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

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Manuscript/eDFT.tex
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@ -1300,10 +1300,10 @@ individual energies do not vary in the same way depending on the state
considered and the value of the weights. considered and the value of the weights.
We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of 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 the ground and first excited-state increase with respect to the
first-excited-state weight $\ew{1}$, thus showing that, \manu{in this first-excited-state weight $\ew{1}$, thus showing that, in this
case}, we case, we
``deteriorate'' these states by optimizing the orbitals \manu{for the ``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 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 as $\ew{2}$ increases. The variations in the ensemble
weights are essentially linear or quadratic. They are induced by the weights are essentially linear or quadratic. They are induced by the
eLDA functional, as readily seen from 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. 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. 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, $\bw = (0,0)$]. 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. which is significantly improved by using equal weights.
The effect on the double excitation is less pronounced. The effect on the double excitation is less pronounced.
Overall, one clearly sees that, with 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 excitation energy only deviates
from the FCI value by a few tenth of percent. from the FCI value by a few tenth of percent.
Moreover, we note that, in the strong correlation regime 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 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 $60 \%$. 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 This also applies to the double excitation, the discrepancy
@ -1447,58 +1447,53 @@ systematically improved, as the strength of electron correlation
increases, when increases, when
taking into account taking into account
the correlation ensemble derivative, this is not 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. 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 For 3-boxium, in the zero-weight limit, its contribution is
\trashEF{also} significantly larger \manu{for the single significantly larger for the single
excitation as compared to the double excitation}; the reverse is observed in the equal-weight triensemble excitation as compared to the double excitation; the reverse is observed in the equal-weight triensemble
case. case.
However, for 5- and 7-boxium, the correlation ensemble derivative hardly 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. 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 This non-systematic behavior in terms of the number of electrons might
be a consequence of how we constructed \trashEF{the weight-dependent be a consequence of how we constructed eLDA.
functional} \manu{eLDA}.
Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of 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. the eLDA functional is based on a \textit{two-electron} finite uniform electron gas.
\manu{Incorporating an $N$-dependence in the functional through the Incorporating an $\nEl$-dependence in the functional through the
curvature of the Fermi hole, in the spirit of Ref. \cite{Loos_2017a}, would be 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.} 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
\manu{Manu: I am sorry to insist but I have a real problem with what follows. If %single excitation, a zero-weight calculation with the ensemble derivative
we look at the N=3 results, one has the impression that, indeed, for the %is almost equivalent to an equal-weight calculation without the
single excitation, a zero-weight calculation with the ensemble derivative %derivative. This is not the case for $N=5$ or 7, maybe because our
is almost equivalent to an equal-weight calculation without the %derivative is based on two electrons. }\\
derivative. This is not the case for $N=5$ or 7, maybe because our %{\it
derivative is based on two electrons. }\\ %Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble
{\it %derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble %compared to the zero-weight calculations).
derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as %%\manu{Manu: well, this is not
compared to the zero-weight calculations). %%really the case for the double excitation, right? I would remove this
%\manu{Manu: well, this is not %%sentence or mention the single excitation explicitly.}
%really the case for the double excitation, right? I would remove this %This could explain why equiensemble calculations are clearly more
%sentence or mention the single excitation explicitly.} %accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
This could explain why equiensemble calculations are clearly more %for a given method, equiensemble orbitals partially remove the burden
accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative: %of modelling properly 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{Manu: I propose to rephrase this part as follows:}\\ \titou{
\\ Interestingly, for the single excitation in 3-boxium, the magnitude of the correlation ensemble
\manu{
Interestingly, for the single excitation in the 3-boxium, the magnitude of the correlation ensemble
derivative is substantially reduced when switching from a zero-weight to derivative is substantially reduced when switching from a zero-weight to
an equal-weight calculation, while giving similar excitation energies, an equal-weight calculation, while giving similar excitation energies,
even in the strongly correlated regime. A possible interpretation is even in the strongly correlated regime. A possible interpretation is
that, at least for the single excitation, equiensemble orbitals partially remove the burden that, at least for the single excitation, equiensemble orbitals partially remove the burden
of modelling properly the correlation ensemble derivative. of modelling properly the correlation ensemble derivative.
This conclusion does not hold for larger This conclusion does not hold for larger
$N=5$ or $N=7$ numbers of numbers of electrons ($N=5$ or $7$), possibly because eLDA extracts density-functional correlation ensemble
electrons, possibly because eLDA extracts density-functional correlation ensemble derivatives from a two-electron uniform electron gas, as mentioned previously.
derivatives from a two-electron gas, as mentioned previously. For the double excitation, the ensemble derivative remains important, even in
For the
double excitation, the ensemble derivative remains important, even in
the equiensemble case. the equiensemble case.
To summarize, in all cases, the equiensemble calculation To summarize, in all cases, the equiensemble calculation
is always more accurate than a zero-weight is always more accurate than a zero-weight

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Notebooks/eDFT_FUEG.nb
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