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done iteration for T2

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Pierre-Francois Loos 2020-03-10 22:56:06 +01:00
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Manuscript/eDFT.tex
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@ -1255,7 +1255,7 @@ drastically.
%If, for some reason, $W_0\approx W_1\approx W_{01}=W$, then the error %If, for some reason, $W_0\approx W_1\approx W_{01}=W$, then the error
%reduces to $-W/2$, which is weight-independent (it fits for example with %reduces to $-W/2$, which is weight-independent (it fits for example with
%what you see in the weakly correlated regime). Such an assumption depends on the nature of the %what you see in the weakly correlated regime). Such an assumption depends on the nature of the
%excitation, not only on the correlation strength, right? Neverthless, %excitation, not only on the correlation strength, right? Nevertheless,
%when looking at your curves, this assumption cannot be made when the %when looking at your curves, this assumption cannot be made when the
%correlation is strong. It is not clear to me which integral ($W_{01}?$) %correlation is strong. It is not clear to me which integral ($W_{01}?$)
%drives the all thing.\\} %drives the all thing.\\}
@ -1325,7 +1325,7 @@ 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 discutable. \cite{Loos_2019} In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more \titou{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
@ -1446,8 +1446,11 @@ The influence of the correlation ensemble derivative becomes substantial in the
\titou{For 3-boxium, in the zero-weight limit, its contribution is also significantly larger in the case of the single \titou{For 3-boxium, in the zero-weight limit, its contribution is also significantly larger in the case of the single
excitation; the reverse is observed in the equal-weight triensemble excitation; the reverse is observed in the equal-weight triensemble
case. case.
Howver, 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.
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.
Therefore, it might be more appropriate to model the derivative discontinuity in few-electron systems.}
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).
@ -1457,7 +1460,7 @@ compared to the zero-weight calculations).
This could explain why equiensemble calculations are clearly more 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 modeling properly the ensemble correlation derivative. of modelling properly the ensemble correlation derivative.
%\manu{Manu: I %\manu{Manu: I
%do not like this statement. As I wrote above, the ensemble derivative is %do not like this statement. As I wrote above, the ensemble derivative is
%still substantial in the strongly correlated limit of the equi %still substantial in the strongly correlated limit of the equi