new equations for excitation energies

Data/DD_n5.dat

@@ -2,7 +2,10 @@
 0.0457	0.0450	0.0435	0.0409	0.0362	0.0241	0.0033	
 0.0861	0.0838	0.0793	0.0712	0.0571	0.0377	0.0196	
 0.1044	0.1036	0.1022	0.0997	0.0953	0.0830	0.0540	
-0.1447	0.1424	0.1380	0.1300	0.1162	0.0966	0.0703	
+0.1447	0.1424 	0.1380	0.1300	0.1162	0.0966	0.0703	
+-0.1356 -0.1342 -0.1314 -0.1264 -0.1176 -0.1037 -0.0849
+-0.1356 -0.1342 -0.1314 -0.1263 -0.1174 -0.1031 -0.0834
+-0.1356 -0.1341 -0.1314 -0.1262 -0.1172 -0.1026 -0.0824
 0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	
 0.0172	0.0163	0.0147	0.0117	0.0067	-0.0004	-0.0080	
 0.0416	0.0402	0.0376	0.0329	0.0253	0.0140	0.0005	
@@ -11,6 +14,9 @@
 0.1080	0.1056	0.1011	0.0925	0.0772	0.0538	0.0325	
 0.0092	0.0081	0.0060	0.0022	-0.0035	-0.0081	-0.0047	
 0.1010	0.0986	0.0943	0.0862	0.0719	0.0523	0.0373	
+-0.1160 -0.1154 -0.1141 -0.1115 -0.1069 -0.0991 -0.0870
+-0.1160 -0.1154 -0.1140 -0.1115 -0.1069 -0.0991 -0.0872
+-0.1160 -0.1153 -0.1140 -0.1115 -0.1069 -0.0991 -0.0871
 -0.0196	-0.0188	-0.0174	-0.0148	-0.0106	-0.0044	0.0029	
 -0.0024	-0.0025	-0.0027	-0.0032	-0.0040	-0.0050	-0.0056	
 0.0220	0.0213	0.0201	0.0180	0.0146	0.0093	0.0027	

Manuscript/SI/eDFT-SI.tex

@@ -622,7 +622,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\caption{
 	\label{tab:OptGap_2}
 	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 2-boxium (i.e.,~$\Nel = 2$ electrons in a box of length $L$).
-	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported. 
+	The values of the ensemble correlation derivative $\DD{c}{(I)}$ are also reported. 
 	(DNC = KS calculation does not converge.)
 	}
 	\begin{ruledtabular}
@@ -685,7 +685,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\caption{
 	\label{tab:OptGap_3}
 	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 3-boxium (i.e.,~$\Nel = 3$ electrons in a box of length $L$).
-	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
+	The values of the ensemble correlation derivative $\DD{c}{(I)}$ are also reported.
 	}
 	\begin{ruledtabular}
 	\begin{tabular}{lclddddddd}
@@ -748,7 +748,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\caption{
 	\label{tab:OptGap_4}
 	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 4-boxium (i.e.,~$\Nel = 4$ electrons in a box of length $L$).
-	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
+	The values of the ensemble correlation derivative $\DD{c}{(I)}$ are also reported.
 	}
 	\begin{ruledtabular}
 	\begin{tabular}{lclddddddd}
@@ -810,7 +810,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\caption{
 	\label{tab:OptGap_5}
 	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 5-boxium (i.e.,~$\Nel = 5$ electrons in a box of length $L$).
-	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
+	The values of the ensemble correlation derivative $\DD{c}{(I)}$ are also reported.
 	}
 	\begin{ruledtabular}
 	\begin{tabular}{lclddddddd}
@@ -872,7 +872,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\caption{
 	\label{tab:OptGap_6}
 	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 6-boxium (i.e.,~$\Nel = 6$ electrons in a box of length $L$).
-	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
+	The values of the ensemble correlation derivative $\DD{c}{(I)}$ are also reported.
 	}
 	\begin{ruledtabular}
 	\begin{tabular}{lclddddddd}
@@ -935,7 +935,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\caption{
 	\label{tab:OptGap_7}
 	Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 7-boxium (i.e.,~$\Nel = 7$ electrons in a box of length $L$).
-	The values of the derivative discontinuity $\DD{c}{(I)}$ are also reported.
+	The values of the ensemble correlation derivative $\DD{c}{(I)}$ are also reported.
 	}
 	\begin{ruledtabular}
 	\begin{tabular}{lclddddddd}

