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34
BSEdyn.bib
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@ -1,11 +1,36 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-07-22 22:53:55 +0200
%% Created for Pierre-Francois Loos at 2020-08-26 16:23:32 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2019c,
Author = {E. Giner and A. Scemama and J. Toulouse and P. F. Loos},
Date-Added = {2020-08-26 16:12:40 +0200},
Date-Modified = {2020-08-26 16:14:16 +0200},
Doi = {10.1063/1.5122976},
Journal = {J. Chem. Phys.},
Pages = {144118},
Title = {Chemically accurate excitation energies with small basis sets},
Volume = {151},
Year = {2019}}
@article{Liu_2020,
Author = {C. Liu and J. Kloppenburg and Y. Yao and X. Ren and H. Appel and Y. Kanai and V. Blum},
Date-Added = {2020-08-26 14:42:08 +0200},
Date-Modified = {2020-08-26 16:12:01 +0200},
Doi = {10.1063/1.5123290},
Journal = {J. Chem. Phys.},
Pages = {044105},
Title = {All-electron ab initio Bethe-Salpeter equation approach to neutral excitations in molecules with numeric atom-centered orbitals},
Volume = {152},
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5123290}}
@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},
@ -16,7 +41,9 @@
Pages = {4646--4651},
Title = {Theoretical 0--0 Energies with Chemical Accuracy},
Volume = {9},
Year = {2018}}
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.8b02058}}
@article{Loos_2020g,
Author = {P. F. Loos and D. Jacquemin},
Date-Added = {2020-07-22 22:42:52 +0200},
@ -26,7 +53,8 @@
Pages = {974},
Title = {Is ADC(3) as Accurate as CC3 for Valence and Rydberg Excitation Energies?},
Volume = {11},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1021/acs.jpclett.9b03652}}
@article{Loos_2020f,
Author = {P. F. Loos and J. Authier},

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BSEdyn.tex
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@ -225,7 +225,7 @@ Our calculations are benchmarked against high-level (coupled-cluster) calculatio
\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{Onida_2002,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).
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,Liu_2020} (see Ref.~\onlinecite{Blase_2018} for a recent review).
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 (\ie, the electron-hole binding energy $\EB$) to the $GW$ HOMO-LUMO gap
\begin{equation}
@ -472,7 +472,7 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
\label{sec:dynW}
%=================================
In the present study, we consider the exact spectral representation of $W$ at the RPA level:
In the present study, we consider the exact spectral representation of $W$ at the RPA level \titou{consistently with the underlying $GW$ calculation}:
\begin{multline}
\label{eq:W-RPA}
W_{ij,ab}(\omega)
@ -825,10 +825,9 @@ One key result of the present investigation is that the dynamical correction is
It is only for the smallest basis set (cc-pVDZ) that one can observe significant differences.
We can then safely conclude that the dynamical correction converges rapidly with respect to the size of the one-electron basis set, a triple-$\zeta$ or an augmented double-$\zeta$ basis being enough to obtain near complete basis set limit values.
This is quite a nice feature as it means that one does not need to compute the dynamical correction in a very large basis to get a meaningful estimate of its magnitude.
%The second important observation extracted from the results gathered in Table \ref{tab:N2} is that the dTDA is a rather satisfactory approximation, especially for the singlet states where one observes a maximum discrepancy of $0.03$ eV between the ``full'' and dTDA excitation energies.
%The story is different for the triplet states for which deviations of the order of $0.3$ eV is observed between the two sets, the dTDA of excitation energies being, as we shall see later on, more accurate.
%Indeed, although the dynamical correction systematically red-shift the excitation energies (as anticipated in Sec.~\ref{sec:dynW}), taking into account the coupling between the resonant and anti-resonant parts of the BSE Hamiltonian [see Eq.~\eqref{eq:BSE-final}] yields a systematic blue-shift of the correction, the overall correction remaining negative but by a smaller amount.
%This outcome is similar to the conclusions of several benchmark studies \cite{Jacquemin_2017b,Rangel_2017} which clearly concluded that static BSE triplet excitations are notably too low in energy and that the use of the TDA is able to partly reduce this error.
