diff --git a/BSE-PES.tex b/BSE-PES.tex
index 70db2e4..a47086e 100644
--- a/BSE-PES.tex
+++ b/BSE-PES.tex
@@ -185,28 +185,7 @@ However, we also observe, in some cases, unphysical irregularities on the ground
 %\label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-%The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
-%For example, chemoluminescence, fluorescence and other related processes are associated with geometric relaxation of excited states, and structural changes upon electronic excitation. \cite{Navizet_2011}
-%Reliable predictions of these mechanisms which have attracted much experimental and theoretical interest lately require exploring the ground- and excited-state PES. 
-%From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Gonzales_2012, Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Andersson_1990,Andersson_1992,Roos_1996,Angeli_2001}) cannot be afforded.
-%For such systems, one has to rely on more approximate, yet computationally cheaper approaches. \cite{Grimme_2004a,Ghosh_2018}
-
-%For the last two decades, time-dependent density-functional theory (TD-DFT) \cite{Casida,Runge_1984} has been the go-to method to compute absorption and emission spectra in large molecular systems.
-%At a relatively low computational cost, TD-DFT can provide accurate vertical and adiabatic transition energies for low-lying excited states of organic molecules with a typical error of $0.2$--$0.4$ eV. \cite{Loos_2019b}
-%Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundation relies on the Runge-Gross theorem. \cite{Runge_1984}
-%The Kohn-Sham (KS) formalism of TD-DFT transfers the complexity of the many-body problem to the exchange-correlation (xc) functional thanks to a judicious mapping between a time-dependent non-interacting reference system and its interacting analog which have both the exact same electronic density.
-%In TD-DFT, the excitation energies are obtained as the poles of the ground-state frequency-dependent linear response function, which is obtained by solving a Dyson equation in which the exchange and correlation effects are cast in the exchange-correlation (xc) kernel.
-%This xc kernel plays for the excited states the same role as the xc functional for the ground state.
-%Hence, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state. 
-%One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first derivatives of the excited-state energy with respect to the nuclear displacements), which enables the exploration of ground- and excited-state PES. \cite{Furche_2002}
-
-%Although the exact spectrum of the interacting system can be obtained in principle, both the ground-state xc potential and the xc kernel (which enters the definition of the response function) have to be approximated in practice.
-%This has major consequences, and it is well documented that, TD-DFT fails badly to model, for examples, charge transfer,\cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} and doubly excited states. \cite{Levine_2006,Tozer_2000,Elliott_2011}
-%An important point from a practical point of view is that the TD-DFT xc kernel is usually considered as static instead of being frequency dependent. \cite{Romaniello_2009a,Sangalli_2011}
-%One key consequence of this so-called adiabatic approximation (based on the assumption that the density varies slowly with time) is that double excitations are completely absent from the TD-DFT spectra. \cite{Levine_2006,Tozer_2000,Huix-Rotllant_2010,Elliott_2011}
-%Yet another problem is the choice of the xc functionals as the quality of excitation energies are substantially dependent on this choice.
-
-With a similar computational cost to time-dependent density-functional theory (TD-DFT), \cite{Casida,Runge_1984} the many-body Green's function Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is a valuable alternative with early \textit{ab initio} calculations in condensed matter physics dated back to the end of the 90's. \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999}
+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} is a valuable alternative with early \textit{ab initio} calculations in condensed matter physics dated back to the end of the 90's. \cite{Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999}
 In the past few years, BSE has gained momentum for the study of molecular systems \cite{Ma_2009,Pushchnig_2002,Tiago_2003,Tiago_2008,Sai_2008,Palumno_2009,Rocca_2010,Sharifzadeh_2012,Cudazzo_2012,Boulanger_2014,Ljungberg_2015,Hirose_2015,Cocchi_2015,Ziaei_2017,Abramson_2017} and is now a serious candidate 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}
@@ -409,13 +388,15 @@ However, to the best of our knowledge, such alternative plasmon expression does
 %\section{Computational details}
 %\label{sec:comp_details}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (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 have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very 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 point to compute the BSE neutral excitations.
 These will be labeled as BSE@{\GOWO}.
 In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation.
 Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018, Veril_2018}.
 Finally, the infinitesimal $\eta$ has been set to zero for all calculations.
 The numerical integration required to compute the correlation energy along the adiabatic path [see Eq.~\eqref{eq:EcBSE}] has been performed with a 21-point Gauss-Legendre quadrature.
+For comparison purposes, we have also computed the PES at the MP2, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} levels of theory using DALTON. \cite{dalton}
+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 has not been enforced in our calculations in order to provide a fairer comparison between methods.
 However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI}).
@@ -428,12 +409,10 @@ 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 diatomic closed-shell molecules around their equilibrium geometry: \ce{H2}, \ce{LiH}, \ce{LiF}, \ce{N2}, \ce{CO}, \ce{BF}, \ce{F2}, and \ce{HCl}.
+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{N2}, \ce{CO}, \ce{BF}, \ce{F2}, and \ce{HCl}.
 The PES of these molecules for various methods and Dunning's triple-$\zeta$ basis cc-pVTZ are represented in Fig.~\ref{fig:PES}, and the computed equilibrium distances are gathered in Table \ref{tab:Req}.
 Additional graphs for other basis sets can be found in the {\SI}.
 
-For comparison purposes, we have also computed the PES at the MP2, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} levels of theory using DALTON. \cite{dalton}
-
 %%% FIG 1 %%%
 \begin{figure*}
 	\includegraphics[width=0.45\linewidth]{H2_GS_VTZ}
@@ -445,7 +424,7 @@ For comparison purposes, we have also computed the PES at the MP2, CC2 \cite{Chr
 	\includegraphics[width=0.45\linewidth]{F2_GS_VTZ}
 	\includegraphics[width=0.45\linewidth]{HCl_GS_VTZ}
 \caption{
-PES of the ground state of diatomic molecules around their respective equilibrium geometry obtained at various levels of theory with the cc-pVTZ basis set. 
+Ground-state PES of diatomic molecules around their respective equilibrium geometry obtained at various levels of theory with the 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}
 }
@@ -539,7 +518,13 @@ Equilibrium distances (in bohr) of the ground state of diatomic molecules obtain
 %\section{Conclusion}
 %\label{sec:conclusion}
 %%%%%%%%%%%%%%%%%%%%%%%%
-\titou{A nice conclusion here saying that what we have done is pretty awesome.}
+In this Letter, we have shown that calculating the BSE correlation energy in the ACFDT framework yield extremely accurate PES around equilibrium.
+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.
+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.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Supporting Information}