From ded51515753be509eed2b112887bf9c4da1f065d Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Tue, 28 Jul 2020 10:22:14 +0200 Subject: [PATCH] saving work --- RapportStage/Rapport.bib | 840 +++++++++++++++++++-------------------- RapportStage/Rapport.tex | 30 +- 2 files changed, 427 insertions(+), 443 deletions(-) diff --git a/RapportStage/Rapport.bib b/RapportStage/Rapport.bib index 7da3ff0..95b3a18 100644 --- a/RapportStage/Rapport.bib +++ b/RapportStage/Rapport.bib @@ -1,197 +1,213 @@ +%% This BibTeX bibliography file was created using BibDesk. +%% http://bibdesk.sourceforge.net/ + +%% Created for Pierre-Francois Loos at 2020-07-28 10:22:06 +0200 + + +%% Saved with string encoding Unicode (UTF-8) + + + +@book{BenderBook, + Author = {C. M. Berder and S. A. Orszag}, + Date-Added = {2020-07-28 09:59:40 +0200}, + Date-Modified = {2020-07-28 09:59:40 +0200}, + Publisher = {Springer}, + Title = {Advanced Mathematical Methods for Scientists and Engineers: Asymptotics Methods and Perturbation Theory}, + Year = {1978}} + @article{Gill_1986, - Title = {Deceptive convergence in {Møller-Plesset} perturbation energies}, - Volume = {132}, - doi = {10.1016/0009-2614(86)80686-8}, - pages = {16--22}, - number = {1}, - Journal = {Chem. Phys. Lett.}, - author = {Gill, Peter M. W. and Radom, Leo}, + Author = {Gill, Peter M. W. and Radom, Leo}, Date = {1986-11-28}, -} + Doi = {10.1016/0009-2614(86)80686-8}, + Journal = {Chem. Phys. Lett.}, + Number = {1}, + Pages = {16--22}, + Title = {Deceptive convergence in {M{\o}ller-Plesset} perturbation energies}, + Volume = {132}, + Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(86)80686-8}} @article{Gill_1988, - Title = {Why does unrestricted {Møller-Plesset} perturbation theory converge so slowly for spin‐contaminated wave functions?}, - Volume = {89}, - doi = {10.1063/1.455312}, - pages = {7307--7314}, - number = {12}, - Journal = {J. Chem. Phys.}, - author = {Gill, Peter M. W. and Pople, John A. and Radom, Leo and Nobes, Ross H.}, + Author = {Gill, Peter M. W. and Pople, John A. and Radom, Leo and Nobes, Ross H.}, Date = {1988-12-15}, -} + Doi = {10.1063/1.455312}, + Journal = {J. Chem. Phys.}, + Number = {12}, + Pages = {7307--7314}, + Title = {Why does unrestricted {M{\o}ller-Plesset} perturbation theory converge so slowly for spin‐contaminated wave functions?}, + Volume = {89}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.455312}} @article{Sergeev_2005, - Title = {On the nature of the {Møller-Plesset} critical point}, - Volume = {123}, - doi = {10.1063/1.1991854}, - pages = {064105}, - number = {6}, - Journal = {J. Chem. Phys.}, - author = {Sergeev, Alexey V. and Goodson, David Z. and Wheeler, Steven E. and Allen, Wesley D.}, + Author = {Sergeev, Alexey V. and Goodson, David Z. and Wheeler, Steven E. and Allen, Wesley D.}, Date = {2005-08-08}, -} + Doi = {10.1063/1.1991854}, + Journal = {J. Chem. Phys.}, + Number = {6}, + Pages = {064105}, + Title = {On the nature of the {M{\o}ller-Plesset} critical point}, + Volume = {123}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1991854}} @article{Sergeev_2006, - Title = {Singularities of {Møller-Plesset} energy functions}, - Volume = {124}, - doi = {10.1063/1.2173989}, - pages = {094111}, - number = {9}, - Journal = {J. Chem. Phys.}, - author = {Sergeev, Alexey V. and Goodson, David Z.}, + Author = {Sergeev, Alexey V. and Goodson, David Z.}, Date = {2006-03-07}, -} + Doi = {10.1063/1.2173989}, + Journal = {J. Chem. Phys.}, + Number = {9}, + Pages = {094111}, + Title = {Singularities of {M{\o}ller-Plesset} energy functions}, + Volume = {124}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.2173989}} @article{Stillinger_2000, - Title = {{Møller-Plesset} convergence issues in computational quantum chemistry}, - Volume = {112}, - doi = {10.1063/1.481608}, - pages = {9711--9715}, - number = {22}, - Journal = {J. Chem. Phys.}, - author = {Stillinger, Frank H.}, + Author = {Stillinger, Frank H.}, Date = {2000-05-31}, -} - + Doi = {10.1063/1.481608}, + Journal = {J. Chem. Phys.}, + Number = {22}, + Pages = {9711--9715}, + Title = {{M{\o}ller-Plesset} convergence issues in computational quantum chemistry}, + Volume = {112}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.481608}} @article{Olsen_1996, - Title = {Surprising cases of divergent behavior in {Møller-Plesset} perturbation theory}, - Volume = {105}, - doi = {10.1063/1.472352}, - pages = {5082--5090}, + Author = {Olsen, Jeppe and Christiansen, Ove and Koch, Henrik and J{\o}rgensen, Poul}, + Date = {1996-09-01}, + Doi = {10.1063/1.472352}, Journal = {J. Chem. Phys.}, - author = {Olsen, Jeppe and Christiansen, Ove and Koch, Henrik and Jørgensen, Poul}, - Date = {1996-09-01} -} + Pages = {5082--5090}, + Title = {Surprising cases of divergent behavior in {M{\o}ller-Plesset} perturbation theory}, + Volume = {105}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.472352}} @article{Olsen_2000, - Title = {Divergence in {Møller-Plesset} theory: {A} simple explanation based on a two-state model}, - Volume = {112}, - doi = {10.1063/1.481611}, - pages = {9736--9748}, - number = {22}, - Journal = {J. Chem. Phys.}, - author = {Olsen, Jeppe and Jørgensen, Poul and Helgaker, Trygve and Christiansen, Ove}, + Author = {Olsen, Jeppe and J{\o}rgensen, Poul and Helgaker, Trygve and Christiansen, Ove}, Date = {2000-05-31}, -} + Doi = {10.1063/1.481611}, + Journal = {J. Chem. Phys.