more refs, intro and clean up methodo
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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 2019-11-15 10:58:08 +0100
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%% Created for Pierre-Francois Loos at 2019-11-17 22:41:55 +0100
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%% Saved with string encoding Unicode (UTF-8)
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@article{Vosko_1980,
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Author = {Vosko, S. H. and Wilk, L. and Nusair, M.},
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Date-Added = {2019-11-17 21:47:25 +0100},
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Date-Modified = {2019-11-17 21:48:12 +0100},
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Doi = {10.1139/p80-159},
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Journal = {Can. J. Phys.},
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Pages = {1200--1211},
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Title = {Accurate Spin-Dependent Electron Liquid Correlation Energies for Local Spin Density Calculations: A Critical Analysis},
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Volume = {58},
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Year = {1980},
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Bdsk-Url-1 = {https://dx.doi.org/10.1139/p80-159}}
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@article{Dirac_1930,
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Author = {Dirac, P. A. M.},
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Date-Added = {2019-11-17 21:46:54 +0100},
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Date-Modified = {2019-11-17 21:49:26 +0100},
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Doi = {10.1017/S0305004100016108},
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Journal = {Proc. Cambridge Philos. Soc.},
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Pages = {376},
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Volume = 26,
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Year = 1930,
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Bdsk-Url-1 = {https://doi.org/10.1017/S0305004100016108}}
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@book{Ulrich_2012,
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Address = {New York},
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Author = {Ullrich, C.},
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@ -21,13 +44,14 @@
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@article{Maitra_2017,
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Author = {N. T. Maitra},
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Date-Added = {2019-11-14 21:04:04 +0100},
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Date-Modified = {2019-11-14 21:04:04 +0100},
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Date-Modified = {2019-11-17 21:52:36 +0100},
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Doi = {10.1088/1361-648X/aa836e},
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Journal = {J. Phys. Cond. Matt.},
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Keywords = {10.1088/1361-648X/aa836e},
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Pages = {423001},
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Title = {Charge Transfer In Time-Dependent Density Functional Theory},
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Volume = {29},
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Year = {2017}}
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Year = {2017},
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Bdsk-Url-1 = {https://doi.org/10.1088/1361-648X/aa836e}}
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@article{Runge_1984,
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Author = {Runge, E. and Gross, E. K. U.},
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@ -346,9 +370,9 @@
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@article{Deur_2019,
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Author = {K. Deur and E. Fromager},
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Date-Added = {2018-12-08 18:03:14 +0100},
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Date-Modified = {2019-11-14 21:15:19 +0100},
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Date-Modified = {2019-11-17 21:51:24 +0100},
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Doi = {10.1063/1.5084312},
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Journal = {arXiv},
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Journal = {J. Chem. Phys.},
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Pages = {094106},
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Title = {Ground and excited energy levels can be extracted exactly from a single ensemble density-functional theory calculation},
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Volume = {150},
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@ -2327,11 +2351,13 @@
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@inbook{Casida,
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Author = {M. E. Casida},
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Date-Added = {2018-10-24 22:38:52 +0200},
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Date-Modified = {2018-10-24 22:38:52 +0200},
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Date-Modified = {2019-11-17 21:55:48 +0100},
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Doi = {10.1142/9789812830586_0005},
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Editor = {D. P. Chong},
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Pages = {155},
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Pages = {155--192},
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Publisher = {World Scientific, Singapore},
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Title = {Recent Advances in Density Functional Methods},
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Series = {Recent Advances in Density Functional Methods},
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Title = {Time-Dependent Density Functional Response Theory for Molecules},
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Year = {1995}}
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@article{Casida_1998,
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@ -3556,12 +3582,14 @@
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@article{Gould_2019,
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Author = {Gould, Tim and Pittalis, Stefano},
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Date-Added = {2018-10-24 22:38:52 +0200},
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Date-Modified = {2019-09-05 12:09:05 +0200},
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Date-Modified = {2019-11-17 21:50:27 +0100},
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Doi = {10.1103/PhysRevLett.123.016401},
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Journal = {Phys. Rev. Lett.},
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Pages = {016401},
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Title = {Density-Driven Correlations in Many-Electron Ensembles: Theory and Application for Excited States},
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Volume = {123},
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Year = {2019}}
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Year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.123.016401}}
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@article{Gould_2013,
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Author = {Gould, Tim and Dobson, John F.},
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@ -30,6 +30,7 @@
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\la}{\lambda}
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\newcommand{\si}{\sigma}
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\newcommand{\cD}{\mathcal{D}}
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% operators
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\newcommand{\hH}{\Hat{H}}
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@ -112,7 +113,7 @@
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\author{Clotilde \surname{Marut}}
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\affiliation{\LCPQ}
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\author{Emmanuel Fromager}
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\author{Emmanuel \surname{Fromager}}
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\affiliation{\LCQ}
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\author{Pierre-Fran\c{c}ois \surname{Loos}}
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\email{loos@irsamc.ups-tlse.fr}
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@ -123,7 +124,7 @@ We report a first generation of local, weight-dependent exchange-correlation den
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These density-functional approximations for ensembles (eDFAs) incorporate information about both ground and excited states in the context of density-functional theory for ensembles (eDFT).
