ready for submission

TrUEGs.tex

@@ -31,7 +31,7 @@
 \newcommand{\VRIJE}{Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modelling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands}
 
 % COUPLING CONSTANT:
-\newcommand{\cc}{{\color{blue} \mu}}
+\newcommand{\cc}{\mu}
 
 \begin{document}	
 
@@ -40,8 +40,6 @@
 \author{Pierre-Fran\c{c}ois \surname{Loos}}
 \email{loos@irsamc.ups-tlse.fr}
 \affiliation{\LCPQ}
-%\author{Paola \surname{Gori-Giorgi}}
-%\affiliation{\VRIJE}
 \author{Michael \surname{Seidl}}
 \affiliation{\VRIJE}
 
@@ -51,7 +49,7 @@
 %	\includegraphics[width=\linewidth]{TOC}
 %\end{wrapfigure}
 The uniform electron gas (UEG), a hypothetical system with finite homogenous electron density composed by an infinite number of electrons in a box of infinite volume, is the practical pillar of density-functional theory (DFT) and the foundation of the most acclaimed approximation of DFT, the local-density approximation (LDA). 
-In the last thirty years, the knowledge of analytical parametrizations of the infinite UEG (IUEG) exchange-correlation energy has allowed researchers to perform countless of approximate electronic structure calculations for atoms, molecules, and solids.
+In the last thirty years, the knowledge of analytical parametrizations of the infinite UEG (IUEG) exchange-correlation energy has allowed researchers to perform a countless number of approximate electronic structure calculations for atoms, molecules, and solids.
 Recently, it has been shown that the traditional concept of the IUEG is not the unique example of UEGs, and systems, in their lowest-energy state, consisting of electrons that are confined to the surface of a sphere provide a new family of UEGs with more customizable properties.
 Here, we show that, some of the excited states associated with these systems can be classified as transient UEGs (TUEGs) as their electron density is only homogenous for very specific values of the radius of the sphere even though the electronic wave function is not rotationally invariant.
 Concrete examples are provided in the case of two-electron systems.
@@ -66,73 +64,75 @@ Alongside the two Hohenberg-Kohn theorems \cite{Hohenberg_1964} which put densit
 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.
 
 Thanks to the construction of exchange-correlation LDA functionals \cite{Slater_1951,Vosko_1980,Perdew_1981,Perdew_1992,Chachiyo_2016} which can be loosely seen as a one-to-one mapping between a given value of the electron density and the exchange-correlation energy of the UEG, one can then straightforwardly compute, within KS-DFT, the electronic ground-state energy and properties of any molecules or materials with, nonetheless, a certain degree of approximation inherently associated with the approximate nature of the exchange-correlation LDA functional.
-One can also access excited states via the time-dependent version of DFT. \cite{Runge_1984,Casida_1995,Petersilka_1996,UllrichBook}
+Moreover, one can also access excited states via the time-dependent version of DFT. \cite{Runge_1984,Casida_1995,Petersilka_1996,UllrichBook}
 
 As commonly done, the LDA can be refined by adding up new ingredients, such as the gradient of the density $\nabla \rho$ [which defines the generalized gradient approximation (GGA)], \cite{Perdew_1986,Becke_1988,Lee_1988,Perdew_1996} the kinetic energy density $\tau$ (meta-GGA), \cite{Becke_1988b,Sun_2015} exact Hartree-Fock (HF) exchange (yielding the so-called hybrid functionals), \cite{Becke_1993a,Becke_1993b,Adamo_1999} and others. 
 Each of these quantities defines a new rung of the well-known Jacob ladder of DFT \cite{Perdew_2001} that is supposed to bring electronic structure theory calculations from the evil Hartree world to the chemical accuracy heaven.
 
