more work on the intro

Manuscript/sfBSE.bib

@@ -1,13 +1,25 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-12-06 15:06:52 +0100 
+%% Created for Pierre-Francois Loos at 2020-12-06 22:23:42 +0100 
 
 
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+@article{Liu_2020,
+	author = {C. Liu and J. Kloppenburg and Y. Yao and X. Ren and H. Appel and Y. Kanai and V. Blum},
+	date-added = {2020-12-06 22:23:41 +0100},
+	date-modified = {2020-12-06 22:23:41 +0100},
+	doi = {10.1063/1.5123290},
+	journal = {J. Chem. Phys.},
+	pages = {044105},
+	title = {All-electron ab initio Bethe-Salpeter equation approach to neutral excitations in molecules with numeric atom-centered orbitals},
+	volume = {152},
+	year = {2020},
+	Bdsk-Url-1 = {https://doi.org/10.1063/1.5123290}}
+
 @article{Krylov_2008,
 	abstract = { The equation-of-motion coupled-cluster (EOM-CC) approach is a versatile electronic-structure tool that allows one to describe a variety of multiconfigurational wave functions within single-reference formalism. This review provides a guide to established EOM methods illustrated by examples that demonstrate the types of target states currently accessible by EOM. It focuses on applications of EOM-CC to electronically excited and open-shell species. The examples emphasize EOM's advantages for selected situations often perceived as multireference cases [e.g., interacting states of different nature, Jahn-Teller (JT) and pseudo-JT states, dense manifolds of ionized states, diradicals, and triradicals]. I also discuss limitations and caveats and offer practical solutions to some problematic situations. The review also touches on some formal aspects of the theory and important current developments. },
 	author = {Krylov, Anna I.},

Manuscript/sfBSE.tex

@@ -44,19 +44,26 @@ Accurately predicting ground- and excited-state energies (hence excitation energ
 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}
+Originally developed in the framework of nuclear physics, \cite{Salpeter_1951} and popularized in condensed-matter physics, \cite{Sham_1966,Strinati_1984,Delerue_2000} one of the new emerging method in the computational chemistry landscape is the Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999,Blase_2018,Blase_2020} from many-body perturbation theory (MBPT) \cite{Onida_2002,Martin_2016} which, based on an underlying $GW$ calculation to compute accurate charged excitations, \cite{Hedin_1965,Golze_2019} is able to provide accurate optical (\ie, neutral) excitations for molecular systems at a rather modest computational cost.\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,Blase_2018,Liu_2020,Blase_2020}
+
+Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
+Like adiabatic time-dependent density-functional theory (TD-DFT), \cite{Runge_1984,Casida_1995,Petersilka_1996} the static BSE formalism 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}
+Indeed both adiabatic TD-DFT and static BSE can only access single excitation with respect to the reference determinant.
 
 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{Krylov_2001a} configuration interaction (CI), \cite{Krylov_2001b} TD-DFT, \cite{Shao_2003,Wang_2004,Li_2011a,Bernard_2012,Zhang_2015} and others \cite{Krylov_2002,Levchenko_2004,Manohar_2008,Casanova_2008,Casanova_2009a,Casanova_2009b,Mayhall_2014a,Mayhall_2014b,Bell_2013,Dutta_2013,Mayhall_2014c,Lefrancois_2015,Mato_2018} with successful applications in bond breaking processes, \cite{Golubeva_2007} radical chemistry, \cite{Slipchenko_2002,Wang_2005,Slipchenko_2003,Rinkevicius_2010,Ibeji_2015,Hossain_2017,Orms_2018,Luxon_2018} and the photochemistry of conical intersections \cite{Casanova_2012,Gozem_2013,Nikiforov_2014,Lefrancois_2016} to mention a few. 
+The idea behind the spin-flip formalism is rather simple: instead of starting the calculation from the singlet ground state, one can start from the lowest triplet state.
+In such a way, one can access the singlet ground state and the doubly-excited state via a spin-flip deexcitation and excitation (respectively), the difference of these excitation energies providing an estimate of the double excitation  where one promotes two electrons from the singlet ground state. 
+One obvious issue of spin-flip methods is that not all double excitations are accessible in such a way.
+Moreover, spin-flip methods are usually hampered by spin-contamination.
+
+Nowadays, spin-flip techniques are widely available for many types of methods such as equation-of-motion coupled cluster (EOM-CC), \cite{Krylov_2001a,Levchenko_2004,Manohar_2008,Casanova_2009a,Dutta_2013} configuration interaction (CI), \cite{Krylov_2001b,Krylov_2002,Mato_2018,Casanova_2008,Casanova_2009b} TD-DFT, \cite{Shao_2003,Wang_2004,Li_2011a,Bernard_2012,Zhang_2015} algebraic-diagrammatic construction (ACD),\cite{Lefrancois_2015} and others \cite{Mayhall_2014a,Mayhall_2014b,Bell_2013,Mayhall_2014c} with successful applications in bond breaking processes, \cite{Golubeva_2007} radical chemistry, \cite{Slipchenko_2002,Wang_2005,Slipchenko_2003,Rinkevicius_2010,Ibeji_2015,Hossain_2017,Orms_2018,Luxon_2018} and the photochemistry of conical intersections \cite{Casanova_2012,Gozem_2013,Nikiforov_2014,Lefrancois_2016} 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}
+The BSE calculations will be based on the spin unrestricted version of $GW$ 
+To the best of our knowledge, the  present study is the first to apply spin-flip formalism to the BSE method.
+Moreover, we also go beyond the static approximation and takes into account dynamical effects via our recently developed perturbative method which builds on the seminal work of Strinati, \cite{Strinati_1982,Strinati_1984,Strinati_1988} Romaniello and collaborators, \cite{Romaniello_2009b,Sangalli_2011} and Rohlfing and coworkers. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b,Lettmann_2019}
 
 
 Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
@@ -337,7 +344,7 @@ Within the $GW$ approximation, the BSE kernel is
 	\\
 	- \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6) 
 \end{multline}
-where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980, Strinati_1982, Strinati_1984, Strinati_1988}
+where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980,Strinati_1982,Strinati_1984,Strinati_1988}
 Within the static approximation which consists in neglecting the frequency dependence of the dynamically-screened Coulomb potential, the spin-conserved and spin-flip BSE optical excitations are obtained by solving the usual Casida-like linear response (eigen)problem:
 \begin{equation}
 \label{eq:LR-BSE}