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@ -132,7 +132,7 @@ In fact, these cases are related to the discontinuities and convergence problems
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One obvious flaw of the one-shot scheme mentioned above is its starting point dependence.
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Indeed, in Eq.~(\ref{eq:G0W0}) we chose to use the HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham orbitals for example.
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Therefore, one could try to optimize the starting point to obtain the best one-shot energies possible. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
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Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2006,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016}
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Alternatively, one could solve this equation self-consistently leading to the eigenvalue-only self-consistent scheme. \cite{Shishkin_2007,Blase_2011,Marom_2012,Kaplan_2016,Wilhelm_2016}
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To do so the energy of the quasi-particle solution of the previous iteration is used to build Eq.~(\ref{eq:G0W0}) and then this equation is solved for $\omega$ again until convergence is reached.
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However, if the quasi-particle solution is not well-defined, self-consistency can be quite difficult, if not impossible, to reach.
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Even if self-consistency has been reached, the starting point dependence has not been totally removed because the results still depend on the starting molecular orbitals. \cite{Marom_2012}
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@ -204,17 +204,17 @@ and the corresponding coupling blocks read
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\end{align}
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Throughout the manuscript $p,q,r,s$ indices are used for 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 $v$ and $w$ will be used for neutral excitations, \ie composite indices $v=(ia)$.
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The usual $GW$ non-linear equation can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021}
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The usual $GW$ non-linear equation
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\begin{equation}
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\label{eq:GWnonlin}
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\left( \bF + \bSig(\omega) \right) \bX = \omega \bX,
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\end{equation}
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with
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can be obtained by applying L\"odwin partitioning technique to Eq.~(\ref{eq:GWlin}) \cite{Lowdin_1963,Bintrim_2021} which gives the following the expression for the self-energy
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\begin{align}
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\bSig(\omega) &= \bV^{\hhp} \left(\omega \mathbb{1} - \bC^{\hhp}\right)^{-1} (\bV^{\hhp})^{\mathsf{T}} \\
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&+ \bV^{\pph} \left(\omega \mathbb{1} - \bC^{\pph})^{-1} (\bV^{\pph}\right)^{\mathsf{T}}, \notag
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\end{align}
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which can be further developed to give
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which can be further developed as
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\begin{equation}
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\label{eq:GW_selfenergy}
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\Sigma_{pq}(\omega)
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@ -310,7 +310,7 @@ Then, one can collect order by order the terms in Eq.~(\ref{eq:flowEquation}) an
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%%%%%%%%%%%%%%%%%%%%%%
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Finally, the SRG formalism exposed above will be applied to $GW$.
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First, one needs to define the diagonal and off-diagonal parts of the $GW$ effective Hamiltonian.
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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 section~\ref{sec:gw}, the diagonal and off-diagonal parts will be defined as
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\begin{align}
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\label{eq:diag_and_offdiag}
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@ -329,7 +329,7 @@ As hinted at the end of section~\ref{sec:gw}, the diagonal and off-diagonal part
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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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Then, the aim of this section is to solve order by order the flow equation [see Eq.~(\ref{eq:flowEquation})] knowing that the initial conditions are
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Then, the aim is to solve order by order the flow equation [see Eq.~(\ref{eq:flowEquation})] knowing that the initial conditions are
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\begin{align}
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\bHd{0}(0) &= \begin{pmatrix}
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\bF{}{} & \bO \\
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@ -349,7 +349,7 @@ Then, the aim of this section is to solve order by order the flow equation [see
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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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Then, the perturbative expansions can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalized quasi-particle equation.
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Once the analytical low-order perturbative expansions are known they can be inserted in Eq.~(\ref{eq:GWlin}) before downfolding to obtain a renormalized quasi-particle equation.
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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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@ -373,26 +373,28 @@ where the $s$ dependence of $\bV^{(0)}$ and $\bV^{(0),\dagger}$ has been droppe
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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 $\bW^{(0)}= \bV^{(0)} \bU$.
