ready for tomorrow

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. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com})