Done with Theory
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@ -154,12 +154,12 @@ Despite its success, the standard Kohn-Sham (KS) formulation of DFT \cite{Kohn_1
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The description of strongly multiconfigurational ground states (often referred to as ``strong correlation problem'') still remains a challenge. \cite{Gori-Giorgi_2010,Gagliardi_2017}
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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 (TD-DFT). \cite{Runge_1984,Casida}
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The standard approach for modeling excited states in DFT is linear-response time-dependent DFT (TDDFT). \cite{Runge_1984,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 TD-DFT, 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 TD-DFT is implemented consists in neglecting memory effects. \cite{Casida}
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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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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 TD-DFT spectrum, thus reducing further the applicability of TD-DFT. \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 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 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, \ie, with the same set of orbitals.
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@ -175,6 +175,8 @@ The main reason is simply the absence of density-functional approximations (DFAs
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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 the present manuscript with the ambition to turn, in the forthcoming future, GOK-DFT into a practical computational method for modeling excited states in molecules and extended systems.
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The present eDFA is specially designed for the computation of single and double excitations within GOK-DFT, and can be seen as a natural extension of the ubiquitous local-density approximation (LDA) for ensemble.
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Consequently, we will refer to this eDFA as eLDA in the remaining of this paper.
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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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In other words, the Coulomb interaction used in this work describes particles which are \emph{strictly} restricted to move within a 1D sub-space of three-dimensional space.
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@ -184,7 +186,7 @@ This description of 1D systems also has interesting connections with the exotic
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In these extreme conditions, where magnetic effects compete with Coulombic forces, entirely new bonding paradigms emerge. \cite{Schmelcher_1990, Schmelcher_1997, Tellgren_2008, Tellgren_2009, Lange_2012, Schmelcher_2012, Boblest_2014, Stopkowicz_2015}
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The paper is organized as follows.
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Section \ref{sec:eDFT} introduces the equations behind GOK-DFT, as well as the different approximations that we apply in order to make the present scheme practical.
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Section \ref{sec:eDFT} introduces the equations behind GOK-DFT.
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In Sec.~\ref{sec:eDFA}, we detail the construction of the weight-dependent local correlation functional specially designed for the computation of single and double excitations within eDFT.
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Computational details needed to reproduce the results of the present work are reported in Sec.~\ref{sec:comp_details}.
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In Sec.~\ref{sec:res}, we illustrate the accuracy of the present eDFA by computing single and double excitations in one-dimensional many-electron systems in the weak, intermediate and strong correlation regimes.
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@ -200,7 +202,7 @@ Atomic units are used throughout.
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\subsection{GOK-DFT}\label{subsec:gokdft}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The GOK ensemble energy~\cite{Gross_1988a,Oliveira_1988,Gross_1988b} is defined as follows:
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The GOK ensemble energy~\cite{Gross_1988a,Oliveira_1988,Gross_1988b} is defined as
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\beq
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\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
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\eeq
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@ -209,12 +211,15 @@ where the $K$th energy level $\E{}{(K)}$ [$K=0$ refers to the ground state] is t
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\hh = \sum_{i=1}^\nEl \qty[ -\frac{1}{2} \nabla_{i}^2 + \vne(\br{i}) ]
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\eeq
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is the one-electron operator describing kinetic and nuclear attraction energies, and $\hat{W}_{\rm ee}$ is the electron repulsion operator.
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The (positive) ensemble weights $\ew{K}$ decrease with increasing index $K$. They are normalized, \ie,
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The (positive) ensemble weights $\ew{K}$ decrease with increasing index $K$.
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They are normalized, \ie,
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\beq
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\ew{0} = 1 - \sum_{K>0} \ew{K},
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\eeq
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so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldots )$ assigned to the excited states can vary independently.
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For simplicity we will assume in the following that the energies are not degenerate. Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states~\cite{}. In GOK-DFT, the ensemble energy is determined variationally as follows:
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For simplicity we will assume in the following that the energies are not degenerate.
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Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states~\cite{}.
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In GOK-DFT, the ensemble energy is determined variationally as follows:
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\beq\label{eq:var_ener_gokdft}
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\E{}{\bw}
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= \min_{\opGam{\bw}}
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@ -226,12 +231,13 @@ where $\Tr$ denotes the trace and the trial ensemble density matrix operator rea
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\beq
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\opGam{\bw}=\sum_{K \geq 0} \ew{K} \dyad*{\Det{(K)}}.
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\eeq
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The determinants (or configuration state functions) $\Det{(K)}$ are all constructed from the same set of (ensemble Kohn--Sham) orbitals that is optimized variationally and the trial ensemble density is simply the weighted sum of the individual densities:
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The determinants (or configuration state functions) $\Det{(K)}$ are all constructed from the same set of ensemble KS orbitals that is variationally optimized.
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The trial ensemble density is simply the weighted sum of the individual densities, \ie,
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\beq\label{eq:KS_ens_density}
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\n{\opGam{\bw}}{}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K)}}{}(\br{}).
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\eeq
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As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange and correlation energies are described with density functionals that are \textit{weight dependent}.
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We focus here on the (exact) Hx part which is defined as follows:
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As readily seen from Eq.~\eqref{eq:var_ener_gokdft}, both Hartree-exchange (Hx) and correlation (c) energies are described with density functionals that are \textit{weight dependent}.
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We focus here on the (exact) Hx part which is defined as
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\beq\label{eq:exact_ens_Hx}
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\E{Hx}{\bw}[\n{}{}]=\sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}[\n{}{}]}{\hWee}{\Det{(K)}[\n{}{}]},
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\eeq
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@ -241,11 +247,12 @@ where the KS wavefunctions fulfill the ensemble density constraint
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\eeq
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The (approximate) description of the correlation part is discussed in Sec.~\ref{sec:eDFA}.
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In practice, one is not much interested in ensemble energies but rather in excitation energies or individual energy levels (for geometry optimizations, for example). The latter can be extracted exactly as follows~\cite{}:
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In practice, the ensemble energy is not the most interesting quantity, and one is more concerned with excitation energies or individual energy levels (for geometry optimizations, for example).
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The latter can be extracted exactly as follows~\cite{}
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\beq\label{eq:indiv_ener_from_ens}
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\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \pdv{\E{}{\bw}}{\ew{K}},
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\eeq
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where, according to the {\it variational} ensemble energy expression of Eq.~\eqref{eq:var_ener_gokdft}, the derivative with respect to $\ew{K}$ can be evaluated from the minimizing KS wavefunctions $\Det{(K)} = \Det{(K),\bw}$ as follows:
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where, according to the {\it variational} ensemble energy expression of Eq.~\eqref{eq:var_ener_gokdft}, the derivative with respect to $\ew{K}$ can be evaluated from the minimizing KS wavefunctions $\Det{(K)} = \Det{(K),\bw}$, \ie,
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\beq\label{eq:deriv_Ew_wk}
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\begin{split}
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\pdv{\E{}{\bw}}{\ew{K}}
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@ -259,7 +266,7 @@ where, according to the {\it variational} ensemble energy expression of Eq.~\eqr
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\Bigg\}_{\n{}{} = \n{\opGam{\bw}}{}}.
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\end{split}
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\eeq
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The Hx contribution to Eq.~\eqref{eq:deriv_Ew_wk} can be rewritten as follows:
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The Hx contribution from Eq.~\eqref{eq:deriv_Ew_wk} can be recast as
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\beq\label{eq:_deriv_wk_Hx}
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\left.
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\pdv{}{\xi_K} \qty(\E{Hx}{\bxi} [\n{}{\bxi,\bxi}]
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@ -270,16 +277,16 @@ where $\bxi \equiv (\xi_1,\xi_2,\ldots,\xi_K,\ldots)$ and
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\beq
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\n{}{\bw,\bxi}(\br{}) = \sum_{K\geq 0} \ew{K} \n{\Det{(K),\bxi}}{}(\br{}).
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\eeq
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Since, for given ensemble weight $\bw$ and $\bxi$ values, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes
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Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes
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from the exact expression in Eq.~\ref{eq:exact_ens_Hx} that
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\beq
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\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}
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\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
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\eeq
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with $\xi_0 = 1 - \sum_{K>0}\xi_K$, and
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\beq
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\E{Hx}{\bw}[\n{}{\bw,\bxi}] = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
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\E{Hx}{\bw}[\n{}{\bw,\bxi}] = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}.
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\eeq
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thus leading, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_Hx}, to the simplified expression
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This yields, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_Hx}, the simplified expression
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\beq\label{eq:deriv_Ew_wk_simplified}
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\begin{split}
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\pdv{\E{}{\bw}}{\ew{K}}
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@ -294,7 +301,7 @@ thus leading, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_H
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}_{\n{}{} = \n{\opGam{\bw}}{}} d\br{}.
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\end{split}
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\eeq
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Since the ensemble energy can be evaluated as follows:
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Since the ensemble energy can be evaluated as
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\beq
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\E{}{\bw} = \sum_{K \geq 0} \ew{K} \mel*{\Det{(K)}}{\hH}{\Det{(K)}} + \E{c}{\bw}[\n{\opGam{\bw}}{}],
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\eeq
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@ -363,14 +370,14 @@ and the ensemble density as follows:
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\bGam{\bw}
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= \sum_{K\geq 0} \ew{K} \bGam{(K)}
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\equiv \eGam{\mu\nu}{\bw}
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= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)}
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= \sum_{K\geq 0} \ew{K} \eGam{\mu\nu}{(K)},
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\eeq
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and
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\beq
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\n{\bGam{\bw}}{}(\br{}) = \sum_{\mu\nu} \AO{\mu}(\br{}) \eGam{\mu\nu}{\bw} \AO{\nu}(\br{}),
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\eeq
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respectively. The exact energy level expression in Eq.~\eqref{eq:exact_ener_level_dets} can be
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rewritten as follows:
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respectively.