\titou{The difference between the values of the $GW$ gap, $\Eg^{\GW}$, obtained with cc-pVQZ and aug-cc-pVQZ can be explained by the additional radial completeness brought the set of diffuse functions in the latter one. \cite{Loos_2019c}}
%%% TABLE I %%%
\begin{squeezetable}
@ -1121,13 +1120,13 @@ Further investigations are required to better evaluate the impact of these consi
To provide further insight into the magnitude of the dynamical correction to valence, Rydberg, and CT excitations, let us consider a simple two-level systems with $i = j = h$ and $a = b = l$, where $(h,l)$ stand for HOMO and LUMO.
The dynamical correction associated with the HOMO-LUMO transition reads
\begin{equation*}
\begin{equation}
\W{hh,ll}{\text{stat}} - \widetilde{W}_{hh,ll}( \Om{1}{} )
% = 4 \sERI{hh}{hl} \sERI{ll}{hl} \frac{ \Om{hl}{1} - 2 \OmRPA{hl}{}}{\OmRPA{hl} ( \OmRPA{hl}{} - \Om{hl}{1} ) },
= - 4 \sERI{hh}{hl} \sERI{ll}{hl} \qty( \frac{1}{\OmRPA{hl}} - \frac{1}{\Om{hl}{1} - \OmRPA{hl}} ),
\end{equation*}
\end{equation}
where the only RPA excitation energy, $\OmRPA{hl} = \e{l} - \e{h} + 2 \ERI{hl}{lh}$, is again the HOMO-LUMO transition, \ie, $m=hl$ [see Eq.~\eqref{eq:sERI}].
For CT excitations with vanishing HOMO-LUMO overlap [\ie, $\ERI{h}{l} \approx 0$], $\sERI{hh}{hl} \approx 0$ and $\sERI{ll}{hl} \approx 0$, so that one can expect the dynamical correction to be weak.
For CT excitations with vanishing HOMO-LUMO overlap [\ie, $\ERI{h}{l} \approx 0$] \titou{and small excitonic binding energy}, $\sERI{hh}{hl} \approx 0$ and $\sERI{ll}{hl} \approx 0$, so that one can expect the dynamical correction to be weak.
Likewise, Rydberg transitions which are characterized by a delocalized LUMO state, that is, a small HOMO-LUMO overlap, are expected to undergo weak dynamical corrections.
The discussion for $\pi \ra \pis$ and $n \ra \pis$ transitions is certainly more complex and molecule-specific symmetry arguments must be invoked to understand the magnitude of the $\sERI{hh}{hl}$ and $\sERI{ll}{hl}$ terms.
@ -1150,6 +1149,9 @@ One key consequence of this static approximation is the absence of higher excita
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 have computed 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.
\titou{Dynamical corrections have been calculated using a renormalized first-order perturbative correction to the static BSE excitation energies following the work of Rohlfing and coworkers. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
Note that, although the present study goes beyond the static approximation of BSE, we do not recover additional excitations as the perturbative treatment accounts for dynamical effects only on excitations already present in the static limit.
However, we hope to report results on a genuinely dynamical approach in the near future.}
In order to assess the accuracy of the present scheme, we have reported a significant number of calculations for various molecular systems.
Our calculations have been benchmarked against high-level CC calculations, allowing to clearly evidence the systematic improvements brought by the dynamical correction for both singlet and triplet excited states.
We have found that, although $n \ra \pis$ and $\pi \ra \pis$ transitions are systematically red-shifted by $0.3$--$0.6$ eV thanks to dynamical effects, their magnitude is much smaller for CT and Rydberg states.

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Cover_Letter/CoverLetter.tex
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@ -25,7 +25,7 @@ In order to assess the accuracy of the present scheme, we report a significant n
Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing to clearly evidence the improvements brought by the dynamical correction.
Because of the novelty of this part and its large impact in the electronic structure community (and beyond), we expect it to be of interest to a wide audience within the chemistry and physics communities.
We suggest Michael Rohlfing, Neepa Maitra, Patrick Rinke, Weito Yang, Wim Klopper, Fabien Bruneval, and Lucia Reining as potential referees.
