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Cover_Letter/CoverLetter.tex
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\justifying
Please find enclosed our manuscript entitled \textit{``Dynamical Kernels for Optical Excitations''}, which we would like you to consider as a Regular Article in the \textit{Journal of Chemical Physics}.
This contribution fits nicely in the section \textit{``Theoretical Methods and Algorithms''}.
This contribution has never been submitted in total nor in parts to any other journal, and has been seen and approved by all authors.
This contribution fits nicely in the section \textit{``Theoretical Methods and Algorithms''} and has never been submitted in total nor in parts to any other journal, and has been seen and approved by the authors.
In the present manuscript, we discuss, in a pedagogical way, the physics of dynamical (i.e., frequency-dependent) kernels for the computation of optical excitations within linear response theory.
In particular, we consider three dynamical kernels, namely i) an a priori built kernel inspired by the dressed TD-DFT kernel of Maitra and coworkers, ii) the dynamical kernel stemming from the BSE formalism derived originally by Strinati, and iii) the second-order BSE kernel derived first by the groups of Weitao Yang and Julien Toulouse.
The principal take-home message of the present paper is that dynamical kernels have much more to give that one would think.
In more scientific terms, dynamical kernels can provide, thanks to their frequency-dependent nature, additional excitations that can be associated to higher-order excitations (such as the infamous double excitations).
However, they sometimes give too much, and generate spurious excitations, i.e., excitation which does not corresponds to any physical excited state.
Using a simple two-model system, prototypical examples of valence, charge-transfer, and Rydberg excitations are studied.
Using a simple two-level system, prototypical examples of valence, charge-transfer, and Rydberg excitations are studied.
From these, we have observed that, overall, the dynamical correction usually improves the static excitation energies, and that, if one has no interest in double excitations, a perturbative treatment is an excellent alternative to a non-linear resolution of the dynamical equations.
We expect this work to be of interest to a wide audience within the chemistry and physics communities.
We suggest Paola Gori-Giorgi, Neepa Maitra, Valerio Olevano, Patrick Rinke, Weitao Yang, Michael Rohlfing, and Lucia Reining as potential referees.
We suggest Paola Gori-Giorgi, Neepa Maitra, Valerio Olevano, Patrick Rinke, Weitao Yang, 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 Kernels for Optical Excitations''}.
\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
{This paper presents a study of three different frequency-dependent kernels proposed in the literature that go beyond the static/adiabatic approximations of TDDFT and GW/BSE approaches to excitations. The presentation is very instructive and coherent, beginning by showing a universal aspect of these kernels (the context of Lowdin partitioning), before examining the performance of each on three simple molecular systems in a minimal basis. Although some of the main points discussed may not be new to experts in the field, the coherent placement of these issues together and clear explanations will be useful to the community. I am likely to recommend publication in JCP, once the following points are addressed. }
\\
\alert{}
\item
{(1) The systems chosen are said to represent examples of valence, charge-transfer, and Rydberg excited states. Some explanation is required for why they represent charge-transfer and Rydberg excited states. For HeH+ example, the internuclear separation is taken to be near equilibrium, if I understand it correctly. What then is the charge-transfer character of the excited states, i.e. how much of a significant change in the charge distribution do the excited states have compared to the ground state? For the He example, if I recall correctly, the lowest double-excitation of He lies in the continuum, i.e. is not a bound state but rather a resonance. So I am not sure whether it is really accurate to say it is a Rydberg excited state. Did the authors check if the excitation energies they are obtaining for this state lie below the ionization threshold? Perhaps the finite basis set makes this resonance appear bound. }
\\
\alert{}
\item
{(2) In the introduction it is stated that only the correlation part of the kernel is frequency-dependent. However I believe this is only true for two-electron systems. For the N electron case the exact-exchange kernel defined within TDDFT is frequency-dependent (see e.g. Hellgren and Gross, PRA 88, 052507 (2013) and references therein, Hesselmann and Goerling JCP 134, 034120 (2011)) }