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The exact expression of the individual energies in Eq.~\eqref{eq:exact_ener_level_dets} can then be rewritten as
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\beq\label{eq:exact_ind_ener_rdm}
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\begin{split}
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\E{}{(I)}
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@ -388,17 +395,18 @@ rewritten as follows:
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\eeq
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where
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\beq
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\bh \equiv h_{\mu\nu} = \int \AO{\mu}(\br{}) \qty[-\frac{1}{2} \nabla_{\br{}}^2 + \vne(\br{}) ]\AO{\nu}(\br{}) d\br{}
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\bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}}
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\eeq
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denote the one-electron integrals matrix.
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The individual Hx energy is obtained from the following trace
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denotes the one-electron integrals matrix.
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The individual Hx energy is obtained from the following trace formula
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\beq
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\Tr[\bGam{(K)} \bG \bGam{(L)}]
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= \sum_{\mu\nu\la\si} \eGam{\mu\nu}{(K)} \eG{\mu\nu\la\si} \eGam{\la\si}{(L)},
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\eeq
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where the two-electron Coulomb-exchange integrals read
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where the antisymmetrized two-electron integrals read
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\beq
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G_{\mu\nu\la\omega}
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\bG
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\equiv G_{\mu\nu\la\si}
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= \dbERI{\mu\nu}{\la\si}
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= \ERI{\mu\nu}{\la\si} - \ERI{\mu\si}{\la\nu},
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\eeq
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@ -498,11 +506,11 @@ w}_K
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%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%
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\subsection{Approximations}\label{subsec:approx}
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%%%%%%%%%%%%%%%
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As Hartree and exchange energies cannot be separated in the
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one-dimension systems considered in the rest of this work, we will substitute the Hartree--Fock
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density-matrix-functional interaction energy,
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As Hartree and exchange energies cannot be separated in the one-dimensional systems considered in the rest of this work, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy,
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\beq\label{eq:eHF-dens_mat_func}
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\WHF[\bGam{}] = \frac{1}{2} \Tr[\bGam{} \bG \bGam{}],
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\eeq
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@ -529,7 +537,6 @@ with
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\eh{\mu\nu}{\bw}
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= \eh{\mu\nu}{} + \int \AO{\mu}(\br{}) \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} \AO{\nu}(\br{}) d\br{}.
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\eeq
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%%%%%%%%%%%%%%%
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\iffalse%%%%%%
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% Manu's derivation %%%%
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@ -652,11 +659,8 @@ and
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\fi%%%%%%%%%%%
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%%%%% end Manu
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%%%%%%%%%%%%%%%%%%%%
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Note that this approximation, where the ensemble density matrix is
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optimized from a non-local exchange potential [rather than a local one,
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as expected from Eq.~\eqref{eq:var_ener_gokdft}] is applicable to real
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(three-dimension) systems. As readily seen from
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Eq.~\eqref{eq:eHF-dens_mat_func}, \textit{ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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Note that this approximation, where the ensemble density matrix is optimized from a non-local exchange potential [rather than a local one, as expected from Eq.~\eqref{eq:var_ener_gokdft}] is applicable to real (three-dimensional) systems.
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As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, \textit{ghost interactions}~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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and curvature~\cite{} will be introduced in the Hx energy:
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\beq
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\begin{split}
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@ -669,17 +673,13 @@ and curvature~\cite{} will be introduced in the Hx energy:
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These errors will be removed when computing individual energies
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according to 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 eLDA will be employed:
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Turning to the density-functional ensemble correlation energy, the following eLDA will be employed:
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\beq\label{eq:eLDA_corr_fun}
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\E{c}{\bw}[\n{}{}] = \int \n{}{}(\br{}) \e{c}{\bw}[\n{}{}(\br{})] d\br{},
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\eeq
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where the correlation energy per particle is {\it weight-dependent}. Its
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construction from a finite uniform electron gas model is discussed
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in detail
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in Sec.~\ref{sec:eDFA}. Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with
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Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression
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within eLDA:
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where the correlation energy per particle is \textit{weight dependent}.
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Its construction from a finite uniform electron gas model is discussed in detail in Sec.~\ref{sec:eDFA}.
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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 eLDA:
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\beq
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\begin{split}
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\E{}{(I)}
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@ -801,7 +801,7 @@ 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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\titou{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
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%\titou{As shown by Gould and Pittalis, comment on density- and and state-driven errors. \cite{Gould_2019}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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@ -810,7 +810,7 @@ Finally, we note that, by construction,
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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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In particular, we investigate systems where $L$ ranges from $\pi/8$ to $8\pi$ and $2 \le \Nel \le 7$.
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\titou{Comment on the quality of these density: density- and functional-driven errors?}
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%\titou{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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