Merge branch 'master' of https://git.irsamc.ups-tlse.fr/loos/eDFT_FUEG

Manuscript/eDFT.tex

@@ -1300,10 +1300,10 @@ individual energies do not vary in the same way depending on the state
 considered and the value of the weights. 
 We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
 the ground and first excited-state increase with respect to the
-first-excited-state weight $\ew{1}$, thus showing that, \manu{in this
-case}, we
-``deteriorate'' these states by optimizing the orbitals \manu{for the
-ensemble, rather than for each state individually}. The reverse actually occurs for the ground state in the triensemble
+first-excited-state weight $\ew{1}$, thus showing that, in this
+case, we
+``deteriorate'' these states by optimizing the orbitals for the
+ensemble, rather than for each state individually. The reverse actually occurs for the ground state in the triensemble
 as $\ew{2}$ increases. The variations in the ensemble
 weights are essentially linear or quadratic. They are induced by the
 eLDA functional, as readily seen from
@@ -1359,7 +1359,7 @@ 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\manu{, in the strong correlation regime,} for the single excitation
+This is especially true, in the strong correlation regime, 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 
@@ -1402,7 +1402,7 @@ the same quality as the one obtained in the linear response formalism
 excitation energy only deviates
 from the FCI value by a few tenth of percent.
 Moreover, we note that, in the strong correlation regime
-(\titou{right} graph of Fig.~\ref{fig:EvsN}), the single excitation
+(right graph of 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 the double excitation, the discrepancy
@@ -1447,58 +1447,53 @@ systematically improved, as the strength of electron correlation
 increases, when
 taking into account  
 the correlation ensemble derivative, this is not
-\trashEF{systematically} \manu{always} the case for larger numbers of electrons.
+always the case for larger numbers of electrons.
 The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
 For 3-boxium, in the zero-weight limit, its contribution is
-\trashEF{also} significantly larger \manu{for the single
-excitation as compared to the double excitation}; the reverse is observed in the equal-weight triensemble
+significantly larger for the single
+excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
 case.
 However, for 5- and 7-boxium, the correlation ensemble derivative hardly
 influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes.
 This non-systematic behavior in terms of the number of electrons might
-be a consequence of how we constructed \trashEF{the weight-dependent
-functional} \manu{eLDA}.
+be a consequence of how we constructed eLDA.
 Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of
-the eLDA functional is based on a \manu{\it two-electron} finite uniform electron gas.
-\manu{Incorporating an $N$-dependence in the functional through the
-curvature of the Fermi hole, in the spirit of Ref. \cite{Loos_2017a}, would be
-valuable in this respect. This is left for future work.}
-\trashEF{Therefore, it might be more appropriate to model the derivative
-discontinuity in few-electron systems.}\\
-\\
-\manu{Manu: I am sorry to insist but I have a real problem with what follows. If
-we look at the N=3 results, one has the impression that, indeed, for the
-single excitation, a zero-weight calculation with the ensemble derivative
-is almost equivalent to an equal-weight calculation without the
-derivative. This is not the case for $N=5$ or 7, maybe because our
-derivative is based on two electrons. }\\
-{\it
-Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble 
-derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
-compared to the zero-weight calculations).
-%\manu{Manu: well, this is not
-%really the case for the double excitation, right? I would remove this
-%sentence or mention the single excitation explicitly.}
-This could explain why equiensemble calculations are clearly more
-accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
-for a given method, equiensemble orbitals partially remove the burden
-of modelling properly the ensemble correlation derivative.
-}\\
-\manu{Manu: I propose to rephrase this part as follows:}\\
-\\
-\manu{
-Interestingly, for the single excitation in the 3-boxium, the magnitude of the correlation ensemble 
+the eLDA functional is based on a \textit{two-electron} finite uniform electron gas.
+Incorporating an $\nEl$-dependence in the functional through the
+curvature of the Fermi hole, in the spirit of Ref.~\onlinecite{Loos_2017a}, would be
+valuable in this respect. This is left for future work.
+%\\
+%\manu{Manu: I am sorry to insist but I have a real problem with what follows. If
+%we look at the N=3 results, one has the impression that, indeed, for the
+%single excitation, a zero-weight calculation with the ensemble derivative
+%is almost equivalent to an equal-weight calculation without the
+%derivative. This is not the case for $N=5$ or 7, maybe because our
+%derivative is based on two electrons. }\\
+%{\it
+%Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble 
+%derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
+%compared to the zero-weight calculations).
+%%\manu{Manu: well, this is not
+%%really the case for the double excitation, right? I would remove this
+%%sentence or mention the single excitation explicitly.}
+%This could explain why equiensemble calculations are clearly more
+%accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
+%for a given method, equiensemble orbitals partially remove the burden
+%of modelling properly the ensemble correlation derivative.
+%}\\
+%\manu{Manu: I propose to rephrase this part as follows:}\\
+%\\
+\titou{
+Interestingly, for the single excitation in 3-boxium, the magnitude of the correlation ensemble 
 derivative is substantially reduced when switching from a zero-weight to
 an equal-weight calculation, while giving similar excitation energies,
 even in the strongly correlated regime. A possible interpretation is
 that, at least for the single excitation, equiensemble orbitals partially remove the burden
 of modelling properly the correlation ensemble derivative. 
 This conclusion does not hold for larger
-$N=5$ or $N=7$ numbers of
-electrons, possibly because eLDA extracts density-functional correlation ensemble
-derivatives from a two-electron gas, as mentioned previously.
-For the
-double excitation, the ensemble derivative remains important, even in
+numbers of electrons ($N=5$ or $7$), possibly because eLDA extracts density-functional correlation ensemble
+derivatives from a two-electron uniform electron gas, as mentioned previously.
+For the double excitation, the ensemble derivative remains important, even in
 the equiensemble case. 
 To summarize, in all cases, the equiensemble calculation 
 is always more accurate than a zero-weight