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@ -4,7 +4,7 @@
doi = {10.1016/0009-2614(86)80686-8}, doi = {10.1016/0009-2614(86)80686-8},
pages = {16--22}, pages = {16--22},
number = {1}, number = {1},
Journal = {Chemical Physics Letters}, Journal = {Chem. Phys. Lett.},
author = {Gill, Peter M. W. and Radom, Leo}, author = {Gill, Peter M. W. and Radom, Leo},
Date = {1986-11-28}, Date = {1986-11-28},
} }
@ -125,7 +125,7 @@
doi = {10.1016/0009-2614(96)00974-8}, doi = {10.1016/0009-2614(96)00974-8},
pages = {369--378}, pages = {369--378},
number = {3}, number = {3},
Journal = {Chemical Physics Letters}, Journal = {Chem. Phys. Lett.},
author = {Christiansen, Ove and Olsen, Jeppe and Jørgensen, Poul and Koch, Henrik and Malmqvist, Per-Åke}, author = {Christiansen, Ove and Olsen, Jeppe and Jørgensen, Poul and Koch, Henrik and Malmqvist, Per-Åke},
urlDate = {2020-07-07}, urlDate = {2020-07-07},
Date = {1996-10-18}, Date = {1996-10-18},
@ -136,7 +136,7 @@
Volume = {47}, Volume = {47},
doi = {10.1016/S0065-3276(04)47011-7}, doi = {10.1016/S0065-3276(04)47011-7},
pages = {193--208}, pages = {193--208},
bookTitle = {Advances in Quantum Chemistry}, bookTitle = {Adv. Quantum Chem.},
publisher = {Academic Press}, publisher = {Academic Press},
author = {Goodson, David Z. and Sergeev, Alexey V.}, author = {Goodson, David Z. and Sergeev, Alexey V.},
Date = {2004-01-01}, Date = {2004-01-01},
@ -149,7 +149,7 @@
doi = {10.1002/wcms.92}, doi = {10.1002/wcms.92},
pages = {743--761}, pages = {743--761},
number = {5}, number = {5},
journaltitle = {{WIREs} Computational Molecular Science}, journaltitle = {{WIREs} Comput. Mol. Sci.},
author = {Goodson, David Z.}, author = {Goodson, David Z.},
date = {2012}, date = {2012},
} }
@ -171,7 +171,7 @@
doi = {10.1002/qua.560200502}, doi = {10.1002/qua.560200502},
pages = {955--1065}, pages = {955--1065},
number = {5}, number = {5},
journalTitle = {International Journal of Quantum Chemistry}, journalTitle = {J. Quantum Chem.},
author = {Fukutome, Hideo}, author = {Fukutome, Hideo},
Date = {1981}, Date = {1981},
} }
@ -331,7 +331,7 @@
Month = may, Month = may,
Number = {5}, Number = {5},
Pages = {394-398}, Pages = {394-398},
Title = {Parity\textendash{}Time-Symmetric Whispering-Gallery Microcavities}, Title = {{Parity\textendash{}Time-Symmetric Whispering-Gallery Microcavities}},
Volume = {10}, Volume = {10},
Year = {2014}, Year = {2014},
Bdsk-Url-1 = {https://doi.org/10.1038/nphys2927}, Bdsk-Url-1 = {https://doi.org/10.1038/nphys2927},
@ -346,7 +346,7 @@
Month = aug, Month = aug,
Number = {7410}, Number = {7410},
Pages = {167-171}, Pages = {167-171},
Title = {Parity\textendash{}Time Synthetic Photonic Lattices}, Title = {{Parity\textendash{}Time Synthetic Photonic Lattices}},
Volume = {488}, Volume = {488},
Year = {2012}, Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1038/nature11298}, Bdsk-Url-1 = {https://doi.org/10.1038/nature11298},
@ -361,7 +361,7 @@
Month = mar, Month = mar,
Number = {3}, Number = {3},
Pages = {192-195}, Pages = {192-195},
Title = {Observation of Parity\textendash{}Time Symmetry in Optics}, Title = {{Observation of Parity\textendash{}Time Symmetry in Optics}},
Volume = {6}, Volume = {6},
Year = {2010}, Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1038/nphys1515}, Bdsk-Url-1 = {https://doi.org/10.1038/nphys1515},
@ -376,7 +376,7 @@
Month = oct, Month = oct,
Number = {4}, Number = {4},
Pages = {040101}, Pages = {040101},
Title = {Experimental Study of Active \emph{LRC} Circuits with PT Symmetries}, Title = {{Experimental Study of Active \emph{LRC} Circuits with PT Symmetries}},
Volume = {84}, Volume = {84},
Year = {2011}, Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.040101}, Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.040101},