}, + Number = {22}, + Pages = {9736--9748}, + Title = {Divergence in {M{\o}ller-Plesset} theory: {A} simple explanation based on a two-state model}, + Volume = {112}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.481611}} @article{Olsen_2019, + Author = {Olsen, Jeppe and J{\o}rgensen, Poul}, + Date = {2019-08-28}, + Doi = {10.1063/1.5110554}, + Journal = {J. Chem. Phys.}, + Number = {8}, + Pages = {084108}, Title = {Convergence patterns and rates in two-state perturbation expansions}, Volume = {151}, - doi = {10.1063/1.5110554}, - pages = {084108}, - number = {8}, - Journal = {J. Chem. Phys.}, - author = {Olsen, Jeppe and Jørgensen, Poul}, - Date = {2019-08-28}, -} + Bdsk-Url-1 = {https://doi.org/10.1063/1.5110554}} @article{Leininger_2000, - Title = {Is {Møller-Plesset} perturbation theory a convergent ab initio method?}, - Volume = {112}, - doi = {10.1063/1.481764}, - pages = {9213--9222}, - number = {21}, - Journal = {J. Chem. Phys.}, - author = {Leininger, Matthew L. and Allen, Wesley D. and Schaefer, Henry F. and Sherrill, C. David}, + Author = {Leininger, Matthew L. and Allen, Wesley D. and Schaefer, Henry F. and Sherrill, C. David}, Date = {2000-05-17}, -} + Doi = {10.1063/1.481764}, + Journal = {J. Chem. Phys.}, + Number = {21}, + Pages = {9213--9222}, + Title = {Is {M{\o}ller-Plesset} perturbation theory a convergent ab initio method?}, + Volume = {112}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.481764}} @article{Moller_1934, + Author = {M{\o}ller, Chr. and Plesset, M. S.}, + Date = {1934-10-01}, + Doi = {10.1103/PhysRev.46.618}, + Journal = {Phys. Rev.}, + Number = {7}, + Pages = {618--622}, Title = {Note on an {Approximation Treatment for Many-Electron Systems}}, Volume = {46}, - doi = {10.1103/PhysRev.46.618}, - pages = {618--622}, - number = {7}, - Journal = {Phys. Rev.}, - author = {Møller, Chr. and Plesset, M. S.}, - Date = {1934-10-01}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.46.618}} @article{Handy_1985, - Title = {On the convergence of the {Møller-Plesset} perturbation series}, - Volume = {68}, - doi = {10.1007/BF00698753}, - pages = {87--100}, - number = {1}, - Journal = {Theoret. Chim. Acta}, - author = {Handy, N. C. and Knowles, P. J. and Somasundram, K.}, + Author = {Handy, N. C. and Knowles, P. J. and Somasundram, K.}, Date = {1985-07-01}, -} + Doi = {10.1007/BF00698753}, + Journal = {Theoret. Chim. Acta}, + Number = {1}, + Pages = {87--100}, + Title = {On the convergence of the {M{\o}ller-Plesset} perturbation series}, + Volume = {68}, + Bdsk-Url-1 = {https://doi.org/10.1007/BF00698753}} @article{Christiansen_1996, - Title = {On the inherent divergence in the {Møller-Plesset} series. {The} neon atom — a test case}, - Volume = {261}, - doi = {10.1016/0009-2614(96)00974-8}, - pages = {369--378}, - number = {3}, - Journal = {Chem. Phys. Lett.}, - author = {Christiansen, Ove and Olsen, Jeppe and Jørgensen, Poul and Koch, Henrik and Malmqvist, Per-Åke}, - urlDate = {2020-07-07}, + Author = {Christiansen, Ove and Olsen, Jeppe and J{\o}rgensen, Poul and Koch, Henrik and Malmqvist, Per-{\AA}ke}, Date = {1996-10-18}, -} + Doi = {10.1016/0009-2614(96)00974-8}, + Journal = {Chem. Phys. Lett.}, + Number = {3}, + Pages = {369--378}, + Title = {On the inherent divergence in the {M{\o}ller-Plesset} series. {The} neon atom --- a test case}, + Urldate = {2020-07-07}, + Volume = {261}, + Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(96)00974-8}} @incollection{Goodson_2004, - Title = {Singularity Structure of {Møller-Plesset Perturbation Theory}}, - Volume = {47}, - doi = {10.1016/S0065-3276(04)47011-7}, - pages = {193--208}, - bookTitle = {Adv. Quantum Chem.}, - publisher = {Academic Press}, - author = {Goodson, David Z. and Sergeev, Alexey V.}, + Author = {Goodson, David Z. and Sergeev, Alexey V.}, + Booktitle = {Adv. Quantum Chem.}, Date = {2004-01-01}, - -} + Doi = {10.1016/S0065-3276(04)47011-7}, + Pages = {193--208}, + Publisher = {Academic Press}, + Title = {Singularity Structure of {M{\o}ller-Plesset Perturbation Theory}}, + Volume = {47}, + Bdsk-Url-1 = {https://doi.org/10.1016/S0065-3276(04)47011-7}} @article{Goodson_2012, - title = {Resummation methods}, - volume = {2}, - doi = {10.1002/wcms.92}, - pages = {743--761}, - number = {5}, - journaltitle = {{WIREs} Comput. Mol. Sci.}, - author = {Goodson, David Z.}, - date = {2012}, -} + Author = {Goodson, David Z.}, + Date = {2012}, + Doi = {10.1002/wcms.92}, + Journaltitle = {{WIREs} Comput. Mol. Sci.}, + Number = {5}, + Pages = {743--761}, + Title = {Resummation methods}, + Volume = {2}, + Bdsk-Url-1 = {https://doi.org/10.1002/wcms.92}} @article{Katz_1962, - Title = {The analytic structure of many-body perturbation theory}, - Volume = {29}, - doi = {10.1016/0029-5582(62)90191-8}, - pages = {353--372}, - Journal = {Nuclear Physics}, - author = {Katz, Amnon}, - urlDate = {2020-07-07}, + Author = {Katz, Amnon}, Date = {1962-01-01}, -} + Doi = {10.1016/0029-5582(62)90191-8}, + Journal = {Nuclear Physics}, + Pages = {353--372}, + Title = {The analytic structure of many-body perturbation theory}, + Urldate = {2020-07-07}, + Volume = {29}, + Bdsk-Url-1 = {https://doi.org/10.1016/0029-5582(62)90191-8}} @article{Fukutome_1981, + Author = {Fukutome, Hideo}, + Date = {1981}, + Doi = {10.1002/qua.560200502}, + Journaltitle = {J. Quantum Chem.}, + Number = {5}, + Pages = {955--1065}, Title = {Unrestricted {Hartree-Fock} theory and its applications to molecules and chemical reactions}, Volume = {20}, - doi = {10.1002/qua.560200502}, - pages = {955--1065}, - number = {5}, - journalTitle = {J. Quantum Chem.}, - author = {Fukutome, Hideo}, - Date = {1981}, -} + Bdsk-Url-1 = {https://doi.org/10.1002/qua.560200502}} @article{Stillinger_1966, + Author = {Stillinger, Frank H.