new graphs and SI

Manuscript/SI/eDFT-SI.tex

@@ -596,25 +596,25 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\includegraphics[height=0.325\linewidth]{EvsL_7}
 	\caption{
 	Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the box length $L$ for various methods.
-	}
+	See main text for additional details.}
 	\label{fig:EvsL}
 \end{figure*}
 %%% %%% %%%
 
 %%% FIG 3 %%%
-\begin{figure*}
+%\begin{figure*}
-	\includegraphics[height=0.325\linewidth]{EvsN_0125}
+%	\includegraphics[height=0.325\linewidth]{EvsN_0125}
-	\includegraphics[height=0.325\linewidth]{EvsN_025}
+%	\includegraphics[height=0.325\linewidth]{EvsN_025}
-	\includegraphics[height=0.325\linewidth]{EvsN_05}
+%	\includegraphics[height=0.325\linewidth]{EvsN_05}
-	\includegraphics[height=0.325\linewidth]{EvsN_1}
+%	\includegraphics[height=0.325\linewidth]{EvsN_1}
-	\includegraphics[height=0.325\linewidth]{EvsN_2}
+%	\includegraphics[height=0.325\linewidth]{EvsN_2}
-	\includegraphics[height=0.325\linewidth]{EvsN_4}
+%	\includegraphics[height=0.325\linewidth]{EvsN_4}
-	\includegraphics[height=0.325\linewidth]{EvsN_8}
+%	\includegraphics[height=0.325\linewidth]{EvsN_8}
-	\caption{
+%	\caption{
-	Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the number of electrons $N$ for various methods and box length $L$.
+%	Error with respect to FCI in single and double excitation energies of $N$-boxium as a function of the number of electrons $N$ for various methods and box length $L$.
-	}
+%	}
-	\label{fig:EvsL}
+%	\label{fig:EvsL}
-\end{figure*}
+%\end{figure*}
 %%% %%% %%%
 
 %%% TABLE II %%%
@@ -701,9 +701,9 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\hline
 			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
 	\hline
-	CIS		&				&	$\Ex{(1)}$		&	0.0163	&	0.0161	&	0.0157	&	0.0149	&	0.0133	&	0.0102	&	0.0057	\\
+%	CIS		&				&	$\Ex{(1)}$		&	0.0163	&	0.0161	&	0.0157	&	0.0149	&	0.0133	&	0.0102	&	0.0057	\\
-	\\
+%	\\
-	TDHF	&				&	$\Ex{(1)}$		&	0.0013	&	0.0013	&	0.0014	&	0.0014	&	0.0013	&	0.0010	&	0.0007	\\
+%	TDHF	&				&	$\Ex{(1)}$		&	0.0013	&	0.0013	&	0.0014	&	0.0014	&	0.0013	&	0.0010	&	0.0007	\\
 	\\
 	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0162	&	0.0157	&	0.0146	&	0.0110	&	-0.0049	&	-0.0344	&	-0.0378	\\
 	\\
@@ -764,10 +764,10 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 	\hline
 			&				&					&	\mc{7}{c}{Deviation from FCI}	\\
 	\hline
-	CIS		&				&	$\Ex{(1)}$		&	0.0203	&	0.0202	&	0.0200	&	0.0195	&	0.0187	&	0.0167	&	0.0116	\\
+%	CIS		&				&	$\Ex{(1)}$		&	0.0203	&	0.0202	&	0.0200	&	0.0195	&	0.0187	&	0.0167	&	0.0116	\\
-	\\
+%	\\
-	TDHF	&				&	$\Ex{(1)}$		&	0.0008	&	0.0008	&	0.0009	&	0.0009	&	0.0008	&	0.0008	&	0.0007	\\
+%	TDHF	&				&	$\Ex{(1)}$		&	0.0008	&	0.0008	&	0.0009	&	0.0009	&	0.0008	&	0.0008	&	0.0007	\\
-	\\
+%	\\
 	TDA-TDLDA&				&	$\Ex{(1)}$		&	0.0203	&	0.0201	&	0.0195	&	0.0181	&	0.0106	&	-0.0178	&	-0.0369	\\
 	\\
 	TDLDA	&				&	$\Ex{(1)}$		&	0.0008	&	0.0007	&	0.0004	&	-0.0006	&	-0.0074	&	-0.0360	&	-0.0653	\\

Manuscript/eDFT.tex

@@ -842,8 +842,7 @@ 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 HF (TDHF), configuration interaction singles (CIS) and TDLDA calculations. \cite{Dreuw_2005}
+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}
-For TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also investigated.
 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.
 
@@ -875,7 +874,7 @@ 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 $L=8\pi$.
+%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.
 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, $\bw = (0,0)$].
@@ -898,18 +897,28 @@ This conclusion is verified for smaller and larger number of electrons (see {\SI
 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 TDHF or TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$.
+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 3 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{EvsN_1}
+	\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 eLDA (solid lines) and eHF (dashed lines) levels of theory. 
+	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}
+%%% %%% %%%
+
+%%% FIG 4 %%%
+\begin{figure*}
+	\includegraphics[width=\linewidth]{EvsN}
 	\caption{
 	\label{fig:EvsN}
 	Error with respect to FCI in single and double excitation energies for $\nEl$-boxium for various methods and number of electrons $\nEl$ at $L=\pi$.
-	Graphs for additional values of $L$ can be found as {\SI}.
 	}
-\end{figure}
+\end{figure*}
 %%% %%% %%%
 
 \titou{Need further discussion on DD.}