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Manu: done with II

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Emmanuel Fromager 2020-03-11 17:29:52 +01:00
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Manuscript/eDFT.tex
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@ -480,7 +480,7 @@ where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatia
} }
\fi%%%%%%%%%%%%%%%%%%%%% \fi%%%%%%%%%%%%%%%%%%%%%
then the density matrix of the then the density matrix of the
determinant $\Det{(K)}$ can be expressed as follows in the AO basis determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
\beq \beq
\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{}, \bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
\eeq \eeq
@ -832,11 +832,12 @@ as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
\eeq \eeq
The ensemble energy is of course expected to vary linearly with the ensemble The ensemble energy is of course expected to vary linearly with the ensemble
weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}]. weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}].
These errors are essentially removed when evaluating the individual energy \manu{
levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}. The explicit linear weight dependence of the ensemble Hx energy is actually restored when evaluating the individual energy
levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.}
Turning to the density-functional ensemble correlation energy, the Turning to the density-functional ensemble correlation energy, the
following ensemble local-density \textit{approximation} (eLDA) will be employed following ensemble local-density approximation (eLDA) will be employed
\beq\label{eq:eLDA_corr_fun} \beq\label{eq:eLDA_corr_fun}
\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{}, \E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
\eeq \eeq
@ -844,7 +845,7 @@ where the ensemble correlation energy per particle
\beq\label{eq:decomp_ens_correner_per_part} \beq\label{eq:decomp_ens_correner_per_part}
\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{}) \e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
\eeq \eeq
is \titou{explicitly} \textit{weight dependent}. is explicitly \textit{weight dependent}.
As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
from a finite uniform electron gas model. from a finite uniform electron gas model.
%\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC. %\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
@ -859,8 +860,10 @@ reads
%Manu, would it be useful to add this equation and the corresponding text? %Manu, would it be useful to add this equation and the corresponding text?
%I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies. %I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies.
%This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative %This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our \titou{final expression of the KS-eLDA energy level} Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with
\titou{\beq\label{eq:EI-eLDA} Eq.~\eqref{eq:eLDA_corr_fun} leads to our final expression of the
KS-eLDA energy levels
\beq\label{eq:EI-eLDA}
\begin{split} \begin{split}
\E{{eLDA}}{(I)} \E{{eLDA}}{(I)}
= =
@ -868,12 +871,12 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
+ \Xi_\text{c}^{(I)} + \Xi_\text{c}^{(I)}
+ \Upsilon_\text{c}^{(I)}, + \Upsilon_\text{c}^{(I)},
\end{split} \end{split}
\eeq} \eeq
where where
\beq\label{eq:ind_HF-like_ener} \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, \titou{and is the analog for ground and excited states (within an ensemble) of the HF energy, and
\begin{gather} \begin{gather}
\begin{split} \begin{split}
\Xi_\text{c}^{(I)} \Xi_\text{c}^{(I)}
@ -881,24 +884,25 @@ is the analog for ground and excited states (within an ensemble) of the HF energ
\\ \\
& &
+ \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ] + \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} d\br{} \left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} =
\n{\bGam{\bw}}{}(\br{})} d\br{},
\\ \\
\end{split} \end{split}
\\ \\
\Upsilon_\text{c}^{(I)} \Upsilon_\text{c}^{(I)}
= \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{}) = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}. \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
\end{gather}} \end{gather}
If, for analysis purposes, we Taylor expand the density-functional If, for analysis purposes, we Taylor expand the density-functional
correlation contributions correlation contributions
around the $I$th KS state density around the $I$th KS state density
$\n{\bGam{(I)}}{}(\br{})$, the sum of $\n{\bGam{(I)}}{}(\br{})$, the
the \titou{second term} on the right-hand side second term on the right-hand side
of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
\beq\label{eq:Taylor_exp_ind_corr_ener_eLDA} \beq\label{eq:Taylor_exp_ind_corr_ener_eLDA}
\titou{\Xi_\text{c}^{(I)}} \Xi_\text{c}^{(I)}
= \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{} = \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}. + \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}.
\eeq \eeq
@ -912,13 +916,13 @@ Let us stress that, to the best of our knowledge, eLDA is the first
density-functional approximation that incorporates ensemble weight density-functional approximation that incorporates ensemble weight
dependencies explicitly, thus allowing for the description of derivative dependencies explicitly, thus allowing for the description of derivative
discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
comment that follows] {\it via} the \titou{third term} on the right-hand side comment that follows] {\it via} the third term on the right-hand side
of Eq.~\eqref{eq:EI-eLDA}. According to the decomposition of of Eq.~\eqref{eq:EI-eLDA}. According to the decomposition of
the ensemble the ensemble
correlation energy per particle in Eq. correlation energy per particle in Eq.
\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast \eqref{eq:decomp_ens_correner_per_part}, the latter can be recast
\begin{equation} \begin{equation}
\titou{\Upsilon_\text{c}^{(I)}} \Upsilon_\text{c}^{(I)}
%&= %&=
%\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{}) %\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
%\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{})) %\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{}))
@ -939,7 +943,7 @@ thus leading to the following Taylor expansion through first order in
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
\beq\label{eq:Taylor_exp_DDisc_term} \beq\label{eq:Taylor_exp_DDisc_term}
\begin{split} \begin{split}
\titou{\Upsilon_\text{c}^{(I)}} \Upsilon_\text{c}^{(I)}
%& = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{}) %& = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{} % \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
%\\ %\\
@ -966,11 +970,10 @@ d\br{}
\end{split} \end{split}
\eeq \eeq
As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the
role of the correlation ensemble derivative \titou{$\Upsilon_\text{c}^{(I)}$} role of the correlation ensemble derivative contribution $\Upsilon_\text{c}^{(I)}$ is, through zeroth order, to substitute the expected
\trashPFL{[last term on the right-hand side of Eq.~\eqref{eq:EI-eLDA}]} is, through zeroth order, to substitute the expected
individual correlation energy per particle for the ensemble one. individual correlation energy per particle for the ensemble one.
Let us finally note that, while the weighted sum of the Let us finally mention that, while the weighted sum of the
individual KS-eLDA energy levels delivers a \textit{ghost-interaction-corrected} (GIC) version of individual KS-eLDA energy levels delivers a \textit{ghost-interaction-corrected} (GIC) version of
the KS-eLDA ensemble energy, \ie, the KS-eLDA ensemble energy, \ie,
\beq\label{eq:Ew-eLDA} \beq\label{eq:Ew-eLDA}
@ -983,7 +986,7 @@ the KS-eLDA ensemble energy, \ie,
\end{split} \end{split}
\eeq \eeq
the excitation energies computed from the KS-eLDA individual energy level the excitation energies computed from the KS-eLDA individual energy level
expressions in Eq. \eqref{eq:EI-eLDA} simply reads expressions in Eq. \eqref{eq:EI-eLDA} can be simplified as follows:
\beq\label{eq:Om-eLDA} \beq\label{eq:Om-eLDA}
\begin{split} \begin{split}
\Ex{eLDA}{(I)} \Ex{eLDA}{(I)}