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Manu: saving work.

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Emmanuel Fromager 2020-02-27 16:02:38 +01:00
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Manuscript/eDFT.tex
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@ -800,7 +800,7 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
\end{split}
\eeq
where
\beq
\beq\label{eq:ind_HF-like_ener}
\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
\eeq
is the analog for ground and excited states (within an ensemble) of the HF energy.
@ -1268,7 +1268,21 @@ electrons}.
\titou{T2: there is a micmac with the derivative discontinuity as it is
only defined at zero weight. We should clean up this.}\manu{I will!}
It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage (with respect to FCI) on the excitation energies obtained at the KS-eLDA and eHF levels [see Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:EI-eHF}, respectively] as a function of the box length $L$ in the case of 5-boxium.
To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage
(with respect to FCI) on the excitation energies obtained at the KS-eLDA
and HF\manu{-like} levels [see Eqs.~\eqref{eq:EI-eLDA} and
\eqref{eq:ind_HF-like_ener}, respectively] as a function of the box
length $L$ in the case of 5-boxium.\\
\manu{Manu: there is something I do not understand. If you want to
evaluate the importance of the ensemble correlation derivatives you
should only remove the following contribution from the $K$th KS-eLDA
excitation energy:
\beq
\int \n{\bGam{\bw}}{}(\br{})
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
\eeq
%rather than $E^{(I)}_{\rm HF}$
}
The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
Its contribution is also significantly larger in the case of the single excitation; the derivative discontinuity hardly influences the double excitation.
Importantly, one realizes that the magnitude of the derivative discontinuity is much smaller in the case of state-averaged calculations (as compared to the zero-weight calculations).