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-\begin{thebibliography}{10} - -\bibitem{Bittner_2012} -S.~Bittner, B.~Dietz, U.~G\"unther, H.~L. Harney, M.~{Miski-Oglu}, A.~Richter, - and F.~Sch\"afer. -\newblock {{PT Symmetry}} and {{Spontaneous Symmetry Breaking}} in a - {{Microwave Billiard}}. -\newblock {\em Phys. Rev. Lett.}, 108(2):024101, January 2012. - -\bibitem{Chong_2011} -Y.~D. Chong, Li~Ge, and A.~Douglas Stone. -\newblock P {{T}} -{{Symmetry Breaking}} and {{Laser}}-{{Absorber Modes}} in - {{Optical Scattering Systems}}. -\newblock {\em Phys. Rev. Lett.}, 106(9):093902, March 2011. - -\bibitem{Chtchelkatchev_2012} -N.~M. Chtchelkatchev, A.~A. Golubov, T.~I. Baturina, and V.~M. Vinokur. -\newblock Stimulation of the {{Fluctuation Superconductivity}} by {{P T - Symmetry}}. -\newblock {\em Phys. Rev. Lett.}, 109(15):150405, October 2012. - -\bibitem{Doppler_2016} -J\"org Doppler, Alexei~A. Mailybaev, Julian B\"ohm, Ulrich Kuhl, Adrian - Girschik, Florian Libisch, Thomas~J. 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Bender}, - Date-Added = {2019-02-17 21:06:07 +0100}, - Date-Modified = {2019-02-17 21:07:28 +0100}, + Date-Added = {2020-08-22 22:22:25 +0200}, + Date-Modified = {2020-08-22 22:22:25 +0200}, Publisher = {World Scientific}, Title = {{{\cal PT}}-Symmetry in Quantum and Classical Physics}, Year = {2019}} -@article{Beygi_2018b, - Author = {Beygi, Alireza and Klevansky, S. P.}, - Date-Added = {2019-02-16 10:26:35 +0100}, - Date-Modified = {2019-02-17 21:08:13 +0100}, - Doi = {10.1103/PhysRevA.98.022105}, - File = {/Users/loos/Zotero/storage/FDYNMNA3/Beygi and Klevansky - 2018 - No-signaling principle and quantum brachistochrone.pdf}, - Issn = {2469-9926, 2469-9934}, - Journal = {Phys. Rev. A}, - Language = {en}, - Month = aug, - Number = {2}, - Pages = {022105}, - Title = {No-Signaling Principle and Quantum Brachistochrone Problem in {{PT}} -Symmetric Fermionic Two- and Four-Dimensional Models}, - Volume = {98}, +@article{Benda_2018, + Author = {Zsuzsanna Benda and Thomas-C. Jagau}, + Date-Added = {2020-08-22 22:03:56 +0200}, + Date-Modified = {2020-08-22 22:04:59 +0200}, + Doi = {10.1021/acs.jpclett.8b03228}, + Journal = {J. Phys. Chem. Lett.}, + Pages = {6978--6984}, + Title = {Locating Exceptional Points on Multidimensional Complex-Valued Potential Energy Surfaces}, + Volume = {9}, Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.98.022105}} - -@article{Cherbal_2012, - Author = {Cherbal, O. and Trifonov, D. A.}, - Date-Added = {2019-02-16 10:26:05 +0100}, - Date-Modified = {2019-02-16 10:26:08 +0100}, - Doi = {10.1103/PhysRevA.85.052123}, - File = {/Users/loos/Zotero/storage/R9WXPZNM/Cherbal and Trifonov - 2012 - Extended PT - and CPT -symmetric representations o.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Language = {en}, - Month = may, - Number = {5}, - Title = {Extended {{PT}} - and {{CPT}} -Symmetric Representations of Fermionic Algebras}, - Volume = {85}, - Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.85.052123}} - -@article{Jones-Smith_2010, - Author = {Jones-Smith, Katherine and Mathur, Harsh}, - Doi = {10.1103/PhysRevA.82.042101}, - Issue = {4}, - Journal = {Phys. Rev. A}, - Month = {Oct}, - Numpages = {8}, - Pages = {042101}, - Publisher = {American Physical Society}, - Title = {Non-Hermitian quantum Hamiltonians with $\mathcal{P}\mathcal{T}$ symmetry}, - Url = {https://link.aps.org/doi/10.1103/PhysRevA.82.042101}, - Volume = {82}, - Year = {2010}, - Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevA.82.042101}, - Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevA.82.042101}} - -@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}, - File = {/Users/loos/Zotero/storage/PIYGTJI7/El-Ganainy et al. - 2018 - Non-Hermitian physics and PT symmetry.pdf}, - Issn = {1745-2473, 1745-2481}, - Journal = {Nat. Phys.}, - Language = {en}, - Month = jan, - Number = {1}, - Pages = {11-19}, - Title = {Non-{{Hermitian}} Physics and {{PT}} Symmetry}, - Volume = {14}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys4323}} - -@article{Peng_2016, - Author = {Peng, Peng and Cao, Wanxia and Shen, Ce and Qu, Weizhi and Wen, Jianming and Jiang, Liang and Xiao, Yanhong}, - Date-Added = {2019-01-30 12:10:10 +0100}, - Date-Modified = {2019-01-30 12:10:10 +0100}, - Doi = {10.1038/nphys3842}, - File = {/Users/loos/Zotero/storage/T4WWWGTW/Peng et al. - 2016 - Anti-parity--time symmetry with flying atoms.pdf}, - Issn = {1745-2473, 1745-2481}, - Journal = {Nat. Phys.}, - Language = {en}, - Month = dec, - Number = {12}, - Pages = {1139-1145}, - Title = {Anti-Parity\textendash{}Time Symmetry with Flying Atoms}, - Volume = {12}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys3842}} - -@article{Zhang_2018, - Author = {Zhang, Xu-Lin and Wang, Shubo and Hou, Bo and Chan, C. 