diff --git a/2021/Lecture_2/ISTPC_Loos_2.tex b/2021/Lecture_2/ISTPC_Loos_2.tex index 9541fff..be8a638 100644 --- a/2021/Lecture_2/ISTPC_Loos_2.tex +++ b/2021/Lecture_2/ISTPC_Loos_2.tex @@ -237,7 +237,7 @@ decoration={snake, Computational aspects } \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]{ Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ Universit\'e de Toulouse, CNRS, UPS, Toulouse, France. @@ -263,6 +263,7 @@ decoration={snake, \end{frame} %----------------------------------------------------- + %----------------------------------------------------- \begin{frame}{Today's program} \begin{itemize} @@ -271,20 +272,21 @@ decoration={snake, \item One-shot $GW$ (\GOWO) \item Partially self-consistent eigenvalue $GW$ (\evGW) \item Quasiparticle self-consistent $GW$ (\qsGW) + \item Other self-energies (GF2, SOSEX, T-matrix, etc) \end{itemize} \bigskip \item \textbf{Neutral excitations} \begin{itemize} - \item Configuration interaction with singles (CIS) - \item Time-dependent Hartree-Fock (TDHF) \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 Bethe-Salpeter equation (BSE) formalism \end{itemize} \bigskip \item \textbf{Total energies} \begin{itemize} - \item Plasmon formula + \item Plasmon (or trace) formula \item Galitski-Migdal formulation \item Adiabatic connection fluctuation-dissipation theorem (ACFDT) \end{itemize} @@ -292,6 +294,11 @@ decoration={snake, \end{frame} %----------------------------------------------------- +%----------------------------------------------------- +\section{Context} +\begin{frame} +\tableofcontents[currentsection] +\end{frame} %----------------------------------------------------- \begin{frame}{Assumptions \& Notations} \begin{block}{Let's talk about notations} @@ -506,6 +513,11 @@ decoration={snake, \end{frame} %----------------------------------------------------- +%----------------------------------------------------- +\section{Charged excitations} +\begin{frame} +\tableofcontents[currentsection] +\end{frame} %----------------------------------------------------- \begin{frame}{Green's function and dynamical screening} \begin{block}{One-body Green's function} @@ -892,6 +904,11 @@ decoration={snake, \end{frame} %----------------------------------------------------- +%----------------------------------------------------- +\section{Neutral excitations} +\begin{frame} +\tableofcontents[currentsection] +\end{frame} %----------------------------------------------------- \begin{frame}{Dynamical vs static kernels} \begin{block}{A non-linear BSE problem \pub{[Strinati, Riv.~Nuovo Cimento 11 (1988) 1]}} @@ -934,6 +951,7 @@ decoration={snake, \end{block} \end{frame} +%----------------------------------------------------- \begin{frame}{L\"owdin partitioning technique} \begin{block}{Folding or dressing process} \begin{equation} @@ -1257,26 +1275,31 @@ decoration={snake, %----------------------------------------------------- %----------------------------------------------------- -\begin{frame}{Correlation energy at the $GW$ level} - \begin{block}{RPA correlation energy: plasmon (or trace) formula} +\section{Total energies} +\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*} \label{eq:Ec-RPA} - \EcRPA - = \frac{1}{2} \qty[ \sum_{p} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ] - = \frac{1}{2} \sum_{m} \qty( \Om{m}{\RPA} - \Om{m}{\TDA} ) + \green{\EcRPA} + = \frac{1}{2} \qty[ \sum_{p} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ] + = \frac{1}{2} \sum_{m} \qty( \orange{\Om{m}{\RPA}} - \orange{\Om{m}{\TDA}} ) \end{equation*} \end{block} \begin{block}{Galitskii-Migdal functional} \begin{equation*} \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} - = 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{block} - \begin{block}{Adiabatic connection} + \begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection} \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{block} @@ -1290,9 +1313,14 @@ decoration={snake, \boxed{ \green{\Ec^\text{ACFDT}} = \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} + $\la$ 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} \begin{block}{Interaction kernel} \begin{equation} @@ -1330,12 +1358,14 @@ decoration={snake, %----------------------------------------------------- \begin{frame}{Gaussian 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} - \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{block} \begin{block}{Quadrature rules} \begin{center} + \small \begin{tabular}{llll} \hline \red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{Name} \\ @@ -1350,7 +1380,6 @@ decoration={snake, \hline \end{tabular} \\ - \bigskip \url{https://en.wikipedia.org/wiki/Gaussian_quadrature} \end{center} \end{block} @@ -1361,7 +1390,7 @@ decoration={snake, \begin{frame}{ACFDT at the RPA/RPAx level} \begin{block}{RPA matrix elements} \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 \orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb} \end{equation} @@ -1376,7 +1405,7 @@ decoration={snake, \begin{block}{RPAx matrix elements} \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 \orange{\B{ia,jb}{\la,\RPAx}} = \la \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ] \end{equation} @@ -1387,8 +1416,13 @@ decoration={snake, \alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ] } \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{frame} %----------------------------------------------------- @@ -1397,7 +1431,7 @@ decoration={snake, \begin{frame}{ACFDT at the BSE level} \begin{block}{BSE matrix elements} \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 \orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)] \end{equation} @@ -1412,10 +1446,10 @@ decoration={snake, \end{block} \begin{block}{$\la$-dependent screening} \begin{equation} - \highlight{W}_{pq,rs}^{\la}(\yo) + \highlight{W}_{pq,rs}^{\la}(\omega) = \ERI{pq}{rs} + 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} \begin{equation} \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} \Procedure{ACFDT for BSE}{} \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 $\green{\Ec} \gets 0$ - \For{$k=1,\ldots,N_\text{grid}$} - \State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}$ + \For{$k=1,\ldots,K$} + \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 Form two-particle density matrix $\bP{}{\violet{\lambda_k}}$ \State $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$ @@ -1447,7 +1481,9 @@ decoration={snake, %----------------------------------------------------- \begin{frame} \begin{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{frame} %----------------------------------------------------- diff --git a/2021/Lecture_2/fig/Sigma.png b/2021/Lecture_2/fig/Sigma.png new file mode 100644 index 0000000..2dde179 Binary files /dev/null and b/2021/Lecture_2/fig/Sigma.png differ