Manu: done with II

Manuscript/eDFT.tex

@@ -480,7 +480,7 @@ where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatia
 }
 \fi%%%%%%%%%%%%%%%%%%%%%
 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
 	\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
 \eeq
@@ -832,11 +832,12 @@ as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
 \eeq
 The ensemble energy is of course expected to vary linearly with the ensemble
 weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}].
-These errors are essentially removed when evaluating the individual energy
-levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.
+\manu{
+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
-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}
 	\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
 \eeq
@@ -844,7 +845,7 @@ where the ensemble correlation energy per particle
 \beq\label{eq:decomp_ens_correner_per_part}
 \e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
 \eeq
-is \titou{explicitly} \textit{weight dependent}. 
+is explicitly \textit{weight dependent}. 
 As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
 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.
@@ -859,8 +860,10 @@ reads
 %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.
 %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} 
-\titou{\beq\label{eq:EI-eLDA}
+Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with
+Eq.~\eqref{eq:eLDA_corr_fun} leads to our final expression of the
+KS-eLDA energy levels 
+\beq\label{eq:EI-eLDA}
 \begin{split}
 	\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)}
 	+ \Upsilon_\text{c}^{(I)},
 \end{split}
-\eeq}
+\eeq
 where
 \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, \titou{and
+is the analog for ground and excited states (within an ensemble) of the HF energy, and
 \begin{gather}
 \begin{split}
 	\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{}) ]
-	\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}
 \\
 	\Upsilon_\text{c}^{(I)} 
 	= \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{}.
-\end{gather}}
+\end{gather}
 
 If, for analysis purposes, we Taylor expand the density-functional
 correlation contributions
 around the $I$th KS state density
-$\n{\bGam{(I)}}{}(\br{})$, the sum of 
-the \titou{second term} on the right-hand side
+$\n{\bGam{(I)}}{}(\br{})$, the 
+second term on the right-hand side
 of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
 $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
 \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{}
 	+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}.
 \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
 dependencies explicitly, thus allowing for the description of derivative
 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
 the ensemble
 correlation energy per particle in Eq.
 \eqref{eq:decomp_ens_correner_per_part}, the latter can be recast
 \begin{equation}
-\titou{\Upsilon_\text{c}^{(I)}}
+\Upsilon_\text{c}^{(I)}
 %&=
 %\int \sum_{K>0} \qty(\delta_{IK} - \ew{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
 $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
 \beq\label{eq:Taylor_exp_DDisc_term}
 \begin{split}
-\titou{\Upsilon_\text{c}^{(I)}}
+\Upsilon_\text{c}^{(I)}
 %& = \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{}
 %\\
@@ -966,11 +970,10 @@ d\br{}
 \end{split}
 \eeq
 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{$\Upsilon_\text{c}^{(I)}$} 
-\trashPFL{[last term on the right-hand side of Eq.~\eqref{eq:EI-eLDA}]} is, through zeroth order, to substitute the expected
+role of the correlation ensemble derivative contribution $\Upsilon_\text{c}^{(I)}$ is, through zeroth order, to substitute the expected
 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
 the KS-eLDA ensemble energy, \ie,
 \beq\label{eq:Ew-eLDA}
@@ -983,7 +986,7 @@ the KS-eLDA ensemble energy, \ie,
 \end{split}
 \eeq
 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}
 \begin{split}
 	\Ex{eLDA}{(I)}