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@ -1295,18 +1295,21 @@ energies, which is in agreement with
the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
individual energies do not vary in the same way depending on the state 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.
\titou{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 first-excited-state weight $\ew{1}$, thus showing that we the ground and first excited-state increase with respect to the
``deteriorate'' these states by optimizing the orbitals. The reverse actually occurs for the ground state in the triensemble first-excited-state weight $\ew{1}$, thus showing that, \manu{in this
as $\ew{2}$ increases.} The variations in the ensemble 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
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
Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first \eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
\titou{excited-state energy} is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble excited-state energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
systematically enhances the weight dependence, due to the lowering of the systematically enhances the weight dependence, due to the lowering of the
ground-state energy in this case, as $\ew{2}$ increases. ground-state energy, as $\ew{2}$ increases.
The reverse is observed for the second \titou{excited state}. The reverse is observed for the second excited state.
%%% FIG 3 %%% %%% FIG 3 %%%
\begin{figure} \begin{figure}
@ -1325,7 +1328,7 @@ For small $L$, the single and double excitations can be labeled as
``pure'', as revealed by a thorough analysis of the FCI wavefunctions. ``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
In other words, each excitation is dominated by a sole, well-defined reference Slater determinant. In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
However, when the box gets larger (\ie, as $L$ increases), there is a strong mixing between the different excitation degrees. However, when the box gets larger (\ie, as $L$ 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 \titou{disputable}. \cite{Loos_2019} In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more disputable. \cite{Loos_2019}
This can be clearly evidenced by the weights of the different configurations in the FCI wave function. This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
% TITOU: shall we keep the paragraph below? % TITOU: shall we keep the paragraph below?
%Therefore, it is paramount to construct a two-weight correlation functional %Therefore, it is paramount to construct a two-weight correlation functional
@ -1353,8 +1356,8 @@ 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 for the single excitation This is especially true\manu{, in the strong correlation regime,} for the single excitation
which is significantly improved by using equal weights, \titou{especially in the strong correlation regime}. 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
equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations. equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
@ -1424,7 +1427,7 @@ It is also interesting to investigate the influence of the
correlation ensemble derivative contribution $\DD{c}{(I)}$ correlation ensemble derivative contribution $\DD{c}{(I)}$
to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}]. to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
In our case, both single ($I=1$) and double ($I=2$) excitations are considered. In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, \titou{for $\nEl = 3$, $5$, and $7$}, the error percentage (with respect to FCI) as a function of the box length $L$ To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, for $\nEl = 3$, $5$, and $7$, the error percentage (with respect to FCI) as a function of the box length $L$
on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}]. on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
%\manu{Manu: there is something I do not understand. If you want to %\manu{Manu: there is something I do not understand. If you want to
%evaluate the importance of the ensemble correlation derivatives you %evaluate the importance of the ensemble correlation derivatives you
@ -1436,21 +1439,37 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}
%\eeq %\eeq
%%rather than $E^{(I)}_{\rm HF}$ %%rather than $E^{(I)}_{\rm HF}$
%} %}
\titou{We first stress that although for $\nEl=3$ both single and double excitation energies are We first stress that although for $\nEl=3$ both single and double excitation energies are
systematically improved, as the strength of electron correlation 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 systematically the case for larger number of electrons.} the correlation ensemble derivative, this is not
\trashPFL{This statement holds in both zero-weight and equal-weight limits.} \trashEF{systematically} \manu{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.
\titou{For 3-boxium, in the zero-weight limit, its contribution is also significantly larger in the case of the single For 3-boxium, in the zero-weight limit, its contribution is
excitation; the reverse is observed in the equal-weight triensemble \trashEF{also} significantly larger \manu{for the single
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 be a consequence of how we constructed the weight-dependent functional. This non-systematic behavior in terms of the number of electrons might
Indeed, as mentioned in Sec.~\ref{sec:eDFA}, the weight dependence of the eLDA functional is based on a two-electron finite uniform electron gas. be a consequence of how we constructed \trashEF{the weight-dependent
Therefore, it might be more appropriate to model the derivative discontinuity in few-electron systems.} functional} \manu{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 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 derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
compared to the zero-weight calculations). compared to the zero-weight calculations).
@ -1461,16 +1480,33 @@ 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: 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 for a given method, equiensemble orbitals partially remove the burden
of modelling properly the ensemble correlation derivative. of modelling properly the ensemble correlation derivative.
%\manu{Manu: I }\\
%do not like this statement. As I wrote above, the ensemble derivative is \manu{Manu: I propose to rephrase this part as follows:}\\
%still substantial in the strongly correlated limit of the equi \\
%triensemble for the double \manu{
%excitation. 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
Note also that, in our case, the second term in 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
the equiensemble case.
To summarize, in all cases, the equiensemble calculation
is always more accurate than a zero-weight
(\ie, a conventional ground-state DFT) one, with or without including the ensemble
derivative correction.
}
\\
Note that the second term in
Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation
potential and the density difference between ground and excited states, potential and the density difference between ground and excited states,
has a negligible effect on the excitation energies \titou{(results not shown)}. has a negligible effect on the excitation energies (results not shown).
%\manu{Manu: Is this %\manu{Manu: Is this
%something that you checked but did not show? It feels like we can see %something that you checked but did not show? It feels like we can see
%this in the Figure but we cannot, right?} %this in the Figure but we cannot, right?}
@ -1494,8 +1530,8 @@ has a negligible effect on the excitation energies \titou{(results not shown)}.
Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime). Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime).
The difference between the solid and dashed curves The difference between the solid and dashed curves
undoubtedly show that, even in the strong correlation regime, the undoubtedly show that, even in the strong correlation regime, the
ensemble correlation derivative has \titou{a rather significant impact} on the double ensemble correlation derivative has a rather significant impact on the double
excitations \titou{(around $10\%$)} with a slight tendency of worsening the excitation energies excitations (around $10\%$) with a slight tendency of worsening the excitation energies
in the case of equal weights, as the number of electrons in the case of equal weights, as the number of electrons
increases. It has a rather large influence on the single increases. It has a rather large influence on the single
excitation energies obtained in the zero-weight limit, showing once excitation energies obtained in the zero-weight limit, showing once