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easy corrections in Results

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Pierre-Francois Loos 2020-04-26 13:11:19 +02:00
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@ -554,6 +554,7 @@ The self-consistent GOK-DFT calculations [see Eqs.~\eqref{eq:eKS} and \eqref{eq:
For more details about the self-consistent implementation of GOK-DFT, we refer the interested reader to Ref.~\onlinecite{Loos_2020} where additional technical details can be found.
For all calculations, we use the aug-cc-pVXZ (X = D, T, Q, and 5) Dunning family of atomic basis sets. \cite{Dunning_1989,Kendall_1992,Woon_1994}
Numerical quadratures are performed with the \texttt{numgrid} library \cite{numgrid} using 194 angular points (Lebedev grid) and a radial precision of $10^{-6}$. \cite{Becke_1988b,Lindh_2001}
This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Moreover, we restrict our study to the case of a two-state ensemble (\ie, $\nEns = 2$) where both the ground state ($I=0$ with weight $1 - \ew{}$) and the first doubly-excited state ($I=1$ with weight $\ew{}$) are considered.
\titou{To ensure the GOK variational principle, one should then have $0 \le \ew{} \le 1/2$.
@ -578,13 +579,12 @@ Applying GOK-DFT in this range of weights would simply consists in switching the
\subsubsection{Weight-independent exchange functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent Slater-Dirac (LDA) local exchange functional, \cite{Dirac_1930, Slater_1951} which is explicitly given by
First, we compute the ensemble energy of the \ce{H2} molecule (at equilibrium bond length, \ie, $\RHH = 1.4$ bohr) using the aug-cc-pVTZ basis set and the weight-independent \titou{LDA Slater exchange functional (\ie, no correlation functional is employed)}, \cite{Dirac_1930, Slater_1951} which is explicitly given by
\begin{align}
\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
&
\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
@ -597,10 +597,6 @@ 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 $1/2$.
@ -614,6 +610,7 @@ However, as a sanity check, we have tried to introduce the single excitations as
\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 $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
\titou{See main text for the definition of the various functionals.}
\label{fig:Ew_H2}
}
\end{figure}
@ -632,8 +629,8 @@ However, as a sanity check, we have tried to introduce the single excitations as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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 $0 \le \ew{} \le 1$.
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 $\E{}{\ew{}}$ as linear as possible for $0 \le \ew{} \le 1$.
\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
@ -647,14 +644,9 @@ construct functionals. Maybe we need to elaborate more on this. For
example, its combination with correlation functionals (as done in the
following) is very interesting. It should be introduced as a kind of
two-step procedure.}
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'' or
``linearized'' (?) seems more
appropriate to me.}
Doing so, we have found that the present weight-dependent exchange functional \titou{(denoted as CC-S for ``curvature-corrected'' Slater functional)}
\begin{equation}
\e{\ex}{\ew{},\text{GIC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
\e{\ex}{\ew{},\text{CC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
\end{equation}
with
\begin{equation}
@ -694,14 +686,14 @@ We shall come back to this point later on.
Third, we add up correlation effects via the VWN5 local correlation functional. \cite{Vosko_1980}
For the sake of clarity, the explicit expression of the VWN5 functional is not reported here but it can be found in Ref.~\onlinecite{Vosko_1980}.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of GIC-S and VWN5 (GIC-SVWN5) exhibit a small curvature (the ensemble energy is slightly concave) and improved excitation energies, especially at small weights, where the GIC-SVWN5 excitation energy is almost spot on.
The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (as shown in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a small curvature (the ensemble energy is slightly concave) and improved excitation energies, especially at small weights, where the CC-SVWN5 excitation energy is almost spot on.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
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.}
@ -826,12 +818,13 @@ We note also that, by construction, we have
\end{equation}
showing that the weight correction is purely linear in eVWN5 and entirely dependent on the FUEG model.
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}).
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, $\ew{} = 1/2$). \manu{Maybe we should refer to Eq.~\eqref{eq:dEdw} for clarity.}
(\ie, $\ew{} = 1/2$).
\titou{These excitation energies can 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
@ -845,18 +838,16 @@ They can then be obtained via GOK-DFT ensemble calculations by performing a line
\Ex{\MOM}{} = \E{}{\ew{}=1} - \E{}{\ew{}=0}.
\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 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}).
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}).
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, it makes me think about the
optimally-tuned range-separated functionals. Maybe we could elaborate
more on this.}
\titou{The present protocol can be related to optimally-tuned range-separated hybrid functionals. T2: more to come.}
%%% TABLE III %%%
\begin{table}
@ -894,15 +885,15 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
& & aug-cc-pVTZ & 21.39 & 27.98 & 24.55 & 27.34 \\
& & aug-cc-pVQZ & 21.38 & 27.97 & 24.55 & 27.34 \\
\\
GIC-S & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\
CC-S & & aug-cc-pVDZ & 26.83 & 26.51 & 26.53 & 26.60 \\
& & aug-cc-pVTZ & 26.88 & 26.59 & 26.61 & 26.67 \\
& & aug-cc-pVQZ & 26.82 & 26.60 & 26.62 & 26.67 \\
\\
GIC-S & VWN5 & aug-cc-pVDZ & 28.54 & 26.94 & 27.48 & 27.10 \\
CC-S & VWN5 & aug-cc-pVDZ & 28.54 & 26.94 & 27.48 & 27.10 \\
& & aug-cc-pVTZ & 28.66 & 27.00 & 27.56 & 27.17 \\
& & aug-cc-pVQZ & 28.64 & 27.00 & 27.56 & 27.17 \\
\\
GIC-S & eVWN5 & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\
CC-S & eVWN5 & aug-cc-pVDZ & 28.78 & 27.10 & 27.56 & 27.27 \\
& & aug-cc-pVTZ & 28.90 & 27.16 & 27.64 & 27.34 \\
& & aug-cc-pVQZ & 28.89 & 27.16 & 27.65 & 27.34 \\
\hline
@ -924,16 +915,15 @@ 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
\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.