\\
\alert{}
\item
{(3) The authors comment that the D-TDDFT kernel (Eq 14) "is known to work best in the weak correlation regime where the true excitations have a clear single and double excitation character" (third last para end of sec IIIB). I found this a bit confusing. To me, weak correlation usually means that the ground-state is well-described by a single Slater determinant, so there are no very low-lying excitations. In this situation excitations of the system can have quite a mixed character, i.e. they need not be largely purely single excitations or largely purely double excitations, but could be, for example 50:50 mixtures of a single and double excitation. Perhaps the language just needs to be clarified in the sentence to reflect that what they mean is that one can define/quantify the single/double excitation character of the state? }
\\
\alert{}
\item
{(4) In that same paragraph, when discussing the accuracy of the method for the excitations, it might be worth noting that in the cited reference 12 (Huix-Rotllant et al), it was observed that the best results seem to arise when using a hybrid kernel for the "static" part. }
\\
\alert{}
\item
{Minor typos:
Sec IIIB: "...same idea was taking further..."- taking should be taken
Sec IV: "factitious" should be fictitious }
\\
\alert{}
\end{itemize}
%%% REVIEWER 2 %%%
\noindent \textbf{\large Authors' answer to Reviewer \#2}
\begin{itemize}
\item
{The authors compare the performance of different ab initio methods to simulate the linear response
spectrum of systems of two interacting electrons, in particular their ability to describe double excitations
and accuracy to predict correct frequency of the single excitations. The three approaches under study use
a dynamical (frequency-dependent) kernel to compute the linear response, since as it is already known an
adiabatic exchange-correlation kernel can only reproduce single excitations.
The poles of the response function are computed using a matrix formulation within the 4-dimensional
vector space that describes a two-level system consisting of one valence and one conduction orbital
(HOMO and LUMO). Transition energies for H2, HeH+ and He using the dressed TDDFT as well as two
different dynamical kernels derived from BSE (including a perturbative treatment) are compared against
the exact solution.
The fact that dynamical(frequency-dependent) xc kernels are able to generate new poles in the linear
response was already observed in previous works which are cited in the manuscript. The authors revisit
this idea and test its effectiveness for different approximations to the dynamical kernel. The study is of
interest to the quantum chemistry community and the TDDFT community because it provides a careful
and useful comparison between different available dynamical kernels for systems for which the exact
spectrum can be computed numerically and used as reference. The manuscript provides a compact
introduction to each of the methods with references to relevant literature on the topic.}
\\
\alert{}
\item
{1. The particularities of the chosen model systems should be better discussed. Since the compared
methods are approximations their performance is likely to depend on the system under study as
well as on the basis set.
For example the dressed TDDFT method is suppose to work best in the case of doubles strongly
coupled to single excitations which from the tables seems not to be the case in these systems
(w1updown and w2updown are not very close in energy) so it's rather remarkable how well
dressed TDDFT predicts the energy of w2updown.}
\\
\alert{}
\item
{2. No explanation or reference is provided accompanying the statement that the model systems that
are chosen are prototypical of valence, charge-transfer and Rydberg excitations and weather these
different excitations can be well represented in the reduced (two-level) space used.}
\\
\alert{}
\item
{3. The molecules H2 and HeH+ are studied at the equilibrium distance, right? It would be
interesting to assess the performance of the different dynamical kernel for stretched molecules.
If you claim the problem is an example of charge-transfer excitations then it would be good to
study the stretched molecules...
Would a larger basis set be needed in such case?}
\\
\alert{}
\item
{4. The statement about the vanishing of the matrix element $<S|H|D>=0$ in H2 could be discussed
better or a reference added.}
\\
\alert{}
\item
{5. The results for He are not discussed at all. Why is the matrix element $<S|H|D>=0$ vanishing for
H2 but not for He? Both systems share the same spatial symmetry (parity) ...
