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Pierre-Francois Loos 2021-03-23 15:52:07 +01:00
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30
TrUEGs.bib
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@ -1,13 +1,41 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2021-01-13 21:07:52 +0100 %% Created for Pierre-Francois Loos at 2021-03-23 15:51:47 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Boyd_1974,
abstract = {A new definition of the Fermi hole in many-electron systems is provided in terms of the distribution function falpha alpha (r12) of the interelectronic distance for electrons with parallel spins. By analogy with the Coulomb hole, the Fermi hole is defined as the difference between the values of f22(r12) derived from the Hartree-Fock and the Hartree wavefunctions. this definition, unlike previous ones, provides a simple picture of the Fermi hole as a function of r12. By assuming that the Hartree and Hartree-Fock orbitals are identical, an analytical formula is derived for the Fermi hole. Explicit calculations are presented for the 23S state of He and the ground state of Be. It is observed that the Fermi hole is remarkably similar in these two cases; and that the effects of Coulomb correlation are more long-ranged than those of Fermi correlation.},
author = {R J Boyd and C A Coulson},
date-added = {2021-03-23 15:48:16 +0100},
date-modified = {2021-03-23 15:49:37 +0100},
doi = {10.1088/0022-3700/7/14/006},
journal = {J. Phys. B: At. Mol. Opt. Phys.},
month = {oct},
number = {14},
pages = {1805--1816},
title = {The Fermi hole in atoms},
volume = {7},
year = 1974,
Bdsk-Url-1 = {https://doi.org/10.1088/0022-3700/7/14/006}}
@article{Giner_2016a,
author = {Giner,Emmanuel and Tenti,Lorenzo and Angeli,Celestino and Malrieu,Jean-Paul},
date-added = {2021-03-23 15:41:50 +0100},
date-modified = {2021-03-23 15:43:41 +0100},
doi = {10.1063/1.4963018},
journal = {J. Chem. Phys.},
number = {12},
pages = {124114},
title = {The ``Fermi hole'' and the correlation introduced by the symmetrization or the anti-symmetrization of the wave function},
volume = {145},
year = {2016},
Bdsk-Url-1 = {https://doi.org/10.1063/1.4963018}}
@article{Becke_1993b, @article{Becke_1993b,
author = {Becke,Axel D.}, author = {Becke,Axel D.},
date-added = {2021-01-13 21:07:34 +0100}, date-added = {2021-01-13 21:07:34 +0100},

420
TrUEGs.nb
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6
TrUEGs.tex
View File

@ -193,7 +193,7 @@ He also provided an estimate $R_\text{UEG} \approx -5.3$ for the $^3P$ ground st
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. 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 $^3P$ 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 $^3P$ 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 creating in the process a so-called Fermi hole. \cite{Boyd_1974,Giner_2016a}
The same physical effect can be obtained by increasing the value of $R$ (\ie, $R \gg 0$). 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} In such a case, the two electrons localize (or ``crystallize'') on opposite side of the sphere to form a Wigner crystal. \cite{Wigner_1934}
Oppositely, when $R \ll 0$, 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} Oppositely, when $R \ll 0$, 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}
@ -201,6 +201,7 @@ For certain $R$ values, the attractive effect stemming from the spatial part of
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. 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 $R$. Therefore, the magnitude of the attractive effect has to be larger to compensate it, which corresponds to more negative values of $R$.
%\titou{What about the nodes? Dyson orbitals? Cf Paola's paper.}
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\textbf{Concluding remarks.} \textbf{Concluding remarks.}
@ -208,7 +209,8 @@ Therefore, the magnitude of the attractive effect has to be larger to compensate
Here, we have introduced the concept of transient UEGs (TUEGs), a novel family of electron gases that exhibit, in very particular conditions, homogenous densities. 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. 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. TUEGs with larger number of electrons certainly exists and we hope to investigate these in the future.
The present concept might be useful in the future development of exchange-correlation functionals within DFT. 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}
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\textbf{Acknowledgements.} \textbf{Acknowledgements.}