minor corrections from T2
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\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
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\usepackage{mathpazo,libertine}
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@ -519,7 +519,7 @@ The total energy of the ground and doubly-excited states are given by the two lo
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& = \frac{\sqrt{\pi}}{2 R} \qty[ \frac{\Gamma\qty(\frac{i+j}{2})}{\Gamma\qty(\frac{i+j+1}{2})} + \frac{ij}{4R} \frac{\Gamma\qty(\frac{i+j-1}{2})}{\Gamma\qty(\frac{i+j+2}{2})} ],
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\end{split}
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\end{equation}
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where $\omega = \theta_1 - \theta_2$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function \cite{NISTbook}, and
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where $\omega = \theta_1 - \theta_2$ is the interelectronic angle, $\Gamma(x)$ is the Gamma function, \cite{NISTbook} and
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\begin{equation}
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\psi_i(\omega) = \sin(\omega/2) \sin^{i-1}(\omega/2), \quad i=1,\ldots,M
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\end{equation}
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@ -584,7 +584,7 @@ Based on these highly-accurate calculations, one can write down, for each state,
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\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
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\end{equation}
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where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
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The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy \cite{Loos_2013a, Loos_2014a}.
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The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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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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@ -1,4 +1,4 @@
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\documentclass[aps,prl,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\documentclass[aip,jcp,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable}
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\usepackage{mathpazo,libertine}
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@ -115,67 +115,69 @@ Their accuracy is illustrated by computing single and double excitations in one-
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{Introduction.---}
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\section{Introduction}
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials \cite{ParrBook}.
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Over the last two decades, density-functional theory (DFT) \cite{Hohenberg_1964, Kohn_1965} has become the method of choice for modeling the electronic structure of large molecular systems and materials. \cite{ParrBook}
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The main reason is that, within DFT, the quantum contributions to the electronic repulsion energy --- the so-called exchange-correlation (xc) energy --- is rewritten as a functional of the electron density $\n{}(\br)$, the latter being a much simpler quantity than the electronic wave function.
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The complexity of the many-body problem is then transferred to the xc functional.
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Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} suffers, in practice, from various deficiencies \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}.
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Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1965} suffers, in practice, from various deficiencies. \cite{Woodcock_2002, Tozer_2003, Tozer_1999, Dreuw_2003, Sobolewski_2003, Dreuw_2004, Tozer_1998, Tozer_2000, Casida_1998, Casida_2000, Tapavicza_2008, Levine_2006}
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The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge.
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Another issue, which is partly connected to the previous one, is the description of electronically-excited states.
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The standard approach for modeling excited states in DFT is linear response time-dependent DFT (TDDFT) \cite{Casida}.
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The standard approach for modeling excited states in DFT is linear response time-dependent DFT (TDDFT). \cite{Casida}
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In this case, the electronic spectrum relies on the (unperturbed) ground-state KS picture, which may break down when electron correlation is strong.
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Moreover, in exact TDDFT, the xc functional is time dependent.
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The simplest and most widespread approximation in state-of-the-art electronic structure programs where TDDFT is implemented consists in neglecting memory effects \cite{Casida}.
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The simplest and most widespread approximation in state-of-the-art electronic structure programs where TDDFT is implemented consists in neglecting memory effects. \cite{Casida}
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In other words, within this so-called adiabatic approximation, the xc functional is assumed to be local in time.
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As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT \cite{Maitra_2004,Cave_2004,Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014}.
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As a result, double electronic excitations are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014}
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When affordable (i.e.~for relatively small molecules), time-independent state-averaged wave function methods \cite{Roos,Andersson_1990,Angeli_2001a,Angeli_2001b,Angeli_2002} can be employed to fix the various issues mentioned above.
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The basic idea is to describe a finite ensemble of states (ground and excited) altogether, i.e.,~with the same set of orbitals.
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Interestingly, a similar approach exists in DFT.
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Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK) \cite{Gross_1988, Gross_1988a, Oliveira_1988}, and is a generalization of Theophilou's variational principle for equi-ensembles \cite{Theophilou_1979}.
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Ensemble DFT (eDFT) was proposed at the end of the 80's by Gross, Oliveira and Kohn (GOK), \cite{Gross_1988, Gross_1988a, Oliveira_1988} and is a generalization of Theophilou's variational principle for equi-ensembles. \cite{Theophilou_1979}
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In eDFT, the (time-independent) xc functional depends explicitly on the weights assigned to the states that belong to the ensemble of interest.
