done iteration for T2
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@ -29,7 +29,7 @@
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\newcommand{\be}[2]{\bar{\eps}_\text{#1}^{#2}}
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\newcommand{\bv}[2]{\bar{f}_\text{#1}^{#2}}
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\newcommand{\n}[1]{n^{#1}}
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\newcommand{\DD}[2]{\Delta_\text{#1}^{#2}}
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\newcommand{\DD}[2]{\Upsilon_\text{#1}^{#2}}
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\newcommand{\LZ}[2]{\Xi_\text{#1}^{#2}}
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@ -623,7 +623,6 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
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\label{tab:OptGap_2}
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Deviation from the FCI quantities (in hartree) of the individual energies, $\E{(I)}$, and the corresponding excitation energies, $\Ex{(I)}$, for the ground ($I=0$), singly-excited ($I=1$) and doubly-excited ($I=2$) states of 2-boxium (i.e.,~$\Nel = 2$ electrons in a box of length $L$).
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The values of the ensemble correlation derivative $\DD{c}{(I)}$ are also reported.
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(DNC = KS calculation does not converge.)
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}
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\begin{ruledtabular}
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\begin{tabular}{lclddddddd}
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@ -643,39 +642,39 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
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\\
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TDHF & & $\Ex{(1)}$ & 0.0019 & 0.0021 & 0.0023 & 0.0027 & 0.0029 & 0.0023 & 0.0011 \\
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\\
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TDA-TDLDA& & $\Ex{(1)}$ & 0.0099 & 0.0088 & 0.0058 & -0.0041 & -0.0316 & -0.0467 & \tabc{DNC} \\
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TDA-TDLDA& & $\Ex{(1)}$ & 0.0099 & 0.0088 & 0.0058 & -0.0041 & -0.0316 & -0.0467 & \\
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\\
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TDLDA & & $\Ex{(1)}$ & 0.0015 & 0.0006 & -0.0018 & -0.0106 & -0.0370 & -0.0518 & \tabc{DNC} \\
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TDLDA & & $\Ex{(1)}$ & 0.0015 & 0.0006 & -0.0018 & -0.0106 & -0.0370 & -0.0518 & \\
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\\
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% eHF & $(0,0)$ & $\E{(0)}$ & 0.0137 & 0.0131 & 0.0119 & 0.0098 & 0.0067 & 0.0033 & 0.0010 \\
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% & & $\E{(1)}$ & 0.0685 & 0.0678 & 0.0664 & 0.0637 & 0.0588 & 0.0500 & 0.0372 \\
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% & & $\E{(2)}$ & -0.0052 & -0.0055 & -0.0058 & -0.0055 & -0.0022 & 0.0086 & 0.0271 \\
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% & & $\Ex{(1)}$ & 0.1082 & 0.1069 & 0.1044 & 0.0998 & 0.0911 & 0.0736 & \tabc{DNC} \\
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% & & $\Ex{(2)}$ & 0.0345 & 0.0336 & 0.0322 & 0.0306 & 0.0302 & 0.0321 & \tabc{DNC} \\
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% & & $\Ex{(1)}$ & 0.1082 & 0.1069 & 0.1044 & 0.0998 & 0.0911 & 0.0736 & \\
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% & & $\Ex{(2)}$ & 0.0345 & 0.0336 & 0.0322 & 0.0306 & 0.0302 & 0.0321 & \\
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% \\
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% eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.0565 & 0.0557 & 0.0540 & 0.0509 & 0.0455 & 0.0372 & 0.0269 \\
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% & & $\E{(1)}$ & 0.0560 & 0.0553 & 0.0538 & 0.0510 & 0.0461 & 0.0384 & 0.0287 \\
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% & & $\E{(2)}$ & 0.0371 & 0.0366 & 0.0357 & 0.0342 & 0.0316 & 0.0277 & 0.0223 \\
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% & & $\Ex{(1)}$ & 0.0529 & 0.0517 & 0.0494 & 0.0456 & 0.0418 & 0.0409 & \tabc{DNC} \\
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% & & $\Ex{(2)}$ & 0.0340 & 0.0330 & 0.0314 & 0.0288 & 0.0274 & 0.0303 & \tabc{DNC} \\
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% & & $\Ex{(1)}$ & 0.0529 & 0.0517 & 0.0494 & 0.0456 & 0.0418 & 0.0409 & \\
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% & & $\Ex{(2)}$ & 0.0340 & 0.0330 & 0.0314 & 0.0288 & 0.0274 & 0.0303 & \\
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% \\
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KS-eLDA & $(0,0)$ & $\E{(0)}$ & -0.0397 & -0.0391 & -0.0380 & -0.0361 & -0.0323 & -0.0236 & \tabc{DNC}\\
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& & $\E{(1)}$ & 0.0215 & 0.0213 & 0.0210 & 0.0200 & 0.0159 & 0.0102 & \tabc{DNC}\\
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& & $\E{(2)}$ & -0.0426 & -0.0425 & -0.0419 & -0.0387 & -0.0250 & -0.0045 & \tabc{DNC}\\
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& & $\Ex{(1)}$ & 0.0612 & 0.0604 & 0.0590 & 0.0561 & 0.0483 & 0.0337 & \tabc{DNC}\\
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& & $\Ex{(2)}$ & -0.0029 & -0.0034 & -0.0039 & -0.0025 & 0.0074 & 0.0191 & \tabc{DNC}\\
