Manu: saving work in the theory section.

Manuscript/eDFT.tex

@@ -210,13 +210,13 @@ where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is t
 is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator. 
 The (positive) ensemble weights $\ew{K}$ decrease with increasing index $K$. 
 They are normalized, \ie,
-\beq
+\beq\label{eq:weight_norm_cond}
 	\ew{0} = 1 - \sum_{K>0} \ew{K},
 \eeq
 so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldots )$ assigned to the excited states can vary independently.  
 For simplicity we will assume in the following that the energies are not degenerate. 
-Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states~\cite{}. 
-In GOK-DFT, the ensemble energy is determined variationally as follows:
+Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states~\cite{Gross_1988b}. 
+In GOK-DFT, the ensemble energy is determined variationally as follows~\cite{Gross_1988b}:
 \beq\label{eq:var_ener_gokdft}
 	\E{}{\bw}
  	= \min_{\opGam{\bw}}
@@ -228,28 +228,43 @@ where $\Tr$ denotes the trace and the trial ensemble density matrix operator rea
 \beq
 	\opGam{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
 \eeq
-The determinants (or configuration state functions) $\Det{(K)}$ are all constructed from the same set of ensemble KS orbitals that is variationally optimized.
-The trial ensemble density is simply the weighted sum of the individual densities, \ie,
+The KS determinants [or configuration state functions~\cite{Gould_2017}]
+$\Det{(K)}$ are all constructed from the same set of ensemble KS
+orbitals that are variationally optimized.
+The trial ensemble density in Eq.~(\ref{eq:var_ener_gokdft}) is simply
+the weighted sum of the individual KS densities, \ie,
 \beq\label{eq:KS_ens_density}
 	\n{\opGam{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K)}}{}(\br{}).
 \eeq
 As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange (Hx) and correlation (c) energies are described with density functionals that are \textit{weight dependent}. 
-We focus here on the (exact) Hx part which is defined as
+We focus in the following on the (exact) Hx part, which is defined as~\cite{Gould_2017}
 \beq\label{eq:exact_ens_Hx}
-	\E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}[\n{}{}]}{\hWee}{\Det{(K)}[\n{}{}]},
+	\E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bw}[\n{}{}]}{\hWee}{\Det{(K),\bw}[\n{}{}]},
 \eeq
 where the KS wavefunctions fulfill the ensemble density constraint
 \beq
-	\sum_{K\geq 0} \ew{K} \n{\Det{(K)}[\n{}{}]}{}(\br{}) = \n{}{}(\br{}).
+	\sum_{K\geq 0} \ew{K} \n{\Det{(K),\bw}[\n{}{}]}{}(\br{}) = \n{}{}(\br{}).
 \eeq 
-The (approximate) description of the correlation part is discussed in Sec.~\ref{sec:eDFA}.
+The (approximate) description of the correlation part is discussed in
+Sec.~\ref{sec:eDFA}.\\
 
