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@ -16,7 +16,7 @@
\newcommand{\alert}[1]{\textcolor{red}{#1}}
\usepackage[normalem]{ulem}
\newcommand{\titou}[1]{\textcolor{red}{#1}}
\newcommand{\manu}[1]{\textcolor{blue}{Manu: #1}}
\newcommand{\manu}[1]{\textcolor{blue}{#1}}
\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
\newcommand{\trashEF}[1]{\textcolor{blue}{\sout{#1}}}
@ -831,26 +831,63 @@ In this case, the individual energies are simply defined as
Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas (IUEG) model (also known as jellium). \cite{ParrBook, Loos_2016}
One major drawback of the jellium paradigm, when it comes to develop eDFAs, is that the ground and excited states are not easily accessible like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
Moreover, because the IUEG model is a metal, it is gapless, which means that both the fundamental and optical gaps are zero.
From this point of view, using finite UEGs (FUEGs), \cite{Loos_2011b, Gill_2012} which have, like an atom, discrete energy levels and non-zero gaps, to construct eDFAs can be seen as more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
However, an obvious drawback of using FUEGs is that the resulting eDFA will inexorably depend on the number of electrons that composed this FUEG (see below).
Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems by combining these FUEGs with the usual IUEG to construct a weigh-dependent LDA functional for ensembles (eLDA).
From this point of view, using finite UEGs (FUEGs), \cite{Loos_2011b,
Gill_2012} which have, like an atom, discrete energy levels and non-zero
gaps, can be seen as more relevant in this context. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
However, an obvious drawback of using FUEGs is that the resulting eDFA
will inexorably depend on the number of electrons in the FUEG (see below).
Here, we propose to construct a weight-dependent eLDA for the
calculations of excited states in 1D systems by combining FUEGs with the
usual IUEG.
As a FUEG, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b}
The most appealing feature of ringium regarding the development of functionals in the context of eDFT is the fact that both ground- and excited-state densities are uniform.
The most appealing feature of ringium regarding the development of
functionals in the context of eDFT is the fact that both ground- and
excited-state densities are uniform, and therefore {\it equal}.
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
This is a necessary condition for being able to model derivative discontinuities.
This is a necessary condition for being able to model the ensemble
correlation derivatives with respect to the weights [last term
on the right-hand side of Eq.~(\ref{eq:exact_ener_level_dets})].
Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous IUEG paradigm. \cite{Loos_2013,Loos_2013a}
\manu{Let us stress that, in a FUEG like ringium, the interacting and
noninteracting densities match individually for all the states within the
ensemble
[these densities are all equal to the uniform density], which means that
so-called density-driven correlation
effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.}
Here, we will consider the most simple ringium system featuring electronic correlation effects, \ie, the two-electron ringium model.
The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
The present weight-dependent eDFA is specifically designed for the
calculation of excited-state energies within GOK-DFT.
In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
(i) the ground state ($I=0$), (ii) the first singly-excited state ($I=1$), and (iii) the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$ is the radius of the ring where the electrons are confined.
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
Generalization to a larger number of states is straightforward and is left for future work.
To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions:
$0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$ [or $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
To ensure the GOK variational principle, \cite{Gross_1988a} the
tri-ensemble weights must fulfil the following conditions:
\manu{\titou{$0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$}. The
constraint in \titou{red} is wrong. If $\ew{2}=0$, you should be allowed
to consider an equi-bi-ensemble
for which $\ew{1}=1/2$. This possibility is excluded with your
inequalities. The correct constraints are given in Ref.~\cite{Deur_2019}
and are the ones you also mentioned, \ie, $0 \le \ew{2} \le 1/3$ and
$\ew{2} \le \ew{1} \le (1-\ew{2})/2$.}
\manu{
Just in case, starting from
\beq
\begin{split}
0\leq \ew{2}\leq \ew{1}\leq (1-\ew{1}-\ew{2})
\\
\end{split}
\eeq
we obtain
\beq
0\leq \ew{2}\leq \ew{1}\leq (1-\ew{2})/2
\eeq
which implies $\ew{2}\leq(1-\ew{2})/2$ or, equivalently, $\ew{2}\leq
1/3$.
}
%%% TABLE 1 %%%
\begin{table*}
\caption{