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Pierre-Francois Loos 2020-07-31 11:16:23 +02:00
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@ -1,7 +1,7 @@
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@book{BenderBook,
Author = {C. M. Berder and S. A. Orszag},
Author = {C. M. Bender and S. A. Orszag},
Date-Added = {2020-07-28 09:59:40 +0200},
Date-Modified = {2020-07-28 09:59:40 +0200},
Publisher = {Springer},
@ -414,7 +414,7 @@
Month = mar,
Number = {1989},
Pages = {20120053-20120053},
Title = {{Observation of a Fast Evolution in a Parity-Time-Symmetric System}},
Title = {{Observation of a Fast Evolution in a Parity\textendash{}Time-Symmetric System}},
Volume = {371},
Year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0053}}
@ -532,11 +532,9 @@
Year = {2011}}
@book{SzaboBook,
Address = {New York},
Author = {A. Szabo and N. S. Ostlund},
Date-Added = {2019-01-22 22:33:30 +0100},
Date-Modified = {2019-01-22 22:33:30 +0100},
Keywords = {qmech},
Publisher = {McGraw-Hill},
Title = {Modern quantum chemistry: {Introduction} to advanced electronic structure},
Year = {1989}}
@ -705,8 +703,6 @@
Author = {Sachdev, Subir},
Date = {2011},
Doi = {10.1017/CBO9780511973765},
Edition = {2},
Location = {Cambridge},
Publisher = {Cambridge University Press},
Title = {{Quantum Phase Transitions}},
Bdsk-Url-1 = {https://doi.org/10.1017/CBO9780511973765}}
@ -779,10 +775,10 @@
Title = {Communication: {Strong}-interaction limit of an adiabatic connection in {Hartree}-{Fock} theory},
Volume = {149},
Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5078565}}
Bdsk-Url-1 = {https://doi.org/10.1063/1.5078565}
}
@book{GiulianiBook,
Address = {Cambridge},
Author = {Giuliani, Gabriele and Vignale, Giovanni},
Doi = {10.1017/CBO9780511619915},
Isbn = {978-0-521-52796-5},
@ -790,27 +786,25 @@
Title = {Quantum {Theory} of the {Electron} {Liquid}},
Urldate = {2020-07-21},
Year = {2005},
Bdsk-Url-1 = {https://doi.org/10.1017/CBO9780511619915}}
}
@book{AngularBook,
Author = {Edmonds, A. R.},
Isbn = {978-0-691-02589-6},
Language = {en},
Month = jan,
Publisher = {Princeton University Press},
Title = {Angular {Momentum} in {Quantum} {Mechanics}},
Year = {1996}}
@book{SlaterBook,
Author = {Slater, John Clarke},
Language = {eng},
Publisher = {New York : McGraw-Hill},
Publisher = {McGraw-Hill},
Title = {{Quantum Theory of Atomic Structure}},
Year = {1960}}
@article{Loos_2009,
Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
Doi = {10.1103/PhysRevA.79.062517},
Journal = {Physical Review A},
Journal = {Phys. Rev. A},
Month = jun,
Number = {6},
Pages = {062517},
@ -820,14 +814,11 @@
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.79.062517}}
@book{Ushveridze_1994,
address = {Bristol [England]; Philadelphia},
title = {Quasi-exactly solvable models in quantum mechanics},
isbn = {978-0-7503-0266-1},
language = {English},
publisher = {Institute of Physics Pub.},
publisher = {Institute of Physics Publishing},
author = {Ushveridze, Alexander G},
year = {1994},
note = {OCLC: 28421899}
}
@article{Lipkin_1965,
@ -836,7 +827,7 @@
doi = {10.1016/0029-5582(65)90862-X},
language = {en},
number = {2},
journal = {Nuclear Physics},
journal = {Nucl. Phys.},
author = {Lipkin, H. J. and Meshkov, N. and Glick, A. J.},
month = feb,
year = {1965},
@ -849,7 +840,7 @@
volume = {46},
doi = {10.1103/PhysRev.46.1002},
number = {11},
journal = {Physical Review},
journal = {Phys. Rev.},
author = {Wigner, E.},
month = dec,
year = {1934},
@ -863,7 +854,7 @@
issn = {0021-9606},
doi = {10.1063/1.1869978},
number = {12},
journal = {The Journal of Chemical Physics},
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author = {Thompson, David C. and Alavi, Ali},
month = mar,
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@ -875,7 +866,7 @@
volume = {75},
doi = {10.1103/PhysRevA.75.062506},
number = {6},
journal = {Physical Review A},
journal = {Phys. Rev. A},
author = {Seidl, Michael},
month = jun,
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@ -887,7 +878,7 @@
volume = {103},
doi = {10.1103/PhysRevLett.103.123008},
number = {12},
journal = {Physical Review Letters},
journal = {Phys. Rev. Let.},
author = {Loos, Pierre-François and Gill, Peter M. W.},
month = sep,
year = {2009},

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@ -692,7 +692,8 @@ Exact & 9.783874 & 0.852781 & 0.391959 & 0.247898 & 0.139471 & 0.064525 & 0.005
\label{fig:SpheriumNrj}
\end{wrapfigure}
There is also another symmetry-broken solution for $R>75/38$ but this one corresponds to a maximum of the HF equations. This solution is associated with another type of symmetry breaking somewhat less known. It corresponds to a configuration where both electrons are on the same side of the sphere, in the same spatial orbital. This solution is called symmetry-broken RHF (sb-RHF). \titou{At a critical value of $R$, placing two electrons in the same orbital on the same side of the sphere increases the repulsion energy more than the kinetic energy of the two electrons in the p\textsubscript{z} orbital.} This configuration breaks the spatial symmetry of charge. Hence this symmetry breaking is associated with a charge density wave, the system oscillates between the situations with the two electrons on each side \cite{GiulianiBook}.
There is also another symmetry-broken solution for $R>75/38$ but this one corresponds to a maximum of the HF equations. This solution is associated with another type of symmetry breaking somewhat less known. It corresponds to a configuration where both electrons are on the same side of the sphere, in the same spatial orbital. This solution is called symmetry-broken RHF (sb-RHF). \antoine{The reasoning is counter-intuitive because the electrons tends to maximize their energy. If the orbitals are symmetric, the maximum is when the two electrons are in the p\textsubscript{z} orbital because it maximizes the kinetic energy. At the critical value of $R$, placing the two electrons in the same symmetry-broken orbital i.e., on the same side of the sphere gives a superior energy than the p\textsubscript{z}\textsuperscript{2} state. This is because it becomes more efficient to maximize the repulsion energy than the kinetic energy for $R>75/38$.}
This configuration breaks the spatial symmetry of charge. Hence this symmetry breaking is associated with a charge density wave, the system oscillates between the situations with the two electrons on each side \cite{GiulianiBook}.
The energy associated with this sb-RHF solution reads
\begin{equation}
E_{\text{sb-RHF}}=\frac{75}{88R^3}+\frac{25}{22R^2}+\frac{91}{66R}.