}, + Date = {1966-11-15}, + Doi = {10.1063/1.1727380}, + Journal = {J. Chem. Phys.}, + Number = {10}, + Pages = {3623--3631}, Title = {{Ground‐State Energy of Two‐Electron Atoms}}, Volume = {45}, - doi = {10.1063/1.1727380}, - pages = {3623--3631}, - number = {10}, - Journal = {J. Chem. Phys.}, - author = {Stillinger, Frank H.}, - Date = {1966-11-15}, -} + Bdsk-Url-1 = {https://doi.org/10.1063/1.1727380}} @article{Bittner_2012, Author = {Bittner, S. and Dietz, B. and G\"unther, U. and Harney, H. L. and Miski-Oglu, M. and Richter, A. and Sch\"afer, F.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:25:31 +0100}, - doi = {10.1103/PhysRevLett.108.024101}, + Doi = {10.1103/PhysRevLett.108.024101}, Journal = {Phys. Rev. Lett.}, Month = jan, Number = {2}, @@ -199,14 +215,13 @@ Title = {{PT Symmetry} and {{Spontaneous Symmetry Breaking}} in a {{Microwave Billiard}}}, Volume = {108}, Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.108.024101}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.108.024101}} @article{Chong_2011, Author = {Chong, Y. D. and Ge, Li and Stone, A. Douglas}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-27 20:43:12 +0100}, - doi = {10.1103/PhysRevLett.106.093902}, + Doi = {10.1103/PhysRevLett.106.093902}, Journal = {Phys. Rev. Lett.}, Month = mar, Number = {9}, @@ -214,14 +229,13 @@ Title = {{PT Symmetry Breaking and Laser Absorber Modes in Optical Scattering Systems}}, Volume = {106}, Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.106.093902}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.106.093902}} @article{Chtchelkatchev_2012, Author = {Chtchelkatchev, N. M. and Golubov, A. A. and Baturina, T. I. and Vinokur, V. M.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:26:10 +0100}, - doi = {10.1103/PhysRevLett.109.150405}, + Doi = {10.1103/PhysRevLett.109.150405}, Journal = {Phys. Rev. Lett.}, Month = oct, Number = {15}, @@ -229,14 +243,13 @@ Title = {{Stimulation of the Fluctuation Superconductivity by PT Symmetry}}, Volume = {109}, Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.109.150405}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.109.150405}} @article{Doppler_2016, Author = {Doppler, J\"org and Mailybaev, Alexei A. and B\"ohm, Julian and Kuhl, Ulrich and Girschik, Adrian and Libisch, Florian and Milburn, Thomas J. and Rabl, Peter and Moiseyev, Nimrod and Rotter, Stefan}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1038/nature18605}, + Doi = {10.1038/nature18605}, Journal = {Nature}, Month = sep, Number = {7618}, @@ -244,14 +257,13 @@ Title = {{Dynamically Encircling an Exceptional Point for Asymmetric Mode Switching}}, Volume = {537}, Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1038/nature18605}, -} + Bdsk-Url-1 = {https://doi.org/10.1038/nature18605}} @article{Guo_2009, Author = {Guo, A. and Salamo, G. J. and Duchesne, D. and Morandotti, R. and {Volatier-Ravat}, M. and Aimez, V. and Siviloglou, G. A. and Christodoulides, D. N.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:26:27 +0100}, - doi = {10.1103/PhysRevLett.103.093902}, + Doi = {10.1103/PhysRevLett.103.093902}, Journal = {Phys. Rev. Lett.}, Month = aug, Number = {9}, @@ -259,14 +271,13 @@ Title = {Observation of {PT Symmetry Breaking in Complex Optical Potentials}}, Volume = {103}, Year = {2009}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.093902}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.093902}} @article{Hang_2013, Author = {Hang, Chao and Huang, Guoxiang and Konotop, Vladimir V.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:26:38 +0100}, - doi = {10.1103/PhysRevLett.110.083604}, + Doi = {10.1103/PhysRevLett.110.083604}, Journal = {Phys. Rev. Lett.}, Month = feb, Number = {8}, @@ -274,14 +285,13 @@ Title = {{PT Symmetry with a System of Three-Level Atoms}}, Volume = {110}, Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.110.083604}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.110.083604}} @article{Liertzer_2012, Author = {Liertzer, M. and Ge, Li and Cerjan, A. and Stone, A. D. and T\"ureci, H. E. and Rotter, S.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:26:53 +0100}, - doi = {10.1103/PhysRevLett.108.173901}, + Doi = {10.1103/PhysRevLett.108.173901}, Journal = {Phys. Rev. Lett.}, Month = apr, Number = {17}, @@ -289,14 +299,13 @@ Title = {{Pump-Induced Exceptional Points in Lasers}}, Volume = {108}, Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.108.173901}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.108.173901}} @article{Longhi_2010, Author = {Longhi, Stefano}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:27:08 +0100}, - doi = {10.1103/PhysRevLett.105.013903}, + Doi = {10.1103/PhysRevLett.105.013903}, Journal = {Phys. Rev. Lett.}, Month = jun, Number = {1}, @@ -304,14 +313,13 @@ Title = {{Optical Realization of Relativistic Non-Hermitian Quantum Mechanics}}, Volume = {105}, Year = {2010}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.105.013903}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.105.013903}} @article{Peng_2014, Author = {Peng, B. and Ozdemir, . K. and Rotter, S. and Yilmaz, H. and Liertzer, M. and Monifi, F. and Bender, C. M. and Nori, F. and Yang, L.