From 4b16e188cb3f963c7fbf2c2fd16fc5a5d849f5e7 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Sun, 10 May 2020 08:50:10 +0200 Subject: [PATCH] Manu: IV A 1 --- Manuscript/FarDFT.bib | 13 +++++++++++++ Manuscript/FarDFT.tex | 24 ++++++++++++++++++++---- 2 files changed, 33 insertions(+), 4 deletions(-) diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib index 06ed905..1d0f720 100644 --- a/Manuscript/FarDFT.bib +++ b/Manuscript/FarDFT.bib @@ -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 + +}, + +} diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index ad5fd68..333f42b 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -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.