bethe-Salpeter equation (BSE) formalism \item Bethe-Salpeter equation (BSE) formalism
\end{itemize} \end{itemize}
\bigskip \bigskip
\item \textbf{Total energies} \item \textbf{Total energies}
\begin{itemize} \begin{itemize}
\item Plasmon formula \item Plasmon (or trace) formula
\item Galitski-Migdal formulation \item Galitski-Migdal formulation
\item Adiabatic connection fluctuation-dissipation theorem (ACFDT) \item Adiabatic connection fluctuation-dissipation theorem (ACFDT)
\end{itemize} \end{itemize}
@ -292,6 +294,11 @@ decoration={snake,
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
%-----------------------------------------------------
\section{Context}
\begin{frame}
\tableofcontents[currentsection]
\end{frame}
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Assumptions \& Notations} \begin{frame}{Assumptions \& Notations}
\begin{block}{Let's talk about notations} \begin{block}{Let's talk about notations}
@ -506,6 +513,11 @@ decoration={snake,
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
%-----------------------------------------------------
\section{Charged excitations}
\begin{frame}
\tableofcontents[currentsection]
\end{frame}
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Green's function and dynamical screening} \begin{frame}{Green's function and dynamical screening}
\begin{block}{One-body Green's function} \begin{block}{One-body Green's function}
@ -892,6 +904,11 @@ decoration={snake,
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
%-----------------------------------------------------
\section{Neutral excitations}
\begin{frame}
\tableofcontents[currentsection]
\end{frame}
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Dynamical vs static kernels} \begin{frame}{Dynamical vs static kernels}
\begin{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}} \begin{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}}
@ -934,6 +951,7 @@ decoration={snake,
\end{block} \end{block}
\end{frame} \end{frame}
%-----------------------------------------------------
\begin{frame}{L\"owdin partitioning technique} \begin{frame}{L\"owdin partitioning technique}
\begin{block}{Folding or dressing process} \begin{block}{Folding or dressing process}
\begin{equation} \begin{equation}
@ -1257,26 +1275,31 @@ decoration={snake,
%----------------------------------------------------- %-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Correlation energy at the $GW$ level} \section{Total energies}
\begin{block}{RPA correlation energy: plasmon (or trace) formula} \begin{frame}
\tableofcontents[currentsection]
\end{frame}
%-----------------------------------------------------
\begin{frame}{Correlation energy at the $GW$ or BSE level}
\begin{block}{RPA@$GW$ correlation energy: plasmon (or trace) formula}
\begin{equation*} \begin{equation*}
\label{eq:Ec-RPA} \label{eq:Ec-RPA}
\EcRPA \green{\EcRPA}
= \frac{1}{2} \qty[ \sum_{p} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ] = \frac{1}{2} \qty[ \sum_{p} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
= \frac{1}{2} \sum_{m} \qty( \Om{m}{\RPA} - \Om{m}{\TDA} ) = \frac{1}{2} \sum_{m} \qty( \orange{\Om{m}{\RPA}} - \orange{\Om{m}{\TDA}} )
\end{equation*} \end{equation*}
\end{block} \end{block}
\begin{block}{Galitskii-Migdal functional} \begin{block}{Galitskii-Migdal functional}
\begin{equation*} \begin{equation*}
\label{eq:GM} \label{eq:GM}
\EcGM \green{\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} = \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}{\eGW{a} - \eGW{i} + \orange{\Om{m}{\RPA}}}
\end{equation*} \end{equation*}
\end{block} \end{block}
\begin{block}{Adiabatic connection} \begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection}
\begin{equation} \begin{equation}
\Ec^\text{ACFDT} = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la \green{\Ec^\text{ACFDT}} = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
\end{equation} \end{equation}
\end{block} \end{block}
@ -1290,9 +1313,14 @@ decoration={snake,
\boxed{ \boxed{
\green{\Ec^\text{ACFDT}} \green{\Ec^\text{ACFDT}}
= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}}) \stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_{k=1}^{K} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})
} }
\end{equation} \end{equation}
$\la$ is the \textbf{strength} of the electron-electron interaction:
\begin{itemize}
\item $\la = 0$ for the \green{non-interacting system}
\item $\la = 1$ for the \alert{physical system}
\end{itemize}
\end{block} \end{block}
\begin{block}{Interaction kernel} \begin{block}{Interaction kernel}
\begin{equation} \begin{equation}
@ -1330,12 +1358,14 @@ decoration={snake,
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Gaussian quadrature} \begin{frame}{Gaussian quadrature}
\begin{block}{Numerical integration by quadrature} \begin{block}{Numerical integration by quadrature}
\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 $k = 1, \ldots, K$.''}
\begin{equation} \begin{equation}
\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}}})} \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}}})}
\end{equation} \end{equation}
\end{block} \end{block}
\begin{block}{Quadrature rules} \begin{block}{Quadrature rules}
\begin{center} \begin{center}
\small
\begin{tabular}{llll} \begin{tabular}{llll}
