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Pierre-Francois Loos 2021-02-08 21:12:26 +01:00
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TrUEGs.tex
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@ -51,6 +51,7 @@ The uniform electron gas (UEG), a hypothetical system with finite homogenous ele
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. 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. 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. 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.
Concrete examples are provided in the case of two-electron systems.
\end{abstract} \end{abstract}
\maketitle \maketitle
@ -70,12 +71,12 @@ Each of these quantities defines a new rung of the well-known Jacob ladder of DF
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} 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. 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 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). In the following, this paradigm is named the infinite UEG (IUEG) for obvious reasons.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{Finite uniform electron gases.} \textbf{Finite uniform electron gases.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Recently, it has been shown that one can create finite UEGs (FUEGs) by placing a finite number of electrons confined to 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} 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} 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 $n = 2(L+1)^2$ (where $L \in \mathbb{N}$) for any $R$ values, and this holds also within the Hartree-Fock approximation. \cite{Loos_2011b} 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 Hartree-Fock 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): 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):
@ -128,7 +129,7 @@ where
c_{\ell} = \frac{2\ell+1}{2} \int_0^\pi P_{\ell}(\cos \gamma) \Psi(r_{12})^2 \sin \gamma d\gamma c_{\ell} = \frac{2\ell+1}{2} \int_0^\pi P_{\ell}(\cos \gamma) \Psi(r_{12})^2 \sin \gamma d\gamma
\end{equation} \end{equation}
and the interelectronic angle $\gamma$ is defined as $r_{12} = R \sqrt{2 - 2 \cos \gamma}$ or, equivalently, $\cos \gamma = \cos \theta_1 \cos \theta_2 + \sin \theta_1 \sin \theta_2 \cos(\phi_1 - \phi_2)$. 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} 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,
\begin{equation} \begin{equation}
P_{\ell}(\cos \gamma) = \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{+\ell} Y_{\ell m}^*(\bm{\Omega}_1) Y_{\ell m}(\bm{\Omega}_2) P_{\ell}(\cos \gamma) = \frac{4\pi}{2\ell+1} \sum_{m=-\ell}^{+\ell} Y_{\ell m}^*(\bm{\Omega}_1) Y_{\ell m}(\bm{\Omega}_2)
\end{equation} \end{equation}
@ -185,7 +186,7 @@ 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} 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. 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. 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). 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}. 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 $R_\text{UEG} \approx -5.3$ for the $^3P$ ground state, a rather good estimate that we refine here to $R_\text{UEG} \approx -5.32527$.