-The UEG, also known as jellium in some context, \cite{Loos_2016} is a hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively charged jelly of infinite volume. \cite{ParrBook,Loos_2016} It can be "created" via a \textit{gedanken} experiment by pouring electrons in an expandable box while keeping the ratio $\rho = N/V$ of the number of electrons $N$ and the volume of the box $V$ constant. In the so-called thermodynamic limit where both $N$ and $V$ goes to infinity but $\rho$ remains finite, the electron density eventually becomes homogeneous. In the following, this paradigm is named the infinite UEG (IUEG) for obvious reasons.
+The UEG, also known as jellium in some context, \cite{Loos_2016} is a hypothetical infinite substance where an infinite number of electrons ``bathe'' in a (uniform) positively charged jelly of infinite volume. \cite{ParrBook,Loos_2016} 
+It can be ``created'' via a \textit{gedanken} experiment by pouring electrons in an expandable box while keeping the ratio $\rho = N/V$ of the number of electrons $N$ and the volume of the box $V$ constant. 
+In the so-called thermodynamic limit where both $N$ and $V$ goes to infinity but $\rho$ remains finite, the electron density eventually becomes homogeneous. In the following, this paradigm is named the infinite UEG (IUEG) for obvious reasons.
  
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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}
-In particular, the spin-unpolarized ground state of $N$ electrons on a sphere has a homogeneous density for $N = 2(\ell_{\rm max}+1)^2$ (where $\ell_{\rm max} \in \mathbb{N}$) for any $R$ values, and this holds also within the HF approximation. \cite{Loos_2011b}
+In particular, the spin-unpolarized ground state of $N$ electrons on a sphere has a homogeneous density for $N = 2(\ell_\text{max}+1)^2$ (where $\ell_\text{max} \in \mathbb{N}$) for any $R$ values, and this holds also within the HF approximation. \cite{Loos_2011b}
 This property comes from the addition theorem of the spherical harmonics \cite{NISTbook} $Y_{\ell m}(\bm{\Omega})$ (which are the spatial orbitals of the system in this particular case):
 \begin{equation}
 	\label{eq:ellMax}
-	\sum_{\ell=0}^{\ell_{\rm max}} \sum_{m=-\ell}^{+\ell} Y_{\ell m}^*(\bm{\Omega}) Y_{\ell m}(\bm{\Omega}) = \frac{(\ell_{\rm max}+1)^2}{2\pi^2}
+	\sum_{\ell=0}^{\ell_{\rm max}} \sum_{m=-\ell}^{+\ell} Y_{\ell m}^*(\bm{\Omega}) Y_{\ell m}(\bm{\Omega}) = \frac{(\ell_\text{max}+1)^2}{2\pi^2},
 \end{equation}
 where $\bm{\Omega}=(\theta,\phi)$ gathers the polar and azimuthal angles, respectively.
 Thanks to this key property, these FUEGs have be employed to construct alternative LDA functionals for both KS-DFT \cite{Loos_2014b,Loos_2017a} and ensemble DFT. \cite{Loos_2020g,Marut_2020}
-Besides, hints of the equivalence of the FUEG and IUEG models have been found in the thermodynamic limit, \ie, when $\ell_{\rm max} \to \infty$. \cite{Bowick_2002,Loos_2011b}
+Besides, hints of the equivalence of the FUEG and IUEG models have been found in the thermodynamic limit, \ie, when $\ell_\text{max} \to \infty$. \cite{Bowick_2002,Loos_2011b}
 