Can you really study Rydberg excitations using a minimal basis?}
\\
\alert{}
\item
{6. It is also not discussed how the exact solution is computed. What is the exact Hamiltonian? Is the
eigensystem computed within the same minimal basis set? How are the excitation energies that
appear as 'exact' in the tables computed?}
\\
\alert{}
\item
{7. It would be illustrative to represent the ground state and the single and double excitations within a
single-particle picture for each system (H2, HeH+,He), maybe with a little figure. For example for
HeH+ is the HOMO consisting of 1 e up and 1 e down (one electron closer to H+ the other
closer to He?) and then a single exc would be promoting one of these electrons to the LUMO and
the double promoting both electrons to the LUMO? That means that within the minimal basis of 2
orbitals both electrons have to do always the same, i.e. you can not describe the excitation of one
electron to the LUMO whereas the other one stays in the HOMO ? (of course this picture only
makes sense in the limit of weak interaction between particles)}
\\
\alert{}
\item
{8. In order to address a larger community it would desirable to discuss briefly which orbitals are taking into account for each system when using the minimal basis.
Maybe that helps understand the results.
What is the difference between using STO-3G and Pople's 6-31G and why is one or the
other chosen?}
\\
\alert{}
\item
{9. How does the dimension of the basis set affect the performance of the different dynamical kernel
under study?}
\\
\alert{}
\item
{10. page 2, beginning of section A:
?two level quantum system made of 2 electrons in its singlet gs? is a bit misleading. The vector
space is 4-dimensional and if you place 2 electrons with opposite spin then the gs, which is a
singlet, consists of both electrons occupying the lowest orbital.}
\\
\alert{}
\item
{11. In page 2, end of second paragraph: when you say 'same numerical examples' do you mean the
same systems (H2, HeH+, He)?}
\\
\alert{}
\item
{Beginning of section B, 'one single and one double excitation'}
\\
\alert{}
\item
{section B, by 'static problem' do you mean 'static kernel'?}
\\
\alert{}
\item
{Table I has different unities than the rest of the tables, which is a bit confusing. Why didn't you
use the same unities everywhere? Is the column called 'exact' in tables 2,3,4 computed from the
values given in table I? It is not just $n*(ev-ec)$ ($n=1$ for single, $n=2$ for double) since the single and
doubles are not a multiple of each other and also that would be $E_{HF}$',right? so how are the exact
transition energies computed? Do they correspond to: omega1 up down= <S|H|S> - <0|H|0> and
omega2 up down= <D|H|D> - <0|H|0> ?}
\\
\alert{}
\item
{15. page 5, second para:
Do you mean by 'static excitations' w1updown and w1upup?}
\\
\alert{}
\end{itemize}
\end{letter}
\end{document}

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Response_Letter/UPS_letterhead.sty Normal file
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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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textwidth 6in
\textheight 9.25in
\oddsidemargin 0.25in
\evensidemargin 0.25in
\topmargin -1.50in
\longindentation 0.50\textwidth
\parindent 5ex
%%%%%%%%%%%%%%%%%%%%%%%%%%% ADDRESS MACRO BELOW %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\address{
\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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\signature{\What~\Who}
\def\opening#1{\ifx\@empty\fromaddress
\thispagestyle{firstpage}
\hspace*{\longindendation}\today\par
\else \thispagestyle{empty}
{\centering\fromaddress \vspace{5\parskip} \\
\today\hspace*{\fill}\par}
\fi
\vspace{3\parskip}