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This weight dependence of the xc functional plays a crucial role in the calculation of excitation energies.
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It actually accounts for the infamous derivative discontinuity contribution to energy gaps.
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\alert{Shall we further discuss the derivative discontinuity? Why is it important and where is it coming from?}
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Despite its formal beauty and the fact that eDFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until recently \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018a,Gould_2018b,Sagredo_2018,Ayers_2018,Deur_2018a,Deur_2018b,Kraisler_2013, Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Senjean_2018,Smith_2016}.
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Despite its formal beauty and the fact that eDFT can in principle tackle near-degenerate situations and multiple excitations, it has not been given much attention until recently. \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018a,Gould_2018b,Sagredo_2018,Ayers_2018,Deur_2018a,Deur_2018b,Kraisler_2013, Kraisler_2014,Alam_2016,Alam_2017,Nagy_1998,Nagy_2001,Nagy_2005,Pastorczak_2013,Pastorczak_2014,Pribram-Jones_2014,Yang_2013a,Yang_2014,Yang_2017,Senjean_2015,Senjean_2016,Senjean_2018,Smith_2016}
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The main reason is simply the absence of density-functional approximations (DFAs) for ensembles in the literature.
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Recent works on this topic are still fundamental and exploratory, as they rely either on simple (but nontrivial) models like the Hubbard dimer \cite{Carrascal_2015,Deur_2017,Deur_2018a,Deur_2018b,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018} or on atoms for which highly accurate or exact-exchange-only calculations have been performed \cite{Yang_2014,Yang_2017}.
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Recent works on this topic are still fundamental and exploratory, as they rely either on simple (but nontrivial) models like the Hubbard dimer \cite{Carrascal_2015,Deur_2017,Deur_2018a,Deur_2018b,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018} or on atoms for which highly accurate or exact-exchange-only calculations have been performed. \cite{Yang_2014,Yang_2017}
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In both cases, the key problem, namely the design of weight-dependent DFAs for ensembles (eDFAs), remains open.
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A first step towards this goal is presented in this Letter with the ambition to turn, in the near future, eDFT into a practical computational method for modeling excited states in molecules and extended systems.
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\alert{Mention WIDFA?}
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In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}.
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Moreover, the present method relies on exact Hartree-Fock (HF) exchange, eschewing the so-called ghost interaction \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}.
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Atomic units are used throughout.\\
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In the following, the present methodology is illustrated on \emph{strict} one-dimensional (1D), spin-polarized electronic systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
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Moreover, the present method relies on exact Hartree-Fock (HF) exchange, eschewing the so-called ghost interaction. \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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Atomic units are used throughout.
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\manu{I added some key equations in the following. Will polish the all
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thing later on.\\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{Generalized KS-eDFT}
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\section{Theory}
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\label{sec:theory}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Generalized KS-eDFT}
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\label{sec:geKS}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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}
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In eDFT, the ensemble energy
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\begin{equation}
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\E{\bw}=(1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
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\E{\bw} = (1-\sum_{I>0}\ew{I})\E{(0)}+\sum_{I>0} \ew{I} \E{(I)}
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\end{equation}
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is obtained variationally as follows,
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In analogy with ground-state generalized KS-DFT, we consider the
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following partitioning of
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the ensemble Levy-Lieb functional
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In analogy with ground-state generalized KS-DFT, we consider the following partitioning of the ensemble Levy-Lieb functional
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\begin{equation}
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F^{\bw}[n]=\underset{\hat{\Gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\Gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
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\end{equation}
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\begin{equation}
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F^{\mathbf{w}}[n]=\underset{\hat{\gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
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F^{\mathbf{w}}[n]=\underset{\hat{\gamma}^{{w}}\rightarrow n}{\rm min}\left\{{\rm Tr}\left[\hat{\gamma}^{{w}}(\hat{T}+\hat{W}_{ee})\right]\right\}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{eKS for excited states.---}
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\subsection{KS-eDFT for excited states}
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\label{sec:KS-eDFT}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, we explain how to perform a self-consistent KS calculation for ensembles (eKS) in the context of excited states.
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In order to take into account both single and double excitations simultaneously, we consider a three-state ensemble including:
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@ -186,7 +188,8 @@ By definition, the ensemble energy is
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\E{\bw} = (1 - \ew{1} - \ew{2}) \E{(0)} + \ew{1} \E{(1)} + \ew{2} \E{(2)}.
