new figs and data

Data/scan_f2_cc-pvdz_cc3.dat

@@ -0,0 +1,24 @@
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Data/scan_f2_cc-pvdz_ccsd.dat

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Data/scan_f2_cc-pvdz_g0w0.dat

@@ -0,0 +1,27 @@
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+1.65   -199.0976218392

Data/scan_f2_cc-pvdz_revgw.dat

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Data/scan_f2_cc-pvdz_rg0w0.dat

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Manuscript/ufGW.bib

@@ -1,7 +1,7 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2022-02-22 14:38:25 +0100 
+%% Created for Pierre-Francois Loos at 2022-02-23 11:35:26 +0100 
 
 
 %% Saved with string encoding Unicode (UTF-8) 

Manuscript/ufGW.tex

@@ -178,6 +178,7 @@ In particular, we showed that each branch of the self-energy $\Sigma$ is associa
 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}
 
 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}
+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.
 
 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}
 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.
@@ -185,7 +186,7 @@ However, none of these solutions is completely satisfying as a static approximat
 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.
 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.
 
-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}
+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. 
 Moreover, we consider a Hartree-Fock (HF) starting point but it can be straightforwardly extended to a Kohn-Sham starting point.
 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 $i \to a$.
 Atomic units are used throughout.
@@ -379,7 +380,7 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
 %	FIGURE 4
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{figure}
-%	\includegraphics[width=\linewidth]{fig4}
+	\includegraphics[width=\linewidth]{fig4}
 	\caption{
 	\label{fig:H2reg}
 	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.
@@ -387,6 +388,18 @@ Therefore, one can conclude that this downfall of $GW$ is a key signature of str
 \end{figure}	
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%	FIGURE 5
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{fig5}
+	\caption{
+	\label{fig:F2}
+	Ground-state potential energy surface of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVDZ basis set.
+}
+\end{figure}	
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 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.
 
 From a general perspective, a regularized $GW$ self-energy reads 
@@ -429,18 +442,25 @@ We have found that $\eta = 1$ is a good compromise that does not alter significa
 
 To further evidence this, Fig.~\ref{fig:H2reg} reports the difference between regularized and non-regularized quasiparticle energies as functions of $\RHH$.
 
+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.
+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}.
+These results are compared to high-level coupled-cluster calculations \cite{Purvis_1982,Christiansen_1995b} extracted from the same work.
+
+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.
+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. 
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Concluding remarks}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 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.
 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.
 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.
-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.
+Moreover, this regularization of the self-energy significantly speeds up the convergence of (partially) self-consistent $GW$ methods.
 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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \acknowledgements{
-The authors thank Pina Romaniello for insightful discussions.
+The authors thank Pina Romaniello and Xavier Blase for insightful discussions.
 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).}
 %%%%%%%%%%%%%%%%%%%%%%%%