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a/Manuscript/fig1.pdf b/Manuscript/fig1.pdf index c41ee34..ec8848f 100644 Binary files a/Manuscript/fig1.pdf and b/Manuscript/fig1.pdf differ diff --git a/Manuscript/fig3a.pdf b/Manuscript/fig3a.pdf index c9e5513..4d345d2 100644 Binary files a/Manuscript/fig3a.pdf and b/Manuscript/fig3a.pdf differ diff --git a/Manuscript/fig3b.pdf b/Manuscript/fig3b.pdf index b7cf5e4..ceab123 100644 Binary files a/Manuscript/fig3b.pdf and b/Manuscript/fig3b.pdf differ diff --git a/Manuscript/fig4.pdf b/Manuscript/fig4.pdf new file mode 100644 index 0000000..8cadb05 Binary files /dev/null and b/Manuscript/fig4.pdf differ diff --git a/Manuscript/fig5.pdf b/Manuscript/fig5.pdf new file mode 100644 index 0000000..0f2603e Binary files /dev/null and b/Manuscript/fig5.pdf differ diff --git a/Manuscript/ufGW.bib b/Manuscript/ufGW.bib index 49f9765..6d0c3eb 100644 --- a/Manuscript/ufGW.bib +++ b/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) diff --git a/Manuscript/ufGW.tex b/Manuscript/ufGW.tex index 906b3e1..ae9a6f1 100644 --- a/Manuscript/ufGW.tex +++ b/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. 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AbsoluteThickness[1.6], + Thickness[0.18], + GrayLevel[0]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + CapForm["Butt"], + AbsoluteThickness[1.6], + Thickness[0.18], + GrayLevel[0]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.1] -> Baseline)], #4}, { + GraphicsBox[{{ + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + CapForm["Butt"], + AbsoluteThickness[1.6], + Thickness[0.18], + GrayLevel[0.5]], { + LineBox[{{0, 10}, {20, 10}}]}}, { + Directive[ + EdgeForm[ + Directive[ + Opacity[0.3], + GrayLevel[0]]], + PointSize[0.5], + CapForm["Butt"], + AbsoluteThickness[1.6], + Thickness[0.18], + GrayLevel[0.5]], {}}}, AspectRatio -> Full, + ImageSize -> {20, 10}, PlotRangePadding -> None, + ImagePadding -> Automatic, + BaselinePosition -> (Scaled[0.1] -> Baseline)], #5}}, + 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