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@ -186,7 +186,7 @@ where $\eta$ is a positive infinitesimal and the screened two-electron integrals
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\end{equation}
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where $\bX$ and $\bY$ are the components of the eigenvectors of the particle-hole direct RPA problem defined as
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\begin{equation}
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\mqty( \bA & \bB \\ -\bA & -\bB ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \boldsymbol{\Omega},
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\mqty( \bA & \bB \\ -\bA & -\bB ) \mqty( \bX \\ \bY ) = \mqty( \bX \\ \bY ) \mqty( \boldsymbol{\Omega} & \bO \\ \bO & -\boldsymbol{\Omega} ),
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\end{equation}
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with
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\begin{subequations}
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@ -196,7 +196,7 @@ with
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B_{ia,jb} & = \eri{ij}{ab}.
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\end{align}
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\end{subequations}
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The diagonal matrix $\boldsymbol{\Omega}$ contains the eigenvalues and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
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The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
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\titou{The TDA case is discussed in Appendix \ref{sec:nonTDA}.}
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Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j,k,l$ and $a,b,c,d$ refers to occupied and virtual orbitals, respectively.
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The indices $\mu$ and $\nu$ are composite indices, \eg $\nu=(ia)$, referring to neutral excitations.
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@ -253,51 +253,52 @@ Various other regularisers are possible and in particular one of us has shown th
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But it would be more rigorous, and more instructive, to obtain this regulariser from first principles by applying the SRG formalism to many-body perturbation theory.
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This is the aim of the rest of this work.
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\ANT{The matrix element expressions should be changed}
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Applying the SRG to $GW$ could gradually remove the coupling between the quasi-particle and the satellites resulting in a renormalized quasi-particle.
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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 $GW$ quasi-particle equation is
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The upfolded $GW$ quasi-particle equation is \cite{Bintrim_2021,Tolle_2023}
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\begin{equation}
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\label{eq:GWlin}
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\begin{pmatrix}
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\bF & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
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\T{(\bV^{\text{2h1p}})} & \bC^{\text{2h1p}} & \bO \\
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\T{(\bV^{\text{2p1h}})} & \bO & \bC^{\text{2p1h}} \\
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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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\bX \\
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\bY^{\text{2h1p}} \\
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\bY^{\text{2p1h}} \\
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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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\bX \\
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\bY^{\text{2h1p}} \\
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\bY^{\text{2p1h}} \\
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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 quasi-particle 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}_{ija,klc} & = \qty[ \qty( \epsilon_{i} + \epsilon_{j} - \epsilon_{a} ) \delta_{jl} \delta_{ac} - \eri{jc}{al} ] \delta_{ik} ,
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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}_{iab,kcd} & = \qty[ \qty( \epsilon_{a} + \epsilon_{b} - \epsilon_{i} ) \delta_{ik} \delta_{ac} + \eri{ak}{ic} ] \delta_{bd},
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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
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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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V^\text{2h1p}_{p,klc} & = \eri{pc}{kl},
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W^\text{2h1p}_{p,i\nu} & = W_{p,i\nu},
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&
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V^\text{2p1h}_{p,kcd} & = \eri{pk}{dc}.
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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\"odwin partitioning technique \cite{Lowdin_1963} to Eq.~\eqref{eq:GWlin} which gives the following expression for the self-energy \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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& = \bV^{\hhp} \qty(\omega \mathbb{1} - \bC^{\hhp})^{-1} \T{(\bV^{\hhp})}
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& = \bW^{\hhp} \qty(\omega \bI - \bC^{\hhp})^{-1} \T{(\bW^{\hhp})}
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\\
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& + \bV^{\pph} \qty(\omega \mathbb{1} - \bC^{\pph})^{-1} \T{(\bV^{\pph})},
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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 give exactly Eq.~(\ref{eq:GW_selfenergy}).
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@ -339,7 +340,7 @@ where $\boldsymbol{\eta}(s)$, the flow generator, is defined as
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\boldsymbol{\eta}(s) = \dv{\bU(s)}{s} \bU^\dag(s) = - \boldsymbol{\eta}^\dag(s).
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\end{equation}
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To solve \titou{this equation} at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.
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To solve the flow equation at a lower cost than the one associated with the diagonalization of the initial Hamiltonian, one must introduce an approximate form for $\boldsymbol{\eta}(s)$.