Manuscript/eDFT.tex

@@ -71,7 +71,7 @@
 \newcommand{\opGam}[1]{\hat{\Gamma}^{#1}}
 \newcommand{\bh}{\boldsymbol{h}}
 \newcommand{\bF}[1]{\boldsymbol{F}^{#1}}
-\newcommand{\Ex}[1]{\Omega^{#1}}
+\newcommand{\Ex}[2]{\Omega_\text{#1}^{#2}}
 
 
 % elements
@@ -302,7 +302,7 @@ where the KS wavefunctions fulfill the ensemble density constraint
 	\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}.\\
+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). 
 As pointed out recently in Ref.~\cite{Deur_2019}, the latter can be extracted
@@ -314,7 +314,7 @@ exactly from a single ensemble calculation as follows:
 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)}
+\E{}{(0)}\equiv\Ex{}{(K)}
 \eeq
 corresponds to the $K$th excitation energy.
 According to the {\it variational} ensemble energy expression of
@@ -825,7 +825,7 @@ as well as {\it curvature}~\cite{Alam_2016,Alam_2017}:
 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}.\\    
+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 approximation (eLDA) will be employed:  
@@ -835,8 +835,8 @@ following ensemble local density approximation (eLDA) will be employed:
 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?}
+%\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
+%What do you think?}
 
 Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within KS-eLDA: 
 \beq\label{eq:EI-eLDA}
@@ -881,8 +881,36 @@ 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 last term on the right-hand side
-of Eq.~\eqref{eq:EI-eLDA}.\\
+of Eq.~\eqref{eq:EI-eLDA}.
 
