From 2fda25163015e4a3f910d50a089d056e304b523e Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 7 Jun 2021 21:48:45 +0200 Subject: [PATCH] ready for tomorrow --- 2021/Lecture_2/ISTPC_Loos_2.tex | 51 +++++++++++++++++++------------- 2021/Lecture_2/fig/H2.pdf | Bin 0 -> 30459 bytes 2021/Lecture_2/fig/H2_QuAcK.png | Bin 0 -> 275547 bytes 2021/Lecture_2/fig/Tmatrix.png | Bin 0 -> 387087 bytes 4 files changed, 30 insertions(+), 21 deletions(-) create mode 100644 2021/Lecture_2/fig/H2.pdf create mode 100644 2021/Lecture_2/fig/H2_QuAcK.png create mode 100644 2021/Lecture_2/fig/Tmatrix.png diff --git a/2021/Lecture_2/ISTPC_Loos_2.tex b/2021/Lecture_2/ISTPC_Loos_2.tex index be8a638..c2e0c87 100644 --- a/2021/Lecture_2/ISTPC_Loos_2.tex +++ b/2021/Lecture_2/ISTPC_Loos_2.tex @@ -284,7 +284,7 @@ decoration={snake, \item Bethe-Salpeter equation (BSE) formalism \end{itemize} \bigskip - \item \textbf{Total energies} + \item \textbf{Correlation energy} \begin{itemize} \item Plasmon (or trace) formula \item Galitski-Migdal formulation @@ -332,17 +332,17 @@ decoration={snake, \bigskip \item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102 \bigskip - \item \orange{Review:} + \item \orange{Reviews \& Books:} \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 + \item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press) \end{itemize} \bigskip \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} %----------------------------------------------------- @@ -572,7 +572,7 @@ decoration={snake, %----------------------------------------------------- \begin{frame}{Computation of the dynamical screening} - \begin{block}{Direct RPA calculation (pseudo-hermitian linear problem)} + \begin{block}{Direct (ph-)RPA calculation (pseudo-hermitian linear problem)} \begin{equation} \begin{pmatrix} \bA{}{\RPA} & \bB{}{\RPA} \\ @@ -899,6 +899,11 @@ decoration={snake, \end{column} \begin{column}{0.35\textwidth} \includegraphics[width=\textwidth]{fig/Sigma} + \\ + \bigskip + \includegraphics[width=\textwidth]{fig/Tmatrix} + \\ + \pub{Martin, Reining \& Ceperley, Interacting Electrons (Cambridge University Press)} \end{column} \end{columns} \end{frame} @@ -1227,21 +1232,21 @@ decoration={snake, \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 ]} + \boxed{\green{f_m} = \frac{2}{3} \orange{\Om{m}{}} \qty[ (\blue{\mu_m^x})^2 + (\blue{\mu_m^y})^2 + (\blue{\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}} + \boxed{\blue{\mu_m^x} = \sum_{ia} \red{(i|x|a)} \orange{(\bX{m}{} + \bY{m}{})_{ia}}} \qquad - (p|x|q) = \int \MO{p}(\br) \,x\, \MO{q}(\br) d\br + \red{(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}}}} + \boxed{\purple{\expval{\hat{S}^2}_m} = \violet{\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) + \violet{\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} @@ -1262,10 +1267,12 @@ decoration={snake, %----------------------------------------------------- %----------------------------------------------------- -\begin{frame}{Open-shell systems} - \begin{block}{Spin-flip formalism} +\begin{frame}{Open-shell systems and double excitations} + \begin{block}{Spin-flip formalism (H2/cc-pVQZ)} \begin{center} - \includegraphics[width=0.5\textwidth]{fig/SFBSE} + \includegraphics[width=0.28\textwidth]{fig/SFBSE} + \includegraphics[width=0.4\textwidth]{fig/H2} + \includegraphics[width=0.3\textwidth]{fig/H2_QuAcK} \\ \bigskip \pub{Monino \& Loos, JCTC 17 (2021) 2852} @@ -1275,7 +1282,7 @@ decoration={snake, %----------------------------------------------------- %----------------------------------------------------- -\section{Total energies} +\section{Correlation energy} \begin{frame} \tableofcontents[currentsection] \end{frame} @@ -1294,12 +1301,12 @@ decoration={snake, \label{eq:GM} \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}{\eGW{a} - \eGW{i} + \orange{\Om{m}{\RPA}}} + = 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\blue{\eGW{a}} - \blue{\eGW{i}} + \orange{\Om{m}{\RPA}}} \end{equation*} \end{block} \begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection} \begin{equation} - \green{\Ec^\text{ACFDT}} = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + \green{\Ec^\text{ACFDT}} = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la \end{equation} \end{block} @@ -1312,7 +1319,7 @@ decoration={snake, \begin{equation} \boxed{ \green{\Ec^\text{ACFDT}} - = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la \stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_{k=1}^{K} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}}) } \end{equation} @@ -1397,7 +1404,7 @@ decoration={snake, \begin{equation} \boxed{ \green{\Ec^\RPA} - = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + = \frac{1}{2} \int_{\red{0}}^{\red{1}} \Tr( \bK{}{} \bP{}{\la}) d\la = \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ] } \end{equation} @@ -1412,14 +1419,14 @@ decoration={snake, \begin{equation} \boxed{ \green{\Ec^\RPAx} - = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + = \frac{1}{2} \int_{\red{0}}^{\red{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} 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 + = \frac{1}{4} \int_{\red{0}}^{\red{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} @@ -1438,7 +1445,7 @@ decoration={snake, \begin{equation} \boxed{ \green{\Ec^\BSE} - = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la + = \frac{1}{2} \int_{\red{0}}^{\red{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} @@ -1469,6 +1476,7 @@ decoration={snake, \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 $\bY{}{\violet{\lambda_k}}$ + \Comment{\alert{This is a $\order*{N^6}$ step done many times!}} \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 @@ -1499,13 +1507,14 @@ decoration={snake, \bigskip \item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102 \bigskip - \item \orange{Review:} + \item \orange{Reviews \& Books:} \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 + \item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press) \end{itemize} \bigskip \item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. 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