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Manuscript/FarDFT.tex
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@ -188,7 +188,7 @@ In other words, memory effects are absent from the xc functional which is assume
Third and more importantly in the present context, a major issue of
TD-DFT actually originates directly from the choice of the (ground-state) xc functional, and more specifically, the possible (not to say likely) substantial variations in the quality of the excitation energies for two different choices of xc functionals.
Because of its popularity, approximate TD-DFT has been studied extensively by the community, and some researchers have quickly unveiled various theoretical and practical deficiencies.
Because of its popularity, approximate TD-DFT has been studied \alert{extensively}, and some researchers have quickly unveiled various theoretical and practical deficiencies.
For example, TD-DFT has problems with charge-transfer \cite{Tozer_1999,Dreuw_2003,Sobolewski_2003,Dreuw_2004,Maitra_2017} and Rydberg \cite{Tozer_1998,Tozer_2000,Casida_1998,Casida_2000,Tozer_2003} excited states (the excitation energies are usually drastically underestimated) due to the wrong asymptotic behaviour of the semi-local xc functional.
The development of range-separated hybrids provides an effective solution to this problem. \cite{Tawada_2004,Yanai_2004}
From a practical point of view, the TD-DFT xc kernel is usually considered as static instead of being frequency dependent.
@ -295,7 +295,7 @@ Their dependence on the density is determined from the ensemble density constrai
\end{equation}
Note that the original decomposition \cite{Gross_1988b} shown in Eq.~\eqref{eq:FGOK_decomp}, where the
conventional (weight-independent) Hartree functional
\beq
\beq \label{eq:Hartree}
\E{\Ha}{}[\n{}{}]=\frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'
\eeq
is separated
@ -390,24 +390,24 @@ We will also adopt the usual decomposition, and write down the weight-dependent
where $\e{\ex}{\bw{}}(\n{}{})$ and $\e{\co}{\bw{}}(\n{}{})$ are the
weight-dependent density-functional exchange and correlation energies
per particle, respectively.
\manu{As shown in Sec.~\ref{subsubsec:weight-dep_corr_func}, the weight
\alert{As shown in Sec.~\ref{subsubsec:weight-dep_corr_func}, the weight
dependence of the correlation energy can be extracted from a FUEG model. In order to make the resulting weight-dependent
correlation functional truly universal, \ie~
correlation functional truly universal, \ie,
independent on the number of electrons in the FUEG, one could use the
curvature of the Fermi hole~\cite{Loos_2017a} as an additional variable in the
density-functional approximation. The development of such a
generalized correlation eLDA is left for future work. Even though a similar strategy could be applied
generalised correlation eLDA is left for future work. Even though a similar strategy could be applied
to the weight-dependent exchange part, we
explore in the present work a different path where the
(system-dependent) exchange functional
parameterization relies on the ensemble energy linearity
parameterisation relies on the ensemble energy linearity
constraint (see Sec.~\ref{subsubsec:weight-dep_x_fun}). Finally, let us
stress that, in order to further
improve the description of the ensemble correlation energy, a
post-treatment of the recently
revealed density-driven
correlations~\cite{Gould_2019,Gould_2019_insights,gould_2020,Fromager_2020} [which, by construction, are absent
from FUEGs] might be necessary. An orbital-dependent correction derived
correlations~\cite{Gould_2019,Gould_2019_insights,gould_2020,Fromager_2020} (which, by construction, are absent
from FUEGs) might be necessary. An orbital-dependent correction derived
in Ref.~\onlinecite{Fromager_2020} might be
used for that purpose. Work is currently in progress in this
direction.\\
@ -505,7 +505,7 @@ linear ensemble energy and, hence, the same value of the excitation energy indep
\includegraphics[width=\linewidth]{fig1}
\caption{
\ce{H2} at equilibrium bond length: deviation from linearity of the ensemble energy $\E{}{\ew{}}$ (in hartree) as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
See main text for the definition of the various functionals' acronyms.
See main text for the definition of the various \alert{functionals'} acronyms.
\label{fig:Ew_H2}
}
\end{figure}
@ -516,7 +516,7 @@ linear ensemble energy and, hence, the same value of the excitation energy indep
\includegraphics[width=\linewidth]{fig2}
\caption{
\ce{H2} at equilibrium bond length: error (with respect to FCI) in the excitation energy $\Ex{}{(2)}$ (in eV) associated with the doubly-excited state as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
See main text for the definition of the various functionals' acronyms.
