minor corrections

TrUEGs.nb

@@ -8596,7 +8596,7 @@ EXyYe2H80wtdt31+meyw3aHp0fEbWg4nyvbNl/qc7DClvTXq8h9Nhz8rP17y
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@@ -13322,7 +13322,7 @@ t+UVKN/E4SZjT2Cwx2V7ANcnWZY=
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TrUEGs.tex

@@ -49,7 +49,7 @@
 %\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 millions of approximate electronic structure calculations for atoms, molecules, and solids.
-Recently, it has been shown that the traditional concept of 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 beautiful new family of UEGs with more customizable properties.
+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.
 \end{abstract}
 
@@ -68,7 +68,7 @@ As commonly done, the LDA can be refined by adding up new ingredients, such as t
 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 is can be ``created`` via a \textit{gedanken} experience 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.
+It is 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).
  
@@ -92,7 +92,7 @@ In this case, the many-body Hamiltonian is simply
 	\label{eq:H}
 	\Hat{H} = - \frac{\nabla_1^2 + \nabla_2^2}{2} + \frac{1}{r_{12}}
 \end{equation}
-where $r_{12} = \abs{\bm{r}_1 - \bm{r}_2}$ is the interelectronic distance, \ie, the electrons interact Coulombically through the sphere.
+where $r_{12} = \abs{\bm{r}_1 - \bm{r}_2}$ is the interelectronic distance, \ie, the electrons interact Coulombically \textit{through} the sphere.
 Following Breit, \cite{Breit_1930} one can write the total electronic wave function as
 \begin{equation}
 \label{eq:Phi}
@@ -119,7 +119,7 @@ After integration over the spin coordinates, the total electronic density is, by
 	\label{eq:rho}
 	\rho(\bm{r}_1) = 2 \int \chi (\bm{r}_1,\bm{r}_2)^2 \Psi(r_{12})^2 d\bm{r}_2 
 \end{equation}
-and, following Seidl, \cite{Seidl_2007} one can decompose the square of the (totally symmetric) interelectronic wave function over the basis set of the Legendre polynomials 
+and, following Seidl, \cite{Seidl_2007} one can decompose the square of the (totally symmetric) interelectronic wave function over the complete basis set composed by the Legendre polynomials 
 \begin{equation}
 	\Psi(r_{12})^2 = \sum_{\ell=0}^{\infty} c_{\ell} P_{\ell}(\cos \gamma) 
 \end{equation}
@@ -184,13 +184,14 @@ As readily seen from these expressions and remembering that $Y_{00}(\bm{\Omega})
 
 For each (ground or excited) $^3P$ states, there exists one and only one value of the radius, $R_\text{UEG}$, for which Eq.~\eqref{eq:condition} is fulfilled, and it 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 $R$ for the $^3P$ ground state and its first and second excited states.
-As one can see, $R_\text{UEG}$ is negative (which corresponds to attractive ``electron'' pair \cite{Seidl_2007,Seidl_2010} or exciton \cite{Pedersen_2010,Loos_2012c}) and gets larger (in absolute value) for excited states.
+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 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 $R_\text{UEG} \approx -5.32527$.
 
 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 $^1P$ states are TUEGs.
+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.
+
 The $^3P$ spin wave function, $\chi_{^3P}$, defined in Eq.~\eqref{eq:spin_3P} has the natural tendency to pull apart the same-spin electron pair in accordance with the Pauli exclusion principle. 
 The same physical effect can be obtained by increasing the value of $R$ (\ie, $R \gg 0$).
 In such a case, the two electrons localize (or ``crystallize'') on opposite side of the sphere to form a Wigner crystal. \cite{Wigner_1934}