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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{Pavlyukh_2017, - Author = {Y. Pavlyukh}, - Date-Added = {2019-01-29 10:04:16 +0100}, - Date-Modified = {2019-01-29 10:04:16 +0100}, - Doi = {10.1038/s41598-017-00355-w}, - Journal = {Nature}, - Pages = {504}, - Title = {Pade resummation of many-body perturbation theory}, - Volume = {7}, - Year = {2017}, - Bdsk-Url-1 = {https://doi.org/10.1038/s41598-017-00355-w}} - -@article{Seidl_2007, - Author = {Seidl, Michael}, - Date-Added = {2019-01-29 08:45:48 +0100}, - Date-Modified = {2019-01-29 08:45:48 +0100}, - Doi = {10.1103/PhysRevA.75.062506}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Language = {en}, - Month = jun, - Number = {6}, - Pages = {062506}, - Shorttitle = {Adiabatic Connection in Density-Functional Theory}, - Title = {Adiabatic Connection in Density-Functional Theory: {{Two}} Electrons on the Surface of a Sphere}, - Volume = {75}, - Year = {2007}, - Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevA.75.062506}} - -@article{Qiu_2017, - Author = {Y. Qiu and T. M. Henderson and J. Zhao and G. E. Scuseria}, - Date-Added = {2019-01-28 12:44:02 +0100}, - Date-Modified = {2019-02-14 20:50:59 +0100}, - Journal = {J. Chem. Phys.}, - Pages = {064111}, - Title = {Projected coupled cluster theory}, - Volume = {147}, - Year = {2017}} - -@article{Jimenez-Hoyos_2012, - Author = {C. A. Jimenez-Hoyos and T. M. Henderson and T. Tsuchimochi and G. E. Scuseria}, - Date-Added = {2019-01-28 12:39:16 +0100}, - Date-Modified = {2019-01-28 12:40:19 +0100}, - Doi = {10.1063/1.4705280}, - Journal = {J. Chem. 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Phys. Lett.}, + Number = {1}, Pages = {16--22}, - Title = {Deceptive convergence in Moller-plesset perturbation energies}, + Title = {Deceptive convergence in {M{\o}ller-Plesset} perturbation energies}, Volume = {132}, Year = {1986}, Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(86)80686-8}} @article{Gill_1988, - Author = {P. M. W. Gill and J. A. Pople and L. Radom}, - Date-Added = {2019-01-28 09:34:27 +0100}, - Date-Modified = {2019-01-28 10:03:30 +0100}, + Author = {Gill, Peter M. W. and Pople, John A. and Radom, Leo and Nobes, Ross H.}, + Date = {1988-12-15}, + Date-Modified = {2020-08-22 22:12:52 +0200}, Doi = {10.1063/1.455312}, Journal = {J. Chem. 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Phys.}, - Language = {en}, - Month = jun, - Number = {12}, - Pages = {1684-1701}, - Shorttitle = {Generalised Adiabatic Connection in Ensemble Density-Functional Theory for Excited States}, - Title = {Generalised Adiabatic Connection in Ensemble Density-Functional Theory for Excited States: Example of the {{H}} {\textsubscript{2}} Molecule}, - Volume = {112}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2013.858191}} - -@article{Giner_2016, - Author = {Giner, E. and David, G. and Scemama, A. and Malrieu, J. P.}, - Date-Added = {2019-01-27 22:00:17 +0100}, - Date-Modified = {2019-01-27 22:00:17 +0100}, - Doi = {10.1063/1.4940781}, - Issn = {1089-7690}, +@article{Sergeev_2005, + Author = {Sergeev, Alexey V. and Goodson, David Z. and Wheeler, Steven E. and Allen, Wesley D.}, + Date-Modified = {2020-08-22 22:13:37 +0200}, + Doi = {10.1063/1.1991854}, Journal = {J. Chem. Phys.}, - Month = {Feb}, Number = {6}, - Pages = {064101}, - Publisher = {AIP Publishing}, - Title = {A simple approach to the state-specific MR-CC using the intermediate Hamiltonian formalism}, - Url = {http://dx.doi.org/10.1063/1.4940781}, - Volume = {144}, - Year = {2016}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4940781}} + Pages = {064105}, + Title = {On the nature of the {M{\o}ller-Plesset} critical point}, + Volume = {123}, + Year = {2005}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1991854}} -@article{Giner_2017a, - Author = {Giner, Emmanuel and Angeli, Celestino and Garniron, Yann and Scemama, Anthony and Malrieu, Jean-Paul}, - Date-Added = {2019-01-27 22:00:17 +0100}, - Date-Modified = {2019-01-27 22:00:17 +0100}, - Doi = {10.1063/1.4984616}, - Issn = {1089-7690}, +@article{Sergeev_2006, + Author = {Sergeev, Alexey V. and Goodson, David Z.}, + Date-Modified = {2020-08-22 22:13:42 +0200}, + Doi = {10.1063/1.2173989}, Journal = {J. Chem. Phys.}, - Month = {Jun}, - Number = {22}, - Pages = {224108}, - Publisher = {AIP Publishing}, - Title = {A Jeziorski-Monkhorst fully uncontracted multi-reference perturbative treatment. I. Principles, second-order versions, and tests on ground state potential energy curves}, - Url = {http://dx.doi.org/10.1063/1.4984616}, - Volume = {146}, - Year = {2017}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4984616}} - -@article{Giner_2017b, - Author = {Giner, E. and Angeli, C. and Scemama, A. and Malrieu, J.