From e450c46ccc88a0bcdac7a64f941e0f46a457055f Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sat, 23 Nov 2019 22:13:29 +0100 Subject: [PATCH] details for FUEG --- Manuscript/FarDFT.bib | 22 +++++++++++++++++++++- Manuscript/FarDFT.tex | 43 +++++++++++++++++++++++-------------------- 2 files changed, 44 insertions(+), 21 deletions(-) diff --git a/Manuscript/FarDFT.bib b/Manuscript/FarDFT.bib index f524953..bbb3834 100644 --- a/Manuscript/FarDFT.bib +++ b/Manuscript/FarDFT.bib @@ -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}, diff --git a/Manuscript/FarDFT.tex b/Manuscript/FarDFT.tex index 6d43bc5..e3420b7 100644 --- a/Manuscript/FarDFT.tex +++ b/Manuscript/FarDFT.tex @@ -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}