modifications in Sec IV
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@ -326,7 +326,7 @@ Then, as performed in Sec.~\ref{sec:srggw}, one can collect order by order the t
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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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A way around this problem is to transform Eq.~\eqref{eq:quasipart_eq} to its upfolded version which elegantly highlights the coupling terms 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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@ -374,8 +374,8 @@ The usual $GW$ non-linear equation can be obtained by applying L\"owdin partitio
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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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Equations \eqref{eq:GWlin} and \eqref{eq:quasipart_eq} \titou{yield exactly the same energies} 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 only $\order{K}$ 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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@ -385,27 +385,41 @@ Therefore, it is natural to define, within the SRG formalism, the diagonal and o
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\label{eq:diag_and_offdiag}
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\bH^\text{d}(s) &=
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\begin{pmatrix}
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\bF & \bO & \bO \\
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\bO & \bC^{\text{2h1p}} & \bO \\
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\bF & \bO & \bO \\
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\bO & \bC^{\text{2h1p}} & \bO \\
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\bO & \bO & \bC^{\text{2p1h}} \\
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\end{pmatrix},
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\\
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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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\bO & \bW^{\text{2h1p}} & \bW^{\text{2p1h}} \\
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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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where we omit the $s$ dependence of the matrices for the sake of brevity.
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Then, the aim is to solve order by order the flow equation [see Eq.~\eqref{eq:flowEquation}] knowing that the initial conditions are
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Then, the aim is to solve, order by order, the flow equation \eqref{eq:flowEquation} knowing that the initial conditions are
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\begin{subequations}
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\begin{align}
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\bHd{0}(0) &= \begin{pmatrix} \bF & \bO \\ \bO & \bC \end{pmatrix},
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\bHd{0}(0) & = \mqty( \bF & \bO \\ \bO & \bC ),
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&
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\bHod{0}(0) & = \bO,
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\\
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\bHd{1}(0) & = \bO,
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&
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\bHod{1}(0) &= \begin{pmatrix} \bO & \bW \\ \bW^{\dagger} & \bO \end{pmatrix},
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\bHod{1}(0) & = \mqty( \bO & \bW \\ \bW^{\dagger} & \bO ),
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\end{align}
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and $\bHod{0}(0) = \bHd{1}(0) = \bO$, where the matrices $\bC$ and $\bV$ collect the 2h1p and 2p1h channels.
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\end{subequations}
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where the supermatrices
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\begin{subequations}
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\begin{align}
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\bC & = \mqty( \bC^{\text{2h1p}} & \bO \\ \bO & \bC^{\text{2p1h}} )
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\\
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\bW & = \mqty( \bW^{\text{2h1p}} & \bW^{\text{2p1h}} )
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\end{align}
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\end{subequations}
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collect the 2h1p and 2p1h channels.
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Once the closed-form expressions of the low-order perturbative expansions are known, they can be inserted in Eq.~\eqref{eq:GWlin} before applying the downfolding process to obtain a renormalized version of the quasiparticle equation.
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In particular, we focus here on the second-order renormalized quasiparticle equation.
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@ -463,13 +477,13 @@ Equation \eqref{eq:F0_C0} implies
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\begin{align}
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\bF^{(1)}(s) &= \bF^{(1)}(0) = \bO, & \bC^{(1)}(s) &= \bC^{(1)}(0) = \bO,
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\end{align}
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and, thanks to the diagonal structure of $\bF^{(0)}$ and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
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and, thanks to the \titou{diagonal structure of $\bF^{(0)}$} and $\bC^{(0)}$, the differential equation for the coupling block in Eq.~\eqref{eq:W1} is easily solved and yields
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\begin{equation}
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W_{p,q\nu}^{(1)}(s) = W_{p,q\nu}^{(1)}(0) e^{- (F_{pp}^{(0)} - C_{q\nu,q\nu}^{(0)})^2 s}
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\end{equation}
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At $s=0$ the elements $W_{p,q\nu}^{(1)}(0)$ are equal to the screened two-electron integrals defined in Eq.~\eqref{eq:GW_sERI}, while for $s\to\infty$, they tend to zero.
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Therefore, $W_{p,q\nu}^{(1)}(s)$ are genuine renormalized two-electron screened integrals.
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It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} following a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
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It is worth noting the close similarity of the first-order elements with the ones derived by Evangelista in Ref.~\onlinecite{Evangelista_2014b} \titou{in a different context} following a second quantization formalism (see also Ref.~\onlinecite{Hergert_2016}).
