Manu: IV A 2

Manuscript/FarDFT.tex

@@ -242,7 +242,7 @@ Let us consider a GOK ensemble of $\nEns$ electronic states with
 individual energies $\E{}{(0)} \le \ldots \le \E{}{(\nEns-1)}$, and
 (normalised) monotonically decreasing weights $\bw \equiv (\ew{1},\ldots,\ew{M-1})$, \ie, $\ew{0}=1-\sum_{I=1}^{\nEns-1} \ew{I}$, and $\ew{0} \ge \ldots \ge \ew{\nEns-1}$.
 The corresponding ensemble energy
-\begin{equation}
+\begin{equation}\label{eq:exp_ens_ener}
 	\E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)}
 \end{equation}
 can be obtained from the GOK variational principle
@@ -576,7 +576,8 @@ the excitation energy associated with the doubly-excited state obtained
 via the derivative of the ensemble energy \manu{with respect to $\ew{2}$
 (and taken at $\ew{2}=\ew{}=\ew{1}$)} revaries 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 $1/3$.
-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.
+Note that the exact xc ensemble functional would yield a perfectly
+linear \manu{ensemble} energy and, hence, the same value of the excitation energy independently of the ensemble weights.
 
 %%% FIG 1 %%%
 \begin{figure}
@@ -605,7 +606,7 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 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), 
+energy (which is mostly due to the ghost-interaction error \cite{Gidopoulos_2002}, but not only \cite{Loos_2020}), 
 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).
 Doing so, we have found that the following weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
 \begin{equation}
@@ -627,13 +628,49 @@ and
 \end{align}
 \end{subequations}
 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}).
+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}).\\
+
+\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.~(\ref{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.
 
 The present procedure can be related to optimally-tuned range-separated hybrid functionals, \cite{Stein_2009} where the range-separation parameters (which control the amount of short- and long-range exact exchange) are determined individually for each system by iteratively tuning them in order to enforce non-empirical conditions related to frontier orbitals (\eg, ionisation potential, electron affinity, etc) or, more importantly here, the piecewise linearity of the ensemble energy for ensemble states described by a fractional number of electrons. \cite{Stein_2009,Stein_2010,Stein_2012,Refaely-Abramson_2012}
-
+\manu{In this context, the analog of the ``IP theorem'' for the first
+(neutral) 
+excitation, for example, would read as follows [see
+Eqs.~(\ref{eq:exp_ens_ener}), (\ref{eq:diff_Ew}), and (\ref{eq:dEdw})]:
+\beq
+2\left(E^{\ew{1}=1/2}-E^{\ew{1}=0}\right)&\overset{0\leq \ew{1}\leq 1/2}{=}&\Eps{1}{\ew{1}} - \Eps{0}{\ew{1}} + \left.
+\pdv{\E{\xc}{\ew{1}}[\n{}{}]}{\ew{1}} \right|_{\n{}{} =
+\n{}{\ew{1}}}.%,\hspace{0.2cm}0\leq \ew{1}\leq 1/2. 
+\eeq
+We enforce this type of {\it exact} constraint (to the
+maximum possible extent) when optimizing the parameters in
+Eq.~(\ref{eq:Cxw}) in order to minimize the curvature of the ensemble energy.}
 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.
+\manuf{again, when reading the text and looking at the figure, I feel
+like $\ew{}$ on the $x$ axis is in fact $\ew{2}$, and $\ew{1}$ is set to
+zero. Nothing to do with the $\ew{1}=\ew{2}=\ew{}$ case expected from
+the computational details. This is very confusing.}
 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: 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.