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TrUEGs.bib
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@ -1,13 +1,27 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/ %% https://bibdesk.sourceforge.io/
%% Created for Pierre-Francois Loos at 2021-11-26 18:15:28 +0100 %% Created for Pierre-Francois Loos at 2022-03-10 10:04:02 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{White_1971,
author = {White, Ronald J. and Stillinger, Frank H.},
date-added = {2022-03-10 10:03:18 +0100},
date-modified = {2022-03-10 10:03:32 +0100},
doi = {10.1103/PhysRevA.3.1521},
journal = {Phys. Rev. A},
month = {May},
pages = {1521--1535},
title = {Electron Correlation in Three-Electron Atoms and Ions},
volume = {3},
year = {1971},
bdsk-url-1 = {https://link-aps-org-s.docadis.univ-tlse3.fr/doi/10.1103/PhysRevA.3.1521},
bdsk-url-2 = {https://doi.org/10.1103/PhysRevA.3.1521}}
@article{Gori-Giorgi_2008, @article{Gori-Giorgi_2008,
abstract = {The correlation energy in density functional theory can be expressed exactly in terms of the change in the probability of finding two electrons at a given distance r12 (intracule density) when the electron--electron interaction is multiplied by a real parameter λ varying between 0 (Kohn--Sham system) and 1 (physical system). In this process{,} usually called adiabatic connection{,} the one-electron density is (ideally) kept fixed by a suitable local one-body potential. While an accurate intracule density of the physical system can only be obtained from expensive wavefunction-based calculations{,} being able to construct good models starting from Kohn--Sham ingredients would highly improve the accuracy of density functional calculations. To this purpose{,} we investigate the intracule density in the λ → ∞ limit of the adiabatic connection. This strong-interaction limit of density functional theory turns out to be{,} like the opposite non-interacting Kohn--Sham limit{,} mathematically simple and can be entirely constructed from the knowledge of the one-electron density. We develop here the theoretical framework and{,} using accurate correlated one-electron densities{,} we calculate the intracule densities in the strong interaction limit for few atoms. Comparison of our results with the corresponding Kohn--Sham and physical quantities provides useful hints for building approximate intracule densities along the adiabatic connection of density functional theory.}, abstract = {The correlation energy in density functional theory can be expressed exactly in terms of the change in the probability of finding two electrons at a given distance r12 (intracule density) when the electron--electron interaction is multiplied by a real parameter λ varying between 0 (Kohn--Sham system) and 1 (physical system). In this process{,} usually called adiabatic connection{,} the one-electron density is (ideally) kept fixed by a suitable local one-body potential. While an accurate intracule density of the physical system can only be obtained from expensive wavefunction-based calculations{,} being able to construct good models starting from Kohn--Sham ingredients would highly improve the accuracy of density functional calculations. To this purpose{,} we investigate the intracule density in the λ → ∞ limit of the adiabatic connection. This strong-interaction limit of density functional theory turns out to be{,} like the opposite non-interacting Kohn--Sham limit{,} mathematically simple and can be entirely constructed from the knowledge of the one-electron density. We develop here the theoretical framework and{,} using accurate correlated one-electron densities{,} we calculate the intracule densities in the strong interaction limit for few atoms. Comparison of our results with the corresponding Kohn--Sham and physical quantities provides useful hints for building approximate intracule densities along the adiabatic connection of density functional theory.},
author = {Gori-Giorgi, Paola and Seidl, Michael and Savin, Andreas}, author = {Gori-Giorgi, Paola and Seidl, Michael and Savin, Andreas},
@ -1508,7 +1522,8 @@
pages = {283001}, pages = {283001},
title = {Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them}, title = {Perturbation Theory in the Complex Plane: Exceptional Points and Where to Find Them},
volume = {33}, volume = {33},
year = {2021}} year = {2021},
bdsk-url-1 = {https://doi.org/10.1088/1361-648X/abe795}}
@article{Moreau_2004, @article{Moreau_2004,
author = {Moreau, Y. and Loos, P.-F. and Assfeld, X.}, author = {Moreau, Y. and Loos, P.-F. and Assfeld, X.},

2
TrUEGs.tex
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@ -266,7 +266,7 @@ Expanding the two HF orbitals of the $^3P$ ground state in a basis of zonal harm
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}. (\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 $sp_z$). 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 $sp_z$). \cite{White_1971}
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. 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} 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}