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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} \end{split}
\eeq \eeq
where 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)}] \E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
\eeq \eeq
is the analog for ground and excited states (within an ensemble) of the HF energy. 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 \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!} 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. 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. 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. 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). 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).