SI typo

FCI/boxints.txt

@@ -1,4 +1,4 @@
-&FCI NORB = 30, NELEC = 7, MS2 = 7, ISYM = 1, NROOT = 1, MAXIT = 200,
+&FCI NORB = 30, NELEC = 2, MS2 = 2, ISYM = 2, NROOT = 6, MAXIT = 200,
  ORBSYM = 1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,
  MEMORY = 300000000 /
 0.4473211247724916	1	1	1	1

FCI/fci.f

@@ -169,7 +169,7 @@ CSUBR INTEGRAL INPUT FOR PORTABLE FULL CI PROGRAM
 cpmg
 c         rs = 2.0d0
 c         R = sqrt(dfloat(na+nb))/2 * rs
-         R = 0.25d0
+         R = 8.0d0
          write(*,*) 'Scaling integrals for R =',R
 cpmg
       CALL FZERO (Z1,NPAIR(1))

Manuscript/SI/eDFT-SI.tex

@@ -581,21 +581,21 @@ The numerical values of the correlation energy for various $R$ are reported in T
 Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
 \begin{equation}
 \label{eq:ec}
-	\e{c}{(I)}(n) = \frac{c_1^{(I)}\,n}{n + c_2^{(I)} \sqrt{n} + c_3^{(I)}},
+	\e{c}{(I)}(n) = \frac{a_1^{(I)}\,n}{n + a_2^{(I)} \sqrt{n} + a_3^{(I)}},
 \end{equation}
-where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
-The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
+where $a_2^{(I)}$ and $a_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
+The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
 Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
 
 
 %%% FIG 2 %%%
 \begin{figure*}
-	\includegraphics[height=0.325\linewidth]{EvsL_2}
-	\includegraphics[height=0.325\linewidth]{EvsL_3}
-	\includegraphics[height=0.325\linewidth]{EvsL_4}
-	\includegraphics[height=0.325\linewidth]{EvsL_5}
-	\includegraphics[height=0.325\linewidth]{EvsL_6}
-	\includegraphics[height=0.325\linewidth]{EvsL_7}
+	\includegraphics[width=0.49\linewidth]{EvsL_2}
+	\includegraphics[width=0.49\linewidth]{EvsL_3}
+	\includegraphics[width=0.49\linewidth]{EvsL_4}
+	\includegraphics[width=0.49\linewidth]{EvsL_5}
+	\includegraphics[width=0.49\linewidth]{EvsL_6}
+	\includegraphics[width=0.49\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.
 	}
@@ -605,13 +605,13 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
 
 %%% FIG 3 %%%
 \begin{figure*}
-	\includegraphics[height=0.325\linewidth]{EvsN_0125}
-	\includegraphics[height=0.325\linewidth]{EvsN_025}
-	\includegraphics[height=0.325\linewidth]{EvsN_05}
-	\includegraphics[height=0.325\linewidth]{EvsN_1}
-	\includegraphics[height=0.325\linewidth]{EvsN_2}
-	\includegraphics[height=0.325\linewidth]{EvsN_4}
-	\includegraphics[height=0.325\linewidth]{EvsN_8}
+	\includegraphics[width=0.49\linewidth]{EvsN_0125}
+	\includegraphics[width=0.49\linewidth]{EvsN_025}
+	\includegraphics[width=0.49\linewidth]{EvsN_05}
+	\includegraphics[width=0.49\linewidth]{EvsN_1}
+	\includegraphics[width=0.49\linewidth]{EvsN_2}
+	\includegraphics[width=0.49\linewidth]{EvsN_4}
+	\includegraphics[width=0.49\linewidth]{EvsN_8}
 	\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 $L$.
 	}

Manuscript/eDFT.tex

@@ -724,13 +724,13 @@ The only remaining piece of information to define at this stage is the weight-de
 \section{Density-functional approximations for ensembles}
 \label{sec:eDFA}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-We decompose the weight-dependent functional as
-\begin{equation}
-	\be{Hxc}{\bw}(\n{}{}) = \be{Hx}{\bw}(\n{}{}) + \be{c}{\bw}(\n{}{}),
-\end{equation}
-where $\be{Hx}{\bw}(\n{}{})$ is a weight-dependent Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} [see Subsec.~\ref{sec:GIC}] and $\be{c}{\bw}(\n{}{})$ is a weight-dependent correlation functional [see Subsec.~\ref{sec:Ec}].
-The construction of these two functionals is described below.
-Note that, because we consider strict 1D systems, one cannot decompose further the Hartree-exchange contribution as each component diverges independently but their sum is finite. \cite{Astrakharchik_2011, Lee_2011a, Loos_2012, Loos_2013, Loos_2013a}
+%We decompose the weight-dependent functional as
+%\begin{equation}
+%	\be{Hxc}{\bw}(\n{}{}) = \be{Hx}{\bw}(\n{}{}) + \be{c}{\bw}(\n{}{}),
+%\end{equation}
+%where $\be{Hx}{\bw}(\n{}{})$ is a weight-dependent Hartree-exchange functional designed to correct the ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} [see Subsec.~\ref{sec:GIC}] and $\be{c}{\bw}(\n{}{})$ is a weight-dependent correlation functional [see Subsec.~\ref{sec:Ec}].
+%The construction of these two functionals is described below.
+%Note that, because we consider strict 1D systems, one cannot decompose further the Hartree-exchange contribution as each component diverges independently but their sum is finite. \cite{Astrakharchik_2011, Lee_2011a, Loos_2012, Loos_2013, Loos_2013a}
 
 Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (UEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
 One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
@@ -748,31 +748,31 @@ As mentioned previously, we consider a three-state ensemble including the ground
 All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
 We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{Ghost-interaction correction}
-\label{sec:GIC}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-The GIC weight-dependent Hartree-exchange functional is defined as  
-\begin{multline}
-	\be{Hx}{\bw}(\n{}{\bw}) = (1-\sum_{I>0} \ew{I}) \be{Hx}{}(\n{}{(0)}) + \sum_{I>0} \ew{I} \be{Hx}{}(\n{}{(I)}) 
-	\\
-	- \be{Hx}{(I)}(\n{}{\bw}),
-\end{multline}
-where
-\begin{equation}
-	\be{Hx}{}(\n{}{}) = \iint \frac{\n{}{}(\br_1) \n{}{}(\br_2) - \n{}{}(\br_1,\br_2)^2}{r_{12}} d\br_1 d\br_2,
-\end{equation}
-and 
-\begin{equation}
-	\n{}{(I)}(\omega) = (\pi R)^{-1} \cos[(I+1) \omega/2]
-\end{equation}
-is the first-order density matrix with $\omega$ the interelectronic angle.
-It yields
-\begin{equation}
-	\be{Hx}{}(\n{}{}) = \n{}{} \qty[ a_1 \ew{1} (\ew{1} - 1) +  a_2 \ew{1} \ew{2} + a_3 \ew{2} (\ew{2} - 1)],
-\end{equation}
-with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%\subsection{Ghost-interaction correction}
+%\label{sec:GIC}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+%The GIC weight-dependent Hartree-exchange functional is defined as  
+%\begin{multline}
+%	\be{Hx}{\bw}(\n{}{\bw}) = (1-\sum_{I>0} \ew{I}) \be{Hx}{}(\n{}{(0)}) + \sum_{I>0} \ew{I} \be{Hx}{}(\n{}{(I)}) 
+%	\\
+%	- \be{Hx}{(I)}(\n{}{\bw}),
+%\end{multline}
+%where
+%\begin{equation}
+%	\be{Hx}{}(\n{}{}) = \iint \frac{\n{}{}(\br_1) \n{}{}(\br_2) - \n{}{}(\br_1,\br_2)^2}{r_{12}} d\br_1 d\br_2,
+%\end{equation}
+%and 
+%\begin{equation}
+%	\n{}{(I)}(\omega) = (\pi R)^{-1} \cos[(I+1) \omega/2]
+%\end{equation}
+%is the first-order density matrix with $\omega$ the interelectronic angle.
+%It yields
+%\begin{equation}
+%	\be{Hx}{}(\n{}{}) = \n{}{} \qty[ a_1 \ew{1} (\ew{1} - 1) +  a_2 \ew{1} \ew{2} + a_3 \ew{2} (\ew{2} - 1)],
+%\end{equation}
+%with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
  
