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Manu: IV A 2

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Emmanuel Fromager 2020-05-10 11:17:49 +02:00
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Manuscript/FarDFT.tex
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@ -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 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 $\ew{0} \ge \ldots \ge \ew{\nEns-1}$. (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 The corresponding ensemble energy
\begin{equation} \begin{equation}\label{eq:exp_ens_ener}
\E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)} \E{}{\bw} = \sum_{I=0}^{\nEns-1} \ew{I} \E{}{(I)}
\end{equation} \end{equation}
can be obtained from the GOK variational principle 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 $\ew{2}$ 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}). (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 $1/3$. 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 %%% %%% FIG 1 %%%
\begin{figure} \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 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). 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) Doing so, we have found that the following weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
\begin{equation} \begin{equation}
@ -627,13 +628,49 @@ and
\end{align} \end{align}
\end{subequations} \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}). 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 $0.025$. 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. 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} 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. 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. 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. 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. We shall come back to this point later on.