-P.}, - Date-Added = {2019-01-27 22:00:17 +0100}, - Date-Modified = {2019-01-27 22:00:17 +0100}, - Doi = {10.1016/j.comptc.2017.03.001}, - Issn = {2210-271X}, - Journal = {Comput. Theor. Chem.}, - Month = {Sep}, - Pages = {134--140}, - Publisher = {Elsevier BV}, - Title = {Orthogonal Valence Bond Hamiltonians incorporating dynamical correlation effects}, - Url = {http://dx.doi.org/10.1016/j.comptc.2017.03.001}, - Volume = {1116}, - Year = {2017}, - Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.comptc.2017.03.001}} - -@article{Giner_2019, - Author = {E. Giner and B. Pradines and A. Fert{{\'e}} and R. Assaraf and A. Savin and J. Toulouse}, - Date-Added = {2019-01-27 22:00:17 +0100}, - Date-Modified = {2019-01-27 22:00:17 +0100}, - Doi = {10.1063/1.5052714}, - Journal = {J. Chem. Phys.}, - Pages = {194301}, - Title = {Curing basis-set convergence of wave-function theory using density-functional theory: A systematically improvable approach}, - Volume = {149}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.5052714}} - -@article{Jones_2005, - Author = {H. F. 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Phys.}, - Month = may, - Number = {19}, - Pages = {194108}, - Title = {An Assessment of Low-Lying Excitation Energies and Triplet Instabilities of Organic Molecules with an {\emph{Ab Initio}} {{Bethe}}-{{Salpeter}} Equation Approach and the {{Tamm}}-{{Dancoff}} Approximation}, - Volume = {146}, - Year = {2017}, - Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4983126}} - -@article{Bruneval_2016, - Author = {Bruneval, Fabien and Rangel, Tonatiuh and Hamed, Samia M. and Shao, Meiyue and Yang, Chao and Neaton, Jeffrey B.}, - Date-Added = {2019-01-26 22:21:53 +0100}, - Date-Modified = {2019-01-26 22:21:53 +0100}, - Doi = {10.1016/j.cpc.2016.06.019}, - File = {/Users/loos/Zotero/storage/CHGLK37T/Bruneval_2016a.pdf}, - Issn = {00104655}, - Journal = {Comput. Phys. Commun.}, - Month = nov, - Pages = {149--161}, - Shorttitle = {Molgw 1}, - Title = {Molgw 1: {{Many}}-Body Perturbation Theory Software for Atoms, Molecules, and Clusters}, - Volume = {208}, - Year = {2016}, - Bdsk-Url-1 = {https://dx.doi.org/10.1016/j.cpc.2016.06.019}} - -@article{vanSchilfgaarde_2006, - Author = {{van Schilfgaarde}, M. and Kotani, Takao and Faleev, S.}, - Date-Added = {2019-01-26 22:21:30 +0100}, - Date-Modified = {2019-01-26 22:21:30 +0100}, - Doi = {10.1103/PhysRevLett.96.226402}, - File = {/Users/loos/Zotero/storage/6S8FDHP8/vanSchilfgaarde_2006.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = jun, - Number = {22}, - Pages = {226402}, - Title = {Quasiparticle {{Self}}-{{Consistent G W Theory}}}, - Volume = {96}, + Number = {9}, + Pages = {094111}, + Title = {Singularities of {M{\o}ller-Plesset} energy functions}, + Volume = {124}, Year = {2006}, - Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevLett.96.226402}} + Bdsk-Url-1 = {https://doi.org/10.1063/1.2173989}} -@article{vanSetten_2013, - Author = {{van Setten}, M. J. and Weigend, F. and Evers, F.}, - Date-Added = {2019-01-26 22:21:30 +0100}, - Date-Modified = {2019-01-26 22:21:30 +0100}, - Doi = {10.1021/ct300648t}, - File = {/Users/loos/Zotero/storage/SLTIFNCK/vanSetten_2013.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = jan, - Number = {1}, - Pages = {232--246}, - Shorttitle = {The {{{\emph{GW}}}} -{{Method}} for {{Quantum Chemistry Applications}}}, - Title = {The {{{\emph{GW}}}} -{{Method}} for {{Quantum Chemistry Applications}}: {{Theory}} and {{Implementation}}}, - Volume = {9}, - Year = {2013}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct300648t}} - -@article{vanSetten_2015, - Author = {{van Setten}, Michiel J. and Caruso, Fabio and Sharifzadeh, Sahar and Ren, Xinguo and Scheffler, Matthias and Liu, Fang and Lischner, Johannes and Lin, Lin and Deslippe, Jack R. and Louie, Steven G. and Yang, Chao and Weigend, Florian and Neaton, Jeffrey B. and Evers, Ferdinand and Rinke, Patrick}, - Date-Added = {2019-01-26 22:21:30 +0100}, - Date-Modified = {2019-01-26 22:21:30 +0100}, - Doi = {10.1021/acs.jctc.5b00453}, - File = {/Users/loos/Zotero/storage/4DNTPV2H/vanSetten_2015.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = dec, - Number = {12}, - Pages = {5665--5687}, - Shorttitle = {{{{\emph{GW}}}} 100}, - Title = {{{{\emph{GW}}}} 100: {{Benchmarking}} {{{\emph{G}}}} {\textsubscript{0}} {{{\emph{W}}}} {\textsubscript{0}} for {{Molecular Systems}}}, - Volume = {11}, - Year = {2015}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.5b00453}} - -@article{vanSetten_2018, - Author = {{van Setten}, Michiel J. and Costa, Ramon and Vi{\~n}es, Francesc and Illas, Francesc}, - Date-Added = {2019-01-26 22:21:30 +0100}, - Date-Modified = {2019-01-26 22:21:30 +0100}, - Doi = {10.1021/acs.jctc.7b01192}, - File = {/Users/loos/Zotero/storage/2KY9G5DS/vanSetten_2018.