We suggest Michael Rohlfing, Neepa Maitra, Patrick Rinke, Weitao Yang, Wim Klopper, Fabien Bruneval, and Lucia Reining as potential referees.
We look forward to hearing from you soon.
\closing{Sincerely, the authors.}

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\documentclass[10pt]{letter}
\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e,hyperref}
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\definecolor{darkgreen}{HTML}{009900}
\begin{document}
\begin{letter}%
{To the Editors of the Journal of Chemical Physics}
\opening{Dear Editors,}
\justifying
Please find attached a revised version of the manuscript entitled
\begin{quote}
\textit{``Dynamical Correction to the Bethe-Salpeter Equation Beyond the Plasmon-Pole Approximation''}.
\end{quote}
We thank the reviewers for their constructive comments.
Our detailed responses to their comments can be found below.
For convenience, changes are highlighted in red in the revised version of the manuscript.
We look forward to hearing from you.
\closing{Sincerely, the authors.}
%%% REVIEWER 1 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#1}
\begin{itemize}
\item
{The manuscript by Loos and Blase discusses high-quality excited-state
calculations for small prototypical molecules. The authors apply
the GW-BSE approach which, starting from HF orbitals and energy levels,
constructs single-particle and charge-neutral excitations via the
equation of motion of corresponding Green functions. While the group
has already published several papers in that direction, their present
focus is on the dynamics of the dielectric screening, in particular
for the Bethe-Salpeter equation (BSE).
The merit of the present paper is the comprehensive discussion of
quite a large number of transitions in prototypical molecules, thus
providing benchmark data for others. The data appear to be of very
high quality.
I believe the paper is worth publishing.}
\\
\alert{We thank the reviewer for supporting the publication of the present manuscript.}
\item
{Nonetheless I have a general comment:
Concerning the significance of screenign dynamics, I am not really sure
if the authors go beyond what others have already done. They go beyond
plasmon-pole modelling, which is a very good concept since plasmon-pole
models have always been questioned concerning their reliability - however,
the authors do not compare their results with data obtained from
plasmon-pole modelling. If they have such data readily available,
they might want to include them in the manuscript.}
\\
\alert{Sadly, it is hard to compare Rohlfing's results with ours as the molecules considered in Rohlfing's works are much larger than ours, and the calculations are performed in non-conventional basis sets [CHECK].
We agree that, ultimately, it would be very interesting to test the performances of both approaches on the very same systems with the very same settings, but we feel this is outside the scope of the present study.}
\item
{Concerning the other aspect, i.e. renormalization of the BSE
excitation energies due to dynamics, the authors seem to have done
what others did before, again in a perturbative manner (see Eq. (42)),
but no more, at least not in this
manuscript. Of course, their method will allow for highly interesting
effects in the future (multiple solutions, re-structuring the composition
of excitations, satellite structure, etc.),
but not in this paper. Or have I misunderstood something?
In any case, I believe the authors should more clearly state how their
(present) concept relates to previous work in the literature, and
more clearly distinguish between present data and future perspectives.
It's all in the manuscript, but somehow 'distributed', and it would
be helpful to have all this in one paragraph. }
\\
\alert{The reviewer is right.
The present perturbative treatment cannot access additional excitations, and we are definitely interested in pursuing in this direction in the near future.
This is clearly stated in the original version of our manuscript near the end of the Introduction:
``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 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.''
To answer the reviewer's comment we have added a new paragraph to the concluding section, which reads: ``Dynamical corrections have been calculated using a renormalized first-order perturbative correction to the static BSE excitation energies following the work of Rohlfing and coworkers.
Note that, although the present study goes beyond the static approximation of BSE, we do not recover additional excitations as the perturbative treatment accounts for dynamical effects only on excitations already present in the static limit.
However, we hope to report results on a genuinely dynamical approach in the near future.''}
\item
{The authors emphasize that their $W$ is evaluated within RPA for
many more than just one frequency (avoiding plasmon-pole modelling etc.).