@ -391,7 +391,7 @@
Month = aug, Month = aug,
Number = {2}, Number = {2},
Pages = {021806}, Pages = {021806},
Title = {PT-Symmetry in Honeycomb Photonic Lattices}, Title = {{PT-Symmetry in Honeycomb Photonic Lattice}s},
Volume = {84}, Volume = {84},
Year = {2011}, Year = {2011},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.021806}, Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.021806},
@ -406,7 +406,7 @@
Month = apr, Month = apr,
Number = {4}, Number = {4},
Pages = {042903}, Pages = {042903},
Title = {Enhanced Magnetic Resonance Signal of Spin-Polarized {{Rb}} Atoms near Surfaces of Coated Cells}, Title = {{Enhanced Magnetic Resonance Signal of Spin-Polarized {{Rb}} Atoms near Surfaces of Coated Cells}},
Volume = {81}, Volume = {81},
Year = {2010}, Year = {2010},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.042903}, Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.042903},
@ -421,7 +421,7 @@
Month = mar, Month = mar,
Number = {1989}, Number = {1989},
Pages = {20120053-20120053}, Pages = {20120053-20120053},
Title = {Observation of a Fast Evolution in a Parity-Time-Symmetric System}, Title = {{Observation of a Fast Evolution in a Parity-Time-Symmetric System}},
Volume = {371}, Volume = {371},
Year = {2013}, Year = {2013},
Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0053}, Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0053},
@ -436,7 +436,7 @@
Month = dec, Month = dec,
Number = {1}, Number = {1},
Pages = {2182}, Pages = {2182},
Title = {Observation of an Anti-PT-Symmetric Exceptional Point and Energy-Difference Conserving Dynamics in Electrical Circuit Resonators}, Title = {{Observation of an Anti-PT-Symmetric Exceptional Point and Energy-Difference Conserving Dynamics in Electrical Circuit Resonators}},
Volume = {9}, Volume = {9},
Year = {2018}, Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1038/s41467-018-04690-y}, Bdsk-Url-1 = {https://doi.org/10.1038/s41467-018-04690-y},
@ -452,7 +452,7 @@
Month = jan, Month = jan,
Number = {1}, Number = {1},
Pages = {11-19}, Pages = {11-19},
Title = {Non-Hermitian Physics and PT Symmetry}, Title = {{Non-Hermitian Physics and PT Symmetry}},
Volume = {14}, Volume = {14},
Year = {2018}, Year = {2018},
Bdsk-Url-1 = {https://doi.org/10.1038/nphys4323}, Bdsk-Url-1 = {https://doi.org/10.1038/nphys4323},
@ -468,7 +468,7 @@
Month = nov, Month = nov,
Number = {44}, Number = {44},
Pages = {444016}, Pages = {444016},
Title = {The Physics of Exceptional Points}, Title = {{The Physics of Exceptional Points}},
Volume = {45}, Volume = {45},
Year = {2012}, Year = {2012},
Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/45/44/444016}, Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/45/44/444016},
@ -483,7 +483,7 @@
Month = apr, Month = apr,
Number = {7}, Number = {7},
Pages = {1167-1178}, Pages = {1167-1178},
Title = {Avoided Level Crossing and Exceptional Points}, Title = {{Avoided Level Crossing and Exceptional Points}},
Volume = {23}, Volume = {23},
Year = {1990}, Year = {1990},
Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/23/7/022}, Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/23/7/022},
@ -496,14 +496,14 @@
doi = {10.1038/nphys3864}, doi = {10.1038/nphys3864},
Journal = {Nat. Phys.}, Journal = {Nat. Phys.},
Pages = {823--824}, Pages = {823--824},
Title = {Circling Exceptional Points}, Title = {{Circling Exceptional Points}},
Volume = {12}, Volume = {12},
Year = {2016}, Year = {2016},
Bdsk-Url-1 = {https://doi.org/10.1038/nphys3864}, Bdsk-Url-1 = {https://doi.org/10.1038/nphys3864},
} }
@article{Heiss_1999, @article{Heiss_1999,
Author = {W. D. Heiss}, Author = {W. D.},
Date-Added = {2018-12-06 20:49:57 +0100}, Date-Added = {2018-12-06 20:49:57 +0100},
