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Manuscript/FarDFT.bib
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@ -9282,3 +9282,16 @@ eprint = {
}
@Article{TDDFTfromager2013,
author = {Emmanuel Fromager and Stefan Knecht and Hans J. {Aa. Jensen}},
title = {Multi-configuration time-dependent density-functional theory based on range separation},
year = {2013},
journal = {J. Chem. Phys.},
volume = {138},
pages = {084101},
URL = {
https://doi.org/10.1063/1.4792199
},
}

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Manuscript/FarDFT.tex
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@ -547,18 +547,34 @@ This procedure is applied to various two-electron systems in order to extract ex
\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 LDA Slater exchange functional (\ie, no correlation functional is employed), \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 \manu{conventional (weight-independent)} LDA Slater exchange functional (\ie, no correlation functional is employed), \cite{Dirac_1930, Slater_1951} which is explicitly given by
\begin{align}
\label{eq:Slater}
\e{\ex}{\text{S}}(\n{}{}) & = \Cx{} \n{}{1/3},
&
\Cx{} & = -\frac{3}{4} \qty(\frac{3}{\pi})^{1/3}.
\end{align}
In the case of \ce{H2}, the ensemble is composed by the $\Sigma_g^+$ ground state of electronic configuration $1\sigma_g^2$, the lowest singly-excited state of the same symmetry as the ground state with configuration $1\sigma_g 2\sigma_g$, and the lowest doubly-excited state of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$, and has an autoionising resonance nature \cite{Bottcher_1974}).
In the case of \ce{H2}, the ensemble is composed by the $\Sigma_g^+$
ground state of electronic configuration $1\sigma_g^2$, the lowest
singly-excited state of the same symmetry as the ground state with
configuration $1\sigma_g 2\sigma_g$, and the lowest doubly-excited state
of configuration $1\sigma_u^2$ (which is also of symmetry $\Sigma_g^+$,
and has an autoionising resonance nature \cite{Bottcher_1974}).
\manu{As mentioned previously, the lower-lying
singly-excited states like $1\sigma_g3\sigma_g$ and
$1\sigma_g4\sigma_g$, which should in principle be part of the ensemble
(see Fig.~3 in Ref.~\onlinecite{TDDFTfromager2013}),
have been excluded, for simplicity.}
The deviation from linearity of the ensemble energy $\E{}{\ew{}}$ is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
The deviation from linearity of the ensemble energy $\E{}{\ew{}}$
\manu{[we recall that $\ew{1}=\ew{2}=\ew{}$]} is depicted in Fig.~\ref{fig:Ew_H2} as a function of weight $0 \le \ew{} \le 1/3$ (blue curve).
Because the Slater exchange functional defined in Eq.~\eqref{eq:Slater} 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 associated with the doubly-excited state obtained via the derivative of the ensemble energy varies significantly with $\ew{}$ (see blue curve in Fig.~\ref{fig:Om_H2}).
As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that
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.