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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress,onecolumn]{revtex4-1}
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts,siunitx}
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\usepackage[version=4]{mhchem}
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@ -148,26 +148,26 @@ Here comes the introduction.
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In the case of {\GOWO}, the quasiparticle equation reads
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\begin{equation}
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\label{eq:qp_eq}
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\eps{p}{\HF} + \SigC{p}(\omega) - \omega = 0
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\eps{p}{} + \SigC{p}(\omega) - \omega = 0
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\end{equation}
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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
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where $\eps{p}{}$ is the one-electron energy of the HF spatial orbital $\MO{p}(\br)$ and the correlation part of the frequency-dependent self-energy is
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\begin{equation}
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\label{eq:SigC}
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\SigC{p}(\omega)
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= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{\HF} + \Om{m}{\RPA}}
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+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}}
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= \sum_{im} \frac{2\ERI{pi}{m}^2}{\omega - \eps{i}{} + \Om{m}{}}
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+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{} - \Om{m}{}}
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\end{equation}
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where
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\begin{equation}
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\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
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\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}
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\end{equation}
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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:
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are the screened two-electron repulsion integrals where $\Om{m}{}$ and $\bX{m}{}$ are the $m$th RPA eigenvalue and eigenvector obtained by solving the following (linear) RPA eigenvalue system within the Tamm-Dancoff approximation:
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\begin{equation}
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\bA{}{\RPA} \cdot \bX{m}{\RPA} = \Om{m}{\RPA} \bX{m}{\RPA}
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\bA{}{\RPA} \cdot \bX{m}{} = \Om{m}{\RPA} \bX{m}{}
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\end{equation}
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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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A_{ia,jb}^{} = (\eps{a}{} - \eps{i}{}) \delta_{ij} \delta_{ab} + \ERI{ia}{bj}
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\end{equation}
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and
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\begin{equation}
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@ -183,7 +183,7 @@ with the following sum rules:
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\begin{align}
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\sum_{s} Z_{p,s} & = 1
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&
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\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
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\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{}
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\end{align}
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As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
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@ -195,7 +195,7 @@ with
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\label{eq:Hp}
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\bH^{(p)} =
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\begin{pmatrix}
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\eps{p}{\HF} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}}
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\eps{p}{} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}}
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\\
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\T{(\bV{p}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO
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\\
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@ -203,27 +203,38 @@ with
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\end{pmatrix}
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\end{equation}
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and where the expressions of the 2h1p and 2p1h blocks reads
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\begin{subequations}
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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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\label{eq:C2h1p}
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C^\text{2h1p}_{IJA,KCL} & = \qty[ \qty( \eps{I}{} + \eps{J}{} - \eps{A}{}) \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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\label{eq:C2p1h}
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C^\text{2p1h}_{IAB,KCD} & = \qty[ \qty( \eps{A}{} + \eps{B}{} - \eps{I}{}) \delta_{IK} \delta_{AC} + 2 \ERI{AI}{KC} ] \delta_{BD}
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\end{align}
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\end{subequations}
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with the following expressions for the coupling blocks:
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\begin{subequations}
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\begin{align}
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\label{eq:V2h1p}
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V^\text{2h1p}_{p,KLC} & = \sqrt{2} \ERI{pK}{CL}
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&
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\\
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\label{eq:V2p1h}
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V^\text{2p1h}_{p,KCD} & = \sqrt{2} \ERI{pD}{KC}
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\end{align}
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\end{subequations}
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Here, we use lower case letters for the electronic configurations belonging to the reference model state and upper case letters for the external determinants (\ie, the perturbers).
