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BSEdyn.bib
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@ -1,13 +1,22 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-05-21 08:48:23 +0200
%% Created for Pierre-Francois Loos at 2020-05-21 09:37:18 +0200
%% Saved with string encoding Unicode (UTF-8)
@phdthesis{Rebolini_PhD,
Author = {E. Rebolini},
Date-Added = {2020-05-21 09:33:45 +0200},
Date-Modified = {2020-05-21 09:36:46 +0200},
School = {Universit{\'e} Pierre et Marie Curie --- Paris VI},
Title = {Range-Separated Density-Functional Theory for Molecular Excitation Energies},
Url = {https://tel.archives-ouvertes.fr/tel-01027522},
Year = {2014}}
@article{Baumeier_2012a,
Author = {Baumeier, Bj\"{o}rn and Andrienko, Denis and Rohlfing, Michael},
Date-Added = {2020-05-20 22:01:43 +0200},

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BSEdyn.tex
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@ -223,7 +223,7 @@ Moreover, we investigate quantitatively the effect of the Tamm-Dancoff approxima
\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_2017a,Jacquemin_2017b,Krause_2017,Gui_2018} (see Ref.~\onlinecite{Blase_2018} for a recent review).
In recent years, it has been shown to be a valuable tool for computational theoretical chemists with a large number of systematic benchmark studies on large molecular systems appearing in the scientific literature \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,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}
@ -242,25 +242,27 @@ Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-
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.
In complete analogy with the ubiquitous adiabatic approximation in TD-DFT, one key consequence of the static approximation is that double (and higher) excitations are completely absent from the BSE spectrum.
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_2018a,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_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.}
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}
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 $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b}
With such as scheme, they have been able to compute the excited states of biological chromophores, showing that taking into account the electron-hole dynamical screening is important for an accurate description of the lowest $n \ra \pi^*$ excitations. \cite{Ma_2009a,Ma_2009b,Baumeier_2012b}
Indeed, studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that \textit{``the influence of dynamical screening on the excitation energies is about $0.1$ eV for the lowest $\pi \ra \pis$ transitions, but for the lowest $n \ra \pis$ transitions the influence is larger, up to $0.25$ eV.''} \cite{Ma_2009b}
A similar conclusion was reached in Ref.~\onlinecite{Ma_2009a}.
Zhang \textit{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.
Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Salpeter kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016,Rebolini_PhD}
In these two latter studies, they also followed a (non-self-consistent) 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.
It is important to note that, although all the studies mentioned above 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, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations (and, in particular, non-interacting double excitations).
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.
Finally, let us also mentioned the work of Romaniello and coworkers, \cite{Romaniello_2009b,Sangalli_2011} in which the authors genuinely accessed additional 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_2009a,Ma_2009b,Baumeier_2012b} by proposing a renormalized first-order perturbative correction to the static neutral excitation energy.
In the present study, we extend the work of Rohlfing and coworkers \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} by proposing a renormalized first-order perturbative correction to the static BSE excitation energies.
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.
@ -270,7 +272,7 @@ Unless otherwise stated, atomic units are used.
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
In this Section, we describe the theoretical foundations leading to the dynamical Bethe-Salpeter equation, following the seminal work by Strinati, \cite{Strinati_1988} presenting in a second step the perturbative implementation \cite{Rohlfing_2000,Ma_2009} of the dynamical correction as compared to the standard adiabatic calculations. More details of the derivation are provided in ...
In this Section, we describe the theoretical foundations leading to the dynamical Bethe-Salpeter equation, following the seminal work by Strinati, \cite{Strinati_1988} presenting in a second step the perturbative implementation \cite{Rohlfing_2000,Ma_2009a,Ma_2009b} of the dynamical correction as compared to the standard adiabatic calculations. More details of the derivation are provided in ...
%================================
\subsection{General dynamical BSE theory}
%=================================
@ -618,7 +620,6 @@ Searching iteratively for the lowest eigenstates, via the Davidson algorithm for
Constructing the static and dynamic BSE Hamiltonians is much more expensive as it requires the complete diagonalization of the $(\Nocc \Nvir \times \Nocc \Nvir)$ RPA linear response matrix [see Eq.~\eqref{eq:LR-RPA}], which corresponds to a $\order*{\Nocc^3 \Nvir^3} = \order*{\Norb^6}$ computational cost.
Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is the computational bottleneck in our current implementation.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}