118 lines
7.5 KiB
TeX
118 lines
7.5 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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{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 support.
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His/her comments are addressed below.}
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\begin{enumerate}
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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{This is a good point. For clarity, we complemented the
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theory section as follows.\\
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{\it
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``As shown in Sec. IV A 4, the weight dependence of the correlation
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energy can be extracted from a FUEG model. In order to make the
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resulting weight-dependent correlation functional truly universal, i.e.
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independent on the number of electrons in the FUEG, one could use the
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curvature of the Fermi hole [88] as an additional variable in the
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density-functional approximation. The development of such a generalized
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correlation eLDA is left for future work. Even though a similar
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strategy could be applied to the weight-dependent exchange part, we
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explore in the present work a different path where the
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(system-dependent) exchange functional parameterization relies on the
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ensemble energy linearity constraint (see Sec. IV A 2). Finally, let us
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stress that, in order to further improve the description of the ensemble
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correlation energy, a post-treatment of the recently revealed
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density-driven correlations [62,92-94] (which, by construction, are absent from FUEGs) might be necessary. An orbital-dependent correction derived in Ref. 92 might be used for that purpose. Work is currently in progress in this direction.
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''}}
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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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{Not clear what density is used in equation 21: From the discussion that follows this appears to not be the ensemble density (which is weight dependent and I would expect it to be used here) but the density only for the ground state Slater determinant. The authors should explain why this is so. }
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\\
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\alert{The density $n$ used in Eqs.~(21), (9), (10) can be any density, and, mathematically, there is no need to specify its origin.
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In the case of Eq.~(21), we simply define the well-known Dirac exchange functional.
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By definition of a density functional, we do not need 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 using this functional (or any other ones) in our work we surely apply it to the {\it minimizing} ensemble density $n^w(r)$ [see Eqs. (5) and (11)] and the notation is 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 3.}
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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{As clearly stated in the original manuscript, GOK variational principle states that the expectation value of the ensemble energy admits a lower bond which is linear with respect to each of the ensemble weight $w_I$ and is the exact ensemble energy of the studied system [Eq.~(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 $I$th excited state associated to this specific weight [Eq.~(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 linear in $w$ because of the approximate nature of the xc functionals.
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Indeed, by inserting the ensemble density in the Hartree interaction functional [Eq.~(9)], one introduces, in the ensemble energy, spurious quadratic curvature with respect to the weight.
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Some of those terms are responsible of the unphysical phenomenon called ghost interaction, as explained in the manuscript.
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Therefore, the ensemble Khon-Sham gap is, somehow, "weight contaminated" and does not possess the correct weight dependence [see first two terms of the right-hand side of Eq.~(16)].
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By taking its derivative with respect to the weight, the weight-dependent xc functional is expected to compensate those spurious quadratic terms in order to retrieve a linear behavior of the exact ensemble energy: only a weight-dependent xc functional can achieve such a feat.
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In other words, the xc functional does not have to be linear with respect to $w$.
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We have clarified this point in the revised manuscript.
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We emphasize that the exact ensemble xc functional has the ideal weight dependency, and would make the corresponding ensemble energy perfectly linear, hence leading to weight-independent excitation energies.
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As shown in the present manuscript, the use of an approximate weight-dependent xc functional reduces the ensemble energy curvature as well as the variation of the excitation energies with respect to $w$.
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This illustrates why the construction of reliable weight-dependent xc functionals is a challenging task in eDFT.
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As a final comment, we insist after Eq.~(28) on the fact that, in the exact theory, the xc ensemble density functional has no reason to vary (for a fixed density $n$) linearly with the ensemble weights.
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We refer to Ref.~92 where exact expressions for the individual xc energies are derived.}
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\end{enumerate}
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\end{letter}
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\end{document}
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