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In this work, we consider Wegner's canonical generator \cite{Wegner_1994}
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\begin{equation}
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\boldsymbol{\eta}^\text{W}(s) = \comm{\bH^\text{d}(s)}{\bH(s)} = \comm{\bH^\text{d}(s)}{\bH^\text{od}(s)},
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@ -387,9 +388,9 @@ As hinted at the end of Sec.~\ref{sec:gw}, the diagonal and off-diagonal parts a
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& \\
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\bH^\text{od}(s) &=
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\begin{pmatrix}
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\bO & \bV^{\text{2h1p}} & \bV^{\text{2p1h}} \\
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(\bV^{\text{2h1p}})^{\mathrm{T}} & \bO & \bO \\
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(\bV^{\text{2p1h}})^{\mathrm{T}} & \bO & \bO \\
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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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\end{pmatrix}
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\end{align}
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where we have omitted the $s$ dependence of the matrix elements for the sake of brevity.
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@ -408,8 +409,8 @@ Then, the aim is to solve order by order the flow equation [see Eq.~\eqref{eq:fl
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\bO & \bO
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\end{pmatrix} &
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\bHod{1}(0) &= \begin{pmatrix}
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\bO & \bV{}{} \\
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\bV{}{\dagger} & \bO \notag
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\bO & \bW{}{} \\
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\bW^{\dagger} & \bO \notag
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\end{pmatrix} \notag
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\end{align}
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where we have defined the matrices $\bC$ and $\bV$ that collects the 2h1p and 2p1h channels for the sake of conciseness.
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@ -417,7 +418,7 @@ Once the analytical low-order perturbative expansions are known they can be inse
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In particular, in this manuscript, the focus will be on the second-order renormalized quasi-particle equation.
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%///////////////////////////%
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\subsection{Zeroth-order matrix elements}
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\subsection{Zeroth-order matrix elements \ANT{This subsection is false, I'll rewrite it soon}}
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%///////////////////////////%
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There is only one zeroth order term in the right-hand side of the flow equation
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@ -429,32 +430,31 @@ and performing the block matrix products gives the following system of equations
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\begin{align}
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\dv{\bF^{(0)}}{s} &= \bO \\
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\dv{\bC^{(0)}}{s} &= \bO \\
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\dv{\bV^{(0),\dagger}}{s} &= 2 \bC^{(0)}\bV^{(0),\dagger}\bF^{(0)} - \bV^{(0),\dagger}(\bF^{(0)})^2 - (\bC^{(0)})^2\bV^{(0),\dagger} \\
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\dv{\bV^{(0)}}{s} &= 2 \bF^{(0)}\bV^{(0)}\bC^{(0)} - (\bF^{(0)})^2\bV^{(0)} - \bV^{(0)}(\bC^{(0)})^2
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\dv{\bW^{(0),\dagger}}{s} &= 2 \bC^{(0)}\bW^{(0),\dagger}\bF^{(0)} - \bW^{(0),\dagger}(\bF^{(0)})^2 - (\bC^{(0)})^2\bW^{(0),\dagger} \\
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\dv{\bW^{(0)}}{s} &= 2 \bF^{(0)}\bW^{(0)}\bC^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)}(\bC^{(0)})^2
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\end{align}
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\end{subequations}
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where the $s$ dependence of $\bV^{(0)}$ and $\bV^{(0),\dagger}$ has been dropped in the last two equations.
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where the $s$ dependence of $\bW^{(0)}$ and $\bW^{(0),\dagger}$ has been dropped in the last two equations.
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$\bF^{(0)}$ and $\bC^{(0)}$ do not depend on $s$ as a consequence of the first two equations.
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The last equation can be solved by introducing $\bU$ the matrix that diagonalizes $\bC^{(0)} = \bU \bD^{(0)} \bU^{-1}$ such that the differential equation for $\bV^{(0)}$ becomes
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\begin{equation}
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\label{eq:eqdiffW0}
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\dv{\bW^{(0)}}{s} = 2 \bF^{(0)}\bW^{(0)} \bD^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)} (\bD^{(0)})^2
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\end{equation}
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where $\bW^{(0)}(s)= \bV^{(0)}(s) \bU$.
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The matrix elements of $\bU$ and $\bD^{(0)}$ are
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\begin{align}
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U_{p\nu,q\mu} &= \delta_{pq} \bX_{\nu\mu} \\
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D_{p\nu,q\mu}^{(0)} &= \left(\epsilon_p + \sgn(\epsilon_p-\epsilon_\text{F})\Omega_\nu\right)\delta_{pq}\delta_{\nu\mu}
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\end{align}
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where $\epsilon_\text{F}$ is the Fermi level.