\titou{For this particular geometry, the doubly-excited state becomes the
lowest excited state 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 CC-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).
One clearly sees that the correction brought by GIC-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.
In other words, the ghost-interaction ``hole'' \manu{see my previous
comments on curvature} depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
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.
In other words, the \titou{curvature ``hole''} depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-independent xc functionals in Ref.~\onlinecite{Senjean_2015}.
Table \ref{tab:BigTab_H2st} reports, for the aug-cc-pVTZ basis set (which delivers basis set converged results), the same set of calculations as in Table \ref{tab:BigTab_H2}.
As a reference value, we computed a FCI/aug-cc-pV5Z excitation energy of $8.69$ eV, which compares well with previous studies. \cite{Senjean_2015}
@ -948,7 +938,7 @@ individual energies (that you state-average then), like in our previous
work. I guess the latter option is what you did. We need to explain more
what we do!!!}
%\bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}.
As expected from the linearity of the ensemble energy, the GIC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
As expected from the linearity of the ensemble energy, the CC-S functional coupled or not with a correlation functional yield extremely stable excitation energies as a function of the weight, with only a few tenths of eV difference between the zero- and equi-weights limits.
Nonetheless, the excitation energy is still off by $3$ eV.
The fundamental theoretical reason of such a poor agreement is not clear.
The fact that HF exchange yields better excitation energies hints at the effect of self-interaction error.
@ -975,9 +965,9 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
S & & 5.31 & 5.60 & 5.46 & 5.56 \\
S & VWN5 & 5.34 & 5.57 & 5.46 & 5.52 \\
S & eVWN5 & 5.53 & 5.76 & 5.56 & 5.72 \\
GIC-S & & 5.55 & 5.56 & 5.56 & 5.56 \\
GIC-S & VWN5 & 5.58 & 5.53 & 5.57 & 5.52 \\
GIC-S & eVWN5 & 5.77 & 5.72 & 5.66 & 5.72 \\
CC-S & & 5.55 & 5.56 & 5.56 & 5.56 \\
CC-S & VWN5 & 5.58 & 5.53 & 5.57 & 5.52 \\
CC-S & eVWN5 & 5.77 & 5.72 & 5.66 & 5.72 \\
\hline
B & LYP & & & & 5.28 \\
B3 & LYP & & & & 5.55 \\
@ -1012,16 +1002,15 @@ Nonetheless, it can be nicely described with a Gaussian basis set containing eno
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 GIC-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}).
In other words, the ghost-interaction hole \manu{see my previous
comments on curvature} is deeper.
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}).
In other words, the \titou{curvature hole} is deeper.
The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
The GIC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the GIC-SeVWN5 excitation energy for $\ew{} = 0$ is only $8$ millihartree (or $0.22$ eV) off the reference value.
%\bruno{But also with GIC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...}
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.
%\bruno{But also with CC-SVWN5, as in the rest of this article, so one could wonder about the usefulness of the eVWN5 functional...}
As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}.
This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the GIC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
%%% TABLE V %%%
@ -1042,9 +1031,9 @@ Excitation energies (in hartree) associated with the lowest double excitation of
S & & 1.062 & 2.056 & 1.547 & 2.030 \\
S & VWN5 & 1.163 & 2.104 & 1.612 & 2.079 \\
S & eVWN5 & 1.174 & 2.108 & 1.615 & 2.083 \\
GIC-S & & 1.996 & 2.044 & 1.988 & 2.030 \\
GIC-S & VWN5 & 2.107 & 2.097 & 2.060 & 2.079 \\
GIC-S & eVWN5 & 2.118 & 2.100 & 2.063 & 2.083 \\
CC-S & & 1.996 & 2.044 & 1.988 & 2.030 \\
CC-S & VWN5 & 2.107 & 2.097 & 2.060 & 2.079 \\
CC-S & eVWN5 & 2.118 & 2.100 & 2.063 & 2.083 \\
\hline
B & LYP & & & & 2.147 \\
B3 & LYP & & & & 2.150 \\
@ -1072,8 +1061,8 @@ Excitation energies (in hartree) associated with the lowest double excitation of
% HF & VWN5 & & & & \\
% S & & 1.72 & 4.00 & 2.86 & 3.99 \\
% S & VWN5 & & & & \\
% GIC-S & & 3.99 & 3.99 & 3.99 & 3.99 \\
% GIC-S & VWN5 & 4.05 & 4.03 & 4.04 & 4.03 \\
% 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] \\

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