-The case with $N=2$ electrons ($\ell_{\rm max} = 0$) is of particular interest \cite{Seidl_2007,Loos_2009a,Loos_2018b} as it has been shown to be extremely useful for testing electronic structure methods \cite{Mitas_2006,Seidl_2007,Loos_2009a,Pedersen_2010,Loos_2012c,Schindlmayr_2013,Loos_2015b,Sun_2016,Loos_2018b} and is, furthermore, exactly solvable for a countably infinite set of $R$ values. \cite{Loos_2009c,Loos_2010e,Loos_2012}
+The case with $N=2$ electrons ($\ell_\text{max} = 0$) is of particular interest \cite{Seidl_2007,Loos_2009a,Loos_2018b} as it has been shown to be extremely useful for testing electronic structure methods \cite{Mitas_2006,Seidl_2007,Loos_2009a,Pedersen_2010,Loos_2012c,Schindlmayr_2013,Loos_2015b,Sun_2016,Loos_2018b} and is, furthermore, exactly solvable for a countably infinite set of $R$ values. \cite{Loos_2009c,Loos_2010e,Loos_2012}
 In this case, the many-body hamiltonian reads
 \begin{equation}
 	\label{eq:H}
-	\Hat{H} = \frac{\hat{\ell}_1^2 + \hat{\ell}_2^2}{2m\,R^2} + \lambda\,\frac{e^2}{r_{12}}.
+	\Hat{H} = \frac{\hat{\ell}_1^2 + \hat{\ell}_2^2}{2m\,R^2} + \lambda\frac{e^2}{r_{12}}.
 \end{equation}
-The squares $\hat{\ell}_n^2=-\hbar^2\hat{A}_n$ of the (orbital) angular momentum operators are essentially the angular parts $\hat{A}_n$ of the Laplacian,
+The squares $\hat{\ell}_i^2=-\hbar^2\hat{L}_i$ of the (orbital) angular momentum operators are essentially the angular parts $\hat{L}_i$ of the Laplacian,
 \begin{equation}
-\hat{A} = \frac1{\sin\theta}\,\frac{\partial}{\partial\theta}\,\sin\theta\,\frac{\partial}{\partial\theta}
+\hat{L} = \frac{1}{\sin\theta}\pdv{}{\theta}\sin\theta \pdv{}{\theta} + \frac{1}{\sin^2\theta}\pdv[2]{}{\phi},
- \;+\;\frac1{\sin^2\theta}\,\frac{\partial^2}{\partial\phi^2},
 \end{equation}
 while $r_{12}$ is the \textit{spatial distance} between the two electrons, \ie, the electrons interact Coulombically \textit{through} the sphere,
 \begin{equation}
-r_{12} = R \sqrt{2(1 - \cos\gamma)} \equiv r_{12}(\gamma).
+	r_{12} = R \sqrt{2(1 - \cos\gamma)} \equiv r_{12}(\gamma).
 \end{equation}
 Here, $\gamma=\gamma(\Omega_1,\Omega_2)$ is the angle between the two electrons on the sphere
 (viewed from the spherical center),
 \begin{equation}
 	\label{eq:cosGamma}
-\cos\gamma = \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 \cos(\phi_1 - \phi_2).
+	\cos\gamma = \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 \cos(\phi_1 - \phi_2).
 \end{equation}
 In Eq.~\eqref{eq:H}, we have introduced a coupling constant $\lambda$ which in the real universe (where the electrons, each with charge $-e$, \textit{repel} each other) has the value $\lambda=1$. However, in addition to this realistic situation, we here also wish to consider cases with $\lambda\ne1$, including the non-interacting case $\lambda=0$. In particular, we shall consider the interesting case $\lambda<0$ when the electrons \textit{attract} each other.
 
 In atomic units ($m=e^2=\hbar=1$) where $R$ is given in units of the bohr radius $a_0=0.529$\AA, our hamiltonian reads
 \begin{equation}
 	\label{eq:H2}
-	\Hat{H} = \frac1{R^2}\left\{ - \frac{\hat{A}_1+\hat{A}_2}2 \; + \; \cc \cdot \frac{1}{\sqrt{2(1 - \cos\gamma)}} \right\},
+	\Hat{H} = \frac1{R^2}\qty{ - \frac{\hat{L}_1+\hat{L}_2}2 + \cc \frac{1}{\sqrt{2(1 - \cos\gamma)}} },
 \end{equation}
-with an effective coupling constant,
+with an effective (dimensionless) coupling constant,
 \begin{equation}
 	\label{eq:cc}
 \cc = \lambda R.
 \end{equation}
 Obviously, different interaction strengths $\lambda\ne0$ at a fixed radius $R>0$ are equivalent to different radii $R\ne0$ at 
 a fixed interaction strength $\lambda>0$ (where a negative sign of $R$ has no geometric meaning but simply describes attractive 
-electrons): Just as in the infinite UEG, the limit of high (low) density corresponds to the limit of weak (strong) interaction.
+electrons). 
+Just as in the IUEG, the limit of high (low) density corresponds to the limit of weak (strong) interaction.
 