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = feb, - Number = {2}, - Pages = {877--883}, - Title = {Assessing {{{\emph{GW}}}} {{Approaches}} for {{Predicting Core Level Binding Energies}}}, - Volume = {14}, - Year = {2018}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.7b01192}} - -@article{Dorey_2009, - Annote = {We study a three-parameter family of -symmetric Hamiltonians, related via the ODE/IM correspondence to the Perk--Schultz models. We show that real eigenvalues merge and become complex at quadratic and cubic exceptional points, and explore the corresponding Jordan block structures by exploiting the quasi-exact solvability of a subset of the models. The mapping of the phase diagram is completed using a combination of numerical, analytical and perturbative approaches. Among other things this reveals some novel properties of the Bender--Dunne polynomials, and gives new insight into a phase transition to infinitely many complex eigenvalues that was first observed by Bender and Boettcher. A new exactly solvable limit, the inhomogeneous complex square well, is also identified.}, - Author = {Patrick Dorey and Clare Dunning and Anna Lishman and Roberto Tateo}, - Date-Added = {2019-01-25 22:48:41 +0100}, - Date-Modified = {2019-01-25 22:50:25 +0100}, - Doi = {10.1088/1751-8113/42/46/465302}, - Journal = {Journal of Physics A: Mathematical and Theoretical}, - Month = {oct}, - Number = {46}, - Pages = {465302}, - Publisher = {{IOP} Publishing}, - Title = {{{\cal PT}} symmetry breaking and exceptional points for a class of inhomogeneous complex potentials}, - Url = {https://doi.org/10.1088%2F1751-8113%2F42%2F46%2F465302}, - Volume = {42}, - Year = 2009, - Bdsk-Url-1 = {https://doi.org/10.1088%2F1751-8113%2F42%2F46%2F465302}, - Bdsk-Url-2 = {https://doi.org/10.1088/1751-8113/42/46/465302}} - -@article{Dorey_2001, - Author = {P. E. Dorey and C. Dunning and R. Tateo}, - Date-Added = {2019-01-25 22:47:51 +0100}, - Date-Modified = {2019-01-25 22:52:34 +0100}, - Doi = {10.1088/0305-4470/34/28/305}, - Journal = {J. Phys. A}, - Pages = {5679-5704}, - Title = {Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics}, - Volume = {34}, - Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/34/28/305}} - -@article{Olsen_2000, - Author = {Olsen, Jeppe and J\o{}rgensen, Poul and Helgaker, Trygve and Christiansen, Ove}, - Date-Added = {2019-01-25 14:35:53 +0100}, - Date-Modified = {2019-01-25 14:35:53 +0100}, - Doi = {10.1063/1.481611}, - Issn = {0021-9606, 1089-7690}, - Journal = {The Journal of Chemical Physics}, - Month = jun, +@article{Stillinger_2000, + Author = {Stillinger, Frank H.}, + Date = {2000-05-31}, + Date-Modified = {2020-08-22 22:14:50 +0200}, + Doi = {10.1063/1.481608}, + Journal = {J. Chem. Phys.}, Number = {22}, - Pages = {9736-9748}, - Shorttitle = {Divergence in {{M\o{}ller}}\textendash{{Plesset}} Theory}, - Title = {Divergence in {{M\o{}ller}}\textendash{{Plesset}} Theory: {{A}} Simple Explanation Based on a Two-State Model}, + Pages = {9711--9715}, + Title = {{M{\o}ller-Plesset} convergence issues in computational quantum chemistry}, Volume = {112}, Year = {2000}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.481611}} + Bdsk-Url-1 = {https://doi.org/10.1063/1.481608}} @article{Olsen_1996, - Author = {Olsen, Jeppe and Christiansen, Ove and Koch, Henrik and Jo/rgensen, Poul}, - Date-Added = {2019-01-25 14:35:53 +0100}, - Date-Modified = {2019-01-25 14:35:53 +0100}, + Author = {Olsen, Jeppe and Christiansen, Ove and Koch, Henrik and J{\o}rgensen, Poul}, + Date = {1996-09-01}, Doi = {10.1063/1.472352}, - Issn = {0021-9606, 1089-7690}, - Journal = {The Journal of Chemical Physics}, - Month = sep, - Number = {12}, - Pages = {5082-5090}, - Title = {Surprising Cases of Divergent Behavior in {{Mo}}/Ller\textendash{{Plesset}} Perturbation Theory}, + Journal = {J. Chem. Phys.}, + Pages = {5082--5090}, + Title = {Surprising cases of divergent behavior in {M{\o}ller-Plesset} perturbation theory}, Volume = {105}, - Year = {1996}, Bdsk-Url-1 = {https://doi.org/10.1063/1.472352}} -@article{Abrams_2004, - Author = {Abrams, Micah L. and Sherrill, C. David}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.1804498}, - Issn = {0021-9606, 1089-7690}, +@article{Olsen_2000, + 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.