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& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & \tabc{DNC} \\
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& & $\DD{c}{(1)}$ & 0.0064 & 0.0056 & 0.0043 & 0.0022 & -0.0007 & -0.0037 & \tabc{DNC}\\
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& & $\DD{c}{(2)}$ & 0.0159 & 0.0147 & 0.0126 & 0.0093 & 0.0046 & -0.0009 & \tabc{DNC}\\
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KS-eLDA & $(0,0)$ & $\E{(0)}$ & -0.0397 & -0.0391 & -0.0380 & -0.0361 & -0.0323 & -0.0236 & \\
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& & $\E{(1)}$ & 0.0215 & 0.0213 & 0.0210 & 0.0200 & 0.0159 & 0.0102 & \\
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& & $\E{(2)}$ & -0.0426 & -0.0425 & -0.0419 & -0.0387 & -0.0250 & -0.0045 & \\
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& & $\Ex{(1)}$ & 0.0612 & 0.0604 & 0.0590 & 0.0561 & 0.0483 & 0.0337 & \\
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& & $\Ex{(2)}$ & -0.0029 & -0.0034 & -0.0039 & -0.0025 & 0.0074 & 0.0191 & \\
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& & $\DD{c}{(0)}$ & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & \\
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& & $\DD{c}{(1)}$ & 0.0064 & 0.0056 & 0.0043 & 0.0022 & -0.0007 & -0.0037 & \\
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& & $\DD{c}{(2)}$ & 0.0159 & 0.0147 & 0.0126 & 0.0093 & 0.0046 & -0.0009 & \\
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\\
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KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0031 & 0.0036 & 0.0044 & 0.0054 & 0.0042 & -0.0025 & \tabc{DNC}\\
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& & $\E{(1)}$ & 0.0090 & 0.0087 & 0.0083 & 0.0076 & 0.0070 & 0.0071 & \tabc{DNC}\\
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& & $\E{(2)}$ & -0.0005 & -0.0009 & -0.0015 & -0.0023 & -0.0030 & -0.0026 & \tabc{DNC}\\
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& & $\Ex{(1)}$ & 0.0058 & 0.0052 & 0.0039 & 0.0022 & 0.0028 & 0.0096 & \tabc{DNC}\\
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& & $\Ex{(2)}$ & -0.0036 & -0.0045 & -0.0058 & -0.0077 & -0.0072 & 0.0000 & \tabc{DNC}\\
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& & $\DD{c}{(0)}$ & -0.0074 & -0.0067 & -0.0055 & -0.0036 & -0.0010 & 0.0019 & \tabc{DNC}\\
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& & $\DD{c}{(1)}$ & -0.0010 & -0.0011 & -0.0014 & -0.0017 & -0.0021 & -0.0022 & \tabc{DNC}\\
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& & $\DD{c}{(2)}$ & 0.0084 & 0.0079 & 0.0069 & 0.0053 & 0.0031 & 0.0003 & \tabc{DNC}\\
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KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0031 & 0.0036 & 0.0044 & 0.0054 & 0.0042 & -0.0025 & \\
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& & $\E{(1)}$ & 0.0090 & 0.0087 & 0.0083 & 0.0076 & 0.0070 & 0.0071 & \\
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& & $\E{(2)}$ & -0.0005 & -0.0009 & -0.0015 & -0.0023 & -0.0030 & -0.0026 & \\
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& & $\Ex{(1)}$ & 0.0058 & 0.0052 & 0.0039 & 0.0022 & 0.0028 & 0.0096 & \\
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& & $\Ex{(2)}$ & -0.0036 & -0.0045 & -0.0058 & -0.0077 & -0.0072 & 0.0000 & \\
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& & $\DD{c}{(0)}$ & -0.0074 & -0.0067 & -0.0055 & -0.0036 & -0.0010 & 0.0019 & \\
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& & $\DD{c}{(1)}$ & -0.0010 & -0.0011 & -0.0014 & -0.0017 & -0.0021 & -0.0022 & \\
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& & $\DD{c}{(2)}$ & 0.0084 & 0.0079 & 0.0069 & 0.0053 & 0.0031 & 0.0003 & \\
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\end{tabular}
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\end{ruledtabular}
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\end{table*}
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@ -132,9 +132,9 @@
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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 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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We report a first generation of local, weight-dependent correlation density-functional approximations 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 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 density-functional approximations for ensembles, 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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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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@ -199,14 +199,14 @@ describe near-degenerate situations and multiple-electron excitation
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processes, it has not
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been given much attention until quite recently. \cite{Franck_2014,Borgoo_2015,Kazaryan_2008,Gould_2013,Gould_2014,Filatov_2015,Filatov_2015b,Filatov_2015c,Gould_2017,Deur_2017,Gould_2018,Gould_2019,Sagredo_2018,Ayers_2018,Deur_2018,Deur_2019,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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One of the reason is the lack, not to say the absence, of reliable
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density-functional approximations for ensembles (eDFAs).
