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Manuscript/BSE_JPCL.bib

@@ -14434,3 +14434,129 @@
 	Year = {2016},
 	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.93.235113},
 	Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.93.235113}}
+
+%%% PCM and QM/MM
+
+@article{Baumeier_2014,
+author = {Baumeier, Björn and Rohlfing, Michael and Andrienko, Denis},
+title = {Electronic Excitations in Push–Pull Oligomers and Their Complexes with Fullerene from Many-Body Green’s Functions Theory with Polarizable Embedding},
+journal = { J. Chem. Theory  Comput. },
+volume = {10},
+number = {8},
+pages = {3104-3110},
+year = {2014},
+doi = {10.1021/ct500479f},
+    note ={PMID: 26588281},
+URL = {   https://doi.org/10.1021/ct500479f},
+eprint = {  https://doi.org/10.1021/ct500479f}
+}
+
+
+@article{Duchemin_2016,
+author = {Duchemin,Ivan  and Jacquemin,Denis  and Blase,Xavier },
+title = {Combining the GW formalism with the polarizable continuum model: A state-specific non-equilibrium approach},
+journal = { J. Chem. Phys. },
+volume = {144},
+number = {16},
+pages = {164106},
+year = {2016},
+doi = {10.1063/1.4946778},
+URL = {   https://doi.org/10.1063/1.4946778},
+eprint = {   https://doi.org/10.1063/1.4946778}
+}
+
+
+@article{Li_2016,
+author = {Li, Jing and D’Avino, Gabriele and Duchemin, Ivan and Beljonne, David and Blase, Xavier},
+title = {Combining the Many-Body GW Formalism with Classical Polarizable Models: Insights on the Electronic Structure of Molecular Solids},
+journal = { J. Phys. Chem. Lett. },
+volume = {7},
+number = {14},
+pages = {2814-2820},
+year = {2016},
+doi = {10.1021/acs.jpclett.6b01302},
+    note ={PMID: 27388926},
+URL = {  https://doi.org/10.1021/acs.jpclett.6b01302},
+eprint = {   https://doi.org/10.1021/acs.jpclett.6b01302}
+}
+
+@article{Varsano_2016,
+	doi = {10.1088/0953-8984/29/1/013002},
+	url = {https://doi.org/10.1088%2F0953-8984%2F29%2F1%2F013002},
+	year = 2016,
+	month = {nov},
+	publisher = {{IOP} Publishing},
+	volume = {29},
+	number = {1},
+	pages = {013002},
+	author = {Daniele Varsano and Stefano Caprasecca and Emanuele Coccia},
+	title = {Theoretical description of protein field effects on electronic excitations of biological chromophores},
+	journal = {J. Phys.: Cond. Matt.}, 
+}
+@Article{Duchemin_2018,
+author ="Duchemin, Ivan and Guido, Ciro A. and Jacquemin, Denis and Blase, Xavier",
+title  ="The Bethe–Salpeter formalism with polarisable continuum embedding: reconciling linear-response and state-specific features",
+journal  ="Chem. Sci.",
+year  ="2018",
+volume  ="9",
+issue  ="19",
+pages  ="4430-4443",
+publisher  ="The Royal Society of Chemistry",
+doi  ="10.1039/C8SC00529J",
+url  ="http://dx.doi.org/10.1039/C8SC00529J" 
+}
+
+@article{Tirimbo_2020,
+author = {Tirimb\`{o},G.  and Sundaram,V.  and \c{C}aylak,O.  and Scharpach,W.  and Sijen,J.  and Junghans,C.  and Brown,J.  and Ruiz,F. Zapata  and Renaud,N.  and Wehner,J.  and Baumeier,B. },
+title = {Excited-state electronic structure of molecules using many-body Green’s functions: Quasiparticles and electron–hole excitations with VOTCA-XTP},
+journal = { J. Chem. Phys. },
+volume = {152},
+number = {11},
+pages = {114103},
+year = {2020},
+doi = {10.1063/1.5144277},
+URL = { https://doi.org/10.1063/1.5144277},
+eprint = {  https://doi.org/10.1063/1.5144277}
+}
+
+@Article{Li_2019,
+author ="Li, Jing and Duchemin, Ivan and Roscioni, Otello Maria and Friederich, Pascal and Anderson, Marie and Da Como, Enrico and Kociok-K\"{o}hn, Gabriele and Wenzel, Wolfgang and Zannoni, Claudio and Beljonne, David and Blase, Xavier and D{'}Avino, Gabriele",
+title  ="Host dependence of the electron affinity of molecular dopants",
+journal  ="Mater. Horiz.",
+year  ="2019",
+volume  ="6",
+issue  ="1",
+pages  ="107-114",
+publisher  ="The Royal Society of Chemistry",
+doi  ="10.1039/C8MH00921J",
+url  ="http://dx.doi.org/10.1039/C8MH00921J" 
+}
+
+
+@article{Cammi_2005,
+author = {Cammi,R.  and Corni,S.  and Mennucci,B.  and Tomasi,J. },
+title = {Electronic excitation energies of molecules in solution: State specific and linear response methods for nonequilibrium continuum solvation models},
+journal = {The Journal of Chemical Physics},
+volume = {122},
+number = {10},
+pages = {104513},
+year = {2005},
+doi = {10.1063/1.1867373},
+URL = {   https://doi.org/10.1063/1.1867373},
+eprint = {  https://doi.org/10.1063/1.186737}
+}
+
+@article{Huu_2020,
+  title = {Antiadiabatic View of Fast Environmental Effects on Optical Spectra},
+  author = {Phan Huu, D. K. Andrea and Dhali, Rama and Pieroni, Carlotta and Di Maiolo, Francesco and Sissa, Cristina and Terenziani, Francesca and Painelli, Anna},
+  journal = {Phys. Rev. Lett.},
+  volume = {124},
+  issue = {10},
+  pages = {107401},
+  numpages = {5},
+  year = {2020},
+  month = {Mar},
+  publisher = {American Physical Society},
+  doi = {10.1103/PhysRevLett.124.107401},
+  url = {https://link.aps.org/doi/10.1103/PhysRevLett.124.107401}
+}