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1126/science.1258004}, + Doi = {10.1126/science.1258004}, Journal = {Science}, Month = oct, Number = {6207}, @@ -319,14 +327,13 @@ Title = {{Loss-Induced Suppression and Revival of Lasing}}, Volume = {346}, Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1126/science.1258004}, -} + Bdsk-Url-1 = {https://doi.org/10.1126/science.1258004}} @article{Peng_2014a, Author = {Peng, Bo and \"Ozdemir, {\c S}ahin Kaya and Lei, Fuchuan and Monifi, Faraz and Gianfreda, Mariagiovanna and Long, Gui Lu and Fan, Shanhui and Nori, Franco and Bender, Carl M. and Yang, Lan}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1038/nphys2927}, + Doi = {10.1038/nphys2927}, Journal = {Nat. Phys.}, Month = may, Number = {5}, @@ -334,14 +341,13 @@ Title = {{Parity\textendash{}Time-Symmetric Whispering-Gallery Microcavities}}, Volume = {10}, Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys2927}, -} + Bdsk-Url-1 = {https://doi.org/10.1038/nphys2927}} @article{Regensburger_2012, Author = {Regensburger, Alois and Bersch, Christoph and Miri, Mohammad-Ali and Onishchukov, Georgy and Christodoulides, Demetrios N. and Peschel, Ulf}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1038/nature11298}, + Doi = {10.1038/nature11298}, Journal = {Nature}, Month = aug, Number = {7410}, @@ -349,14 +355,13 @@ Title = {{Parity\textendash{}Time Synthetic Photonic Lattices}}, Volume = {488}, Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1038/nature11298}, -} + Bdsk-Url-1 = {https://doi.org/10.1038/nature11298}} @article{Ruter_2010, Author = {R\"uter, Christian E. and Makris, Konstantinos G. and {El-Ganainy}, Ramy and Christodoulides, Demetrios N. and Segev, Mordechai and Kip, Detlef}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1038/nphys1515}, + Doi = {10.1038/nphys1515}, Journal = {Nat. Phys.}, Month = mar, Number = {3}, @@ -364,14 +369,13 @@ Title = {{Observation of Parity\textendash{}Time Symmetry in Optics}}, Volume = {6}, Year = {2010}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys1515}, -} + Bdsk-Url-1 = {https://doi.org/10.1038/nphys1515}} @article{Schindler_2011, Author = {Schindler, Joseph and Li, Ang and Zheng, Mei C. and Ellis, F. M. and Kottos, Tsampikos}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:27:21 +0100}, - doi = {10.1103/PhysRevA.84.040101}, + Doi = {10.1103/PhysRevA.84.040101}, Journal = {Phys. Rev. A}, Month = oct, Number = {4}, @@ -379,14 +383,13 @@ Title = {{Experimental Study of Active \emph{LRC} Circuits with PT Symmetries}}, Volume = {84}, Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.040101}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.040101}} @article{Szameit_2011, Author = {Szameit, Alexander and Rechtsman, Mikael C. and {Bahat-Treidel}, Omri and Segev, Mordechai}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:27:31 +0100}, - doi = {10.1103/PhysRevA.84.021806}, + Doi = {10.1103/PhysRevA.84.021806}, Journal = {Phys. Rev. A}, Month = aug, Number = {2}, @@ -394,14 +397,13 @@ Title = {{PT-Symmetry in Honeycomb Photonic Lattice}s}, Volume = {84}, Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.021806}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.021806}} @article{Zhao_2010, Author = {Zhao, K. F. and Schaden, M. and Wu, Z.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:27:45 +0100}, - doi = {10.1103/PhysRevA.81.042903}, + Doi = {10.1103/PhysRevA.81.042903}, Journal = {Phys. Rev. A}, Month = apr, Number = {4}, @@ -409,14 +411,13 @@ Title = {{Enhanced Magnetic Resonance Signal of Spin-Polarized {{Rb}} Atoms near Surfaces of Coated Cells}}, Volume = {81}, Year = {2010}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.042903}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.81.042903}} @article{Zheng_2013, Author = {Zheng, C. and Hao, L. and Long, G. L.}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1098/rsta.2012.0053}, + Doi = {10.1098/rsta.2012.0053}, Journal = {Philos. Trans. R. Soc. Math. Phys. Eng. Sci.}, Month = mar, Number = {1989}, @@ -424,14 +425,13 @@ Title = {{Observation of a Fast Evolution in a Parity-Time-Symmetric System}}, Volume = {371}, Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0053}, -} + Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0053}} @article{Choi_2018, Author = {Choi, Youngsun and Hahn, Choloong and Yoon, Jae Woong and Song, Seok Ho}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-21 16:29:02 +0100}, - doi = {10.1038/s41467-018-04690-y}, + Doi = {10.1038/s41467-018-04690-y}, Journal = {Nat. Commun.}, Month = dec, Number = {1}, @@ -439,14 +439,13 @@ Title = {{Observation of an Anti-PT-Symmetric Exceptional Point and Energy-Difference Conserving Dynamics in Electrical Circuit Resonators}}, Volume = {9}, 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}} @article{El-Ganainy_2018, Author = {El-Ganainy, Ramy and Makris, Konstantinos G. and Khajavikhan, Mercedeh and Musslimani, Ziad H. and Rotter, Stefan and Christodoulides, Demetrios N.}, Date-Added = {2019-01-30 12:10:10 +0100}, Date-Modified = {2019-01-30 12:10:10 +0100}, - doi = {10.1038/nphys4323}, + Doi = {10.1038/nphys4323}, Journal = {Nat. Phys.}, Language = {en}, Month = jan, @@ -455,14 +454,13 @@ Title = {{Non-Hermitian Physics and PT Symmetry}}, Volume = {14}, Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys4323}, -} + Bdsk-Url-1 = {https://doi.org/10.1038/nphys4323}} @article{Heiss_2012, Author = {Heiss, W D}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1088/1751-8113/45/44/444016}, + Doi = {10.1088/1751-8113/45/44/444016}, Issn = {1751-8113, 1751-8121}, Journal = {J. Phys. Math. Theor.}, Month = nov, @@ -471,14 +469,13 @@ Title = {{The Physics of Exceptional Points}}, Volume = {45}, 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}} @article{Heiss_1990, Author = {Heiss, W D and Sannino, A L}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-20 22:03:11 +0100}, - doi = {10.1088/0305-4470/23/7/022}, + Doi = {10.1088/0305-4470/23/7/022}, Journal = {J. Phys. Math. Gen.