 Following Breit, \cite{Breit_1930} one can write the total electronic wave function as
 \begin{equation}
 \label{eq:PhiOLD}
-	\Phi(\bm{x}_1,\bm{x}_2) = \Xi(s_1,s_2) \, \chi(\Omega_1,\Omega_2) \, \Psi(\gamma)
+	\Phi(\bm{x}_1,\bm{x}_2) = \Xi(s_1,s_2) \, \chi(\Omega_1,\Omega_2) \, \Psi(\gamma),
 \end{equation}
 where $\Xi$, $\chi$ and $\Psi$ are the spin, the non-interacting angular and the interelectronic angular wave functions, respectively, and $\bm{x}_i = (s_i,\Omega_i)$ is a composite coordinate gathering the spin coordinate $s_i$ and the spatial (angular) coordinate $\Omega_i$ associated with the $i$th electron.
 
-In the non-interacting limit $\lambda=\cc=0$, we have $\Psi(\gamma)=1$. In cases with finite interaction, $\cc\ne0$, this interelectronic wavefunction $\Psi(\gamma)=\Psi_{\cc}(\gamma)$, depending on the value of $\cc$, is either known analytically {\color{red} (Ref.?)} or must be computed numerically.
+In the non-interacting limit $\lambda=\cc=0$, we have $\Psi(\gamma)=1$. In cases with finite interaction, $\cc\ne0$, this interelectronic wavefunction $\Psi(\gamma)=\Psi_{\cc}(\gamma)$, depending on the value of $\cc$, is either known analytically \cite{Loos_2009c,Loos_2010e,Loos_2012} or must be computed numerically.
   
 The singlet and triplet spin wave functions read \cite{BetheBook}
 \begin{subequations}
@@ -150,10 +150,10 @@ The singlet and triplet spin wave functions read \cite{BetheBook}
 Since the factor $\Psi(\gamma)$ in Eq.~\eqref{eq:PhiOLD} is symmetrical (upon swapping the coordinates of the two electrons), see Eq.~\eqref{eq:cosGamma}, the symmetry of the non-interacting angular wave functions $\chi(\Omega_1,\Omega_2)$ must be opposite to the one of the spin factor $\Xi(s_1,s_2)$. Consequently (as in the helium atom), the ground-state is a singlet,
 \begin{equation}
 	\label{eq:spin_1S}
-	\chi_{^1S} (\Omega_1,\Omega_2) = Y_{00}(\Omega_1)Y_{00}(\Omega_2) = \frac1{4\pi}
+	\chi_{^1S} (\Omega_1,\Omega_2) = Y_{00}(\Omega_1)Y_{00}(\Omega_2) = \frac1{4\pi},
 \end{equation}
 (in the usual notation $^{2S+1}L$ with the quantum number $L$ of total orbital angular momentum
-$\hat{\bm{L}}=\hat{\bm{\ell}}_1+\hat{\bm{\ell}}_2$ of the two electrons).
+$\hat{\ell}_1+\hat{\ell}_2$ of the two electrons).
 Similarly, for the $^3P$ and $^1P$ two-electron states, the angular non-interacting wave functions are \cite{Breit_1930,Seidl_2007,Loos_2009a,Loos_2009c,Loos_2010e}
 \begin{subequations}
 \begin{align}
@@ -172,12 +172,11 @@ Similarly, for the $^3P$ and $^1P$ two-electron states, the angular non-interact
 By definition, \cite{DavidsonBook} the total electronic density (as a function of the solid angle $\Omega$ on the sphere) is given by the integral
 \begin{equation}
 	\label{eq:rho}
-	\rho(\Omega_1) = 2 \int \chi (\Omega_1,\Omega_2)^2 \Psi(\gamma)^2 d\Omega_2 
+	\rho(\Omega_1) = 2 \int \chi (\Omega_1,\Omega_2)^2 \Psi(\gamma)^2 d\Omega_2,
-%	\rho(\bm{r}_1) = 2 \int \chi (\bm{r}_1,\bm{r}_2)^2 \Psi(r_{12})^2 d\bm{r}_2 
 \end{equation}
 (in the notation $d\Omega_2=\sin\theta_2\,d\theta_2\,d\phi_2$), where we have already integrated over the spin coordinates.
 
