Done for T2
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@ -213,12 +213,6 @@ Although the Hohenberg--Kohn theorems \cite{Hohenberg_1964} are here to provide
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However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct ground-state functionals as shown in Refs.~\onlinecite{Loos_2014a,Loos_2014b,Loos_2017a}, where the authors proposed generalised LDA exchange and correlation functionals.
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Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
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%\manu{It goes much too fast here. One should make a clear distinction
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%between your previous work with Peter on the ground-state theory. Then
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%we should refer to our latest work where GOK-DFT is applied
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%to ringium. In the present work we extend the approach to glomium. As we
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%did in our previous work we should motivate the use of FUEGs for
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%developing weight-dependent functionals.}
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Very recently, \cite{Loos_2020} two of the present authors have taken advantages of these FUEGs to construct a local, weight-dependent correlation functional specifically designed for one-dimensional many-electron systems.
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Unlike any standard functional, this first-rung functional automatically incorporates ensemble derivative contributions thanks to its natural weight dependence, \cite{Levy_1995, Perdew_1983} and has shown to deliver accurate excitation energies for both single and double excitations.
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Extending this methodology to more realistic (atomic and molecular) systems, we combine here these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA correlation functional for ensembles, which is specifically designed to compute double excitations within GOK-DFT.
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@ -271,15 +265,6 @@ Turning to GOK-DFT, the extension of the Hohenberg--Kohn theorem to ensembles al
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where $\vne(\br{})$ is the external potential and $\F{}{\bw}[\n{}{}]$ is the universal ensemble functional
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(the weight-dependent analog of the Hohenberg--Kohn universal functional for ensembles).
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In the KS formulation, this functional can be decomposed as
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%\begin{equation}
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% \F{}{\bw}[\n{}{}]
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% = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
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% = \Tr[ \hgam{\bw} \hT ] + \Tr[ \hgam{\bw} \hWee ],
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%\end{equation}
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%\manu{The above equation is wrong (the correlation is missing) and the
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%notations are ambiguous. I should also say that Tim does not like the
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%original separation into H and xc. I propose the following reformulation
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%to get everyone satisfied. I also reorganized the theory for clarity.
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\begin{equation}\label{eq:FGOK_decomp}
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\F{}{\bw}[\n{}{}]
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= \Tr{ \hgamdens{\bw} \hT }+ \E{\Ha}{}[\n{}{}]+\E{\xc}{\bw}[\n{}{}],
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@ -343,7 +328,6 @@ where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
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\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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=
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\int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}'
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%\fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})}
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+ \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}.
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\end{equation}
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The ensemble density can be obtained directly (and exactly, if no
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@ -397,70 +381,7 @@ weight-dependent density-functional exchange and correlation energies
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per particle, respectively.
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The explicit construction of these functionals is discussed at length in Sec.~\ref{sec:res}.
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%%%%%%%%%%%%%%%%
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%%%%%%% Manu: stuff that I removed from the first version %%%%%
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\iffalse%%%%
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\begin{equation}
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\begin{split}
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\label{eq:dEdw}
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\pdv{\E{}{\bw}}{\ew{I}}
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& = \E{}{(I)} - \E{}{(0)}
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\\
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& = \Eps{I}{\bw} - \Eps{0}{\bw} + \left. \pdv{\E{\xc}{\bw}[\n{}{}]}{\ew{I}} \right|_{\n{}{} = \n{}{\bw}},
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\end{split}
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\end{equation}
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where
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\begin{align}
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\label{eq:nw}
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\n{}{\bw}(\br{}) & = \sum_{I=0}^{\nEns-1} \ew{I} \n{\Det{I}{\bw}}{}(\br{}),
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\\
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\label{eq:nI}
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\n{\Det{I}{\bw}}{}(\br{}) & = \sum_{p}^{\nOrb} \ON{p}{(I)} [\MO{p}{\bw}(\br{})]^2
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\end{align}
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are the ensemble and individual one-electron densities, respectively,
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and
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\begin{equation}
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\label{eq:exc_def}
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\begin{split}
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\E{\Hxc}{\bw}[\n{}{}]
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& = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}]
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\\
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& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'
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+ \int \e{\xc}{\bw}(\n{}{}(\br{})) \n{}{}(\br{}) d\br{}
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\end{split}
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\end{equation}
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is the ensemble Hartree-exchange-correlation (Hxc) functional.