Manuscript/BSE_JPCL.tex

@@ -294,7 +294,7 @@ The neglect of the vertex leads to the so-called $GW$ approximation of the self-
 \end{equation}
 that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with
 \begin{gather}
-	W(1,2) = v(1,2) + \int d34 \, v(1,2)   \chi_0(3,4) W(4,2),
+	W(1,2) = v(1,2) + \int d34 \, v(1,2)   \chi_0(3,4) W(4,2),  \label{eq:defW}
 	\\
 	\chi_0(1,2) = -i \int d34 \,  G(2,3) G(4,2),
 \end{gather}
@@ -509,9 +509,24 @@ The analysis of the screened Coulomb potential matrix elements in the BSE kernel
 The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011,Baumeier_2012a,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems.\\
 
 %==========================================
-\subsection{Solvent effects} 
+\subsection{ Combining BSE with PCM and QM/MM models } 
 %==========================================
-\titou{T2: introduce discussion about coupling between BSE and solvent models.}
+
+Recent attempts to merge the $GW$ and BSE formalisms with model polarizable environments at the PCM or QM/MM levels
+\cite{Baumeier_2014,Duchemin_2016,Li_2016,Varsano_2016,Duchemin_2018,Li_2019,Tirimbo_2020} paved the way not only to interesting applications but also to a better understanding of the merits of these approaches relying on the use of the  screened Coulomb potential designed to capture  polarization effects at all spatial ranges. As a matter of fact,
+dressing the bare Coulomb potential with the reaction field matrix 
+$$
+v({\bf r},{\bf r}') \longrightarrow v({\bf r},{\bf r}') + v^{\text{reac}}({\bf r},{\bf r}'; \omega)
+$$
+in the relation between the screened Coulomb potential $W$ and the independent-electron susceptibility (equation \ref{eq:defW}) allows to perform $GW$ and BSE calculations in a polarizable environment (a solvent, a donor/acceptor interface, a semiconducting or metallic substrate, etc.) with the same complexity as in the gas phase. The reaction field matrix $v^{\text{reac}}({\bf r},{\bf r}'; \omega)$ describes the potential generated in ${\bf r}'$ by the charge rearrangements in the polarizable environment induced by a source charge located in ${\bf r}$, with $\bf r$ and ${\bf r}'$ in the quantum mechanical (QM) subsystem of interest. The reaction field is dynamical since the dielectric properties of the environment, such as the macroscopic dielectric constant $\epsilon_M(\omega)$, are in principle frequency dependent.   Once the reaction field matrix known, with  typically   $\mathcal{O}(N^3)$ operations, the full spectrum of $GW$ quasiparticle energies and BSE neutral excitations can be renormalized by the effect of the environment.
+
+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.
+~\citenum{Huu_2020}). \\
+
+
+
  We now leave the description of successes to discuss difficulties and Perspectives.\\
 
 %==========================================