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final corrections b4 submission

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Pierre-Francois Loos 2022-03-10 09:59:58 +01:00
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TrUEGs.tex
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@ -85,7 +85,7 @@ This property comes from the addition theorem of the spherical harmonics \cite{N
\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}, \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} \end{equation}
where $\bm{\Omega}=(\theta,\phi)$ gathers the polar and azimuthal angles, respectively. 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} Thanks to this key property, these FUEGs have been 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 $\ell_\text{max} \to \infty$. \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 ($\ell_\text{max} = 0$) 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} 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}
@ -94,9 +94,9 @@ In this case, the many-body hamiltonian reads
\label{eq:H} \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} \end{equation}
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, The squares $\hat{\ell}_i^2=-\hbar^2\hat{\Lambda}_i$ of the (orbital) angular momentum operators are essentially the angular parts $\hat{\Lambda}_i$ of the Laplacian,
\begin{equation} \begin{equation}
\hat{L} = \frac{1}{\sin\theta}\pdv{}{\theta}\sin\theta \pdv{}{\theta} + \frac{1}{\sin^2\theta}\pdv[2]{}{\phi}, \hat{\Lambda} = \frac{1}{\sin\theta}\pdv{}{\theta}\sin\theta \pdv{}{\theta} + \frac{1}{\sin^2\theta}\pdv[2]{}{\phi},
\end{equation} \end{equation}
while $r_{12}$ is the \textit{spatial distance} between the two electrons, \ie, the electrons interact Coulombically \textit{through} the sphere, while $r_{12}$ is the \textit{spatial distance} between the two electrons, \ie, the electrons interact Coulombically \textit{through} the sphere,
\begin{equation} \begin{equation}
@ -113,7 +113,7 @@ In Eq.~\eqref{eq:H}, we have introduced a coupling constant $\lambda$ which in t
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 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} \begin{equation}
\label{eq:H2} \label{eq:H2}
\Hat{H} = \frac1{R^2}\qty{ - \frac{\hat{L}_1+\hat{L}_2}2 + \cc \frac{1}{\sqrt{2(1 - \cos\gamma)}} }, \Hat{H} = \frac1{R^2}\qty{ - \frac{\hat{\Lambda}_1+\hat{\Lambda}_2}2 + \cc \frac{1}{\sqrt{2(1 - \cos\gamma)}} },
\end{equation} \end{equation}
with an effective (dimensionless) coupling constant, with an effective (dimensionless) coupling constant,
\begin{equation} \begin{equation}
@ -153,7 +153,7 @@ Since the factor $\Psi(\gamma)$ in Eq.~\eqref{eq:PhiOLD} is symmetrical (upon sw
\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} \end{equation}
(in the usual notation $^{2S+1}L$ with the quantum number $L$ of total orbital angular momentum (in the usual notation $^{2S+1}L$ with the quantum number $L$ of total orbital angular momentum
$\hat{\ell}_1+\hat{\ell}_2$ of the two electrons). $\hat{L} = \hat{\ell}_1+\hat{\ell}_2$ 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} 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{subequations}
\begin{align} \begin{align}
@ -265,7 +265,10 @@ Expanding the two HF orbitals of the $^3P$ ground state in a basis of zonal harm
\end{equation} \end{equation}
is uniform. is uniform.
At $\cc \approx -7$, however, $\tilde{\rho}^\text{HF}(\theta)$ is \textit{locally} uniform around $\theta = \frac{\pi}{2}$ At $\cc \approx -7$, however, $\tilde{\rho}^\text{HF}(\theta)$ is \textit{locally} uniform around $\theta = \frac{\pi}{2}$
(\ie, in a belt closely above and below 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, by definition, can only include one of the three equivalent $sp$ configurations (\ie, $sp_x$, $sp_y$, and $sp_z$). The fact that this phenomenon appears at larger (absolute) $\cc$ values in the HF approximation is not surprising as, contrary to the repulsive regime (\ie, $\cc > 0$) 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, $\cc < 0$) 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 $\cc$) to overcome this drawback. \cite{Seidl_2010,Burton_2019a,Marie_2020} (\ie, in a belt closely above and below 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, by definition, can only include one (linear combination) of the three equivalent $sp$ configurations (\ie, $sp_x$, $sp_y$, and $sp_z$).
The fact that this phenomenon appears at larger (absolute) $\cc$ values in the HF approximation is not surprising as, contrary to the repulsive regime (\ie, $\cc > 0$) 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, $\cc < 0$) 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 $\cc$) to overcome this drawback. \cite{Seidl_2010,Burton_2019a,Marie_2020}
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\section{Concluding remarks} \section{Concluding remarks}
@ -281,6 +284,7 @@ The three-dimensional version where electrons are confined to the surface of a 3
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Stimulating discussions with Paola Gori-Giorgi are acknowledged. Stimulating discussions with Paola Gori-Giorgi are 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). 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).
This work was also supported by the Netherlands Organisation for Scientific Research (NWO) under Vici grant 724.017.001.
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