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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}, diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 8266c19..b4cdfb7 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -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. diff --git a/Response_Letter/CNRS_logo.pdf b/Response_Letter/CNRS_logo.pdf new file mode 100644 index 0000000..954b3f7 Binary files /dev/null and b/Response_Letter/CNRS_logo.pdf differ diff --git a/Response_Letter/Response_Letter.tex b/Response_Letter/Response_Letter.tex new file mode 100644 index 0000000..4447a1f --- /dev/null +++ b/Response_Letter/Response_Letter.tex @@ -0,0 +1,97 @@ +\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} + + + + + + diff --git a/Response_Letter/UPS_letterhead.sty b/Response_Letter/UPS_letterhead.sty new file mode 100644 index 0000000..2e5f5bb --- /dev/null +++ b/Response_Letter/UPS_letterhead.sty @@ -0,0 +1,70 @@ +%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 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