Merge branch 'master' of https://github.com/pfloos/EPAWTFT
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%% Saved with string encoding Unicode (UTF-8)
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@book{BenderBook,
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Author = {C. M. Berder and S. A. Orszag},
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Author = {C. M. Bender and S. A. Orszag},
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Date-Added = {2020-07-28 09:59:40 +0200},
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Date-Modified = {2020-07-28 09:59:40 +0200},
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Publisher = {Springer},
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@ -414,7 +414,7 @@
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Month = mar,
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Number = {1989},
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Pages = {20120053-20120053},
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Title = {{Observation of a Fast Evolution in a Parity-Time-Symmetric System}},
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Title = {{Observation of a Fast Evolution in a Parity\textendash{}Time-Symmetric System}},
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Volume = {371},
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Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0053}}
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@ -532,11 +532,9 @@
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Year = {2011}}
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@book{SzaboBook,
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Address = {New York},
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Author = {A. Szabo and N. S. Ostlund},
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Date-Added = {2019-01-22 22:33:30 +0100},
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Date-Modified = {2019-01-22 22:33:30 +0100},
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Keywords = {qmech},
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Publisher = {McGraw-Hill},
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Title = {Modern quantum chemistry: {Introduction} to advanced electronic structure},
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Year = {1989}}
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@ -705,8 +703,6 @@
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Author = {Sachdev, Subir},
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Date = {2011},
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Doi = {10.1017/CBO9780511973765},
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Edition = {2},
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Location = {Cambridge},
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Publisher = {Cambridge University Press},
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Title = {{Quantum Phase Transitions}},
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Bdsk-Url-1 = {https://doi.org/10.1017/CBO9780511973765}}
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@ -779,10 +775,10 @@
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Title = {Communication: {Strong}-interaction limit of an adiabatic connection in {Hartree}-{Fock} theory},
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Volume = {149},
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Year = {2018},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5078565}}
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5078565}
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}
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@book{GiulianiBook,
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Address = {Cambridge},
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Author = {Giuliani, Gabriele and Vignale, Giovanni},
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Doi = {10.1017/CBO9780511619915},
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Isbn = {978-0-521-52796-5},
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@ -790,27 +786,25 @@
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Title = {Quantum {Theory} of the {Electron} {Liquid}},
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Urldate = {2020-07-21},
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Year = {2005},
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Bdsk-Url-1 = {https://doi.org/10.1017/CBO9780511619915}}
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}
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@book{AngularBook,
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Author = {Edmonds, A. R.},
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Isbn = {978-0-691-02589-6},
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Language = {en},
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Publisher = {Princeton University Press},
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Title = {Angular {Momentum} in {Quantum} {Mechanics}},
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Year = {1996}}
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@book{SlaterBook,
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Author = {Slater, John Clarke},
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Language = {eng},
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Publisher = {New York : McGraw-Hill},
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Publisher = {McGraw-Hill},
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Title = {{Quantum Theory of Atomic Structure}},
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Year = {1960}}
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@article{Loos_2009,
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Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.},
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Doi = {10.1103/PhysRevA.79.062517},
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Journal = {Physical Review A},
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Journal = {Phys. Rev. A},
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Month = jun,
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Number = {6},
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Pages = {062517},
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@ -820,14 +814,11 @@
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Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.79.062517}}
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@book{Ushveridze_1994,
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address = {Bristol [England]; Philadelphia},
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title = {Quasi-exactly solvable models in quantum mechanics},
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isbn = {978-0-7503-0266-1},
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language = {English},
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publisher = {Institute of Physics Pub.},
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publisher = {Institute of Physics Publishing},
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author = {Ushveridze, Alexander G},
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year = {1994},
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note = {OCLC: 28421899}
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}
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@article{Lipkin_1965,
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@ -836,7 +827,7 @@
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doi = {10.1016/0029-5582(65)90862-X},
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language = {en},
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number = {2},
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journal = {Nuclear Physics},
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journal = {Nucl. Phys.},
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author = {Lipkin, H. J. and Meshkov, N. and Glick, A. J.},
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month = feb,
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year = {1965},
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@ -849,7 +840,7 @@
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volume = {46},
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doi = {10.1103/PhysRev.46.1002},
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number = {11},
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journal = {Physical Review},
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journal = {Phys. Rev.},
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author = {Wigner, E.},
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month = dec,
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@ -863,7 +854,7 @@
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issn = {0021-9606},
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doi = {10.1063/1.1869978},
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number = {12},
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journal = {The Journal of Chemical Physics},
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@ -875,7 +866,7 @@
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volume = {75},
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doi = {10.1103/PhysRevA.75.062506},
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number = {6},
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journal = {Physical Review A},
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journal = {Phys. Rev. A},
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author = {Seidl, Michael},
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month = jun,
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year = {2007},
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@ -887,7 +878,7 @@
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volume = {103},
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doi = {10.1103/PhysRevLett.103.123008},
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number = {12},
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journal = {Physical Review Letters},
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author = {Loos, Pierre-François and Gill, Peter M. W.},
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month = sep,
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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
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\label{fig:SpheriumNrj}
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\end{wrapfigure}
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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}.
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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 $R>75/38$.}
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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}.
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The energy associated with this sb-RHF solution reads
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\begin{equation}
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E_{\text{sb-RHF}}=\frac{75}{88R^3}+\frac{25}{22R^2}+\frac{91}{66R}.
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