\hline \hline
\red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{Name} \\ \red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{Name} \\
@ -1350,7 +1380,6 @@ decoration={snake,
\hline \hline
\end{tabular} \end{tabular}
\\ \\
\bigskip
\url{https://en.wikipedia.org/wiki/Gaussian_quadrature} \url{https://en.wikipedia.org/wiki/Gaussian_quadrature}
\end{center} \end{center}
\end{block} \end{block}
@ -1361,7 +1390,7 @@ decoration={snake,
\begin{frame}{ACFDT at the RPA/RPAx level} \begin{frame}{ACFDT at the RPA/RPAx level}
\begin{block}{RPA matrix elements} \begin{block}{RPA matrix elements}
\begin{equation} \begin{equation}
\orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\la\ERI{ia}{bj} \orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\violet{\eHF{a}} - \violet{\eHF{i}}) + 2\la\ERI{ia}{bj}
\qquad \qquad
\orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb} \orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb}
\end{equation} \end{equation}
@ -1376,7 +1405,7 @@ decoration={snake,
\begin{block}{RPAx matrix elements} \begin{block}{RPAx matrix elements}
\begin{equation} \begin{equation}
\orange{\A{ia,jb}{\la,\RPAx}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \la \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ] \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} ]
\qquad \qquad
\orange{\B{ia,jb}{\la,\RPAx}} = \la \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ] \orange{\B{ia,jb}{\la,\RPAx}} = \la \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
\end{equation} \end{equation}
@ -1387,8 +1416,13 @@ decoration={snake,
\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ] \alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
} }
\end{equation} \end{equation}
If exchange added to kernel, i.e., $\bK{}{} = \bK{}{\x}$, then \pub{[Angyan et al. JCTC 7 (2011) 3116]}
\begin{equation}
\green{\Ec^\RPAx}
= \frac{1}{4} \int_0^1 \Tr( \bK{}{\x} \bP{}{\la}) d\la
\alert{=} \frac{1}{4} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
\end{equation}
\end{block} \end{block}
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
@ -1397,7 +1431,7 @@ decoration={snake,
\begin{frame}{ACFDT at the BSE level} \begin{frame}{ACFDT at the BSE level}
\begin{block}{BSE matrix elements} \begin{block}{BSE matrix elements}
\begin{equation} \begin{equation}
\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) ] \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) ]
\qquad \qquad
\orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)] \orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)]
\end{equation} \end{equation}
@ -1412,10 +1446,10 @@ decoration={snake,
\end{block} \end{block}
\begin{block}{$\la$-dependent screening} \begin{block}{$\la$-dependent screening}
\begin{equation} \begin{equation}
\highlight{W}_{pq,rs}^{\la}(\yo) \highlight{W}_{pq,rs}^{\la}(\omega)
= \ERI{pq}{rs} = \ERI{pq}{rs}
+ 2 \sum_m \violet{\ERI{pq}{m}^{\la}} \violet{\ERI{rs}{m}^{\la}} + 2 \sum_m \violet{\ERI{pq}{m}^{\la}} \violet{\ERI{rs}{m}^{\la}}
\qty[ \frac{1}{\yo - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\la,\RPA}} - i \eta} ] \qty[ \frac{1}{\omega - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\omega + \orange{\Om{m}{\la,\RPA}} - i \eta} ]
\end{equation} \end{equation}
\begin{equation} \begin{equation}
\violet{\ERI{pq}{m}^{\la}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\la,\RPA}+\bY{m}{\la,\RPA}})_{ia} \violet{\ERI{pq}{m}^{\la}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\la,\RPA}+\bY{m}{\la,\RPA}})_{ia}
@ -1430,10 +1464,10 @@ decoration={snake,
\begin{algorithmic} \begin{algorithmic}
\Procedure{ACFDT for BSE}{} \Procedure{ACFDT for BSE}{}
\State Compute $GW$ quasiparticle energies $\blue{\beGW}$ and interaction kernel $\bK{}{}$ \State Compute $GW$ quasiparticle energies $\blue{\beGW}$ and interaction kernel $\bK{}{}$
\State Get Gauss-Legendre weights and roots $\{\purple{w_k},\violet{\lambda_k}\}_{1\le k \le N_\text{grid}}$ \State Get Gauss-Legendre weights and roots $\{\purple{w_k},\violet{\lambda_k}\}_{1\le k \le K}$
\State $\green{\Ec} \gets 0$ \State $\green{\Ec} \gets 0$
\For{$k=1,\ldots,N_\text{grid}$} \For{$k=1,\ldots,K$}
\State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}$ \State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}(\omega = 0)$
\State Perform BSE calculation at $\la = \violet{\lambda_k}$ to get $\bX{}{\violet{\lambda_k}}$ and $\bY{}{\violet{\lambda_k}}$ \State Perform BSE calculation at $\la = \violet{\lambda_k}$ to get $\bX{}{\violet{\lambda_k}}$ and $\bY{}{\violet{\lambda_k}}$
\State Form two-particle density matrix $\bP{}{\violet{\lambda_k}}$ \State Form two-particle density matrix $\bP{}{\violet{\lambda_k}}$
\State $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$ \State $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$
@ -1447,7 +1481,9 @@ decoration={snake,
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame} \begin{frame}
\begin{center} \begin{center}
\includegraphics[width=0.8\textwidth]{fig/TOC_BSE} \includegraphics[width=0.7\textwidth]{fig/TOC_BSE}
\\
\pub{Loos et al. JPCL 11 (2020) 3536}
\end{center} \end{center}
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------

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