changes xav and ivan

BSE-PES.bib

@@ -264,9 +264,10 @@
 	Author = {J. Li and I. Duchemin and X. Blase and V. Olevano},
 	Date-Added = {2020-01-04 20:06:04 +0100},
 	Date-Modified = {2020-02-05 20:59:40 +0100},
-	Journal = {arXiv:physics.chem-ph},
-	Pages = {1812.00932},
-	Title = {Ground-State Correlation Energy of Beryllium Dimer by the Bethe-Salpeter Equation},
+	Journal = { SciPost Phys. },
+	Volume = {8},
+	Pages = { 020 },
+	Title = {Ground-State Correlation Energy of Beryllium Dimer by the Bethe--Salpeter Equation},
 	Year = {2020}}
 
 @article{Salpeter_1951,
@@ -9522,7 +9523,7 @@
 	Title = {Hybrid and Constrained Resolution-of-Identity Techniques for Coulomb Integrals},
 	Volume = {13},
 	Year = {2017},
-	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b01215}}
+	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b01215}  }
 
 @article{Rojas_1995,
 	Author = {H. N. Rojas and R. W. Godby and R. J. Needs},
@@ -9534,16 +9535,16 @@
 	Title = {Space-Time Method for Ab Initio Calculations of Self-Energies and Dielectric Response Functions of Solids},
 	Volume = {74},
 	Year = {1995},
-	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.74.1827}}
+	Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.74.1827}  }
 
 @article{Duchemin_2020,
 	Author = {Ivan Duchemin and Xavier Blase},
 	Date-Added = {2019-10-23 10:00:45 +0200},
 	Date-Modified = {2020-01-26 11:17:42 +0100},
-	Journal = {arXiv:1912.06459},
+	Journal = {J. Chem. Theory Comput (accepted) },
 	Title = {Robust Analytic Continuation Approach to Many-Body GW Calculations},
-	Year = {2019},
-	Bdsk-Url-1 = {https://doi.org/10.1063/1.5090605}}
+	Year = {2020},
+	Bdsk-Url-1 = {}  }
 
 @article{Weigend_2003a,
 	Author = {Weigend, Florian and Furche, Filipp and Ahlrichs, Reinhart},

BSE-PES.tex

@@ -186,7 +186,7 @@ The combination of the many-body Green's function $GW$ approximation and the Bet
 The BSE formalism can also be employed to compute ground-state correlation energies thanks to the adiabatic-connection fluctuation-dissipation theorem (ACFDT).
 Here, we study the topology of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium bond length. 
 Thanks to comparisons with state-of-art computational approaches, we show that ACFDT@BSE is surprisingly accurate, and can even compete with coupled cluster methods in terms of total energies and equilibrium bond distances for the considered systems.  
-However, we sometimes observe unphysical irregularities on the ground-state PES in relation with the appearance of satellite resonances with a weight similar to that of the $GW$ quasiparticle peak.\\
+However, we sometimes observe unphysical irregularities on the ground-state PES in relation with \sout{the appearance of satellite resonances with a weight similar to that of the $GW$ quasiparticle peak} \xavier{discontinuities of a few $GW$ quasiparticle energies, questioning their identification with the solution of the quasiparticle equation with the largest spectral weight.    }   \\
 \bigskip
 \begin{center}
 	\boxed{\includegraphics[width=0.5\linewidth]{TOC}}
@@ -559,7 +559,7 @@ The error (in \%) compared to the reference CC3 values are reported in square br
 Let us start with the two smallest molecules, \ce{H2} and \ce{LiH}.
 Their 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 both 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 to FCI in the case of \ce{H2}.
+RPA@HF and RPA@{\GOWO}@HF yield almost identical results, and both significantly \sout{underestimate} \xavier{overestimate} the FCI \xavier{correlation} energy, while RPAx@HF and BSE@{\GOWO}@HF slightly \xavier{over- and undershoot ??} the FCI energy, respectively, RPAx@HF being the best match to FCI in the case of \ce{H2}.
 Interestingly, the BSE@{\GOWO}@HF scheme yields a more accurate equilibrium bond length than any other method irrespectively of the basis set (see Table in the {\SI}).
 For example, BSE@{\GOWO}@HF/cc-pVQZ is only off by $0.003$ bohr as compared to FCI/cc-pVQZ, 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.
@@ -573,7 +573,7 @@ The cases of \ce{LiF} and \ce{HCl} (see Fig.~\ref{fig:PES-LiF-HCl}) are chemical
 For these partially 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 is considered herein.
+This observation is not only true for \ce{LiF} and \ce{HCl}, but holds for every single systems that is considered herein. \xavier{This is consistent with the study by Maggio and Kresse of the homogeneous electron gas (HEG)\cite{Maggio_2016} showing that BSE slightly overestimates the correlation energy as compared to QMC reference data. Similarly, the much larger overestimation of  the correlation energy we observe at the  RPA@$GW$ level was also observed for the HEG. Care must be taken however in drawing comparisons since the HEG studies were performed starting with LDA input eigenstates.  } 
 
