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Manu: saving work.

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Manuscript/FarDFT.tex
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@ -534,8 +534,23 @@ First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bo
&
\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
\end{align}
\manu{no correlation functional is employed?}
In the case of \ce{H2}, the ensemble is composed by the $1\sigma_g^2$ ground state and the lowest doubly-excited state of configuration $1\sigma_u^2$, which 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
ignoring the singly-excited one. One can always argue that we explore
stationary points (and not minima) but an obvious and important question that any
referee working on GOK-DFT would ask is: How would your results
be changed if you were incorporating the single excitation in your
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 weight $0 \le \ew{} \le 1$.
\manu{Many acronyms that have not been explained are used in the
caption. The corresponding methods are also not explained. We need to
update the theory section or mention briefly in the text how the GIC
correction works.}
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 $1/2$.
@ -562,8 +577,14 @@ 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 energy (which is mostly due to the ghost-interaction error, but not only), one can easily reverse-engineer (for this particular system, geometry, and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
Second, in order to remove this spurious curvature of the ensemble
energy (which is mostly due to the ghost-interaction error, but not only
\manu{I would be more explicit. We can also cite Ref. \cite{Loos_2020}}), one can easily reverse-engineer (for this particular system, geometry, and basis set) a local exchange functional to make $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
Doing so, we have found that the present weight-dependent exchange functional (denoted as GIC-S in the following as its main purpose is to correct for the ghost-interaction error)
\manu{As mentioned in our previous work, the individual-state Hartree
energies (which have nothing to do with the ghost-interaction) also have a quadratic-in-$\ew{}$ pre-factor. I am not a big fan
of the acronym GIC-S (why S?). Something like ``curvature-corrected'' seems more
appropriate to me.}
\begin{equation}
\e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
\end{equation}
@ -611,6 +632,12 @@ 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''. GIC-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}
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.
@ -734,7 +761,9 @@ showing that the weight correction is purely linear in eVWN5 and entirely depend
As shown in Fig.~\ref{fig:Ew_H2}, the GIC-SeVWN5 is slightly less concave than its GIC-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, $\ew{} = 1/2$).
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, $\ew{} = 1/2$). \manu{Maybe we should refer to Eq.~(\ref{eq:dEdw}) for clarity.}
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
@ -749,14 +778,17 @@ They can then be obtained via GOK-DFT ensemble calculations by performing a line
\end{equation}
The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the GIC-SeVWN5 functional at zero weight are the most accurate with an improvement of $0.25$ eV as compared to GIC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
The GIC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/2$) are less satisfactory, but still remains in good agreement with FCI, with again a small improvement as compared to GIC-SVWN5.
The GIC-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 GIC-SVWN5.
It is also important to mention that the GIC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\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.
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 $1$.
\manu{That is a good point. Maybe I was too hard with you when referring
to GIC-S as ``semi-empirical''. Actually, I see here an analogy with the
optimally-tuned range-separated functionals. Maybe we should elaborate
more on this.}
%%% TABLE III %%%
\begin{table}
@ -824,7 +856,9 @@ 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 lowest excited state.
For this particular geometry, the doubly-excited state becomes the
\manu{``is the true ...''?} lowest excited state \manu{with the same symmetry as
the ground state}.
We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a GIC-S functional for this system at $\RHH = 3.7$ bohr.
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).