micro modif in conclusion and start clean up results

Manuscript/eDFT.tex

@@ -1115,7 +1115,7 @@ To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $
 The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented
 in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while
 fulfilling the restrictions on the ensemble weights to ensure the GOK
-variational principle [\ie, $0 \le \manu{\ew{2}} \le 1/3$ and \manu{$\ew{2} \le \ew{1} \le (1-\ew{2})/2$}].
+variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
 To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI-eLDA}].
 \manu{Manu: Just to be sure. What you refer to as the GIC ensemble
 energy is
@@ -1212,18 +1212,18 @@ In other words, each excitation is dominated by a sole, well-defined reference S
 However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
 In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
 This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
-Therefore, it is paramount to construct a two-weight \manu{correlation} functional
-(\manu{\ie, a triensemble functional}, as we have done here) which
-allows the mixing of \trashEF{single and double} \manu{singly- and doubly-excited} configurations.
+Therefore, it is paramount to construct a two-weight correlation functional
+(\ie, a triensemble functional, as we have done here) which
+allows the mixing of singly- and doubly-excited configurations.
 Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
-\titou{Shall we add results for $\ew{2} = 0$ to illustrate
-this?}\manu{Well, neglecting the second excited state is not the same as
-considering the $w_2=0$ limit. I thought you were referring to an
-approximation where the triensemble calculation is performed with
-the biensemble functional. This is not the same as taking $w_2=0$
-because, in this limit, you may still have a derivative discontinuity
-correction. The latter is absent if you truly neglect the second excited
-state in your ensemble functional. This should be clarified.}\\
+\titou{Shall we add results for the biensemble to illustrate this?}
+%\manu{Well, neglecting the second excited state is not the same as
+%considering the $w_2=0$ limit. I thought you were referring to an
+%approximation where the triensemble calculation is performed with
+%the biensemble functional. This is not the same as taking $w_2=0$
+%because, in this limit, you may still have a derivative discontinuity
+%correction. The latter is absent if you truly neglect the second excited
+%state in your ensemble functional. This should be clarified.}\\
 \manu{Are the results in the supp mat? We could just add "[not
 shown]" if not. This is fine as long as you checked that, indeed, the
 results deteriorate ;-)}
@@ -1237,18 +1237,11 @@ When the box gets larger, they start to deviate.
 For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
 TDA-TDLDA slightly corrects this trend thanks to error compensation. 
 Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
-This is especially true for the single excitation\manu{Manu: in the
-light of your comments about the mixed singly-excited/doubly-excited
-character of the first and second excited states when correlation is
-strong, I would refer to the
-"first excitation" rather than the "single excitation" (to be corrected 
-everywhere in the discussion if adopted)} which is significantly
-improved by using state-averaged weights\manu{Manu: you mean equal-weight?
-State-averaged does not mean equal-weight, don't you think? In the state-averaged CASSCF
-you do not have to use equal weights, even though most people do}.
-The effect on the \trashEF{double} \manu{second?} excitation is less pronounced.
-Overall, one clearly sees that, with \trashEF{state-averaged}
-\manu{equal} weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
+This is especially true for the single excitation
+which is significantly improved by using equal weights.
+The effect on the double excitation is less pronounced.
+Overall, one clearly sees that, with 
+equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
 This conclusion is verified for smaller and larger numbers of electrons
 (see {\SI}).\\
 \manu{Manu: now comes the question that is, I believe, central in this
@@ -1273,33 +1266,31 @@ end of the section, or something like that ...}
 
 For the same set of methods, Fig.~\ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI) as a function of $\nEl$ for three values of $L$ ($\pi/8$, $\pi$, and $8\pi$).
 We draw similar conclusions as above: irrespectively of the number of
-electrons, the eLDA functional with \trashEF{state-averaged} equal
+electrons, the eLDA functional with equal
 weights is able to accurately model single and double excitations, with
-a very significant improvement brought by the \trashEF{state-averaged}
-\manu{equiensemble} KS-eLDA orbitals as compared to their zero-weight
-\manu{(\ie, conventional ground-state)} analogs.
-\manu{As a rule of thumb, in the weak and intermediate correlation regimes, we
-see that the \trashEF{single} \manu{first
-excitation} obtained from \manu{equiensemble} KS-eLDA is of
+a very significant improvement brought by the 
+equiensemble KS-eLDA orbitals as compared to their zero-weight
+(\ie, conventional ground-state) analogs.
+As a rule of thumb, in the weak and intermediate correlation regimes, we
+see that the single
+excitation obtained from equiensemble KS-eLDA is of
 the same quality as the one obtained in the linear response formalism
-(such as TDLDA). On the other hand, the \trashEF{double} second
+(such as TDLDA). On the other hand, the double
 excitation energy only deviates
-from the FCI value by a few tenth of percent} \trashEF{for these two box
-lengths}.
+from the FCI value by a few tenth of percent.
 Moreover, we note that, in the strong correlation regime (left graph of
-Fig.~\ref{fig:EvsN}), the \trashEF{single} \manu{first} excitation
+Fig.~\ref{fig:EvsN}), the single excitation
 energy obtained at the equiensemble KS-eLDA level remains in good
 agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
-This also applies to \trashEF{double} \manu{the second} excitation
-\manu{(which has a strong doubly-excited character)}, the discrepancy
-between FCI and \manu{equiensemble} KS-eLDA remaining of the order of a few percents in the strong correlation regime.
+This also applies to the double excitation, the discrepancy
+between FCI and equiensemble KS-eLDA remaining of the order of a few percents in the strong correlation regime.
 These observations nicely illustrate the robustness of the
-\trashEF{present state-averaged} GOK-DFT scheme in any correlation regime for both single and double excitations.
+GOK-DFT scheme in any correlation regime for both single and double excitations.
 This is definitely a very pleasing outcome, which additionally shows
 that, even though we have designed the eLDA functional based on a
 two-electron model system, the present methodology is applicable to any
-1D electronic system, \manu{\ie, a system that has more than two
-electrons}. 
+1D electronic system, \ie, a system that has more than two
+electrons. 
 
 %%% FIG 4 %%%
 \begin{figure}
@@ -1395,23 +1386,14 @@ calculations.}\manu{to be updated ...}
 
 Let us finally stress that the present methodology can be extended
 straightforwardly to other types of ensembles like, for example, the
-$N$-centered ones\cite{Senjean_2018,Senjean_2020}, thus allowing for the design an LDA-type functional for the
+$\nEl$-centered ones, \cite{Senjean_2018,Senjean_2020} thus allowing for the design of a LDA-type functional for the
 calculation of ionization potentials, electron affinities, and
 fundamental gaps. 
 Like in the present
 eLDA, such a functional would incorporate the infamous derivative
-discontinuity contribution to the gap through its explicit weight
+discontinuity contribution to the fundamental gap through its explicit weight
 dependence. We hope to report on this in the near future.
 
-
-\trashEF{This can be done by constructing a functional for the one- and
-three-electron ground-state systems, and combining them with the
-two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and
-\eqref{eq:ecw}.}\manu{I find the sentence too technical for a
-conclusion.} 
-
-
-
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 \section*{Supplementary material}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -1419,14 +1401,11 @@ See {\SI} for the additional details about the construction of the functionals,
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{acknowledgements}
-PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
-This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.} 
+The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.
+This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.} 
 \end{acknowledgements}
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