117 lines
7.4 KiB
TeX
117 lines
7.4 KiB
TeX
\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}
|
|
|
|
{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 support.
|
|
His/her comments are addressed below.}
|
|
|
|
|
|
\begin{enumerate}
|
|
|
|
\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{This is a good point. For clarity, we complemented the
|
|
theory section as follows.\\
|
|
{\it
|
|
``As shown in Sec. IV A 4, the weight dependence of the correlation
|
|
energy can be extracted from a FUEG model. In order to make the
|
|
resulting weight-dependent correlation functional truly universal, i.e.
|
|
independent on the number of electrons in the FUEG, one could use the
|
|
curvature of the Fermi hole [88] as an additional variable in the
|
|
density-functional approximation. The development of such a generalized
|
|
cor- relation eLDA is left for future work. Even though a similar
|
|
strategy could be applied to the weight-dependent exchange part, we
|
|
explore in the present work a different path where the
|
|
(system-dependent) exchange functional parameterization relies on the
|
|
ensemble energy linearity constraint (see Sec. IV A 2). Finally, let us
|
|
stress that, in order to further improve the description of the ensemble
|
|
correlation energy, a post-treatment of the recently revealed
|
|
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.
|
|
''}}
|
|
|
|
\item
|
|
{In the captions of Figures 1 and 2 replace "functional's" with "functionals'"}
|
|
\\
|
|
\alert{This has been fixed.}
|
|
|
|
\item
|
|
{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. }
|
|
\\
|
|
\alert{The density $n$ used in Eq.~(21), (9), (10) can be any density and does not represent any specific density.
|
|
In the case of Eq.~(21), we simply present the well-known Dirac-exchange density functional and, by definition of a density functional, does 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 $n(r)$.
|
|
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 4.}
|
|
|
|
\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{enumerate}
|
|
|
|
\end{letter}
|
|
\end{document}
|
|
|
|
|
|
|
|
|
|
|
|
|