98 lines
6.0 KiB
TeX
98 lines
6.0 KiB
TeX
\documentclass[10pt]{letter}
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\usepackage{UPS_letterhead,xcolor,mhchem,mathpazo,ragged2e,hyperref}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\begin{document}
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\begin{letter}%
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{To the Members of the Faraday Discussions Scientific Committee,}
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\opening{Dear Members of the Faraday Discussions Scientific Committee,}
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\justifying
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Please find attached a revised version of the manuscript entitled
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\begin{quote}
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\textit{``Weight Dependence of Local Exchange-Correlation Functionals in Ensemble Density-Functional Theory: Double Excitations in Two-Electron Systems''}.
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\end{quote}
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We thank the reviewer for his/her constructive comments.
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Our detailed responses to his/her comments can be found below.
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We look forward to hearing from you.
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\closing{Sincerely, the authors.}
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%%% REVIEWER 1 %%%
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\noindent \textbf{\large Authors' answer to Reviewer \#1}
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\begin{itemize}
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\item
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{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:}
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\\
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\alert{We thank the reviewer for his/her kind comments.
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His/her comments are addressed below.}
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\item
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{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.}
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\\
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\alert{bla bla bla}
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\item
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{In the captions of Figures 1 and 2 replace "functional's" with "functionals'"}
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\\
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\alert{This has been fixed.}
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\item
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{The density $n(r)$ used in equation 21, 9, 10 and more doesn't represent any specific density.
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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 $n(r)$.
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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. }
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\item
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{Change "Third, we add up correlation effects" to "Third, we include correlation effects"}
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\\
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\alert{ This has been fixed.}
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\item
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{Change "studied in excruciated details" to "studied extensively"}
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\\
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\alert{ This has been fixed.}
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\item
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{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 "$n^w(r)$".
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I don?t believe they defined "$n(r)$" in the paper so I don?t know which density it represents. }
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\\
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\alert{See our response to 6.}
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\item
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{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.}
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\\
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\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).
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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).
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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.
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In practice, the ensemble energy is rarely w-linear (linear in w ?) because of the use of approximate xc-functionals.
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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.
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Some of those terms are responsible of the unphysical phenomenon called ghost-interaction errors.
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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.
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(two first terms of the right-hand side of equation16)
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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.
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At the best of my knowledge, I cannot see any reason why the xc-functional should be w-linear.
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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.
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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.
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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.
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That is why the construction of "good" weight-dependant xc-functionals is a really challenging matter in eDFT.}
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\end{itemize}
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\end{letter}
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\end{document}
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