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Manuscript/FarDFT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-14 21:16:19 +0100 %% Created for Pierre-Francois Loos at 2019-11-15 10:58:08 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@ -8224,13 +8224,12 @@
@article{Gidopoulos_2002, @article{Gidopoulos_2002,
Author = {Gidopoulos, N. I. and Papaconstantinou, P. G. and Gross, E. K. U.}, Author = {Gidopoulos, N. I. and Papaconstantinou, P. G. and Gross, E. K. U.},
Date-Modified = {2019-11-15 10:58:08 +0100},
Doi = {10.1103/PhysRevLett.88.033003}, Doi = {10.1103/PhysRevLett.88.033003},
File = {/Users/loos/Zotero/storage/RRB3BXVQ/Gidopoulos et al. - 2002 - Spurious Interactions, and Their Correction, in th.pdf},
Issn = {0031-9007, 1079-7114},
Journal = {Phys. Rev. Lett.}, Journal = {Phys. Rev. Lett.},
Language = {en},
Month = jan, Month = jan,
Number = {3}, Number = {3},
Pages = {033003},
Title = {Spurious {{Interactions}}, and {{Their Correction}}, in the {{Ensemble}}-{{Kohn}}-{{Sham Scheme}} for {{Excited States}}}, Title = {Spurious {{Interactions}}, and {{Their Correction}}, in the {{Ensemble}}-{{Kohn}}-{{Sham Scheme}} for {{Excited States}}},
Volume = {88}, Volume = {88},
Year = {2002}, Year = {2002},

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Manuscript/FarDFT.tex
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@ -154,7 +154,7 @@ In order to go beyond the adiabatic approximation, a dressed TD-DFT approach has
In this approach the exchange-correlation kernel is made frequency dependent \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states. In this approach the exchange-correlation kernel is made frequency dependent \cite{Romaniello_2009a,Sangalli_2011}, which allows to treat doubly-excited states.
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} 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}
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{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} 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}
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. 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.
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. 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.
We believe that it is due to the lack of accurate approximations for eDFT. We believe that it is due to the lack of accurate approximations for eDFT.
@ -188,7 +188,7 @@ One of the key feature of eDFT in the present context is that one can easily ext
\pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{(I)} \pdv{\E{}{\bw}}{\ew{I}} = \E{}{(I)} - \E{}{(0)} = \Ex{(I)}
\end{equation} \end{equation}
where we used the fact that $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$. where we used the fact that $\ew{0} = 1 - \sum_{I \ne 0} \ew{I}$.
For such an ensemble, one can define a universal ensemble functional $\F{}{\bw}[\n{}{}]$ such that In GOK-DFT, one defines a universal (weight-dependent) ensemble functional $\F{}{\bw}[\n{}{}]$ such that
\begin{equation} \begin{equation}
\E{}{\bw} = \min_{\n{}{}} \qty[ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} ] \E{}{\bw} = \min_{\n{}{}} \qty[ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} ]
\end{equation} \end{equation}
@ -197,7 +197,7 @@ In the KS formulation of eDFT, the universal ensemble functional (the weight-dep
\begin{equation} \begin{equation}
\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}] \F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
\end{equation} \end{equation}
where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}[\n{}{}]$ are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively with where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}{}[\n{}{}]$ are the noninteracting ensemble kinetic energy functional and ensemble Hartree-exchange-correlation (Hxc) functional, respectively with
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\E{\Hxc}{\bw}[\n{}{}] \E{\Hxc}{\bw}[\n{}{}]
@ -206,6 +206,9 @@ where $\Ts{\bw}[\n{}{}]$ and $\E{\Hxc}[\n{}{}]$ are the noninteracting ensemble
& = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'+ \E{\xc}{\bw}[\n{}{}] & = \frac{1}{2} \iint \frac{\n{}{}(\br{}) \n{}{}(\br{}')}{\abs{\br{}-\br{}'}} d\br{} d\br{}'+ \E{\xc}{\bw}[\n{}{}]
\end{split} \end{split}
\end{equation} \end{equation}
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 $\E{\xc}{\bw}[\n{}{}]$.
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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%%% FUNCTIONAL %%% %%% FUNCTIONAL %%%
@ -412,7 +415,7 @@ This is a crucial point as we intend to incorporate into standard functionals (w
Finally, we note that, by construction, Finally, we note that, by construction,
\begin{equation} \begin{equation}
\left. \pdv{\be{xc}{\ew{}}[\n{}{}]}{\ew{J}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{xc}{(J)}[\n{}{\ew{}}(\br)] - \be{xc}{(0)}[\n{}{\ew{}}(\br)]. \left. \pdv{\be{xc}{\ew{}}[\n{}{}]}{\ew{I}}\right|_{\n{}{} = \n{}{\ew{}}(\br)} = \be{xc}{(I)}[\n{}{\ew{}}(\br)] - \be{xc}{(0)}[\n{}{\ew{}}(\br)].
\end{equation} \end{equation}
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