-$\bullet$ For the singlet ground state, we have $\chi_{^1S} (\Omega_1,\Omega_2) = \frac{1}{4\pi}$, see Eq.~\eqref{eq:spin_1S}. Consequently, 
+For the singlet ground state, we have $\chi_{^1S} (\Omega_1,\Omega_2) = \frac{1}{4\pi}$ [see Eq.~\eqref{eq:spin_1S}]. Consequently, 
 \begin{equation}
 \rho_{^1S}(\Omega_1) = \frac{2}{(4\pi)^2} \int \Psi(\gamma)^2 d\Omega_2 .
 \end{equation}
@@ -187,21 +186,17 @@ Obviously, this integral cannot depend on $\Omega_1$, implying that the ground s
 \end{equation}
 for any value $\cc \in {\mathbb{R}}$ of the interaction constant. This result is also true for any excited states with $^1S$ symmetry.
 
-$\bullet$ For the other electronic states corresponding to higher total orbital angular momentum $L$, such as the lowest singlet and triplet $P$ states, \cite{Loos_2010e} the electron density is typically nonuniform, except in the very unlikely conditions discussed in the following section. \cite{Seidl_2007} {\color{green}
+For the other electronic states corresponding to higher total orbital angular momentum $L$, such as the lowest singlet and triplet $P$ states, \cite{Loos_2010e} the electron density is typically nonuniform, except in the very unlikely conditions discussed in the following section. \cite{Seidl_2007} 
-%We wish to point out that nonuniform densities are not in contradiction with the form of our hamiltonian $\hat{H}$ in Eq.~\eqref{eq:H} 
+%We wish to point out that nonuniform densities do not contradict the form of our hamiltonian $\hat{H}$ in Eq.~\eqref{eq:H} although $\hat{H}$ has no preferred direction in space.
-We wish to point out that nonuniform densities do not contradict the form of our hamiltonian $\hat{H}$ in Eq.~\eqref{eq:H} 
-although $\hat{H}$ has no preferred direction in space.
-{\color{red} Ref.?}
-}
 
 %Remark: Discuss the question of having no preferred direction in space!
 
 %%% FIG 1 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{3P}
+	\includegraphics[width=\linewidth]{fig1}
 	\caption{
-	$c_0 - 2 c_1/3 + c_2/5$ as a function of the radius of the sphere $R$ for various states of $^3P$ symmetry.
+	$c_0 - 2 c_1/3 + c_2/5$ as a function of $\mu$ for various states of $^3P$ symmetry.
-	The zero associated with each state (which corresponds to the value of $R$ for which the electron density is uniform) is located by a marker.
+	The zero associated with each state (which corresponds to the value of $\mu$ for which the electron density is uniform) is located by a marker.
 	\label{fig:3P}}
 \end{figure}
 %%% %%% %%% %%%
@@ -211,7 +206,7 @@ although $\hat{H}$ has no preferred direction in space.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 As evidenced by Eq.~\eqref{eq:rho}, the electron density $\rho(\Omega)$ on the spherical surface is affected by both the non-interacting angular wave function $\chi(\Omega_1,\Omega_2)$ and the interelectronic one, $\Psi(\gamma)$. For particular values of $\cc$, a subtle interplay between these two quantities may result in a uniform density $\rho(\Omega)=\frac{2}{4\pi}$ as we shall illustrate now explicitly for the example of the $^3P$ and $^1P$ two-electron states.
 
