From f7c8f7f6b54a85c716644119c657cb0ac9a4bc35 Mon Sep 17 00:00:00 2001 From: Emmanuel Fromager Date: Tue, 25 Feb 2020 16:16:13 +0100 Subject: [PATCH] Manu: polished III A --- Manuscript/eDFT.tex | 57 +++++++++++++++++++++++++++++++++++++-------- 1 file changed, 47 insertions(+), 10 deletions(-) diff --git a/Manuscript/eDFT.tex b/Manuscript/eDFT.tex index 524228d..3ab3184 100644 --- a/Manuscript/eDFT.tex +++ b/Manuscript/eDFT.tex @@ -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{