new figs and data
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Data/scan_f2_cc-pvdz_cc3.dat
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Data/scan_f2_cc-pvdz_revgw.dat
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Manuscript/fig4.pdf
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Manuscript/fig5.pdf
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2022-02-22 14:38:25 +0100
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%% Created for Pierre-Francois Loos at 2022-02-23 11:35:26 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@ -178,6 +178,7 @@ In particular, we showed that each branch of the self-energy $\Sigma$ is associa
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Multiple solution issues in $GW$ appears frequently, \cite{vanSetten_2015,Maggio_2017,Duchemin_2020} especially for orbitals that are energetically far from the Fermi level, such as in core ionized states. \cite{Golze_2018,Golze_2020}
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In addition to obvious irregularities on potential energy surfaces that hampers the accurate determination of properties such as equilibrium bond lengths and harmonic vibrational frequencies, \cite{Loos_2020e,Berger_2021} one direct consequence of these discontinuities is the difficulty to converge (partially) self-consistent $GW$ calculations as the self-consistent procedure jumps erratically from one solution to the other even if convergence accelerator techniques such as DIIS are employed. \cite{Pulay_1980,Pulay_1982,Veril_2018}
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Note in passing that the present issues do not only appear in $GW$ as the $T$-matrix \cite{Romaniello_2012,Zhang_2017,Li_2021b,Loos_2022} and second-order Green's function (or second Born) formalisms \cite{SzaboBook,Casida_1989,Casida_1991,Stefanucci_2013,Ortiz_2013,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Hirata_2015,Hirata_2017} exhibit the same drawbacks.
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It was shown that these problems can be tamed by using a static Coulomb-hole plus screened-exchange (COHSEX) \cite{Hedin_1965,Hybertsen_1986,Hedin_1999,Bruneval_2006} self-energy \cite{Berger_2021} or by considering a fully self-consistent $GW$ scheme, \cite{Stan_2006,Stan_2009,Rostgaard_2010,Caruso_2012,Caruso_2013,Caruso_2013a,Caruso_2013b,Koval_2014,Wilhelm_2018} where one considers not only the quasiparticle solution but also the satellites at each iteration. \cite{DiSabatino_2021}
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However, none of these solutions is completely satisfying as a static approximation of the self-energy can induce significant loss in accuracy and fully self-consistent calculations can be quite challenging in terms of implementation and cost.
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@ -185,7 +186,7 @@ However, none of these solutions is completely satisfying as a static approximat
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In the present article, via an upfolding process of the non-linear $GW$ equation, \cite{Bintrim_2021a} we provide further physical insights into the origin of these discontinuities by highlighting, in particular, the role of intruder states.
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Inspired by regularized electronic structure theories, \cite{Lee_2018a,Evangelista_2014b} these new insights allow us to propose a cheap and efficient regularization scheme in order to avoid these issues.
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Here, we consider the one-shot {\GOWO} \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes.\cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Gui_2018,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016}
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Here, we consider the one-shot {\GOWO} \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} for the sake of simplicity but the same analysis can be performed in the case of (partially) self-consistent schemes such as ev$GW$, \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016} where one updates only the quasiparticle energies or qs$GW$, \cite{Gui_2018,Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016} where both quasiparticle energies and orbitals are updated at each iteration.
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Moreover, we consider a Hartree-Fock (HF) starting point but it can be straightforwardly extended to a Kohn-Sham starting point.
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Throughout this article, $p$ and $q$ are general (spatial) orbitals, $i$, $j$, $k$, and $l$ denotes occupied orbitals, $a$, $b$, $c$, and $d$ are vacant orbitals, while $m$ labels single excitations $i \to a$.
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Atomic units are used throughout.
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@ -379,7 +380,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
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% FIGURE 4
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{figure}
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% \includegraphics[width=\linewidth]{fig4}
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\includegraphics[width=\linewidth]{fig4}
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\caption{
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\label{fig:H2reg}
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Difference between regularized and non-regularized quasiparticle energies $\reps{p}{\GW} - \eps{p}{\GW}$ as functions of the internuclear distance $\RHH$ (in \si{\angstrom}) of \ce{H2} at the {\GOWO}@HF/6-31G level.
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@ -387,6 +388,18 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% FIGURE 5
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{figure}
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\includegraphics[width=\linewidth]{fig5}
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\caption{
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\label{fig:F2}
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Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set.
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}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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One way to alleviate the issues discussed above and to massively improve the convergence properties of self-consistent $GW$ calculations is to resort to a regularization of the self-energy without altering too much the quasiparticle energies.
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From a general perspective, a regularized $GW$ self-energy reads
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@ -429,18 +442,25 @@ We have found that $\eta = 1$ is a good compromise that does not alter significa
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To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized and non-regularized quasiparticle energies as functions of $\RHH$.
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As a final example, we report in Fig.~\ref{fig:F2} the ground-state potential energy surface of the \ce{F2} molecule obtained at various levels of theory with the cc-pVDZ basis.
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In particular, we compute, with and without regularization, the total energy at the Bethe-Salpeter equation (BSE) level \cite{Salpeter_1951,Strinati_1988,Blase_2018,Blase_2020} within the adiabatic connection fluctuation dissipation formalism \cite{Maggio_2016,Holzer_2018b,Loos_2020e} following the same protocol detailed in Ref.~\onlinecite{Loos_2020e}.
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These results are compared to high-level coupled-cluster calculations \cite{Purvis_1982,Christiansen_1995b} extracted from the same work.
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As already shown in Ref.~\onlinecite{Loos_2020e}, the potential energy surface of \ce{F2} at the BSE@{\GOWO}@HF (blue curve) is very ``bumpy'' and it is clear that the regularization scheme (black curve) allows to smooth it out without significantly altering the overall accuracy.
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Morevoer, while it is extremely challenging to perform self-consistent $GW$ calculations without regularization, it is now straightforward to compute the BSE@ev$GW$@HF potential energy surface.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Concluding remarks}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In the present article, we have provided mathematical and physical explanations behind the appearance of multiple solutions and discontinuities in various physical quantities computed within the $GW$ approximation.
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More precisely, we have evidenced that intruder states are the main cause behind these issues and that this downfall of $GW$ is a key signature of strong correlation.
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A simple and efficient regularization procedure inspired by the similarity renormalization group has been proposed to remove these discontinuities without altering too much the quasiparticle energies.
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Note in passing that the present issues do not only appear in $GW$ as the $T$-matrix \cite{Romaniello_2012,Zhang_2017,Li_2021b,Loos_2022} and second-order Green's function (or second Born) formalisms \cite{SzaboBook,Casida_1989,Casida_1991,Stefanucci_2013,Ortiz_2013,Phillips_2014,Phillips_2015,Rusakov_2014,Rusakov_2016,Hirata_2015,Hirata_2017} exhibit the same problems.
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Moreover, this regularization of the self-energy significantly speeds up the convergence of (partially) self-consistent $GW$ methods.
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We hope that these new physical insights and technical developments will broaden the applicability of Green's function methods in the molecular electronic structure community and beyond.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\acknowledgements{
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The authors thank Pina Romaniello for insightful discussions.
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The authors thank Pina Romaniello and Xavier Blase for insightful discussions.
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This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
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%%%%%%%%%%%%%%%%%%%%%%%%
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