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Weight-dependent correlation functional}
@@ -782,10 +782,10 @@ with $a_1 = 2 \ln 2 - 1/3$, $a_2 = 8/3$ and $a_3 = 32/15$.
 Based on highly-accurate calculations (see {\SI} for additional details), one can write down, for each state, an accurate analytical expression of the reduced (i.e., per electron) correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
 \begin{equation}
 \label{eq:ec}
-	\e{c}{(I)}(\n{}{}) = \frac{c_1^{(I)}\,\n{}{}}{\n{}{} + c_2^{(I)} \sqrt{\n{}{}} + c_3^{(I)}},
+	\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}\,\n{}{}}{\n{}{} + a_2^{(I)} \sqrt{\n{}{}} + a_3^{(I)}},
 \end{equation}
-where the $c_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
-The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
+where the $a_k^{(I)}$'s are state-specific fitting parameters provided in Table \ref{tab:OG_func}.
+The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
 Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
 Combining these, one can build a three-state weight-dependent correlation eDFA:
 \begin{equation}
@@ -797,11 +797,11 @@ Combining these, one can build a three-state weight-dependent correlation eDFA:
 \begin{table*}
 	\caption{
 	\label{tab:OG_func}
-	Parameters of the correlation DFAs defined in Eq.~\eqref{eq:ec}.}
+	Parameters of the weight-dependent correlation DFAs defined in Eq.~\eqref{eq:ec}.}
 %	\begin{ruledtabular}
 		\begin{tabular}{lcddd}
 			\hline\hline
-			State					&	$I$		&	\tabc{$c_1^{(I)}$}	&	\tabc{$c_2^{(I)}$}	&	\tabc{$c_3^{(I)}$}	\\
+			State					&	$I$		&	\tabc{$a_1^{(I)}$}	&	\tabc{$a_2^{(I)}$}	&	\tabc{$a_3^{(I)}$}	\\
 			\hline
 			Ground state			&	$0$		&	-0.0137078			&	0.0538982			&	0.0751740			\\
 			Singly-excited state	&	$1$		&	-0.0238184			&	0.00413142			&	0.0568648			\\
@@ -826,15 +826,15 @@ where
 \end{equation}
 The local-density approximation (LDA) correlation functional,
 \begin{equation}
-	\e{c}{\text{LDA}}(\n{}{}) = c_1^\text{LDA} \, F\qty[1,\frac{3}{2},c_3^\text{LDA}, \frac{c_1^\text{LDA}(1-c_3^\text{LDA})}{c_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{align}
-	c_1^\text{LDA} & = - \frac{\pi^2}{360},
+	a_1^\text{LDA} & = - \frac{\pi^2}{360},
 	&
-	c_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
+	a_2^\text{LDA} & = \frac{3}{4} - \frac{\ln{2\pi}}{2},
 	& 
-	c_3^\text{LDA} & = 2.408779.
+	a_3^\text{LDA} & = 2.408779.
 \end{align}
 Equation \eqref{eq:becw} can be recast
 \begin{equation}
@@ -880,21 +880,20 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
 	\end{cases}
 \end{equation} 
 with $ \mu = 1,\ldots,\Nbas$ and $\Nbas = 30$ for all calculations.
-For the self-consistent calculations (such as HF, KS or eKS), the convergence threshold has been set to $\tau = 10^{-7}$.
-For KS and eKS calculations, a Gauss-Legendre quadrature is employed to compute numerical integrals.
+For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold has been set to $\tau = 10^{-5}$.
+For KS-DFT and KS-eDFT calculations, a Gauss-Legendre quadrature is employed to compute numerical integrals.
 
 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 TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also tested.
-
-Concerning the eKS 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 $\bw = (1/3,1/3)$.
+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 $\bw = (1/3,1/3)$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and discussion}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e., $\Nel = 5$).
-Similar graphs are obtained for the other $\Nel$ values and they can be found --- alongside the numerical data associated with each method --- in the {\SI}.
+In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (i.e., $\Nel = 5$).
+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 (i.e., 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 $L=8\pi$.
@@ -905,24 +904,23 @@ This is especially true for the single excitation which is significantly improve
 The effect on the double excitation is less pronounced.
 Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.
 This conclusion is verified for smaller and larger number of electrons (see {\SI}).
-\alert{Shall I test the one-electron system for self-interaction?}
 
 %%% FIG 1 %%%
 \begin{figure}
 	\includegraphics[width=\linewidth]{EvsL_5}
 	\caption{
 	\label{fig:EvsL}
-	Error with respect to FCI in single and double excitation energies for 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}
 %%% %%% %%%
 
-Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies, for the same methods, as a function of $\Nel$ and fixed $L$ (in this case $L=\pi$).
+Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (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 $L=\pi$, an error of the same order as CIS or TDA-TDLDA.
-Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only a few percent.
+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 $L=\pi$.
+Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only few percents.
 
 %%% FIG 2 %%%
 \begin{figure}
@@ -957,9 +955,10 @@ This eDFA delivers accurate excitation energies for both single and double excit
 Generalization to a larger number of states is straightforward and will be investigated in future work.
 Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos_2010,Loos_2010d,Loos_2017a} of the present eDFA is currently under development to model excited states in molecules and solids.
 
-Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap.
+Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap.\cite{Senjean_2018}
 This can be done by constructing DFAs for the one- and three-electron ground state systems, and combining them with the two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and \eqref{eq:ecw}.
-However, as shown by Senjean and Fromager, \cite{Senjean_2018} one must modify the weights accordingly in order to maintain a constant density.
+We hope to report on this in the near future.
+%However, as shown by Senjean and Fromager, \cite{Senjean_2018} one must modify the weights accordingly in order to maintain a constant density.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Supplementary material}
@@ -969,6 +968,7 @@ See {\SI} for the additional details about the construction of the functionals,
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{acknowledgements}
 E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Grant No.~ANR-14-CE06-0014-01) for funding.
+This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the ``Programme des Investissements d'Avenir''.
 \end{acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%