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BSE-PES.tex
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BSE-PES.tex
@ -524,7 +524,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
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\caption{
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\caption{
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Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
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Equilibrium bond length (in bohr) of the ground state of diatomic molecules obtained at various levels of theory and basis sets.
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The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
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The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
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The values in parenthesis have been obtained by fitting a Morse potential to the PES.
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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.
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}
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}
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\label{tab:Req}
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\label{tab:Req}
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\begin{ruledtabular}
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\begin{ruledtabular}
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@ -564,31 +564,40 @@ The values in parenthesis have been obtained by fitting a Morse potential to the
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Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds.
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Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}, which are both held by covalent bonds.
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Their corresponding PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
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Their corresponding PES computed with the cc-pVQZ basis are reported in Fig.~\ref{fig:PES-H2-LiH}.
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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.
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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.
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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}.
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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}.
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Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
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Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set.
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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.
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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.
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The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
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The RPA-based schemes are much less accurate, with even shorter equilibrium bond lengths.
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This is a general trend that, as one can see, is magnified in larger systems.
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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.
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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.
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In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, and they are almost perfectly parallel.
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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.
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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).
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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).
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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.
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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.
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For these ionic bonds, the performance of BSE@{\GOWO}@HF are terrific with an almost perfect match to the CC3 curve.
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For these ionic bonds, the performance of BSE@{\GOWO}@HF is terrific with an almost perfect match to the CC3 curve.
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%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}.
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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.
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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.
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Interestingly, while CCSD and CC2 systematically underestimates the total energy, the BSE@{\GOWO}@HF energy is always lower than the reference CC3 energy.
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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.
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This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that we have studied here.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
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Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
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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.
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These value are reported in parenthesis in Table \ref{tab:Req}.
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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).
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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).
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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.
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The conclusions drawn for the previous systems also apply to these diatomic molecules.
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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.
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One can notice some irregularities in the PES of \ce{BF} with the cc-pVDZ et cc-pVTZ basis sets (see {\SI}).
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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.
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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).
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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).
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Similarly to what we have observed for \ce{LiF} and \ce{BF}, there is an irregularities near the minimum of the {\GOWO}-based curves.
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Similarly to what we have observed for \ce{LiF} and \ce{BF}, there are irregularities near the minimum of the {\GOWO}-based curves.
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However, BSE@{\GOWO}@HF is the closest to the CC3 curve
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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}
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%\section{Conclusion}
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@ -605,14 +614,13 @@ In the perspective of developing analytical nuclear gradients within the BSE@$GW
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We hope to be able to report on this in the near future.
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We hope to be able to report on this in the near future.
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\begin{acknowledgements}
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\section*{Acknowledgements}
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%PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and
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%PFL would like to thank Julien Toulouse for enlightening discussions about RPA, and
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XB is indebted to Valerio Olevano for numerous discussions.
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XB is indebted to Valerio Olevano for numerous discussions.
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This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
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This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A0040801738) and CALMIP (Toulouse) under allocation 2019-18005.
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Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
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Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
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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''.}
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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''.}
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\end{acknowledgements}
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\section*{Supporting Information}
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\section*{Supporting Information}
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