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\end{equation}
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$\E{(I)}$'s are individual energies, while $\ew{1}$ and $\ew{2}$ are the weights assigned to the to the single excitation and double excitation, respectively.
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\alert{To ensure the GOK variational principle \cite{Gross_1988a}, the weights must fulfil the following conditions: $0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$. T2: I don't understand the asymmetry of the weights in this equation.}
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To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions:
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$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
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Note that, in order to extract individual energies from a single eKS calculation (see below), the weights must remain independent.
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By construction, the excitation energies are
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\begin{equation}
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@ -240,9 +243,10 @@ The self-consistent process described above is carried on until $\max \abs{\bF{\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{Extracting individual energies.---}
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\subsection{Extracting individual energies}
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\label{sec:ind_energy}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Following Deur and Fromager \cite{Deur_2018b}, it is possible to extract individual energies, $\E{(I)}$, from the ensemble energy [Eq.~\eqref{eq:Ew}] as follows:
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Following Deur and Fromager, \cite{Deur_2018b} it is possible to extract individual energies, $\E{(I)}$, from the ensemble energy [Eq.~\eqref{eq:Ew}] as follows:
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\begin{multline}
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\E{(I)} = \Tr(\bGamma{(I)} \, \bHc) + \frac{1}{2} \Tr(\bGamma{(I)} \, \bG \, \bGamma{(I)})
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\\
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@ -261,15 +265,16 @@ The only remaining piece of information to define at this stage is the weight-de
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\alert{Mention LIM?}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{Uniform electron gases.---}
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\subsection{Uniform electron gases}
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\label{sec:UEG}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas model (also known as jellium) \cite{ParrBook, Loos_2016}.
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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}.
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From this point of view, using finite uniform electron gases \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs is much more relevant \cite{Loos_2014a, Loos_2014b, Loos_2017a}.
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Most of the standard local and semi-local DFAs rely on the infinite uniform electron gas model (also known as jellium). \cite{ParrBook, Loos_2016}
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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}
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From this point of view, using finite uniform electron gases \cite{Loos_2011b, Gill_2012} (which have, like an atom, discrete energy levels) to construct eDFAs is much more relevant. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
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Here, we propose to construct a weight-dependent eDFA for the calculations of excited states in 1D systems.
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Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires \cite{Meyer_2009, Deshpande_2010}.
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Despite their simplicity, 1D models are scrutinized as paradigms for quasi-1D materials \cite{Schulz_1993, Fogler_2005a} such as carbon nanotubes \cite{Bockrath_1999, Ishii_2003, Deshpande_2008} or nanowires. \cite{Meyer_2009, Deshpande_2010}
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As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e.~a circle) \cite{Loos_2012, Loos_2013a, Loos_2014b}.
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As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (i.e.,~a circle). \cite{Loos_2012, Loos_2013a, Loos_2014b}
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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.
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As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
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This is a necessary condition for being able to model derivative discontinuities.
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@ -294,7 +299,8 @@ This is a necessary condition for being able to model derivative discontinuities
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%%% %%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{Density-functional approximations for ensembles.---}
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\subsection{Density-functional approximations for ensembles}
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\label{sec:eDFA}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The present weight-dependent eDFA is specifically designed for the calculation of excitation energies within eDFT.
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As mentioned previously, we consider a three-state ensemble including the ground state ($I=0$), the first singly-excited state ($I=1$), and the first doubly-excited state ($I=2$) of the (spin-polarized) two-electron ringium system.
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@ -307,7 +313,7 @@ Based on highly-accurate calculations (see {\SI} for additional details), one ca
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\e{c}{(I)}(\n{}) = \frac{a^{(I)}\,\n{}}{\n{} + b^{(I)} \sqrt{\n{}} + c^{(I)}},
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\end{equation}
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where $b^{(I)}$ and $c^{(I)}$ are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
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The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy \cite{Loos_2013a, Loos_2014a}.
