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BSEdyn.bib
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@ -6,7 +6,20 @@
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2018,
Author = {Loos, Pierre-Fran{\c c}ois and Galland, Nicolas and Jacquemin, Denis},
Date-Added = {2020-07-24 13:26:39 +0200},
Date-Modified = {2020-07-24 13:26:46 +0200},
Doi = {10.1021/acs.jpclett.8b02058},
Eprint = {https://doi.org/10.1021/acs.jpclett.8b02058},
Journal = {J. Phys. Chem. Lett.},
Number = {16},
Pages = {4646--4651},
Title = {Theoretical 0--0 Energies with Chemical Accuracy},
Url = {https://doi.org/10.1021/acs.jpclett.8b02058},
Volume = {9},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.8b02058}}
@article{Loos_2020g,
Author = {P. F. Loos and D. Jacquemin},

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BSEdyn.tex
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@ -195,8 +195,7 @@
\begin{document}
\title{Dynamical Correction to the Bethe-Salpeter Equation}
%\title{ \textcolor{red}{Assessing} Dynamical Correction to the Bethe-Salpeter Equation}
\title{Dynamical Correction to the Bethe-Salpeter Equation Beyond the Plasmon-Pole Approximation}
\author{Pierre-Fran\c{c}ois \surname{Loos}}
\email{loos@irsamc.ups-tlse.fr}
@ -227,7 +226,7 @@ Our calculations are benchmarked against high-level (coupled-cluster) calculatio
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 (or optical) excitations of a given electronic system.
In recent years, it has been shown to be a valuable tool for computational chemists with a large number of systematic benchmark studies on large families of molecular systems appearing in the literature \cite{Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_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
Qualitatively, 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 \titou{(\ie, the electron-hole binding energy) $\EB$} to the $GW$ HOMO-LUMO gap
\begin{equation}
\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \eps_{\HOMO}^{\GW},
\end{equation}
@ -239,7 +238,7 @@ in order to approximate the optical gap
\begin{equation}
\EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB,
\end{equation}
where
where \trashPFL{$\EB$ is the electron-hole binding energy and}
\begin{equation} \label{eq:Egfun}
\EgFun = I^N - A^N
\end{equation}
@ -267,8 +266,7 @@ Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Sa
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 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.
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 \titou{accounts for dynamical effects only on excitations already present in the static limit.} However, it does permit to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher 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. \cite{Loos_2020f}
Based on a simple two-level model 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.
@ -283,7 +281,7 @@ Following a similar strategy, Romaniello \textit{et al.} \cite{Romaniello_2009b}
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.
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. \xavier{By comparison with higher-level calculations coupled-cluster (CC) we show that ... RESULTS AS IN CONCLUSION ... }
%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.
@ -529,7 +527,7 @@ One can verify that, in the static limit where $\Om{m}{\RPA} \to \infty$, the ma
\label{eq:Wstat}
\W{ij,ab}{\text{stat}}
\equiv W_{ij,ab}(\omega = 0)
= \ERI{ij}{ab} - 4 \sum_m \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta},
= \ERI{ij}{ab} - 4 \sum_m \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} },
\end{equation}
evidencing that the standard static BSE problem is recovered from the present dynamical formalism in this limit.
@ -550,7 +548,7 @@ The analysis of the (off-diagonal) screened Coulomb potential matrix elements mu
\\
\times \qty[ \frac{1}{\Om{ij}{S} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ba}{S} - \Om{m}{\RPA} + i\eta} ],
\end{multline}
reveals strong divergences even for low-lying excitations with, for example, $\Om{ba}{S} - \Om{m}{\RPA} = \Om{S}{} - \Om{m}{\RPA} - ( \eGW{a} - \eGW{b} ) \approx 0$.
reveals strong divergences even for low-lying excitations when, for example, $\Om{ba}{S} - \Om{m}{\RPA} = \Om{S}{} - \Om{m}{\RPA} - ( \eGW{a} - \eGW{b} ) \approx 0$.
Such divergences may explain that in previous studies dynamical effects were only accounted for at the TDA level. \cite{Strinati_1988,Rohlfing_2000,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Zhang_2013,Rebolini_2016}
To avoid confusions here, enforcing the TDA for the dynamical correction (which corresponds to neglecting the dynamical correction originating from the anti-resonant part of the BSE Hamiltonian) will be labeled as dTDA in the following.
Going beyond the dTDA is outside the scope of the present study but shall be addressed eventually.
@ -1109,18 +1107,43 @@ Errors associated with these excitation energies (with respect to CC3) are repre
As expected the static BSE excitation energies are much more accurate for these larger molecules with a MAE of $0.32$ eV, a MSE of $0.30$ eV, and a RMSE of $0.38$ eV.
Here again, the dynamical correction improves the accuracy of BSE by lowering the MAE, MSE, and RMSE to $0.23$, $0.00$, and $0.29$ eV, respectively.
Rydberg states are again very slightly affected by dynamical effects, while the dynamical corrections associated with the $n \ra \pis$ and $\pi \ra \pis$ transitions are much larger and of the same magnitude ($0.3$--$0.6$ eV) for both types of transitions.
