lots of changes

Manuscript/eDFT.tex

@@ -244,11 +244,7 @@ where the KS wavefunctions fulfill the ensemble density constraint
 \eeq 
 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 
-\beq
-	\Ex{(I)} = \E{}{(I)} - \E{}{(0)},
-\eeq
-or individual energy levels (for geometry optimizations, for example). 
+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{}
 \beq\label{eq:indiv_ener_from_ens}
 	\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \pdv{\E{}{\bw}}{\ew{K}},
@@ -324,7 +320,8 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
 	\right|_{\n{}{} = \n{\opGam{\bw}}{}}.
 \end{split}
 \eeq
-\titou{The last term in Eq.~\eqref{eq:exact_ener_level_dets} corresponds to the derivative discontinuity (DD).}
+\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}
@@ -525,6 +522,7 @@ expression of Eq.~\eqref{eq:var_ener_gokdft}:
 	\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
 \beq\label{eq:commut_F_AO}
@@ -680,11 +678,14 @@ Turning to the density-functional ensemble correlation energy, the following eLD
 \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}. 
+\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 eLDA: 
-\beq\label{eq:EI}
+\beq\label{eq:EI-eLDA}
 \begin{split}
-	\E{}{(I)} 
-	& \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
+	\E{\titou{eLDA}}{(I)} 
+	& \titou{\approx} \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{}
 	\\
@@ -698,6 +699,13 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
 \eeq
 \titou{T2: I think we should specify what those terms are physically... Maybe earlier in the manuscript?}
 
+\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}
+	\E{eHF}{(I)} \approx \Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}].
+\eeq
+}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
 \label{sec:eDFA}
@@ -787,16 +795,19 @@ where
 \end{equation}
 The local-density approximation (LDA) correlation functional,
 \begin{equation}
-	\e{c}{\text{LDA}}(\n{}{}) = a_1^\text{LDA} \, F\qty[1,\frac{3}{2},a_3^\text{LDA}, \frac{a_1^\text{LDA}(1-a_3^\text{LDA})}{a_2^\text{LDA}} {\n{}{}}^{-1}],
+	\e{c}{\text{LDA}}(\n{}{}) 
+	= a_1^\text{LDA} F\qty[1,\frac{3}{2},a_3^\text{LDA}, \frac{a_1^\text{LDA}(1-a_3^\text{LDA})}{a_2^\text{LDA}} {\n{}{}}^{-1}],
 \end{equation}
 specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
+\begin{subequations}
 \begin{align}
 	a_1^\text{LDA} & = - \frac{\pi^2}{360},
-	&
+	\\
 	a_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
-	& 
+	\\ 
 	a_3^\text{LDA} & = 2.408779.
 \end{align}
+\end{subequations}
 Equation \eqref{eq:becw} can be recast
 \begin{equation}
 \label{eq:eLDA}
@@ -843,8 +854,7 @@ For KS-DFT and KS-eDFT calculations, a Gauss-Legendre quadrature is employed to
 In order to test the present eLDA functional we have performed various sets of calculations.
 To get reference excitation energies for both the single and double excitations, we have performed full configuration interaction (FCI) calculations with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
 For the single excitations, we have also performed time-dependent LDA (TDLDA) calculations, and the quality of the Tamm-Dancoff approximation (TDA) has been also investigated.\cite{Dreuw_2005}
-Concerning the KS-eDFT calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
-In order to test the influence of correlation effects on excitation energies, we have also performed ensemble HF (labeled as eHF) calculations.
+Concerning the KS-eDFT and eHF calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and discussion}
@@ -853,18 +863,20 @@ In order to test the influence of correlation effects on excitation energies, we
 