{\raggedright \toname \\ \toaddress \par}\vspace{3\parskip}
\noindent #1\par\raggedright\parindent 5ex\par
}
%I do not know what does the macro below
%\long\def\closing#1{\par\nobreak\vspace{\parskip}
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%\else \fromsig \fi\strut}
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dynker.bib
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-29 21:09:55 +0200
%% Created for Pierre-Francois Loos at 2020-09-08 20:27:46 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Staroverov_1998,
Abstract = {A technique for reducing computational effort in multireference second-order perturbation theory with very large complete active space {\v Z}CAS. SCF reference functions is proposed. This is achieved by construction of an effective Hamiltonian within the space of configurations dominating the reference function expansion. The method is tested on the standard problems of singlet\textendash{}triplet {\v Z}1A1\textendash{}3B1. separation in the CH 2 radical and vertical excitation energies in formaldehyde. Numerical results show that good accuracy can be obtained even with substantially reduced model spaces. q 1998 Elsevier Science B.V. All rights reserved.},
Author = {Staroverov, Viktor N. and Davidson, Ernest R.},
Date-Added = {2020-09-01 13:18:30 +0200},
Date-Modified = {2020-09-01 13:18:30 +0200},
Doi = {10.1016/S0009-2614(98)01092-6},
File = {/Users/loos/Zotero/storage/EBHP4UN5/Staroverov and Davidson - 1998 - The reduced model space method in multireference s.pdf},
Issn = {00092614},
Journal = {Chem. Phys. Lett.},
Language = {en},
Month = nov,
Number = {5-6},
Pages = {435-444},
Title = {The Reduced Model Space Method in Multireference Second-Order Perturbation Theory},
Volume = {296},
Year = {1998},
Bdsk-Url-1 = {https://doi.org/10.1016/S0009-2614(98)01092-6}}
@article{Rawlings_1983,
Author = {Rawlings, Diane C and Davidson, Ernest R},
Date-Added = {2020-09-01 13:18:01 +0200},
Date-Modified = {2020-09-01 13:18:20 +0200},
Doi = {10.1016/0009-2614(83)80080-3},
File = {/Users/loos/Zotero/storage/W9ZT5W8N/RAILINGS and DAVIDSON - 1983 - THE RAYLEIGH-SCHRijDINGER BK METHOD APPLIED TO THE.pdf},
Journal = {Chem. Phys. Lett.},
Number = {5},
Pages = {4},
Title = {{{The Rayleigh}}-{{Schrodinger Bk Method Applied To The Lower Electronic States Of Pyrrole}}},
Volume = {98},
Year = {1983},
Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(83)80080-3}}
@article{Davidson_1981,
Author = {Davidson, Ernest R. and McMurchie, Larry E. and Day, Sheryl J.},
Date-Added = {2020-09-01 13:17:55 +0200},
Date-Modified = {2020-09-01 13:17:55 +0200},
Doi = {10.1063/1.440954},
File = {/Users/loos/Zotero/storage/BWUPHNX4/Davidson et al. - 1981 - The iBi sub iKi sub method Applica.pdf},
Issn = {0021-9606, 1089-7690},
Journal = {J. Chem. Phys.},
Language = {en},
Month = may,
Number = {10},
Pages = {5491--5496},
Shorttitle = {The {{{\emph{B}}}} {\textsubscript{ }}{{{\textsubscript{{\emph{K}}}}}}{\textsubscript{ }} Method},
Title = {The {{{\emph{B}}}} {\textsubscript{ }}{{{\textsubscript{{\emph{K}}}}}}{\textsubscript{ }} Method: {{Application}} to Methylene},
Volume = {74},
Year = {1981},
Bdsk-Url-1 = {https://doi.org/10.1063/1.440954}}
@article{Nitzsche_1978b,
Abstract = {A b initio SCF-CI calculations have been performed on several states of N-methylacetamide near the ground-state equilibrium geometry. The 3 m * state is predicted to lie about 0.5-0.6 eV above the 3 n \textasciitilde{} s*tate. The 3 n \textasciitilde{} s*tate is expected to lie about 0.2-0.3 eV below the experimental l n r * state at 5.5 eV. The I K K * , IK\textasciitilde{}P,,and ln3p configurations are predicted to be strongly mixed, giving three states of large oscillator strength in the region of the broad V band of the absorption spectrum.},
