wiring notes

Manuscript/MRGW.tex

@@ -86,7 +86,8 @@
 \newcommand{\bI}{\boldsymbol{1}}
 \newcommand{\bH}{\boldsymbol{H}}
 \newcommand{\bvc}{\boldsymbol{v}}
-\newcommand{\bSig}[1]{\boldsymbol{\Sigma}^{#1}}
+\newcommand{\bSig}[2]{\boldsymbol{\Sigma}_{#1}^{#2}}
+\newcommand{\bSigC}[1]{\boldsymbol{\Sigma}_{#1}^{\text{c}}}
 \newcommand{\be}{\boldsymbol{\epsilon}}
 \newcommand{\bOm}[1]{\boldsymbol{\Omega}^{#1}}
 \newcommand{\bA}[2]{\boldsymbol{A}_{#1}^{#2}}
@@ -125,6 +126,7 @@
 	\affiliation{\LCPQ}
 
 \begin{abstract}
+Here comes the abstract.
 %\bigskip
 %\begin{center}
 %	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
@@ -134,44 +136,61 @@
 
 \maketitle
 
+
+%%%%%%%%%%%%%%%%%%%%%%
+\section{Introduction}
+%%%%%%%%%%%%%%%%%%%%%%
+Here comes the introduction.
+
+%%%%%%%%%%%%%%%%%
+\section{Theory}
+%%%%%%%%%%%%%%%%%
+In the case of {\GOWO}, the quasiparticle equation reads
 \begin{equation}
 \label{eq:qp_eq}
 	\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
 \end{equation}
+where $\eps{p}{\HF}$ is the HF one-electron energy of the spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is
 \begin{equation}
 \label{eq:SigC}
 	\SigC{p}(\omega) 
 	= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}}
 	+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}}
 \end{equation}
+where 
 \begin{equation}
 	\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
 \end{equation}
+are the screened two-electron repulsion integrals where $\Om{m}{\RPA}$ and $\bX{m}{\RPA}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system:
 \begin{equation}
 	\bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA}
 \end{equation}
+with
 \begin{equation}
 	A_{ia,jb}^{\RPA} = (\eps{a}{\HF} - \eps{i}{\HF}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
 \end{equation}
+and
 \begin{equation}
 	\ERI{pq}{ia} = \iint \MO{p}(\br_1) \MO{q}(\br_1) \frac{1}{\abs{\br_1 - \br_2}} \MO{i}(\br_2) \MO{a}(\br_2) d\br_1 \dbr_2
 \end{equation}
 
+The spectral weight of the solution $\eps{p,s}{\GW}$ is given by
 \begin{equation}
 \label{eq:Z}
 	0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
 \end{equation} 
+with the following sum rules:
 \begin{align}
 	\sum_{s} Z_{p,s} & = 1
 	&
 	\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
 \end{align}
 
-
+As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
-
 \begin{equation}
 	\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
 \end{equation}
+with
 \begin{equation}
 \label{eq:Hp}
 	\bH^{(p)} = 
@@ -183,83 +202,96 @@
 		\T{(\bV{p}{\text{2p1h}})}	&	\bO				&	\bC{}{\text{2p1h}}	
 	\end{pmatrix}
 \end{equation}
+and where the expressions of the 2h1p and 2p1h blocks reads
 \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{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} 
+	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}
+with the following expressions for the coupling blocks:
 \begin{align}
-	V^\text{2h1p}_{p,klc} & = \sqrt{2} \ERI{pk}{cl}
+	V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
 	&
-	V^\text{2p1h}_{p,kcd} & = \sqrt{2} \ERI{pd}{kc}
+	V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
 \end{align}
-
+By solving the secular equation
 \begin{equation}
 	\det[ \bH^{(p)} - \omega \bI ] = 0 
 \end{equation}
+we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie, 
 \begin{equation}
 	\SigC{p}(\omega) 
 	= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} 
 	+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} 
 \end{equation}
+with
 \begin{equation}
 \label{eq:Z_proj}
 	Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
 \end{equation}
 
