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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
%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
%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
%correlation is strong. It is not clear to me which integral ($W_{01}?$)
%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.
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.
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.
% TITOU: shall we keep the paragraph below?
%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
excitation; the reverse is observed in the equal-weight triensemble
case.
Howver, 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}.
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 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
derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
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
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 modeling 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
%still substantial in the strongly correlated limit of the equi