I assume that they employ the same 'high-quality' screening dynamics for
evaluating their GW energy levels, correct? They should clearly state
that somewhere (unless I missed it). }
\\
\alert{Yes, this is correct. We have clarified this point above Eq.~(25).}
\item
{The authors use two different types of basis sets, cc-pVXZ and
aux-cc-pVXZ, both of which can be improved towards convergence by
choosig X=D, T, Q (etc.). However, Tab. I shows that the nitrogen $GW$
quasiparticle gap does not converge towards the same value for the two
families (cc-pVQZ: 20.05 eV, aux-cc-pQVZ: 19.00 eV).
This seems to be a substantial difference. Why is that? }
\\
\alert{This is completely normal.
To reach the complete basis set limit, one must have a basis set which is angularly complete as well as radially complete.
When X goes up, one improves the angular completeness of the basis set, while diffuse functions takes care of the radial completeness.
For excited states and HOMO-LUMO gaps, because of the special importance of diffuse functions in these cases, the limit that one reaches with these two families of basis functions is indeed different.
We refer the reviewer to [J. Chem. Phys. 151, 144118 (2019)] where the present observation is clearly illustrated.
We have mentioned this fact in the revised version near the beginning of the Results section.}
\item
{Near the end of Sec. IV the authors state that CT or Rydberg states
observe weak dTDA effects because the single-particle overlap (e.g.,
between HOMO and LUMO) is small. I agree, but it should also be
mentioned that the excitonic binding itself is also small, for the
same reason.
}
\\
\alert{Thank you for pointing that out.
We have mentioned this in the revised version of the manuscript.}
\end{itemize}
%%% REVIEWER 2 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#2}
\begin{itemize}
\item
{In the context of computational spectroscopy, dynamical corrections to the Bethe-Salpeter equation (BSE) are investigated for small organic molecules. The authors present their implementation of dynamical corrections that have been derived before. Their work goes beyond previous work in that it does not use the plasmon pole approximation for the screened Coulomb interaction and is thus potentially more accurate. The dynamical BSE calculations are benchmarked for a test set of molecules for which high-level coupled cluster reference calculations are available.
The work is rigorous, carefully done and well presented. The dynamical corrections can be quiet sizeable and improve the BSE results considerably. With the full screened Coulomb interaction, the corrections become larger than previously reported for the plasmon pole model. Still, the dynamical corrections do not add significant computational effort, which makes the BSE scheme highly competitive for optical excitations of weakly to moderately correlated molecules. The methodology and the results will be highly interesting for the core readership of The Journal of Chemical Physics and I recommend publication after the small comments below have been addressed.}
\\
\alert{}
\item
{Throughout the manuscript I kept wondering, if the dynamical corrections had already been derived before.
This is not particularly clear and I encourage the authors to clarify this.
For example, in Section E are equations 32 to 42 new or were they already derived by Rohlfing and others? }
\\
\alert{Originally, the dynamical correction has been derived by Strinati [see Ref.~(2) for a detailed derivation].}
\item
{In the introduction, the authors advocate the dynamical BSE corrections for double excitations.
Yet, in the results section of the manuscript, no double excitations are reported. All result tables include a column for $\Omega_S^\text{stat}$, which I understand is from a standard, static BSE calculation.
For double excitations, however, I expect static BSE to not give a solution (or state) at all.
Thus, the corresponding table entry would be blank. In fact, in the methodological write up, the authors point to these additional solutions in several places, e.g. just after eq. 32 "Note that due to its non-linear nature, eq. 32 may provide more than one solution for each value of S."
So I wonder, does the perturbative approach taken for the dynamical corrections provide access to double excitations?
In eq. 42, which is the final expression, all quantities carry an S index. This implies that dynamical corrections are only calculated for states that are already part of the static solution and no new states can be found with this approach. What is the potential then for applying dynamical corrections to double excitations? This should be clarified. }
\\
\alert{}
\item
{Equation 41 contains a derivative of $A^{(1)}(\Omega_S)$ with respect to $\Omega_S$.
Is this derivative easy to compute?
Is $A^{(1)}(\Omega_S)$ available in analytic form or is the derivative performed numerically? }
\\
\alert{Yes, this derivative is straightforward to compute, and it is computed at no extra cost basically.