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Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$
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is the weight-dependent KS energy of state $I$, and $\eps{p}{\bw}$ is the KS orbital energy associated with $\MO{p}{\bw}(\br{})$ [$\ON{p}{(I)}$ being its occupancy for the state $I$].
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The latters are determined by solving the ensemble KS equation
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\begin{equation}
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\label{eq:eKS}
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\qty{ \hHc(\br{}) + \fdv{\E{\Hxc}{\bw}[\n{}{\bw}]}{\n{}{}(\br{})}} \MO{p}{\bw}(\br{}) = \eps{p}{\bw} \MO{p}{\bw}(\br{}),
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\end{equation}
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built with KS orbitals $\lbrace \MO{p}{\bw}(\br{}) \rbrace_{1 \le p \le
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\nOrb}$,
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where $\hHc(\br{}) = -\nabla^2/2 + \vne(\br{})$, and
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\begin{equation}
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\fdv{\E{\Hxc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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= \fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})} + \fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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\end{equation}
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is the Hxc potential, with
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\begin{subequations}
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\begin{align}
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\fdv{\E{\Ha}{}[\n{}{}]}{\n{}{}(\br{})}
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& = \frac{1}{2} \int \frac{\n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{}',
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\\
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\fdv{\E{\xc}{\bw}[\n{}{}]}{\n{}{}(\br{})}
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& = \left. \pdv{\e{\xc}{\bw{}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{}{}(\br{})} \n{}{}(\br{}) + \e{\xc}{\bw{}}(\n{}{}(\br{})).
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\end{align}
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\end{subequations}
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%%%%% end stuff removed by Manu %%%%%%
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\fi%%%%
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%\section{Some thougths illustrated with the Hubbard dimer model}
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%\section{Some thoughts illustrated with the Hubbard dimer model}
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%
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%The definition of an ensemble density functional relies on the concavity
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%of the ensemble energy with respect to the external potential. In the
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@ -560,28 +481,6 @@ Numerical quadratures are performed with the \texttt{numgrid} library \cite{numg
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This study deals only with spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
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Moreover, we restrict our study to the case of a three-state ensemble (\ie, $\nEns = 3$) where the ground state ($I=0$ with weight $1 - \ew{1} - \ew{2}$), a singly-excited state ($I=1$ with weight $\ew{1}$), as well as the lowest doubly-excited state ($I=2$ with weight $\ew{2}$) are considered.
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Assuming that the singly-excited state is lower in energy than the doubly-excited state (which is not always the case as one would notice later), one should have $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1 - \ew{2})/2$ to ensure the GOK variational principle.
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%Taking a generic two-electron system as an example, the individual one-electron densities read
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%\begin{subequations}
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%\begin{align}
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% \n{}{(0)} & = 2 \HOMO{2},
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% \\
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% \n{}{(1)} & = \HOMO{2} + \LUMO{2},
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% \\
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% \n{}{(2)} & = 2 \LUMO{2},
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%\end{align}
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%\end{subequations}
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%and they can be combined to produce the ensemble density
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%\begin{equation}
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% \label{eq:nw1w2}
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% \n{}{(\ew{1},\ew{2})}
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% = (1 - \ew{1} - \ew{2}) \n{}{(0)}
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% + \ew{1} \n{}{(1)} + \ew{2} \n{}{(2)}.
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%\end{equation}
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%For analysis purposes, Eq.~\eqref{eq:nw1w2} can be conveniently recast as a single-weight quantity
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%\begin{equation}
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% \n{}{\eW} = (1 - \eW) \n{}{(0)} + \eW \n{}{(2)},
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%\end{equation}
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%with $\eW = \ew{1}/2 + \ew{2}$ and $0 \le \eW \le 1/2$.
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Unless otherwise stated, we set the same weight to the two excited states (\ie, $\ew{} \equiv \ew{1} = \ew{2}$).
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In this case, the ensemble energy will be written as a single-weight quantity, $\E{}{\ew{}}$.
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The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiweight ensemble (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 1/3$) are considered in the following.