}, Month = apr, Number = {7}, @@ -486,58 +483,53 @@ Title = {{Avoided Level Crossing and Exceptional Points}}, Volume = {23}, 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}} @article{Heiss_2016, Author = {Heiss, Dieter}, Date-Added = {2019-01-20 22:03:11 +0100}, Date-Modified = {2019-01-27 20:41:40 +0100}, - doi = {10.1038/nphys3864}, + Doi = {10.1038/nphys3864}, Journal = {Nat. Phys.}, Pages = {823--824}, Title = {{Circling Exceptional Points}}, Volume = {12}, Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys3864}, -} + Bdsk-Url-1 = {https://doi.org/10.1038/nphys3864}} @article{Heiss_1999, Author = {W. D.}, Date-Added = {2018-12-06 20:49:57 +0100}, Date-Modified = {2018-12-06 20:51:10 +0100}, - doi = {10.1007/s100530050339}, + Doi = {10.1007/s100530050339}, Journal = {Eur. Phys. J. D}, Pages = {1}, Title = {Phases of wave functions and level repulsion}, Volume = {7}, Year = {1999}, - Bdsk-Url-1 = {https://doi.org/10.1007/s100530050339}, -} + Bdsk-Url-1 = {https://doi.org/10.1007/s100530050339}} @article{Yarkony_1996, Author = {D. R. Yarkony}, Date-Added = {2018-12-06 20:45:42 +0100}, Date-Modified = {2018-12-06 20:47:03 +0100}, - doi = {10.1103/RevModPhys.68.985}, + Doi = {10.1103/RevModPhys.68.985}, Journal = {Rev. Mod. Phys.}, Pages = {985}, Title = {Diabolical conical intersections}, Volume = {68}, Year = {1996}, - Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.68.985}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.68.985}} @article{Berry_1984, Author = {M. V. Berry}, - doi = {10.1103/RevModPhys.35.496}, + Doi = {10.1103/RevModPhys.35.496}, Journal = {Proc. Royal Soc. A}, Pages = {45}, Title = {{Quantal Phase Factors Accompanying Adiabatic Changes}}, Volume = {392}, Year = {1984}, - Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.35.496}, -} + Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.35.496}} @book{MoiseyevBook, Author = {N. Moiseyev}, @@ -545,8 +537,7 @@ Date-Modified = {2018-10-16 13:35:43 +0200}, Publisher = {Cambridge University Press}, Title = {{Non-Hermitian Quantum Mechanics}}, - Year = {2011}, -} + Year = {2011}} @book{SzaboBook, Address = {New York}, @@ -556,293 +547,282 @@ Keywords = {qmech}, Publisher = {McGraw-Hill}, Title = {Modern quantum chemistry: {Introduction} to advanced electronic structure}, - Year = {1989}, -} + Year = {1989}} @book{JensenBook, Author = {F. Jensen}, Publisher = {Wiley}, Title = {Introduction to computational chemistry}, - Year = {2017}, -} + Year = {2017}} @article{Lepetit_1988, - title = {Origins of the poor convergence of many‐body perturbation theory expansions from unrestricted {Hartree-Fock} zeroth‐order descriptions}, - volume = {89}, - doi = {10.1063/1.455170}, - pages = {998--1008}, - number = {2}, + Author = {Lepetit, M. B. and P{\'e}lissier, M. and Malrieu, J. P.}, + Date = {1988-07-15}, + Doi = {10.1063/1.455170}, Journal = {J. Chem. Phys.}, - author = {Lepetit, M. B. and Pélissier, M. and Malrieu, J. P.}, - date = {1988-07-15}, -} - + Number = {2}, + Pages = {998--1008}, + Title = {Origins of the poor convergence of many‐body perturbation theory expansions from unrestricted {Hartree-Fock} zeroth‐order descriptions}, + Volume = {89}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.455170}} @article{Cremer_1996, - title = {{Sixth-Order Møller-Plesset Perturbation Theory On the Convergence of the MPn Series}}, - volume = {100}, - doi = {10.1021/jp952815d}, - pages = {6173--6188}, - number = {15}, - journal = {J. Phys. Chem.}, - author = {Cremer, Dieter and He, Zhi}, - date = {1996-01-01}, -} + Author = {Cremer, Dieter and He, Zhi}, + Date = {1996-01-01}, + Doi = {10.1021/jp952815d}, + Journal = {J. Phys. Chem.}, + Number = {15}, + Pages = {6173--6188}, + Title = {{Sixth-Order M{\o}ller-Plesset Perturbation Theory On the Convergence of the MPn Series}}, + Volume = {100}, + Bdsk-Url-1 = {https://doi.org/10.1021/jp952815d}} @article{Baker_1971, - title = {{Singularity Structure of the Perturbation Series for the Ground-State Energy of a Many-Fermion System}}, - volume = {43}, - doi = {10.1103/RevModPhys.43.479}, - pages = {479--531}, - number = {4}, + Author = {{Baker}, {GEORGE} A.}, + Date = {1971-10-01}, + Doi = {10.1103/RevModPhys.43.479}, Journal = {Rev. Mod. Phys.}, - author = {{Baker}, {GEORGE} A.}, - date = {1971-10-01}, -} + Number = {4}, + Pages = {479--531}, + Title = {{Singularity Structure of the Perturbation Series for the Ground-State Energy of a Many-Fermion System}}, + Volume = {43}, + Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.43.479}} @article{Cejnar_2005, - title = {Thermodynamic analogy for quantum phase transitions at zero temperature}, - volume = {71}, - doi = {10.1103/PhysRevC.71.011304}, - pages = {011304}, - number = {1}, + Author = {Cejnar, Pavel and Heinze, Stefan and Dobe{\v s}, Jan}, + Date = {2005-01-26}, + Doi = {10.1103/PhysRevC.71.011304}, Journal = {Phys. Rev. C}, - author = {Cejnar, Pavel and Heinze, Stefan and Dobeš, Jan}, - date = {2005-01-26}, -} + Number = {1}, + Pages = {011304}, + Title = {Thermodynamic analogy for quantum phase transitions at zero temperature}, + Volume = {71}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevC.71.011304}} @article{Stransky_2018, - title = {Exceptional points near first- and second-order quantum phase transitions}, - volume = {97}, - doi = {10.1103/PhysRevE.97.012112}, - pages = {012112}, - number = {1}, + Author = {Str{\'a}nsk{\'y}, Pavel and Dvo{\v r}{\'a}k, Martin and Cejnar, Pavel}, + Date = {2018-01-11}, + Doi = {10.1103/PhysRevE.97.012112}, Journal = {Phys. Rev. E}, - author = {Stránský, Pavel and Dvořák, Martin and Cejnar, Pavel}, - date = {2018-01-11}, -} + Number = {1}, + Pages = {012112}, + Title = {Exceptional points near first- and second-order quantum phase transitions}, + Volume = {97}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevE.97.012112}} @article{Cejnar_2007, - title = {{Coulomb Analogy for Non-Hermitian Degeneracies near Quantum Phase Transitions}}, - volume = {99}, - doi = {10.1103/PhysRevLett.99.100601}, - pages = {100601}, - number = {10}, + Author = {Cejnar, Pavel and Heinze, Stefan and Macek, Michal}, + Date = {2007-09-07}, + Doi = {10.1103/PhysRevLett.99.100601}, Journal = {Phys. Rev. Lett.