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Manuscript/FarDFT.bib

@@ -1,13 +1,57 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2020-04-24 09:46:46 +0200 
+%% Created for Pierre-Francois Loos at 2020-05-03 21:33:16 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@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},
+	Date-Modified = {2020-05-03 21:28:50 +0200},
+	Doi = {10.1103/PhysRevLett.109.226405},
+	Journal = {Phys. Rev. X},
+	Pages = {226405},
+	Title = {Quasiparticle Spectra from a Nonempirical Optimally Tuned Range-Separated Hybrid Density Functional},
+	Volume = {109},
+	Year = {2012}}
+
+@article{Stein_2012,
+	Author = {Tamar Stein and Jochen Autschbach and Niranjan Govind and Leeor Kronik and Roi Baer},
+	Date-Added = {2020-05-03 21:26:03 +0200},
+	Date-Modified = {2020-05-03 21:27:21 +0200},
+	Doi = {10.1021/jz3015937},
+	Journal = {J. Phys. Chem. Lett.},
+	Pages = {3740},
+	Title = {Curvature and Frontier Orbital Energies in Density Functional Theory},
+	Volume = {3},
+	Year = {2012}}
+
+@article{Stein_2010,
+	Author = {Tamar Stein and Helen Eisenberg and Leeor Kronik and Roi Baer},
+	Date-Added = {2020-05-03 21:24:45 +0200},
+	Date-Modified = {2020-05-03 21:25:40 +0200},
+	Doi = {10.1103/PhysRevLett.105.266802},
+	Journal = {Phys. Rev. Lett.},
+	Pages = {266802},
+	Title = {Fundamental Gaps in Finite Systems from Eigenvalues of a Generalized Kohn-Sham Method},
+	Volume = {105},
+	Year = {2010}}
+
+@article{Stein_2009,
+	Author = {Tamar Stein and Leeor Kronik and Roi Baer},
+	Date-Added = {2020-05-03 21:19:44 +0200},
+	Date-Modified = {2020-05-03 21:29:11 +0200},
+	Doi = {10.1021/ja8087482},
+	Journal = {J. Am. Chem. Soc.},
+	Pages = {2818},
+	Title = {Reliable Prediction of Charge Transfer Excitations in Molecular Complexes Using Time-Dependent Density Functional Theory},
+	Volume = {131},
+	Year = {2009}}
+
 @article{Paragi_2001,
 	Author = {G. Paragi and I. K. Gyemnnt and V. E. VanDoren},
 	Date-Added = {2020-04-24 09:39:54 +0200},
@@ -17,7 +61,8 @@
 	Pages = {153--161},
 	Title = {Investigation of exchange-correlation potentials in ensemble density functional theory: parameter fitting and excitation energy},
 	Volume = {571},
-	Year = {2001}}
+	Year = {2001},
+	Bdsk-Url-1 = {https://doi.org/10.1016/S0166-1280(01)00561-9}}
 
 @article{Nagy_1996,
 	Author = {\'A. Nagy},
@@ -28,7 +73,8 @@
 	Pages = {389--394},
 	Title = {Local ensemble exchange potential},
 	Volume = {29},
-	Year = {1996}}
+	Year = {1996},
+	Bdsk-Url-1 = {https://doi.org/10.1088/0953-4075/29/3/007}}
 
 @article{Casida_2012,
 	Author = {Casida, M.E. and Huix-Rotllant, M.},

Manuscript/FarDFT.tex

@@ -208,10 +208,7 @@ weight-dependent density-functional approximation for ensembles (eDFA)
 has never been developed for atoms and molecules from first principles.
 The present contribution paves the way towards this goal.
 
-When one talks about constructing functionals, the local-density
+The local-density approximation (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}
-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 ground-state functionals as shown in Refs.~\onlinecite{Loos_2014a,Loos_2014b,Loos_2017a}, where the authors proposed generalised LDA exchange and correlation functionals.
 