}, - Month = nov, - Number = {19}, - Pages = {9211-9219}, - Shorttitle = {Full Configuration Interaction Potential Energy Curves for the {{X 1$\Sigma$g}}+, {{B 1$\Delta$g}}, and {{B}}${'}$ {{1$\Sigma$g}}+ States of {{C2}}}, - Title = {Full Configuration Interaction Potential Energy Curves for the {{X 1$\Sigma_g^+$}}, {{B 1$\Delta_g$}}, and {{B}}${'}$ {{1$\Sigma_g^+$}} States of {{C$_2$}}: {{A}} Challenge for Approximate Methods}, - Volume = {121}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1804498}} - -@article{Abrams_2005, - Author = {Abrams, Micah L. and Sherrill, C. David}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1016/j.cplett.2005.06.107}, - Issn = {0009-2614}, - Journal = {Chem. Phys. Lett.}, - Month = {Aug}, - Number = {1-3}, - Pages = {121--124}, - Publisher = {Elsevier BV}, - Title = {Important configurations in configuration interaction and coupled-cluster wave functions}, - Url = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}, - Volume = {412}, - Year = {2005}, - Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cplett.2005.06.107}} - -@article{Ahmed_2004, - Author = {Ahmed, Khalil and {Balint-Kurti}, Gabriel G. and Western, Colin M.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.1806820}, - File = {/Users/loos/Zotero/storage/IFFHAPDR/Ahmed et al. - 2004 - iAb Initioi calculations and vibrational ener.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = nov, - Number = {20}, - Pages = {10041-10051}, - Title = {{\emph{Ab }}{{{\emph{Initio}}}} Calculations and Vibrational Energy Level Fits for the Lower Singlet Potential-Energy Surfaces of {{C3}}}, - Volume = {121}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1806820}} - -@article{dalton, - Author = {Aidas, Kestutis and Angeli, Celestino and Bak, Keld L. and Bakken, Vebj{\o}rn and Bast, Radovan and Boman, Linus and Christiansen, Ove and Cimiraglia, Renzo and Coriani, Sonia and Dahle, P{\aa}l and Dalskov, Erik K. and Ekstr{\"o}m, Ulf and Enevoldsen, Thomas and Eriksen, Janus J. and Ettenhuber, Patrick and Fern{\'a}ndez, Berta and Ferrighi, Lara and Fliegl, Heike and Frediani, Luca and Hald, Kasper and Halkier, Asger and H{\"a}ttig, Christof and Heiberg, Hanne and Helgaker, Trygve and Hennum, Alf Christian and Hettema, Hinne and Hjerten{\ae}s, Eirik and H{\o}st, Stinne and H{\o}yvik, Ida-Marie and Iozzi, Maria Francesca and Jans{\'\i}k, Branislav and Jensen, Hans J{\o}rgen Aa. and Jonsson, Dan and J{\o}rgensen, Poul and Kauczor, Joanna and Kirpekar, Sheela and Kj{\ae}rgaard, Thomas and Klopper, Wim and Knecht, Stefan and Kobayashi, Rika and Koch, Henrik and Kongsted, Jacob and Krapp, Andreas and Kristensen, Kasper and Ligabue, Andrea and Lutn{\ae}s, Ola B. and Melo, Juan I. and Mikkelsen, Kurt V. and Myhre, Rolf H. and Neiss, Christian and Nielsen, Christian B. and Norman, Patrick and Olsen, Jeppe and Olsen, J{\'o}gvan Magnus H. and Osted, Anders and Packer, Martin J. and Pawlowski, Filip and Pedersen, Thomas B. and Provasi, Patricio F. and Reine, Simen and Rinkevicius, Zilvinas and Ruden, Torgeir A. and Ruud, Kenneth and Rybkin, Vladimir V. and Sa{\l}ek, Pawel and Samson, Claire C. M. and de Mer{\'a}s, Alfredo S{\'a}nchez and Saue, Trond and Sauer, Stephan P. A. and Schimmelpfennig, Bernd and Sneskov, Kristian and Steindal, Arnfinn H. and Sylvester-Hvid, Kristian O. and Taylor, Peter R. and Teale, Andrew M. and Tellgren, Erik I. and Tew, David P. and Thorvaldsen, Andreas J. and Th{\o}gersen, Lea and Vahtras, Olav and Watson, Mark A. and Wilson, David J. D. and Ziolkowski, Marcin and {\AA}gren, Hans}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1002/wcms.1172}, - Issn = {1759-0884}, - Journal = {WIREs Comput. Mol. Sci.}, - Number = {3}, - Pages = {269--284}, - Title = {The Dalton Quantum Chemistry Program System}, - Url = {http://dx.doi.org/10.1002/wcms.1172}, - Volume = {4}, - Year = {2014}, - Bdsk-Url-1 = {http://dx.doi.org/10.1002/wcms.1172}} - -@article{Alam_2016, - Author = {Alam, Md. 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O.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1007/BF01113393}, - Issn = {0040-5744, 1432-2234}, - Journal = {Theor. Chem. Acc.}, - Month = jan, - Number = {4-5}, - Pages = {403-413}, - Shorttitle = {The Excited States of Pyrazine}, - Title = {The Excited States of Pyrazine: {{A}} Basis Set Study}, - Volume = {87}, - Year = {1994}, - Bdsk-Url-1 = {https://doi.org/10.1007/BF01113393}} - -@article{Fulscher_1992, - Author = {Fulscher, Markus P. and Andersson, Kerstin and Roos, Bjoern O.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/j100202a026}, - Issn = {0022-3654, 1541-5740}, - Journal = {J. Phys. Chem.}, - Month = nov, - Number = {23}, - Pages = {9204-9212}, - Shorttitle = {Toward an Accurate Molecular Orbital Theory for Excited States}, - Title = {Toward an Accurate Molecular Orbital Theory for Excited States: The Azabenzenes}, - Volume = {96}, - Year = {1992}, - Bdsk-Url-1 = {https://doi.org/10.1021/j100202a026}} - -@article{Galvez_2002, - Author = {G\'alvez, F. J. and Buend\i\'a, E. and Sarsa, A.