saving work

Manuscript/FarDFT.tex

@@ -565,7 +565,7 @@ This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\
 Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), the first singly-excited state ($I=1$ with weight $\ew{1}$), as well as the first doubly-excited state ($I=2$ with weight $\ew{2}$) are considered. 
 To ensure the GOK variational principle, one should then have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$. 
 
-Taking a two-electron system as an example, the individual one-electron densities reads
+Taking a generic two-electron system as an example, the individual one-electron densities read
 \begin{subequations}
 \begin{align}
 	\n{}{(0)} & = 2 \HOMO{2},
@@ -582,23 +582,25 @@ and they can be combined to produce the ensemble density
 	= (1 - \ew{1} - \ew{2}) \n{}{(0)}
 	+ \ew{1} \n{}{(1)} + \ew{2} \n{}{(2)}.
 \end{equation}
-Equation \eqref{eq:nw1w2} can be conveniently recast as a single-weight quantity
+For analysis purposes, Eq.~\eqref{eq:nw1w2} can be conveniently recast as a single-weight quantity
 \begin{equation}
 	\n{}{\eW} = (1 - \eW) \n{}{(0)} +  \eW \n{}{(2)},
 \end{equation}
 with $\eW = \ew{1}/2 + \ew{2}$ and $0 \le \eW \le 1/2$.
 Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{1} = \ew{2}$), and we consider the zero-weight limit $\eW = 0$ (\ie, $\ew{1} = \ew{2} = 0$), and the equiweight ensemble $\eW = 1/2$ (\ie, $\ew{1} = \ew{2} = 1/3$).
 Nonetheless, we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals.
-Indeed, the limit $\ew{2} = 1$ (which corresponds to a pure excited state) is of particular interest as it is a genuine saddle point of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
+Indeed, the pure-state limits (\ie, $\ew{1} = 1 \land \ew{2} = 0$ or $\ew{1} = 0 \land \ew{2} = 1$) are of particular interest as they are genuine saddle points of the KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
 %Although the range $1/2 < \ew{} \leq 1$ stands a little bit beyond the theory discussed previously, we look at these solutions for analysis purposes mainly.
 %These solutions of the density matrix operator functional in Eq.~\eqref{eq:min_KS_DM} correspond to stationary points rather than minimising ones.
 %Applying GOK-DFT in this range of weights would simply consists in switching the ground and excited states if true minimisations of the ensemble energy were performed.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\section{Results}
+\section{Results and Discussion}
 \label{sec:res}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+In this Section, we propose a two-step procedure to design, first, a weight-dependent, system-dependent local exchange functional in order to remove the curvature of the ensemble energy.
+Second, we describe the construction of a universal, weight-dependent local correlation functional based on FUEGs.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Hydrogen molecule at equilibrium}
@@ -626,16 +628,16 @@ In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground sta
 %ensemble? In one way or another
 %we have to look at this, even within the simplest weight-independent
 %approximation.}
-The ensemble energy $\E{}{\eW{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the composite weight $0 \le \eW{} \le 1/2$.
+The ensemble energy $\E{}{\eW}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of the composite weight $0 \le \eW \le 1/2$.
 Because this exchange functional does not depend on the ensemble weight, there is no contribution from the ensemble derivative term [last term in Eq.~\eqref{eq:dEdw}].
-As anticipated, $\E{}{\eW{}}$ is far from being linear, which means that the excitation energy obtained via the derivative of the ensemble energy varies significantly with the weight of the double excitation (see Fig.~\ref{fig:Om_H2}).
-Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\eW{} = 0$ to $1/2$.
-Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\ew{}$.
+As anticipated, $\E{}{\eW}$ is far from being linear, which means that the excitation energy associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\eW$ (see Fig.~\ref{fig:Om_H2}).
+Taking as a reference the full configuration interaction (FCI) value of $28.75$ eV obtained with the aug-mcc-pV8Z basis set, \cite{Barca_2018a} one can see that the excitation energy varies by more than $8$ eV from $\eW = 0$ to $1/2$.
+Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of $\eW$.
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{Ew_H2}
 	\caption{
-	\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\eW}$ (in hartree) as a function of the weight of the double excitation $\eW = \ew{1}/2 + \ew{2}$ and $\ew{1} = \ew{2}$ for various functionals and the aug-cc-pVTZ basis set.
+	\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\eW}$ (in hartree) as a function of the composite weight $\eW = \ew{1}/2 + \ew{2}$ and $\ew{1} = \ew{2}$ for various functionals and the aug-cc-pVTZ basis set.
 	See main text for the definition of the various functional's acronyms.
 	\label{fig:Ew_H2}
 	}
@@ -644,7 +646,7 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
 \begin{figure}
 	\includegraphics[width=\linewidth]{Om_H2}
 	\caption{
-	\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) \titou{for the double excitation} as a function of the composite weight $\eW$ for various functionals and the aug-cc-pVTZ basis set.
+	\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) of the doubly-excited state $\Ex{}{(2)}$ as a function of the composite weight $\eW = \ew{1}/2 + \ew{2}$ and $\ew{1} = \ew{2}$ for various functionals and the aug-cc-pVTZ basis set.
 	\label{fig:Om_H2}
 	}
 \end{figure}
@@ -693,16 +695,16 @@ makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the e
 The parameters $\alpha$, $\beta$, and $\gamma$ entering Eq.~\eqref{eq:Cxw} have been obtained via a least-square fit of the non-linear component of the ensemble energy computed between $\eW = 0$ and $\eW = 1/2$ by steps of $0.025$. 
 Note that this procedure makes, by construction, the ensemble energy also linear with respect to $\ew{1}$ and $\ew{2}$.
 
