moving big chunck of text around to make it clearer
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@ -250,71 +250,15 @@ One of the main results of the present manuscript is the derivation, from first
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Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
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Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
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Therefore, by suppressing this dependence the static approximation relies on the fact that there is one well-defined quasiparticle solution.
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Therefore, by suppressing this dependence, the static approximation relies on the fact that there is well-defined quasiparticle solutions.
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If it is not the case, the qs$GW$ self-consistent scheme inevitably oscillates between solutions with large spectral weights. \cite{Forster_2021}
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The satellites causing convergence problems are the above-mentioned intruder states.
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One can deal with them by introducing \textit{ad hoc} regularizers.
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The $\ii \eta$ term \titou{that is usually added in the denominators of the self-energy} [see Eq.~\eqref{eq:GW_selfenergy}] is the usual imaginary-shift regularizer used in various other theories \titou{...} by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
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Various other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
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But it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
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This is one of the aims of the present work.
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Applying the SRG to $GW$ could gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle.
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However, to do so one needs to identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
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The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version and the coupling terms will elegantly appear in the process.
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The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022}
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\begin{equation}
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\label{eq:GWlin}
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\begin{pmatrix}
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\bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
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\T{(\bW^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO \\
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\T{(\bW^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}} \\
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\end{pmatrix}
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\begin{pmatrix}
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\bZ^{\text{1h/1p}} \\
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\bZ^{\text{2h1p}} \\
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\bZ^{\text{2p1h}} \\
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\end{pmatrix}
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=
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\begin{pmatrix}
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\bZ^{\text{1h/1p}} \\
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\bZ^{\text{2h1p}} \\
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\bZ^{\text{2p1h}} \\
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\end{pmatrix}
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\boldsymbol{\epsilon},
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\end{equation}
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where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, the 2h1p and 2p1h matrix elements are
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\begin{subequations}
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\begin{align}
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C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
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\\
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C^\text{2p1h}_{a\nu,b\mu} & = \left(\epsilon_a + \Omega_\nu\right)\delta_{ab}\delta_{\nu\mu},
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\end{align}
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\end{subequations}
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and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
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\begin{align}
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W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu},
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&
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W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}.
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\end{align}
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The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021}
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\begin{equation}
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\begin{split}
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\bSig(\omega)
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& = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1} \T{(\bW^{\hhp})}
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\\
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& + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1} \T{(\bW^{\pph})},
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\end{split}
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\end{equation}
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which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
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Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other one is not.
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The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter.
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We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Refs.~\onlinecite{Tolle_2022,Scott_2023}).
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As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $\bW^\text{2h1p}$ and $\bW^\text{2p1h}$ are coupling the 1h and 1p configuration to the 2h1p and 2p1h configurations.
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Therefore, these blocks will be the target of the SRG transformation but before going into more detail we review the SRG formalism.
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\titou{The $\ii \eta$ term that is usually added in the denominators of the self-energy} [see Eq.~\eqref{eq:GW_selfenergy}] is similar to the usual imaginary-shift regularizer employed in various other theories affected by the intruder-state problem. \cite{Surjan_1996,Forsberg_1997,Monino_2022,Battaglia_2022}
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Several other regularizers are possible \cite{Stuck_2013,Rostam_2017,Lee_2018a,Evangelista_2014b,Shee_2021} and in particular, it was shown in Ref.~\onlinecite{Monino_2022} that a regularizer inspired by the SRG had some advantages over the imaginary shift.
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Nonetheless, it would be more rigorous, and more instructive, to obtain this regularizer from first principles by applying the SRG formalism to many-body perturbation theory.
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This is the central aim of the present work.
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%%%%%%%%%%%%%%%%%%%%%%
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\section{The similarity renormalization group}
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@ -366,7 +310,7 @@ For $s=0$, the initial problem is
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\begin{equation}
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\bH(0) = \bH^\text{d}(0) + \lambda \bH^\text{od}(0),
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\end{equation}
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where $\lambda$ is the usual perturbation parameter and the off-diagonal part of the Hamiltonian has been defined as the pertrubation.
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where $\lambda$ is the usual perturbation parameter and the off-diagonal part of the Hamiltonian has been defined as the perturbation.