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They are specially designed for the computation of double excitations within eDFT, and can be seen as a natural extension of the ubiquitous local-density approximation (LDA) to ensembles.
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The resulting eDFAs, dubbed eLDA, which are 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 on the prototypical H$_2$ molecule.
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Their accuracy is illustrated by computing the double excitation in the prototypical H$_2$ molecule.
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\end{abstract}
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\maketitle
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@ -154,15 +155,22 @@ In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has
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In this approach the exchange-correlation kernel is made frequency dependent \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states.
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Maybe surprisingly, a possible way of accessing double excitations is to resort to a time-\textit{independent} formalism. \cite{Yang_2017,Sagredo_2018,Deur_2019}
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Density-functional theory for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988,Gross_1988a,Oliveira_1988} is a viable alternative currently under active development which follow such a strategy. \cite{Gidopoulos_2002,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,Smith_2016,Senjean_2018}
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In the assumption of monotonically decreasing weights, eDFT has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, \cite{Gross_1988a} and excitation energies can be quite easily extracted from the total ensemble energy.
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Density-functional theory for ensembles (eDFT) \cite{Theophilou_1979,Gross_1988,Gross_1988a,Oliveira_1988} is a viable alternative currently under active development which follows such a strategy. \cite{Gidopoulos_2002,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,Smith_2016,Senjean_2018}
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In the assumption of monotonically decreasing weights, eDFT has the undeniable advantage to be based on a rigorous variational principle for ground and excited states, \cite{Gross_1988a} and excitation energies can be quite easily extracted from the total ensemble energy. \cite{Deur_2019}
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Although the formal foundation of eDFT has been set three decades ago, \cite{Gross_1988,Gross_1988a,Oliveira_1988} the practical developments of eDFT have been rather slow.
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We believe that it is due to the lack of accurate approximations for eDFT.
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In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation has never been developed for atoms and molecules in the context of eDFT.
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In particular, to the best of our knowledge, an explicitly weight-dependent density-functional approximation for ensemble (eDFA) has never been developed for atoms and molecules in the context of eDFT.
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The present contribution is a first step towards this goal.
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When one talks about constructing functionals, the local-density approximation (LDA) has always a special place.
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The LDA, as we know it, is based on the uniform electron gas (UEG) also known as jellium, an hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively-charged jelly. \cite{Loos_2016}
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Although the Hohenberg-Kohn theorems \cite{Hohenberg_1964} are here to provide firm theoretical grounds to DFT, modern KS-DFT rests largely on the presumed similarity between this hypothetical UEG and the electronic behavior in a real system. \cite{Kohn_1965}
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However, Loos and Gill have recently shown that there exists other UEGs which contain finite numbers of electrons (more like in a molecule), \cite{Loos_2011b,Gill_2012} and that they can be exploited to construct LDA functionals. \cite{Loos_2014a,Loos_2014b,Loos_2017a}
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Electrons restricted to remain on the surface of a $\cD$-sphere (where $\cD$ is the dimensionality of the surface of the sphere) are an example of finite UEGs (FUEGs). \cite{Loos_2011b}
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Here, we combine these FUEGs with the usual infinite UEG (IUEG) to construct a weigh-dependent LDA functional for ensembles (eLDA).
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The paper is organised as follows.
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In Sec.~\ref{sec:theo}, ...