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The value of $a^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
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Combining these, one can build a three-state weight-dependent correlation eDFA:
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\begin{equation}
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@ -316,7 +322,7 @@ Combining these, one can build a three-state weight-dependent correlation eDFA:
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{LDA-centered functional.---}
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\subsection{LDA-centered functional}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
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\begin{equation}
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@ -331,7 +337,7 @@ The local-density approximation (LDA) correlation functional,
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\begin{equation}
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\e{c}{\text{LDA}}(\n{}) = a^\text{LDA} \, F\qty[1,\frac{3}{2},c^\text{LDA}, \frac{a^\text{LDA}(1-c^\text{LDA})}{b^\text{LDA}} {\n{}}^{-1}],
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\end{equation}
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specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function \cite{NISTbook}, and
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specifically designed for 1D systems in Ref.~\onlinecite{Loos_2013} as been used, where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
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\begin{align}
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a^\text{LDA} & = - \frac{\pi^2}{360},
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&
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@ -356,7 +362,7 @@ Finally, we note that, by construction,
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\begin{equation}
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\left. \pdv{\be{c}{\bw}[\n{}]}{\ew{J}}\right|_{\n{} = \n{\bw}(\br)} = \be{c}{(J)}[\n{\bw}(\br)] - \be{c}{(0)}[\n{\bw}(\br)].
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\end{equation}
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\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors \cite{Gould_2018b}.}
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\alert{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2018b}}
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%%% FIG 1 %%%
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\begin{figure}
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\includegraphics[width=\linewidth]{EvsL_5}
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@ -381,7 +387,7 @@ Finally, we note that, by construction,
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%%% %%% %%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\textit{Computational details.---}
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\section{Computational details}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Having defined the eLDA functional in the previous section [see Eq.~\eqref{eq:eLDA}], we now turn to its validation.
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Our testing playground for the validation of the eLDA functional is the ubiquitous ``electrons in a box'' model where $\Nel$ electrons are confined in a 1D box of length $L$, a family of systems that we call $\Nel$-boxium.
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@ -389,7 +395,7 @@ In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ an
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\alert{Comment on the quality of these density: density- and functional-driven errors?}
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These inhomogeneous systems have non-trivial electronic structure properties which can be tuned by varying the box length.
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For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime \cite{Rogers_2017,Rogers_2016}.
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For small $L$, the system is weakly correlated, while strong correlation effects dominate in the large-$L$ regime. \cite{Rogers_2017,Rogers_2016}
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We use as basis functions the (orthonormal) orbitals of the one-electron system, i.e.
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\begin{equation}
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\AO{\mu}(x) =
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@ -405,13 +411,13 @@ For KS and eKS calculations, a Gauss-Legendre quadrature is employed to compute
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In order to test the present eLDA functional we have performed various sets of calculations.
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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}.
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For the single excitations, we have also performed time-dependent HF (TDHF), configuration interaction singles (CIS) and TDLDA calculations \cite{Dreuw_2005}.
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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 tested.
|
||||
|
||||
Concerning the eKS 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 $\bw = (1/3,1/3)$.
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|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\textit{Results and discussion.---}
|
||||
\section{Results and discussion}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In Fig.~\ref{fig:EvsL}, we report the error (in \%) in excitation energies (compared to FCI) for various methods and box sizes in the case of 5-boxium (i.e.~$\Nel = 5$).
|
||||
Similar graphs are obtained for the other $\Nel$ values and they can be found --- alongside the numerical data associated with each method --- in the {\SI}.
|
||||
@ -448,7 +454,7 @@ It would highlight the contribution of the derivative discontinuity.}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\textit{Concluding remarks.---}
|
||||
\section{Concluding remarks}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In the present Letter, we have constructed a weight-dependent three-state DFA in the context of ensemble DFT.
|
||||
This eDFA delivers accurate excitation energies for both single and double excitations.
|
||||
@ -457,11 +463,17 @@ Using similar ideas, a three-dimensional version \cite{Loos_2009,Loos_2009c,Loos
|
||||
|
||||
Similar to the present excited-state methodology for ensembles, one can easily design a local eDFA for the calculations of the ionization potential, electron affinity, and fundamental gap.
|
||||
This can be done by constructing DFAs for the one- and three-electron ground state systems, and combining them with the two-electron DFA in complete analogy with Eqs.~\eqref{eq:ec} and \eqref{eq:ecw}.
|
||||
However, as shown by Senjean and Fromager \cite{Senjean_2018}, one must modify the weights accordingly in order to maintain a constant density.
|
||||
However, as shown by Senjean and Fromager, \cite{Senjean_2018} one must modify the weights accordingly in order to maintain a constant density.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\textit{Acknowledgements.---}
|
||||
\section*{Supplementary material}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
See {\SI} for the additional details about the construction of the functionals, raw data and additional graphs.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
E.~F.~thanks the \textit{Agence Nationale de la Recherche} (MCFUNEX project, Grant No.~ANR-14-CE06-0014-01) for funding.
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
\bibliography{eDFT}
|
||||
|
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