+\titou{The GIC KS-eLDA ensemble energy is thus given by
+\beq\label{eq:Ew-eLDA}
+\E{GIC-eLDA}{\bw}=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)},
+\eeq
+while the uncorrected KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_with_HF_ener_fun} can be recast as 
+\beq\label{eq:Ew-GIC-eLDA}
+\E{eLDA}{\bw}=\E{GIC-eLDA}{\bw}+\WHF[
+\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}].
+\eeq
+%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
+The corresponding excitation energies are
+\beq\label{eq:Om-eLDA}
+\begin{split}
+	\Ex{eLDA}{(I)} 
+	& = 
+        \Ex{HF}{(I)}
+	\\
+	& + \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} 
+	\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{} 
+	\\
+	& + \int \n{\bGam{\bw}}{}(\br{})
+	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{},
+\end{split}
+\eeq
+with $\Ex{HF}{(I)} = \E{HF}{(I)} - \E{HF}{(0)}$, where the last term is the ensemble correlation derivative contribution to the excitation energy.
+}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
 \label{sec:eDFA}
@@ -1116,15 +1144,7 @@ The deviation from linearity of the three-state ensemble energy $\E{}{(\ew{1},\e
 in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while
 fulfilling the restrictions on the ensemble weights to ensure the GOK
 variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
-To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI-eLDA}].
-\titou{Manu will move this to the theory section later on
-\beq
-\E{GIC-eLDA}{\bw}=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)},
-\eeq
-\beq
-\E{eLDA}{\bw}=\E{GIC-eLDA}{\bw}+\WHF[
-\bGam{\bw}]-\sum_{I\geq0}\ew{I}\WHF[ \bGam{(I)}]
-\eeq}
+To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above \titou{[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
 As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
 ensemble energy becomes less and less linear as $L$
 gets larger, while the GIC makes the ensemble energy almost
@@ -1150,7 +1170,7 @@ linear.
 It is important to note that, even though the GIC removes the explicit
 quadratic terms from the ensemble energy, a non-negligible curvature
 remains in the GIC-eLDA ensemble energy due to the optimization of the
-ensemble KS orbitals in the presence of GIE [see Eq.~\eqref{eq:min_with_HF_ener_fun}].
+ensemble KS orbitals in the presence of GIE \titou{[see Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:Ew-eLDA}]}.
 %However, this orbital-driven error is small (in our case at
 %least) \trashEF{as the correlation part of the ensemble KS potential $\delta
 %\E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared
@@ -1187,7 +1207,7 @@ Interesting also to see that the reverse occurs in the tri-ensemble.}
 	\includegraphics[width=\linewidth]{EvsL_5}
 	\caption{
 	\label{fig:EvsL}
-	Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{(1)}$ (bottom) and double excitation $\Ex{(2)}$ (top) of 5-boxium for various methods and box length $L$.
+	Excitation energies (multiplied by $L^2$) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) of 5-boxium for various methods and box length $L$.
 	Graphs for additional values of $\nEl$ can be found as {\SI}.
 	}
 \end{figure}
@@ -1200,11 +1220,12 @@ In other words, each excitation is dominated by a sole, well-defined reference S
 However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
 In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
 This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
+% TITOU: shall we keep the paragraph below?
 Therefore, it is paramount to construct a two-weight correlation functional
 (\ie, a triensemble functional, as we have done here) which
 allows the mixing of singly- and doubly-excited configurations.
 Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
-\titou{Shall we add results for the biensemble to illustrate this?}
+\titou{Titou will add results for the biensemble to illustrate this.}
 %\manu{Well, neglecting the second excited state is not the same as
 %considering the $w_2=0$ limit. I thought you were referring to an
 %approximation where the triensemble calculation is performed with
@@ -1212,13 +1233,13 @@ Using a single-weight (\ie, a biensemble) functional where only the ground state
 %because, in this limit, you may still have a derivative discontinuity
 %correction. The latter is absent if you truly neglect the second excited
 %state in your ensemble functional. This should be clarified.}\\
-\manu{Are the results in the supp mat? We could just add "[not
-shown]" if not. This is fine as long as you checked that, indeed, the
-results deteriorate ;-)}
-\manu{Should we add that, in the bi-ensemble case, the ensemble
-correlation derivative $\partial \epsilon^\bw_{\rm c}(n)/\partial w_2$
-is neglected (if this is really what you mean (?)). I guess that this is the reason why
-the second excitation energy would not be well described (?)}
+%\manu{Are the results in the supp mat? We could just add "[not
+%shown]" if not. This is fine as long as you checked that, indeed, the
+%results deteriorate ;-)}
+%\manu{Should we add that, in the bi-ensemble case, the ensemble
+%correlation derivative $\partial \epsilon^\bw_{\rm c}(n)/\partial w_2$
+%is neglected (if this is really what you mean (?)). I guess that this is the reason why
+%the second excitation energy would not be well described (?)}
 
 As shown in Fig.~\ref{fig:EvsL}, all methods provide accurate estimates of the excitation energies in the weak correlation regime (\ie, small $L$).
 When the box gets larger, they start to deviate.
@@ -1286,7 +1307,7 @@ electrons.
 	\includegraphics[width=\linewidth]{EvsL_5_HF}
 	\caption{
 	\label{fig:EvsLHF}
-	Error with respect to FCI (in \%) associated with the single excitation $\Ex{(1)}$ (bottom) and double excitation $\Ex{(2)}$ (top) as a function of the box length $L$ for 5-boxium at the KS-eLDA (solid lines) and eHF (dashed lines) levels. 
+	Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 5-boxium at the KS-eLDA (solid lines) and eHF (dashed lines) levels. 
 	Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and state-averaged (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported. 
 	}
 \end{figure}
@@ -1299,7 +1320,7 @@ 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.\\
+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