Author = {Nitzsche, Larry E. and Davidson, Ernest R.},
Date-Added = {2020-09-01 13:17:49 +0200},
Date-Modified = {2020-09-01 13:17:49 +0200},
Doi = {10.1021/ja00491a013},
File = {/Users/loos/Zotero/storage/DVYMZD27/Nitzsche and Davidson - 1978 - Ab initio calculation of some vertical excitation .pdf},
Issn = {0002-7863},
Journal = {J. Am. Chem. Soc.},
Language = {en},
Month = nov,
Number = {23},
Pages = {7201--7204},
Title = {Ab Initio Calculation of Some Vertical Excitation Energies of {{N}}-Methylacetamide},
Volume = {100},
Year = {1978},
Bdsk-Url-1 = {https://doi.org/10.1021/ja00491a013}}
@article{LiManni_2013,
Abstract = {A new multiconfigurational quantum chemical method, SplitGAS, is presented. The configuration interaction expansion, generated from a generalized active space, GAS, wave function is split in two parts, a principal part containing the most relevant configurations and an extended part containing less relevant, but not negligible, configurations. The partition is based on an orbital criterion. The SplitGAS method has been employed to study the HF, N2, and Cr2 molecules. The results on these systems, especially on the challenging, multiconfigurational Cr2 molecule, are satisfactory. While SplitGAS is comparable with the GASSCF method in terms of memory requirements, it performs better than the complete active space method followed by second-order perturbation theory, CASPT2, in terms of equilibrium bond length, dissociation energy, and vibrational properties.},
Author = {Li Manni, Giovanni and Ma, Dongxia and Aquilante, Francesco and Olsen, Jeppe and Gagliardi, Laura},
Date-Added = {2020-09-01 13:17:43 +0200},
Date-Modified = {2020-09-01 13:17:43 +0200},
Doi = {10.1021/ct400046n},
File = {/Users/loos/Zotero/storage/TKY5BQKF/Li Manni et al. - 2013 - SplitGAS Method for Strong Correlation and the Cha.pdf},
Issn = {1549-9618, 1549-9626},
Journal = {J. Chem. Theory Comput.},
Language = {en},
Month = aug,
Number = {8},
Pages = {3375-3384},
Title = {{{SplitGAS Method}} for {{Strong Correlation}} and the {{Challenging Case}} of {{Cr}} {\textsubscript{2}}},
Volume = {9},
Year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1021/ct400046n}}
@article{Gershgorn_1968,
Author = {Gershgorn, Z. and Shavitt, I.},
Date-Added = {2020-09-01 13:17:05 +0200},
Date-Modified = {2020-09-01 13:17:05 +0200},
Doi = {10.1002/qua.560020603},
File = {/Users/loos/Zotero/storage/VPC9E83K/Gershgorn and Shavitt - 1968 - An application of perturbation theory ideas in con.pdf},
Issn = {0020-7608, 1097-461X},
Journal = {Int. J. Quantum Chem.},
Language = {en},
Month = nov,
Number = {6},
Pages = {751--759},
Title = {An Application of Perturbation Theory Ideas in Configuration Interaction Calculations},
Volume = {2},
Year = {1968},
Bdsk-Url-1 = {https://doi.org/10.1002/qua.560020603}}
@article{Malrieu_1985,
Abstract = {The theory of effective Hamiltonians is well established. However, limitations appear in its applicability for many problems in molecular physics and quantum chemistry. The standard effective Hamiltonians may become strongly non-Hermitian when there is a large coupling between the model space, in which they are defined, and the outer space Moreover, in the presence of intruder states, discontinuities appear in the matrix elements of these effective Hamiltonians as a function of the internuclear distances. To solve these difficulties, a new class of effective Hamiltonians (called intermediate Hamiltonians) is presented; only one part of their roots are exact eigen-energies of the full Hamiltonian. The theory of these intermediate Hamiltonians is presented by means of a new wave-operator R which is the analogue of the wave-operator Omega in the theory of effective Hamiltonians. Solutions are obtained by a generalised degenerate perturbation theory (GDPT) and by iterative procedures. Two model systems are numerically solved which demonstrate the good convergence properties of GDPT with respect to standard degenerate perturbation theory (DPT). Continuity of the solutions is also checked in the presence of an intruder state.},