Date-Modified = {2018-12-06 20:51:10 +0100}, Date-Modified = {2018-12-06 20:51:10 +0100},
doi = {10.1007/s100530050339}, doi = {10.1007/s100530050339},
@ -533,7 +533,7 @@
doi = {10.1103/RevModPhys.35.496}, doi = {10.1103/RevModPhys.35.496},
Journal = {Proc. Royal Soc. A}, Journal = {Proc. Royal Soc. A},
Pages = {45}, Pages = {45},
Title = {Quantal Phase Factors Accompanying Adiabatic Changes}, Title = {{Quantal Phase Factors Accompanying Adiabatic Changes}},
Volume = {392}, Volume = {392},
Year = {1984}, Year = {1984},
Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.35.496}, Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.35.496},
@ -544,7 +544,7 @@
Date-Added = {2018-10-16 13:34:32 +0200}, Date-Added = {2018-10-16 13:34:32 +0200},
Date-Modified = {2018-10-16 13:35:43 +0200}, Date-Modified = {2018-10-16 13:35:43 +0200},
Publisher = {Cambridge University Press}, Publisher = {Cambridge University Press},
Title = {Non-Hermitian Quantum Mechanics}, Title = {{Non-Hermitian Quantum Mechanics}},
Year = {2011}, Year = {2011},
} }
@ -555,7 +555,7 @@
Date-Modified = {2019-01-22 22:33:30 +0100}, Date-Modified = {2019-01-22 22:33:30 +0100},
Keywords = {qmech}, Keywords = {qmech},
Publisher = {McGraw-Hill}, Publisher = {McGraw-Hill},
Title = {Modern quantum chemistry: Introduction to advanced electronic structure}, Title = {Modern quantum chemistry: {Introduction} to advanced electronic structure},
Year = {1989}, Year = {1989},
} }
@ -567,7 +567,7 @@
} }
@article{Lepetit_1988, @article{Lepetit_1988,
title = {Origins of the poor convergence of many‐body perturbation theory expansions from unrestricted Hartree-Fock zeroth‐order descriptions}, title = {Origins of the poor convergence of many‐body perturbation theory expansions from unrestricted {Hartree-Fock} zeroth‐order descriptions},
volume = {89}, volume = {89},
doi = {10.1063/1.455170}, doi = {10.1063/1.455170},
pages = {998--1008}, pages = {998--1008},
@ -579,24 +579,23 @@
@article{Cremer_1996, @article{Cremer_1996,
title = {Sixth-Order Møller-Plesset Perturbation Theory On the Convergence of the MPn Series}, title = {{Sixth-Order Møller-Plesset Perturbation Theory On the Convergence of the MPn Series}},
volume = {100}, volume = {100},
doi = {10.1021/jp952815d}, doi = {10.1021/jp952815d},
pages = {6173--6188}, pages = {6173--6188},
number = {15}, number = {15},
journaltitle = {The Journal of Physical Chemistry}, journal = {J. Phys. Chem.},
shortjournal = {J. Phys. Chem.},
author = {Cremer, Dieter and He, Zhi}, author = {Cremer, Dieter and He, Zhi},
date = {1996-01-01}, date = {1996-01-01},
} }
@article{Baker_1971, @article{Baker_1971,
title = {Singularity Structure of the Perturbation Series for the Ground-State Energy of a Many-Fermion System}, title = {{Singularity Structure of the Perturbation Series for the Ground-State Energy of a Many-Fermion System}},
volume = {43}, volume = {43},
doi = {10.1103/RevModPhys.43.479}, doi = {10.1103/RevModPhys.43.479},
pages = {479--531}, pages = {479--531},
number = {4}, number = {4},
shortjournal = {Rev. Mod. Phys.}, Journal = {Rev. Mod. Phys.},
author = {{Baker}, {GEORGE} A.}, author = {{Baker}, {GEORGE} A.},
date = {1971-10-01}, date = {1971-10-01},
} }
@ -607,7 +606,7 @@
doi = {10.1103/PhysRevC.71.011304}, doi = {10.1103/PhysRevC.71.011304},
pages = {011304}, pages = {011304},
number = {1}, number = {1},
shortjournal = {Phys. Rev. C}, Journal = {Phys. Rev. C},
author = {Cejnar, Pavel and Heinze, Stefan and Dobeš, Jan}, author = {Cejnar, Pavel and Heinze, Stefan and Dobeš, Jan},
date = {2005-01-26}, date = {2005-01-26},
} }
@ -618,30 +617,29 @@
doi = {10.1103/PhysRevE.97.012112}, doi = {10.1103/PhysRevE.97.012112},
pages = {012112}, pages = {012112},
number = {1}, number = {1},