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In Sec.~\ref{sec:theo}, the theory behind eDFT is presented.
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Section \ref{sec:func} provides details about the construction of the weight-dependent exchange-correlation functional.
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The results of our calculations for the prototypical \ce{H2} molecule are reported and discussed in Sec.~\ref{sec:res}.
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Finally, we draw our conclusions in Sec.~\ref{sec:ccl}.
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@ -185,17 +193,17 @@ built from an ensemble of $\Nens$ electronic states with individual energies $\E
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\end{align}
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One of the key feature of eDFT in the present context is that one can easily extract individual excitation energies from the ensemble energy via differentiation with respect to individual weights:
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\begin{equation}
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\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{(I)}
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\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{(I)},
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\end{equation}
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where we used the fact that $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$.
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In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}{\bw}[\n{}{}]$ such that
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\begin{equation}
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\E{}{\bw} = \min_{\n{}{}} \qty[ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} ]
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\E{}{\bw} = \min_{\n{}{}} \qty{ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} },
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\end{equation}
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where $\vext(\br{})$ is the external potential.
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In the KS formulation of eDFT, the universal ensemble functional (the weight-dependent analog of the Hohenberg-Kohn universal functional for ensembles) is decomposed as
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\begin{equation}
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\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
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\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}],
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\end{equation}
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where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}{}[\n{}{}]$ are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively with
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\begin{equation}
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@ -203,18 +211,17 @@ where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}{}[\n{}{}]$ are the noninteracting ensembl
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\E{\Hxc}{\bw}[\n{}{}]
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& = \E{\Ha}{}[\n{}{}] + \E{\xc}{\bw}[\n{}{}]
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\\
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& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'+ \E{\xc}{\bw}[\n{}{}]
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& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'+ \int \e{\xc}{\bw}[\n{}{}(\br{})] \n{}{}(\br{}) d\br{}.
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\end{split}
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\end{equation}
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Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIC) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\E{\xc}{\bw}[\n{}{}]$.
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The present work deals with the explicit construction of $\E{\xc}{\bw}[\n{}{}]$ at the LDA level in the case of the two-state ensemble (\ie, $\Nens = 2$).
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Note that the weight-independent Hartree functional $\E{\Ha}{}[\n{}{}]$ causes the infamous ghost-interaction error (GIC) \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017} in eDFT, which is supposed to be cancelled by the weight-dependent xc functional $\e{\xc}{\bw}[\n{}{}]$.
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%%%%%%%%%%%%%%%%%%
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%%% FUNCTIONAL %%%
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%%%%%%%%%%%%%%%%%%
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\section{Functional}
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\label{sec:func}
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The present work deals with the explicit construction of $\e{\xc}{\bw}[\n{}{}]$ at the LDA level in the case of the two-state ensemble (\ie, $\Nens = 2$) where both the ground state ($I=0$) and the first doubly-excited state ($I=1$) are considered.
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We adopt the usual decomposition, and write down the weight-dependent exchange-correlation functional as
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\begin{equation}
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@ -224,21 +231,20 @@ where $\e{x}{\ew{}}(\n{}{})$ and $\e{c}{\ew{}}(\n{}{})$ are the weight-dependent
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The construction of these two functionals is described below.
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Here, we restrict our study to spin-unpolarized systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$.
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The present weight-dependent eDFA is specifically designed for the calculation of double excitations within eDFT.
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As mentioned previously, we consider a two-state ensemble including the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the two-electron glomium system.
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All these states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$ where $R$ is the radius of the glome where the electrons are confined.
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We refer the interested reader to Refs.~\onlinecite{Loos_2011b} for more details about this paradigm.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Weight-dependent exchange functional}
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\label{sec:Ex}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The reduced (\ie, per electron) HF energy for these two states is
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We consider the ground- and doubly-excited states of the two-electron glomium system in its singlet ground state.
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These two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$ where $R$ is the radius of the glome where the electrons are confined.
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We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
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The reduced (\ie, per electron) Hartree-Fock (HF) energy for these two states is
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\begin{subequations}
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\begin{align}
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\e{HF}{(0)}(\n{}{}) & = \frac{4}{3\pi^{1/3}} \n{}{1/3},
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\e{HF}{(0)}(\n{}{}) & = \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
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\\
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\e{HF}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3} + \frac{176}{105\pi^{1/3}} \n{}{1/3}.
