saving work
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sfBSE.rty
94
sfBSE.rty
@ -4,6 +4,7 @@
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\trashXB}[1]{\textcolor{darkgreen}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\mc}{\multicolumn}
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@ -23,11 +24,9 @@
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\Hx}{\text{Hx}}
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\newcommand{\xc}{\text{xc}}
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\newcommand{\Ha}{\text{H}}
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\newcommand{\co}{\text{c}}
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\newcommand{\ex}{\text{x}}
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\newcommand{\co}{\text{x}}
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%
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\newcommand{\Norb}{N_\text{orb}}
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@ -37,21 +36,14 @@
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% operators
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\newcommand{\hH}{\Hat{H}}
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\newcommand{\ha}{\Hat{a}}
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% methods
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\newcommand{\KS}{\text{KS}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\RPAx}{\text{RPAx}}
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\newcommand{\TDHF}{\text{TDHF}}
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\newcommand{\dRPAx}{\text{dRPAx}}
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\newcommand{\BSE}{\text{BSE}}
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\newcommand{\TDABSE}{\text{BSE(TDA)}}
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\newcommand{\dBSE}{\text{dBSE}}
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\newcommand{\pBSE}{\text{pBSE}}
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\newcommand{\TDAdBSE}{\text{dBSE(TDA)}}
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\newcommand{\GW}{GW}
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\newcommand{\GF}{\text{GF2}}
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\newcommand{\stat}{\text{stat}}
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\newcommand{\dyn}{\text{dyn}}
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\newcommand{\TDA}{\text{TDA}}
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@ -65,23 +57,52 @@
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\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
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% orbital energies
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\newcommand{\e}[2]{\eps_{#1}^{#2}}
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\newcommand{\e}[1]{\eps_{#1}}
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\newcommand{\eHF}[1]{\eps^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\eps^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\eps^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\eps^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\eps^{GW}_{#1}}
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\newcommand{\eGF}[1]{\eps^{\text{GF2}}_{#1}}
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\newcommand{\eevGW}[1]{\eps^\text{\evGW}_{#1}}
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\newcommand{\eGnWn}[2]{\eps^\text{\GnWn{#2}}_{#1}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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\newcommand{\tOm}[2]{\Tilde{\Omega}_{#1}^{#2}}
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\newcommand{\homu}{\frac{{\omega}_1}{2}}
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% Matrix elements
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\newcommand{\Z}[1]{Z_{#1}}
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\newcommand{\A}[2]{A_{#1}^{#2}}
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\newcommand{\tA}[2]{\Tilde{A}_{#1}^{#2}}
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\newcommand{\B}[2]{B_{#1}^{#2}}
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\renewcommand{\S}[1]{S_{#1}}
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\newcommand{\ABSE}[2]{A_{#1}^{#2,\text{BSE}}}
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\newcommand{\BBSE}[2]{B_{#1}^{#2,\text{BSE}}}
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\newcommand{\ARPA}[2]{A_{#1}^{#2,\text{RPA}}}
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\newcommand{\BRPA}[2]{B_{#1}^{#2,\text{RPA}}}
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\newcommand{\ARPAx}[2]{A_{#1}^{#2,\text{RPAx}}}
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\newcommand{\BRPAx}[2]{B_{#1}^{#2,\text{RPAx}}}
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\newcommand{\G}[1]{G_{#1}}
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\newcommand{\LBSE}[1]{L_{#1}}
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\newcommand{\XiBSE}[1]{\Xi_{#1}}
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\newcommand{\Po}[1]{P_{#1}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\Sig}[1]{\Sigma_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^{\GW}_{#1}}
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\newcommand{\SigGF}[1]{\Sigma^{\GF}_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\ERI}[2]{(#1|#2)}
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\newcommand{\sERI}[2]{[#1|#2]}
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%% bold in Table
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\newcommand{\bb}[1]{\textbf{#1}}
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\newcommand{\rb}[1]{\textbf{\textcolor{red}{#1}}}
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\newcommand{\gb}[1]{\textbf{\textcolor{darkgreen}{#1}}}
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% excitation energies
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\newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}}
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\newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}}
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@ -94,19 +115,27 @@
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% Matrices
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\newcommand{\bO}{\mathbf{0}}
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\newcommand{\bH}{\mathbf{H}}
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\newcommand{\bR}{\mathbf{R}}
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\newcommand{\bS}{\mathbf{S}}
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\newcommand{\bX}{\mathbf{X}}
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\newcommand{\bY}{\mathbf{Y}}
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\newcommand{\bV}{\mathbf{V}}
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\newcommand{\bI}{\mathbf{1}}
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\newcommand{\bb}{\mathbf{b}}
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\newcommand{\bA}{\mathbf{A}}
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\newcommand{\bB}{\mathbf{B}}
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\newcommand{\bC}{\mathbf{C}}