See main text for the definition of the various \alert{functionals'} acronyms.
\label{fig:Om_H2}
}
\end{figure}
@ -553,24 +553,6 @@ makes the ensemble energy $\E{}{(0,\ew{2})}$ almost perfectly linear (by constru
It also allows to ``flatten the curve'' making 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}).\\
%
%\manuf{One point is not clear to me at all. If I understood correctly,
%the optimization of $\alpha$, $\beta$, and $\gamma$ is done for
%$\ew{1}=0$. So, once the optimisation is done, we have a coefficient
%$\Cx{\ew{2}}$ that is a function of $\ew{2}$. Then, how do you obtain
%a coefficient $\Cx{\ew{}}$ that is supposed to describe a {\it
%different} ensemble defined as $\ew{1}=\ew{2}=\ew{}$ (it says in the
%computational details that, ultimately, this is what we are looking at)? Did you just
%replace $\ew{2}$ by $\ew{}$? This should be clarified. Another point: in
%order to apply Eq.~\eqref{eq:dEdw} for computing excitation energies,
%you need $\ew{1}$ and $\ew{2}$ to be independent variables before
%differentiating (and taking the value of the derivatives at
%$\ew{1}=\ew{2}=\ew{}$). I do not see how you can do this (and generate
%the results in Fig.~\ref{fig:Om_H2}) if the only ensemble functional you
%have depends on $\ew{}$ and not on both $\ew{1}$ and $\ew{2}$. Regarding
%Fig.~\ref{fig:Om_H2}, I would suspect
%that you took $\ew{1}=0$, which is questionable and not clear at all from
%the text.}
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 $0.025$.
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 behaviour in the whole range of weights in order to obtain accurate excitation energies.
@ -591,7 +573,8 @@ We enforce this type of \textit{exact} constraint (to the maximum possible exten
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 $\Cx{\ew{2}}$ reduces to $\Cx{}$ in these two limits.
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 $\ew{2} = 0$, the behaviour of Eq.~\eqref{eq:Cxw} is linear: this is the main feature that 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.
\titou{Nonetheless, the CC-S functional also includes quadratic terms in order to compensate the spurious curvature of the ensemble energy originating, mainly, from the Hartree term [see Eq.~\eqref{eq:Hartree}].}
%We shall come back to this point later on.
%%% FIG 3 %%%
\begin{figure}
@ -608,7 +591,7 @@ We shall come back to this point later on.
\subsubsection{Weight-independent correlation functional}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Third, we include correlation effects via the conventional VWN5 local correlation functional. \cite{Vosko_1980}
Third, we \titou{include} correlation effects via the conventional 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 (weight-independent) Slater and VWN5 functionals (SVWN5) yield a highly convex ensemble energy (green curve in Fig.~\ref{fig:Ew_H2}), while the combination of CC-S and VWN5 (CC-SVWN5) exhibit a smaller curvature and improved excitation energies (red curve in Figs.~\ref{fig:Ew_H2} and \ref{fig:Om_H2}), especially at small weights, where the CC-SVWN5 excitation energy is almost spot on.
@ -758,10 +741,6 @@ In particular, we report the excitation energies obtained with GOK-DFT
in the zero-weight limit (\ie, $\ew{} = 0$) and for equi-weights (\ie, $\ew{} = 1/3$).
These excitation energies are computed using
Eq.~\eqref{eq:dEdw}.
