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Manuscript/sfBSE.tex

@@ -38,7 +38,27 @@
 \section{Introduction}
 \label{sec:intro}
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-\alert{Here comes the introduction.}
+
+Due to the ubiquitous influence of processes involving electronic excited states in physics, chemistry, and biology, their faithful description from first principles has been one of the grand challenges faced by theoretical chemists since the dawn of computational chemistry. 
+Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community.
+An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. \cite{Roos_1996,Piecuch_2002,Dreuw_2005,Krylov_2006,Sneskov_2012,Gonzales_2012,Laurent_2013,Adamo_2013,Ghosh_2018,Blase_2020,Loos_2020a,Casanova_2020}
+The fact that none of these methods is successful in every chemical scenario has encouraged chemists to carry on the development of new excited-state methodologies, their main goal being to get the most accurate excitation energies (and properties) at the lowest possible computational cost in the most general context. \cite{Loos_2020a}
+
+Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996} the Bethe-Salpeter equation (BSE) formalism within the static approximation \cite{Salpeter_1951,Strinati_1988} is plagued by the lack of double (and higher) excitations, which are, for example, ubiquitous in conjugated molecules like polyenes. \cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
+
+One way to access double excitations is via the spin-flip formalism established by Krylov in 2001, \cite{Krylov_2001a,Krylov_2001b,Krylov_2002} with earlier attempts by Bethe, \cite{Bethe_1931} as well as Shibuya and McKoy. \cite{Shibuya_1970}
+Nowadays, spin-flip techniques are widely available for many types of methods such as equation-of-motion coupled cluster (EOM-CC), \cite{} configuration interaction (CI), \cite{} TD-DFT, \cite{} and others \cite{} with successful applications in bond breaking processes, \cite{} radical chemistry, \cite{} and the photochemistry of conical intersections \cite{} to mention a few. 
+We refer the interested reader to Refs.~\onlinecite{Krylov_2006,Krylov_2008,Casanova_2020} for a more detailed review of spin-flip methods.
+
+The idea behind the spin-flip formalism is quite simple: instead of starting the calculation  from the singlet ground state, one can start from the lowest triplet state.
+With a similar idea.
+One of the main issue of spin-flip methods is the spin contamination.
+
+Here we apply the spin-flip formalism to the BSE formalism in order to access, in particular, double excitations.
+We also go beyond the static approximation  \cite{Strinati_1982,Strinati_1984,Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
+Following Strinati's footsteps, Rohlfing and coworkers have developed an efficient way of taking into account, thanks to first-order perturbation theory, the dynamical effects via a plasmon-pole approximation combined with the Tamm-Dancoff approximation (TDA). \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
+
+
 Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
 
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@@ -281,7 +301,7 @@ These are obtained via the diagonalization of an effective Fock matrix which inc
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 \section{Unrestricted Bethe-Salpeter equation formalism}
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-Like its TD-DFT cousin, the BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018} 
+Like its TD-DFT cousin, \cite{Runge_1984,Casida_1995,Petersilka_1996,Dreuw_2005} the BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018} 
 Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
 In a nutshell, BSE builds on top of a $GW$ calculation by adding up excitonic effects (\ie, the electron-hole binding energy) to the $GW$ fundamental gap which is itself a corrected version of the KS gap. 
 The purpose of the underlying $GW$ calculation is to provide quasiparticle energies and a dynamically-screened Coulomb potential that are used to build the BSE Hamiltonian from which the vertical excitations of the system are extracted.