minor corrections T2
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@ -1297,10 +1297,10 @@ individual energies do not vary in the same way depending on the state
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considered and the value of the weights.
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We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
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the ground and first excited-state increase with respect to the
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first-excited-state weight $\ew{1}$, thus showing that, \manu{in this
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case}, we
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``deteriorate'' these states by optimizing the orbitals \manu{for the
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ensemble, rather than for each state individually}. The reverse actually occurs for the ground state in the triensemble
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first-excited-state weight $\ew{1}$, thus showing that, in this
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case, we
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``deteriorate'' these states by optimizing the orbitals for the
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ensemble, rather than for each state individually. The reverse actually occurs for the ground state in the triensemble
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as $\ew{2}$ increases. The variations in the ensemble
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weights are essentially linear or quadratic. They are induced by the
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eLDA functional, as readily seen from
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@ -1356,7 +1356,7 @@ When the box gets larger, they start to deviate.
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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.
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TDA-TDLDA slightly corrects this trend thanks to error compensation.
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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, $\bw = (0,0)$].
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This is especially true\manu{, in the strong correlation regime,} for the single excitation
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This is especially true, in the strong correlation regime, for the single excitation
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which is significantly improved by using equal weights.
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The effect on the double excitation is less pronounced.
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Overall, one clearly sees that, with
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@ -1399,7 +1399,7 @@ the same quality as the one obtained in the linear response formalism
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excitation energy only deviates
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from the FCI value by a few tenth of percent.
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Moreover, we note that, in the strong correlation regime
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(\titou{right} graph of Fig.~\ref{fig:EvsN}), the single excitation
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(right graph of Fig.~\ref{fig:EvsN}), the single excitation
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energy obtained at the equiensemble KS-eLDA level remains in good
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agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
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This also applies to the double excitation, the discrepancy
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@ -1444,58 +1444,53 @@ systematically improved, as the strength of electron correlation
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increases, when
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taking into account
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the correlation ensemble derivative, this is not
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\trashEF{systematically} \manu{always} the case for larger numbers of electrons.
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always the case for larger numbers of electrons.
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The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
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For 3-boxium, in the zero-weight limit, its contribution is
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\trashEF{also} significantly larger \manu{for the single
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excitation as compared to the double excitation}; the reverse is observed in the equal-weight triensemble
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significantly larger for the single
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excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
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case.
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However, for 5- and 7-boxium, the correlation ensemble derivative hardly
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influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes.
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This non-systematic behavior in terms of the number of electrons might
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be a consequence of how we constructed \trashEF{the weight-dependent
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functional} \manu{eLDA}.
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be a consequence of how we constructed eLDA.
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Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of
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the eLDA functional is based on a \manu{\it two-electron} finite uniform electron gas.
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\manu{Incorporating an $N$-dependence in the functional through the
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curvature of the Fermi hole, in the spirit of Ref. \cite{Loos_2017a}, would be
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valuable in this respect. This is left for future work.}
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\trashEF{Therefore, it might be more appropriate to model the derivative
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discontinuity in few-electron systems.}\\
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\\
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\manu{Manu: I am sorry to insist but I have a real problem with what follows. If
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we look at the N=3 results, one has the impression that, indeed, for the
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single excitation, a zero-weight calculation with the ensemble derivative
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is almost equivalent to an equal-weight calculation without the
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derivative. This is not the case for $N=5$ or 7, maybe because our
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derivative is based on two electrons. }\\
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{\it
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Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble
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derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
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compared to the zero-weight calculations).
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%\manu{Manu: well, this is not
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%really the case for the double excitation, right? I would remove this
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%sentence or mention the single excitation explicitly.}
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This could explain why equiensemble calculations are clearly more
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accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
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for a given method, equiensemble orbitals partially remove the burden
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of modelling properly the ensemble correlation derivative.
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}\\
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\manu{Manu: I propose to rephrase this part as follows:}\\
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\\
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\manu{
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Interestingly, for the single excitation in the 3-boxium, the magnitude of the correlation ensemble
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the eLDA functional is based on a \textit{two-electron} finite uniform electron gas.
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Incorporating an $\nEl$-dependence in the functional through the
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curvature of the Fermi hole, in the spirit of Ref.~\onlinecite{Loos_2017a}, would be
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valuable in this respect. This is left for future work.
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%\\
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%\manu{Manu: I am sorry to insist but I have a real problem with what follows. If
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%we look at the N=3 results, one has the impression that, indeed, for the
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%single excitation, a zero-weight calculation with the ensemble derivative
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%is almost equivalent to an equal-weight calculation without the
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%derivative. This is not the case for $N=5$ or 7, maybe because our
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%derivative is based on two electrons. }\\
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%{\it
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%Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble
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%derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
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%compared to the zero-weight calculations).
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%%\manu{Manu: well, this is not
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%%really the case for the double excitation, right? I would remove this
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%%sentence or mention the single excitation explicitly.}
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%This could explain why equiensemble calculations are clearly more
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%accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
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%for a given method, equiensemble orbitals partially remove the burden
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%of modelling properly the ensemble correlation derivative.
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%}\\
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%\manu{Manu: I propose to rephrase this part as follows:}\\
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%\\
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\titou{
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Interestingly, for the single excitation in 3-boxium, the magnitude of the correlation ensemble
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derivative is substantially reduced when switching from a zero-weight to
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an equal-weight calculation, while giving similar excitation energies,
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even in the strongly correlated regime. A possible interpretation is
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that, at least for the single excitation, equiensemble orbitals partially remove the burden
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of modelling properly the correlation ensemble derivative.
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This conclusion does not hold for larger
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$N=5$ or $N=7$ numbers of
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electrons, possibly because eLDA extracts density-functional correlation ensemble
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derivatives from a two-electron gas, as mentioned previously.
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For the
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double excitation, the ensemble derivative remains important, even in
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numbers of electrons ($N=5$ or $7$), possibly because eLDA extracts density-functional correlation ensemble
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derivatives from a two-electron uniform electron gas, as mentioned previously.
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For the double excitation, the ensemble derivative remains important, even in
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the equiensemble case.
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To summarize, in all cases, the equiensemble calculation
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is always more accurate than a zero-weight
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15853
Notebooks/eDFT_FUEG.nb
15853
Notebooks/eDFT_FUEG.nb
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