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density-functional approximations for ensembles.
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The most recent works dealing with this particular issue are still fundamental and
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exploratory, as they rely either on simple (but nontrivial) model
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systems
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\cite{Carrascal_2015,Deur_2017,Deur_2018,Deur_2019,Senjean_2015,Senjean_2016,Senjean_2018,Sagredo_2018,Senjean_2020,Fromager_2020,Gould_2019}
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or atoms. \cite{Yang_2014,Yang_2017,Gould_2019_insights}
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Despite all these efforts, it is still unclear how weight dependencies
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can be incorporated into eDFAs. This problem is actually central not
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can be incorporated into density-functional approximations. This problem is actually central not
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only in GOK-DFT but also in conventional (ground-state) DFT as the infamous derivative
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discontinuity problem that ocurs when crossing an integral number of
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electrons can be recast into a weight-dependent ensemble
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@ -218,7 +218,7 @@ with the ambition to turn, in the forthcoming future, GOK-DFT into a
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Starting from the ubiquitous local-density approximation (LDA), we
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design a weight-dependent ensemble correction based on a finite uniform
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electron gas from which density-functional excitation energies can be
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extracted. The present eDFA, which can be seen as a natural
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extracted. The present density-functional approximation for ensembles, which can be seen as a natural
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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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@ -288,7 +288,7 @@ where $\Tr$ denotes the trace and the trial ensemble density matrix operator rea
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The KS determinants [or configuration state functions~\cite{Gould_2017}]
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$\Det{(K)}$ are all constructed from the same set of ensemble KS
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orbitals that are variationally optimized.
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The trial ensemble density in Eq.~(\ref{eq:var_ener_gokdft}) is simply
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The trial ensemble density in Eq.~\eqref{eq:var_ener_gokdft} is simply
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the weighted sum of the individual KS 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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@ -312,7 +312,7 @@ exactly from a single ensemble calculation as follows:
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\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} )
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\pdv{\E{}{\bw}}{\ew{K}},
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\eeq
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where, according to the normalization condition of Eq.~(\ref{eq:weight_norm_cond}),
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where, according to the normalization condition of Eq.~\eqref{eq:weight_norm_cond},
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\beq
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\pdv{\E{}{\bw}}{\ew{K}}= \E{}{(K)} -
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\E{}{(0)}\equiv\Ex{}{(K)}
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@ -348,7 +348,7 @@ auxiliary double-weight ensemble density reads
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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 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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from the exact expression in Eq.~\eqref{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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\eeq
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@ -371,7 +371,7 @@ This yields, according to Eqs.~\eqref{eq:deriv_Ew_wk} and \eqref{eq:_deriv_wk_Hx
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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, according to Eqs.~(\ref{eq:var_ener_gokdft}) and (\ref{eq:exact_ens_Hx}), the ensemble energy can be evaluated as
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Since, according to Eqs.~\eqref{eq:var_ener_gokdft} and \eqref{eq:exact_ens_Hx}, 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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@ -396,7 +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.~\eqref{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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@ -431,8 +431,8 @@ can be recast as follows:
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\pdv{\E{c}{\bw}[\n{\Det{(0)}}{}]}{\ew{I}}
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\right|_{\bw=0}.