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For finite values of $s$, we have the following perturbation expansion of the Hamiltonian
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\begin{equation}
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\label{eq:perturbation_expansionH}
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@ -380,9 +324,62 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
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\label{sec:srggw}
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%%%%%%%%%%%%%%%%%%%%%%
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Finally, the SRG formalism exposed above is applied to $GW$.
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The first step is to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
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As hinted at the end of Sec.~\ref{sec:gw}, it is natural to define the diagonal and off-diagonal parts as
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By applying the SRG to $GW$, our aim is to gradually remove the coupling between the quasiparticle and the satellites resulting in a renormalized quasiparticle equation.
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However, to do so, one must identify the coupling terms in Eq.~\eqref{eq:quasipart_eq}, which is not straightforward.
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The way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms.
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The upfolded $GW$ quasiparticle equation is \cite{Bintrim_2021,Tolle_2022}
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\begin{equation}
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\label{eq:GWlin}
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\begin{pmatrix}
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\bF & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
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(\bW^{\text{2h1p}})^\dag & \bC^{\text{2h1p}} & \bO \\
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(\bW^{\text{2p1h}})^\dag & \bO & \bC^{\text{2p1h}} \\
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\end{pmatrix}
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\begin{pmatrix}
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\bZ^{\text{1h/1p}} \\
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\bZ^{\text{2h1p}} \\
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\bZ^{\text{2p1h}} \\
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\end{pmatrix}
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=
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\begin{pmatrix}
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\bZ^{\text{1h/1p}} \\
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\bZ^{\text{2h1p}} \\
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\bZ^{\text{2p1h}} \\
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\end{pmatrix}
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\boldsymbol{\epsilon},
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\end{equation}
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where $\boldsymbol{\epsilon}$ is a diagonal matrix collecting the quasiparticle and satellite energies, the 2h1p and 2p1h matrix elements are
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\begin{subequations}
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\begin{align}
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C^\text{2h1p}_{i\nu,j\mu} & = \left(\epsilon_i - \Omega_\nu\right)\delta_{ij}\delta_{\nu\mu},
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\\
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C^\text{2p1h}_{a\nu,b\mu} & = \left(\epsilon_a + \Omega_\nu\right)\delta_{ab}\delta_{\nu\mu},
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\end{align}
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\end{subequations}
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and the corresponding coupling blocks read [see Eq.~(\ref{eq:GW_sERI})]
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\begin{align}
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W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu},
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&
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W^\text{2p1h}_{p,a\nu} & = W_{p,a\nu}.
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\end{align}
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The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} yielding \cite{Bintrim_2021}
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\begin{equation}
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\begin{split}
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\bSig(\omega)
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& = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1} (\bW^{\hhp})^\dag
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\\
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& + \bW^{\pph} \qty(\omega \bI - \bC^{\pph})^{-1} (\bW^{\pph})^\dag,
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\end{split}
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\end{equation}
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which can be further developed to recover exactly Eq.~\eqref{eq:GW_selfenergy}.
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Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} have exactly the same solutions but one is linear and the other is not.
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The price to pay for this linearity is that the size of the matrix in the former is $\order{K^3}$ while it is $\order{K}$ in the latter.
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We refer to Ref.~\onlinecite{Bintrim_2021} for a detailed discussion of the up/downfolding processes of the $GW$ equations (see also Refs.~\onlinecite{Tolle_2022,Scott_2023}).
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As can be readily seen in Eq.~\eqref{eq:GWlin}, the blocks $\bW^\text{2h1p}$ and $\bW^\text{2p1h}$ are coupling the 1h and 1p configuration to the 2h1p and 2p1h configurations.
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Therefore, it is natural to define, within the SRG formalism, the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian as
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\begin{subequations}
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\begin{align}
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\label{eq:diag_and_offdiag}
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@ -396,8 +393,8 @@ As hinted at the end of Sec.~\ref{sec:gw}, it is natural to define the diagonal
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\bH^\text{od}(s) &=
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\begin{pmatrix}
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\bO & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
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(\bW^{\text{2h1p}})^{\mathrm{T}} & \bO & \bO \\
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(\bW^{\text{2p1h}})^{\mathrm{T}} & \bO & \bO \\
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(\bW^{\text{2h1p}})^\dag & \bO & \bO \\
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(\bW^{\text{2p1h}})^\dag & \bO & \bO \\
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\end{pmatrix},
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\end{align}
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\end{subequations}
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