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\e{HF}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3} + \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
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\end{align}
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\end{subequations}
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These two energies can be conveniently decomposed as
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@ -250,15 +256,15 @@ with
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\begin{align}
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\kin{s}{(0)}(\n{}{}) & = 0,
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&
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\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{4/3}}{2} \n{}{2/3}.
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\kin{s}{(1)}(\n{}{}) & = \frac{3\pi^{2}}{2} \qty(\frac{\n{}{}}{\pi})^{2/3},
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\\
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\e{H}{(0)}(\n{}{}) & = \frac{8}{3\pi^{1/3}} \n{}{1/3},
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\e{H}{(0)}(\n{}{}) & = \frac{8}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
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&
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\e{H}{(1)}(\n{}{}) & = \frac{352}{105\pi^{1/3}} \n{}{1/3}.
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\e{H}{(1)}(\n{}{}) & = \frac{352}{105} \qty(\frac{\n{}{}}{\pi})^{1/3},
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\\
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\e{x}{(0)}(\n{}{}) & = - \frac{4}{3\pi^{1/3}} \n{}{1/3},
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\e{x}{(0)}(\n{}{}) & = - \frac{4}{3} \qty(\frac{\n{}{}}{\pi})^{1/3},
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&
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\e{x}{(1)}(\n{}{}) & = - \frac{176}{105\pi^{1/3}} \n{}{1/3}.
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\e{x}{(1)}(\n{}{}) & = - \frac{176}{105} \qty(\frac{\n{}{}}{\pi})^{1/3}.
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\end{align}
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\end{subequations}
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@ -270,10 +276,11 @@ we obtain
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\begin{align}
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\Cx{(0)} & = - \frac{4}{3} \qty( \frac{1}{\pi} )^{1/3},
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&
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\Cx{(1)} & = - \frac{176}{105} \qty( \frac{1}{\pi} )^{1/3}
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\Cx{(1)} & = - \frac{176}{105} \qty( \frac{1}{\pi} )^{1/3}.
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\end{align}
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We can now combine these two exchange functionals to create a weight-dependent exchange functional
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We can now combine these two exchange functionals to create a weight-dependent exchange functional for a two-state ensemble
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\begin{equation}
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\label{eq:exw}
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\begin{split}
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\e{x}{\ew{}}(\n{}{})
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& = (1-\ew{}) \e{x}{(0)}(\n{}{}) + \ew{} \e{x}{(1)}(\n{}{})
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@ -283,10 +290,9 @@ We can now combine these two exchange functionals to create a weight-dependent e
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\end{equation}
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with
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\begin{equation}
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\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}
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\Cx{\ew{}} = (1-\ew{}) \Cx{(0)} + \ew{} \Cx{(1)}.
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\end{equation}
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Amazingly, the weight dependence of the exchange functional can be transfered to the $\Cx{}$ coefficient.
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This is obvious but kind of nice.
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Quite remarkably, the weight dependence of the exchange functional can be transferred to the $\Cx{}$ coefficient, which is expected from a theoretical point of view but also a nice property from a more practical aspect.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -294,15 +300,16 @@ This is obvious but kind of nice.
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\label{sec:Ec}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Based on highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
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Based on highly-accurate calculations, \cite{Loos_2009a,Loos_2009c,Loos_2010e} one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
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\begin{equation}
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\label{eq:ec}
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\e{c}{(I)}(\n{}{}) = \frac{a_1^{(I)}}{1 + a_2^{(I)} \n{}{-1/6} + a_3^{(I)} \n{}{-1/3}},
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\end{equation}
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where the $a_k^{(I)}$'s are state-specific fitting parameters, which are provided in Table \ref{tab:OG_func}.
