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Pierre-Francois Loos 2019-11-15 11:54:24 +01:00
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Manuscript/FarDFT.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% 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
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@ -8224,13 +8224,12 @@
@article{Gidopoulos_2002,
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},
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.},
Language = {en},
Month = jan,
Number = {3},
Pages = {033003},
Title = {Spurious {{Interactions}}, and {{Their Correction}}, in the {{Ensemble}}-{{Kohn}}-{{Sham Scheme}} for {{Excited States}}},
Volume = {88},
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.
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.
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.
@ -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)}
\end{equation}
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}
\E{}{\bw} = \min_{\n{}{}} \qty[ \F{}{\bw}[\n{}{}] + \int \vext(\br{}) \n{}{}(\br{}) d\br{} ]
\end{equation}
@ -197,7 +197,7 @@ In the KS formulation of eDFT, the universal ensemble functional (the weight-dep
\begin{equation}
\F{}{\bw}[\n{}{}] = \Ts{\bw}[\n{}{}] + \E{\Hxc}{\bw}[\n{}{}]
\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{split}
\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{}{}]
\end{split}
\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 %%%
@ -412,7 +415,7 @@ This is a crucial point as we intend to incorporate into standard functionals (w
Finally, we note that, by construction,
\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}
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