response letter with clotilde comments

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FarDFT.nb

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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-05-10 19:45:36 +0200
%% Created for Pierre-Francois Loos at 2020-05-19 16:23:00 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Fromager_2020,
Archiveprefix = {arXiv},
Author = {Emmanuel Fromager},
Date-Added = {2020-05-13 07:41:14 +0200},
Date-Modified = {2020-05-13 07:41:14 +0200},
Eprint = {2001.08605},
Primaryclass = {physics.chem-ph},
Title = {Individual correlations in ensemble density-functional theory: State-driven/density-driven decomposition without additional Kohn-Sham systems},
Year = {2020}}
@article{Refaely-Abramson_2012,
Author = {Sivan Refaely-Abramson and Sahar Sharifzadeh and Niranjan Govind and Jochen Autschbach and Jeffrey B. Neaton and Roi Baer and Leeor Kronik},
Date-Added = {2020-05-03 21:27:34 +0200},
@ -252,14 +262,14 @@
Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package},
Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}}
@article{Fromager_2020,
@article{Loos_2020,
Archiveprefix = {arXiv},
Author = {Emmanuel Fromager},
Author = {P. F. Loos and E. Fromager},
Date-Added = {2020-04-08 14:13:18 +0200},
Date-Modified = {2020-04-08 14:13:18 +0200},
Eprint = {2001.08605},
Date-Modified = {2020-05-13 07:41:52 +0200},
Eprint = {2003.05553},
Primaryclass = {physics.chem-ph},
Title = {Individual correlations in ensemble density-functional theory: State-driven/density-driven decomposition without additional Kohn-Sham systems},
Title = {A weight-dependent local correlation density-functional approximation for ensembles},
Year = {2020}}
@article{Bottcher_1974,
@ -328,15 +338,6 @@
Volume = {112},
Year = {2008}}
@article{Loos_2020,
Author = {P. F. Loos and E. Fromager},
Date-Added = {2020-04-07 10:59:44 +0200},
Date-Modified = {2020-04-07 11:01:30 +0200},
Journal = {J. Chem. Phys.},
Pages = {arXiv:2003.05553},
Title = {A weight-dependent local correlation density-functional approximation for ensembles},
Year = {submitted}}
@article{Lindh_2001,
Author = {R. Lindh and P.-A. Malmqvist and L. Gagliardi},
Date-Added = {2020-03-30 09:59:22 +0200},
@ -5267,10 +5268,10 @@
Volume = {41},
Year = {1990}}
@article{Loos_2018,
@article{Loos_2018b,
Author = {P. F. Loos and A. Scemama and A. Blondel and Y. Garniron and M. Caffarel and D. Jacquemin},
Date-Added = {2018-10-24 22:38:52 +0200},
Date-Modified = {2020-04-10 22:19:17 +0200},
Date-Modified = {2020-05-19 14:01:04 +0200},
Doi = {10.1021/acs.jctc.8b00406},
Journal = {J. Chem. Theory Comput.},
Pages = {4360},

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@ -188,7 +188,7 @@ In other words, memory effects are absent from the xc functional which is assume
Third and more importantly in the present context, a major issue of
TD-DFT actually originates directly from the choice of the (ground-state) xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Because of its popularity, approximate TD-DFT has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies.
Because of its popularity, approximate TD-DFT has been studied extensively by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies.
For example, TD-DFT has problems with charge-transfer \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} and Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the semi-local xc functional.
The development of range-separated hybrids provides an effective solution to this problem. \cite{Tawada_2004,Yanai_2004}
From a practical point of view, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
@ -483,7 +483,7 @@ linear ensemble energy and, hence, the same value of the excitation energy indep
\includegraphics[width=\linewidth]{fig1}
\caption{
\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\ew{}}$ (in hartree) as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
See main text for the definition of the various functional's acronyms.
See main text for the definition of the various functionals' acronyms.
\label{fig:Ew_H2}
}
\end{figure}
@ -494,7 +494,7 @@ linear ensemble energy and, hence, the same value of the excitation energy indep
\includegraphics[width=\linewidth]{fig2}
\caption{
\ce{H2} at equilibrium bond length: error (with respect to FCI) in the excitation energy $\Ex{}{(2)}$ (in eV) associated with the doubly-excited state as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
See main text for the definition of the various functional's acronyms.
See main text for the definition of the various functionals' acronyms.
\label{fig:Om_H2}
}
\end{figure}
@ -585,7 +585,7 @@ We shall come back to this point later on.
\subsubsection{Weight-independent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Third, we add up correlation effects via the conventional VWN5 local correlation functional. \cite{Vosko_1980}
Third, we include correlation effects via the conventional VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the (weight-independent) Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on.