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.1503776}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = oct, - Number = {13}, - Pages = {6071-6082}, - Title = {Excited States of Beryllium Atom from Explicitly Correlated Wave Functions}, - Volume = {117}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1503776}} - -@article{Garniron_2018, - Author = {Y. Garniron and A. Scemama and E. Giner and M. Caffarel and P. F. Loos}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.5044503}, - Journal = {J. Chem. Phys.}, - Pages = {064103}, - Title = {Selected Configuration Interaction Dressed by Perturbation}, - Volume = {149}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}} - -@article{Garniron_2017a, - Author = {Garniron, Yann and Giner, Emmanuel and Malrieu, Jean-Paul and Scemama, Anthony}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.4980034}, - Issn = {1089-7690}, - Journal = {J. Chem. Phys.}, - Month = {Apr}, - Number = {15}, - Pages = {154107}, - Publisher = {AIP Publishing}, - Title = {Alternative definition of excitation amplitudes in multi-reference state-specific coupled cluster}, - Url = {http://dx.doi.org/10.1063/1.4980034}, - Volume = {146}, - Year = {2017}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4980034}} - -@article{Garniron_2017b, - Author = {Garniron, Yann and Scemama, Anthony and Loos, Pierre-Fran{\c c}ois and Caffarel, Michel}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.4992127}, - Issn = {1089-7690}, - Journal = {J. Chem. Phys.}, - Month = {Jul}, - Number = {3}, - Pages = {034101}, - Publisher = {AIP Publishing}, - Title = {Hybrid stochastic-deterministic calculation of the second-order perturbative contribution of multireference perturbation theory}, - Url = {http://dx.doi.org/10.1063/1.4992127}, - Volume = {147}, - Year = {2017}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4992127}} - -@article{Garziano_2016, - Author = {Garziano, Luigi and Macr\`i, Vincenzo and Stassi, Roberto and Di Stefano, Omar and Nori, Franco and Savasta, Salvatore}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevLett.117.043601}, - File = {/Users/loos/Zotero/storage/L994UR4E/Garziano et al. - 2016 - One Photon Can Simultaneously Excite Two or More A.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = jul, - Number = {4}, - Title = {One {{Photon Can Simultaneously Excite Two}} or {{More Atoms}}}, - Volume = {117}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.117.043601}} - -@article{Gatti_2007, - Author = {Gatti, Matteo and Olevano, Valerio and Reining, Lucia and Tokatly, Ilya V.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevLett.99.057401}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = aug, - Number = {5}, - Shorttitle = {Transforming {{Nonlocality}} into a {{Frequency Dependence}}}, - Title = {Transforming {{Nonlocality}} into a {{Frequency Dependence}}: {{A Shortcut}} to {{Spectroscopy}}}, - Volume = {99}, - Year = {2007}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.99.057401}} - -@article{Giner_2013, - Author = {Giner, Emmanuel and Scemama, Anthony and Caffarel, Michel}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1139/cjc-2013-0017}, - Issn = {1480-3291}, - Journal = {Can. J. Chem.}, - Month = {Sep}, - Number = {9}, - Pages = {879--885}, - Publisher = {Canadian Science Publishing}, - Title = {Using perturbatively selected configuration interaction in quantum Monte Carlo calculations}, - Url = {http://dx.doi.org/10.1139/cjc-2013-0017}, - Volume = {91}, - Year = {2013}, - Bdsk-Url-1 = {http://dx.doi.org/10.1139/cjc-2013-0017}} - -@article{Giner_2015, - Author = {Emmanuel Giner and Anthony Scemama and Michel Caffarel}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.4905528}, - Issn = {1089-7690}, - Journal = {J. Chem. Phys.}, - Month = {Jan}, - Number = {4}, - Pages = {044115}, - Publisher = {AIP Publishing}, - Title = {Fixed-node diffusion Monte Carlo potential energy curve of the fluorine molecule F$_2$ using selected configuration interaction trial wavefunctions}, - Url = {http://dx.doi.org/10.1063/1.4905528}, - Volume = {142}, - Year = {2015}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4905528}} - -@article{Giner_2015b, - Author = {Giner, Emmanuel and Angeli, Celestino}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.4931639}, - File = {/Users/loos/Zotero/storage/8FL9BQ3F/Giner and Angeli - 2015 - Metal-ligand delocalization and spin density in th.