%\manuf{OK but, again, how do you compute the exchange ensemble
%derivative for both excited states when it seems like the functional in
%Eqs.~\eqref{eq:ensemble_Slater_func} and \eqref{eq:Cxw}
%only depends on $\ew{}$ rather than $\ew{1}$ AND $\ew{2}$.}
For comparison, we also report results obtained with the linear interpolation method (LIM). \cite{Senjean_2015,Senjean_2016}
The latter simply consists in extracting the excitation energies (which are
@ -774,16 +753,6 @@ follows:
\Ex{\LIM}{(2)} & = 3 \qty[\E{}{\bw{}=(1/3,1/3)} - \E{}{\bw{}=(1/2,0)}] + \frac{1}{2} \Ex{\LIM}{(1)}. \label{eq:LIM2}
\end{align}
\end{subequations}
%\manu{
%$\frac{1}{2}\Ex{\LIM}{(1)}=\frac{1}{2}\left(E_1-E_0\right)$\\
%$\E{}{\bw{}=(1/3,1/3)}=\frac{1}{3}\left(E_0+E_1+E_2\right)$\\
%$\E{}{\bw{}=(1/2,0)}=\frac{1}{2}\left(E_0+E_1\right)$\\
%$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
%\E{}{\bw{}=(1/2,0)}]=-\frac{1}{2}\left(E_0+E_1\right)+E_2$
%\\
%$3 \qty[\E{}{\bw{}=(1/3,1/3)} -
%\E{}{\bw{}=(1/2,0)}]+\frac{1}{2} \Ex{\LIM}{(1)}=E_2-E_0$
%}\\
For a general expression with multiple (and possibly degenerate) states, we refer the reader to Eq.~(106) of Ref.~\onlinecite{Senjean_2015}, where LIM is shown to interpolate linearly the ensemble energy between equi-ensembles.
Note that two calculations are needed to get the first LIM excitation energy, with an additional equi-ensemble calculation for each higher excitation energy.
@ -896,10 +865,6 @@ We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again
It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
The weight dependence of $\Cx{\ew{2}}$ is illustrated in
Fig.~\ref{fig:Cxw} (green curve).
%\manuf{Again, it would be nice to say
%explicitly if you construct a functional, function of $\ew{1}$ and
%$\ew{2}$ (how then) or just $\ew{}$ (how to compute the separate
%derivatives then?)}
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 curvature ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
@ -907,15 +872,6 @@ Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-ind
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}
For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the excitation energy, the closest match being reached with HF exchange and VWN5 correlation at equi-weights.
%\manu{We did not mention HF exchange neither in the theory section nor
%in the computational details. We should be clear about this. Is this an
%ad-hoc correction, like in our previous work on ringium? Is HF exchange
%used for the full ensemble energy (i.e. the HF interaction energy is
%computed with the ensemble density matrix and therefore with
%ghost-interaction errors) or for
%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!!!}
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.
As a direct consequence of this linearity, LIM and MOM
do not provide any noticeable improvement on the excitation

46
Response_Letter/Response_Letter.tex
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@ -64,11 +64,10 @@ density-driven correlations [62,92-94] (which, by construction, are absent from
\item
{Not clear what density is used in equation 21: From the discussion that follows this appears to not be the ensemble density (which is weight dependent and I would expect it to be used here) but the density only for the ground state Slater determinant. The authors should explain why this is so. }
\\
\alert{The density $n$ used in Eq.~(21), (9), (10) can be any density and does not represent any specific density.
In the case of Eq.~(21), we simply present the well-known Dirac-exchange density functional and, by definition of a density functional, does not need to specify in its formulation to which density it is applied but only that it is a mathematical object applying to any density $n(r)$.
Of course, when we will use this functional or any other one in
our work we will surely apply it to the {\it minimizing} ensemble density
$n^w(r)$ [see Eqs. (5) and (11)] and the notation will be carefully modified accordingly.}
\alert{The density $n$ used in Eqs.~(21), (9), (10) can be any density, and, mathematically, there is no need to specify its origin.
In the case of Eq.~(21), we simply define the well-known Dirac exchange functional.
By definition of a density functional, we do not need to specify in its formulation to which density it is applied but only that it is a mathematical object applying to any density $n(r)$.
Of course, when using this functional (or any other ones) in our work we surely apply it to the {\it minimizing} ensemble density $n^w(r)$ [see Eqs. (5) and (11)] and the notation is carefully modified accordingly.}
\item
{Change "Third, we add up correlation effects" to "Third, we include correlation effects"}
@ -84,33 +83,28 @@ $n^w(r)$ [see Eqs. (5) and (11)] and the notation will be carefully modified acc
\item
{They need to be a bit more consistent with their notation as in equation 9 and elsewhere "$n(r)$" should be the ensemble (weight-dependent) density "$n^w(r)$".
I don?t believe they defined "$n(r)$" in the paper so I don?t know which density it represents. }
I don't believe they defined "$n(r)$" in the paper so I don?t know which density it represents. }
\\
\alert{See our response to 4.}
\alert{See our response to 3.}
\item
{Even if it sounds trivial, they should explain why the exact xc functional should have linear dependence in the excitation energies as a function of the weight value.}
\\
\alert{GOK variational principle states that the expectation value of the ensemble energy admits/possesses a lower bond which is linear with respect to each of the ensemble-weights $w_i$ and is the exact ensemble energy of the studied system (equation 1).