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Note that the matrix $\bU$ is also used in the downfolding process of Eq.~\eqref{eq:GWlin}. \cite{Bintrim_2021}
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% The last equation can be solved by introducing $\bU$ the matrix that diagonalizes $\bC^{(0)} = \bU \bD^{(0)} \bU^{-1}$ such that the differential equation for $\bV^{(0)}$ becomes
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% \begin{equation}
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% \label{eq:eqdiffW0}
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% \dv{\bW^{(0)}}{s} = 2 \bF^{(0)}\bW^{(0)} \bD^{(0)} - (\bF^{(0)})^2\bW^{(0)} - \bW^{(0)} (\bD^{(0)})^2
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% \end{equation}
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% where $\bW^{(0)}(s)= \bV^{(0)}(s) \bU$.
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% The matrix elements of $\bU$ and $\bD^{(0)}$ are
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% \begin{align}
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% U_{p\nu,q\mu} &= \delta_{pq} \bX_{\nu\mu} \\
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% D_{p\nu,q\mu}^{(0)} &= \left(\epsilon_p + \sgn(\epsilon_p-\epsilon_\text{F})\Omega_\nu\right)\delta_{pq}\delta_{\nu\mu}
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% \end{align}
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% where $\epsilon_\text{F}$ is the Fermi level.
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% Note that the matrix $\bU$ is also used in the downfolding process of Eq.~\eqref{eq:GWlin}. \cite{Bintrim_2021}
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Thanks to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, Eq.~\eqref{eq:eqdiffW0} can be easily solved and give
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Thanks to the diagonal structure of $\bF^{(0)}$ and $\bC^{(0)}$, the last equation can be easily solved and give
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\begin{equation}
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W_{p,q\nu}^{(0)}(s) = W_{p,q\nu}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{q\nu,q\nu}^{(0)})^2 s}
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\end{equation}
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Due to the initial conditions $\bV^{(0)}(0) = \bO$, we have $\bW^{(0)}(s)=\bO$ and therefore $\bV^{(0)}(s)=\bO=\bV^{(0)}(0) $.
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Therefore, the zeroth order Hamiltonian is
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The initial condition $\bW^{(0)}(0) = \bO$ implies $\bW^{(0)}(s)=\bO$ and therefore the zeroth order Hamiltonian is
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\begin{equation}
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\bH^{(0)}(s) = \bH^{(0)}(0),
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\end{equation}
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@ -468,7 +468,7 @@ Knowing that $\bHod{0}(s)=\bO$, the first order flow equation is
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\begin{equation}
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\dv{\bH^{(1)}}{s} = \comm{\comm{\bHd{0}}{\bHod{1}}}{\bHd{0}}
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\end{equation}
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which gives the same system of equations as in the previous subsection except that $\bV^{(0)}$ and $\bV^{(0),\dagger}$ should be replaced by $\bV^{(1)}$ and $\bV^{(1),\dagger}$.
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which gives the same system of equations as in the previous subsection except that $\bW^{(0)}$ and $\bW^{(0),\dagger}$ should be replaced by $\bW^{(1)}$ and $\bW^{(1),\dagger}$.
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Once again the two first equations are easily solved
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\begin{align}
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@ -496,21 +496,21 @@ with
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\begin{align}
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\widetilde{\bF}(s) &= \bF^{(0)}+\bF^{(2)}(s)\\
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\label{eq:srg_sigma}
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\widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \mathbb{1} - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}
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\widetilde{\bSig}(\omega; s) &= \bV^{(1)}(s) \left(\omega \bI - \bC^{(0)}\right)^{-1} (\bV^{(1)}(s))^{\dagger}
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\end{align}
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As can be readily seen above, $\bF^{(2)}$ is the only second-order block of the effective Hamiltonian contributing to the second-order SRG quasi-particle equation.
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Collecting every second-order terms in the flow equation and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
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\begin{equation}
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\begin{multline}
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\label{eq:diffeqF2}
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\dv{\bF^{(2)}}{s} = \bF^{(0)}\bV^{(1)}\bV^{(1),\dagger} + \bV^{(1)}\bV^{(1),\dagger}\bF^{(0)} - 2 \bV^{(1)}\bC^{(0)}\bV^{(1),\dagger} .
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\end{equation}
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\dv{\bF^{(2)}}{s} = \bF^{(0)}\bW^{(1)}\bW^{(1),\dagger} + \bW^{(1)}\bW^{(1),\dagger}\bF^{(0)} \\ - 2 \bW^{(1)}\bC^{(0)}\bW^{(1),\dagger} .
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\end{multline}
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This can be solved by simple integration along with the initial condition $\bF^{(2)}=\bO$ to give
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\begin{multline}
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F_{pq}^{(2)}(s) = \sum_{r\mu} \frac{\Delta_{pr\mu}+ \Delta_{qr\mu}}{\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2} W_{p,r\mu} W^{\dagger}_{r\mu,q} \times \\
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\left(1 - e^{-(\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2) s}\right),
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F_{pq}^{(2)}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W^{\dagger}_{r\nu,q} \times \\
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\left(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}\right),
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\end{multline}
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with $\Delta_{prv} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_v$.