@@ -561,7 +558,7 @@ For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family
 Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-7}$. \cite{Becke_1988b,Lindh_2001}
 
 This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
-Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), the first singly-excited state ($I=1$ with weight $\ew{1}$), as well as the first doubly-excited state ($I=2$ with weight $\ew{2}$) are considered. 
+Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), the lowest singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered. 
 Assuming that the singly-excited state is lower in energy than the doubly-excited state (which is not always the case as one would notice later), one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle.
 %Taking a generic two-electron system as an example, the individual one-electron densities read
 %\begin{subequations}
@@ -585,12 +582,14 @@ Assuming that the singly-excited state is lower in energy than the doubly-excite
 %	\n{}{\eW} = (1 - \eW) \n{}{(0)} +  \eW \n{}{(2)},
 %\end{equation}
 %with $\eW = \ew{1}/2 + \ew{2}$ and $0 \le \eW \le 1/2$.
-Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$), and we consider the zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$).
+Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$).
 In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
+The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following.
+(Note that the zero-weight limit corresponds to a conventional ground-state KS calculation.)
 
-Let us mention now that we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals.
+Let us mention now that we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals by considering the extended range of weights $0 \le \ew{2} \le 1$.
 However, let us stress that we will not compute excitation energies with these ensembles inconsistent with GOK theory.
-The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is nonetheless of particular interest as it is a genuine saddle point of the restricted KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is nonetheless of particular interest as it is, like the (ground-state) zero-weight limit, a genuine saddle point of the restricted KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
 %Although the range $1/2 < \ew{} \leq 1$ stands a little bit beyond the theory discussed previously, we look at these solutions for analysis purposes mainly.
 %These solutions of the density matrix operator functional in Eq.~\eqref{eq:min_KS_DM} correspond to stationary points rather than minimising ones.
 %Applying GOK-DFT in this range of weights would simply consists in switching the ground and excited states if true minimisations of the ensemble energy were performed.
@@ -615,24 +614,15 @@ This procedure is applied to various two-electron systems in order to extract ex
 
 First, we compute the ensemble energy of the \ce{H2} molecule at equilibrium bond length (\ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent LDA Slater exchange functional (\ie, no correlation functional is employed), \cite{Dirac_1930, Slater_1951} which is explicitly given by 
 \begin{align}
+\label{eq:Slater}
 	\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
 	&
 	\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
 \end{align}
-In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state, the lowest singly-excited state $1\sigma_g 1\sigma_u$ of the same symmetry as the ground state (\ie, $\Sigma_g^+$), and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}).
+In the case of \ce{H2}, the ensemble is composed by the $\Sigma_g^+$ ground state of electronic configuration $1\sigma_g^2$, the lowest singly-excited state of the same symmetry as the ground state with configuration $1\sigma_g 2\sigma_g$, and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}).
-%\manu{At equilibrium, I expect the singly-excited configuration
-%$1\sigma_g2\sigma_g$ to be lower in energy. From the point of view of
-%GOK-DFT I do not see how we can reach the doubly-excited state while
-%ignoring the singly-excited one. One can always argue that we explore
-%stationary points (and not minima) but an obvious and important question that any
-%referee working on GOK-DFT would ask is: How would your results
-%be changed if you were incorporating the single excitation in your
-%ensemble? In one way or another
-%we have to look at this, even within the simplest weight-independent
-%approximation.}
 
 The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
-Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
+Because the Slater exchange functional defined in Eq.~\eqref{eq:Slater} does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
 As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}).
 Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $7$ eV from $\ew{} = 0$ to $1/3$.
 Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of the ensemble weights.
@@ -675,7 +665,7 @@ one can easily reverse-engineer (for this particular system, geometry, basis set
 %example, its combination with correlation functionals (as done in the
 %following) is very interesting. It should be introduced as a kind of
 %two-step procedure.}
-Doing so, we have found that the present weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
+Doing so, we have found that the following weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
 \begin{equation}
 	\e{\ex}{\ew{},\text{CC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
 \end{equation}
@@ -699,9 +689,11 @@ It also makes the excitation energy much more stable (with respect to $\ew{}$),
 The parameters $\alpha$, $\beta$, and $\gamma$ entering Eq.~\eqref{eq:Cxw} have been obtained via a least-square fit of the non-linear component of the ensemble energy computed between $\ew{2} = 0$ and $\ew{2} = 1$ by steps of $0.025$. 
 Although this range of weights is inconsistent with GOK theory, we have found that it is important, from a practical point of view, to ensure a correct behavior on the whole range of weights in order to obtain accurate excitation energies.
 
