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Manuscript/BSE_JPCL.tex

@@ -177,7 +177,8 @@
 
 \let\oldmaketitle\maketitle
 \let\maketitle\relax
-	\title{A Chemist Guide to the Bethe-Salpeter Equation}
+	\title{A Chemist Guide to the Bethe-Salpeter Equation \\
+	\xavier{ Challenges  for the Bethe-Salpeter Equation Formalism : from Physics to Chemistry } }
 \date{\today}
 
 \begin{tocentry}
@@ -522,7 +523,7 @@ in the relation between the screened Coulomb potential $W$ and the independent-e
 
 A remarkable property \cite{Duchemin_2018} of the BSE formalism combined with a polarizable environment is that the scheme described here above, with electron-electron and electron-hole interactions renormalized by the reaction field, allows to capture both linear-response (LR)  and state-specific (SS) contributions \cite{Cammi_2005} to the solvatochromic shift of the optical lines, allowing to treat on the same footing Frenkel and CT excitations. This is an important advantage as compared e.g. to TD-DFT calculations where LR and SS effects have to be explored with different formalisms. 
 
-To date, the effect of the environment on fast electronic excitations is only included by considering the low-frequency optical response of the polarizable medium (e.g. considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant for water in the optical range), neglecting the variations with frequency of the dielectric constant in the optical range. Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, namely in situations where neither the adiabatic or antiadiabatic limits are expected to be valid (for a recent discussion, see Ref.
+To date, the effect of the environment on fast electronic excitations is only included by considering the low-frequency optical response of the polarizable medium (e.g. considering the $\epsilon_{\infty} \simeq 1.78$ macroscopic dielectric constant for water in the optical range), neglecting the variations with frequency of the dielectric constant in the optical range. Generalization to fully frequency-dependent polarizable properties of the environment would allow to explore systems where the relative dynamics of the solute and the solvent are not decoupled, namely in situations where neither the adiabatic nor antiadiabatic limits are expected to be valid (for a recent discussion, see Ref.
 ~\citenum{Huu_2020}). \\
 
 
@@ -542,7 +543,7 @@ In practice, the main bottleneck for standard BSE calculations as compared to TD
 %%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states. 
 %%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019} 
 The field of low-scaling $GW$ calculations is however witnessing significant advances. 
-While the sparsity of ..., \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
+\xavier{While the sparsity of e.g. the overlap matrix in the atomic basis allows to reduce the scaling in the large size limit,} \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
 Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017} 
 The separability of occupied and virtual states summations lying at the heart of these approaches are now blooming in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in random-phase approximation (RPA) and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020}   
 These ongoing developments pave the way to applying the $GW$@BSE formalism to systems comprising several hundred atoms on standard laboratory clusters.
@@ -556,9 +557,7 @@ stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1
 contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range. \cite{Sears_2011}
 While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter as a mean to act on the triplet instability, \cite{Sears_2011} BSE calculations do not offer this pragmatic way-out since the screened Coulomb potential that builds the kernel does not offer any parameter to tune.
 
-benchmarks \cite{Jacquemin_2017b,Rangel_2017}
-
-a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$.  \cite{Holzer_2018b}\\
+Benchmark calculations  \cite{Jacquemin_2017b,Rangel_2017} clearly concluded that triplets are notably located at too low an energy within BSE and that the use of the TDA was able to partly reduce this error. However, the error remains rather unsatisfactory for reliable applications. An alternative   cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$.  \cite{Holzer_2018b}\\
 
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 \subsection{The challenge of analytical nuclear gradients}