}, - author = {Cejnar, Pavel and Heinze, Stefan and Macek, Michal}, - date = {2007-09-07}, -} + Number = {10}, + Pages = {100601}, + Title = {{Coulomb Analogy for Non-Hermitian Degeneracies near Quantum Phase Transitions}}, + Volume = {99}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.99.100601}} @article{Heiss_1988, - title = {{Exceptional points of a Hamiltonian and phase transitions in finite systems}}, - volume = {329}, - doi = {10.1007/BF01283767}, - pages = {133--138}, - number = {2}, + Author = {Heiss, W. D.}, + Date = {1988-06-01}, + Doi = {10.1007/BF01283767}, Journal = {Z. Physik A - Atomic Nuclei}, - author = {Heiss, W. D.}, - date = {1988-06-01}, -} + Number = {2}, + Pages = {133--138}, + Title = {{Exceptional points of a Hamiltonian and phase transitions in finite systems}}, + Volume = {329}, + Bdsk-Url-1 = {https://doi.org/10.1007/BF01283767}} @article{Heiss_2002, - title = {Universal relationship between a quantum phase transition and instability points of classical systems}, - volume = {66}, - doi = {10.1103/PhysRevE.66.016217}, - pages = {016217}, - number = {1}, + Author = {Heiss, W. D. and M{\"u}ller, M.}, + Date = {2002-07-26}, + Doi = {10.1103/PhysRevE.66.016217}, Journal = {Phys. Rev. E}, - author = {Heiss, W. D. and Müller, M.}, - date = {2002-07-26}, -} + Number = {1}, + Pages = {016217}, + Title = {Universal relationship between a quantum phase transition and instability points of classical systems}, + Volume = {66}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevE.66.016217}} @article{Sindelka_2017, - title = {Excited-state quantum phase transitions studied from a non-{Hermitian} perspective}, - volume = {95}, - doi = {10.1103/PhysRevA.95.010103}, - pages = {010103}, - number = {1}, + Author = {{\v S}indelka, Milan and Santos, Lea F. and Moiseyev, Nimrod}, + Date = {2017-01-24}, + Doi = {10.1103/PhysRevA.95.010103}, Journal = {Phys. Rev. A}, - author = {Šindelka, Milan and Santos, Lea F. and Moiseyev, Nimrod}, - date = {2017-01-24}, -} - + Number = {1}, + Pages = {010103}, + Title = {Excited-state quantum phase transitions studied from a non-{Hermitian} perspective}, + Volume = {95}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.95.010103}} @article{Cejnar_2015, - title = {Excited-state quantum phase transitions in finite many-body systems}, - volume = {90}, - doi = {10.1088/0031-8949/90/11/114015}, - pages = {114015}, - number = {11}, + Author = {Cejnar, Pavel and Str{\'a}nsk{\'y}, Pavel and Kloc, Michal}, + Date = {2015-10}, + Doi = {10.1088/0031-8949/90/11/114015}, Journal = {Phys. Scr.}, - author = {Cejnar, Pavel and Stránský, Pavel and Kloc, Michal}, - date = {2015-10}, -} + Number = {11}, + Pages = {114015}, + Title = {Excited-state quantum phase transitions in finite many-body systems}, + Volume = {90}, + Bdsk-Url-1 = {https://doi.org/10.1088/0031-8949/90/11/114015}} @article{Cejnar_2009, - title = {Quantum phase transitions in the interacting boson model}, - volume = {62}, - doi = {10.1016/j.ppnp.2008.08.001}, - pages = {210--256}, - number = {1}, + Author = {Cejnar, Pavel and Jolie, Jan}, + Date = {2009-01-01}, + Doi = {10.1016/j.ppnp.2008.08.001}, Journal = {Prog. Part. Nucl. Phys.}, - author = {Cejnar, Pavel and Jolie, Jan}, - date = {2009-01-01}, -} + Number = {1}, + Pages = {210--256}, + Title = {Quantum phase transitions in the interacting boson model}, + Volume = {62}, + Bdsk-Url-1 = {https://doi.org/10.1016/j.ppnp.2008.08.001}} @article{Borisov_2015, - title = {{Multiply Degenerate Exceptional Points and Quantum Phase Transitions}}, - volume = {54}, - doi = {10.1007/s10773-014-2493-y}, - pages = {4293--4305}, - number = {12}, + Author = {Borisov, Denis I. and Ru{\v z}i{\v c}ka, Franti{\v s}ek and Znojil, Miloslav}, + Date = {2015-12-01}, + Doi = {10.1007/s10773-014-2493-y}, Journal = {Int J Theor Phys}, - author = {Borisov, Denis I. and Ružička, František and Znojil, Miloslav}, - date = {2015-12-01}, -} + Number = {12}, + Pages = {4293--4305}, + Title = {{Multiply Degenerate Exceptional Points and Quantum Phase Transitions}}, + Volume = {54}, + Bdsk-Url-1 = {https://doi.org/10.1007/s10773-014-2493-y}} @article{Caprio_2008, - title = {Excited state quantum phase transitions in many-body systems}, - volume = {323}, - doi = {10.1016/j.aop.2007.06.011}, - pages = {1106--1135}, - number = {5}, + Author = {Caprio, M. A. and Cejnar, P. and Iachello, F.}, + Date = {2008-05-01}, + Doi = {10.1016/j.aop.2007.06.011}, Journal = {Ann Phys (N Y)}, - author = {Caprio, M. A. and Cejnar, P. and Iachello, F.