+The present procedure can be related to optimally-tuned range-separated hybrid functionals, where the range-separation parameters (which control the amount of short- and long-range exact exchange) are determined individually for each system by iteratively tuning them in order to enforce non-empirical conditions related to frontier orbitals (\eg, ionisation potential, electron affinity, etc) or, more importantly here, the piecewise linearity of the ensemble energy for ensemble states described by a fractional number of electrons. \cite{Stein_2009,Stein_2010,Stein_2012,Refaely-Abramson_2012}
+
 As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{1} = \ew{2} = 0$ and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
 Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which are genuine saddle points of the KS equations, as mentioned above.
-Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature one needs to catch in order to get accurate excitation energies in the zero-weight limit.
+Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature that one needs to catch in order to get accurate excitation energies in the zero-weight limit.
 We shall come back to this point later on.
 
 \begin{figure}
@@ -719,18 +711,12 @@ We shall come back to this point later on.
 
 Third, we add up correlation effects via the 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 Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a small curvature (the ensemble energy is slightly concave) and improved excitation energies, especially at small weights, where the CC-SVWN5 excitation energy is almost spot on. 
+The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies, especially at small weights, where the CC-SVWN5 excitation energy is almost spot on. 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsubsection{Weight-dependent correlation functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-%\manu{It seems crucial to me to distinguish what follows from the
-%previous results, which are more ``semi-empirical''. CC-S is fitted on
-%a specific system. I would personally add a subsection on glomium in the
-%theory section. I would also not dedicate specific subsections to the
-%previous results.}
-
 Fourth, in the spirit of our recent work, \cite{Loos_2020} we design a universal, weight-dependent correlation functional. 
 To build this correlation functional, we consider the singlet ground state, the first singly-excited state, as well as the first doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome). \cite{Loos_2009a,Loos_2009c,Loos_2010e}
 Notably, these three states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
@@ -888,13 +874,11 @@ which also require three separate calculations at a different set of ensemble we
 %We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
 %They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
 
-The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
+The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
 The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5.
 It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
 \bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{} = 1/4$ and $\ew{} = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
-Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between $\ew{} = 0$ and $\ew{} = 1/2$.
+Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits.
-
-\titou{This two-step procedure can be related to optimally-tuned range-separated hybrid functionals. T2: more to come.}
 
 %%% TABLE III %%%
 \begin{table}
@@ -963,7 +947,7 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 
 To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
 Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state.
-Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble as defined in Sec.~\ref{sec:H2}
+Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble defined in Sec.~\ref{sec:H2}
 %In other words, we set the weight of the single excitation to zero (\ie, $\ew{1} = 0$) and we have thus $\ew = \ew{2}$ for the rest of this example.
 We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr.
 It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
@@ -1045,8 +1029,10 @@ Nonetheless, it can be nicely described with a Gaussian basis set containing eno
 Consequently, we consider for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
 The excitation energies associated with this double excitation computed with various methods and combinations of xc functionals are gathered in Table \ref{tab:BigTab_He}.
 
-Before analysing the results, we would like to point out that, there is a large number of singly-excited states, between the $1s2s$ state and the $2s^2$.
+Before analysing the results, we would like to highlight the fact that there is a large number of singly-excited states lying in between the $1s2s$ and $2s^2$ states.
 Therefore, the present ensemble is not consistent with GOK theory.
+However, it is impossible, from a practical point of view, to take into account all these single excited states.
+We then restrict ourselves to a triensemble keeping in mind the possible theoretical loopholes of such a choice.
 
 The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.912\,574$, $\beta = +2.715\,267$, and $\gamma = +2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
 The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.