 In practice, the ensemble energy is not the most interesting quantity, and one is more concerned with excitation energies or individual energy levels (for geometry optimizations, for example). 
-The latter can be extracted exactly as follows~\cite{}
+As pointed out recently in Ref.~\cite{Deur_2019}, the latter can be extracted
+exactly from a single ensemble calculation as follows:
 \beq\label{eq:indiv_ener_from_ens}
-	\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \pdv{\E{}{\bw}}{\ew{K}},
+	\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} )
+\pdv{\E{}{\bw}}{\ew{K}},
 \eeq
-where, according to the {\it variational} ensemble energy expression of Eq.~\eqref{eq:var_ener_gokdft}, the derivative with respect to $\ew{K}$ can be evaluated from the minimizing KS wavefunctions $\Det{(K)} = \Det{(K),\bw}$, \ie,
+where, according to the normalization condition of Eq.~(\ref{eq:weight_norm_cond}), 
+\beq
+\pdv{\E{}{\bw}}{\ew{K}}= \E{}{(K)} -
+\E{}{(0)}\equiv\Ex{(K)}
+\eeq
+corresponds to the $K$th excitation energy.
+According to the {\it variational} ensemble energy expression of
+Eq.~\eqref{eq:var_ener_gokdft}, the derivative with respect to $\ew{K}$
+can be evaluated from the minimizing weight-dependent KS wavefunctions
+$\Det{(K)} \equiv \Det{(K),\bw}$ as follows:
 \beq\label{eq:deriv_Ew_wk}
 \begin{split}
 	\pdv{\E{}{\bw}}{\ew{K}} 
@@ -270,16 +285,17 @@ The Hx contribution from Eq.~\eqref{eq:deriv_Ew_wk} can be recast as
 	- \E{Hx}{\bw}[\n{}{\bw,\bxi}] )
 	\right|_{\bxi=\bw},
 \eeq
-where $\bxi \equiv (\xi_1,\xi_2,\ldots,\xi_K,\ldots)$ and
+where $\bxi \equiv (\xi_1,\xi_2,\ldots,\xi_K,\ldots)$ and the
+auxiliary double-weight ensemble density reads
 \beq
 	\n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}).
 \eeq 
 Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes
-from the exact expression in Eq.~\ref{eq:exact_ens_Hx} that 
+from the exact expression in Eq.~(\ref{eq:exact_ens_Hx}) that 
 \beq
-	\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
+	\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}
 \eeq  
-with $\xi_0 = 1 - \sum_{K>0}\xi_K$, and
+and
 \beq
 	\E{Hx}{\bw}[\n{}{\bw,\bxi}] = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}.
 \eeq  
@@ -298,13 +314,13 @@ This yields, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_Hx
 	}_{\n{}{} = \n{\opGam{\bw}}{}} d\br{}.
 \end{split}
 \eeq
-Since the ensemble energy can be evaluated as 
+Since, according to Eqs.~(\ref{eq:var_ener_gokdft}) and (\ref{eq:exact_ens_Hx}), the ensemble energy can be evaluated as 
 \beq
 	\E{}{\bw} = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}}{\hH}{\Det{(K)}} + \E{c}{\bw}[\n{\opGam{\bw}}{}],
 \eeq 
 with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGam{\bw}}{} = \n{}{\bw,\bw}$], 
 we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
-\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\cite{} for the $I$th energy level:
+\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\cite{Fromager_2020} for the $I$th energy level:
 \beq\label{eq:exact_ener_level_dets}
 \begin{split}
 	\E{}{(I)} 
@@ -320,8 +336,27 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 	\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
+Note that, when $\bw=0$, the ensemble correlation functional becomes the
+conventional (ground-state) correlation functional $E_{\rm c}[n]$ and
+the ensemble density reduces to the ground-state one $n^{(0)}$. As a
+result,  
+Eq.~(\ref{eq:exact_ener_level_dets}) can be simplified as follows:
+\beq
+\begin{split}
+	\E{}{(I)} 
+	& = \mel*{\Det{(I)}}{\hH}{\Det{(I)}} + \E{c}{}[\n{}{(0)}]
+	\\
+	& + \int \fdv{\E{c}{}[\n{}{(0)}]}{\n{}{}(\br{})}
+	\qty[ \n{\Det{(I)}}{}(\br{}) - \n{\opGam{\bw}}{}(\br{}) ] d\br{}
+	\\
+	&+
+	\sum_{K>0} \qty(\delta_{IK} - \ew{K} )
+	\left.
+	\pdv{\E{c}{\bw}[\n{}{}]}{\ew{K}}
+	\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
+\end{split}
+\eeq 
 \titou{Manu, shall we mention that the last term in Eq.~\eqref{eq:exact_ener_level_dets} corresponds to the derivative discontinuity (DD)?}
-\titou{We also need to defined the excitation energies $\Ex{(I)} = \E{}{(I)} - \E{}{(0)}$ at the right place...}
 
 %%%%%%%%%%%%%%%%
 \subsection{One-electron reduced density matrix formulation}