}, - date = {2008-05-01}, -} + Number = {5}, + Pages = {1106--1135}, + Title = {Excited state quantum phase transitions in many-body systems}, + Volume = {323}, + Bdsk-Url-1 = {https://doi.org/10.1016/j.aop.2007.06.011}} @article{Macek_2019, - title = {Excited-state quantum phase transitions in systems with two degrees of freedom. {III}. {Interacting boson systems}}, - volume = {99}, - doi = {10.1103/PhysRevC.99.064323}, - pages = {064323}, - number = {6}, + Author = {Macek, Michal and Str{\'a}nsk{\'y}, Pavel and Leviatan, Amiram and Cejnar, Pavel}, + Date = {2019-06-21}, + Doi = {10.1103/PhysRevC.99.064323}, Journal = {Phys. Rev. C}, - author = {Macek, Michal and Stránský, Pavel and Leviatan, Amiram and Cejnar, Pavel}, - date = {2019-06-21}, -} - + Number = {6}, + Pages = {064323}, + Title = {Excited-state quantum phase transitions in systems with two degrees of freedom. {III}. {Interacting boson systems}}, + Volume = {99}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevC.99.064323}} @book{Sachdev_2011, - location = {Cambridge}, - edition = {2}, - title = {{Quantum Phase Transitions}}, - publisher = {Cambridge University Press}, - author = {Sachdev, Subir}, - date = {2011}, - doi = {10.1017/CBO9780511973765}, -} - + 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}} @article{Cejnar_2016, - title = {Quantum phase transitions in the collective degrees of freedom: nuclei and other many-body systems}, - volume = {91}, - doi = {10.1088/0031-8949/91/8/083006}, - pages = {083006}, - number = {8}, + Author = {Cejnar, Pavel and Str{\'a}nsk{\'y}, Pavel}, + Date = {2016-07}, + Doi = {10.1088/0031-8949/91/8/083006}, Journal = {Phys. Scr.}, - author = {Cejnar, Pavel and Stránský, Pavel}, - date = {2016-07}, -} + Number = {8}, + Pages = {083006}, + Title = {Quantum phase transitions in the collective degrees of freedom: nuclei and other many-body systems}, + Volume = {91}, + Bdsk-Url-1 = {https://doi.org/10.1088/0031-8949/91/8/083006}} @article{Coulson_1949, - title = {{XXXIV}. {Notes} on the molecular orbital treatment of the hydrogen molecule}, - volume = {40}, - doi = {10.1080/14786444908521726}, - pages = {386--393}, - number = {303}, - journaltitle = {The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science}, - author = {Coulson, C. A. and Fischer, I.}, - date = {1949-04-01}, -} + Author = {Coulson, C. A. and Fischer, I.}, + Date = {1949-04-01}, + Doi = {10.1080/14786444908521726}, + Journaltitle = {The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science}, + Number = {303}, + Pages = {386--393}, + Title = {{XXXIV}. {Notes} on the molecular orbital treatment of the hydrogen molecule}, + Volume = {40}, + Bdsk-Url-1 = {https://doi.org/10.1080/14786444908521726}} @article{Burton_2019, - title = {Complex adiabatic connection: {A} hidden non-{Hermitian} path from ground to excited states}, - volume = {150}, - doi = {10.1063/1.5085121}, - number = {4}, - journal = {J. Chem. Phys.}, - author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-François}, - month = jan, - year = {2019}, - pages = {041103}, -} + Author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-Fran{\c c}ois}, + Doi = {10.1063/1.5085121}, + Journal = {J. Chem. Phys.}, + Month = jan, + Number = {4}, + Pages = {041103}, + Title = {Complex adiabatic connection: {A} hidden non-{Hermitian} path from ground to excited states}, + Volume = {150}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.5085121}} @article{Burton_2019a, - title = {Parity-{Time} {Symmetry} in {Hartree}–{Fock} {Theory}}, - volume = {15}, - doi = {10.1021/acs.jctc.9b00289}, - number = {8}, - journal = {J. Chem. Theory Comput.}, - author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-François}, - month = aug, - year = {2019}, - pages = {4374--4385}, -} + Author = {Burton, Hugh G. A. and Thom, Alex J. W. and Loos, Pierre-Fran{\c c}ois}, + Doi = {10.1021/acs.jctc.9b00289}, + Journal = {J. Chem. Theory Comput.}, + Month = aug, + Number = {8}, + Pages = {4374--4385}, + Title = {Parity-{Time} {Symmetry} in {Hartree}--{Fock} {Theory}}, + Volume = {15}, + Year = {2019}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.9b00289}} @article{Hiscock_2014, - title = {Holomorphic {Hartree}–{Fock} {Theory} and {Configuration} {Interaction}}, - volume = {10}, - doi = {10.1021/ct5007696}, - number = {11}, - journal = {J. Chem. Theory Comput.}, - author = {Hiscock, Hamish G. and Thom, Alex J. W.}, - month = nov, - year = {2014}, - pages = {4795--4800}, -} - + Author = {Hiscock, Hamish G. and Thom, Alex J. W.}, + Doi = {10.1021/ct5007696}, + Journal = {J. Chem. Theory Comput.}, + Month = nov, + Number = {11}, + Pages = {4795--4800}, + Title = {Holomorphic {Hartree}--{Fock} {Theory} and {Configuration} {Interaction}}, + Volume = {10}, + Year = {2014}, + Bdsk-Url-1 = {https://doi.org/10.1021/ct5007696}} @article{Seidl_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}, -} - + Author = {Seidl, Michael and Giarrusso, Sara and Vuckovic, Stefan and Fabiano, Eduardo and Gori-Giorgi, Paola}, + Doi = {10.1063/1.5078565}, + Journal = {J. Chem. Phys.}, + Month = dec, + Number = {24}, + Pages = {241101}, + 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}} @book{GiulianiBook, - address = {Cambridge}, - title = {Quantum {Theory} of the {Electron} {Liquid}}, - isbn = {978-0-521-52796-5}, - urldate = {2020-07-21}, - publisher = {Cambridge University Press}, - author = {Giuliani, Gabriele and Vignale, Giovanni}, - year = {2005}, - doi = {10.1017/CBO9780511619915}, -} + Address = {Cambridge}, + Author = {Giuliani, Gabriele and Vignale, Giovanni}, + Doi = {10.1017/CBO9780511619915}, + Isbn = {978-0-521-52796-5}, + Publisher = {Cambridge University Press}, + Title = {Quantum {Theory} of the {Electron} {Liquid}}, + Urldate = {2020-07-21}, + Year = {2005}, + Bdsk-Url-1 = {https://doi.org/10.1017/CBO9780511619915}} @book{AngularBook, - title = {Angular {Momentum} in {Quantum} {Mechanics}}, - isbn = {978-0-691-02589-6}, - language = {en}, - month = jan, - year = {1996}, - author = {Edmonds, A. R.