shortjournal = {Phys. Rev. E}, Journal = {Phys. Rev. E},
author = {Stránský, Pavel and Dvořák, Martin and Cejnar, Pavel}, author = {Stránský, Pavel and Dvořák, Martin and Cejnar, Pavel},
date = {2018-01-11}, date = {2018-01-11},
} }
@article{Cejnar_2007, @article{Cejnar_2007,
title = {Coulomb Analogy for Non-Hermitian Degeneracies near Quantum Phase Transitions}, title = {{Coulomb Analogy for Non-Hermitian Degeneracies near Quantum Phase Transitions}},
volume = {99}, volume = {99},
doi = {10.1103/PhysRevLett.99.100601}, doi = {10.1103/PhysRevLett.99.100601},
pages = {100601}, pages = {100601},
number = {10}, number = {10},
shortjournal = {Phys. Rev. Lett.}, Journal = {Phys. Rev. Lett.},
author = {Cejnar, Pavel and Heinze, Stefan and Macek, Michal}, author = {Cejnar, Pavel and Heinze, Stefan and Macek, Michal},
date = {2007-09-07}, date = {2007-09-07},
} }
@article{Heiss_1988, @article{Heiss_1988,
title = {Exceptional points of a Hamiltonian and phase transitions in finite systems}, title = {{Exceptional points of a Hamiltonian and phase transitions in finite systems}},
volume = {329}, volume = {329},
doi = {10.1007/BF01283767}, doi = {10.1007/BF01283767},
pages = {133--138}, pages = {133--138},
number = {2}, number = {2},
journaltitle = {Zeitschrift für Physik A Atomic Nuclei}, Journal = {Z. Physik A - Atomic Nuclei},
shortjournal = {Z. Physik A - Atomic Nuclei},
author = {Heiss, W. D.}, author = {Heiss, W. D.},
date = {1988-06-01}, date = {1988-06-01},
} }
@ -652,18 +650,18 @@
doi = {10.1103/PhysRevE.66.016217}, doi = {10.1103/PhysRevE.66.016217},
pages = {016217}, pages = {016217},
number = {1}, number = {1},
shortjournal = {Phys. Rev. E}, Journal = {Phys. Rev. E},
author = {Heiss, W. D. and Müller, M.}, author = {Heiss, W. D. and Müller, M.},
date = {2002-07-26}, date = {2002-07-26},
} }
@article{Sindelka_2017, @article{Sindelka_2017,
title = {Excited-state quantum phase transitions studied from a non-Hermitian perspective}, title = {Excited-state quantum phase transitions studied from a non-{Hermitian} perspective},
volume = {95}, volume = {95},
doi = {10.1103/PhysRevA.95.010103}, doi = {10.1103/PhysRevA.95.010103},
pages = {010103}, pages = {010103},
number = {1}, number = {1},
shortjournal = {Phys. Rev. A}, Journal = {Phys. Rev. A},
author = {Šindelka, Milan and Santos, Lea F. and Moiseyev, Nimrod}, author = {Šindelka, Milan and Santos, Lea F. and Moiseyev, Nimrod},
date = {2017-01-24}, date = {2017-01-24},
} }
@ -675,7 +673,7 @@
doi = {10.1088/0031-8949/90/11/114015}, doi = {10.1088/0031-8949/90/11/114015},
pages = {114015}, pages = {114015},
number = {11}, number = {11},
shortjournal = {Phys. Scr.}, Journal = {Phys. Scr.},
author = {Cejnar, Pavel and Stránský, Pavel and Kloc, Michal}, author = {Cejnar, Pavel and Stránský, Pavel and Kloc, Michal},
date = {2015-10}, date = {2015-10},
} }
@ -686,18 +684,18 @@
doi = {10.1016/j.ppnp.2008.08.001}, doi = {10.1016/j.ppnp.2008.08.001},
pages = {210--256}, pages = {210--256},
number = {1}, number = {1},
shortjournal = {Progress in Particle and Nuclear Physics}, Journal = {Prog. Part. Nucl. Phys.},
author = {Cejnar, Pavel and Jolie, Jan}, author = {Cejnar, Pavel and Jolie, Jan},
date = {2009-01-01}, date = {2009-01-01},
} }
@article{Borisov_2015, @article{Borisov_2015,
title = {Multiply Degenerate Exceptional Points and Quantum Phase Transitions}, title = {{Multiply Degenerate Exceptional Points and Quantum Phase Transitions}},
volume = {54}, volume = {54},
doi = {10.1007/s10773-014-2493-y}, doi = {10.1007/s10773-014-2493-y},
pages = {4293--4305}, pages = {4293--4305},
number = {12}, number = {12},
shortjournal = {Int J Theor Phys}, Journal = {Int J Theor Phys},