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with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$.
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At $s=0$, this second-order correction is null while for $s\to\infty$ it tends towards the following static limit
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\begin{equation}
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@ -524,7 +524,7 @@ y
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Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~\eqref{eq:GW_renorm} defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} which matrix elements read as
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\begin{equation}
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\label{eq:sym_qsGW}
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\Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\mu} \left( \frac{\Delta_{pr\mu}}{\Delta_{pr\mu}^2 + \eta^2} +\frac{\Delta_{qr\mu}}{\Delta_{qr\mu}^2 + \eta^2} \right) W_{p,r\mu} W^{\dagger}_{r\mu,q}.
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\Sigma_{pq}^{\text{qs}}(\eta) = \frac{1}{2} \sum_{r\nu} \left( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \eta^2} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \eta^2} \right) W_{p,r\nu} W^{\dagger}_{r\nu,q}.
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\end{equation}
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Yet, both approximation are closely related as they share the same diagonal terms when $\eta=0$.
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Also, note that the SRG static approximation is naturally Hermitian as opposed to the symmetrized case where it is enforced by symmetrization.
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@ -536,7 +536,7 @@ Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
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\begin{multline}
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\label{eq:SRG_qsGW}
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\Sigma_{pq}^{\text{SRG}}(s) = \sum_{r\mu} \frac{\Delta_{pr\mu}+ \Delta_{qr\mu}}{\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2} W_{p,r\mu} W_{q,r\mu} \times \\ \left(1 - e^{-(\Delta_{pr\mu}^2 + \Delta_{qr\mu}^2) s}\right)
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\Sigma_{pq}^{\text{SRG}}(s) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} W_{q,r\nu} \times \\ \left(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}\right)
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\end{multline}
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which depends on one regularising parameter $s$ analogously to $\eta$ in the usual case.
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The fact that the $s\to\infty$ static limit does not always converge when used in a qs$GW$ calculation could have been predicted because in this limit even the intruder states have been included in $\tilde{\bF}$.
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@ -612,7 +612,7 @@ This behavior as a function of $s$ can be \ant{approximately} streamlined by app
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Through second order, the principal IP is
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\begin{equation}
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\label{eq:2nd_order_IP}
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I_k(2) = - \epsilon_k - \sum_{i\nu} \frac{W_{k,i\nu}^2}{\epsilon_k - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{W_{k,a\nu}^2}{\epsilon_k - \epsilon_a - \Omega_\nu} + \order{3}
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I_k = - \epsilon_k - \sum_{i\nu} \frac{W_{k,i\nu}^2}{\epsilon_k - \epsilon_i + \Omega_\nu} - \sum_{a\nu} \frac{W_{k,a\nu}^2}{\epsilon_k - \epsilon_a - \Omega_\nu} + \order{3}
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\end{equation}
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where $k$ is the index of the highest molecular MO (HOMO).
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The first term is the zeroth order IP and the two following terms come from the 2h1p and 2p1h coupling, respectively.
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@ -695,7 +695,7 @@ The matrix elements of the $GW$ self-energy within the TDA are the same as in Eq
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\label{eq:GWnonTDA_sERI}
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W_{p,q\nu} = \sum_{ia}\eri{pi}{qa}\qty( \bX_{\nu})_{ia},
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\end{equation}
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where $\bX$ ais the eigenvector matrix of the TDA particle-hole dRPA problem defined as
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where $\bX$ is the eigenvector matrix of the TDA particle-hole dRPA problem defined as
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\begin{equation}
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\label{eq:full_dRPA}
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\bA \bX = \bX \boldsymbol{\Omega}
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@ -704,7 +704,7 @@ with
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\begin{align}
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A_{ia,jb} &= (\epsilon_i - \epsilon_a) \delta_{ij}\delta_{ab} + \eri{ib}{aj} \\
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\end{align}
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$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues. Note that $\boldsymbol{\Omega}$ has the same size as without the TDA because in RPA we consider only the positive excitations of the full dRPA problem.
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$\boldsymbol{\Omega}$ is the diagonal matrix of eigenvalues.
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Defining an unfold version of this equation that does not require a diagonalization of the RPA problem before unfolding is a tricky task (see supplementary material of Ref.~\onlinecite{Bintrim_2021}).
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However, because we will eventually downfold again the upfolded matrix, we can use the following matrix \cite{Tolle_2022}
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