minor corrections

TrUEGs.bib

@@ -1,13 +1,27 @@
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-%% Created for Pierre-Francois Loos at 2021-05-23 09:33:22 +0200 
+%% Created for Pierre-Francois Loos at 2021-05-23 14:27:26 +0200 
 
 
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+@article{Gori-Giorgi_2008,
+	abstract = {The correlation energy in density functional theory can be expressed exactly in terms of the change in the probability of finding two electrons at a given distance r12 (intracule density) when the electron--electron interaction is multiplied by a real parameter λ varying between 0 (Kohn--Sham system) and 1 (physical system). In this process{,} usually called adiabatic connection{,} the one-electron density is (ideally) kept fixed by a suitable local one-body potential. While an accurate intracule density of the physical system can only be obtained from expensive wavefunction-based calculations{,} being able to construct good models starting from Kohn--Sham ingredients would highly improve the accuracy of density functional calculations. To this purpose{,} we investigate the intracule density in the λ → ∞ limit of the adiabatic connection. This strong-interaction limit of density functional theory turns out to be{,} like the opposite non-interacting Kohn--Sham limit{,} mathematically simple and can be entirely constructed from the knowledge of the one-electron density. We develop here the theoretical framework and{,} using accurate correlated one-electron densities{,} we calculate the intracule densities in the strong interaction limit for few atoms. Comparison of our results with the corresponding Kohn--Sham and physical quantities provides useful hints for building approximate intracule densities along the adiabatic connection of density functional theory.},
+	author = {Gori-Giorgi, Paola and Seidl, Michael and Savin, Andreas},
+	date-added = {2021-05-23 14:27:13 +0200},
+	date-modified = {2021-05-23 14:27:22 +0200},
+	doi = {10.1039/B803709B},
+	issue = {23},
+	journal = {Phys. Chem. Chem. Phys.},
+	pages = {3440-3446},
+	title = {Intracule densities in the strong-interaction limit of density functional theory},
+	volume = {10},
+	year = {2008},
+	Bdsk-Url-1 = {http://dx.doi.org/10.1039/B803709B}}
+
 @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},

TrUEGs.tex

@@ -57,7 +57,7 @@ Concrete examples are provided in the case of two-electron systems.
 \maketitle
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\textbf{Uniform electron gases.}
+\section{Uniform electron gases}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Alongside the two Hohenberg-Kohn theorems \cite{Hohenberg_1964} which put density-functional theory (DFT) on firm mathematical grounds and the Kohn-Sham (KS) formalism \cite{Kohn_1965} that makes DFT practically feasible, the uniform electron gas (UEG) \cite{Loos_2016} is one of the many pieces of the puzzle that have made DFT \cite{ParrBook} so successful in the past thirty years.
 Indeed, apart from very few exceptions, most density-functional approximations are based, at some level at least, on the UEG via the so-called local-density approximation (LDA) \cite{Thomas_1927,Fermi_1927,Dirac_1930,Slater_1951,Ceperley_1980} which assumes that the electron density $\rho$ of an atom, a molecule, or a solid is locally uniform and has identical ``properties'' to the UEG with the same electron density.
@@ -74,7 +74,7 @@ In the so-called thermodynamic limit where both $n$ and $V$ goes to infinity but
 In the following, this paradigm is named the infinite UEG (IUEG) for obvious reasons.
  
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-\textbf{Finite uniform electron gases.}
+\section{Finite uniform electron gases}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Recently, it has been shown that one can create finite UEGs (FUEGs) by placing a finite number of electrons onto the surface of a sphere of radius $R$. \cite{Tempere_2002,Tempere_2007,Seidl_2007,Loos_2009a,Loos_2009c,Loos_2010e,Loos_2011b,Gill_2012,Loos_2018b} 
 Of course, FUEGs only appear for well-defined electron numbers and electronic states. \cite{Rogers_2016,Rogers_2017}
@@ -148,7 +148,7 @@ For the other electronic states corresponding to higher total angular momentum,
 %%% %%% %%% %%%
 
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-\textbf{Transient uniform electron gases.}
+\section{Transient uniform electron gases}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 As evidenced by Eq.~\eqref{eq:rho}, both the angular and interelectronic wave functions contribute to the electron density and a subtle interplay between these two quantities may result in a uniform density.
 To illustrate this, let us consider explicit examples.
@@ -218,11 +218,11 @@ Actually, it is even possible for a system to be a TUEG within the exact treatme
 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.
 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}
+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{Gori-Giorgi_2008,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 value of $R$) to overcome this drawback. \cite{Seidl_2010,Burton_2019a,Marie_2020}
 
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-\textbf{Concluding remarks.}
+\section{Concluding remarks}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 Here, we have introduced the concept of transient UEGs (TUEGs), a novel family of electron gases that exhibit, in very particular conditions, homogenous densities.
 Using the electrons-on-a-sphere model, we have presented an example of such TUEGs created thanks to the competing effects of the Pauli exclusion principle and the creation of an attractive electron pair. 
@@ -231,11 +231,13 @@ As a final remark, we would like to mention that a very similar analysis can be
 The three-dimensional version where electrons are confined to the surface of a 3-sphere (or glome) could be of particular interest, especially in the context of the development of new exchange-correlation functionals within DFT.\cite{Sun_2015,Agboola_2015,Loos_2017a}
 
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-\textbf{Acknowledgements.}
+\section*{Acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).
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+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\section*{References}
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 \bibliography{TrUEGs}
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