author = {Borisov, Denis I. and Ružička, František and Znojil, Miloslav}, author = {Borisov, Denis I. and Ružička, František and Znojil, Miloslav},
date = {2015-12-01}, date = {2015-12-01},
} }
@ -708,18 +706,18 @@
doi = {10.1016/j.aop.2007.06.011}, doi = {10.1016/j.aop.2007.06.011},
pages = {1106--1135}, pages = {1106--1135},
number = {5}, number = {5},
shortjournal = {Annals of Physics}, Journal = {Ann Phys (N Y)},
author = {Caprio, M. A. and Cejnar, P. and Iachello, F.}, author = {Caprio, M. A. and Cejnar, P. and Iachello, F.},
date = {2008-05-01}, date = {2008-05-01},
} }
@article{Macek_2019, @article{Macek_2019,
title = {Excited-state quantum phase transitions in systems with two degrees of freedom. {III}. Interacting boson systems}, title = {Excited-state quantum phase transitions in systems with two degrees of freedom. {III}. {Interacting boson systems}},
volume = {99}, volume = {99},
doi = {10.1103/PhysRevC.99.064323}, doi = {10.1103/PhysRevC.99.064323},
pages = {064323}, pages = {064323},
number = {6}, number = {6},
shortjournal = {Phys. Rev. C}, Journal = {Phys. Rev. C},
author = {Macek, Michal and Stránský, Pavel and Leviatan, Amiram and Cejnar, Pavel}, author = {Macek, Michal and Stránský, Pavel and Leviatan, Amiram and Cejnar, Pavel},
date = {2019-06-21}, date = {2019-06-21},
} }
@ -728,7 +726,7 @@
@book{Sachdev_2011, @book{Sachdev_2011,
location = {Cambridge}, location = {Cambridge},
edition = {2}, edition = {2},
title = {Quantum Phase Transitions}, title = {{Quantum Phase Transitions}},
publisher = {Cambridge University Press}, publisher = {Cambridge University Press},
author = {Sachdev, Subir}, author = {Sachdev, Subir},
date = {2011}, date = {2011},
@ -742,13 +740,13 @@
doi = {10.1088/0031-8949/91/8/083006}, doi = {10.1088/0031-8949/91/8/083006},
pages = {083006}, pages = {083006},
number = {8}, number = {8},
shortjournal = {Phys. Scr.}, Journal = {Phys. Scr.},
author = {Cejnar, Pavel and Stránský, Pavel}, author = {Cejnar, Pavel and Stránský, Pavel},
date = {2016-07}, date = {2016-07},
} }
@article{Coulson_1949, @article{Coulson_1949,
title = {{XXXIV}. Notes on the molecular orbital treatment of the hydrogen molecule}, title = {{XXXIV}. {Notes} on the molecular orbital treatment of the hydrogen molecule},
volume = {40}, volume = {40},
doi = {10.1080/14786444908521726}, doi = {10.1080/14786444908521726},
pages = {386--393}, pages = {386--393},
@ -763,7 +761,7 @@
volume = {150}, volume = {150},
doi = {10.1063/1.5085121}, doi = {10.1063/1.5085121},
number = {4}, number = {4},
journal = {The Journal of Chemical Physics}, journal = {J. Chem. Phys.},
author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-François}, author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-François},
month = jan, month = jan,
year = {2019}, year = {2019},
@ -775,7 +773,7 @@
volume = {15}, volume = {15},
doi = {10.1021/acs.jctc.9b00289}, doi = {10.1021/acs.jctc.9b00289},
number = {8}, number = {8},
journal = {Journal of Chemical Theory and Computation}, journal = {J. Chem. Theory Comput.},
author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-François}, author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-François},
month = aug, month = aug,
year = {2019}, year = {2019},
@ -787,9 +785,22 @@
volume = {10}, volume = {10},
doi = {10.1021/ct5007696}, doi = {10.1021/ct5007696},
number = {11}, number = {11},
journal = {Journal of Chemical Theory and Computation}, journal = {J. Chem. Theory Comput.},
author = {Hiscock, Hamish G. and Thom, Alex J. W.}, author = {Hiscock, Hamish G. and Thom, Alex J. W.},
month = nov, month = nov,
year = {2014}, year = {2014},
pages = {4795--4800}, pages = {4795--4800},
} }
@article{seidl_communication_2018,
title = {Communication: {Strong}-interaction limit of an adiabatic connection in {Hartree}-{Fock} theory},