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\eeq
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As readily seen from Eqs.~(\ref{eq:dens_func_Hamilt}) and
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(\ref{eq:corr_LZ_shift}), introducing any constant shift $\delta
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As readily seen from Eqs.~\eqref{eq:dens_func_Hamilt} and
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\eqref{eq:corr_LZ_shift}, introducing any constant shift $\delta
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\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})\rightarrow \delta
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\E{c}{}[\n{\Det{(0)}}{}]/\delta n({\bf r})+C$ into the correlation
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potential leaves the density-functional Hamiltonian $\hat{H}[n]$ (and
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@ -441,7 +441,7 @@ this context,
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the correlation derivative discontinuities induced by the
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excitation process~\cite{Levy_1995} will be fully described by the ensemble
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correlation derivatives [second term on the right-hand side of
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Eq.~(\ref{eq:excited_ener_level_gs_lim})].
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Eq.~\eqref{eq:excited_ener_level_gs_lim}].
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%%%%%%%%%%%%%%%%
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\subsection{One-electron reduced density matrix formulation}
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@ -449,7 +449,7 @@ Eq.~(\ref{eq:excited_ener_level_gs_lim})].
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For implementation purposes, we will use in the rest of this work
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(one-electron reduced) density matrices
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as basic variables, rather than Slater determinants.
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As the theory is applied later on to {\it spin-polarized}
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As the theory is applied later on to \textit{spin-polarized}
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systems, we drop spin indices in the density matrices, for convenience.
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If we expand the
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ensemble KS orbitals (from which the determinants are constructed) in an atomic orbital (AO) basis,
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@ -668,7 +668,7 @@ following approximation:
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\Tr[\bgam{\bw} \bh ] + \WHF[ \bgam{\bw}] + \E{c}{\bw}[\n{\bgam{\bw}}{}]
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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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The minimizing ensemble density matrix in Eq.~\eqref{eq:min_with_HF_ener_fun} fulfills the following
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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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@ -806,14 +806,14 @@ and
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\fi%%%%%%%%%%%
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%%%%% end Manu
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%%%%%%%%%%%%%%%%%%%%
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Note that, within the approximation of Eq.~(\ref{eq:min_with_HF_ener_fun}), the ensemble density matrix is
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Note that, within the approximation of Eq.~\eqref{eq:min_with_HF_ener_fun}, the ensemble density matrix is
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optimized with a non-local exchange potential rather than a
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density-functional local one, as expected from
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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 \titou{(GIE)} \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
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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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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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@ -824,7 +824,7 @@ as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
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\end{split}
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\eeq
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The ensemble energy is of course expected to vary linearly with the ensemble
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weights [see Eq.~(\ref{eq:exact_GOK_ens_ener})].
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weights [see Eq.~\eqref{eq:exact_GOK_ens_ener}].
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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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@ -889,11 +889,11 @@ around the $I$th KS state density
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$\n{\bGam{(I)}}{}(\br{})$, the sum of
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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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\titou{$\dn{\bGam{\bw}}{(I)}(\br{}) = \n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$}:
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$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
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\beq\label{eq:Taylor_exp_ind_corr_ener_eLDA}
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\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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+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(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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@ -905,7 +905,7 @@ 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
|
||||
comment that follows] {\it via} the \titou{third} on the right-hand side
|
||||
comment that follows] {\it via} the \titou{third term} on the right-hand side
|
||||
of Eq.~\eqref{eq:EI-eLDA}. According to the decomposition of
|
||||
the ensemble
|
||||
correlation energy per particle in Eq.
|
||||
@ -929,16 +929,15 @@ correlation energy per particle in Eq.
|
||||
%\sum_{K>0}\delta_{IK}\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})}
|
||||
\end{equation}
|
||||
thus leading to the following Taylor expansion through first order in
|
||||
\titou{$\dn{\bGam{\bw}}{(I)}(\br{})$}
|
||||
%$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
|
||||
$\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$:
|
||||
\beq\label{eq:Taylor_exp_DDisc_term}
|
||||
\begin{split}
|
||||
\titou{\Upsilon_\text{c}^{(I)}}
|
||||
%& = \int \sum_{K>0} \qty(\delta_{IK} - \ew{K} ) \n{\bGam{\bw}}{}(\br{})
|
||||
% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
|
||||
%\\
|
||||
&=-\int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
|
||||
+\int \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
|
||||
&=
|
||||
\int \qty[ \be{c}{(I)}(\n{\bGam{(I)}}{}(\br{})) - \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) ] \n{\bGam{(I)}}{}(\br{}) d\br{}
|
||||
\\
|
||||
&+\int \Bigg[
|
||||
\n{\bGam{(I)}}{}(\br{})
|
||||
@ -952,16 +951,16 @@ thus leading to the following Taylor expansion through first order in
|
||||
&+\be{c}{(I)}(\n{\bGam{(I)}}{}(\br{}))
|
||||
-
|
||||
\e{c}{\bw}(\n{\bGam{(I)}}{}(\br{}))\Bigg]
|
||||
\dn{\bGam{\bw}}{(I)}(\br{})
|
||||
\qty[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]
|
||||
d\br{}
|
||||
\\
|
||||
&
|
||||
+ \titou{\order{[\dn{\bGam{\bw}}{(I)}(\br{})]^2}}.