This latter observation is quite different from the outcomes reached by Rohlfing and coworkers in previous works \cite{Ma_2009a,Ma_2009b} (see Sec.~\ref{sec:intro}) where they observed i) smaller corrections (maybe due to the plasmon-pole approximation), and ii) that $n \ra \pis$ transitions are more affected by the dynamical screening than $\pi \ra \pis$ transitions.
This latter observation is somehow different from the outcomes reached by Rohlfing and coworkers in previous works \cite{Ma_2009a,Ma_2009b} (see Sec.~\ref{sec:intro}) where they observed i) smaller corrections, and ii) that $n \ra \pis$ transitions are more affected by the dynamical screening than $\pi \ra \pis$ transitions.
\titou{The larger size of the molecules considered in Refs.~\onlinecite{Ma_2009a,Ma_2009b} may play a role on the magnitude of the corrections, even though we do not observe here a significant reduction going from small systems (\ce{N2}, \ce{CO}, \ldots) to larger ones (acrolein, butadiene, \ldots).
We emphasize further that previous calculations \cite{Ma_2009a,Ma_2009b} were performed within the plasmon-pole approximation for modeling the dynamical behaviour of the screened Coulomb potential, while we go beyond this approximation in the present study [see Eq.~\eqref{eq:wtilde}].
Finally, while errors were defined with respect to experimental data in Refs.~\onlinecite{Ma_2009a,Ma_2009b}, we consider here as reference high-level CC calculations performed with the very same geometries and basis sets.
As pointed out in previous works, \cite{Loos_2018,Loos_2019b,Loos_2020g} a direct comparison between theoretical transition energies and experimental data is a delicate task, as many factors (such as zero-point vibrational energies and geometrical relaxation) must be taken into account for fair comparisons.
Further investigations are required to better evaluate the impact of these considerations on the influence of dynamical screening.}
As a final comment, let us discuss the two singlet states of butadiene reported in Table \ref{tab:BigMol}.\cite{Maitra_2004,Cave_2004,Saha_2006,Watson_2012,Shu_2017,Barca_2018a,Barca_2018b,Loos_2019}
As discussed in Sec.~\ref{sec:intro}, these corresponds to a bright state of $^1B_u$ symmetry with a clear single-excitation character, and a dark $^1A_g$ state including a substantial fraction of double excitation character (roughly $30\%$).
Although they are both of $\pi \ra \pis$ nature, they are very slightly altered by dynamical screening with corrections of $-0.12$ and $-0.03$ eV for the $^1B_u$ and $^1A_g$ states, respectively.
The small correction on the $^1A_g$ state might be explained by its rather diffuse nature (similar to a Rydberg states). \cite{Boggio-Pasqua_2004}
\xavier{DISCUSSION: To provide further insight into the magnitude of the dynamical correction to valence, Rydberg or CT excitations, let's consider a simple 2-level systems with (ij=hh) and (ab=ll) where (h,l) stand for HOMO and LUMO. The dynamical correction to the $HOMO \rightarrow LUMO$ transition reads:}
\begin{equation*}
\W{hh,ll}{\text{stat}} - \widetilde{W}_{hh,ll}( \Om{1}{} )
= 4 \sERI{hh}{hl} \sERI{ll}{hl} \left( \frac{ \Om{hl}{1} - 2 \OmRPA{hl}{}}{\OmRPA{hl}{} ( \OmRPA{hl}{} - \Om{hl}{1} ) } \right),
\end{equation*}
where the only RPA excitation energy is again the $HOMO-LUMO$ transition with m=hl (see equation 26). For CT excitations with vanishing (hl) overlap, $[hh|hl] \simeq 0$ and similarly $[ll|hl] \simeq 0$ so that one can expect the dynamical correction to be weak. Similarly, Rydberg transitions characterized by a delocalized LUMO state, namely a small HOMO-LUMO (hl) overlap, are expected to undergo weak dynamical corrections. The discussion on $\pi-\pi^*$ and $n-\pi^*$ transitions is certainly more complex and molecule-specific symmetry arguments must be invoked to understand the magnitude of the $[hh|hl]$ and $[ll|hl]$ terms.
[ Taking the simple case of a $\pi$-conjugated planar molecule located in the (xy) plane, then $\pi-\pi^*$ transition will yield $[hh|hl]$ and $[ll|hl]$ terms built from $(p_z^2 | p_z^2)$ contributions on each atom. On the contrary, $n-\pi^*$ transitions will involve $(sp_z|ss)$ or $(sp_z|p_z
^2)$ local contribution that are small by planar symmetry. Bah bon ben ... should lead to small corrections to $n-\pi^*$ ...]
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%
This is the conclusion
The BSE formalism is quickly gaining momentum in the electronic structure community thanks to its attractive computational scaling with system size and its overall accuracy for modeling single excitations of various natures.
It now stands as a genuine cost-effective excited-state method and is regarded as a valuable alternative to the popular TD-DFT method.