volume = {149},
doi = {10.1063/1.5078565},
number = {24},
journal = {J. Chem. Phys.},
author = {Seidl, Michael and Giarrusso, Sara and Vuckovic, Stefan and Fabiano, Eduardo and Gori-Giorgi, Paola},
month = dec,
year = {2018},
pages = {241101},
}

8
RapportStage/Rapport.tex
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@ -324,7 +324,7 @@ E_{\text{MP}_{n}}= \sum_{k=0}^n E^{(k)}
But as mentioned before \textit{a priori} there are no reasons that $E_{\text{MP}_{n}}$ is always convergent when $n$ goes to infinity. In fact, it is known that when the Hartree-Fock wave function is a bad approximation of the exact wave function, for example for multi-reference states, the M{\o}ller-Plesset method will give bad results \cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. A smart way to investigate the convergence properties of the M{\o}ller-Plesset series is to transform the coupling parameter $\lambda$ into a complex variable. By doing so the Hamiltonian and the energy become functions of this variable. The energy becomes a multivalued function on $n$ Riemann sheets. As mentioned above by searching the singularities of the function $E(\lambda)$ we can get information on the convergence properties of the M{\o}ller-Plesset perturbation theory. Those singularities of the energy are exactly the exceptional points connecting the electronic states mentioned in the introduction. The direct computation of the terms of the series is quite easy up to the 4th order and the 5th and 6th order can be obtained at high cost \cite{JensenBook}. But to deeply understand the behavior of the M{\o}ller-Plesset series and how it is connected to the singularities, we need to have access to high order terms of the series. For small systems we can have access to the whole series using Full Configuration Interaction (FCI). If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI basis set we get the exact energies (in this finite basis set) and expanding in $\lambda$ allows to to get the M{\o}ller-Plesset perturbation series at every order. But as mentioned before \textit{a priori} there are no reasons that $E_{\text{MP}_{n}}$ is always convergent when $n$ goes to infinity. In fact, it is known that when the Hartree-Fock wave function is a bad approximation of the exact wave function, for example for multi-reference states, the M{\o}ller-Plesset method will give bad results \cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. A smart way to investigate the convergence properties of the M{\o}ller-Plesset series is to transform the coupling parameter $\lambda$ into a complex variable. By doing so the Hamiltonian and the energy become functions of this variable. The energy becomes a multivalued function on $n$ Riemann sheets. As mentioned above by searching the singularities of the function $E(\lambda)$ we can get information on the convergence properties of the M{\o}ller-Plesset perturbation theory. Those singularities of the energy are exactly the exceptional points connecting the electronic states mentioned in the introduction. The direct computation of the terms of the series is quite easy up to the 4th order and the 5th and 6th order can be obtained at high cost \cite{JensenBook}. But to deeply understand the behavior of the M{\o}ller-Plesset series and how it is connected to the singularities, we need to have access to high order terms of the series. For small systems we can have access to the whole series using Full Configuration Interaction (FCI). If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI basis set we get the exact energies (in this finite basis set) and expanding in $\lambda$ allows to to get the M{\o}ller-Plesset perturbation series at every order.