 For \ce{HCl}, the data reported in Table \ref{tab:Req} show that the BSE@{\GOWO}@HF equilibrium bond length is again in very good agreement with its CC3 counterpart. 
 In contrast to CCSD which is known to provide slightly shorter bond lengths, ACFDT@BSE underestimates the bond lengths by a few hundredths of bohr.
@@ -583,7 +583,7 @@ Including a broadening via an increase of the $\eta$ value entering in the expre
 %Note that these irregularities would be genuine discontinuities in the case of {\evGW}. \cite{Veril_2018}
 When irregularities are present in the PES, we have fitted a Morse potential of the form $M(R) = D_0\qty{1-\exp[-\alpha\qty(R-\Req)]}^2$ to the PES in order to provide an estimate of the equilibrium bond length.
 These values are reported in parenthesis in Table \ref{tab:Req}.
-For the smooth PES where one can obtain both the genuine minimum and the fitted minimum (\ie, based on the Morse curve), this procedure has been shown to be very accurate with an error of the order of $10^{-3}$ bohr in most cases.
+For the smooth PES where one can obtain both the genuine minimum and the fitted minimum (\ie, based on the Morse curve), this procedure has been shown to be very accurate with an error of the order of $10^{-3}$ bohr in most cases. \xavier{ We note that these irregularities are much smaller than the differences between the BSE and the other RPA-like techniques (RPA, RPAx, RPA@$GW$)  leaving BSE unambiguously more accurate. }
 
 Let us now look at the isoelectronic series \ce{N2}, \ce{CO}, and \ce{BF}, which have a decreasing bond order (from triple to single bond).
 The conclusions drawn for the previous systems also apply to these molecules. 
@@ -596,7 +596,10 @@ As a final example, we consider the \ce{F2} molecule, a notoriously difficult ca
 Similarly to what is 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 error on the correlation energy of $1\%$ and an estimated bond length of $2.640$ bohr (via a Morse fit) at the BSE@{\GOWO}@HF/cc-pVQZ level.
 Note that, for this system, triplet (and then singlet) instabilities appear for quite short bond lengths.
-However, around the equilibrium structure, we have not encountered any instabilities.
+However, around the equilibrium structure, we have not encountered any instabilities. \xavier{ This is an important outcome of the present study that the difficulties encountered at large interatomic distance, namely close to the dissociation limit, does not prevent the BSE approach to be potentially extremely useful and accurate in the vicinity of the equilibrium distance. Preliminary results indicate that no singlet instabilities could be seen in the vicinity of the lowest excited states minima. }
+
+\xavier{ Although we considered here only a limited set of compounds,
+our correlation energy MAE (MSE) with BSE of xxx (xxx) as compared to CC3 are significantly smaller than the one obtained with MP2 and CCSD; xxx (xxx) } 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 %\section{Conclusion}
@@ -608,8 +611,8 @@ To do so, we have shown that calculating the BSE correlation energy computed wit
 We have illustrated this for 8 diatomic molecules for which we have also computed reference ground-state energies using coupled cluster methods (CC2, CCSD, and CC3).
 For the larger systems considered here, we have observed that BSE@{\GOWO} recovers $99\%$ of the CC3 correlation energy.
 Moreover, because triplet states do not contribute to the ACFDT correlation energy and singlet instabilities do not appear for weakly-correlated systems around their equilibrium structure, the present scheme does not suffer from singlet nor triplet instabilities.
-However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak.
-This shortcoming, which is entirely due to the quasiparticle nature of the underlying $GW$ calculation, has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
+However, we have also observed that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of \sout{a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak} \xavier{discontinuities as a function of bond length of a few $GW$ quasiparticle energies. Such an unphysical behaviour stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight  in cases where several solutions can be found.} 
+This shortcoming \sout{, which is entirely due to the quasiparticle nature of the underlying $GW$ calculation,} has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
 We believe that this central issue must be resolved if one wants to expand the applicability of the present method.
 %In the perspective of developing analytical nuclear gradients within the BSE@$GW$ formalism, we are currently investigating the accuracy of the ACFDT@BSE scheme for excited-state PES.
 %We hope to be able to report on this in the near future.