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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,v),(q,w)} &= \delta_{pq} \bX_{v,w} \\
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D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_F)\Omega_v\right)\delta_{pq}\delta_{vw}
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\end{align}
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where $\epsilon_F$ is the Fermi level.
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Note that the matrix $\bU$ is also used in the downfolding process of Eq.~(\ref{eq:GWlin}). \cite{Bintrim_2021}
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Due to the diagonal structure of $\bF^{(0)}$ and $\bD^{(0)}$, these equations can be easily solved and give
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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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\begin{equation}
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W_{p,(q,v)}^{(0)}(s) = W_{p,(q,v)}^{(0)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(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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The two first equations of the system are trivial and finally, we have
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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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\bH^{(0)}(s) = \bH^{(0)}(0),
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\end{equation}
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which shows that the zero-th order matrix elements are independent of $s$.
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The matrix elements of $\bU$ and $\bD$ are
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\begin{align}
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U_{(p,v),(q,w)}^{(0)} &= \delta_{pq} \bX_{v,w} \\
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D_{(p,v),(q,w)}^{(0)} &= \left(\epsilon_p + \text{sign}(\epsilon_p-\epsilon_F)\Omega_v\right)\delta_{pq}\delta_{vw}
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\end{align}
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where $\epsilon_F$ is the Fermi level.
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\ie it is independent of $s$.
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%///////////////////////////%
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\subsubsection{First order matrix elements}
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@ -411,9 +413,9 @@ Once again the two first equations are easily solved
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and the first order coupling elements are given by (up to a multiplication by $\bU^{-1}$)
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\begin{align}
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W_{p,(q,v)}^{(1)}(s) &= W_{p,(q,v)}^{(1)}(0) e^{- (F_{pp}^{(0)} - D_{(q,v),(q,v)}^{(0)})^2 s} \\
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W_{p,(q,v)}^{(1)}(s) &= \left( \sum_{ia}\eri{pi}{qa}\qty( \bX_{v})_{ia} \right) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s}
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&= W_{p,(q,v)}^{(1)}(0) e^{- (\epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v)^2 s} \notag
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\end{align}
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Note that at $s=0$ the elements $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~(\ref{eq:GW_sERI}) and that for $s\to\infty$ they go to zero.
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At $s=0$ the elements $W_{p,(q,v)}^{(1)}(0)$ are equal to the two-electron screened integrals defined in Eq.~(\ref{eq:GW_sERI}) while for $s\to\infty$ they go to zero.
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Therefore, $W_{p,(q,v)}^{(1)}(s)$ are renormalized two-electrons screened integrals.
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Note the close similarity of the first-order element expressions with the ones of Evangelista in Ref.~\onlinecite{Evangelista_2014b} obtained in a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
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@ -434,7 +436,7 @@ with
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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 and performing the block matrix products results in the following differential equation for $\bF^{(2)}$
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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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\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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@ -443,35 +445,40 @@ This can be solved by simple integration along with the initial condition $\bF^{
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\begin{equation}
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F_{pq}^{(2)}(s) = \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right).
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\end{equation}
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with $\Delta_{pqv} = \epsilon_p - \epsilon_q - \text{sign}(\epsilon_q-\epsilon_F)\Omega_v$.
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with $\Delta_{prv} = \epsilon_p - \epsilon_r - \text{sign}(\epsilon_r-\epsilon_F)\Omega_v$.
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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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\label{eq:static_F2}
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F_{pq}^{(2)}(\infty) = \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
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F_{pq}^{(2)}(\infty) = \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W^{\dagger}_{(r,v),q}.
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\end{equation}
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Note that in the $s\to\infty$ limit the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
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Therefore, the SRG flow gradually transforms the dynamic degrees of freedom of $\bSig(\omega)$ in static ones, starting from the ones that have the largest denominators in Eq.~(\ref{eq:static_F2}).