} -} + Author = {Edmonds, A. R.}, + Isbn = {978-0-691-02589-6}, + Language = {en}, + Month = jan, + Title = {Angular {Momentum} in {Quantum} {Mechanics}}, + Year = {1996}} @book{SlaterBook, - title = {{Quantum Theory of Atomic Structure}}, - language = {eng}, - publisher = {New York : McGraw-Hill}, - author = {Slater, John Clarke}, - year = {1960}, -} - + Author = {Slater, John Clarke}, + Language = {eng}, + Publisher = {New York : McGraw-Hill}, + Title = {{Quantum Theory of Atomic Structure}}, + Year = {1960}} @article{Loos_2009, - title = {Ground state of two electrons on a sphere}, - volume = {79}, - doi = {10.1103/PhysRevA.79.062517}, - number = {6}, - journal = {Physical Review A}, - author = {Loos, Pierre-François and Gill, Peter M. W.}, - month = jun, - year = {2009}, - pages = {062517}, -} \ No newline at end of file + Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.}, + Doi = {10.1103/PhysRevA.79.062517}, + Journal = {Physical Review A}, + Month = jun, + Number = {6}, + Pages = {062517}, + Title = {Ground state of two electrons on a sphere}, + Volume = {79}, + Year = {2009}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.79.062517}} diff --git a/RapportStage/Rapport.tex b/RapportStage/Rapport.tex index 9ecfb90..8ef7274 100644 --- a/RapportStage/Rapport.tex +++ b/RapportStage/Rapport.tex @@ -308,36 +308,40 @@ This decomposition is obviously non-unique and motivated by several factors as d Accordingly to Eq.~\eqref{eq:SchrEq-PT}, the energy can then be written as a power series of $\lambda$ \begin{equation} \label{eq:Elambda} - E(\lambda) = \sum_{k=0}^\infty \lambda^k E^{(k)} + E(\lambda) = \sum_{k=0}^\infty \lambda^k E^{(k)}. \end{equation} -However it is not guaranteed that the series \eqref{eq:Elambda} has a radius of convergence $\abs{\lambda_0} < 1$. +However, it is not guaranteed that the series \eqref{eq:Elambda} has a radius of convergence $\abs{\lambda_0} < 1$. In other words, the series might well be divergent for the physical system at $\lambda = 1$. One can prove that the actual value of the radius of convergence $\abs{\lambda_0}$ can be obtained by looking for the singularities of $E(\lambda)$ in the complex $\lambda$ plane. This is due to the following theorem \cite{Goodson_2012}: \begin{quote} - \textit{``The Taylor series about a point $z_0$ of a function over the complex $z$ plane will converge at a value $z_1$ if the function is non-singular at all values of $z$ in the circular region centered at $z_0$ with radius $\abs{z_1 − z_0}$. If the function has a singular point $z_s$ such that $\abs{z_s − z_0} < \abs{z_1 − z_0}$, then the series will diverge when evaluated at $z_1$.''} + \textit{``The Taylor series about a point $z_0$ of a function over the complex $z$ plane will converge at a value $z_1$ if the function is non-singular at all values of $z$ in the circular region centered at $z_0$ with radius $\abs{z_1-z_0}$. If the function has a singular point $z_s$ such that $\abs{z_s-z_0} < \abs{z_1-z_0}$, then the series will diverge when evaluated at $z_1$.''} \end{quote} -This theorem means that the radius of convergence of the perturbation series is equal to the distance to the origin of the closest singularity of $E(\lambda)$. To illustrate this result we consider the simple function \eqref{eq:DivExample}. This function is smooth for $x \in \mathbb{R}$ and infinitely differentiable in $\mathbb{R}$. One would expect that the Taylor series of such a function would be convergent for all $x \in \mathbb{R}$, however this series is divergent for $x\geq1$. This is because the function has four singularities in the complex plane ($x = e^{i\pi/4}, e^{-i\pi/4}, e^{i3\pi/4}, e^{-i3\pi/4}$) with a modulus equal to 1. This simple example emphasizes the importance of the singularities in the complex plane to understand the convergence properties on the real axis. - +This theorem means that the radius of convergence of the perturbation series is equal to the distance to the origin of the closest singularity of $E(\lambda)$. To illustrate this result we consider the simple function \cite{BenderBook} \begin{equation} \label{eq:DivExample} -f(x)=\frac{1}{1+x^4} + f(x)=\frac{1}{1+x^4} \end{equation} +This function is smooth for $x \in \mathbb{R}$ and infinitely differentiable in $\mathbb{R}$. One would expect that the Taylor series of such a function would be convergent $\forall x \in \mathbb{R}$. However this series is divergent for $x \ge 1$. This is because the function has four singularities in the complex plane ($x = e^{i\pi/4}$, $e^{-i\pi/4}$, $e^{i3\pi/4}$, and $e^{-i3\pi/4}$) with a modulus equal to $1$. This simple example emphasizes the importance of the singularities in the complex plane to understand the convergence properties on the real axis. + \subsection{The Hartree-Fock Hamiltonian} -In the Born-Oppenheimer approximation, the equation \eqref{eq:ExactHamiltonian} gives the exact electronic Hamiltonian for a chemical system with $n$ electrons and $N$ nuclei with respective charge $Z_k$. The first term is the kinetic energy of the electrons, the two following terms account respectively for the electron-nuclei attraction and the electron-electron repulsion. - +Within the Born-Oppenheimer approximation, \begin{equation}\label{eq:ExactHamiltonian} - \bH=\sum\limits_{i=1}^{n}\left[ -\frac{1}{2}\grad_i^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_i-\vb{R}_k|}+\sum\limits_{i