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BSEdyn.bib
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%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-06-01 08:13:45 +0200 %% Created for Pierre-Francois Loos at 2020-06-06 09:02:31 +0200
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Wambach_1988,
Author = {J. Wambach},
Date-Added = {2020-06-06 09:01:23 +0200},
Date-Modified = {2020-06-06 09:02:27 +0200},
Doi = {10.1088/0034-4885/51/7/002},
Journal = {Rep. Prog. Phys.},
Pages = {989},
Title = {Damping of small-amplitude nuclear collective motion},
Volume = {51},
Year = {1988}}
@article{Sakkinen_2012, @article{Sakkinen_2012,
Author = {N. Sakkinen and M. Manninen and R. van Leeuwen}, Author = {N. Sakkinen and M. Manninen and R. van Leeuwen},
Date-Added = {2020-06-01 08:10:19 +0200}, Date-Added = {2020-06-01 08:10:19 +0200},

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BSEdyn.tex
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\begin{document} \begin{document}
\title{ \textcolor{red}{Assessing} Dynamical Corrections to the Bethe-Salpeter Equation} \title{Dynamical Correction to the Bethe-Salpeter Equation}
%\title{ \textcolor{red}{Assessing} Dynamical Correction to the Bethe-Salpeter Equation}
\author{Pierre-Fran\c{c}ois \surname{Loos}} \author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr} \email{loos@irsamc.ups-tlse.fr}
@ -240,7 +241,7 @@ 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. 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. 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 $\EgOpt < \EgFun$. 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. Due to the smaller amount of screening in molecules as compared to solids, a faithful description of 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. 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. 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.
@ -248,7 +249,7 @@ Indeed, a frequency-dependent kernel has the ability to create additional poles
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} 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} 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} 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} Double excitations play also a significant role in the correct location of the excited states of polyenes that are closely related to rhodopsin, a biological pigment found in the rods of the retina and 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,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_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,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019}
@ -268,12 +269,12 @@ These higher excitations would be explicitly present in the BSE Hamiltonian by `
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. 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} 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. 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. \cite{Wambach_1988}
Finally, let us mention efforts to borrow ingredients from BSE in order to go beyond the adiabatic approximation of TD-DFT. Finally, let us mention efforts to borrow ingredients from BSE in order to go beyond the adiabatic approximation of TD-DFT.
For example, Huix-Rotllant and Casida \cite{Casida_2005,Huix-Rotllant_2011} proposed a nonadiabatic correction to the xc kernel by using the formalism of superoperators, which includes as a special case the dressed TD-DFT method of Maitra and coworkers. \cite{Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012} For example, Huix-Rotllant and Casida \cite{Casida_2005,Huix-Rotllant_2011} proposed a nonadiabatic correction to the xc kernel using the formalism of superoperators, which includes as a special case the dressed TD-DFT method of Maitra and coworkers. \cite{Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012}
Following a similar strategy, Romaniello \textit{et al.} \cite{Romaniello_2009b} took advantages of the dynamically-screened Coulomb potential from BSE to obtain a dynamic TD-DFT kernel. Following a similar strategy, Romaniello \textit{et al.} \cite{Romaniello_2009b} took advantages of the dynamically-screened Coulomb potential from BSE to obtain a dynamic TD-DFT kernel.
In this regard, MBPT provides key insights about what is missing in adiabatic TD-DFT, as discussed at length by Casida and Huix-Rotllant in Ref.~\onlinecite{Casida_2016}. In this regard, MBPT provides key insights about what is missing in adiabatic TD-DFT, as discussed in details by Casida and Huix-Rotllant in Ref.~\onlinecite{Casida_2016}.
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. 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. Importantly, our correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly.