theory
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%% This BibTeX bibliography file was created using BibDesk.
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%% http://bibdesk.sourceforge.net/
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%% Created for Pierre-Francois Loos at 2020-03-09 08:48:49 +0100
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%% Created for Pierre-Francois Loos at 2020-03-10 16:50:31 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@ -8418,6 +8418,7 @@
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@article{Levy_2014,
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Author = {Levy, Mel and Zahariev, Federico},
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Date-Modified = {2020-03-10 16:48:46 +0100},
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Doi = {10.1103/PhysRevLett.113.113002},
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File = {/Users/loos/Zotero/storage/4C87D4GM/Levy and Zahariev - 2014 - Ground-State Energy as a Simple Sum of Orbital Ene.pdf},
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Issn = {0031-9007, 1079-7114},
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@ -8425,6 +8426,7 @@
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Language = {en},
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Month = sep,
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Number = {11},
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Pages = {113002},
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Shorttitle = {Ground-{{State Energy}} as a {{Simple Sum}} of {{Orbital Energies}} in {{Kohn}}-{{Sham Theory}}},
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Title = {Ground-{{State Energy}} as a {{Simple Sum}} of {{Orbital Energies}} in {{Kohn}}-{{Sham Theory}}: {{A Shift}} in {{Perspective}} through a {{Shift}} in {{Potential}}},
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Volume = {113},
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@ -48,6 +48,7 @@
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\newcommand{\be}[2]{\overline{\eps}_\text{#1}^{#2}}
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\newcommand{\bv}[2]{\overline{f}_\text{#1}^{#2}}
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\dn}[2]{\Delta n_{#1}^{#2}}
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\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
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\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
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@ -132,10 +133,10 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{abstract}
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We report a first generation of local, weight-dependent correlation density-functional approximations (DFAs) that incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
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These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within \titou{Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for excited states)}, and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA).
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These density-functional approximations for ensembles (eDFAs) are specially designed for the computation of single and double excitations within Gross--Oliveira--Kohn (GOK) DFT (\textit{i.e.}, eDFT for excited states), and can be seen as a natural extension of the ubiquitous local-density approximation for ensemble (eLDA).
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The resulting eDFAs, based on both finite and infinite uniform electron gas models, automatically incorporate the infamous derivative discontinuity contributions to the excitation energies through their explicit ensemble weight dependence.
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Their accuracy is illustrated by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
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\titou{Although the present weight-dependent functional has been specifically designed for one-dimensional systems, the methodology proposed here is directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.}
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Although the present weight-dependent functional has been specifically designed for one-dimensional systems, the methodology proposed here is directly applicable to the construction of weight-dependent functionals for realistic three-dimensional systems, such as molecules and solids.
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\end{abstract}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -171,7 +172,7 @@ functional of the time-dependent density $\n{}{} \equiv \n{}{}(\br,t)$ and, as
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such, it should incorporate memory effects. Standard implementations of TDDFT rely on
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the adiabatic approximation where these effects are neglected. \cite{Dreuw_2005} In other
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words, the xc functional is assumed to be local in time. \cite{Casida,Casida_2012}
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As a result, double electronic excitations \titou{(where two electrons are simultaneously promoted by a single photon)} are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
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As a result, double electronic excitations (where two electrons are simultaneously promoted by a single photon) are completely absent from the TDDFT spectrum, thus reducing further the applicability of TDDFT. \cite{Maitra_2004,Cave_2004,Mazur_2009,Romaniello_2009a,Sangalli_2011,Mazur_2011,Huix-Rotllant_2011,Elliott_2011,Maitra_2012,Sundstrom_2014,Loos_2019}
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When affordable (\ie, for relatively small molecules), time-independent
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state-averaged wave function methods
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@ -222,9 +223,9 @@ extension of the LDA, will be referred to as eLDA in the remaining of this paper
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As a proof of concept, we apply this general strategy to
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ensemble correlation energies (that we combine with
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ensemble exact exchange energies) in the particular case of
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\emph{strict} one-dimensional (1D) \trashPFL{and}
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\emph{strict} one-dimensional (1D)
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spin-polarized systems. \cite{Loos_2012, Loos_2013a, Loos_2014a, Loos_2014b}
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In other words, the Coulomb interaction used in this work \titou{corresponds} to
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In other words, the Coulomb interaction used in this work corresponds to
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particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
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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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%Early models of 1D atoms using this interaction have been used to study the effects of external fields upon Rydberg atoms \cite{Burnett_1993, Mayle_2007} and the dynamics of surface-state electrons in liquid helium. \cite{Nieto_2000, Patil_2001}
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@ -254,7 +255,7 @@ Atomic units are used throughout.
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In this section we give a brief review of GOK-DFT and discuss the
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extraction of individual energy levels \cite{Deur_2019,Fromager_2020} with a particular focus on exact
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individual exchange energies.
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Let us start by introducing the GOK ensemble energy: \cite{Gross_1988a}
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Let us start by introducing the GOK ensemble energy \cite{Gross_1988a}
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\beq\label{eq:exact_GOK_ens_ener}
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\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
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\eeq
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@ -395,7 +396,7 @@ we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
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Note that, when $\bw=0$, the ensemble correlation functional reduces to the
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conventional (ground-state) correlation functional $E_{\rm c}[n]$. As a
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result, the regular KS-DFT expression is recovered from
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Eq.~(\ref{eq:exact_ener_level_dets}) for the ground-state energy:
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Eq.~(\ref{eq:exact_ener_level_dets}) for the ground-state energy
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\beq
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\E{}{(0)}=\mel*{\Det{(0)}}{\hH}{\Det{(0)}} +
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\E{c}{}[\n{\Det{(0)}}{}],
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@ -472,7 +473,7 @@ where $\bx{}=(\omega,\br{})$ is a composite coordinate gathering spin and spatia
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}
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\fi%%%%%%%%%%%%%%%%%%%%%
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then the density matrix of the
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determinant $\Det{(K)}$ can be expressed as follows in the AO basis:
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determinant $\Det{(K)}$ can be expressed as follows in the AO basis
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\beq
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\bGam{(K)} \equiv \eGam{\mu\nu}{(K)} = \sum_{\SO{p}{} \in (K)} \cMO{\mu p}{} \cMO{\nu p}{},
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\eeq
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@ -668,7 +669,7 @@ following approximation:
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}.
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\eeq
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The minimizing ensemble density matrix in Eq.~(\ref{eq:min_with_HF_ener_fun}) fulfills the following
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stationarity condition:
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stationarity condition
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\beq\label{eq:commut_F_AO}
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\bF{\bw} \bGam{\bw} \bS = \bS \bGam{\bw} \bF{\bw},
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\eeq
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@ -812,7 +813,7 @@ Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
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applicable to not-necessarily spin polarized and real (higher-dimensional) systems.
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As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
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ensemble density matrix into the HF interaction energy functional
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introduces unphysical \textit{ghost interaction} errors \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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introduces unphysical \textit{ghost interaction} errors \titou{(GIE)} \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
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\beq\label{eq:WHF}
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\begin{split}
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@ -828,17 +829,15 @@ These errors are essentially removed when evaluating the individual energy
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levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.
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Turning to the density-functional ensemble correlation energy, the
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following ensemble local density \textit{approximation} (eLDA) will be employed:
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following ensemble local-density \textit{approximation} (eLDA) will be employed
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\beq\label{eq:eLDA_corr_fun}
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\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
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\eeq
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\manu{
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where the ensemble correlation energy per particle
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\beq\label{eq:decomp_ens_correner_per_part}
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\e{c}{\bw}(\n{}{})=\sum_{K\geq 0}w_K\be{c}{(K)}(\n{}{})
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\eeq
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}
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is \textit{weight dependent}.
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is \titou{explicitly} \textit{weight dependent}.
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As shown in Sec.~\ref{sec:eDFA}, the latter can be constructed
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from a finite uniform electron gas model.
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%\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
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@ -853,40 +852,48 @@ reads
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%Manu, would it be useful to add this equation and the corresponding text?
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%I think it is useful for the discussion later on when we talk about the different contributions to the excitation energies.
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%This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
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Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within KS-eLDA:
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\beq\label{eq:EI-eLDA}
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Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our \titou{final expression of the KS-eLDA energy level}
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\titou{\beq\label{eq:EI-eLDA}
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\begin{split}
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\E{{eLDA}}{(I)}
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& =
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=
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\E{HF}{(I)}
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\\
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%\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
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& + \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
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+ \Xi_\text{c}^{(I)}
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+ \Upsilon_\text{c}^{(I)},
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\end{split}
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\eeq}
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where
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\beq\label{eq:ind_HF-like_ener}
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\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
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\eeq
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is the analog for ground and excited states (within an ensemble) of the HF energy, \titou{and
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\begin{gather}
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\begin{split}
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\Xi_\text{c}^{(I)}
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& = \int \e{c}{\bw}(\n{\bGam{\bw}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
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\\
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&
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+ \int \n{\bGam{\bw}}{}(\br{}) \qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{\bw}}{}(\br{}) ]
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\left. \pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}} \right|_{\n{}{} = \n{\bGam{\bw}}{}(\br{})} d\br{}
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\\
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& + \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
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\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{},
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\end{split}
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\eeq
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where
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\beq\label{eq:ind_HF-like_ener}
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\E{HF}{(I)}=\Tr[\bGam{(I)} \bh] + \frac{1}{2} \Tr[\bGam{(I)} \bG \bGam{(I)}]
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\eeq
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is the analog for ground and excited states (within an ensemble) of the HF energy.
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\\
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\Upsilon_\text{c}^{(I)}
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= \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
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\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}.
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\end{gather}}
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If, for analysis purposes, we Taylor expand the density-functional
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correlation contributions
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around the $I$th KS state density
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$\n{\bGam{(I)}}{}(\br{})$, the sum of
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the second and third terms on the right-hand side
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the \titou{second term} on the right-hand side
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of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
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$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
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\titou{$\dn{\bGam{\bw}}{(I)}(\br{}) = \n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$}:
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\beq\label{eq:Taylor_exp_ind_corr_ener_eLDA}
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\int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
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+\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
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\titou{\Xi_\text{c}^{(I)}}
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= \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
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+ \titou{\order{[\dn{\bGam{\bw}}{(I)}(\br{})]^2}}.
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\eeq
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Therefore, it can be identified as
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an individual-density-functional correlation energy where the density-functional
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@ -898,46 +905,40 @@ Let us stress that, to the best of our knowledge, eLDA is the first
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density-functional approximation that incorporates ensemble weight
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dependencies explicitly, thus allowing for the description of derivative
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discontinuities [see Eq.~\eqref{eq:excited_ener_level_gs_lim} and the
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comment that follows] {\it via} the last term on the right-hand side
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of Eq.~\eqref{eq:EI-eLDA}. \manu{According to the decomposition of
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comment that follows] {\it via} the \titou{third} on the right-hand side
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of Eq.~\eqref{eq:EI-eLDA}. According to the decomposition of
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the ensemble
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correlation energy per particle in Eq.
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\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast as
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follows:
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\beq
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\begin{split}
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&
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\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
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\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
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\\
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&=
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\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
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\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{}))
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-
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\be{c}{(0)}(\n{\bGam{\bw}}{}(\br{}))
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\Big)
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d\br{}
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\\
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&=\int
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\Big(\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
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\eqref{eq:decomp_ens_correner_per_part}, the latter can be recast
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\begin{equation}
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\titou{\Upsilon_\text{c}^{(I)}}
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%&=
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%\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
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%\Big(\be{c}{(K)}(\n{\bGam{\bw}}{}(\br{}))
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%-
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%\be{c}{(0)}(\n{\bGam{\bw}}{}(\br{}))
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%\Big)
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%d\br{}
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%\\
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=\int
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\qty[\be{c}{(I)}(\n{\bGam{\bw}}{}(\br{}))
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-
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\e{c}{\bw}(\n{\bGam{\bw}}{}(\br{}))
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\Big)\,\n{\bGam{\bw}}{}(\br{})
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] \n{\bGam{\bw}}{}(\br{})
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d\br{},
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%\sum_{K>0}\delta_{IK}\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})}
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\end{split}
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\eeq
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\end{equation}
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thus leading to the following Taylor expansion through first order in
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$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
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\titou{$\dn{\bGam{\bw}}{(I)}(\br{})$}
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%$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
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\beq\label{eq:Taylor_exp_DDisc_term}
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\begin{split}
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&
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\int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
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\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
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\\
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\titou{\Upsilon_\text{c}^{(I)}}
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%& = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
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% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
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%\\
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&=-\int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
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\\
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&+\int \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
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+\int \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
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\\
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&+\int \Bigg[
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\n{\bGam{(I)}}{}(\br{})
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@ -950,24 +951,22 @@ $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
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\\
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&+\be{c}{(I)}(\n{\bGam{(I)}}{}(\br{}))
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-
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\e{c}{\bw}(\n{\bGam{(I)}}{}(\br{}))\Bigg]\times
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\Big(\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})\Big)
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\e{c}{\bw}(\n{\bGam{(I)}}{}(\br{}))\Bigg]
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\dn{\bGam{\bw}}{(I)}(\br{})
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d\br{}
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\\
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&
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+\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
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+ \titou{\order{[\dn{\bGam{\bw}}{(I)}(\br{})]^2}}.
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\end{split}
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\eeq
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As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the
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role of the correlation ensemble derivative [last term on the right-hand side of Eq.
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\eqref{eq:EI-eLDA}] is, through zeroth order, to substitute the expected
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role of the correlation ensemble derivative \titou{$\Upsilon_\text{c}^{(I)}$} \trashPFL{[last term on the right-hand side of Eq.
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\eqref{eq:EI-eLDA}]} is, through zeroth order, to substitute the expected
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individual correlation energy per particle for the ensemble one.
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}
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\manu{
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Let us finally note that, while the weighted sum of the
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individual KS-eLDA energy levels delivers a \manu{\it ghost-interaction-corrected (GIC)} version of
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the KS-eLDA ensemble energy:
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individual KS-eLDA energy levels delivers a \textit{ghost-interaction-corrected} (GIC) version of
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the KS-eLDA ensemble energy, \ie,
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\beq\label{eq:Ew-eLDA}
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\begin{split}
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\E{GIC-eLDA}{\bw}&=\sum_{I\geq0}\ew{I}\E{{eLDA}}{(I)}
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@ -982,15 +981,15 @@ expressions in Eq. \eqref{eq:EI-eLDA} simply reads
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\beq\label{eq:Om-eLDA}
|
||||
\begin{split}
|
||||
\Ex{eLDA}{(I)}
|
||||
=&
|
||||
&=
|
||||
\Ex{HF}{(I)}
|
||||
+ \int
|
||||
\\
|
||||
&+ \int
|
||||
\qty[\e{c}{{\bw}}(\n{}{})+n\pdv{\e{c}{{\bw}}(\n{}{})}{\n{}{}}]
|
||||
_{\n{}{} =
|
||||
\n{\bGam{\bw}}{}(\br{})}
|
||||
\\
|
||||
&\times\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{}
|
||||
+ \DD{c}{(I)},
|
||||
\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{}
|
||||
\\ & + \DD{c}{(I)},
|
||||
\end{split}
|
||||
\eeq
|
||||
where the HF-like excitation energies $\Ex{HF}{(I)} = \E{HF}{(I)} -
|
||||
@ -1001,7 +1000,6 @@ where the HF-like excitation energies $\Ex{HF}{(I)} = \E{HF}{(I)} -
|
||||
\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
||||
\eeq
|
||||
is the eLDA correlation ensemble derivative contribution to the $I$th excitation energy.
|
||||
}
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Density-functional approximations for ensembles}
|
||||
@ -1013,7 +1011,7 @@ is the eLDA correlation ensemble derivative contribution to the $I$th excitation
|
||||
\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}
|
||||
Most of the standard local and semi-local \titou{density-functional approximations (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,
|
||||
@ -1046,9 +1044,9 @@ The present weight-dependent eDFA is specifically designed for the
|
||||
calculation of excited-state energies within GOK-DFT.
|
||||
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.
|
||||
\titou{To ensure the GOK variational principle, \cite{Gross_1988a} the
|
||||
To ensure the GOK variational principle, \cite{Gross_1988a} the
|
||||
triensemble weights must fulfil the following conditions: \cite{Deur_2019}
|
||||
$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.}
|
||||
$0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$, where $\ew{1}$ and $\ew{2}$ are the weights associated with the singly- and doubly-excited states, respectively.
|
||||
All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$, where $R$ is the radius of the ring on which 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.
|
||||
@ -1209,7 +1207,7 @@ In order to test the present eLDA functional we perform various sets of calculat
|
||||
To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
|
||||
For the single excitations, we also perform time-dependent LDA (TDLDA)
|
||||
calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}].
|
||||
\titou{Its Tamm-Dancoff approximation version (TDA-TDLDA) is also considered.} \cite{Dreuw_2005}
|
||||
Its Tamm-Dancoff approximation version (TDA-TDLDA) is also considered. \cite{Dreuw_2005}
|
||||
|
||||
Concerning the ensemble calculations, two sets of weight are tested: the zero-weight
|
||||
(ground-state) limit where $\bw = (0,0)$ and the
|
||||
@ -1225,28 +1223,25 @@ equi-tri-ensemble (or equal-weight state-averaged) limit where $\bw = (1/3,1/3)$
|
||||
\includegraphics[width=\linewidth]{EvsW_n5}
|
||||
\caption{
|
||||
\label{fig:EvsW}
|
||||
Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
|
||||
Deviation from linearity of the weight-dependent KS-eLDA ensemble energy $\E{eLDA}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost-interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
|
||||
}
|
||||
\end{figure*}
|
||||
%%% %%% %%%
|
||||
|
||||
First, we discuss the linearity of the \manu{computed} (approximate)
|
||||
ensemble \manu{energies}.
|
||||
First, we discuss the linearity of the computed (approximate)
|
||||
ensemble energies.
|
||||
To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
|
||||
The deviation from linearity of the three-state ensemble energy
|
||||
$\E{}{(\ew{1},\ew{2})}$ (\ie, the deviation from the
|
||||
\trashEF{hypothetical} \manu{linearly-interpolated} ensemble energy
|
||||
\trashEF{obtained by linear interpolation}) is represented
|
||||
in Fig.~\ref{fig:EvsW} as a function of \trashEF{both} $\ew{1}$
|
||||
\trashEF{and} \manu{or} $\ew{2}$ while
|
||||
linearly-interpolated ensemble energy) is represented
|
||||
in Fig.~\ref{fig:EvsW} as a function of $\ew{1}$ or $\ew{2}$ while
|
||||
fulfilling the restrictions on the ensemble weights to ensure the GOK
|
||||
variational principle [\ie, $0 \le \ew{2} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
|
||||
To illustrate the magnitude of the ghost interaction error (GIE), we report the KS-eLDA ensemble energy with and without ghost interaction correction (GIC) as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
|
||||
To illustrate the magnitude of the \titou{GIE}, we report the KS-eLDA ensemble energy with and without \titou{GIC} as explained above {[see Eqs.~\eqref{eq:Ew-GIC-eLDA} and \eqref{eq:Ew-eLDA}]}.
|
||||
As one can see in Fig.~\ref{fig:EvsW}, without GIC, the
|
||||
ensemble energy becomes less and less linear as $L$
|
||||
gets larger, while the GIC \trashEF{makes the ensemble energy almost
|
||||
linear} reduces \manu{the curvature of the ensemble energy
|
||||
drastically}.
|
||||
gets larger, while the GIC reduces the curvature of the ensemble energy
|
||||
drastically.
|
||||
%\manu{This
|
||||
%is a strong statement I am not sure about. The nature of the excitation
|
||||
%should also be invoked I guess (charge transfer or not, etc ...). If we look at the GIE:
|
||||
@ -1295,7 +1290,6 @@ ensemble KS orbitals in the presence of GIE {[see Eqs.~\eqref{eq:min_with_HF_ene
|
||||
%%% %%% %%%
|
||||
|
||||
Figure \ref{fig:EIvsW} reports the behavior of the three KS-eLDA individual energies as functions of the weights.
|
||||
\manu{
|
||||
Unlike in the exact theory, we do not obtain
|
||||
straight horizontal lines when plotting these
|
||||
energies, which is in agreement with
|
||||
@ -1315,7 +1309,6 @@ excitation energy is reduced as the correlation increases. On the other hand, sw
|
||||
systematically enhances the weight dependence, due to the lowering of the
|
||||
ground-state energy in this case, as $w_2$ increases.
|
||||
The reverse is observed for the second excitation energy.
|
||||
}
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure}
|
||||
@ -1331,7 +1324,7 @@ The reverse is observed for the second excitation energy.
|
||||
Figure \ref{fig:EvsL} reports the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
|
||||
Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
|
||||
For small $L$, the single and double excitations can be labeled as
|
||||
``pure'', \manu{as revealed by a thorough analysis of the FCI wavefunctions}.
|
||||
``pure'', as revealed by a thorough analysis of the FCI wavefunctions.
|
||||
In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
|
||||
However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
|
||||
In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
|
||||
@ -1405,8 +1398,7 @@ the same quality as the one obtained in the linear response formalism
|
||||
excitation energy only deviates
|
||||
from the FCI value by a few tenth of percent.
|
||||
Moreover, we note that, in the strong correlation regime
|
||||
(left\manu{Manu: you mean right?} graph of
|
||||
Fig.~\ref{fig:EvsN}), the single excitation
|
||||
(\titou{right} graph of Fig.~\ref{fig:EvsN}), the single excitation
|
||||
energy obtained at the equiensemble KS-eLDA level remains in good
|
||||
agreement with FCI and is much more accurate than the TDLDA and TDA-TDLDA excitation energies which can deviate by up to $60 \%$.
|
||||
This also applies to the double excitation, the discrepancy
|
||||
@ -1431,11 +1423,9 @@ electrons.
|
||||
%%% %%% %%%
|
||||
|
||||
It is also interesting to investigate the influence of the
|
||||
\manu{correlation ensemble derivative contribution} $\DD{c}{(I)}$
|
||||
\manu{to the $I$th excitation energy} [see Eq.~\eqref{eq:DD-eLDA}].
|
||||
\manu{In
|
||||
our case, both single ($I=1$) and double ($I=2$) excitations are
|
||||
considered}.
|
||||
correlation ensemble derivative contribution $\DD{c}{(I)}$
|
||||
to the $I$th excitation energy [see Eq.~\eqref{eq:DD-eLDA}].
|
||||
In our case, both single ($I=1$) and double ($I=2$) excitations are considered.
|
||||
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$
|
||||
on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
|
||||
%\manu{Manu: there is something I do not understand. If you want to
|
||||
@ -1474,12 +1464,13 @@ still substantial in the strongly correlated limit of the equi
|
||||
triensemble for the double
|
||||
excitation.
|
||||
}
|
||||
\titou{Note also that, in our case, the second term in
|
||||
Note also that, in our case, the second term in
|
||||
Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation
|
||||
potential and the density difference between ground and excited states,
|
||||
has a negligible effect on the excitation energies.}\manu{Manu: Is this
|
||||
something that you checked but did not show? It feels like we can see
|
||||
this in the Figure but we cannot, right?}
|
||||
has a negligible effect on the excitation energies \titou{(results not shown)}.
|
||||
%\manu{Manu: Is this
|
||||
%something that you checked but did not show? It feels like we can see
|
||||
%this in the Figure but we cannot, right?}
|
||||
%\manu{Manu: well, we
|
||||
%would need the exact derivative value to draw such a conclusion. We can
|
||||
%only speculate. Let us first see how important the contribution in
|
||||
@ -1503,8 +1494,8 @@ undoubtedly show that, even in the strong correlation regime, the
|
||||
ensemble correlation derivative has a small impact on the double
|
||||
excitations \manu{Manu: well, the impact is larger than the one on the single
|
||||
excitation in the equiensemble} with a slight tendency of worsening the excitation energies
|
||||
in the case of equal weights, \manu{as the number of electrons
|
||||
increases. It has} a rather large influence on the single
|
||||
in the case of equal weights, as the number of electrons
|
||||
increases. It has a rather large influence on the single
|
||||
excitation energies obtained in the zero-weight limit, showing once
|
||||
again that the usage of equal weights has the benefit of significantly reducing the magnitude of the ensemble correlation derivative.
|
||||
|
||||
@ -1530,12 +1521,12 @@ from the finite uniform electron gas eLDA is
|
||||
combined with eLDA, delivers accurate excitation energies for both
|
||||
single and double excitations, especially when an equiensemble is used.
|
||||
In the latter case, the same weights are assigned to each state belonging to the ensemble.
|
||||
\titou{The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
|
||||
The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
|
||||
We have observed that, although the ensemble correlation discontinuity has a
|
||||
non-negligible effect on the excitation energies (especially for the
|
||||
single excitations), its magnitude can be significantly reduced by
|
||||
performing equiweight calculations instead of zero-weight
|
||||
calculations.}
|
||||
calculations.
|
||||
|
||||
Let us finally stress that the present methodology can be extended
|
||||
straightforwardly to other types of ensembles like, for example, the
|
||||
@ -1554,7 +1545,7 @@ See {\SI} for the additional details about the construction of the functionals,
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\begin{acknowledgements}
|
||||
\titou{The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.}
|
||||
The authors thank Bruno Senjean and Clotilde Marut for stimulating discussions.
|
||||
This work has been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}
|
||||
\end{acknowledgements}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
10128
Notebooks/eDFT_FUEG.nb
10128
Notebooks/eDFT_FUEG.nb
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