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Manuscript/FarDFT.bib

@@ -1,13 +1,63 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-04-10 22:33:34 +0200 
+%% Created for Pierre-Francois Loos at 2020-04-24 09:46:46 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Paragi_2001,
+	Author = {G. Paragi and I. K. Gyemnnt and V. E. VanDoren},
+	Date-Added = {2020-04-24 09:39:54 +0200},
+	Date-Modified = {2020-04-24 09:45:22 +0200},
+	Doi = {10.1016/S0166-1280(01)00561-9},
+	Journal = {J. Mol. Struct. THEOCHEM},
+	Pages = {153--161},
+	Title = {Investigation of exchange-correlation potentials in ensemble density functional theory: parameter fitting and excitation energy},
+	Volume = {571},
+	Year = {2001}}
+
+@article{Nagy_1996,
+	Author = {\'A. Nagy},
+	Date-Added = {2020-04-24 09:39:07 +0200},
+	Date-Modified = {2020-04-24 09:42:26 +0200},
+	Doi = {10.1088/0953-4075/29/3/007},
+	Journal = {J. Phys. B: At. Mol. Opt. Phys.},
+	Pages = {389--394},
+	Title = {Local ensemble exchange potential},
+	Volume = {29},
+	Year = {1996}}
+
+@article{Casida_2012,
+	Author = {Casida, M.E. and Huix-Rotllant, M.},
+	Date-Added = {2020-04-24 09:29:21 +0200},
+	Date-Modified = {2020-04-24 09:29:21 +0200},
+	Journal = {Annu. Rev. Phys. Chem.},
+	Pages = {287},
+	Title = {Progress in Time-Dependent Density-Functional Theory},
+	Volume = {63},
+	Year = {2012}}
+
+@article{Vignale_2008,
+	Author = {Vignale, Giovanni},
+	Date-Added = {2020-04-24 09:29:15 +0200},
+	Date-Modified = {2020-04-24 09:29:15 +0200},
+	Doi = {10.1103/PhysRevA.77.062511},
+	Issue = {6},
+	Journal = {Phys. Rev. A},
+	Month = {Jun},
+	Numpages = {9},
+	Pages = {062511},
+	Publisher = {American Physical Society},
+	Title = {Real-time resolution of the causality paradox of time-dependent density-functional theory},
+	Url = {https://link.aps.org/doi/10.1103/PhysRevA.77.062511},
+	Volume = {77},
+	Year = {2008},
+	Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.77.062511},
+	Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.77.062511}}
+
 @article{Loos_2020a,
 	Author = {P. F. Loos and A. Scemama and D. Jacquemin},
 	Date-Added = {2020-04-10 22:11:02 +0200},
@@ -2146,19 +2196,18 @@
 	Volume = {143},
 	Year = {2015},
 	Bdsk-Url-1 = {https://doi.org/10.1063/1.4932595}}
-	
-@article{Toulouse_2004,
-  title={Short-range exchange-correlation energy of a uniform electron gas with modified electron--electron interaction},
-  author={Toulouse, Julien and Savin, Andreas and Flad, Heinz-J{\"u}rgen},
-  journal={Int. J. Quantum Chem.},
-  volume={100},
-  number={6},
-  pages={1047--1056},
-  year={2004},
-  publisher={Wiley Online Library},
-  url={https://doi.org/10.1002/qua.20259}
-}
 
+@article{Toulouse_2004,
+	Author = {Toulouse, Julien and Savin, Andreas and Flad, Heinz-J{\"u}rgen},
+	Journal = {Int. J. Quantum Chem.},
+	Number = {6},
+	Pages = {1047--1056},
+	Publisher = {Wiley Online Library},
+	Title = {Short-range exchange-correlation energy of a uniform electron gas with modified electron--electron interaction},
+	Url = {https://doi.org/10.1002/qua.20259},
+	Volume = {100},
+	Year = {2004},
+	Bdsk-Url-1 = {https://doi.org/10.1002/qua.20259}}
 
 @article{Blunt_2017,
 	Author = {Blunt, N. S. and Neuscamman, Eric},
@@ -2674,10 +2723,10 @@
 	Volume = {6},
 	Year = 2010}
 
-@inbook{Casida,
+@inbook{Casida_1995,
 	Author = {M. E. Casida},
 	Date-Added = {2018-10-24 22:38:52 +0200},
-	Date-Modified = {2019-11-17 21:55:48 +0100},
+	Date-Modified = {2020-04-24 09:29:52 +0200},
 	Doi = {10.1142/9789812830586_0005},
 	Editor = {D. P. Chong},
 	Pages = {155--192},
@@ -8687,7 +8736,7 @@
 @article{Nagy_2001,
 	Abstract = {Recently an optimized potential method (OPM) has been derived for ensembles of excited states. Here an alternative OPM is proposed. The ensemble Kohn\textendash{}Sham potential in the generalized version of the Krieger\textendash{}Li\textendash{}Iafrate approximation to the OPM method is obtained.},
 	Author = {Nagy, \'A.},
-	Date-Modified = {2018-12-11 14:00:17 +0100},
+	Date-Modified = {2020-04-24 09:44:33 +0200},
 	Doi = {10.1088/0953-4075/34/12/305},
 	File = {/Users/loos/Zotero/storage/N7CH5INL/Nagy - 2001 - An alternative optimized potential method for ense.pdf},
 	Issn = {0953-4075, 1361-6455},