The situation is very similar to the computation of the derivative of the $GW$ self-energy $\Sigma$ which is involved in the calculation of the spectral weight $Z$.
In the revised version of the manuscript, we have mentioned that this derivative can be computed easily at no extra cost.}
\item
{Following up on my previous point, why does equation 40 have to be renormalised?
It could be solve iteratively, as is done for the quasiparticle equation in $GW$.
Then the aforementioned derivative would not have to be calculated. }
\\
\alert{Yes, the referee is right. There's two possibilities to solve this equation: renormalization or self-consistency.
As mentioned on page 6 of our original manuscript: "Moreover, we have observed that an iterative, self-consistent resolution [where the dynamically-corrected excitation energies are re-injected in Eq. (39)] yields basically the same results as its (cheaper) renormalized version."
In other words, we have tested both strategies and they basically yield similar results.
However, as mentioned, the renormalized version is much cheaper as one does not have to recompute all these quantities.}
\item
{First paragraph of the introduction: "In recent years, it has been shown to be a valuable tool for computational chemists with a large number os systematic benchmark studies on large families of molecular systems appearing the literature [11-20] (see Ref. 21 for a recent review)." Maybe one could reference also the following, recent all-electron BSE implementation and benchmark study here: C. Liu, J. Kloppenburg, Y. Yao, X. Ren, H. Appel, Y. Kanai, and V. Blum, J. Chem. Phys. 152, 044105 (2020) }
\\
\alert{The reference has been added in due place.}
\item
{Results for N2 are reported in Table II.
Not far off the equilibrium bond length, N2 exhibits a conical intersection in its $^5 \Pi_u$ and $^1\Delta_u$ states.
These should also be visible in the optical transition energies.
Such a conical intersection would be a good test for a theory that goes beyond standard BSE and can tackle more correlated systems, as recently demonstrated, for example, for dynamical configuration interaction (DCI) theory, which also includes GW and BSE elements (see e.g. M. Dvorak, D. Golze, and P. Rinke, Physical Review Materials 3, 070801(R) (2019)). }
\\
\alert{}
\item
{At the end of the Results and Discussion section the authors present an equation for a two level model (that does not have an equation number) to estimate when dynamical corrections are large and when not.
This is very interesting!
Can more be said than dynamical effects depend on the wave function overlap between the occupied and the unoccupied state?
For example, would dynamical effects be larger in systems with more screening?
Or are they more pronounced, if $W$ has more structure in its frequency dependence?
}
\\
\alert{}
\end{itemize}
\end{letter}
\end{document}

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%ANU etterhead Yves
%version 1.0 12/06/08
%need to be improved
\RequirePackage{graphicx}
%%%%%%%%%%%%%%%%%%%%% DEFINE USER-SPECIFIC MACROS BELOW %%%%%%%%%%%%%%%%%%%%%
\def\Who {Pierre-Fran\c{c}ois Loos}
\def\What {Dr}
\def\Where {Universit\'e Paul Sabatier}
\def\Address {Laboratoire de Chimie et Physique Quantiques}
\def\CityZip {Toulouse, France}
\def\Email {loos@irsamc.ups-tlse.fr}
\def\TEL {+33 5 61 55 73 39}
\def\URL {} % NOTE: use $\sim$ for tilde
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MARGINS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\includegraphics[height=0.7in]{CNRS_logo.pdf} \hspace*{\fill}\includegraphics[height=0.7in]{UPS_logo.pdf}
\\
\hrulefill
\\
{\small \What~\Who\hspace*{\fill} Telephone:\ \TEL
\\
\Where\hspace*{\fill} Email:\ \Email
\\
\Address\hspace*{\fill}
\\
\CityZip\hspace*{\fill} \URL}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%% OTHER MACROS BELOW %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\thispagestyle{firstpage}
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\vspace{3\parskip}
{\raggedright \toname \\ \toaddress \par}\vspace{3\parskip}
\noindent #1\par\raggedright\parindent 5ex\par
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%I do not know what does the macro below
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