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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 Members of the Faraday Discussions Scientific Committee,}
\opening{Dear Members of the Faraday Discussions Scientific Committee,}
\justifying
Please find attached a revised version of the manuscript entitled
\begin{quote}
\textit{``Weight Dependence of Local Exchange-Correlation Functionals in Ensemble Density-Functional Theory: Double Excitations in Two-Electron Systems''}.
\end{quote}
We thank the reviewer for his/her constructive comments.
Our detailed responses to his/her comments can be found below.
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 authors describe the ensemble formulation of DFT or the Gross-Oliveira-Kohn DFT (GOK-DFT) in its Kohn-Sham formulation as a viable method for excited state calculations. They provide a very clear summary of the theory, followed by the main work of the paper which is the investigation of weight-dependent LDA-type xc functionals for eDFT calculations. The provide important insights on small systems with 2 electrons and functionals that are tailored for double excitations in these systems. The manuscript makes an important contribution to the field of DFT and should be accepted for publication. However, I would be grateful if the authors modify the paper slightly to address the following minor points and corrections:}
\\
\alert{We thank the reviewer for his/her kind comments.
His/her comments are addressed below.}
\item
{They should comment about what is needed (or even if it is possible) to develop a weight-dependent universal xc functional for eDFT calculations instead of application-specific functionals as presented in this paper.}
\\
\alert{bla bla bla}
\item
{In the captions of Figures 1 and 2 replace "functional's" with "functionals'"}
\\
\alert{This has been fixed.}
\item
{The density $n(r)$ used in equation 21, 9, 10 and more doesn't represent any specific density.
In the case of equation 21, we simply present the well-known Dirac-exchange density functional and, by definition of a density functional, we don't have to specify in its formulation to which density it is applied but only that it is a mathematical object applying to any density $n(r)$.
Of course, when we will use this functional or any other one in our work we will surely apply it to the ensemble Density $n^w(r)$ and the notation will be carefully modified accordingly. }
\item
{Change "Third, we add up correlation effects" to "Third, we include correlation effects"}
\\
\alert{ This has been fixed.}
\item
{Change "studied in excruciated details" to "studied extensively"}
\\
\alert{ This has been fixed.}
\item
{They need to be a bit more consistent with their notation as in equation 9 and elsewhere "$n(r)$" should be the ensemble (weight-dependent) density "$n^w(r)$".
I don?t believe they defined "$n(r)$" in the paper so I don?t know which density it represents. }
\\
\alert{See our response to 6.}
\item
{Even if it sounds trivial, they should explain why the exact xc functional should have linear dependence in the excitation energies as a function of the weight value.}
\\
\alert{GOK variational principle states that the expectation value of the ensemble energy admits/possesses a lower bond which is linear with respect to each of the ensemble-weights $w_i$ and is the exact ensemble energy of the studied system (equation 1).
Moreover, by construction, one can easily see that the slope of the exact ensemble energy with respect to a specific weight $w_i$ corresponds to the excitation energy of the system defined between the ground state and the ith-excited state associated to this specific weight (equation 4).
It is important that the reader keeps in mind that the exact excitation energies are based on pure-state energies and, therefore, do not depend on the weights of the ensemble.
In practice, the ensemble energy is rarely w-linear (linear in w ?) because of the use of approximate xc-functionals.
Indeed, by inserting the ensemble density in the Hartree interaction functional (equation 9), it introduces spurious quadratic curvature with respect to the weight in the ensemble energy.
Some of those terms are responsible of the unphysical phenomenon called ghost-interaction errors.
Therefore, the ensemble-Khon-Sham gap obtained at the end of the ensemble-HF-calculation is, somehow, "weight-contaminated" and doesn't possess the right weight-dependence.
(two first terms of the right-hand side of equation16)
By taking its first derivative with regard to the weight, the xc-functional is expected to compensate those parasite-quadratic terms in order to retrieve the linear behavior of the exact ensemble energy and one can understand that only a weight-dependant xc-functional could do so.
At the best of my knowledge, I cannot see any reason why the xc-functional should be w-linear.
The important idea is that the linearity must be in the ensemble energy but the main constraint on the xc-functional should be that it is weight-dependant.
We emphasize that only the exact ensemble-xc-functional would have the ideal weight-dependency that would make the corresponding ensemble energy reproduce perfectly the linear behavior of the exact ensemble energy and lead to weight-independant excitation energies, that is exact excitation energies.
The use of an approximate weight-dependant xc-functional could reduce the ensemble energy curvature and give less weight-dependant excitation energies but it is reasonable to admit that it also could make things worse it the weight-dependency of the functional is poorly chosen.
That is why the construction of "good" weight-dependant xc-functionals is a really challenging matter in eDFT.}
\end{itemize}
\end{letter}
\end{document}

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