+In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space.
+Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$)
+\begin{equation}
+\label{eq:Hp_qia}
+	\bH^{(p,qia)} = 
+	\begin{pmatrix}
+		\eps{p}{\HF}				&	V_{p,qia}					&	\bV{p}{\text{2h1p}}		&	\bV{p}{\text{2p1h}}
+		\\
+		V_{qia,p}					&	C_{qia,qia}					&	\bC{qia}{\text{2h1p}}	&	\bC{qia}{\text{2p1h}}
+		\\
+		\T{(\bV{p}{\text{2h1p}})}	&	\T{(\bC{qia}{\text{2h1p}})}	&	\bC{}{\text{2h1p}}		&	\bO
+		\\
+		\T{(\bV{p}{\text{2p1h}})}	&	\T{(\bC{qia}{\text{2p1h}})}	&	\bO						&	\bC{}{\text{2p1h}}	
+	\end{pmatrix}
+\end{equation}
+with new blocks defined as
+\begin{gather}
+	V_{p,qia} = V_{qia,p} \sqrt{2} = \ERI{pq}{ia}
+	\\
+	C_{qia,qia} = \text{sgn}(\eps{q}{\HF} - \mu) \qty[ \qty(\eps{q}{\HF} + \eps{a}{\HF} - \eps{i}{\HF} ) + 2 \ERI{ia}{ia} ]
+	\\
+	C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
+	\\
+	C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
+\end{gather}
+
+Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy
 \begin{equation}
 \label{eq:Hp}
-	\bH^{(P,QIA)} = 
+	\bSigC{p,qia}(\omega)  = 
 	\begin{pmatrix}
-		\eps{P}{\HF}				&	W_{P,QIA}					&	\bW{P}{\text{2h1p}}		&	\bW{P}{\text{2p1h}}
+		\eps{p}{\HF} + \SigC{p}(\omega)				&	V_{p,qia} + \SigC{p,qia}(\omega)		
 		\\
-		W_{QIA,P}					&	D_{QIA,QIA}					&	\bD{QIA}{\text{2h1p}}	&	\bD{QIA}{\text{2p1h}}
+		V_{qia,p} + \SigC{qia,p}(\omega)			&	C_{qia,qia}	+ \SigC{qia}(\omega)				
 		\\
-		\T{(\bW{P}{\text{2h1p}})}	&	\T{(\bD{QIA}{\text{2h1p}})}	&	\bD{}{\text{2h1p}}		&	\bO
-		\\
-		\T{(\bW{P}{\text{2p1h}})}	&	\T{(\bD{QIA}{\text{2p1h}})}	&	\bO						&	\bD{}{\text{2p1h}}	
 	\end{pmatrix}
 \end{equation}
 with
 \begin{gather}
-	W_{P,QIA} = \sqrt{2} \ERI{PQ}{IA}
+	\SigC{p}(\omega) 
+	  = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} 
+	  + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} 
 	\\
-	D_{QIA,QIA} = \text{sgn}(\eps{Q}{\HF} - \mu) \qty[ \qty(\eps{Q}{\HF} + \eps{A}{\HF} - \eps{I}{\HF} ) + 2 \ERI{IA}{IA} ]
+	\SigC{qia}(\omega) 
+	  = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} 
+	  + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} 	
 	\\
-	D_{QIA,klc}^\text{2h1p} = - 2 \ERI{IA}{cl} \delta_{Qk}
+	\SigC{p,qia}(\omega) 
+	  = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})} 
+	  + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})} 
 	\\
-	D_{QIA,klc}^\text{2p1h} = + 2 \ERI{IA}{kc} \delta_{Qd}
+	\SigC{qia,p}(\omega) 
+	  = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})} 
+	  + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})} 
 \end{gather}
+Of course, the present procedure can be generalized to any number of states.
 
-\begin{equation}
+%%%%%%%%%%%%%%%%%%%%%%
-\label{eq:Hp}
+\section{Conclusion}
-	\bH^{(P,QIA)} = 
+%%%%%%%%%%%%%%%%%%%%%%
-	\begin{pmatrix}
+Here comes the conclusion.
-		\eps{P}{\HF} + \SigC{P}(\omega)				&	W_{P,QIA} + \SigC{P,QIA}(\omega)		
+d
-		\\
-		W_{QIA,P} + \SigC{QIA,P}(\omega)					&	D_{QIA,QIA}	+ \SigC{QIA}(\omega)				
-		\\
-	\end{pmatrix}
-\end{equation}
-with
-\begin{align}
-	\SigC{P}(\omega) 
-	& = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})} 
-	  + \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})} 
-	\\
-	\SigC{QIA}(\omega) 
-	& = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})} 
-	  + \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})} 	
-	\\
-	\SigC{P,QIA}(\omega) 
-	& = \bW{P}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2h1p}})} 
-	  + \bW{P}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bD{QIA}{\text{2p1h}})} 
-	\\
-	\SigC{QIA,P}(\omega) 
-	& = \bD{QIA}{\text{2h1p}} \cdot \qty(\omega \bI - \bD{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2h1p}})} 
-	  + \bD{QIA}{\text{2p1h}} \cdot \qty(\omega \bI - \bD{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bW{P}{\text{2p1h}})} 
-\end{align}
-
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
 This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}