|
||||
+ \order{[\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})]^2}.
|
||||
\end{split}
|
||||
\eeq
|
||||
As readily seen from Eqs. \eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and \eqref{eq:Taylor_exp_DDisc_term}, the
|
||||
role of the correlation ensemble derivative \titou{$\Upsilon_\text{c}^{(I)}$} \trashPFL{[last term on the right-hand side of Eq.
|
||||
\eqref{eq:EI-eLDA}]} is, through zeroth order, to substitute the expected
|
||||
role of the correlation ensemble derivative \titou{$\Upsilon_\text{c}^{(I)}$}
|
||||
\trashPFL{[last term on the right-hand side of Eq.~\eqref{eq:EI-eLDA}]} is, through zeroth order, to substitute the expected
|
||||
individual correlation energy per particle for the ensemble one.
|
||||
|
||||
Let us finally note that, while the weighted sum of the
|
||||
@ -992,8 +991,8 @@ _{\n{}{} =
|
||||
\\ & + \DD{c}{(I)},
|
||||
\end{split}
|
||||
\eeq
|
||||
where the HF-like excitation energies $\Ex{HF}{(I)} = \E{HF}{(I)} -
|
||||
\E{HF}{(0)}$ are determined from a single set of ensemble KS orbitals and
|
||||
where the HF-like excitation energies, $\Ex{HF}{(I)} = \E{HF}{(I)} -
|
||||
\E{HF}{(0)}$, are determined from a single set of ensemble KS orbitals and
|
||||
\beq\label{eq:DD-eLDA}
|
||||
\DD{c}{(I)}
|
||||
= \int \n{\bGam{\bw}}{}(\br{})
|
||||
@ -1011,28 +1010,28 @@ is the eLDA correlation ensemble derivative contribution to the $I$th excitation
|
||||
\label{sec:paradigm}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
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,
|
||||
Most of the standard local and semi-local density-functional approximations rely on the infinite uniform electron gas model (also known as jellium). \cite{ParrBook, Loos_2016}
|
||||
One major drawback of the jellium paradigm, when it comes to develop density-functional approximations for ensembles, 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 infinite uniform electron gas 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 finite uniform electron gases, \cite{Loos_2011b,
|
||||
Gill_2012} which have, like an atom, discrete energy levels and non-zero
|
||||
gaps, can be seen as more relevant in this context. \cite{Loos_2014a, Loos_2014b, Loos_2017a}
|
||||
However, an obvious drawback of using FUEGs is that the resulting eDFA
|
||||
will inexorably depend on the number of electrons in the FUEG (see below).
|
||||
However, an obvious drawback of using finite uniform electron gases is that the resulting density-functional approximation for ensemble
|
||||
will inexorably depend on the number of electrons in the finite uniform electron gas (see below).
|
||||
Here, we propose to construct a weight-dependent eLDA for the
|
||||
calculations of excited states in 1D systems by combining FUEGs with the
|
||||
usual IUEG.
|
||||
calculations of excited states in 1D systems by combining finite uniform electron gases with the
|
||||
usual infinite uniform electron gas.
|
||||
|
||||
As a FUEG, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b}
|
||||
As a finite uniform electron gas, we consider the ringium model in which electrons move on a perfect ring (\ie, a circle) but interact \textit{through} the ring. \cite{Loos_2012, Loos_2013a, Loos_2014b}
|
||||
The most appealing feature of ringium regarding the development of
|
||||
functionals in the context of eDFT is the fact that both ground- and
|
||||
excited-state densities are uniform, and therefore {\it equal}.
|
||||
As a result, the ensemble density will remain constant (and uniform) as the ensemble weights vary.
|
||||
This is a necessary condition for being able to model the ensemble
|
||||
correlation derivatives with respect to the weights [last term
|
||||
on the right-hand side of Eq.~(\ref{eq:exact_ener_level_dets})].
|
||||
Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous IUEG paradigm. \cite{Loos_2013,Loos_2013a}
|
||||
Let us stress that, in a FUEG like ringium, the interacting and
|
||||
on the right-hand side of Eq.~\eqref{eq:exact_ener_level_dets}].
