Denis corrections

BSE-PES.tex

@@ -185,7 +185,7 @@
 The combination of the many-body Green's function $GW$ approximation and the Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) for computing vertical transition energies and oscillator strengths in molecular systems. 
 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.  
+Thanks to comparisons with state-of-art computational approaches (CC3), we show that ACFDT@BSE is surprisingly accurate, and can even compete with lower-order coupled cluster methods (CC2 and CCSD) 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 difficulties in the identification of a few $GW$ quasiparticle energies.
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 \bigskip
@@ -204,26 +204,27 @@ However, we sometimes observe unphysical irregularities on the ground-state PES
 
 With a similar computational scaling as time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida} the many-body Green's function Bethe-Salpeter equation (BSE) formalism 
 \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} is a valuable alternative that has gained momentum in the past few years for studying molecular systems.\cite{Ma_2009,Pushchnig_2002,Tiago_2003,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} 
-It now stands as a cost-effective computational method that can model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmarks. \cite{Jacquemin_2015,Bruneval_2015,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018}
+It now stands as a cost-effective computational method that can model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV for spin-conserving transitions according to large and systematic benchmarks. \cite{Jacquemin_2015,Bruneval_2015,Hung_2016,Hung_2017,Krause_2017,Jacquemin_2017,Blase_2018}
 One of the main advantages of BSE compared to TD-DFT is that it allows a faithful description of charge-transfer states. \cite{Lastra_2011,Blase_2011b,Baumeier_2012,Duchemin_2012,Cudazzo_2013,Ziaei_2016}
-Moreover, when performed on top of a (partially) self-consistent {\evGW} calculation, \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} BSE@{\evGW} has been shown to be weakly dependent on its starting point (\ie, on the exchange-correlation functional selected for the underlying DFT calculation). \cite{Jacquemin_2015,Gui_2018}
+Moreover, when performed on top of a (partially) self-consistent {\evGW} calculation, \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} BSE@{\evGW} has been shown to be weakly dependent on its starting point (\eg, on the exchange-correlation functional selected for the underlying DFT calculation). \cite{Jacquemin_2015,Gui_2018}
 However, similar to adiabatic TD-DFT, \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011} the static version of BSE cannot describe multiple excitations. \cite{Romaniello_2009a,Sangalli_2011,Loos_2019}
 
 A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients (\ie, the first derivatives of the energy with respect to the nuclear displacements) for both the ground and excited states, \cite{Furche_2002} preventing efficient studies of excited-state processes (\eg, chemoluminescence and fluorescence) associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Bernardi_1996,Olivucci_2010,Navizet_2011,Robb_2007}
 While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
-\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} In this seminar work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the Kohn-Sham (KS) LDA forces as its ground-state contribution.
+\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003} 
+In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the Kohn-Sham (KS) LDA forces as its ground-state contribution.
 
 In contrast to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state analog, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
 Consequently, the BSE ground-state formalism remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020}
 In the largest available benchmark study \cite{Holzer_2018} encompassing the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, \cite{Harding_2008} and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018} 
 Such a modified BSE polarization propagator was inspired by a previous study on the homogeneous electron gas (HEG). \cite{Maggio_2016} 
-With such an approximation, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either KS-DFT or $GW$ eigenvalues in the construction of the propagator becomes further relevant, increasing accordingly the number of possible definitions for the ground-state correlation energy. 
+Within RPAsX, amounting to neglect excitonic effects in the electron-hole propagator, the question of using either KS-DFT or $GW$ eigenvalues in the construction of the propagator becomes further relevant, increasing accordingly the number of possible definitions for the ground-state correlation energy. 
 Finally, renormalizing or not the Coulomb interaction by the interaction strength $\IS$ in the Dyson equation for the interacting polarizability (see below) leads to two different versions of the BSE correlation energy, \cite{Holzer_2018} emphasizing further the lack of general agreement around the definition of the ground-state BSE energy.
 
 Here, in analogy to the random-phase approximation (RPA)-type formalisms \cite{Furche_2008,Toulouse_2009,Toulouse_2010,Angyan_2011,Ren_2012} and similarly to Refs.~\onlinecite{Olsen_2014,Maggio_2016,Holzer_2018}, the ground-state BSE energy is calculated in the adiabatic connection framework. 
-Embracing this definition, the purpose of the present study is to investigate the quality of ground-state PES near equilibrium obtained within the BSE approach for several diatomic molecules.
+Embracing this definition, the purpose of the present Letter is to investigate the quality of ground-state PES near equilibrium obtained within the BSE approach for several diatomic molecules.
 The location of the minimum on the ground-state PES is of particular interest.
-This study is a preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
+This study is a first necessary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
 Thanks to comparisons with both similar and state-of-art computational approaches, we show that the ACFDT@BSE@$GW$ approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods in terms of absolute energies and equilibrium distances.
 However, we also observe that, in some cases, unphysical irregularities on the ground-state PES, which are due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
 
@@ -419,7 +420,7 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS}
 \end{equation} 
 is the correlation part of the two-electron density matrix at interaction strength $\IS$, and $\Tr$ denotes the matrix trace.
 Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF in the following, has been named ``XBS'' for ``extended Bethe Salpeter'' by Holzer \textit{et al.} \cite{Holzer_2018}
-Contrary to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintained as $\IS$ varies.
+In contrast to DFT where the electron density is fixed along the adiabatic path, in the present formalism, the density is not maintained as $\IS$ varies.
 Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be, in principle, added.
 However, as commonly done within RPA and RPAx, \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. 
 
