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42
BSEdyn.bib
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@ -1,13 +1,36 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-05-26 10:43:21 +0200
%% Created for Pierre-Francois Loos at 2020-05-29 10:22:08 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Casida_2005,
Author = {M. E. Casida},
Date-Added = {2020-05-29 10:21:26 +0200},
Date-Modified = {2020-05-29 10:22:06 +0200},
Doi = {10.1063/1.1836757},
Journal = {J. Chem. Phys.},
Pages = {054111},
Title = {Propagator corrections to adiabatic time- dependent density-functional theory linear response theory},
Volume = {122},
Year = {2005}}
@article{Casida_2016,
Author = {M. E. Casida and M. {Huix-Rotllant}},
Date-Added = {2020-05-29 10:10:35 +0200},
Date-Modified = {2020-05-29 10:19:41 +0200},
Doi = {10.1007/128_2015_632},
Journal = {Top. Curr. Chem.},
Pages = {1--60},
Title = {{{Many-Body Perturbation Theory (MBPT) and Time-Dependent Density-Functional Theory (TD-DFT): MBPT Insights About What Is Missing In, and Corrections To, the TD-DFT Adiabatic Approximation}}},
Volume = {368},
Year = {2016},
Bdsk-Url-1 = {https://doi.org/10.1007/128_2015_632}}
@phdthesis{Rebolini_PhD,
Author = {E. Rebolini},
Date-Added = {2020-05-21 09:33:45 +0200},
@ -4075,23 +4098,6 @@
Year = {2004},
Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2004.03.051}}
@article{Cave_2004a,
Author = {Cave, Robert J. and Zhang, Fan and Maitra, Neepa T. and Burke, Kieron},
Date-Added = {2020-01-01 21:36:51 +0100},
Date-Modified = {2020-01-01 21:36:51 +0100},
Doi = {10.1016/j.cplett.2004.03.051},
File = {/Users/loos/Zotero/storage/6L9X6HT4/Cave et al. - 2004 - A dressed TDDFT treatment of the 21Ag states of bu.pdf},
Issn = {00092614},
Journal = {Chem. Phys. Lett.},
Language = {en},
Month = may,
Number = {1-3},
Pages = {39-42},
Title = {A Dressed {{TDDFT}} Treatment of the {{21Ag}} States of Butadiene and Hexatriene},
Volume = {389},
Year = {2004},
Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2004.03.051}}
@inbook{Ceperley_1979,
Author = {D. M. Ceperley and M. H. Kalos},
Date-Added = {2020-01-01 21:36:51 +0100},

26
BSEdyn.tex
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@ -245,6 +245,12 @@ Most of BSE implementations rely on the so-called static approximation, which ap
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}
% double excitations are important as well for open-shell ground state cf Pina and Miquel
%There are also important in the lowest lying excited states of polyenes (such as butadiene) because they strongly mix with the
%for example, the lowest-lying singlet state of polyenes is not a simple highest occupied molecular orbital?lowest unoccupied molecular orbital ??HOMO-LUMO?? one-electron excitation but has a HOMO^2-LUMO^2 double excitation character
%A frequency-dependent xc kernel could create extra poles in the response function, which would describe states with a multiple-excitation character
%The poles of the true response function give the excitation energies of the interacting system, where the excited states can be a mixture of single, double, and higher-multiple ex- citations, whereas the poles of the KS response function are just at single KS excitation energies
%Therefore ??s has fewer poles than ??.
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.}
@ -259,10 +265,18 @@ In these two latter studies, they also followed a (non-self-consistent) perturba
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 additional 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 BSE excitation energies.
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}
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.
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 exchange-correlation (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}
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 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.
@ -322,7 +336,7 @@ is the BSE kernel that takes into account the self-consistent variation of the H
\begin{equation}
v_\text{H}(1) = - i \int d2 \, v(1,2) G(2,2^+),
\end{equation}
[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the exchange-correlation self-energy $ \Sigma_\text{xc}$ with respect to the variation of $G$.
[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the xc self-energy $ \Sigma_\text{xc}$ with respect to the variation of $G$.
In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add (respectively) an electron to the $N$-electron ground state $\ket{N}$ in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator.
The resolution of the dynamical BSE equation \cite{Strinati_1988} starts with the expansion of $L_0$ and $L$ [see Eqs.~\eqref{eq:L0} and \eqref{eq:L}] over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$).
@ -404,7 +418,7 @@ with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
\subsection{Dynamical BSE within the $GW$ approximation}
%=================================
Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
Adopting now the $GW$ approximation for the xc self-energy, \ie,
\begin{equation}
\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
\end{equation}
@ -455,7 +469,7 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
\subsection{Dynamical screening}
%=================================
In the present study, we consider the exact spectral representation of $W$ at the random-phase approximation (RPA) level:
In the present study, we consider the exact spectral representation of $W$ at the RPA level:
\begin{multline}
\label{eq:W}
W_{ij,ab}(\omega)