minors

BSE-PES.tex

@@ -179,11 +179,11 @@
 %	\centering
 %  \includegraphics[width=\linewidth]{TOC}
 %\end{wrapfigure}
-The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalisms have shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of molecular systems. 
+The combined many-body Green's function $GW$ and Bethe-Salpeter equation (BSE) formalism has shown to be a promising alternative to time-dependent density-functional theory (TD-DFT) in order to compute vertical transition energies of 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 topological features of the ground-state potential energy surfaces (PES) of several diatomic molecules near their equilibrium distance. 
 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.  
-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 the appearance of satellite resonances with a weight similar to that of the $GW$ quasiparticle peak.
 \end{abstract}
 
 \maketitle
@@ -194,14 +194,14 @@ However, we sometimes observe unphysical irregularities on the ground-state PES,
 %%%%%%%%%%%%%%%%%%%%%%%%
 
 With a similar computational cost to 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 the study of 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} 
+\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 computationally inexpensive method that can effectively model excited states \cite{Gonzales_2012,Loos_2020a} with a typical error of $0.1$--$0.3$ eV according to large and systematic benchmark calculations. \cite{Jacquemin_2015,Bruneval_2015,Blase_2016,Jacquemin_2016,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-consistently {\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 xc functional selected for the underlying DFT calculation). \cite{Jacquemin_2016,Gui_2018}
+Moreover, when performed on top of a (partially) self-consistently {\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_2016,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}
 
 A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytic 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 applications to the study of chemoluminescence, fluorescence and other related processes \cite{Navizet_2011} associated with geometric relaxation of ground and excited states, and structural changes upon electronic excitation. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
-While calculations of the $GW$ quasiparticle energies ionic gradients is becoming popular,
+While calculations of the $GW$ quasiparticle energy ionic gradients is becoming 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 study devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the KS-DFT (LDA) forces as its ground-state analog.
 
 Contrary to TD-DFT which relies on Kohn-Sham density-functional theory (KS-DFT) \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state counterpart, the BSE ground-state energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
@@ -216,7 +216,7 @@ Embracing this definition, the purpose of the present study is to investigate th
 The location of the minima on the ground-state PES is of particular interest.
 This study is a first preliminary step towards the development of analytical nuclear gradients within the BSE@$GW$ formalism.
 Thanks to comparison 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, 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}
+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}
 
 %The paper is organized as follows. 
 %In Sec.~\ref{sec:theo}, we introduce the equations behind the BSE formalism.
@@ -268,7 +268,7 @@ Finally, the static approximation is enforced, \ie, $\W{}{}(1,2) = \W{}{}(\{\br{
 %\label{sec:BSE_basis}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-For a closed-shell system, to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$) in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} 
+For a closed-shell system in a finite basis, to compute the singlet BSE excitation energies (within the static approximation) of the physical system (\ie, $\IS = 1$), one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} 
 \begin{equation}
 \label{eq:LR}
 	\begin{pmatrix}
@@ -319,7 +319,7 @@ where $\eGW{p}$ are the $GW$ quasiparticle energies,
 	\\
 	+ \sum_m^{\Nocc \Nvir} \sERI{ia}{m} \sERI{jb}{m} \qty(\frac{1}{\omega - \OmRPA{m}{\IS} + i \eta} + \frac{1}{\omega + \OmRPA{m}{\IS} - i \eta})
 \end{multline}
-are the elements of the screened Coulomb operator $\W{}{\IS}$, 
+are the elements of the screened Coulomb operator, 
 \begin{equation}
 	\sERI{pq}{m} =  \sum_i^{\Nocc} \sum_a^{\Nvir} \ERI{pq}{ia} (\bX{\IS}_m + \bY{\IS}_m)_{ia}
 \end{equation}
@@ -343,7 +343,7 @@ where $\eHF{p}$ are the HF orbital energies.
 %\subsection{Ground-state BSE energy}
 %\label{sec:BSE_energy}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-The key quantity to define in the present context is the total ground-state BSE energy $\EBSE$.
+The key quantity to define in the present context is the total BSE ground-state energy $\EBSE$.
 Although this choice is not unique, \cite{Holzer_2018} we propose here to define it as
 \begin{equation}
 \label{eq:EtotBSE}
@@ -378,8 +378,8 @@ is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\tA{ia,jb}{\IS}
 	\end{pmatrix}
 \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 here, has be labeled ``XBS'' for ``extended Bethe Salpeter'' by Holzer 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 maintain with respect to $\IS$.
+Note that the present definition of the BSE correlation energy [see Eq.~\eqref{eq:EcBSE}], which we refer to as BSE@$GW$@HF here, has be labeled ``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.
 Therefore, an additional contribution to Eq.~\eqref{eq:EcBSE} originating from the variation of the Green's function along the adiabatic connection should be added.
 However, as commonly done within RPA and RPAx (\ie, RPA with exchange), \cite{Toulouse_2009, Toulouse_2010, Holzer_2018} we shall neglect it in the present study. 
 