|
||||
Moreover, it has been shown that, in the thermodynamic limit, the ringium model is equivalent to the ubiquitous infinite uniform electron gas paradigm. \cite{Loos_2013,Loos_2013a}
|
||||
Let us stress that, in a finite uniform electron gas like ringium, the interacting and
|
||||
noninteracting densities match individually for all the states within the
|
||||
ensemble
|
||||
(these densities are all equal to the uniform density), which means that
|
||||
@ -1040,7 +1039,7 @@ so-called density-driven correlation
|
||||
effects~\cite{Gould_2019,Gould_2019_insights,Senjean_2020,Fromager_2020} are absent from the model.
|
||||
Here, we will consider the most simple ringium system featuring electronic correlation effects, \ie, the two-electron ringium model.
|
||||
|
||||
The present weight-dependent eDFA is specifically designed for the
|
||||
The present weight-dependent density-functional approximation 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.
|
||||
@ -1075,7 +1074,7 @@ Generalization to a larger number of states is straightforward and is left for f
|
||||
\begin{table*}
|
||||
\caption{
|
||||
\label{tab:OG_func}
|
||||
Parameters of the weight-dependent correlation DFAs defined in Eq.~\eqref{eq:ec}.}
|
||||
Parameters of the weight-dependent correlation density-functional approximations defined in Eq.~\eqref{eq:ec}.}
|
||||
% \begin{ruledtabular}
|
||||
\begin{tabular}{lcddd}
|
||||
\hline\hline
|
||||
@ -1102,8 +1101,8 @@ Based on highly-accurate calculations (see {\SI} for additional details), one ca
|
||||
\end{equation}
|
||||
where the $a_k^{(I)}$'s are state-specific fitting parameters provided in Table \ref{tab:OG_func}.
|
||||
The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
||||
Equation \eqref{eq:ec} provides three state-specific correlation DFAs based on a two-electron system.
|
||||
Combining these, one can build the following three-state weight-dependent correlation eDFA:
|
||||
Equation \eqref{eq:ec} provides three state-specific correlation density-functional approximations based on a two-electron system.
|
||||
Combining these, one can build the following three-state weight-dependent correlation density-functional approximation:
|
||||
\begin{equation}
|
||||
\label{eq:ecw}
|
||||
%\e{c}{\bw}(\n{}{})
|
||||
@ -1114,10 +1113,10 @@ Combining these, one can build the following three-state weight-dependent correl
|
||||
\subsection{LDA-centered functional}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
One of the main driving force behind the popularity of DFT is its ``universal'' nature, as xc density functionals can be applied to any electronic system.
|
||||
Obviously, the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the FUEG.
|
||||
However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron FUEG (the impurity) is embedded in the IUEG (the bath).
|
||||
Obviously, the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} does not have this feature as it does depend on the number of electrons constituting the finite uniform electron gas.
|
||||
However, one can partially cure this dependency by applying a simple embedding scheme in which the two-electron finite uniform electron gas (the impurity) is embedded in the infinite uniform electron gas (the bath).
|
||||
The weight-dependence of the correlation functional is then carried exclusively by the impurity [\ie, the functional defined in Eq.~\eqref{eq:ecw}], while the remaining correlation effects are provided by the bath (\ie, the usual LDA correlation functional).
|
||||
Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} as follows:
|
||||
Following this simple strategy, which can be further theoretically justified by the generalized adiabatic connection formalism for ensembles (GACE) originally derived by Franck and Fromager, \cite{Franck_2014} we propose to \emph{shift} the two-electron-based density-functional approximation for ensemble defined in Eq.~\eqref{eq:ecw} as follows:
|
||||
\begin{equation}
|
||||
\label{eq:becw}
|
||||
\Tilde{\epsilon}_{\rm c}^\bw(n)\rightarrow{\e{c}{\bw}(\n{}{})} = (1-\ew{1}-\ew{2}) \be{c}{(0)}(\n{}{}) + \ew{1} \be{c}{(1)}(\n{}{}) + \ew{2} \be{c}{(2)}(\n{}{}),
|
||||
@ -1143,7 +1142,7 @@ where $F(a,b,c,x)$ is the Gauss hypergeometric function, \cite{NISTbook} and
|
||||
a_3^\text{LDA} & = 2.408779.
