Manu: sving work in II Cwq

Manuscript/eDFT.tex

@@ -200,7 +200,7 @@ Atomic units are used throughout.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 The GOK ensemble energy~\cite{Gross_1988a,Oliveira_1988,Gross_1988b} is defined as
-\beq
+\beq\label{eq:exact_GOK_ens_ener}
 	\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
 \eeq
 where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is the eigenvalue of the electronic Hamiltonian $\hH = \hh + \hWee$, where 
@@ -576,28 +576,35 @@ w}_K
 \subsection{Approximations}\label{subsec:approx}
 %%%%%%%%%%%%%%%
 
-As Hartree and exchange energies cannot be separated in the one-dimensional systems considered in the rest of this work, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy,
+
+In the following, GOK-DFT will be applied
+to one-dimension
+spin-polarized systems where 
+Hartree and exchange energies cannot be separated. 
+For that reason, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy,
 \beq\label{eq:eHF-dens_mat_func}
 	\WHF[\bGam{}] = \frac{1}{2} \Tr[\bGam{} \bG \bGam{}], 
 \eeq
 for the Hx density-functional energy in the variational energy
-expression of Eq.~\eqref{eq:var_ener_gokdft}:  
-\beq
+expression of Eq.~\eqref{eq:var_ener_gokdft}, thus leading to the
+following approximation:  
+\beq\label{eq:min_with_HF_ener_fun}
 	\bGam{\bw} 
-	\approx \argmin_{\bgam{\bw}} 
+	\rightarrow \argmin_{\bgam{\bw}} 
 	\qty{ 
 	\Tr[\bgam{\bw} \bh ] + \WHF[ \bgam{\bw}] + \E{c}{\bw}[\n{\bgam{\bw}}{}]
 	}.
 \eeq
-\titou{Manu, I don't really understand the $\approx$ sign in the previous equation}
-The minimizing ensemble density matrix fulfills the following
-stationarity condition
+The minimizing ensemble density matrix in Eq.~(\ref{eq:min_with_HF_ener_fun}) fulfills the following
+stationarity condition:
 \beq\label{eq:commut_F_AO}
 	\bF{\bw} \bGam{\bw} \bS = \bS \bGam{\bw} \bF{\bw},
 \eeq
-where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the metric and the ensemble Fock-like matrix reads
+where $\bS \equiv \eS{\mu\nu} = \braket*{\AO{\mu}}{\AO{\nu}}$ is the
+overlap matrix and the ensemble Fock-like matrix reads
 \beq
-	\bF{\bw} \equiv \eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} + \sum_{\la\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw}
+	\bF{\bw} \equiv \eF{\mu\nu}{\bw} = \eh{\mu\nu}{\bw} +
+\sum_{\la\si} \eG{\mu\nu\la\si} \eGam{\la\si}{\bw},
 \eeq 
 with
 \beq
@@ -726,9 +733,15 @@ and
 \fi%%%%%%%%%%%
 %%%%% end Manu 
 %%%%%%%%%%%%%%%%%%%%
-Note that this approximation, where the ensemble density matrix is optimized from a non-local exchange potential [rather than a local one, as expected from Eq.~\eqref{eq:var_ener_gokdft}] is applicable to real (three-dimensional) systems. 
-As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, \textit{ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
-and curvature~\cite{} will be introduced in the Hx energy:
+Note that, within the approximation of Eq.~(\ref{eq:min_with_HF_ener_fun}), the ensemble density matrix is
+optimized with a non-local exchange potential rather than a
+density-functional local one, as expected from
+Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
+applicable to not-necessarily spin polarized and real (higher-dimension) systems. 
+As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
+ensemble density matrix into the HF interaction energy functional
+introduces unphysical \textit{ghost interaction} errors~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
+as well as {\it curvature}~\cite{Alam_2016,Alam_2017}:
 \beq\label{eq:WHF}
 \begin{split}
 	\WHF[\bGam{\bw}] 
@@ -737,14 +750,19 @@ and curvature~\cite{} will be introduced in the Hx energy:
 	& + \sum_{L>K\geq 0} \ew{K} \ew{L}\Tr[\bGam{(K)} \bG \bGam{(L)}].
 \end{split}
 \eeq
-These errors will be removed when computing individual energies according to Eq.~\eqref{eq:exact_ind_ener_rdm}.    
+The ensemble energy is of course expected to vary linearly with the ensemble
+weights [see Eq.~(\ref{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}.\\    
 
-Turning to the density-functional ensemble correlation energy, the following eLDA functional will be employed:  
+Turning to the density-functional ensemble correlation energy, the
+following ensemble local density approximation (eLDA) will be employed:  
 \beq\label{eq:eLDA_corr_fun}
-	\E{c}{\bw}[\n{}{}] = \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
-where the correlation energy per particle is \textit{weight dependent}. 
-Its construction from a finite uniform electron gas model is discussed in detail in Sec.~\ref{sec:eDFA}. 
+where the correlation energy per particle $\e{c}{\bw}(\n{}{})$ is \textit{weight dependent}. 
+As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed, for
+example, 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.
 What do you think?}
 
@@ -752,8 +770,10 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
 \beq\label{eq:EI-eLDA}
 \begin{split}
 	\E{\titou{eLDA}}{(I)} 
-	& \titou{\approx} \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)}]
 	& + \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
 	\\
 	&
@@ -765,7 +785,17 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
 \end{split}
 \eeq
 \titou{T2: I think we should specify what those terms are physically... Maybe earlier in the manuscript?}
-
+where
+\beq
+\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
+\eeq   
+is the analog for ground {\it and} excited states of the HF energy. 
+Let us stress that, to the best of our knowledge, eLDA is the first
+density-functional approximation that incorporates 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 last term on the right-hand side
+of Eq.~\eqref{eq:EI-eLDA}.\\
 \titou{In order to test the influence of the derivative discontinuity on the excitation energies, it is useful to perform ensemble HF (labeled as eHF) calculations in which the correlation effects are removed.
 In this case, the individual energies are simply defined as
 \beq\label{eq:EI-eHF}