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details for FUEG

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22
Manuscript/FarDFT.bib
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@ -1,13 +1,33 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2019-11-21 21:56:23 +0100
%% Created for Pierre-Francois Loos at 2019-11-23 21:56:54 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Avery_1993,
Author = {J. Avery},
Date-Added = {2019-11-23 21:56:19 +0100},
Date-Modified = {2019-11-23 21:56:52 +0100},
Doi = {10.1021/j100112a048},
Journal = {J. Phys. Chem.},
Pages = {2406--2412},
Title = {Selected applications of hyperspherical harmonics in quantum theory},
Volume = {97},
Year = {1993}}
@book{AveryBook,
Address = {Dordrecht},
Author = {J. Avery},
Date-Added = {2019-11-23 21:56:19 +0100},
Date-Modified = {2019-11-23 21:56:19 +0100},
Publisher = {Kluwer Academic},
Title = {Hyperspherical harmonics: applications in quantum theory},
Year = {1989}}
@article{Loos_2019,
Author = {Loos, Pierre-Fran{\c c}ois and Boggio-Pasqua, Martial and Scemama, Anthony and Caffarel, Michel and Jacquemin, Denis},
Date-Added = {2019-11-21 21:56:17 +0100},

43
Manuscript/FarDFT.tex
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@ -257,14 +257,17 @@ The construction of these two functionals is described below.
Here, we restrict our study to spin-unpolarised systems, \ie, $\n{\uparrow}{} = \n{\downarrow}{} = \n{}{}/2$ (where $\n{\uparrow}{}$ and $\n{\downarrow}{}$ are the spin-up and spin-down electron densities).
Extension to spin-polarised systems will be reported in future work.
Although this choice is far from being unique, we consider here the singlet ground state and the first singlet doubly-excited state of a two-electron FUEG which consists of two electrons confined to the surface of a 3-sphere (also known as a glome).\cite{Loos_2009a,Loos_2009c,Loos_2010e}
These two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$, where $R$ is the radius of the 3-sphere onto which the electrons are confined.
Indeed, the orbitals for an electron on a 3-sphere of unit radius are the normalized hyperspherical harmonics $Y_{\ell\mu}$, where $\ell$ is the principal quantum number and $\mu$ is a composite index of the remaining two quantum numbers. \cite{AveryBook, Avery_1993}
As mentioned above, we confine our attention to paramagnetic (or unpolarized) systems, and in particular to the simple two-electron system in which the orbital with $\ell = 0$ is doubly-occupied by one spin-up and one spin-down electron, thus yielding an electron density that is uniform on the 3-sphere.
Note that the present paradigm is equivalent to the IUEG model in the thermodynamic limit. \cite{Loos_2011b}
We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent exchange functional}
\label{sec:Ex}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\titou{T2: More details required to understand what is glomium.}
We consider the ground- and doubly-excited states of the two-electron glomium system in its singlet ground state. \cite{Loos_2009a,Loos_2009c,Loos_2010e}
These two states have the same (uniform) density $\n{}{} = 2/(2\pi^2 R^3)$ where $R$ is the radius of the glome onto which the electrons are confined.
We refer the interested reader to Refs.~\onlinecite{Loos_2011b,Loos_2017a} for more details about this paradigm.
The reduced (\ie, per electron) Hartree-Fock (HF) energies for these two states are
\begin{subequations}
@ -364,18 +367,18 @@ Combining these, we build a two-state weight-dependent correlation functional:
& \tabc{Ground state} & \tabc{Doubly-excited state} \\
$R$ & \tabc{$I=0$} & \tabc{$I=1$} \\
\hline
$0$ & $0.023818$ & $0.014463$ \\
$0.1$ & $0.023392$ & $0.014497$ \\
$0.2$ & $0.022979$ & $0.014523$ \\
$0.5$ & $0.021817$ & $0.014561$ \\
$1$ & $0.020109$ & $0.014512$ \\
$2$ & $0.017371$ & $0.014142$ \\
$5$ & $0.012359$ & $0.012334$ \\
$10$ & $0.008436$ & $0.009716$ \\
$20$ & $0.005257$ & $0.006744$ \\
$50$ & $0.002546$ & $0.003584$ \\
$100$ & $0.001399$ & $0.002059$ \\
$150$ & $0.000972$ & $0.001458$ \\
$0$ & $0.023\,818$ & $0.014\,463$ \\
$0.1$ & $0.023\,392$ & $0.014\,497$ \\
$0.2$ & $0.022\,979$ & $0.014\,523$ \\
$0.5$ & $0.021\,817$ & $0.014\,561$ \\
$1$ & $0.020\,109$ & $0.014\,512$ \\
$2$ & $0.017\,371$ & $0.014\,142$ \\
$5$ & $0.012\,359$ & $0.012\,334$ \\
$10$ & $0.008\,436$ & $0.009\,716$ \\
$20$ & $0.005\,257$ & $0.006\,744$ \\
$50$ & $0.002\,546$ & $0.003\,584$ \\
$100$ & $0.001\,399$ & $0.002\,059$ \\
$150$ & $0.000\,972$ & $0.001\,458$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
@ -387,13 +390,13 @@ Combining these, we build a two-state weight-dependent correlation functional:
Parameters of the correlation functionals for each individual state defined in Eq.~\eqref{eq:ec}.
The values of $a_1$ are obtained to reproduce the exact high density correlation energy of each individual state, while $a_2$ and $a_3$ are fitted on the numerical values reported in Table \ref{tab:Ref}.}
\begin{ruledtabular}
\begin{tabular}{lcc}
\begin{tabular}{ldd}
& \tabc{Ground state} & \tabc{Doubly-excited state} \\
& \tabc{$I=0$} & \tabc{$I=1$} \\
\hline
$a_1$ & $-0.0238184$ & $-0.0144633$ \\
$a_2$ & $+0.00540994$ & $-0.0506019$ \\
$a_3$ & $+0.0830766$ & $+0.0331417$ \\
$a_1$ & -0.023\,818\,4 & -0.014\,463\,3 \\
$a_2$ & +0.005\,409\,94 & -0.050\,601\,9 \\
$a_3$ & +0.083\,076\,6 & +0.033\,141\,7 \\
\end{tabular}
\end{ruledtabular}
\end{table}