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@ -590,9 +489,6 @@ The zero-weight limit (\ie, $\ew{} \equiv \ew{1} = \ew{2} = 0$), and the equiwei
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Let us mention now that we will sometimes ``violate'' the GOK variational principle in order to build our weight-dependent functionals by considering the extended range of weights $0 \le \ew{2} \le 1$.
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However, let us stress that we will not compute excitation energies with these ensembles inconsistent with GOK theory.
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The pure-state limit, $\ew{1} = 0 \land \ew{2} = 1$, is nonetheless of particular interest as it is, like the (ground-state) zero-weight limit, a genuine saddle point of the restricted KS equations, and match perfectly the results obtained with the maximum overlap method (MOM) developed by Gilbert, Gill and coworkers. \cite{Gilbert_2008,Barca_2018a,Barca_2018b}
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%Although the range $1/2 < \ew{} \leq 1$ stands a little bit beyond the theory discussed previously, we look at these solutions for analysis purposes mainly.
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%These solutions of the density matrix operator functional in Eq.~\eqref{eq:min_KS_DM} correspond to stationary points rather than minimising ones.
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%Applying GOK-DFT in this range of weights would simply consists in switching the ground and excited states if true minimisations of the ensemble energy were performed.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -627,6 +523,7 @@ As anticipated, $\E{}{\ew{}}$ is far from being linear, which means that the exc
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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 $1/3$.
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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.
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%%% FIG 1 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{Ew_H2}
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\caption{
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@ -635,15 +532,18 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
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\label{fig:Ew_H2}
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}
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\end{figure}
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%%% %%% %%%
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%%% FIG 2 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{Om_H2}
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\caption{
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\ce{H2} at equilibrium bond length: error (with respect to FCI) in excitation energy (in eV) of the doubly-excited state $\Ex{}{(2)}$ as a function of $\ew{}$ for various functionals and the aug-cc-pVTZ basis set.
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\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.
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See main text for the definition of the various functional's acronyms.
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\label{fig:Om_H2}
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}
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\end{figure}
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%%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsubsection{Weight-dependent exchange functional}
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@ -652,19 +552,6 @@ Note that the exact xc ensemble functional would yield a perfectly linear energy
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Second, in order to remove some of this spurious curvature of the ensemble
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energy (which is mostly due to the ghost-interaction error, \cite{Loos_2020} but not only),
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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).
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%\manu{Something that seems important to me: you may require linearity in
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%the range $0\leq \ew{}\leq 1/2$. The excitation energy you would obtain
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%is simply the one of LIM, right? I suspect that by considering the
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%endpoint $\ew{}=1$ you change the excitation energy substantially. How
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%different are the results? At first sight, it seems like MOM gives you
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%the excitation energy that drives the
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%parameterization of the functional. Regarding the excitation energy, the
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%parameterized functional does not bring any additional information,
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%right? Maybe I miss something. Of course it gives ideas about how to
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%construct functionals. Maybe we need to elaborate more on this. For
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%example, its combination with correlation functionals (as done in the
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%following) is very interesting. It should be introduced as a kind of
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%two-step procedure.}
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Doing so, we have found that the following weight-dependent exchange functional (denoted as CC-S for ``curvature-corrected'' Slater functional)
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\begin{equation}
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\e{\ex}{\ew{},\text{CC-S}}(\n{}{}) = \Cx{\ew{}} \n{}{1/3},
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@ -687,15 +574,16 @@ and
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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}).
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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}).
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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 $0.025$.
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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 behavior on the whole range of weights in order to obtain accurate excitation energies.
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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.
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The present procedure can be related to optimally-tuned range-separated hybrid functionals, 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}
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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}
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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.
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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.
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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.
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We shall come back to this point later on.
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%%% FIG 3 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{Cxw}
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\caption{
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@ -703,6 +591,7 @@ We shall come back to this point later on.
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\label{fig:Cxw}
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}
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\end{figure}
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%%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -720,8 +609,6 @@ The combination of the Slater and VWN5 functionals (SVWN5) yield a highly convex
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Fourth, in the spirit of our recent work, \cite{Loos_2020} we design a universal, weight-dependent correlation functional.