@@ -436,7 +437,7 @@ Even for weakly correlated systems, triplet instabilities are much more common,
 %\section{Computational details}
 %\label{sec:comp_details}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point, which is a very adequate choice in the case of the (small) systems that we have considered here.
+All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point, which is an adequate choice in the case of the (small) systems that we have considered here.
 Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting points to compute the BSE neutral excitations.
 In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation.
 Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
@@ -451,11 +452,11 @@ As shown in Refs.~\onlinecite{Hattig_2005c,Budzak_2017}, CC3 provides extremely
 All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018,Veril_2018}
 As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian Gaussian functions. 
 Unless otherwise stated, the frozen-core approximation is not applied in order to provide a fair comparison between methods.
-We have, however, found that the conclusions drawn in the present study hold within the frozen-core approximation (see the {\SI} for additional information).
+We have, however, found that our conclusions hold within the frozen-core approximation (see the {\SI} for 	 information).
 
 Because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum for each quadrature point, we perform several complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
 This step is, by far, the computational bottleneck in our current implementation.
-However, we are currently pursuing different avenues to lower the cost of this step by computing the two-electron density matrix of Eq.~\eqref{eq:2DM} via a quadrature in frequency space. \cite{Duchemin_2019,Duchemin_2020}
+However, we are currently pursuing different avenues to lower the formal scaling and practical cost of this step by computing the two-electron density matrix of Eq.~\eqref{eq:2DM} via a quadrature in frequency space. \cite{Duchemin_2019,Duchemin_2020}
 
 %%% FIG 1 %%%
 \begin{figure*}
@@ -512,7 +513,7 @@ Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various
 In order to illustrate the performance of the BSE-based adiabatic connection formulation, we compute the ground-state PES of several closed-shell diatomic molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{HCl}, \ce{N2}, \ce{CO}, \ce{BF}, and \ce{F2}.
 The PES of these molecules are represented in Figs.~\ref{fig:PES-H2-LiH}, \ref{fig:PES-LiF-HCl}, \ref{fig:PES-N2-CO-BF}, and \ref{fig:PES-F2}, while the computed equilibrium distances and correlation energies are gathered in Table \ref{tab:Req}.
 Both of these properties are computed with Dunning's cc-pVQZ basis set. 
-Graphs and tables for additional basis sets can be found in the {\SI}.
+Graphs and tables for the corresponding double- and triple-$\zeta$ basis sets can be found in the {\SI}.
 
 %%% TABLE I %%%
 \begin{squeezetable}
@@ -559,19 +560,19 @@ 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 overestimate the FCI correlation 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 overestimate the FCI correlation energy, while RPAx@HF and BSE@{\GOWO}@HF slightly over- and undershoot the FCI energy, respectively, RPAx@HF yielding 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.
 This is a general trend that is magnified in larger systems as the ones discussed below.
 
-Despite the shallow nature of its PES, the scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
+Despite the shallow nature of its PES, the scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2 energies.
 In this case, RPAx@HF and BSE@{\GOWO}@HF nestle the CCSD and CC3 energy curves, theses surfaces running almost perfectly parallel to one another.
 Here again, the BSE@{\GOWO}@HF/cc-pVQZ equilibrium bond length is extremely accurate ($3.017$ bohr) as compared to CC3/cc-pVQZ ($3.019$ bohr).
 
 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-column elements.
 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.
+Maybe surprisingly, BSE@{\GOWO}@HF is on par with both CC2 and CCSD, and outperforms RPAx@HF by a big margin, 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. 
 Moreover, this is consistent with the study by Maggio and Kresse on the HEG showing that BSE slightly overestimates the correlation energy as compared to QMC reference data. \cite{Maggio_2016} 
@@ -593,7 +594,8 @@ The conclusions drawn for the previous systems also apply to these molecules.
 In particular, as shown in Fig.~\ref{fig:PES-N2-CO-BF}, the performance of BSE@{\GOWO}@HF is outstanding with an error of the order of $1\%$ on the correlation energy.
 Importantly, it 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 the {\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 $2.065$, $2.134$, and $2.385$ 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.
+The PES of \ce{N2} and \ce{CO} are smooth though, and yield accurate equilibrium bond lengths once more. 
+Indeed, at the BSE@{\GOWO}@HF/cc-pVQZ level of theory, we obtain $2.065$, $2.134$, and $2.385$ 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.
 
 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 is observed for \ce{LiF} and \ce{BF}, there are irregularities near the minimum of the {\GOWO}-based curves.