res first draft

BSE-PES.tex

@@ -524,7 +524,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
 \caption{
 Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets. 
 The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
-The values in parenthesis have been obtained by fitting a Morse potential to the PES.
+When irregularities appear in the PES, the values are reported in parenthesis and they have been obtained by fitting a Morse potential to the PES.
 }
 \label{tab:Req}
 	\begin{ruledtabular}
@@ -564,31 +564,40 @@ The values in parenthesis have been obtained by fitting a Morse potential to the
 Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds.
 Their corresponding PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
 For \ce{H2}, we take as reference the full configuration interaction (FCI) energies \cite{QP2} and we also report the MP2 curve and its third-order variant (MP3), which improves upon MP2 towards FCI.
-RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over and undershoot the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
+RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and significantly underestimate the FCI energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over- and under-shoot the FCI energy, respectively, RPAx@HF being the best match in the case of \ce{H2}.
 Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
 For example, with the cc-pVQZ basis set, BSE@{\GOWO}@HF is only off by $0.003$ bohr compared to FCI, while RPAx@HF, MP2, and CC2 underestimate the bond length by $0.008$, $0.011$, and $0.011$ bohr, respectively.
 The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
+This is a general trend that, as one can see, is magnified in larger systems.
 
 Albeit the shallow nature of the \ce{LiH} PES, the scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
-In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, and they are almost perfectly parallel.
+In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, theses curves running almost perfectly parallel to one another.
 Here again, the BSE@{\GOWO}@HF equilibrium bond length (obtained with cc-pVQZ) is extremely accurate ($3.017$ bohr) as compared to FCI ($3.019$ bohr).
 
-The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are interesting as they corresponds to strongly polarized bonds towards the halogen atoms which are much more electronegative than the first row elements.
-For these ionic bonds, the performance of BSE@{\GOWO}@HF are terrific with an almost perfect match to the CC3 curve.
-%For \ce{LiF}, the two curves starting to deviate a few tenths of bohr after the equilibrium geometry, but they remain tightly bound for much longer in the case of \ce{HCl}.
-Maybe surprisingly, BSE@{\GOWO}@HF is on par with both CC2 and CCSD, and outperforms RPAx@HF by a big margin for these two molecules exhibiting charge transfer.
-However, in the case of \ce{LiF}, the attentive reader would have observed a small glitch in the $GW$-based curves very close to their minimum.
+The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are chemically interesting as they correspond to strongly polarized bonds towards the halogen atoms which are much more electronegative than the first row elements.
+For these ionic bonds, the performance of BSE@{\GOWO}@HF is terrific with an almost perfect match to the CC3 curve.
+Maybe surprisingly, BSE@{\GOWO}@HF is on par with both CC2 and CCSD, and outperforms RPAx@HF by a big margin for these two molecules exhibiting charge transfer, the latter fact being also observed for the other diatomics discussed below.
+Interestingly, while CCSD and CC2 systematically underestimates the total energy, the BSE@{\GOWO}@HF energy is always lower than the reference CC3 energy.
+This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that we have studied here.
+
+For \ce{HCl}, the data reported in Table \ref{tab:Req} show that the BSE@{\GOWO}@HF equilibrium bond lengths are again in very good agreement with the CC3 reference values.
+Compared to CCSD which is known to provide slightly too short bond lengths, ACFDT@BSE usually underestimates the bond lengths by a few hundredths of bohr.
+However, in the case of \ce{LiF}, the attentive reader would have observed a small ``glitch'' in the $GW$-based curves very close to their minimum.
 As observed in Refs.~\onlinecite{vanSetten_2015,Maggio_2017,Loos_2018} and explained in details in Refs.~\onlinecite{Veril_2018,Duchemin_2020}, these irregularities, which makes particularly tricky the location of the minima, are due to ``jumps'' between distinct solutions of the $GW$ quasiparticle equation.
 Including a broadening via the increasing the value of $\eta$ in the $GW$ self-energy and the screened Coulomb operator soften the problem, but does not remove it completely.
 Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
-
+In the case of irregularities in the PES, in order to provide an estimate of the equilibrium bond length, we have fitted a Morse potential to the PES.
+These value are reported in parenthesis in Table \ref{tab:Req}.
 
 Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple bond to single bond).
-In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, and systematically outperforms both CC2 and CCSD.
+The conclusions drawn for the previous systems also apply to these diatomic molecules. 
+In particular, the performance of BSE@{\GOWO}@HF are outstanding, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, and systematically outperforms both CC2 and CCSD.
+One can notice some irregularities in the PES of \ce{BF} with the cc-pVDZ et cc-pVTZ basis sets (see {\SI}).
+The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium bond lengths once again: at the BSE@{\GOWO}@HF/cc-pVQZ level of theory, we obtain \gb{$2.070$}, \gb{$2.130$}, and \gb{$2.383$} bohr for \ce{N2}, \ce{CO}, and \ce{BF}, respectively, which has to be compared with the CC3/cc-pVQZ values of $2.075$, $2.136$ and $2.390$ bohr, respectively, for the same set of molecules.
 
-The \ce{F2} molecule is a notoriously difficult case to treat due to the weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence its relatively long equilibrium bond length ($2.663$ bohr at the CC3/cc-pVQZ level).
-Similarly to what we have observed for \ce{LiF} and \ce{BF}, there is an irregularities near the minimum of the {\GOWO}-based curves.
-However, BSE@{\GOWO}@HF is the closest to the CC3 curve
+As a final example, we consider the \ce{F2} molecule, a notoriously difficult case to treat due to the weakness of its covalent bond (see Fig.~\ref{fig:PES-F2}), hence its relatively long equilibrium bond length ($2.663$ bohr at the CC3/cc-pVQZ level).
+Similarly to what we have observed for \ce{LiF} and \ce{BF}, there are irregularities near the minimum of the {\GOWO}-based curves.
+However, BSE@{\GOWO}@HF is the closest to the CC3 curve, with an estimated bond length of $2.640$ bohr (via a Morse fit) at the BSE@{\GOWO}@HF/cc-pVQZ level.
 
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 %\section{Conclusion}
@@ -605,14 +614,13 @@ In the perspective of developing analytical nuclear gradients within the BSE@$GW
 We hope to be able to report on this in the near future.
 
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-\begin{acknowledgements}
+\section*{Acknowledgements}
+%%%%%%%%%%%%%%%%%%%%%%%%
 %PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and 
 XB is indebted to Valerio Olevano for numerous discussions.
 This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
 Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
 This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.} 
-\end{acknowledgements}
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 \section*{Supporting Information}