 First, we discuss the linearity of the ensemble energy.
 To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
-The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$.
-To illustrate the magnitude of the ghost interaction error (GIE), we have reported the ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI}].
-As one can see, the linearity of the ensemble energy deteriorates when $L$ gets larger. 
+The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented 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, $0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$).
+To illustrate the magnitude of the ghost interaction error (GIE), we have reported the ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI-eLDA}].
+As one can see, the linearity of the GOC-free ensemble energy deteriorates when $L$ gets larger, while the GOC makes the ensemble energy almost perfectly linear.
 In other words, the GIE increases as the correlation gets stronger.
-Moreover, because the GIE can easily computed via Eq.~\eqref{eq:WHF} even for real, three-dimensional systems, this provides a cheap way of quantifying strong correlation in a given electronic system.
+Because the GIE can be easily computed via Eq.~\eqref{eq:WHF} even for real, three-dimensional systems, this provides a cheap way of quantifying strong correlation in a given electronic system.
+It is important to note that, even though the GIC removes the explicit quadratic terms from the ensemble energy, a weak non-linearity remains in the ensemble energy due to the optimization of the ensemble KS orbitals in presence of GIE.
+However, this ``density-driven''-type of error is extremely small (in our case at least).
 
 %%% FIG 1 %%%
 \begin{figure*}
 	\includegraphics[width=\linewidth]{EvsW_n5}
 	\caption{
 	\label{fig:EvsW}
-	Weight dependence of the ensemble energy $\E{}{(\ew{1},\ew{2})}$ with and without ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
+	Weight dependence of the ensemble energy $\E{}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
 	}
 \end{figure*}
 %%% %%% %%%
@@ -874,8 +886,7 @@ In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$)
 Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
 In the weakly correlated regime (\ie, small $L$), all methods provide accurate estimates of the excitation energies.
 When the box gets larger, they start to deviate.
-%For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$.
-TDLDA yields larger errors at large $L$ by underestimating the excitation energies.
+For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
 TDA-TDLDA slightly corrects this trend thanks to error compensation. 
 Concerning the eLDA functional, our results clearly evidence that the equi-weights [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
 This is especially true for the single excitation which is significantly improved by using state-averaged weights.
@@ -894,11 +905,13 @@ This conclusion is verified for smaller and larger number of electrons (see {\SI
 \end{figure}
 %%% %%% %%%
 
-Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\nEl$ and fixed $L$ (in this case $L=\pi$).
-The graphs associated with other $L$ values are reported as {\SI}.
-Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations. 
-As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$.
-Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only few percents.
+It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
+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 eLDA and eHF levels [see Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:EI-eHF}, respectively] as a function of the box length $L$ in the case of 5-boxium.
+\titou{The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
+Its contribution is also significantly larger in the case of the single excitation; the derivative discontinuity hardly influence the double excitation.
+Importantly, one realises that the magnitude of the derivative discontinuity is much smaller in the case of state-averaged calculations (as compared to the zero-weight calculations).
+This could explain why equiensemble calculations are clearly more accurate as it reduces the influence of the derivative discontinuity: for a given method, state-averaged orbitals partially remove the burden of modelling properly the derivative discontinuity.
+}
 
 %%% FIG 3 %%%
 \begin{figure}
@@ -911,6 +924,12 @@ Even for larger boxes, the discrepancy between FCI and eLDA for double excitatio
 \end{figure}
 %%% %%% %%%
 
+Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\nEl$ and fixed $L$ (in this case $L=\pi$).
+The graphs associated with other $L$ values are reported as {\SI}.
+Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations. 
+As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$.
+Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only few percents.
+
 %%% FIG 4 %%%
 \begin{figure*}
 	\includegraphics[width=\linewidth]{EvsN}
@@ -921,8 +940,6 @@ Even for larger boxes, the discrepancy between FCI and eLDA for double excitatio
 \end{figure*}
 %%% %%% %%%
 
-\titou{Need further discussion on DD.}
-
 \titou{For small $L$, the single and double excitations are ``pure''. In other words, the excitation is dominated by a single reference Slater determinant.
 However, when the box gets larger, there is a strong mixing between different degree of excitations.
 In particular, the single and double excitations strongly mix.