bit more intro

BSEdyn.bib

@@ -1,13 +1,25 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-05-18 21:44:02 +0200 
+%% Created for Pierre-Francois Loos at 2020-05-18 22:14:10 +0200 
 
 
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+@article{Loos_2020b,
+	Author = {P. F. Loos and F. Lipparini and M. Boggio-Pasqua and A. Scemama and D. Jacquemin},
+	Date-Added = {2020-05-18 22:13:24 +0200},
+	Date-Modified = {2020-05-18 22:13:54 +0200},
+	Doi = {10.1021/acs.jctc.9b01216},
+	Journal = {J. Chem. Theory Comput.},
+	Pages = {1711},
+	Title = {A Mountaineering Strategy to Excited States: Highly-Accurate Energies and Benchmarks for Medium Size Molecules,},
+	Volume = {16},
+	Year = {2020},
+	Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b01216}}
+
 @article{Albrecht_1997,
 	Author = {Albrecht, Stefan and Onida, Giovanni and Reining, Lucia},
 	Date-Added = {2020-05-18 21:40:28 +0200},
@@ -1527,8 +1539,8 @@
 @article{Loos_2020,
 	Author = {Pierre-Fran{\c c}ois Loos and Anthony Scemama and Ivan Duchemin and Denis Jacquemin and Xavier Blase},
 	Date-Added = {2020-05-18 21:40:28 +0200},
-	Date-Modified = {2020-05-18 21:40:28 +0200},
+	Date-Modified = {2020-05-18 22:14:08 +0200},
-	Title = {{ Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies }},
+	Title = {Pros and Cons of the Bethe-Salpeter Formalism for Ground-State Energies},
 	Year = {2020}}
 
 @article{Loos_2020a,

BSEdyn.tex

@@ -211,7 +211,7 @@ This is the abstract
 \label{sec:intro}
 %%%%%%%%%%%%%%%%%%%%%%%%
 The Bethe-Salpeter equation (BSE) formalism \cite{Salpeter_1951,Strinati_1988} is to the $GW$ approximation \cite{Hedin_1965,Golze_2019} of many-body perturbation theory (MBPT) \cite{Martin_2016} what time-dependent density-functional theory (TD-DFT) \cite{Runge_1984,Casida_1995} is to Kohn-Sham density-functional theory (KS-DFT), \cite{Hohenberg_1964,Kohn_1965} an affordable way of computing the neutral excitations of a given electronic system. 
-In recent years, it has shown to be useful for molecular systems with a large number of systematic benchmark studies appearing in the scientific literature \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} (see Ref.~ \onlinecite{Blase_2018} for a recent review).
+In recent years, it has shown to be useful for molecular systems with a large number of systematic benchmark studies appearing in the scientific literature \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} (see Ref.~\onlinecite{Blase_2018} for a recent review).
 
 Taking the optical gap (\ie, the lowest optical excitation energy) as an example, BSE builds on top of a $GW$ calculation by adding up excitonic effects $\EB$ to the $GW$ HOMO-LUMO gap
 \begin{equation}
@@ -229,13 +229,34 @@ where $\EgFun = I^\Nel - A^\Nel$ is the the fundamental gap, \cite{Bredas_2014}
 Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ 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 $\EgOpt < \EgFun$.
 
+Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit.
+One key consequence of this approximation is that double (and higher) excitations are completely absent from the BSE spectra. 
+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} 
+They are, moreover, a real challenge for high-level computational methods. \cite{Loos_2018,Loos_2019,Loos_2020b}
+
+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_2009,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.}
+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_2009}
+With such as scheme, they have been able to compute the excited state of biological chromophores, showing that taking into account the electron-hole screening is important for an accurate description of the lowest $n \rightarrow \pi^*$ excitations. \cite{Ma_2009}
+Zhang \text{et al.} \cite{Zhang_2013}, as well as Rebolini and Toulouse \cite{Rebolini_2016} (in a range-separated context) have separately studied the frequency-dependent second-order Bethe-Salpeter kernel showing a modest improvement over its static counterpart.
+In the two latter studies, they also followed a perturbative approach within the TDA with a renormalization of the first-order perturbative correction.
+
+It is important to note that, although these studies are clearly going beyond the static approximation of BSE, they are not able to recover double excitations as the perturbative treatment makes ultimately the BSE kernel static. 
+However, it does permit to recover additional relaxation effects coming from the higher excitations which would be present by ``unfolding'' the dynamical BSE kernel in order to recover a linear eigenvalue problem.
+Finally, let us also mentioned the work of Romaniello and coworkers, \cite{Romaniello_2009b,Sangalli_2011} in which the authors genuinely accessed double excitations by solving the non-linear, frequency-dependent eigenvalue problem. 
+However, it is based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations.
+
+In the present study, we extend the work of Rohlfing and coworkers \cite{Rohlfing_2000,Ma_2009} by proposing a renormalized first-order perturbative correction to the static neutral excitation energy.
+Importantly, our correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.
+Moreover, we investigate quantitatively the effect of the TDA by computing both the resonant and anti-resonant dynamical corrections to the BSE excitation energies.
+Unless otherwise stated, atomic units are used.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \section{Theory}
 \label{sec:theory}
 %%%%%%%%%%%%%%%%%%%%%%%%
 %================================
-\subsection{Theory for physics}
+\subsection{Theory for physicists}
 %=================================
 
 The resolution of the dynamical Bethe-Salpeter equation (dBSE) [Strinati]