-As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the two ghost-interaction-free limits at $\ew{} = 0$ and $\ew{} = 1$, as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
-Maybe surprisingly, one would have noticed that we are not only using data from $0 \le \ew{} \le 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is deterred by the GOK variational principle. \cite{Gross_1988a} 
-However, it is important to ensure that the weight-dependent functional does not alter the $\ew{} = 1$ limit, which is a genuine saddle point of the KS equations, as mentioned above.
-Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
+As readily seen from Eq.~\eqref{eq:Cxw} and graphically illustrated in Fig.~\ref{fig:Cxw} (red curve), the weight-dependent correction does not affect the three ghost-interaction-free limits at $\ew{1} = \ew{2} = 0$, $\ew{1} = 1 \land \ew{2} = 0$, and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\eW}$ reduces to $\Cx{}$ in these three limits thanks to the factor $\eW (1 - \eW)$.
+%Maybe surprisingly, one would have noticed that we are not only using data from $0 \le \ew{} \le 1/2$, but we also consider ensemble energies for $1/2 < \ew{} \le 1$, which is deterred by the GOK variational principle. \cite{Gross_1988a} 
+However, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which is a genuine saddle point of the KS equations, as mentioned above.
+Finally, let us mention that, around $\eW = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
 We shall come back to this point later on.
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{Cxw}
 	\caption{
-	$\Cx{\ew{}}/\Cx{\ew{}=0}$ as a function of $\ew{}$ [see Eq.~\eqref{eq:Cxw}] computed with the aug-cc-pVTZ basis set for the \ce{He} atom (blue) and the \ce{H2} molecule at $\RHH = 1.4$ bohr (red) and $\RHH = 3.7$ bohr (green).
+	$\Cx{\eW}/\Cx{}$ as a function of $\eW$ [see Eq.~\eqref{eq:Cxw}] computed with the aug-cc-pVTZ basis set for the \ce{He} atom (blue) and the \ce{H2} molecule at $\RHH = 1.4$ bohr (red) and $\RHH = 3.7$ bohr (green).
 	\label{fig:Cxw}
 	}
 \end{figure}
@@ -720,14 +722,14 @@ The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex
 \subsubsection{Weight-dependent correlation functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-\manu{It seems crucial to me to distinguish what follows from the
-previous results, which are more ``semi-empirical''. CC-S is fitted on
-a specific system. I would personally add a subsection on glomium in the
-theory section. I would also not dedicate specific subsections to the
-previous results.}
+%\manu{It seems crucial to me to distinguish what follows from the
+%previous results, which are more ``semi-empirical''. CC-S is fitted on
+%a specific system. I would personally add a subsection on glomium in the
+%theory section. I would also not dedicate specific subsections to the
+%previous results.}
 
-Fourth, in the spirit of our recent work, \cite{Loos_2020} we have designed a weight-dependent correlation functional. 
-To build this weight-dependent correlation functional, we consider the singlet ground state and the first singlet doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
+Fourth, in the spirit of our recent work, \cite{Loos_2020} we design a universal, weight-dependent correlation functional. 
+To build this correlation functional, we consider the singlet ground state, the first singly-excited state as well as the first doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome). \cite{Loos_2009a,Loos_2009c,Loos_2010e}
 Notably, these two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
 Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}  
 As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
@@ -748,7 +750,7 @@ The reduced (\ie, per electron) Hartree-Fock (HF) energies for these three state
 \end{align}
 \end{subequations}
 