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\newcommand{\bc}{\mathbf{c}}
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\newcommand{\bx}{\mathbf{x}}
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\newcommand{\bvc}{\mathbf{v}}
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\newcommand{\bSig}{\mathbf{\Sigma}}
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\newcommand{\bSigX}{\mathbf{\Sigma}^\text{x}}
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\newcommand{\bSigC}{\mathbf{\Sigma}^\text{c}}
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\newcommand{\bSigGW}{\mathbf{\Sigma}^{GW}}
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\newcommand{\be}{\mathbf{\epsilon}}
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\newcommand{\beGW}{\mathbf{\epsilon}^{GW}}
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\newcommand{\beGnWn}[1]{\mathbf{\epsilon}^\text{\GnWn{#1}}}
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\newcommand{\bde}{\mathbf{\Delta\epsilon}}
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\newcommand{\bdeHF}{\mathbf{\Delta\epsilon}^\text{HF}}
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\newcommand{\bdeGW}{\mathbf{\Delta\epsilon}^{GW}}
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\newcommand{\bOm}[1]{\mathbf{\Omega}^{#1}}
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\newcommand{\bA}[1]{\mathbf{A}^{#1}}
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\newcommand{\btA}[1]{\Tilde{\mathbf{A}}^{#1}}
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\newcommand{\bB}[1]{\mathbf{B}^{#1}}
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\newcommand{\bX}[2]{\mathbf{X}_{#1}^{#2}}
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\newcommand{\bY}[2]{\mathbf{Y}_{#1}^{#2}}
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\newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}}
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\newcommand{\bK}{\mathbf{K}}
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\newcommand{\bP}[1]{\mathbf{P}^{#1}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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@ -115,9 +144,7 @@
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\newcommand{\kcal}{kcal/mol}
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% orbitals, gaps, etc
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\newcommand{\updw}{\uparrow\downarrow}
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\newcommand{\upup}{\uparrow\uparrow}
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\newcommand{\eps}{\epsilon}
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\newcommand{\eps}{\varepsilon}
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\newcommand{\IP}{I}
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\newcommand{\EA}{A}
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\newcommand{\HOMO}{\text{HOMO}}
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@ -127,6 +154,11 @@
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\newcommand{\EgOpt}{\Eg^\text{opt}}
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\newcommand{\EB}{E_B}
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\newcommand{\si}{\sigma}
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\newcommand{\sip}{\sigma'}
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% addresses
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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50
sfBSE.tex
50
sfBSE.tex
@ -48,11 +48,57 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th
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\subsection{The dynamical screening}
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Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
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\begin{multline}
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W_{pq\si,rs\sip}(\omega) = \ERI{pq\si}{rs\sip}
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\\
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+ \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} \qty[ \frac{1}{\omega - \OmRPA{m} + i \eta} - \frac{1}{\omega + \OmRPA{m} - i \eta} ]
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\end{multline}
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\begin{equation}
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\ERI{pq\si}{rs\sip} = \iint \MO{p\si}(\br) \MO{q\si}(\br) \frac{1}{\abs{\br - \br'}} \MO{r\sip}(\br') \MO{s\sip}(\br') d\br d\br'
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\end{equation}
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\begin{equation}
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\sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\RPA}+\bY{m}{\RPA})_{ia\sip}
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\end{equation}
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\begin{equation}
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\label{eq:LR-RPA}
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\begin{pmatrix}
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\bA{\RPA} & \bB{\RPA} \\
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-\bB{\RPA} & -\bA{\RPA} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{m}{\RPA} \\
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\bY{m}{\RPA} \\
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\end{pmatrix}
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=
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\OmRPA{m}
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\begin{pmatrix}
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\bX{m}{\RPA} \\
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\bY{m}{\RPA} \\
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\end{pmatrix},
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\end{equation}
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with
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-A}
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\A{ia\si,jb\sip}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a} - \e{i}) + 2 \ERI{ia\si}{jb\sip},
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\\
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\label{eq:LR_RPA-B}
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\B{ia\si,jb\sip}{\RPA} & = 2 \ERI{ia\si}{bj\sip},
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\end{align}
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\end{subequations}
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\subsection{The $GW$ self-energy}
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The quasiparticle energies $\eGW{p}$ are obtained by solving the
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The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
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\begin{equation}
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\omega = \eHF{p} + \Z{p} \SigGW{p}(\omega)
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\omega = \eHF{p\sigma} + \SigGW{p\sigma}(\omega)
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Computational details}
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