Moreover, by construction, one can easily see that the slope of the exact ensemble energy with respect to a specific weight $w_i$ corresponds to the excitation energy of the system defined between the ground state and the ith-excited state associated to this specific weight (equation 4).
\alert{As clearly stated in the original manuscript, GOK variational principle states that the expectation value of the ensemble energy admits a lower bond which is linear with respect to each of the ensemble weight $w_I$ and is the exact ensemble energy of the studied system [Eq.~(1)].
Moreover, by construction, one can easily see that the slope of the exact ensemble energy with respect to a specific weight $w_I$ corresponds to the excitation energy of the system defined between the ground state and the $I$th excited state associated to this specific weight [Eq.~(4)].
It is important that the reader keeps in mind that the exact excitation energies are based on pure-state energies and, therefore, do not depend on the weights of the ensemble.
In practice, the ensemble energy is rarely w-linear (linear in w ?) because of the use of approximate xc-functionals.
Indeed, by inserting the ensemble density in the Hartree interaction functional (equation 9), it introduces spurious quadratic curvature with respect to the weight in the ensemble energy.
Some of those terms are responsible of the unphysical phenomenon called ghost-interaction errors.
Therefore, the ensemble-Khon-Sham gap obtained at the end of the ensemble-HF-calculation is, somehow, "weight-contaminated" and doesn't possess the right weight-dependence.
(two first terms of the right-hand side of equation16)
By taking its first derivative with regard to the weight, the xc-functional is expected to compensate those parasite-quadratic terms in order to retrieve the linear behavior of the exact ensemble energy and one can understand that only a weight-dependant xc-functional could do so.
At the best of my knowledge, I cannot see any reason why the xc-functional should be w-linear.
The important idea is that the linearity must be in the ensemble energy but the main constraint on the xc-functional should be that it is weight-dependant.
We emphasize that only the exact ensemble-xc-functional would have the ideal weight-dependency that would make the corresponding ensemble energy reproduce perfectly the linear behavior of the exact ensemble energy and lead to weight-independant excitation energies, that is exact excitation energies.
The use of an approximate weight-dependant xc-functional could reduce the ensemble energy curvature and give less weight-dependant excitation energies but it is reasonable to admit that it also could make things worse it the weight-dependency of the functional is poorly chosen.
That is why the construction of "good" weight-dependant
xc-functionals is a really challenging matter in eDFT.\\
As a final comment, we insist after Eq. (28) on the fact that, in the
exact theory, the xc ensemble density functional has no reason to vary (for a fixed density $n$)
linearly with the ensemble weights. We refer to Ref. 92 where exact
expressions for the individual xc energies are derived.}
In practice, the ensemble energy is rarely linear in $w$ because of the approximate nature of the xc functionals.
Indeed, by inserting the ensemble density in the Hartree interaction functional [Eq.~(9)], one introduces, in the ensemble energy, spurious quadratic curvature with respect to the weight.
Some of those terms are responsible of the unphysical phenomenon called ghost interaction, as explained in the manuscript.
Therefore, the ensemble Khon-Sham gap is, somehow, "weight contaminated" and does not possess the correct weight dependence [see first two terms of the right-hand side of Eq.~(16)].
By taking its derivative with respect to the weight, the weight-dependent xc functional is expected to compensate those spurious quadratic terms in order to retrieve a linear behavior of the exact ensemble energy: only a weight-dependent xc functional can achieve such a feat.
In other words, the xc functional does not have to be linear with respect to $w$.
We have clarified this point in the revised manuscript.
We emphasize that the exact ensemble xc functional has the ideal weight dependency, and would make the corresponding ensemble energy perfectly linear, hence leading to weight-independent excitation energies.
As shown in the present manuscript, the use of an approximate weight-dependent xc functional reduces the ensemble energy curvature as well as the variation of the excitation energies with respect to $w$.
This illustrates why the construction of reliable weight-dependent xc functionals is a challenging task in eDFT.
As a final comment, we insist after Eq.~(28) on the fact that, in the exact theory, the xc ensemble density functional has no reason to vary (for a fixed density $n$) linearly with the ensemble weights.
We refer to Ref.~92 where exact expressions for the individual xc energies are derived.}
\end{enumerate}
\end{letter}