working on fig1
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@ -231,9 +231,10 @@ These additional solutions with large weights are the previously mentioned intru
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One obvious drawback of the one-shot scheme mentioned above is its starting point dependence.
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Indeed, in Eq.~\eqref{eq:G0W0} we choose to rely on HF orbital energies but this is arbitrary and one could have chosen Kohn-Sham energies (and orbitals) instead.
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As commonly done, one can even ``tune'' the starting point to obtain the best possible one-shot $GW$ quasiparticle energies. \cite{Korzdorfer_2012,Marom_2012,Bruneval_2013,Gallandi_2015,Caruso_2016, Gallandi_2016}
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Alternatively, one may solve this set of quasiparticle equations self-consistently leading to the ev$GW$ scheme.
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\titou{The solutions $\epsilon_p$ are used to build Eq.~\eqref{eq:G0W0} instead of the HF ones and then these equations are solved for $\omega$ again.}
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This procedure is iterated until convergence on the quasiparticle energies is reached.
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Alternatively, one may solve this set of quasiparticle equations self-consistently leading to the ev$GW$ scheme.
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\ant{To do so the quasiparticle energies are used to define a new RPA problem leading to updated two-electron screened integrals.
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Then the diagonal elements of the self-energy are updated as well and Eq.~\eqref{eq:G0W0} is solved again to obtain new quasiparticle energies.}
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This procedure is then iterated until convergence on the quasiparticle energies is reached.
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However, if one of the quasiparticle equations does not have a well-defined quasiparticle solution, reaching self-consistency can be challenging, if not impossible.
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Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}
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@ -265,7 +266,6 @@ One can deal with them by introducing \textit{ad hoc} regularizers.
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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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\PFL{SRG is energy-dependent.}
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%%%%%%%%%%%%%%%%%%%%%%
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\section{The similarity renormalization group}
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@ -530,6 +530,16 @@ Note that, in the limit $s\to\infty$, the dynamic part of the self-energy [see E
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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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%%% FIG 1 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\linewidth]{fig1.pdf}
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\caption{
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Add caption
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\label{fig:fig1}}
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\end{figure*}
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%%% %%% %%% %%%
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%///////////////////////////%
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\subsection{Alternative form of the static self-energy}
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% ///////////////////////////%
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@ -543,14 +553,21 @@ This yields a $s$-dependent static self-energy which matrix elements read
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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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Note that the SRG static approximation is naturally Hermitian as opposed to the usual case [see Eq.~(\ref{eq:sym_qsGW})] where it is enforced by symmetrization.
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Another important difference is that the SRG regularizer is energy dependent while the imaginary shift is the same for every state.
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Yet, these approximations are closely related because for $\eta=0$ and $s\to\infty$ they share the same diagonal terms.
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It is well-known that in traditional qs$GW$ calculation increasing the imaginary shift $\ii\eta$ to ensure convergence in difficult cases is most often unavoidable.
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Similarly, in SRG-qs$GW$ one might need to decrease the value of $s$ to ensure convergence.
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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, 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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It is instructive to plot both regularizing functions, this is done in Fig.~\ant{???}.
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The surfaces correspond to a value of the regularizing parameter value of 1.
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The SRG surface is much smoother than its qs counterpart.
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In fact the SRG regularization has less work to do because for $\eta=0$ there is a single singularity at $x=y=0$.
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On the other hand the function $f_{\text{qs}}(x,y;0)$ is singular on two entire axis, $x=0$ and $y=0$.
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The smoothness of the SRG surface might induce a smoother convergence of SRG-qs$GW$ compared to qs$GW$.
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The convergence properties and the accuracy of both static approximations will be quantitatively gauged in the results section.
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In addition,
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To conclude this section, the case of discontinuities will be briefly discussed.
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Indeed, it has been previously 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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@ -598,14 +615,24 @@ Then the accuracy of the IP yielded by the traditional and SRG schemes will be s
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%%% FIG 1 %%%
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\begin{figure}
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\centering
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\includegraphics[width=\linewidth]{fig1.pdf}
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\includegraphics[width=\linewidth]{fig2.pdf}
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\caption{
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Principal IP of the water molecule in the aug-cc-pVTZ cartesian basis set as a function of the flow parameter $s$ for the SRG-qs$GW$ method with and without TDA.
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Reference values (HF, qs$GW$ with and without TDA) are also reported as dashed lines.
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\label{fig:fig1}}
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\label{fig:fig2}}
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\end{figure}
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%%% %%% %%% %%%
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%%% FIG 2 %%%
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\begin{figure*}
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\centering
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\includegraphics[width=\linewidth]{fig3.pdf}
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\caption{
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Add caption
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\label{fig:fig3}}
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\end{figure*}
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%%% %%% %%% %%%
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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 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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@ -657,15 +684,7 @@ However, as we will see in the next subsection these are just particular molecul
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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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\begin{figure*}
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\centering
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\includegraphics[width=\linewidth]{fig2.pdf}
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\caption{
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Add caption
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\label{fig:fig2}}
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\end{figure*}
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Statistical analysis}
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