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The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
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Equation \eqref{eq:ec} provides two state-specific correlation DFAs based on a two-electron system.
|
||||
Combining these, one can build a two-state weight-dependent correlation eDFA:
|
||||
The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2011b}
|
||||
Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
|
||||
|
||||
Combining these, we build a two-state weight-dependent correlation functional:
|
||||
\begin{equation}
|
||||
\label{eq:ecw}
|
||||
\e{c}{\ew{}}(\n{}{}) = (1-\ew{}) \e{c}{(0)}(\n{}{}) + \ew{} \e{c}{(1)}(\n{}{}).
|
||||
|
|
@ -310,9 +317,9 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
|
|||
|
||||
%%% FIG 1 %%%
|
||||
\begin{figure}
|
||||
% \includegraphics[width=\linewidth]{Ec}
|
||||
\includegraphics[width=\linewidth]{fig1}
|
||||
\caption{
|
||||
Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = ...$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
|
||||
Reduced (i.e., per electron) correlation energy $\e{c}{(I)}$ [see Eq.~\eqref{eq:ec}] as a function of $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
|
||||
The data gathered in Table \ref{tab:Ref} are also reported.
|
||||
}
|
||||
\label{fig:Ec}
|
||||
|
|
@ -323,7 +330,7 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
|
|||
\begin{table}
|
||||
\caption{
|
||||
\label{tab:Ref}
|
||||
$-\e{c}{(I)}$ as a function of the radius of the glome $R$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
|
||||
$-\e{c}{(I)}$ as a function of the radius of the glome $R = 1/(\pi \n{}{})^{1/3}$ for the ground state ($I=0$), and the first doubly-excited state ($I=1$) of the (spin-unpolarized) two-electron glomium system.
|
||||
}
|
||||
\begin{ruledtabular}
|
||||
\begin{tabular}{ldd}
|
||||
|
|
@ -342,21 +349,11 @@ Combining these, one can build a two-state weight-dependent correlation eDFA:
|
|||
$20$ & & \\
|
||||
$50$ & & \\
|
||||
$100$ & & \\
|
||||
$150$ & & \\
|
||||
\end{tabular}
|
||||
\end{ruledtabular}
|
||||
\end{table}
|
||||
|
||||
Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant
|
||||
\begin{equation}
|
||||
\label{eq:ec}
|
||||
\e{c}{(I)}(n) = \frac{a^{(I)}\,n}{n + b^{(I)} \sqrt{n} + c^{(I)}},
|
||||
\end{equation}
|
||||
where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript.
|
||||
The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a}
|
||||
Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}.
|
||||
|
||||
|
||||
|
||||
%%% TABLE 1 %%%
|
||||
\begin{table*}
|
||||
\caption{
|
||||
|
|
@ -376,7 +373,12 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
|
|||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\subsection{LDA-centered functional}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
In order to make the two-electron-based eDFA defined in Eq.~\eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
|
||||
|
||||
\titou{Here, I shall explain our embedding scheme where we consider that a two-electron system (the impurity) is embedded in a larger system (the bath).
|
||||
Here the bath is the IUEG while the impurity is our two-electron systems.
|
||||
The weight-dependence only comes from the impurity, while the remaining effect originates from the bath.}
|
||||
|
||||
In order to make the two-electron-based eDFA defined in Eqs.~\eqref{eq:exw} and \eqref{eq:ecw} more universal and to ``center'' it on the jellium reference (as commonly done in DFT), we propose to \emph{shift} it as follows:
|
||||
\begin{equation}
|
||||
\label{eq:becw}
|
||||
\be{xc}{\ew{}}(\n{}{}) = (1-\ew{}) \be{xc}{(0)}(\n{}{}) + \ew{} \be{c}{(1)}(\n{}{}),
|
||||
|
|
@ -389,10 +391,10 @@ The local-density approximation (LDA) exchange-correlation functional is
|
|||
\begin{equation}
|
||||
\e{xc}{\LDA}(\n{}{}) = \e{x}{\LDA}(\n{}{}) + \e{c}{\LDA}(\n{}{}).
|
||||
\end{equation}
|
||||
where we use here the Dirac exchange functional and the VWN5 correlation functional
|
||||
where we use here the Dirac exchange functional \cite{Dirac_1930} and the VWN5 correlation functional \cite{Vosko_1980}
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\e{x}{\LDA}(\n{}{}) & = \Cx{\LDA} \n{}{1/3}
|
||||
\e{x}{\LDA}(\n{}{}) & = \Cx{\LDA} \n{}{1/3},
|
||||
\\
|
||||
\e{c}{\LDA}(\n{}{}) & \equiv \e{c}{\text{VWN5}}(\n{}{}).
|
||||
\end{align}
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user