Manuscript/FarDFT.tex

@@ -143,17 +143,15 @@
 	\affiliation{\LCPQ}
 
 \begin{abstract}
-Gross--Oliveira--Kohn (GOK) ensemble density-functional theory (GOK-DFT)
+\titou{Gross--Oliveira--Kohn (GOK) ensemble density-functional theory (GOK-DFT)
-is a time-\textit{independent} formalism \manu{extension of DFT?} which
+is a time-\textit{independent} extension of density-functional theory (DFT) which
-allows to compute excitation energies \manu{I would say excited-state
+allows to compute excited-state
-energies (or energy levels)} via the derivative of the ensemble energy
+energies via the derivatives of the ensemble energy with
-with respect to the weight of each excited state \manu{and then, to be
+respect to the ensemble weights.}
-consistent: {\it ``via the derivatives of the ensemble energy with
+Contrary to the time-dependent version of DFT (TD-DFT), double excitations can be easily computed within GOK-DFT.
-respect to the ensemble weights''}}.
-Contrary to the time-dependent version of density-functional theory (TD-DFT), double excitations can be easily computed within GOK-DFT.
 However, to take full advantage of this formalism, one must have access to a \textit{weight-dependent} exchange-correlation functional in order to model the infamous ensemble derivative contribution to the excitation energies.
-In the present article, we discuss the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) specifically designed for the computation of double excitations within GOK-DFT.
+In the present article, we discuss the construction of first-rung (\textit{i.e.}, local) weight-dependent exchange-correlation density-functional approximations for two-electron atomic and molecular systems (He and H$_2$) specifically designed for the computation of double excitations within GOK-DFT.
-A specific protocol is proposed to obtain accurate energies associated with double excitations.
+\titou{In the spirit of optimally-tuned range-separated hybrid functionals,} a specific protocol is proposed to obtain accurate energies associated with double excitations.
 \end{abstract}
 
 \maketitle
@@ -162,37 +160,28 @@ A specific protocol is proposed to obtain accurate energies associated with doub
 %%% INTRODUCTION %%%
 %%%%%%%%%%%%%%%%%%%%
 \section{Introduction}
-Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.\cite{Casida,Ulrich_2012,Loos_2020a}
+Time-dependent density-functional theory (TD-DFT) has been the dominant force in the calculation of excitation energies of molecular systems in the last two decades.\cite{Casida_1995,Ulrich_2012,Loos_2020a}
-At a relatively low \manu{Tim suggested me to refer to TD-DFT as a
+\titou{At a moderate computational cost} (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
-method with a moderate computational cost which I
+\titou{Importantly, within the widely-used adiabatic approximation, setting up a TD-DFT calculation for a given system is an
-think is fair. It is more involved than a regular DFT or GOK-DFT
-calculation} computational cost (at least compared to the other excited-state \textit{ab initio} methods), TD-DFT can provide accurate transition energies for low-lying excited states of organic molecules (see, for example, Ref.~\onlinecite{Dreuw_2005} and references therein).
-Importantly, setting up a TD-DFT calculation for a given system is an
 almost pain-free process from a user perspective as the only (yet
-essential) input variable is the choice of the so-called
+essential) input variable is the choice of the 
-\manu{I would say ``ground-state functional'' and mention the widely used adiabatic approximation} exchange-correlation (xc) functional.
+ground-state exchange-correlation (xc) functional.}
 
 Similar to density-functional theory (DFT), \cite{Hohenberg_1964,Kohn_1965,ParrBook} TD-DFT is an in-principle exact theory which formal foundations relie on the Runge-Gross theorem. \cite{Runge_1984}
-The Kohn-Sham (KS) formalism \manu{formulation?} of TD-DFT transfers the
+The Kohn-Sham (KS) \titou{formulation} of TD-DFT transfers the
 complexity of the many-body problem to the xc functional thanks to a
 judicious mapping between a time-dependent non-interacting reference
-system and its interacting analog which have both the exact
+system and its interacting analog \titou{which have both
-\manu{exactly the?} same one-electron density.
+exactly the same one-electron density.}
 
