minor corrections

Manuscript/eDFT.tex

@@ -63,15 +63,15 @@
 \newcommand{\EFCI}{E_\text{FCI}}
 
 % matrices/operator
-\newcommand{\br}[1]{\bm{r}_{#1}}
-\newcommand{\bw}{{\bm{w}}}
-\newcommand{\bG}{\bm{G}}
-\newcommand{\bS}{\bm{S}}
-\newcommand{\bGam}[1]{\bm{\Gamma}^{#1}}
-\newcommand{\bgam}[1]{\bm{\gamma}^{#1}}
+\newcommand{\br}[1]{\boldsymbol{r}_{#1}}
+\newcommand{\bw}{{\boldsymbol{w}}}
+\newcommand{\bG}{\boldsymbol{G}}
+\newcommand{\bS}{\boldsymbol{S}}
+\newcommand{\bGam}[1]{\boldsymbol{\Gamma}^{#1}}
+\newcommand{\bgam}[1]{\boldsymbol{\gamma}^{#1}}
 \newcommand{\opGam}[1]{\hat{\Gamma}^{#1}}
-\newcommand{\bh}{\bm{h}}
-\newcommand{\bF}[1]{\bm{F}^{#1}}
+\newcommand{\bh}{\boldsymbol{h}}
+\newcommand{\bF}[1]{\boldsymbol{F}^{#1}}
 \newcommand{\Ex}[1]{\Omega^{#1}}
 
 
@@ -110,9 +110,9 @@
 %%% added by Manu %%%
 \newcommand{\beq}{\begin{equation}}
 \newcommand{\eeq}{\end{equation}}
-\newcommand{\bmk}{\bm{\kappa}} % orbital rotation vector
-\newcommand{\bmg}{\bm{\Gamma}} % orbital rotation vector
-\newcommand{\bxi}{\bm{\xi}} 
+\newcommand{\bmk}{\boldsymbol{\kappa}} % orbital rotation vector
+\newcommand{\bmg}{\boldsymbol{\Gamma}} % orbital rotation vector
+\newcommand{\bxi}{\boldsymbol{\xi}} 
 \newcommand{\bfx}{{\bf{x}}}
 \newcommand{\bfr}{{\bf{r}}}
 \DeclareMathOperator*{\argmax}{arg\,max}
@@ -702,28 +702,34 @@ Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} le
 \label{sec:eDFA}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Paradigm}
+\label{sec:paradigm}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
 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 cannot be easily identified like in a molecule. \cite{Gill_2012, Loos_2012, Loos_2014a, Loos_2014b, Agboola_2015, Loos_2017a}
+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, one of the drawbacks of using FUEGs is that the resulting eDFA will inexorably depend on the number of electrons (see below).
+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).
 
-As a FUEG, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
+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. 
 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. 
+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}
+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.
 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 $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
 
-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.
-
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Weight-dependent correlation functional}
 \label{sec:Ec}
@@ -765,8 +771,8 @@ Combining these, one can build a three-state weight-dependent correlation eDFA:
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{LDA-centered functional}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc functionals can be applied to any electronic system.
-The two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the FUEG.
+One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc density functionals can be applied to any electronic system.
+Obviously, the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the FUEG.
 However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
 The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional).
 Following this simple strategy, which is further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows:
@@ -830,14 +836,15 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
 	\end{cases}
 \end{equation} 
 with $ \mu = 1,\ldots,\Nbas$ and $\Nbas = 30$ for all calculations.
-For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold has been set to $\tau = 10^{-5}$.
-For KS-DFT and KS-eDFT calculations, a Gauss-Legendre quadrature is employed to compute numerical integrals.
+For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ been set to $10^{-5}$.
+For KS-DFT and KS-eDFT calculations, a Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form.
 
 In order to test the present eLDA functional we have performed various sets of calculations.
 To get reference excitation energies for both the single and double excitations, we have performed full configuration interaction (FCI) calculations with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
 For the single excitations, we have also performed time-dependent HF (TDHF), configuration interaction singles (CIS) and TDLDA calculations. \cite{Dreuw_2005}
 For TDLDA, the validity of the Tamm-Dancoff approximation (TDA) has been also investigated.
 Concerning the KS-eDFT calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
+In order to test the influence of correlation effects on excitation energies, we have also performed ensemble HF (labeled as eHF) calculations.
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Results and discussion}
@@ -850,7 +857,7 @@ When the box gets larger, they start to deviate.
 For the single excitation, TDHF is extremely accurate over the whole range of $L$ values, while CIS is slightly less accurate and starts to overestimate the excitation energy by a few percent at $L=8\pi$.
 TDLDA yields larger errors at large $L$ by underestimating the excitation energies.
 TDA-TDLDA slightly corrects this trend thanks to error compensation. 
-Concerning the eLDA functional, our results clearly evidences that the equi-weights [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
+Concerning the eLDA functional, our results clearly evidence that the equi-weights [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
 This is especially true for the single excitation which is significantly improved by using state-averaged weights.
 The effect on the double excitation is less pronounced.
 Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.