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To build this correlation functional, we consider the singlet ground state, the first singly-excited state, as well as the first doubly-excited state of a two-electron FUEGs which consists of two electrons confined to the surface of a 3-sphere (also known as a glome). \cite{Loos_2009a,Loos_2009c,Loos_2010e}
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Notably, these three states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
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%Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalised hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}
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%As mentioned above, we confine our attention to paramagnetic (or unpolarised) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron in its ground state, thus yielding an electron density that is uniform on the 3-sphere.
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Note that the present paradigm is equivalent to the conventional IUEG model in the thermodynamic limit. \cite{Loos_2011b}
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We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
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@ -747,14 +634,13 @@ Thanks to highly-accurate calculations \cite{Loos_2009a,Loos_2009c,Loos_2010e} a
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where $a_2^{(I)}$ and $a_3^{(I)}$ are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
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The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2011b}
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Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
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Combining these, we build a three-state weight-dependent correlation functional:
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\begin{equation}
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\label{eq:ecw}
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\e{\co}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{\co}{(0)}(\n{}{}) + \ew{1} \e{\co}{(1)}(\n{}{}) + \ew{2} \e{\co}{(2)}(\n{}{})
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\e{\co}{\bw}(\n{}{}) = (1-\ew{1}-\ew{2}) \e{\co}{(0)}(\n{}{}) + \ew{1} \e{\co}{(1)}(\n{}{}) + \ew{2} \e{\co}{(2)}(\n{}{}).
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\end{equation}
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%%% FIG 1 %%%
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%%% FIG 4 %%%
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\begin{figure}
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\includegraphics[width=0.8\linewidth]{fig1}
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\caption{
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@ -871,12 +757,10 @@ which require three independent calculations, as well as the MOM excitation ener
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\end{align}
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\end{subequations}
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which also require three separate calculations at a different set of ensemble weights.
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%We point out that the MOM excitation energy is obtained by the difference in energy between the doubly-excited state at $\ew{} = 1$ and the ground state at $\ew{} = 0$.
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%They can then be obtained via GOK-DFT ensemble calculations by performing a linear interpolation between $\ew{} = 0$ and $\ew{} = 1$, \ie,
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The results gathered in Table \ref{tab:BigTab_H2} show that the GOK-DFT excitation energies obtained with the CC-SeVWN5 functional at zero weights are the most accurate with an improvement of $0.25$ eV as compared to CC-SVWN5, which is due to the ensemble derivative contribution of the eVWN5 functional.
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The CC-SeVWN5 excitation energies at equi-weights (\ie, $\ew{} = 1/3$) are less satisfactory, but still remain in good agreement with FCI.
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It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes accordingly for $\ew{} = 1$ (\textit{vide supra}).
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It is also important to mention that the CC-S functional does not alter the MOM excitation energy as the correction vanishes in this limit (\textit{vide supra}).
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\bruno{Note that by construction, for ensemble energies that are quadratic with respect to the weight (which is almost always the case in this paper), LIM and MOM can be reduced to a single calculation at $\ew{} = 1/4$ and $\ew{} = 1/2$, respectively, instead of performing an interpolation between two different calculations.}
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Finally, although we had to design a system-specific, weight-dependent exchange functional to reach such accuracy, we have not used any high-level reference data (such as FCI) to tune our functional, the only requirement being the linearity of the ensemble energy (obtained with LDA exchange) between the ghost-interaction-free pure-state limits.
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@ -947,18 +831,17 @@ Excitation energies (in eV) associated with the lowest double excitation of \ce{
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To investigate the weight dependence of the xc functional in the strong correlation regime, we now consider the \ce{H2} molecule in a stretched geometry ($\RHH = 3.7$ bohr).
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Note that, for this particular geometry, the doubly-excited state becomes the lowest excited state with the same symmetry as the ground state.
|
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Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble defined in Sec.~\ref{sec:H2}
|
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%In other words, we set the weight of the single excitation to zero (\ie, $\ew{1} = 0$) and we have thus $\ew = \ew{2}$ for the rest of this example.
|
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Although we could safely restrict ourselves to a biensemble composed by the ground state and the doubly-excited state, we eschew doing this and we still consider the same triensemble defined in Sec.~\ref{sec:H2}.
|
||||
We then follow the same protocol as in Sec.~\ref{sec:H2}, and considering again the aug-cc-pVTZ basis set, we design a CC-S functional for this system at $\RHH = 3.7$ bohr.
|
||||
It yields $\alpha = +0.019\,226$, $\beta = -0.017\,996$, and $\gamma = -0.022\,945$ [see Eq.~\eqref{eq:Cxw}].