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%///////////////////////////%
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\subsection{Second-order matrix elements}
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@ -490,7 +504,7 @@ with
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\end{align}
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\end{subequations}
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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 quasiparticle 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
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Collecting every second-order term in the flow equation and performing the block matrix products results in the following differential equation
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\begin{multline}
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\label{eq:diffeqF2}
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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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@ -500,7 +514,7 @@ which can be solved by simple integration along with the initial condition $\bF^
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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} \\
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\times \qty[1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s}],
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\end{multline}
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with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ \titou{(where $\epsilon_F$ is the ...)}.
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with $\Delta_{pr\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
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At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends towards the following static limit
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\begin{equation}
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@ -508,23 +522,24 @@ At $s=0$, the second-order correction vanishes while, for $s\to\infty$, it tends
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F_{pq}^{(2)}(\infty) = \sum_{r\nu} \frac{\Delta_{pr\nu}+ \Delta_{qr\nu}}{\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2} W_{p,r\nu} \titou{W_{q,r\nu}}.
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\end{equation}
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Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see Eq.~\eqref{eq:srg_sigma}] tends to zero.
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Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}(s)$.
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\titou{Therefore, the SRG flow continuously transforms the dynamic part $\widetilde{\bSig}(\omega; s)$ into a static correction $\widetilde{\bF}(s)$.}
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This transformation is done gradually starting from the states that have the largest denominators in Eq.~\eqref{eq:static_F2}.
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%///////////////////////////%
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\subsection{Alternative form of the static self-energy}
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% ///////////////////////////%
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Interestingly, the static limit, \ie the $s\to\infty$ 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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\PFL{This part has to be rewritten because it is too confusing.}
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Interestingly, the static limit, \ie the $s\to\infty$ limit of Eq.~\eqref{eq:GW_renorm}, defines an alternative qs$GW$ approximation to the one defined by Eq.~\eqref{eq:sym_qsgw} with matrix elements
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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\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}} ) W_{p,r\nu} \titou{W_{q,r\nu}}.
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\Sigma_{pq}^{\titou{\text{qs}}}(\eta) = \frac{1}{2} \sum_{r\nu} \qty( \frac{\Delta_{pr\nu}}{\Delta_{pr\nu}^2 + \titou{\eta^2}} +\frac{\Delta_{qr\nu}}{\Delta_{qr\nu}^2 + \titou{\eta^2}} ) W_{p,r\nu} \titou{W_{q,r\nu}}.
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\end{equation}
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This alternative static form will be refered to as SRG-qs$GW$ in the following.
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This alternative static form will be referred to as SRG-qs$GW$ in the following.
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Both approximations 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 usual 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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However, as we shall discuss further in Sec.~\ref{sec:results}, the convergence of the qs$GW$ scheme using \titou{$\widetilde{\bF}(\infty)$} 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 traditional qs$GW$ calculation 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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@ -532,9 +547,9 @@ Therefore, we will define the SRG-qs$GW$ static effective Hamiltonian as
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\label{eq:SRG_qsGW}
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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 \qty[(1 - e^{-(\Delta_{pr\nu}^2 + \Delta_{qr\nu}^2) s} ],
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\end{multline}
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which depends on one regularising parameter $s$ analogously to $\eta$ in the usual qs$GW$.
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which depends on one regularizing parameter $s$ analogously to $\eta$ in the usual qs$GW$.
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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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Therefore, one should use a value of $s$ large enough to include almost every state 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, 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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@ -545,7 +560,7 @@ However, doing a change of variable such that
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e^{-s} &= 1-e^{-t} & 1 - e^{-s} &= e^{-t}
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\end{align}
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hence choosing a finite value of $t$ is well-designed to avoid discontinuities in the dynamic.
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In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regulariser introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
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In fact, the dynamic part after the change of variable is closely related to the SRG-inspired regularizer introduced by Monino and Loos in Ref.~\onlinecite{Monino_2022}.
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%=================================================================%
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\section{Computational details}
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@ -570,8 +585,8 @@ However, in order to perform a black-box comparison of the methods these paramet
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\label{sec:results}
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%=================================================================%
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The results section is divided in two parts.
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The first step will be to analyse in depth the behavior of the two static self-energy approximations in a few test cases.
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The results section is divided into two parts.