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By solving the secular equation
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\begin{equation}
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\det[ \bH^{(p)} - \omega \bI ] = 0
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\end{equation}
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we recover the dynamical expression of the self-energy \eqref{eq:SigC}, \ie,
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\begin{equation}
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\begin{multline}
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\SigC{p}(\omega)
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= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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\\
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{equation}
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\end{multline}
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with
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\begin{equation}
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\label{eq:Z_proj}
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@ -236,9 +247,9 @@ Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $
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\label{eq:Hp_qia}
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\bH^{(p,qia)} =
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\begin{pmatrix}
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\eps{p}{\HF} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}}
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\eps{p}{} & V_{p,qia} & \bV{p}{\text{2h1p}} & \bV{p}{\text{2p1h}}
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\\
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V_{qia,p} & C_{qia,qia} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}}
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V_{qia,p} & \eps{qia}{} & \bC{qia}{\text{2h1p}} & \bC{qia}{\text{2p1h}}
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\\
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\T{(\bV{p}{\text{2h1p}})} & \T{(\bC{qia}{\text{2h1p}})} & \bC{}{\text{2h1p}} & \bO
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\\
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@ -247,51 +258,68 @@ Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $
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\end{equation}
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with new blocks defined as
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\begin{gather}
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V_{p,qia} = V_{qia,p} \sqrt{2} = \ERI{pq}{ia}
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V_{p,qia} = V_{qia,p} = \sqrt{2} \ERI{pq}{ia}
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\\
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C_{qia,qia} = \text{sgn}(\eps{q}{\HF} - \mu) \qty[ \qty(\eps{q}{\HF} + \eps{a}{\HF} - \eps{i}{\HF} ) + 2 \ERI{ia}{ia} ]
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\eps{qia}{} \equiv C_{qia,qia} = \text{sgn}(\eps{q}{} - \mu) \qty[ \qty(\eps{q}{} + \eps{a}{} - \eps{i}{} ) + 2 \ERI{ia}{ia} ]
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\\
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C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
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\\
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C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
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\end{gather}
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The expressions of $\bC{p}{\text{2h1p}}$, $\bC{p}{\text{2p1h}}$, $\bV{}{\text{2h1p}}$, and $\bV{}{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has remove the contribution from the 2h1p or 2p1h configuration $qia$.
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While $\eps{p}{\HF}$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps{qia}{} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration.
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Therefore, when $\eps{p}{\HF}$ and $\eps{qia}{}$ becomes of similar mangitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems.
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Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy
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Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy matrix
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\begin{equation}
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\label{eq:Hp}
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\bSigC{p,qia}(\omega) =
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\begin{pmatrix}
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\eps{p}{\HF} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega)
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\eps{p}{} + \SigC{p}(\omega) & V_{p,qia} + \SigC{p,qia}(\omega)
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\\
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V_{qia,p} + \SigC{qia,p}(\omega) & C_{qia,qia} + \SigC{qia}(\omega)
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V_{qia,p} + \SigC{qia,p}(\omega) & \eps{qia}{} + \SigC{qia}(\omega)
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\\
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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{gather}
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\SigC{p}(\omega)
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= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\begin{split}
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\SigC{p}(\omega)
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& = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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\\
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& + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{split}
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\\
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\begin{split}
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\SigC{qia}(\omega)
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= \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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+ \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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& = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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\\
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& + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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\end{split}
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\\
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\begin{split}
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\SigC{p,qia}(\omega)
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= \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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& = \bV{p}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2h1p}})}
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\\
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& + \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bC{qia}{\text{2p1h}})}
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\end{split}
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\\
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\begin{split}
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\SigC{qia,p}(\omega)
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= \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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+ \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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& = \bC{qia}{\text{2h1p}} \cdot \qty(\omega \bI - \bC{}{\text{2h1p}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2h1p}})}
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\\
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& + \bC{qia}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
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\end{split}
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\end{gather}
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\end{subequations}
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Of course, the present procedure can be generalized to any number of states.
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%%%%%%%%%%%%%%%%%%%%%%
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\section{Conclusion}
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%%%%%%%%%%%%%%%%%%%%%%
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Here comes the conclusion.
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d
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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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).}
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