-However, TD-DFT is far from being perfect as, in practice, drastic
+\titou{However, TD-DFT is far from being perfect as, in practice, drastic approximations must be made.
-approximations must be made for the xc functional. \manu{At this point I
+First, within the linear-response approximation, the electronic spectrum relies on the (unperturbed) pure-ground-state KS picture, \cite{Runge_1984, Casida_1995, Casida_2012} which may not be adequate in certain situations (such as strong correlation). 
-would mention the time dependence of the functional which is treated at
+Second, the time dependence of the functional is usually treated at the local approximation level within the standard adiabatic approximation.
-the local approximation level within the standard adiabatic
+In other words, memory effects are absent from the xc functional which is assumed to be local in time
-approximation. In other words, memory effects are absent from the
+(the xc energy is in fact an xc action, not an energy functional). \cite{Vignale_2008}
-functional (which is an action functional, not an energy functional). I
+Third and more importantly in the present context, a major issue of TD-DFT actually originates directly from the choice of the 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.}
-guess the choice of functional you discuss in the following refers to
-standard (frequency-independent) ground-state functionals. As you refer
-to exact TD-DFT first, the different levels of approximation should be
-clearly highlighted. You also discuss only the linear response regime
-without referring explicitly to it.}
-One of its issues actually originates directly from the choice of the 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.
-Moreover, because it is so popular, it has been studied in excruciated details by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies of approximate TD-DFT.
 
+\titou{Because 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.}
 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.
@@ -210,32 +199,28 @@ In the assumption of monotonically decreasing weights, eDFT for excited states h
 In short, GOK-DFT (\ie, eDFT for neutral excitations) is the density-based analog of state-averaged wave function methods, and excitation energies can then be easily extracted from the total ensemble energy. \cite{Deur_2019}
 Although the formal foundations of GOK-DFT have been set three decades ago, \cite{Gross_1988a,Gross_1988b,Oliveira_1988} its practical developments have been rather slow.
 We believe that it is partly due to the lack of accurate approximations for GOK-DFT.
-In particular, to the best of our knowledge, an explicitly
+\titou{In particular, to the best of our knowledge, albeit several attempts have been made, \cite{Nagy_1996,Paragi_2001} an explicitly
 weight-dependent density-functional approximation for ensembles (eDFA)
-has never been developed for atoms and molecules \manu{I would add
+has never been developed for atoms and molecules from first principles.
-``from first principles'' to be on the safe side. I remember this work
+The present contribution paves the way towards this goal.}
-by Nagy [J. Phys. B: At. Mol. Opt. Phys. 29 (1996) 389–394] where she
-did build an exchange functional for a bi-ensemble. Her approach was
-semi-empirical as she used experimental excitation energies. It would be
-fair to cite her paper. There is also this paper [Journal of Molecular
-Structure (Theochem) 571 (2001) 153-161] that I never really understood
-but they tried something}.
-The present contribution is a small \manu{too modest I think. ``paves
-the way towards ...'' or something like that} step towards this goal.
 
 When one talks about constructing functionals, the local-density
-approximation (LDA) is never far away \manu{too ``oral'' style I think}.
+approximation (LDA) is never far away.
+%\manu{too ``oral'' style I think}. Let's be fun Manu!
 The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
 Although the Hohenberg--Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behaviour in a real system. \cite{Kohn_1965}
-However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a}
+However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct \titou{ground-state functionals as shown in Refs.~\onlinecite{Loos_2014a,Loos_2014b,Loos_2017a}, where the authors proposed generalised LDA exchange and correlation functionals.}
+
 Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
-\manu{It goes much too fast here. One should make a clear distinction
+%\manu{It goes much too fast here. One should make a clear distinction
-between your previous work with Peter on the ground-state theory. Then
+%between your previous work with Peter on the ground-state theory. Then
-we should refer to our latest work where GOK-DFT is applied
+%we should refer to our latest work where GOK-DFT is applied
-to ringium. In the present work we extend the approach to glomium. As we
+%to ringium. In the present work we extend the approach to glomium. As we
-did in our previous work we should motivate the use of FUEGs for
+%did in our previous work we should motivate the use of FUEGs for
-developing weight-dependent functionals.}
+%developing weight-dependent functionals.}
-In particular, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT, and automatically incorporates the infamous ensemble derivative contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}
+\titou{Very recently, \cite{Loos_2020} two of the present authors have taken advantages of these FUEGs to construct a local, weight-dependent correlation functional specifically designed for one-dimensional many-electron systems.
+Unlike any standard functional, this first-rung functional incorporates derivative discontinuities thanks to its natural weight dependence, and has shown to deliver accurate excitation energies for both single and double excitations.
+Extending this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT, and automatically incorporates the infamous ensemble derivative contribution to the excitation energies through its explicit ensemble weight dependence. \cite{Levy_1995, Perdew_1983}}
 
 The paper is organised as follows.
 In Sec.~\ref{sec:theo}, the theory behind GOK-DFT is presented.