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The weight dependence of $\Cx{\ew{}}$ is illustrated in Fig.~\ref{fig:Cxw} (green curve).
|
||||
|
||||
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 \titou{curvature ``hole''} depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
|
||||
%In other words, the curvature ``hole'' depicted in Fig.~\ref{fig:Cxw} is thus much more shallow at stretched geometry.
|
||||
Note that this linearity at $\RHH = 3.7$ bohr was also observed using weight-independent xc functionals in Ref.~\onlinecite{Senjean_2015}.
|
||||
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 eVWN5 correlation at equi-weights.
|
||||
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
|
||||
@ -968,7 +851,6 @@ For $\RHH = 3.7$ bohr, it is much harder to get an accurate estimate of the exci
|
||||
%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!!!}
|
||||
%\bruno{? I don't see it, for me HF is really bad here, especially due to its very hight dependence on the weight ! Maybe you are just referring to MOM ?}.
|
||||
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.
|
||||
Nonetheless, the excitation energy is still off by $3$ eV.
|
||||
The fundamental theoretical reason of such a poor agreement is not clear.
|
||||
@ -976,7 +858,6 @@ The fact that HF exchange yields better excitation energies hints at the effect
|
||||
|
||||
For additional comparison, we provide the excitation energy calculated by short-range multiconfigurational DFT in Ref.~\onlinecite{Senjean_2015}, using the (weight-independent) srLDA functional \cite{Toulouse_2004} and setting the range-separation parameter to $\mu = 0.4$ bohr$^{-1}$.
|
||||
The excitation energy improves by $1$ eV compared to the weight-independent SVWN5 functional, thus showing that treating the long-range part of the electron-electron repulsion by wave function theory plays a significant role.
|
||||
%\bruno{I'm a bit surprise that the ensemble correction to the correlation functional does not improve things at all... Is the derivative discontinuity, computed with this functional, almost 0 here ?}
|
||||
|
||||
%%% TABLE IV %%%
|
||||
\begin{table}
|
||||
@ -1031,13 +912,13 @@ The excitation energies associated with this double excitation computed with var
|
||||
|
||||
Before analysing the results, we would like to highlight the fact that there is a large number of singly-excited states lying in between the $1s2s$ and $2s^2$ states.
|
||||
Therefore, the present ensemble is not consistent with GOK theory.
|
||||
However, it is impossible, from a practical point of view, to take into account all these single excited states.
|
||||
However, it is impossible, from a practical point of view, to take into account all these single excitations.
|
||||
We then restrict ourselves to a triensemble keeping in mind the possible theoretical loopholes of such a choice.
|
||||
|
||||
The parameters of the CC-S weight-dependent exchange functional (computed with the smaller aug-cc-pVTZ basis) are $\alpha = +1.912\,574$, $\beta = +2.715\,267$, and $\gamma = +2.163\,422$ [see Eq.~\eqref{eq:Cxw}], the curvature of the ensemble energy being more pronounced in \ce{He} than in \ce{H2} (blue curve in Fig.~\ref{fig:Cxw}).
|
||||
The results reported in Table \ref{tab:BigTab_He} evidence this strong weight dependence of the excitation energies for HF or LDA exchange.
|
||||
|
||||
The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy for $\ew{} = 0$ is only $18$ millihartree off the reference value.
|
||||
The CC-S exchange functional attenuates significantly this dependence, and when coupled with the eVWN5 weight-dependent correlation functional, the CC-SeVWN5 excitation energy at $\ew{} = 0$ is only $18$ millihartree off the reference value.
|
||||
As in the case of \ce{H2}, the excitation energies obtained at zero-weight are more accurate than at equi-weight, while the opposite conclusion were made in Ref.~\onlinecite{Loos_2020}.
|
||||
This motivates further the importance of developing weight-dependent functionals that yields linear ensemble energies in order to get rid of the weight-dependency of the excitation energy.
|
||||
As a final comment, let us stress again that the present protocol does not rely on high-level calculations as the sole requirement for constructing the CC-S functional is the linearity of the ensemble energy with respect to the weight of the double excitation.
|
||||
|
||||
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