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
Note that the strategy described in Eq.~(\ref{eq:becw}) is general and
|
||||
Note that the strategy described in Eq.~\eqref{eq:becw} is general and
|
||||
can be applied to real (higher-dimensional) systems. In order to make the
|
||||
connection with the GACE formalism \cite{Franck_2014,Deur_2017} more explicit, one may
|
||||
recast Eq.~\eqref{eq:becw} as
|
||||
@ -1166,7 +1165,7 @@ or, equivalently,
|
||||
\end{equation}
|
||||
where the $K$th correlation excitation energy (per electron) is integrated over the
|
||||
ensemble weight $\xi_K$ at fixed (uniform) density $\n{}{}$.
|
||||
Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the LDA in the present eDFA.
|
||||
Equation \eqref{eq:eLDA_gace} nicely highlights the centrality of the LDA in the present density-functional approximation for ensemble.
|
||||
In particular, ${\e{c}{(0,0)}(\n{}{})} = \e{c}{\text{LDA}}(\n{}{})$.
|
||||
Consequently, in the following, we name this correlation functional ``eLDA'' as it is a natural extension of the LDA for ensembles.
|
||||
Finally, we note that, by construction,
|
||||
@ -1196,7 +1195,7 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
|
||||
\end{equation}
|
||||
with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
|
||||
The convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw}
|
||||
\bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~(\ref{eq:commut_F_AO})] is set
|
||||
\bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~\eqref{eq:commut_F_AO}] is set
|
||||
to $10^{-5}$. For comparison, regular HF and KS-DFT calculations
|
||||
are performed with the same threshold.
|
||||
In order to compute the various density-functional
|
||||
@ -1237,7 +1236,7 @@ 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 \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}]}.
|
||||
To illustrate the magnitude of the ghost-interaction error, we report the KS-eLDA ensemble energy with and without 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 reduces the curvature of the ensemble energy
|
||||
@ -1263,7 +1262,7 @@ drastically.
|
||||
It is important to note that, even though the GIC removes the explicit
|
||||
quadratic terms from the ensemble energy, a non-negligible curvature
|
||||
remains in the GIC-eLDA ensemble energy due to the optimization of the
|
||||
ensemble KS orbitals in the presence of GIE {[see Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:Ew-eLDA}]}.
|
||||
ensemble KS orbitals in the presence of ghost-interaction error {[see Eqs.~\eqref{eq:min_with_HF_ener_fun} and \eqref{eq:Ew-eLDA}]}.
|
||||
%However, this orbital-driven error is small (in our case at
|
||||
%least) \trashEF{as the correlation part of the ensemble KS potential $\delta
|
||||
%\E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared
|
||||
@ -1296,19 +1295,18 @@ energies, which is in agreement with
|
||||
the curvature of the GIC-eLDA ensemble energy discussed previously. Interestingly, the
|
||||
individual energies do not vary in the same way depending on the state
|
||||
considered and the value of the weights.
|
||||
We see for example that, within the biensemble [$w_2=0$], the energy of
|
||||
the ground state increases with the first-excited-state weight $w_1$, thus showing that we
|
||||
``deteriorate'' this state a little by optimizing the orbitals also for
|
||||
the first excited state. The reverse actually occurs in the triensemble
|
||||
as $w_2$ increases. The variations in the ensemble
|
||||
\titou{We see for example that, within the biensemble (\ie, $\ew{2}=0$), the energies of
|
||||
the ground and first excited-state increase with respect to the first-excited-state weight $\ew{1}$, thus showing that we
|
||||
``deteriorate'' these states by optimizing the orbitals. The reverse actually occurs for the ground state in the triensemble
|
||||
as $\ew{2}$ increases.} The variations in the ensemble
|
||||
weights are essentially linear or quadratic. They are induced by the
|
||||
eLDA functional, as readily seen from
|
||||
Eqs.~(\ref{eq:Taylor_exp_ind_corr_ener_eLDA}) and
|
||||
Eqs.~\eqref{eq:Taylor_exp_ind_corr_ener_eLDA} and
|
||||
\eqref{eq:Taylor_exp_DDisc_term}. In the biensemble, the weight dependence of the first
|
||||
excitation energy is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
|
||||
\titou{excited-state energy} is reduced as the correlation increases. On the other hand, switching from a bi- to a triensemble
|
||||
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.
|
||||
ground-state energy in this case, as $\ew{2}$ increases.
|
||||
The reverse is observed for the second \titou{excited state}.
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure}
|
||||
@ -1326,7 +1324,7 @@ Similar graphs are obtained for the other $\nEl$ values and they can be found in
|
||||
For small $L$, the single and double excitations can be labeled as
|
||||
``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.
|
||||
However, when the box gets larger (\ie, as $L$ 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}
|
||||
This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
|
||||
% TITOU: shall we keep the paragraph below?
|
||||
@ -1356,7 +1354,7 @@ For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yie
|
||||
TDA-TDLDA slightly corrects this trend thanks to error compensation.
|
||||
Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
|
||||
This is especially true for the single excitation
|
||||
which is significantly improved by using equal weights.
|
||||
which is significantly improved by using equal weights, \titou{especially in the strong correlation regime}.
|
||||
The effect on the double excitation is less pronounced.