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Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qs$GW$ approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}).
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Yet, both are closely related as they share the same diagonal terms.
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Also, note that the hermiticity is naturally enforced in the SRG static approximation as opposed to the symmetrized case.
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Therefore, the SRG flow transforms the dynamic part of $\bSig(\omega)$ into a static correction.
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This transformation is done gradually starting from the states that have the largest denominators in Eq.~(\ref{eq:static_F2}).
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Interestingly, the static limit, \ie $s\to\infty$ limit, of Eq.~(\ref{eq:GW_renorm}) defines an alternative qs$GW$ approximation to the one defined by Eq.~(\ref{eq:sym_qsgw}) which matrix elements read as
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\begin{equation}
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\label{eq:static_F2}
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\Sigma_{pq}^{\text{qs}GW}(\eta) = \sum_{r,v} \left( \frac{\Delta_{prv}}{\Delta_{prv}^2 + \eta^2} +\frac{\Delta_{qrv}}{\Delta_{qrv}^2 + \eta^2} \right) W_{p,(r,v)} W^{\dagger}_{(r,v),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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However, as will be discussed in more detail in the results section, the convergence of the qs$GW$ scheme using $\widetilde{\bF}(\infty)$ is very poor.
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This is similar to the symmetric case when the imaginary shift $\ii \eta$ is set to zero.
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Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable. \ant{ref fabien ?}
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Indeed, in qs$GW$ calculation using the symmetrized static form, increasing $\eta$ to ensure convergence in difficult cases is most often unavoidable.
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Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
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\begin{align}
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\label{eq:SRG_qsGW}
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\Sigma_{pq}^{\text{SRG}}(s) &= \epsilon_p \delta_{pq} + \sum_{r,v} \frac{\left(\epsilon_{p} + \epsilon_{q} - 2 (\epsilon_r \pm \Omega_v)\right) W_{p,(r,v)} W_{q,(r,v)}}{(\epsilon_p - \epsilon_r \pm \Omega_v)^2 + (\epsilon_q - \epsilon_r \pm \Omega_v)^2} \notag \\
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&\times \left(1 - e^{-(\epsilon_p - \epsilon_r \pm \Omega_v)^2s} e^{-(\epsilon_q - \epsilon_r \pm \Omega_v)^2s}\right)
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\Sigma_{pq}^{\text{SRG}}(s) = \frac{1}{2} \sum_{r,v} \frac{\Delta_{prv}+ \Delta_{qrv}}{\Delta_{prv}^2 + \Delta_{qrv}^2} W_{p,(r,v)} W_{q,(r,v)}\left(1 - e^{-(\Delta_{prv}^2 + \Delta_{qrv}^2) s}\right)
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\end{align}
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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 well 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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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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Therefore, we should use a value of $s$ large enough to include almost every states but small enough to avoid intruder states.
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To conclude this section, we will discuss the case of discontinuities.
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Indeed, we have previously said that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
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So is it possible to remove discontinuities by using the SRG machinery developed above?
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Indeed, previously we mentioned that intruder states are responsible for both the poor convergence of qs$GW$ and discontinuities in physical quantities at the $\GOWO$ level.
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So is it possible to use the SRG machinery developed above to remove discontinuities?
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In fact, not directly because discontinuities are due to intruder states in the dynamic part while we have seen just above that a finite value of $s$ is well-designed to avoid the intruder states in the static part.
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However, doing a change of variable such that
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\begin{align}
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@ -483,7 +490,7 @@ In fact, the dynamic part after the change of variable is closely related to the
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%=================================================================%
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\section{Computational details}
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\label{sec:comp_det}
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% =================================================================%
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%=================================================================%
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The two qs$GW$ variants considered in this work have been implemented in an in-house program.
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The $GW$ implementation closely follows the one of mol$GW$. \cite{Bruneval_2016}
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