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The first step will be to analyze in depth the behavior of the two static self-energy approximations in a few test cases.
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Then the accuracy of the IP yielded by the traditional and SRG schemes will be statistically gauged over a test set of molecules.
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%%%%%%%%%%%%%%%%%%%%%%
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@ -592,7 +607,7 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s
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This section starts by considering a prototypical molecular system, \ie the water molecule, in the aug-cc-pVTZ cartesian basis set.
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Figure~\ref{fig:fig1} shows the error of various methods for the principal IP with respect to (w.r.t.) the CCSD(T) reference value.
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The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well-understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
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The HF IP (dashed black line) is overestimated, this is a consequence of the missing correlation, a result which is now well understood. \cite{Lewis_2019} \ANT{I should maybe search for another ref as well.}
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\PFL{Check Szabo\&Ostlund, section on Koopman's theorem.}
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The usual qs$GW$ scheme (dashed blue line) brings a quantitative improvement as the IP is now within \SI{0.3}{\electronvolt} of the reference.
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%The Neon atom is a well-behaved system and could be converged without regularisation parameter while for water $\eta$ was set to 0.01 to help convergence.
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@ -618,27 +633,27 @@ Therefore, for small $s$ only the last term of Eq.~\eqref{eq:2nd_order_IP} will
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As soon as $s$ is large enough to decouple the 2h1p block as well the IP will start to decrease and eventually go below the $s=0$ initial value as observed in Fig.~\ref{fig:fig1}.
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In addition, the qs$GW$ and SRG-qs$GW$ methods based on a TDA screening are also considered in Fig.~\ref{fig:fig1}.
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The TDA IPs are now underestimated unlike their RPA counterparts.
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For both static self-energies, the TDA leads to a slight increase of the absolute error.
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This trend will be investigated in more details in the next subsection.
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The TDA IPs are now underestimated, unlike their RPA counterparts.
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For both static self-energies, the TDA leads to a slight increase in the absolute error.
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This trend will be investigated in more detail in the next subsection.
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Now the flow parameter dependence of the SRG-qs$GW$ method will be investigated in three less well-behaved molecular systems.
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The left panel of Fig.~\ref{fig:fig2} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated .
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The left panel of Fig.~\ref{fig:fig2} shows the results for the Lithium dimer, \ce{Li2} is an interesting case because the HP IP is actually underestimated.
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On the other hand, the qs-$GW$ and SRG-qs$GW$ IPs are overestimated
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Indeed, we can see that the positive increase of the SRG-qs$GW$ IP is proportionally more important than for water.
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In addition, the plateau is reached for larger values of $s$ in comparison to Fig.~\ref{fig:fig1}.
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Both TDA results are worse than their RPA counterparts but in this case the SRG-qs$GW_\TDA$ is more accurate than the qs$GW_\TDA$.
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Now turning to the Lithium hydrid heterodimer, see middle panel of Fig.~\ref{fig:fig2}.
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In this case the qs$GW$ IP is actually worse than the HF one which is already pretty accurate.
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Now turning to the lithium hydride heterodimer, see the middle panel of Fig.~\ref{fig:fig2}.
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In this case, the qs$GW$ IP is actually worse than the HF one which is already pretty accurate.
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However, the SRG-qs$GW$ can improve slightly the accuracy with respect to HF.
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Finally, the Beryllium oxyde is considered as a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
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Finally, the beryllium oxide is considered a prototypical example of a molecular system difficult to converge because of intruder states. \cite{vanSetten_2015,Veril_2018,Forster_2021}
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The SRG-qs$GW$ could be converged without any problem even for large values of $s$.
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Once again, a plateau is attained and the corresponding value is slightly more accurate than its qs$GW$ counterpart.
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Note that for \ce{LiH} and \ce{BeO} the TDA actually improves the accuracy compared to RPA-based qs$GW$ schemes.
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However, as we will see in the next subsection these are just particular molecular systems and in average the RPA polarizability performs better than the TDA one.
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Also the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig2} but this is the other way around.
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However, as we will see in the next subsection these are just particular molecular systems and on average the RPA polarizability performs better than the TDA one.
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Also, the SRG-qs$GW_\TDA$ is better than qs$GW_\TDA$ in the three cases of Fig.~\ref{fig:fig2} but this is the other way around.
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Therefore, it seems that the effect of the TDA can not be systematically predicted.
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%%% FIG 2 %%%
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