-To evaluate the integral of Eq.~\eqref{eq:rho} in the general case, we follow Ref.\cite{Seidl_2007} and decompose the square of the interelectronic wave function over the complete basis set composed by the Legendre polynomials $P_{\ell}(x)$,
+To evaluate the integral of Eq.~\eqref{eq:rho} in the general case, we follow Ref.~\onlinecite{Seidl_2007} and decompose the square of the interelectronic wave function over the complete basis set composed by the Legendre polynomials $P_{\ell}(x)$,
 \begin{equation}
 	\label{eq:expansion_c}
 	\Psi(\gamma)^2 = \sum_{\ell=0}^{\infty} c_{\ell} P_{\ell}(\cos \gamma).
@@ -222,12 +217,12 @@ Using this expression along with the angular functions \eqref{eq:spin_3P} and \e
 	\label{eq:3P}
 	\rho_{^3P}(\Omega) & = 
 	\qty(c_0 - \frac{c_2}{5}) Y_{00}(\Omega)^2 
-	+ \qty(c_0 - \frac{2 c_1}{3} + \frac{c_2}{5}) Y_{10}(\Omega)^2
+	+ \qty(c_0 - \frac{2 c_1}{3} + \frac{c_2}{5}) Y_{10}(\Omega)^2,
 	\\
 	\label{eq:1P}
 	\rho_{^1P}(\Omega) & = 
 	\qty(c_0 - \frac{c_2}{5}) Y_{00}(\Omega)^2 
-	+ \qty(c_0 + \frac{2 c_1}{3} + \frac{c_2}{5}) Y_{10}(\Omega)^2
+	+ \qty(c_0 + \frac{2 c_1}{3} + \frac{c_2}{5}) Y_{10}(\Omega)^2.
 \end{align}
 \end{subequations}
 Since $Y_{00}(\bm{\Omega})^2 = \frac1{4\pi}$, these densities are uniform if and only if the component associated with $Y_{10}(\bm{\Omega})^2$ vanishes,
@@ -240,7 +235,7 @@ From our numerical wave functions $\Psi(\gamma)$, the coefficients $c_{\ell}$ of
 	c_{\ell} = \frac{2\ell+1}{2} \int_0^\pi P_{\ell}(\cos \gamma) \Psi(\gamma)^2 \sin \gamma d\gamma.
 \end{equation}
 
-For each (ground or excited) $^3P$ states, there exists one and only one value of the "radius" $\cc$ in Eq.~\eqref{eq:cc}, $\cc=\cc_\text{UEG}<0$, for which these coefficients satisfy Eq.~\eqref{eq:condition}. $\cc_\text{UEG}$ can be computed numerically with great precision thanks to explicitly correlated calculations. \cite{Loos_2009a} Figure \ref{fig:3P} shows the behavior of $c_0 - 2 c_1/3 + c_2/5$ as a function of $\cc$ for the $^3P$ ground state and its first and second excited states. As one can see, $\cc_\text{UEG}$ 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.
+For each (ground or excited) $^3P$ states, there exists one and only one value of $\cc$ in Eq.~\eqref{eq:cc}, $\cc=\cc_\text{UEG}<0$, for which these coefficients satisfy Eq.~\eqref{eq:condition}. $\cc_\text{UEG}$ can be computed numerically with great precision thanks to explicitly correlated calculations. \cite{Loos_2009a} Figure \ref{fig:3P} shows the behavior of $c_0 - 2 c_1/3 + c_2/5$ as a function of $\cc$ for the $^3P$ ground state and its first and second excited states. As one can see, $\cc_\text{UEG}$ 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 a nonuniform electron density except for a unique $\cc$ value, we name these ``ephemeral'' systems as transient UEGs (TUEGs). Note that this feature was first discovered in Appendix A of Ref.~\onlinecite{Seidl_2007}. There, also an estimate $\cc_\text{UEG} \approx -5.3$ was provided for the $^3P$ ground state, a rather good estimate that we refine here to $\cc_\text{UEG} \approx -5.32527$.
 
