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