boosting seriously the intro

BSEdyn.tex

@@ -240,14 +240,16 @@ where
 is the the fundamental gap, \cite{Bredas_2014} $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N+1} - E_0^N$ being the ionization potential and the electron affinity of the $N$-electron system.
 Here, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ corresponds to its ground-state energy.
 Because the excitonic effect corresponds physically to the stabilization implied by the attraction of the excited electron and its hole left behind, we have $\EgOpt < \EgFun$.
+Due to the smaller amount of screening in molecules as compared to solids, a faithful description of the excitonic effects is paramount in molecular systems.
 
 Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
 In complete analogy with the ubiquitous adiabatic approximation in TD-DFT where the exchange-correlation (xc) kernel is made static, one key consequence of the static approximation within BSE is that double (and higher) excitations are completely absent from the BSE spectrum.
 Indeed, a frequency-dependent kernel has the ability to create additional poles in the response function, which describe states with a multiple-excitation character, and, in particular, double excitations.
-Although these double excitations are usually experimentally dark (which means that they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms, \cite{Boggio-Pasqua_2007} and are particularly important in the faithful description of the ground state of open-shell molecules. \cite{Casida_2005,Romaniello_2009a,Huix-Rotllant_2011,Loos_2020c}
-They are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
+Although these double excitations are usually experimentally dark (which means that they usually cannot be observed in photo-absorption spectroscopy), these states play, indirectly, a key role in many photochemistry mechanisms, \cite{Boggio-Pasqua_2007} as they strongly mix with the bright singly-excited states leading to the formation of satellite peaks. \cite{Helbig_2011,Elliott_2011}
+They are particularly important in the faithful description of the ground state of open-shell molecules, \cite{Casida_2005,Romaniello_2009a,Huix-Rotllant_2011,Loos_2020c}
+and they are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018a,Loos_2019,Loos_2020b,Loos_2020c}
 Double excitations play also a significant role in the correct location of the excited states of polyenes that are closely related to rhodopsin which is involved in the visual transduction. \cite{Olivucci_2010,Robb_2007,Manathunga_2016}
-In butadiene, for example, while the bright $1 ^1B_u$ state has a clear ($\HOMO \ra \LUMO$) single-excitation character, the dark $2 ^1A_g$ state includes a substantial fraction of doubly-excited character from the $\HOMO^2 \ra \LUMO^2$ double excitation (roughly $30\%$), yet dominant contributions from the $\HOMO-1 \ra \LUMO$ and $\HOMO \ra \LUMO+1$ single excitations. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
+In butadiene, for example, while the bright $1 ^1B_u$ state has a clear $\HOMO \ra \LUMO$ single-excitation character, the dark $2 ^1A_g$ state includes a substantial fraction of doubly-excited character from the $\HOMO^2 \ra \LUMO^2$ double excitation (roughly $30\%$), yet dominant contributions from the $\HOMO-1 \ra \LUMO$ and $\HOMO \ra \LUMO+1$ single excitations. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
 
 Going beyond the static approximation is tricky and very few groups have dared to take the plunge. \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
 Nonetheless, it is worth mentioning the seminal work of Strinati, \cite{Strinati_1988} who \titou{bla bla bla.}
@@ -264,7 +266,7 @@ However, it does permit to recover, for transitions with a dominant single-excit
 These higher excitations would be explicitly present in the BSE Hamiltonian by ``unfolding'' the dynamical BSE kernel, and one would recover a linear eigenvalue problem with, nonetheless, a much larger dimension.
 
 Based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations, Romaniello and coworkers \cite{Romaniello_2009b,Sangalli_2011} evidenced that one can genuinely access additional excitations by solving the non-linear, frequency-dependent eigenvalue problem. 
-For this particular system, it was shown that a BSE kernel based on the random-phase approximation (RPA) produces indeed double excitations, but also unphysical excitations. \cite{Romaniello_2009b} 
+For this particular system, it was shown that a BSE kernel based on the random-phase approximation (RPA) produces indeed double excitations but also unphysical excitations. \cite{Romaniello_2009b} 
 The appearance of these spurious excitations was attributed to the self-screening problem. \cite{Romaniello_2009a}
 This was fixed in a follow-up paper by Sangalli \textit{et al.} \cite{Sangalli_2011} thanks to the design of a number-conserving approach based on the second RPA.