 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
@@ -249,13 +244,13 @@ between the non-interacting and the interelectronic (angular) parts (denoted her
 
 The $^3P$ non-interacting angular wave function, $\chi_{^3P}$, defined in Eq.~\eqref{eq:spin_3P} has the natural tendency to pull apart same-spin electrons in accordance with the Pauli exclusion principle, creating in the process a so-called Fermi hole. \cite{Boyd_1974,Giner_2016a}
 The same physical effect can, independently, result from repulsion: In the case $\cc \gg 0$, two strongly repulsive electrons localize (or ``crystallize'') on opposite sides of the sphere to minimize their repulsion and they form a Wigner crystal. \cite{Wigner_1934} 
-Oppositely, when $\cc \ll 0$, the two electrons are strongly attracted to each other, forming a tightly bound pair that moves freely on the sphere. \cite{Seidl_2007,Seidl_2010} For certain values $\cc<0$, the attractive force seems to exactly compensate the "repulsive" effect of the Pauli exclusion principle, thus making the total electron density uniform, hence producing TUEGs. 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, a compensating attraction must be larger, corresponding to stronger negative values of $\cc$.
+Oppositely, when $\cc \ll 0$, the two electrons are strongly attracted to each other, forming a tightly bound pair that moves freely on the sphere. \cite{Seidl_2007,Seidl_2010} For certain values $\cc<0$, the attractive force seems to exactly compensate the ``repulsive'' effect of the Pauli exclusion principle, thus making the total electron density uniform, hence producing TUEGs. 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, a compensating attraction must be larger, corresponding to larger negative values of $\cc$.
 
 %%% FIG 2 %%%
 \begin{figure}
-	\includegraphics[width=\linewidth]{HF}
+	\includegraphics[width=\linewidth]{fig2}
 	\caption{
-	Hartree-Fock electron density $\rho^\text{HF}(\theta)$ as a function of the polar angle $\theta$ for various $R$ values.
+	Hartree-Fock electron density $\tilde{\rho}^\text{HF}(\theta)$ as a function of the polar angle $\theta$ for various $\mu$ values.
 	\label{fig:HF}}
 \end{figure}
 %%% %%% %%% %%%
@@ -266,7 +261,7 @@ Indeed, it is very unlikely that the exact theory and the HF approximation provi
 
 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 $\cc$ values for which the HF electron density,
 \begin{equation}
-\tilde{\rho}^\text{HF}(\theta) = \sin\theta\cdot\int_0^{2\pi}d\phi\,\rho^\text{HF}(\Omega),
+\tilde{\rho}^\text{HF}(\theta) = \int_0^{2\pi}\rho^\text{HF}(\Omega) \sin\theta d\phi,
 \end{equation}
 is uniform.
 At $\cc \approx -7$, however, $\tilde{\rho}^\text{HF}(\theta)$ is \textit{locally} uniform around $\theta = \frac{\pi}{2}$
@@ -278,19 +273,16 @@ At $\cc \approx -7$, however, $\tilde{\rho}^\text{HF}(\theta)$ is \textit{locall
 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. 
 TUEGs with larger number of electrons certainly exist and we hope to investigate these in the future.
-As a final remark, we would like to mention that a very similar analysis can be easily performed for higher-dimensional systems where TUEGs can likely be obtained for different values of the radius of the $D$-dimensional sphere. \cite{Loos_2011b}
+As a final remark, we would like to mention that a very similar analysis can be easily performed for higher-dimensional systems where TUEGs can likely be obtained for different $\mu$ values. \cite{Loos_2011b}
 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}
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section*{Acknowledgements}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-{\color{green} The authors thank Paola Gori-Giorgi for pointing out the uniform density case of Ref.~\onlinecite{Seidl_2007}.}
+Stimulating discussions with Paola Gori-Giorgi are acknowledged.
-%Stimulating discussions with Mike Seidl are also acknowledged.
 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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