From d222e647ad4a0e303596bafe2a021778d0850d73 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 5 Jun 2021 15:27:25 +0200 Subject: [PATCH] almost OK for lecture 2 --- 2021/Lecture_2/ISTPC_Loos_2.tex | 140 ++++++++++++++++++-------------- 1 file changed, 77 insertions(+), 63 deletions(-) diff --git a/2021/Lecture_2/ISTPC_Loos_2.tex b/2021/Lecture_2/ISTPC_Loos_2.tex index 589ce6b..9541fff 100644 --- a/2021/Lecture_2/ISTPC_Loos_2.tex +++ b/2021/Lecture_2/ISTPC_Loos_2.tex @@ -211,10 +211,11 @@ \newcommand{\bX}[2]{\bm{X}_{#1}^{#2}} \newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}} \newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}} -\newcommand{\bK}[2]{\bm{K}_{#1}^{#2}} -\newcommand{\bP}[2]{\bm{P}_{#1}^{#2}} +\newcommand{\bK}[2]{\blue{\bm{K}}_{#1}^{#2}} +\newcommand{\bP}[2]{\red{\bm{P}}_{#1}^{#2}} \newcommand{\yo}{\yellow{\omega}} +\newcommand{\la}{\yellow{\lambda}} \newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}} @@ -917,12 +918,19 @@ decoration={snake, \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} + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{center} + \includegraphics[width=0.7\textwidth]{fig/dyn} + \end{center} + \end{column} + \begin{column}{0.5\textwidth} + Dynamical kernels can give you more than static kernels... Sometimes, too much... + \end{column} + \end{columns} + \bigskip + \center + \pub{Authier \& Loos, JCP 153 (2020) 184105} [see also \pub{Romaniello et al, JCP 130 (2009) 044108}] \end{block} \end{frame} @@ -947,18 +955,17 @@ decoration={snake, \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} + \begin{align} + \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 + \\ \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} + & \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{align} \begin{equation} \boxed{ \underbrace{\Tilde{\bH}_1(\yo)}_{\text{Effective Hamitonian}} @@ -1063,7 +1070,7 @@ decoration={snake, \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)} + \State Compute lowest 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} @@ -1155,7 +1162,7 @@ decoration={snake, \begin{block}{General linear response problem} \begin{algorithmic} \Procedure{Linear response}{} - \State Compute $\red{\bA{}{}}$ matrix at a given level of theory + \State Compute $\red{\bA{}{}}$ matrix at a given level of theory (RPA, RPAx, TD-DFT, BSE, etc) \If{$\TDA$} \State Diagonalize $\red{\bA{}{}}$ to get $\highlight{\Om{m}{\TDA}}$ and $\bX{m}{\TDA}$ \Else @@ -1251,7 +1258,7 @@ decoration={snake, %----------------------------------------------------- \begin{frame}{Correlation energy at the $GW$ level} - \begin{block}{RPA correlation energy: plasmon formula} + \begin{block}{RPA correlation energy: plasmon (or trace) formula} \begin{equation*} \label{eq:Ec-RPA} \EcRPA @@ -1267,6 +1274,12 @@ decoration={snake, = 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\e{a}{} - \e{i}{} + \orange{\Om{m}{\RPA}}} \end{equation*} \end{block} + \begin{block}{Adiabatic connection} + \begin{equation} + \Ec^\text{ACFDT} = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + \end{equation} + \end{block} + \end{frame} %----------------------------------------------------- @@ -1275,9 +1288,9 @@ decoration={snake, \begin{block}{Adiabatic connection} \begin{equation} \boxed{ - \Ec^\text{ACFDT} - = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda - \stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} w_k \Tr( \bK{}{} \bP{}{\lambda_k}) + \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}}) } \end{equation} \end{block} @@ -1290,18 +1303,18 @@ decoration={snake, \btB{}{} & \btA{}{} \end{pmatrix} \qquad - \tA{ia,jb}{} = 2\lambda\ERI{ia}{bj} + \tA{ia,jb}{} = 2\la\ERI{ia}{bj} \qquad - \tB{ia,jb}{} = 2\lambda\ERI{ia}{jb} + \tB{ia,jb}{} = 2\la\ERI{ia}{jb} \end{equation} \end{block} \begin{block}{Correlation part of the two-particle density matrix} \begin{equation} - \bP{}{\lambda} = + \bP{}{\la} = \begin{pmatrix} - \bY{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bY{}{\lambda} \cdot \T{(\bX{}{\lambda})} + \bY{}{\la} \cdot \T{(\bY{}{\la})} & \bY{}{\la} \cdot \T{(\bX{}{\la})} \\ - \bX{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bX{}{\lambda} \cdot \T{(\bX{}{\lambda})} + \bX{}{\la} \cdot \T{(\bY{}{\la})} & \bX{}{\la} \cdot \T{(\bX{}{\la})} \end{pmatrix} - \begin{pmatrix} @@ -1318,14 +1331,14 @@ decoration={snake, \begin{frame}{Gaussian quadrature} \begin{block}{Numerical integration by quadrature} \begin{equation} - \boxed{\int_a^b f(x) w(x) dx \approx \sum_k \underbrace{w_k}_{\text{weights}} f(\underbrace{x_k}_{\text{roots}})} + \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}}})} \end{equation} \end{block} \begin{block}{Quadrature rules} \begin{center} \begin{tabular}{llll} \hline - Interval $[a,b]$ & Weight function $w(x)$ & Orthogonal polynomials & Name \\ + \red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{Name} \\ \hline $[-1,1]$ & $1$ & Legendre $P_n(x)$ & Gauss-Legendre \\ $(-1,1)$ & $(1-x)^\alpha(1+x)^\beta, \quad \alpha,\beta > -1$ & Jacobi $P_n^{\alpha,\beta}(x)$ & Gauss-Jacobi \\ @@ -1348,30 +1361,30 @@ decoration={snake, \begin{frame}{ACFDT at the RPA/RPAx level} \begin{block}{RPA matrix elements} \begin{equation} - \A{ia,jb}{\lambda,\RPA} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\lambda\ERI{ia}{bj} + \orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\la\ERI{ia}{bj} \qquad - \B{ia,jb}{\lambda,\RPA} = 2\lambda\ERI{ia}{jb} + \orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb} \end{equation} \begin{equation} \boxed{ - \Ec^\RPA - = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda - = \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ] + \green{\Ec^\RPA} + = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + = \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ] } \end{equation} \end{block} \begin{block}{RPAx matrix elements} \begin{equation} - \A{ia,jb}{\lambda,\RPAx} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \lambda \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ] + \orange{\A{ia,jb}{\la,\RPAx}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \la \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ] \qquad - \B{ia,jb}{\lambda,\RPAx} = \lambda \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} \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}) ] + \green{\Ec^\RPAx} + = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + \alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ] } \end{equation} \end{block} @@ -1384,28 +1397,28 @@ 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}) + \lambda \qty[2 \ERI{ia}{bj} - W_{ij,ab}^{\lambda}(\omega = 0) ] + \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) ] \qquad - \B{ia,jb}{\lambda,\BSE} = \lambda \qty[2 \ERI{ia}{jb} - W_{ib,ja}^{\lambda}(\omega = 0)] + \orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\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}) ] + \green{\Ec^\BSE} + = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + \alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\BSE}} - \Tr(\orange{\bA{}{\BSE}}) ] } \end{equation} \end{block} - \begin{block}{$\lambda$-dependent screening} + \begin{block}{$\la$-dependent screening} \begin{equation} - \highlight{W}_{pq,rs}^{\lambda}(\yo) + \highlight{W}_{pq,rs}^{\la}(\yo) = \ERI{pq}{rs} - + 2 \sum_m \violet{\ERI{pq}{m}^{\lambda}} \violet{\ERI{rs}{m}^{\lambda}} - \qty[ \frac{1}{\yo - \orange{\Om{m}{\lambda,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\lambda,\RPA}} - i \eta} ] + + 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} ] \end{equation} \begin{equation} - \violet{\ERI{pq}{m}^{\lambda}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\lambda,\RPA}+\bY{m}{\lambda,\RPA}})_{ia} + \violet{\ERI{pq}{m}^{\la}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\la,\RPA}+\bY{m}{\la,\RPA}})_{ia} \end{equation} \end{block} \end{frame} @@ -1416,14 +1429,14 @@ decoration={snake, \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 $\Ec \gets 0$ + \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 $\green{\Ec} \gets 0$ \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 $\Ec \gets \Ec + w_k \Tr( \bK{}{} \bP{}{\lambda_k})$ + \State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}$ + \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 $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$ \EndFor \EndProcedure \end{algorithmic} @@ -1440,15 +1453,15 @@ decoration={snake, %----------------------------------------------------- %----------------------------------------------------- -\begin{frame}{Useful papers} +\begin{frame}{Useful papers/programs} \begin{itemize} - \item \red{molGW:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149 + \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 + \item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102 \bigskip \item \orange{Review:} \begin{itemize} @@ -1459,7 +1472,8 @@ decoration={snake, \item Blase et al. JPCL 11 (2020) 7371 \end{itemize} \bigskip - \item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 + \item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com}) + \end{itemize} \end{frame} %-----------------------------------------------------