@@ -393,7 +393,7 @@ For RPA, these expressions have been provided in Eq.~\eqref{eq:LR_RPA}, and thei
 	\BRPAx{ia,jb}{\IS} & = 2 \IS \ERI{ia}{jb} - \IS \ERI{ia}{bj}.
 \end{align}
 \end{subequations}
-These two types of calculations will be refer to as RPA@HF and RPAx@HF respectively in the following.
+In the following, we will refer to these two types of calculations as RPA@HF and RPAx@HF, respectively.
 Finally, we will also consider the RPA@$GW$@HF scheme which consists in replacing the HF orbital energies in Eq.~\eqref{eq:LR_RPA} by the $GW$ quasiparticles energies.
 
 Several important comments are in order here.
@@ -432,7 +432,7 @@ However, we are currently pursuing different avenues to lower this cost by compu
 %\section{Potential energy surfaces}
 %\label{sec:PES}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed 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}.
+In order to illustrate the performance of the BSE-based adiabatic connection formulation, we have computed 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 for various methods 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 for various basis sets are gathered in Table \ref{tab:Req}.
 Additional graphs for other basis sets can be found in the {\SI}.
 
@@ -440,7 +440,7 @@ Additional graphs for other basis sets can be found in the {\SI}.
 \begin{table*}
 \caption{
 Equilibrium distances (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 black and red bold for visual convenience, respectively.
+The reference CC3 and corresponding BSE@{\GOWO}@HF data are highlighted in bold black and bold red for visual convenience, respectively.
 }
 \label{tab:Req}
 	
@@ -514,9 +514,9 @@ Interestingly though, the BSE@{\GOWO}@HF scheme yields a more accurate equilibri
 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.
 
-The scenario is almost identical for \ce{LiH} for which we report the CC2, CCSD and CC3 energies in addition to MP2.
+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.
-Here again, the BSE@{\GOWO}@HF equilibrium bond length is extremely accurate ($3.017$ bohr) as compared to FCI ($3.019$ bohr).
+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).
 
 %%% FIG 1 %%%
 \begin{figure*}
@@ -532,10 +532,10 @@ Additional graphs for other basis sets and within the frozen-core approximation
 
 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 outperforms both CC2 and CCSD, as well as RPAx@HF by a big margin for these two molecules exhibiting charge transfer.
+%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.
-As observed in Refs.~\cite{vanSetten_2015,Maggio_2017,Loos_2018} and explained in details in Refs.~\cite{Veril_2018,Duchemin_2020}, these irregularities, which makes tricky the location of the minima, are due to ``jumps'' between two distinct solutions of the $GW$ quasiparticle equation.
+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}
 
@@ -544,7 +544,7 @@ Note that these irregularities would be genuine discontinuities in the case of {
 	\includegraphics[height=0.35\linewidth]{LiF_GS_VTZ}
 	\includegraphics[height=0.35\linewidth]{HCl_GS_VTZ}
 \caption{
-Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. 
+Ground-state PES of \ce{LiF} (left) and \ce{HCl} (right) around their respective equilibrium geometry obtained at various levels of theory with the \titou{cc-pVTZ} basis set. 
 Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-LiF-HCl}
 }
@@ -560,7 +560,7 @@ In that case again, the performance of BSE@{\GOWO}@HF are outstanding, as shown
 	\includegraphics[height=0.26\linewidth]{CO_GS_VTZ}
 	\includegraphics[height=0.26\linewidth]{BF_GS_VTZ}
 \caption{
-Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. 
+Ground-state PES of the isoelectronic series \ce{N2} (left), \ce{CO} (center), and \ce{BF} (right) around their respective equilibrium geometry obtained at various levels of theory with the \titou{cc-pVTZ} basis set. 
 Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-N2-CO-BF}
 }
@@ -575,7 +575,7 @@ However, BSE@{\GOWO}@HF is the closest to the CC3 curve
 \begin{figure}
 	\includegraphics[width=\linewidth]{F2_GS_VTZ}
 \caption{
-Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. 
+Ground-state PES of \ce{F2} around its equilibrium geometry obtained at various levels of theory with the \titou{cc-pVTZ} basis set. 
 Additional graphs for other basis sets and within the frozen-core approximation can be found in the {\SI}.
 \label{fig:PES-F2}
 }
@@ -586,14 +586,14 @@ Additional graphs for other basis sets and within the frozen-core approximation
 %\section{Conclusion}
 %\label{sec:conclusion}
 %%%%%%%%%%%%%%%%%%%%%%%%
-In this Letter, we have shown that calculating the BSE correlation energy in the ACFDT framework yield extremely accurate PES around equilibrium.
+In this Letter, we have shown that calculating the BSE correlation energy within the ACFDT framework yield extremely accurate PES around equilibrium.
+(Their accuracy near the dissociation limit remains an open question.)
 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).
 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}
-We believe that this central issue must be resolved if one wants to expand the applicability of the present methods.
+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.
-\titou{We hope to have demonstrated that future developments around $GW$ methods are worthwhile.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Supporting Information}