|
||||
Overall, one clearly sees that, with
|
||||
equal weights, KS-eLDA yields accurate excitation energies for both single and double excitations.
|
||||
@ -1426,7 +1424,7 @@ It is also interesting to investigate the influence of the
|
||||
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 $L$
|
||||
To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, \titou{for $\nEl = 3$, $5$, and $7$}, 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
|
||||
%evaluate the importance of the ensemble correlation derivatives you
|
||||
@ -1438,32 +1436,34 @@ on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{(I)}
|
||||
%\eeq
|
||||
%%rather than $E^{(I)}_{\rm HF}$
|
||||
%}
|
||||
\manu{We first stress that both single and double excitation energies are
|
||||
\titou{We first stress that although for $\nEl=3$ both single and double excitation energies are
|
||||
systematically improved, as the strength of electron correlation
|
||||
increases, when
|
||||
taking into account
|
||||
the correlation ensemble derivative. This statement holds in both
|
||||
zero-weight and equal-weight limits.
|
||||
the correlation ensemble derivative, this is not systematically the case for larger number of electrons.}
|
||||
\trashPFL{This statement holds in both zero-weight and equal-weight limits.}
|
||||
The influence of the correlation ensemble derivative becomes substantial in the strong correlation regime.
|
||||
}
|
||||
\manu{In the zero-weight limit, its contribution is also significantly larger in the case of the single
|
||||
\titou{For 3-boxium, in the zero-weight limit, its contribution is also significantly larger in the case of the single
|
||||
excitation; the reverse is observed in the equal-weight triensemble
|
||||
case.}\trashEF{the correlation ensemble derivative hardly
|
||||
influences the double excitation}.
|
||||
Importantly, one realizes that the magnitude of the correlation ensemble
|
||||
derivative is much smaller in the case of equal-weight calculations (as
|
||||
compared to the zero-weight calculations).\manu{Manu: well, this is not
|
||||
really the case for the double excitation, right? I would remove this
|
||||
sentence or mention the single excitation explicitly.}
|
||||
case.
|
||||
Howver, for 5- and 7-boxium, the correlation ensemble derivative hardly
|
||||
influences the double excitation (except when the correlation is strong), and slightly deteriorates the single excitation in the intermediate and strong correlation regimes}.
|
||||
Importantly, \titou{for the single excitation}, one realizes that the magnitude of the correlation ensemble
|
||||
derivative is \trashPFL{much} smaller in the case of equal-weight calculations (as
|
||||
compared to the zero-weight calculations).
|
||||
%\manu{Manu: well, this is not
|
||||
%really the case for the double excitation, right? I would remove this
|
||||
%sentence or mention the single excitation explicitly.}
|
||||
This could explain why equiensemble calculations are clearly more
|
||||
accurate as it reduces the influence of the ensemble correlation derivative:
|
||||
accurate \titou{for the single excitation} as it reduces the influence of the ensemble correlation derivative:
|
||||
for a given method, equiensemble orbitals partially remove the burden
|
||||
of modeling properly the ensemble correlation derivative.\manu{Manu: I
|
||||
do not like this statement. As I wrote above, the ensemble derivative is
|
||||
still substantial in the strongly correlated limit of the equi
|
||||
triensemble for the double
|
||||
excitation.
|
||||
}
|
||||
of modeling properly the ensemble correlation derivative.
|
||||
%\manu{Manu: I
|
||||
%do not like this statement. As I wrote above, the ensemble derivative is
|
||||
%still substantial in the strongly correlated limit of the equi
|
||||
%triensemble for the double
|
||||
%excitation.
|
||||
%}
|
||||
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,
|
||||
@ -1491,9 +1491,8 @@ has a negligible effect on the excitation energies \titou{(results not shown)}.
|
||||
Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime).
|
||||
The difference between the solid and dashed curves
|
||||
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
|
||||
ensemble correlation derivative has \titou{a rather significant impact} on the double
|
||||
excitations \titou{(around $10\%$)} with a slight tendency of worsening the excitation energies
|
||||
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
|
||||
@ -1516,7 +1515,7 @@ progress in this direction.
|
||||
|
||||
Unlike any standard functional, eLDA incorporates derivative
|
||||
discontinuities through its weight dependence. The latter originates
|
||||
from the finite uniform electron gas eLDA is
|
||||
from the finite uniform electron gas \titou{on which} eLDA is
|
||||
(partially) based on. The KS-eLDA scheme, where exact exchange is
|
||||
combined with eLDA, delivers accurate excitation energies for both
|
||||
single and double excitations, especially when an equiensemble is used.
|
||||
|
||||
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Reference in New Issue
Block a user