add HF section

TrUEGs.bib

@@ -1,13 +1,26 @@
 %% This BibTeX bibliography file was created using BibDesk.
 %% http://bibdesk.sourceforge.net/
 
-%% Created for Pierre-Francois Loos at 2021-04-08 08:22:37 +0200 
+%% Created for Pierre-Francois Loos at 2021-05-23 09:33:22 +0200 
 
 
 %% Saved with string encoding Unicode (UTF-8) 
 
 
 
+@article{Pearson_2009,
+	author = {Jason K. Pearson and Peter M.W. Gill and Jesus M. Ugalde and Russell J. Boyd},
+	date-added = {2021-05-23 09:33:16 +0200},
+	date-modified = {2021-05-23 09:33:22 +0200},
+	doi = {10.1080/00268970902740563},
+	journal = {Mol. Phys.},
+	number = {8-12},
+	pages = {1089-1093},
+	title = {Can correlation bring electrons closer together?},
+	volume = {107},
+	year = {2009},
+	Bdsk-Url-1 = {https://doi.org/10.1080/00268970902740563}}
+
 @book{VignaleBook,
 	address = {Cambridge, England},
 	author = {G. F. Giuliani and G. Vignale},

TrUEGs.tex

@@ -1,5 +1,5 @@
 \documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
-\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,pifont,wrapfig}
+\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,pifont,wrapfig,siunitx}
 
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
@@ -188,7 +188,7 @@ Figure \ref{fig:3P} shows the behavior of $c_0 - 2 c_1/3 + c_2/5$ as a function
 As one can see, $R_\text{UEG}$ is negative (which corresponds to an attractive ``electron'' pair \cite{Seidl_2007,Seidl_2010} or exciton \cite{Pedersen_2010,Loos_2012c}) and gets larger (in absolute value) for excited states.
 Because this model has an inhomogeneous electron density except for a unique $R$ value, we name these ``ephemeral'' systems as transient UEGs (TUEGs).
 Note that this feature was first discovered by Seidl in Appendix A of Ref.~\onlinecite{Seidl_2007}. 
-He also provided an estimate $R_\text{UEG} \approx -5.3$ for the $^3P$ ground state, a rather good estimate that we refine here to $R_\text{UEG} \approx -5.32527$.
+He also provided an estimate $R_\text{UEG} \approx \SI{-5.3}{\bohr}$ for the $^3P$ ground state, a rather good estimate that we refine here to $R_\text{UEG} \approx \SI{-5.32527}{\bohr}$.
 
 For the $^1P$ states, the condition $c_0 + 2 c_1/3 + c_2/5 = 0$ [see Eq.~\eqref{eq:condition}] cannot be fulfilled and, hence, these states never exhibit uniform densities.
 This further highlights the subtle balance that must be accomplished between the spin and spatial parts of the wave function [see Eq.~\eqref{eq:rho}] and can help us rationalizing why the $^3P$ states are TUEGs.
@@ -201,12 +201,25 @@ For certain $R$ values, the attractive effect stemming from the spatial part of
 In higher-energy excited states, the same-spin electrons are further away as compared to the ground state due to the larger number of nodes in the excited-state wave functions.
 Therefore, the magnitude of the attractive effect has to be larger to compensate it, which corresponds to more negative values of $R$.
 
-While for the IUEG and FUEGs, the density is uniform independently of the level of theory, \ie, the system has homogeneous density within the exact theory but also within the HF approximation (unless the spin and/or spatial symmetry is broken \cite{Fukutome_1981,Stuber_2003,VignaleBook}), the value of $R_\text{UEG}$ is, \textit{a priori}, highly dependent of the level of theory for TUEGs.
+%%% FIG 2 %%%
+\begin{figure}
+	\includegraphics[width=\linewidth]{HF}
+	\caption{
+	Hartree-Fock electron density $\rho^\text{HF}(\theta)$ as a function of the polar angle $\theta$ for various $R$ values.
+	\label{fig:HF}}
+\end{figure}
+%%% %%% %%% %%%
+
+For the IUEG and FUEGs, the density is uniform independently of the level of theory, \ie, the system has homogeneous density within the exact theory or any approximate methods (such as the HF approximation) unless the spin and/or spatial symmetry is broken. \cite{Fukutome_1981,Stuber_2003,VignaleBook}
+However, the value of $R_\text{UEG}$ is, \textit{a priori}, highly dependent of the level of theory for TUEGs.
 Indeed, it is very unlikely that the exact theory and the HF approximation provide the same value of $R_\text{UEG}$ as the uniformity stems from the competition between Fermi effects originating from the antisymmetric nature of the wave function (which are well described at the HF level) and correlation effects (which are absent, by definition, at the HF level). 
-Actually, it is even possible for a system to be a TUEG within the exact treatment and being non-uniform for any $R$ values at the HF level.
+Actually, it is even possible for a system to be a TUEG within the exact treatment and being non-uniform for any $R$ values at the HF level, and this seems to be the case for the present two-electron example.
 
-
+Expanding the two HF orbitals of the $^3P$ ground state in a basis of zonal harmonics $Y_{\ell}(\theta) \equiv Y_{\ell 0}(\theta,\phi)$, we have not found any $R$ values for which the HF electron density, $\rho^\text{HF}(\theta)$, is uniform.
-%\titou{What about the nodes? Dyson orbitals? Cf Paola's paper.}
+However, at $R \approx \SI{-7}{\bohr}$, $\rho^\text{HF}$ is \textit{locally} uniform around $\theta = \pi/2$ (\ie, in the $xy$ plane), as shown in Fig.~\ref{fig:HF}.
+We believe that this outcome is a direct consequence of the single-determinant nature of the HF approximation which can only include, by definition, one of the three equivalent $sp$ configurations (\ie, $sp_x$,  $sp_y$, and $sp_z$).
+The fact that this phenomenon appears at larger (absolute) $R$ values in the HF approximation is not surprising as, contrary to the repulsive regime (\ie, $R > 0$) where the electrons are too close to each other at the HF level (compared to the exact picture), \cite{Pearson_2009} in the attractive regime (\ie, $R < 0$) they are too far away from each other. 
+This implies that the interaction strength has to be greater (which is equivalent to a larger absolute $R$ value) to overcome this drawback. \cite{Seidl_2010,Burton_2019a,Marie_2020}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \textbf{Concluding remarks.}