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Manuscript/FarDFT.tex

@@ -586,6 +586,7 @@ Assuming that the singly-excited state is lower in energy than the doubly-excite
 %\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, $\ew{} \equiv \ew{1} = \ew{2}$), and we consider the zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$).
+In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
 
 Let us mention now that we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals.
 However, let us stress that we will not compute excitation energies with these ensembles inconsistent with GOK theory.
@@ -618,7 +619,7 @@ First, we compute the ensemble energy of the \ce{H2} molecule at equilibrium bon
 	&
 	\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
 \end{align}
-In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state, the lowest singly-excited state $1\sigma_g 1\sigma_u$ and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which has an autoionising resonance nature \cite{Bottcher_1974}).
+In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state, the lowest singly-excited state $1\sigma_g 1\sigma_u$ of the same symmetry as the ground state (\ie, $\Sigma_g^+$), and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}).
 %\manu{At equilibrium, I expect the singly-excited configuration
 %$1\sigma_g2\sigma_g$ to be lower in energy. From the point of view of
 %GOK-DFT I do not see how we can reach the doubly-excited state while
@@ -629,10 +630,11 @@ 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 weight $0 \le \ew{} \le 1/3$.
+
+The ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
 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 associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see Fig.~\ref{fig:Om_H2}).
+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 blue curve in 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 $1/3$.
+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 $7$ eV from $\ew{} = 0$ to $1/3$.
 Note that the exact xc ensemble functional would yield a perfectly linear energy and, hence, the same value of the excitation energy independently of the ensemble weights.
 
 \begin{figure}
@@ -657,9 +659,9 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
 \subsubsection{Weight-dependent exchange functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-Second, in order to remove this spurious curvature of the ensemble
+Second, in order to remove some of this spurious curvature of the ensemble
 energy (which is mostly due to the ghost-interaction error, \cite{Loos_2020} but not only), 
-one can easily reverse-engineer (for this particular system, geometry, and basis set) a local exchange functional to make $\E{}{(0,\ew{2})}$ as linear as possible for $0 \le \ew{2} \le 1$ assuming a perfect linearity between the pure-state limits $ \ew{1} = \ew{2} = 0$ (ground state) and $\ew{1} = 0 \land \ew{2} = 1$ (doubly-excited state).
+one can easily reverse-engineer (for this particular system, geometry, basis set, and excitation) a local exchange functional to make $\E{}{(0,\ew{2})}$ as linear as possible for $0 \le \ew{2} \le 1$ assuming a perfect linearity between the pure-state limits $ \ew{1} = \ew{2} = 0$ (ground state) and $\ew{1} = 0 \land \ew{2} = 1$ (doubly-excited state).
 %\manu{Something that seems important to me: you may require linearity in
 %the range $0\leq \ew{}\leq 1/2$. The excitation energy you would obtain
 %is simply the one of LIM, right? I suspect that by considering the
@@ -692,19 +694,20 @@ and
 	\gamma & = - 0.367\,189,
 \end{align}
 \end{subequations}
-makes the ensemble almost perfectly linear (see Fig.~\ref{fig:Ew_H2}), and the excitation energy much more stable (with respect to $\ew{}$) and closer to the FCI reference (see Fig.~\ref{fig:Om_H2}).
+makes the ensemble energy $\E{}{(0,\ew{2})}$ almost perfectly linear (by construction), and removes some of the curvature of $\E{}{\ew{}}$ (see yellow curve in Fig.~\ref{fig:Ew_H2}).
+It also makes the excitation energy much more stable (with respect to $\ew{}$), and closer to the FCI reference (see yellow curve in Fig.~\ref{fig:Om_H2}).
 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 $\ew{2} = 0$ and $\ew{2} = 1$ by steps of $0.025$. 
-\titou{Although this range of weights is inconsistent with GOK theory, we have found that it is important, from a practical point of view, to ensure ...}
+Although this range of weights is inconsistent with GOK theory, we have found that it is important, from a practical point of view, to ensure a correct behavior on the whole range of weights in order to obtain accurate excitation energies.
 
-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 $\ew{1} = \ew{2} = 0$ and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits thanks to the factor $\ew{} (1 - \ew{})$.
+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 $\ew{1} = \ew{2} = 0$ and $\ew{1} = 0 \land \ew{2} = 1$ (\ie, the pure-state limits), as $\Cx{\ew{}}$ reduces to $\Cx{}$ in these two limits.
-Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limit, which is a genuine saddle point of the KS equations, as mentioned above.
+Indeed, it is important to ensure that the weight-dependent functional does not alter these pure-state limits, which are genuine saddle points of the KS equations, as mentioned above.
-Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear.
+Finally, let us mention that, around $\ew{} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature one needs to catch in order to get accurate excitation energies in the zero-weight limit.
 We shall come back to this point later on.
 
 \begin{figure}
 	\includegraphics[width=\linewidth]{Cxw}
 	\caption{
-	$\Cx{\ew{}}/\Cx{}$ 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).
+	$\Cx{\ew{}}/\Cx{}$ 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}
@@ -959,12 +962,11 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
-For this particular geometry, the doubly-excited state becomes the
+Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state.
-lowest excited state with the same symmetry as
+Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble as defined in Sec.~\ref{sec:H2}
-the ground state, so we can safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state.
 %In other words, we set the weight of the single excitation to zero (\ie, $\ew{1} = 0$) and we have thus $\ew = \ew{2}$ for the rest of this example.
 We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr.
-It yields $\alpha = +0.019\,182$, $\beta = -0.015\,453$, and $\gamma = -0.012\,720$ [see Eq.~\eqref{eq:Cxw}].
+It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
 The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve). 
 