-Thanks to highly-accurate calculations and the expressions of the HF energies provided by Eqs.~\eqref{eq:eHF_0} and \eqref{eq:eHF_1}, \cite{Loos_2009a,Loos_2009c,Loos_2010e} 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 \cite{Sun_2016,Loos_2020}
+Thanks to highly-accurate calculations \cite{Loos_2009a,Loos_2009c,Loos_2010e} and the expressions of the HF energies provided by Eqs.~\eqref{eq:eHF_0}, \eqref{eq:eHF_1}, and \eqref{eq:eHF_2}, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following simple Pad\'e approximant \cite{Sun_2016,Loos_2020}
 \begin{equation}
 \label{eq:ec}
 	\e{\co}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
@@ -846,7 +848,7 @@ Also, Eq.~\eqref{eq:becw} can be recast as
 	+ \ew{2} \qty[\e{\co}{(2)}(\n{}{}) - \e{\co}{(0)}(\n{}{})]
 \end{split}
 \end{equation}
-which nicely highlights the centrality of the LDA in the present weight-dependent density-functional approximation for ensembles.
+which nicely highlights the centrality of VWN5 in the present weight-dependent density-functional approximation for ensembles.
 In particular, $\e{\co}{(0,0),\eVWN}(\n{}{}) = \e{\co}{\VWN}(\n{}{})$.
 We note also that, by construction, we have
 \begin{equation}
@@ -855,17 +857,17 @@ We note also that, by construction, we have
 	= \e{\co}{(I)}(n) - \e{\co}{(0)}(n),
 \end{equation}
 showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
+Note that, in the correlation case, one cannot define straightforwardly a composite weight like in the exchange case [see Eq.~\eqref{eq:Cxw}].
 
 As shown in Fig.~\ref{fig:Ew_H2}, the CC-SeVWN5 is slightly less concave than its CC-SVWN5 counterpart and it also improves (not by much) the excitation energy (see Fig.~\ref{fig:Om_H2}).
 
 For a more qualitative picture, Table \ref{tab:BigTab_H2} reports excitation energies for various methods and basis sets.
 In particular, we report the excitation energies obtained with GOK-DFT
-in the zero-weight limit (\ie, $\ew{} = 0$) and for the equi-ensemble
-(\ie, $\ew{} = 1/2$).
-\titou{These excitation energies can computed using Eq.~\eqref{eq:dEdw}.}
+in the zero-weight limit (\ie, $\eW = 0$) and for the equi-ensemble for which $\ew{1} = \ew{2} = 1/3$ (\ie, $\eW = 1/2$).
+These excitation energies are computed using Eq.~\eqref{eq:dEdw}.
+
 For comparison purposes, we also report the linear interpolation method (LIM) excitation energies, \cite{Senjean_2015,Senjean_2016} 
-a pragmatic way of getting weight-independent
-excitation energies defined as
+a pragmatic way of getting weight-independent excitation energies, defined as
 \begin{subequations}
 \begin{align}
 	\Ex{\LIM}{(1)} & = 2 \qty[\E{}{\bw{}=(1/2,0)} - \E{}{\bw{}=(0,0)}],
@@ -882,16 +884,16 @@ which require three independent calculations, as well as the MOM excitation ener
 \end{align}
 \end{subequations}
 which also require three separate calculations.
-We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
-They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
+%We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
+%They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
 
 The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
-The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5.
-It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
+The CC-SeVWN5 excitation energies at equi-weights (\ie, $\eW = 1/2$) are less satisfactory, but still remain in good agreement with FCI, with again a small improvement as compared to CC-SVWN5.
+It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{2} = 1$ (\textit{vide supra}).
 Note that by construction,
 for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper),
 LIM and MOM can be reduced to a single calculation
-at $w = 1/4$ and $w=1/2$, respectively, instead of performing an interpolation between two different calculations.
+at $\ew{} = 1/4$ and $\ew=1/2$, respectively, instead of performing an interpolation between two different calculations.
 Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between $\ew{} = 0$ and $1$.
 
 \titou{The present protocol can be related to optimally-tuned range-separated hybrid functionals. T2: more to come.}