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2021/Lecture_2/ISTPC_Loos_2.tex
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@ -284,7 +284,7 @@ decoration={snake,
\item Bethe-Salpeter equation (BSE) formalism \item Bethe-Salpeter equation (BSE) formalism
\end{itemize} \end{itemize}
\bigskip \bigskip
\item \textbf{Total energies} \item \textbf{Correlation energy}
\begin{itemize} \begin{itemize}
\item Plasmon (or trace) formula \item Plasmon (or trace) formula
\item Galitski-Migdal formulation \item Galitski-Migdal formulation
@ -332,17 +332,17 @@ decoration={snake,
\bigskip \bigskip
\item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102 \item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
\bigskip \bigskip
\item \orange{Review:} \item \orange{Reviews \& Books:}
\begin{itemize} \begin{itemize}
\item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344 \item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344
\item Onida et al. Rev. Mod. Phys. 74 (2002) 601 \item Onida et al. Rev. Mod. Phys. 74 (2002) 601
\item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022 \item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022
\item Golze et al. Front. Chem. 7 (2019) 377 \item Golze et al. Front. Chem. 7 (2019) 377
\item Blase et al. JPCL 11 (2020) 7371 \item Blase et al. JPCL 11 (2020) 7371
\item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press)
\end{itemize} \end{itemize}
\bigskip \bigskip
\item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com}) \item \red{$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{itemize}
\end{frame} \end{frame}
%----------------------------------------------------- %-----------------------------------------------------
@ -572,7 +572,7 @@ decoration={snake,
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Computation of the dynamical screening} \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{equation}
\begin{pmatrix} \begin{pmatrix}
\bA{}{\RPA} & \bB{}{\RPA} \\ \bA{}{\RPA} & \bB{}{\RPA} \\
@ -899,6 +899,11 @@ decoration={snake,
\end{column} \end{column}
\begin{column}{0.35\textwidth} \begin{column}{0.35\textwidth}
\includegraphics[width=\textwidth]{fig/Sigma} \includegraphics[width=\textwidth]{fig/Sigma}
\\
\bigskip
\includegraphics[width=\textwidth]{fig/Tmatrix}
\\
\pub{Martin, Reining \& Ceperley, Interacting Electrons (Cambridge University Press)}
\end{column} \end{column}
\end{columns} \end{columns}
\end{frame} \end{frame}
@ -1227,21 +1232,21 @@ decoration={snake,
\begin{frame}{Properties} \begin{frame}{Properties}
\begin{block}{Oscillator strength (length gauge)} \begin{block}{Oscillator strength (length gauge)}
\begin{equation} \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{equation}
\end{block} \end{block}
\begin{block}{Transition dipole} \begin{block}{Transition dipole}
\begin{equation} \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 \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{equation}
\end{block} \end{block}
\begin{block}{Monitoring possible spin contamination \pub{[Monino \& Loos, JCTC 17 (2021) 2852]}} \begin{block}{Monitoring possible spin contamination \pub{[Monino \& Loos, JCTC 17 (2021) 2852]}}
\begin{equation} \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 \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{equation}
\end{block} \end{block}
\end{frame} \end{frame}
@ -1262,10 +1267,12 @@ decoration={snake,
%----------------------------------------------------- %-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\begin{frame}{Open-shell systems} \begin{frame}{Open-shell systems and double excitations}
\begin{block}{Spin-flip formalism} \begin{block}{Spin-flip formalism (H2/cc-pVQZ)}
\begin{center} \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 \bigskip
\pub{Monino \& Loos, JCTC 17 (2021) 2852} \pub{Monino \& Loos, JCTC 17 (2021) 2852}
@ -1275,7 +1282,7 @@ decoration={snake,
%----------------------------------------------------- %-----------------------------------------------------
%----------------------------------------------------- %-----------------------------------------------------
\section{Total energies} \section{Correlation energy}
\begin{frame} \begin{frame}
\tableofcontents[currentsection] \tableofcontents[currentsection]
\end{frame} \end{frame}
@ -1294,12 +1301,12 @@ decoration={snake,
\label{eq:GM} \label{eq:GM}
\green{\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}{\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{equation*}
\end{block} \end{block}
\begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection} \begin{block}{ACFDT@BSE@$GW$ correlation energy from the adiabatic connection}
\begin{equation} \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{equation}
\end{block} \end{block}
@ -1312,7 +1319,7 @@ decoration={snake,
\begin{equation} \begin{equation}
\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_{\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}}) \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}
@ -1397,7 +1404,7 @@ decoration={snake,
\begin{equation} \begin{equation}
\boxed{ \boxed{
\green{\Ec^\RPA} \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}}) ] = \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
} }
\end{equation} \end{equation}
@ -1412,14 +1419,14 @@ decoration={snake,
\begin{equation} \begin{equation}
\boxed{ \boxed{
\green{\Ec^\RPAx} \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}}) ] \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]} If exchange added to kernel, i.e., $\bK{}{} = \bK{}{\x}$, then \pub{[Angyan et al. JCTC 7 (2011) 3116]}
\begin{equation} \begin{equation}
\green{\Ec^\RPAx} \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}}) ] \alert{=} \frac{1}{4} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
\end{equation} \end{equation}
\end{block} \end{block}
@ -1438,7 +1445,7 @@ decoration={snake,
\begin{equation} \begin{equation}
\boxed{ \boxed{
\green{\Ec^\BSE} \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}}) ] \alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\BSE}} - \Tr(\orange{\bA{}{\BSE}}) ]
} }
\end{equation} \end{equation}
@ -1469,6 +1476,7 @@ decoration={snake,
\For{$k=1,\ldots,K$} \For{$k=1,\ldots,K$}
\State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}(\omega = 0)$ \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 $\bY{}{\violet{\lambda_k}}$
\Comment{\alert{This is a $\order*{N^6}$ step done many times!}}
\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 $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$
\EndFor \EndFor
@ -1499,13 +1507,14 @@ decoration={snake,
\bigskip \bigskip
\item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102 \item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
\bigskip \bigskip
\item \orange{Review:} \item \orange{Reviews \& Books:}
\begin{itemize} \begin{itemize}
\item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344 \item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344
\item Onida et al. Rev. Mod. Phys. 74 (2002) 601 \item Onida et al. Rev. Mod. Phys. 74 (2002) 601
\item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022 \item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022
\item Golze et al. Front. Chem. 7 (2019) 377 \item Golze et al. Front. Chem. 7 (2019) 377
\item Blase et al. JPCL 11 (2020) 7371 \item Blase et al. JPCL 11 (2020) 7371
\item Martin, Reining \& Ceperley \textit{Interacting Electrons} (Cambridge University Press)
\end{itemize} \end{itemize}
\bigskip \bigskip
\item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com}) \item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com})

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