Author = {J P Malrieu and P Durand and J P Daudey},
Date-Added = {2020-09-01 13:10:57 +0200},
Date-Modified = {2020-09-01 13:19:11 +0200},
Doi = {10.1088/0305-4470/18/5/014},
Journal = {J. Phys. A: Math. Theor.},
Month = {apr},
Number = {5},
Pages = {809--826},
Publisher = {{IOP} Publishing},
Title = {Intermediate Hamiltonians as a new class of effective Hamiltonians},
Url = {https://doi.org/10.1088%2F0305-4470%2F18%2F5%2F014},
Volume = {18},
Year = 1985,
Bdsk-Url-1 = {https://doi.org/10.1088%2F0305-4470%2F18%2F5%2F014},
Bdsk-Url-2 = {https://doi.org/10.1088/0305-4470/18/5/014}}
@article{Dvorak_2019b,
Author = {M. Dvorak and D. Golze and P. Rinke},
Date-Added = {2020-09-01 12:56:52 +0200},
Date-Modified = {2020-09-01 12:57:50 +0200},
Doi = {10.1103/PhysRevMaterials.3.070801},
Journal = {Phys. Rev. Mat.},
Pages = {070801(R)},
Title = {Quantum embedding theory in the screened Coulomb interaction: Combining configuration interaction with $GW$/BSE},
Volume = {3},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.99.115134}}
@article{Dvorak_2019a,
Author = {M. Dvorak and P. Rinke},
Date-Added = {2020-09-01 12:55:55 +0200},
Date-Modified = {2020-09-01 12:56:40 +0200},
Doi = {10.1103/PhysRevB.99.115134},
Journal = {Phys. Rev. B},
Pages = {115134},
Title = {Dynamical configuration interaction: Quantum embedding that combines wave functions and Green's functions},
Volume = {99},
Year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.99.115134}}
@article{Gunnarsson_1976,
Author = {Gunnarsson, O. and Lundqvist, B. I.},
Date-Added = {2020-08-28 14:22:38 +0200},
@ -57,9 +202,9 @@
@misc{Loos_2020e,
Archiveprefix = {arXiv},
Author = {Pierre-Francois Loos and Xavier Blase},
Date-Modified = {2020-08-29 20:57:02 +0200},
Date-Modified = {2020-09-08 20:27:45 +0200},
Eprint = {2007.13501},
Howpublished = {(submitted to J. Chem. Phys.)},
Howpublished = {(J. Chem. Phys., in press)},
Primaryclass = {physics.chem-ph},
Title = {{{Dynamical Correction to the Bethe-Salpeter Equation Beyond the Plasmon-Pole Approximation}}},
Year = {2020}}
@ -123,9 +268,9 @@
Bdsk-Url-1 = {https://tel.archives-ouvertes.fr/tel-01027522}}
@article{Blase_2020,
Author = {X. Blase and Y. Duchemin and D. Jacquemin},
Author = {X. Blase and I. Duchemin and D. Jacquemin and P. F. Loos},
Date-Added = {2020-06-22 09:07:38 +0200},
Date-Modified = {2020-08-27 16:37:40 +0200},
Date-Modified = {2020-09-08 20:26:11 +0200},
Doi = {10.1021/acs.jpclett.0c01875},
Journal = {J. Phys. Chem. Lett.},
Pages = {7371},

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dynker.tex
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@ -165,7 +165,7 @@ Note that this \textit{exact} decomposition does not alter, in any case, the val
How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear system of equations of smaller dimension [see Eq.~\eqref{eq:non_lin_sys}].
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Sottile_2003,Garniron_2018,QP2}
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts. \cite{Gershgorn_1968,Malrieu_1985,LiManni_2013,Nitzsche_1978b,Davidson_1981,Rawlings_1983,Staroverov_1998,Sottile_2003,Garniron_2018,QP2,Dvorak_2019a,Dvorak_2019b}
Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension.
However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analog given by Eq.~\eqref{eq:lin_sys}.
Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.