saving wotk

Manuscript/ufGW.tex

@@ -84,10 +84,6 @@
 \newcommand{\rbra}[1]{(#1|}
 \newcommand{\rket}[1]{|#1)}
 
-%% bold in Table
-\newcommand{\bb}[1]{\textbf{#1}}
-\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
-\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
 
 % Matrices
 \newcommand{\bO}{\boldsymbol{0}}
@@ -104,6 +100,7 @@
 \newcommand{\bX}[2]{\boldsymbol{X}_{#1}^{#2}}
 \newcommand{\bY}[2]{\boldsymbol{Y}_{#1}^{#2}}
 \newcommand{\bZ}[2]{\boldsymbol{Z}_{#1}^{#2}}
+\newcommand{\bc}[2]{\boldsymbol{c}_{#1}^{#2}}
 
 % orbitals, gaps, etc
 \newcommand{\IP}{I}
@@ -148,6 +145,11 @@
 \end{abstract}
 
 \maketitle
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+We consider {\GOWO} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Downfold: The non-linear $GW$ problem}
@@ -162,8 +164,8 @@ Within the {\GOWO} approximation, in order to obtain the quasiparticle energies
 where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of the {\GOWO} self-energy reads
 \begin{equation}
 	\SigC{p}(\omega) 
-	= \sum_{im} \frac{\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta}
+	= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA} - i \eta}
-	+ \sum_{am} \frac{\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta}
+	+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA} + i \eta}
 \end{equation}
 Within the Tamm-Dancoff approximation, the screened two-electron integrals are given by
 \begin{equation}
@@ -177,41 +179,67 @@ with
 \begin{equation}
 	A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
 \end{equation}
-As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,\ell}{\GW}$ and their corresponding weight is given by the value of the so-called renormalization factor
+As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{}$ and their corresponding weight is given by the value of the so-called renormalization factor
 \begin{equation}
-	0 \le Z_{p,\ell} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,\ell}{\GW}} ]^{-1} \le 1
+	0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{}} ]^{-1} \le 1
 \end{equation} 
-In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{\GW} \equiv \eps{p,\ell=0}{\GW}$ has a large weight $Z_{\ell} \equiv Z_{p,\ell=0}$
+In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{} \equiv \eps{p,s=0}{}$ has a large weight $Z_{} \equiv Z_{p,=0}$
 Note that we have the following important conservation rules
 \begin{align}
-	\sum_{\ell} Z_{p,\ell} & = 1
+	\sum_{s} Z_{p,s} & = 1
 	&
-	\sum_{\ell} Z_{p,\ell} \eps{p,\ell}{\GW} & = \eps{p}{\HF}
+	\sum_{s} Z_{p,s} \eps{p,s}{} & = \eps{p}{\HF}
 \end{align}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Upfolding: the linear $GW$ problem}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The non-linear quasiparticle equation \eqref{eq:qp_ep} can be transformed into a larger linear problem via an upfolding process:
+The non-linear quasiparticle equation \eqref{eq:qp_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
+is downfolded on the 1h and 1p sectors via their interaction with the 2h1p and 2p1h sectors:
 \begin{equation}
-	C^\text{1h1p}_{iajb,kcld} = \qty[ \qty( \eps{b}{\GW} + \eps{a}{\HF} - \eps{i}{\GW} - \eps{j}{\HF} ) \delta_{jl} \delta_{ac} + 2 \ERI{ja}{cl} ] \delta_{ik} \delta_{bd}
+	\bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{} \bc{}{(p,s)}
 \end{equation}
-\begin{align}
+with
-	V^\text{2h1p}_{p,kld} & = \ERI{pk}{ld}
-	&
-	V^\text{2h1p}_{p,cld} & = \ERI{pc}{ld}
-\end{align}
-
 \begin{equation}
-	\bH = 
+\label{eq:Hp}
+	\bH^{(p)} = 
 	\begin{pmatrix}
-		\Tilde{\bA{}{}}		&	\bV{}{(1)}	&	\bV{}{(2)}
+		\eps{p}{\HF}		&	\bV{p}{\text{2h1p}}	&	\bV{p}{\text{2p1h}}
 		\\
-		\T{(\bV{}{(1)})}	&	\bC{}{}			&	\bO
+		\T{(\bV{p}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}			&	\bO
 		\\
-		\T{(\bV{}{(2)})}	&	\bO				&	\bC{}{}	
+		\T{(\bV{p}{\text{2p1h}})}	&	\bO				&	\bC{}{\text{2p1h}}	
 	\end{pmatrix}
 \end{equation}
+where
+\begin{align}
+	C^\text{2h1p}_{ija,kcl} & = \qty[ \qty( \eps{i}{\HF} + \eps{j}{\HF} - \eps{a}{\HF}) \delta_{jl} \delta_{ac} - 2 \ERI{ja}{cl} ] \delta_{ik} 
+	\\
+	C^\text{2p1h}_{iab,kcd} & = \qty[ \qty( \eps{a}{\HF} + \eps{b}{\HF} - \eps{i}{\HF}) \delta_{ik} \delta_{ac} + 2 \ERI{ai}{kc} ] \delta_{bd} 
+\end{align}
+and the corresponding coupling blocks read
+\begin{align}
+	V^\text{2h1p}_{p,kld} & = \sqrt{2} \ERI{pk}{cl}
+	&
+	V^\text{2p1h}_{p,cld} & = \sqrt{2} \ERI{pd}{kc}
+\end{align}
+The size of this eigenvalue problem is $N = 1 + N^\text{2h1p} + N^\text{2p1h} = 1 + O^2 V + O V^2$.
+Because the renormalization factor corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference space, the weight of a solution $(p,s)$ is given by the  the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie, 
+\begin{equation}
+	Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
+\end{equation}
+
+It is important to understand that diagonalizing $\bH^{(p)}$ in Eq.~\eqref{eq:Hp} is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}.
+The main difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the eigenvectors associated with each quasiparticle and satellites.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section{Introducing regularized $GW$ methods}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+One way to hamper such issues is to resort to regularization of the $GW$ self-energy.
+Of course, the way of regularizing the self-energy is not unique but here we consider 3 different ways directly imported from MP2 theory.
+
+This helps greatly convergence for (partially) self-consistent $GW$ methods.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{