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = sep, - Number = {12}, - Pages = {124305}, - Shorttitle = {Metal-Ligand Delocalization and Spin Density in the {{CuCl}} {\textsubscript{2}} and [{{CuCl}} {\textsubscript{4}} ] {\textsuperscript{2-}} Molecules}, - Title = {Metal-Ligand Delocalization and Spin Density in the {{CuCl}} {\textsubscript{2}} and [{{CuCl}} {\textsubscript{4}} ] {\textsuperscript{2-}} Molecules: {{Some}} Insights from Wave Function Theory}, - Volume = {143}, - Year = {2015}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.4931639}} - -@article{Gordon_1979, - Author = {Robert D. Gordon and Paula Luck}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {http://dx.doi.org/10.1016/0009-2614(79)80276-6}, - Issn = {0009-2614}, + Author = {Christiansen, Ove and Olsen, Jeppe and J{\o}rgensen, Poul and Koch, Henrik and Malmqvist, Per-{\AA}ke}, + Date = {1996-10-18}, + Date-Modified = {2020-08-22 22:14:27 +0200}, + Doi = {10.1016/0009-2614(96)00974-8}, Journal = {Chem. Phys. Lett.}, Number = {3}, - Pages = {480--483}, - Title = {Conformational Changes Accompanying Electronic Excitation of CD$_3$NO}, - Url = {http://www.sciencedirect.com/science/article/pii/0009261479802766}, - Volume = {65}, - Year = {1979}, - Bdsk-Url-1 = {http://www.sciencedirect.com/science/article/pii/0009261479802766}, - Bdsk-Url-2 = {http://dx.doi.org/10.1016/0009-2614(79)80276-6}} - -@article{Gould_, - Author = {Gould, Tim and Pittalis, Stefano}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - File = {/Users/loos/Zotero/storage/R2I9XXUN/Gould and Pittalis - Correlation energies of many-electron ensembles ar.pdf}, - Pages = {5}, - Title = {Correlation Energies of Many-Electron Ensembles Are More than the Sum of Their Parts}} - -@article{Gould_2013, - Author = {Gould, Tim and Dobson, John F.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.4773284}, - File = {/Users/loos/Zotero/storage/IGEZZ6JP/Gould and Dobson - 2013 - The flexible nature of exchange, correlation, and .pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = jan, - Number = {1}, - Pages = {014103}, - Shorttitle = {The Flexible Nature of Exchange, Correlation, and {{Hartree}} Physics}, - Title = {The Flexible Nature of Exchange, Correlation, and {{Hartree}} Physics: {{Resolving}} ``Delocalization'' Errors in a ``Correlation Free'' Density Functional}, - Volume = {138}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.4773284}} - -@article{Gould_2014, - Author = {Gould, Tim and Toulouse, Julien}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevA.90.050502}, - File = {/Users/loos/Zotero/storage/QIMXFUQN/Gould and Toulouse - 2014 - Kohn-Sham potentials in exact density-functional t.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Month = nov, - Number = {5}, - Title = {Kohn-{{Sham}} Potentials in Exact Density-Functional Theory at Noninteger Electron Numbers}, - Volume = {90}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.90.050502}} - -@article{Gould_2017, - Author = {Gould, Tim and Pittalis, Stefano}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevLett.119.243001}, - File = {/Users/loos/Zotero/storage/PJUSVVIV/Gould and Pittalis - 2017 - Hartree and Exchange in Ensemble Density Functiona.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = dec, - Number = {24}, - Shorttitle = {Hartree and {{Exchange}} in {{Ensemble Density Functional Theory}}}, - Title = {Hartree and {{Exchange}} in {{Ensemble Density Functional Theory}}: {{Avoiding}} the {{Nonuniqueness Disaster}}}, - Volume = {119}, - Year = {2017}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.119.243001}} - -@article{Gould_2018, - Author = {Gould, Tim and Kronik, Leeor and Pittalis, Stefano}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.5022832}, - File = {/Users/loos/Zotero/storage/C5DEDGG2/Gould et al. - 2018 - Charge transfer excitations from exact and approxi.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = may, - Number = {17}, - Pages = {174101}, - Title = {Charge Transfer Excitations from Exact and Approximate Ensemble {{Kohn}}-{{Sham}} Theory}, - Volume = {148}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.5022832}} - -@article{Gozem_2012, - Abstract = {The ground state potential energy surface of the retinal chromophore of visual pigments (e.g., bovine rhodopsin) features a low-lying conical intersection surrounded by regions with variable charge-transfer and diradical electronic structures. This implies that dynamic electron correlation may have a large effect on the shape of the force fields driving its reactivity. To investigate this effect, we focus on mapping the potential energy for three paths located along the ground state CASSCF potential energy surface of the penta-2,4-dieniminium cation taken as a minimal model of the retinal chromophore. The first path spans the bond length alternation coordinate and intercepts a conical intersection point. The other two are minimum energy paths along two distinct but kinetically competitive thermal isomerization coordinates. We show that the effect of introducing the missing dynamic electron correlation variationally (with MRCISD) and perturbatively (with the CASPT2, NEVPT2, and XMCQDPT2 methods) leads, invariably, to a stabilization of the regions with charge transfer character and to a significant reshaping of the reference CASSCF potential energy surface and suggesting a change in the dominating isomerization mechanism. The possible impact of such a correction on the photoisomerization of the retinal chromophore is discussed.