\subsection{Alternative partitioning} \subsection{Alternative partitioning}\label{sec:AlterPart}
The M{\o}ller-Plesset partitioning is not the only one possible in electronic structure theory. An other possibility, even more natural than the M{\o}ller-Plesset one, is to take the diagonal elements of $\bH$ as the zeroth-order Hamiltonian. Hence, the off-diagonal elements of $\bH(\lambda)$ are the perturbation operator. The M{\o}ller-Plesset partitioning is not the only one possible in electronic structure theory. An other possibility, even more natural than the M{\o}ller-Plesset one, is to take the diagonal elements of $\bH$ as the zeroth-order Hamiltonian. Hence, the off-diagonal elements of $\bH(\lambda)$ are the perturbation operator.
@ -550,7 +550,9 @@ In this part, we will try to investigate how some parameters of $\bH(\lambda)$ i
\pdv{E}\text{det}[E-\bH(\lambda)]=0 \pdv{E}\text{det}[E-\bH(\lambda)]=0
\end{equation} \end{equation}
We will take the simple case of the M{\o}ller-Plesset partitioning with a restricted Hartree-Fock minimal basis set as our starting point for this analysis. \\ We will take the simple case of the M{\o}ller-Plesset partitioning with a restricted Hartree-Fock minimal basis set as our starting point for this analysis. The electron 1 have a spin $\alpha$ and the electron 2 a spin $\beta$. Hence we can forget the spin part of the spin-orbitals and from now we will work with spatial orbitals. In the restricted formalism the spatial orbitals are the same so the two-electron basis set is: $\psi_1=Y_0(\theta_1)Y_0(\theta_2)$; $\psi_2=Y_0(\theta_1)Y_1(\theta_2)$; $\psi_3=Y_1(\theta_1)Y_0(\theta_2)$; $\psi_4=Y_1(\theta_1)Y_1(\theta_2)$.
The Hamiltonian $\bH(\lambda)$ is block diagonal because of the symmetry of the basis set. $\psi_1$ have the same symmetry as $\psi_4$ and there is an avoided crossing between these two states as we can see in \autoref{fig:RHFMiniBasRCV}.
\begin{figure}[h!] \begin{figure}[h!]
\centering \centering
@ -560,7 +562,7 @@ We will take the simple case of the M{\o}ller-Plesset partitioning with a restri
\label{fig:RHFMiniBasRCV} \label{fig:RHFMiniBasRCV}
\end{figure} \end{figure}
Then we will compare the different partitioning using the same basis set. The \autoref{fig:RadiusPartitioning} shows the evolution of the radius of convergence in function of $R$ for the M{\o}ller-Plesset, the Epstein-Nesbet, the weak correlation and the strong coupling partitioning. We can see that Then we can compare the different partitioning of \autoref{sec:AlterPart} using the same basis set. The \autoref{fig:RadiusPartitioning} shows the evolution of the radius of convergence in function of $R$ for the M{\o}ller-Plesset, the Epstein-Nesbet, the Weak Correlation and the Strong Coupling partitioning. We can see that
\begin{figure}[h!] \begin{figure}[h!]
\centering \centering