 One clearly sees that the correction brought by CC-S is much more gentle than at $\RHH = 1.4$ bohr, which means that the ensemble energy obtained with the LDA exchange functional is much more linear at $\RHH = 3.7$ bohr.
@@ -1040,10 +1042,10 @@ Similar to \ce{H2}, our ensemble contains the ground state of configuration $1s^
 In \ce{He}, the lowest doubly-excited state is an auto-ionising resonance state, extremely high in energy and lies in the continuum. \cite{Madden_1963}
 In Ref.~\onlinecite{Burges_1995}, highly-accurate calculations estimate an excitation energy of $2.126$ hartree for this $1s^2 \rightarrow 2s^2$ transition.
 Nonetheless, it can be nicely described with a Gaussian basis set containing enough diffuse functions.
-Consequently,  we consider for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
+Consequently, we consider for this particular example the d-aug-cc-pVQZ basis set which contains two sets of diffuse functions.
 The excitation energies associated with this double excitation computed with various methods and combinations of xc functionals are gathered in Table \ref{tab:BigTab_He}.
 
-The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.901\,572$, $\beta = +2.523\,660$, and $\gamma = +1.665\,228$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
+The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.912\,574$, $\beta = +2.715\,267$, and $\gamma = +2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
 The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
 
 The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.
@@ -1083,39 +1085,6 @@ Excitation energies (in hartree) associated with the lowest double excitation of
 \fnt[1]{Explicitly-correlated calculations from Ref.~\onlinecite{Burges_1995}.}
 \end{table}
 
-%%% TABLE I %%%
-%\begin{table}
-%\caption{
-%Excitation energies (in eV) associated with the lowest double excitation of \ce{HNO} obtained with the aug-cc-pVDZ basis set for various methods and combinations of xc functionals.
-%\label{tab:BigTab_H2st}
-%}
-%\begin{ruledtabular}
-%\begin{tabular}{llcccc}
-%	\mc{2}{c}{xc functional} 		&	\mc{2}{c}{GOK}	\\
-%	\cline{1-2}							\cline{3-4}
-%	\tabc{x}	&	\tabc{c}		&	 $\ew{} = 0$	&	$\ew{} = 1/2$	&	LIM		&	MOM		\\
-%	\hline
-%	HF			&					&			&			&		&		\\
-%	HF			&	VWN5			&			&			&		&		\\
-%	S			&					&	1.72	&	4.00	&	2.86	&	3.99	\\
-%	S			&	VWN5			&			&			&		&		\\
-%	CC-S		&					&	3.99		&	3.99		&	3.99	&	3.99		\\
-%	CC-S		&	VWN5			&	4.05		&	4.03		&	4.04	&	4.03	\\
-%	\hline
-%	S			&	PW92			&			&			&			&	4.00\fnm[1]	\\
-%	PBE			&	PBE				&			&			&			&	4.13\fnm[1]	\\
-%	SCAN		&	SCAN			&			&			&			&	4.24\fnm[1]	\\
-%	B97M-V		&	B97M-V			&			&			&			&	4.33\fnm[1]	\\
-%	PBE0		&	PBE0			&			&			&			&	4.24\fnm[1]	\\
-%	\hline
-%	\mc{5}{l}{Theoretical best estimate\fnm[2]}							&	4.32	\\
-%\end{tabular}
-%\end{ruledtabular}
-%\fnt[1]{Square gradient minimization (SGM) approach from Ref.~\onlinecite{Hait_2020} obtained with the aug-cc-pVTZ basis set. SGM is theoretically equivalent to MOM.}
-%\fnt[2]{Theoretical best estimate from Ref.~\onlinecite{Loos_2019} obtained at the (extrapolated) FCI/aug-cc-pVQZ level.}
-%\end{table}
-%%% %%% %%% %%%
-
 %%%%%%%%%%%%%%%%%%
 %%% CONCLUSION %%%
 %%%%%%%%%%%%%%%%%%
@@ -1123,7 +1092,7 @@ Excitation energies (in hartree) associated with the lowest double excitation of
 \label{sec:ccl}
 In the present article, we have discussed the construction of first-rung (\ie, local) weight-dependent exchange-correlation density-functional approximations for two-electron systems (\ce{He} and \ce{H2}) specifically designed for the computation of double excitations within GOK-DFT, a time-\textit{independent} formalism thanks to which one can extract excitation energies via the derivative of the ensemble energy with respect to the weight of each excited state.
 
-In the spirit of optimally-tuned range-separated hybrid functionals, we have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing most of the curvature of the ensemble energy).
+In the spirit of optimally-tuned range-separated hybrid functionals, we have found that the construction of a system-specific, weight-dependent local exchange functional can significantly reduce the curvature of the ensemble energy (by removing some of the curvature of the ensemble energy), and improves excitation energies.
 Although the weight-dependent correlation functional developed in this paper (eVWN5) performs systematically better than their weight-independent counterpart (VWN5), the improvement remains rather small.
 To better understand the reasons behind this, it would be particularly interesting to investigate the influence of the self-consistent procedure, 
 \ie, the variation in excitation energy when the \textit{exact} ensemble density (built with the exact individual densities) is used instead