\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
\newcommand{\VRIJE}{Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modelling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands}
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 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.
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.
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}
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 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.
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}
In particular, the spin-unpolarized ground state of $n$ electrons on a sphere has a homogeneous density for $n =2(L+1)^2$ (where $L \in\mathbb{N}$) 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):
where $\bm{\Omega}=(\theta,\phi)$ gathers the polar and azimuthal angles, respectively.
Thanks to this amazing 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 $L \to\infty$. \cite{Bowick_2002,Loos_2011b}
The two-electron case (\ie, $L =0$) 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{Seidl_2007,Loos_2009a,Pedersen_2010,Loos_2012c,Schindlmayr_2013,Sun_2016,Loos_2018b} and is, furthermore, exactly solvable for a countably infinite set of $R$ values. \cite{Loos_2009c,Loos_2010e,Loos_2012}
where $\Xi$, $\chi$ and $\Psi$ are the spin, angular and interelectronic wave functions, respectively, and $\bm{x}_i =(s_i,\bm{r}_i)$ is a composite coordinate gathering the spin coordinate $s_i$ and the spatial coordinate $\bm{r}_i$ associated with the $i$th electron.
The singlet and triplet wave functions read \cite{BetheBook}
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
and the interelectronic angle $\gamma$ is defined as $r_{12}= R \sqrt{2-2\cos\gamma}$ or, equivalently, $\cos\gamma=\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\cos(\phi_1-\phi_2)$.
For the singlet ground state, we have $\chi_{^1S}(\bm{r}_1,\bm{r}_2)=1$, and taking advantage of the addition theorem of the Legendre polynomials, \cite{NISTbook}\ie,
and the orthonormality of the spherical harmonics, one finds that $c_\ell=0$ for $\ell > 0$, demonstrating that the ground state of two electrons on a surface of a sphere has a uniform density $\rho_{^1S}(\bm{r})=2/(4\pi R^2)$ for any $R$ values.
This result is also true for any excited states with $^1S$ symmetry.
For the other electronic states corresponding to higher total angular momentum, such as the lowest singlet and triplet $P$ states, \cite{Loos_2010e} the electron density is, in general, inhomogeneous, except in the very unlikely conditions discussed below. \cite{Seidl_2007}
%%% FIG 1 %%%
\begin{figure}
\includegraphics[width=\linewidth]{3P}
\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.
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.
For the $^3P$ and $^1P$ two-electron states, the angular wave functions are \cite{Breit_1930,Seidl_2007,Loos_2009a,Loos_2009c,Loos_2010e}
After a lengthy derivation, the square of $\chi$ can be expanded in terms of spherical harmonics, and, after integration over the coordinates of the second electron, it yields
As readily seen from these expressions and remembering that $Y_{00}(\bm{\Omega})^2=1/(4\pi)$, the electron density is uniform if and only if the component associated with $Y_{10}(\bm{\Omega})^2$ vanishes, \ie,
\begin{equation}
\label{eq:condition}
c_0 \mp\frac{2 c_1}{3} + \frac{c_2}{5} = 0
\end{equation}
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 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.
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 this 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 same-spin electrons in accordance with the Pauli exclusion principle, creating in the process a so-called Fermi hole. \cite{Boyd_1974,Giner_2016a}
In such a case, the two electrons localize (or ``crystallize'') on opposite side of the sphere to minimize their repsulsion and they form a Wigner crystal. \cite{Wigner_1934}
Oppositely, when $R \ll0$, the two electrons are attracted to each other to form a pair of tightly bound electrons that freely move on the sphere. \cite{Seidl_2007,Seidl_2010}
For certain $R$ values, the attractive effect stemming from the spatial part of the wave function exactly compensates the Pauli exclusion principle originating from the spin part of the wave function to make 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, the magnitude of the attractive effect has to be larger to compensate it, which corresponds to more negative values of $R$.
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}
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).
Expanding the two HF orbitals of the $^3P$ ground state in a basis of zonal harmonics $Y_{\ell}(\theta)\equiv Y_{\ell0}(\theta,\phi)$, 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{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}
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 exists 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}
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}
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).