However, the vast majority of the BSE calculations are performed within the static approximation in which, in complete analogy with the ubiquitous adiabatic approximation in TD-DFT, the dynamical BSE kernel is replaced by its static limit.
One key consequence of this static approximation is the absence of higher excitations from the BSE optical spectrum.
Following Strinati's footsteps, \cite{Strinati_1982,Strinati_1984,Strinati_1988} several groups have explored the BSE formalism beyond the static approximation by retaining (or reviving) the dynamical nature of the screened Coulomb potential \cite{Sottile_2003,Romaniello_2009b,Sangalli_2011} or via a perturbative approach coupled with the plasmon-pole approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
In the present study, we compute exactly the dynamical screening of the Coulomb interaction within the random-phase approximation, going effectively beyond both the usual static approximation and the plasmon-pole approximation.
In order to assess the accuracy of the present scheme, we report a significant number of calculations for various molecular systems.
Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to clearly evidence the systematic improvements brought by the dynamical correction for both singlet and triplet excited states.
%%%%%%%%%%%%%%%%%%%%%%%%
%\section*{Supplementary material}
@ -1128,13 +1151,14 @@ This is the conclusion
%See the {\SI} for a detailed derivation of the dynamical BSE equation.
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
\acknowledgements
%%%%%%%%%%%%%%%%%%%%%%%%
The authors would like to thank Elisa Rebolini, Pina Romaniello, Arjan Berger, and Julien Toulouse for insightful discussions on dynamical kernels.
PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This study has been (partially) supported through the EUR grant NanoX No.~ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
The authors would like to thank Elisa Rebolini, Pina Romaniello, Arjan Berger, and Julien Toulouse for insightful discussions on dynamical kernels.
%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Data availability}
@ -1147,7 +1171,7 @@ The data that support the findings of this study are available within the articl
\label{app:A}
In this Appendix, we derive Eqs.~\eqref{eq:iL0} to \eqref{eq:iL0bis}.
Combining the Fourier transform (with respect to $t_1$) of $L_0(1,3;4,1')$
Combining the Fourier transform (with respect to $t_1$) of $L_0(1,4;1',3)$
\begin{align}
[L_0](\bx_1,4;\bx_{1'},3 \; | \; \omega_1 )
= -i \int dt_1 e^{i \omega_1 t_1 } G(1,3)G(4,1'),
@ -1189,7 +1213,7 @@ Finally, using the Lehman representation of the Green's functions [see Eq.~\eqre
\end{equation}
with $\tau = \tau_{34}$, and where pp and hh label the particle-particle and hole-hole channels (respectively) that are neglected here. \cite{Strinati_1988}
Projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$ selects the first line of the right-hand-side of Eq.~\eqref{eq:A}, yielding Eq.~\eqref{eq:iL0bis}
with $\Om{1}{} \to \Om{s}{}$.
with $\omega_1 \to \Om{s}{}$.
\section{ $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,S}$ in the electron-hole product basis}
\label{app:B}
@ -1232,7 +1256,7 @@ one gets
\end{split}
\end{equation}
%We now act on the $N$-electron ground-state wave function with
Substituting the following identities
Assuming now that $\lbrace \e{p} , \e{q} \rbrace$ are proper addition/removal energies, such as the $GW$ quasiparticle energies, one can use the following relations:
\begin{subequations}
\begin{align}
e^{+i\hH \tau_{65} } \ha^{\dagger}_p \ket{N} &=
@ -1242,7 +1266,7 @@ Substituting the following identities
e^{-i \qty( E^N_0 - \e{q} ) \tau_{65} } \ket{N},
\end{align}
\end{subequations}
into Eq.~\eqref{eq:N65NS} yields
that plugged into Eq.~\eqref{eq:N65NS} yield
%where $\lbrace \e{p/q} \rbrace$ are quasiparticle energies, such as the $GW$ ones, namely proper addition/removal energies.
%Taking the associated bras that we plug into the orbital product basis expansion of $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)]}{N,S}$ one obtains:
\begin{equation}
@ -1255,7 +1279,11 @@ into Eq.~\eqref{eq:N65NS} yields
- & \theta(-\tau_{65}) \mel{N}{ \ha^{\dagger}_q \ha_p }{N,S} e^{ -i \e{q} \tau_{65} } e^{ - i \Om{S}{} t_6 } ],
\end{split}
\end{equation}
leading to Eq.~\eqref{eq:spectral65} with $\Om{S}{} = E^N_S - E^N_0$, $t_6 = \tau_{65}/2 + t^{65}$, and $t_5 = - \tau_{65}/2 + t^{65}$.
leading to Eq.~\eqref{eq:spectral65} with $\Om{S}{} = E^N_S - E^N_0$, $t_6 = \tau_{65}/2 + t^{65}$, and $t_5 = - \tau_{65}/2 + t^{65}$.
% \titou{ This allows to further recover the expression (11.6) provided by Strinati in the TDA (see Ref.~\onlinecite{Strinati_1988}). }
\rule{0.2\textwidth}{0.4pt}
\bibliography{BSEdyn}