}, - Author = {Gozem, Samer and Huntress, Mark and Schapiro, Igor and Lindh, Roland and Granovsky, Alexander A. and Angeli, Celestino and Olivucci, Massimo}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/ct3003139}, - File = {/Users/loos/Zotero/storage/BBENY2RV/Gozem et al. - 2012 - Dynamic Electron Correlation Effects on the Ground.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = nov, - Number = {11}, - Pages = {4069-4080}, - Title = {Dynamic {{Electron Correlation Effects}} on the {{Ground State Potential Energy Surface}} of a {{Retinal Chromophore Model}}}, - Volume = {8}, - Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1021/ct3003139}} - -@article{Gozem_2013, - Abstract = {This work investigates the performance of equation-ofmotion coupled-cluster (EOM-CC) methods for describing the changes in the potential energy surfaces of the penta-2,4-dieniminium cation, a reduced model of the retinal chromophore of visual pigments, due to dynamical electron correlation effects. The groundstate wave function of this model includes charge-transfer and diradical configurations whose weights vary along different displacements and are rapidly changing at the conical intersection between the ground and the first excited states, making the shape of the potential energy surface sensitive to a balanced description of nondynamical and dynamical correlation. Recently, variational (MRCISD) and perturbative (MRPT2) approaches for including dynamical correlation in CASSCF-based calculations were tested along three representative ground state paths. Here, we use the same three paths to compare the performance of single-reference EOM-CC methods against MRCISD and MRCISD+Q. We find that the spin-flip variant of EOM-CCSD with perturbative inclusion of triple excitations (dT or fT) produces potential energy profiles of the two lowest electronic states in quantitative agreement with MRCISD+Q (our highest-quality reference method). The nonparallelity errors and differences in vertical energy differences of the two surfaces along these scans are less than 1.4 kcal/mol (EOM-SF-CCSD(dT) versus MRCISD+Q). For comparison, the largest error of MRCISD versus MRCISD+Q is 1.7 kcal/mol. Our results show that the EOM-CC methods provide an alternative to multireference approaches and may be used to study photochemical systems like the one used in this work.}, - Author = {Gozem, Samer and Krylov, Anna I. and Olivucci, Massimo}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/ct300759z}, - File = {/Users/loos/Zotero/storage/DG2HDM7R/Gozem et al. - 2013 - Conical Intersection and Potential Energy Surface .pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = jan, - Number = {1}, - Pages = {284-292}, - Shorttitle = {Conical {{Intersection}} and {{Potential Energy Surface Features}} of a {{Model Retinal Chromophore}}}, - Title = {Conical {{Intersection}} and {{Potential Energy Surface Features}} of a {{Model Retinal Chromophore}}: {{Comparison}} of {{EOM}}-{{CC}} and {{Multireference Methods}}}, - Volume = {9}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1021/ct300759z}} - -@article{Gozem_2013a, - Author = {Gozem, Samer and Melaccio, Federico and Lindh, Roland and Krylov, Anna I. and Granovsky, Alexander A. and Angeli, Celestino and Olivucci, Massimo}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/ct400460h}, - File = {/Users/loos/Zotero/storage/PNMIDT7G/Gozem et al. - 2013 - Mapping the Excited State Potential Energy Surface.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = oct, - Number = {10}, - Pages = {4495-4506}, - Title = {Mapping the {{Excited State Potential Energy Surface}} of a {{Retinal Chromophore Model}} with {{Multireference}} and {{Equation}}-of-{{Motion Coupled}}-{{Cluster Methods}}}, - Volume = {9}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1021/ct400460h}} - -@article{Gozem_2014, - Abstract = {We report and characterize ground-state and excited-state potential energy profiles using a variety of electronic structure methods along a loop lying on the branching plane associated with a conical intersection (CI) of a reduced retinal model, the penta-2,4-dieniminium cation (PSB3). Whereas the performance of the equation-of-motion coupled-cluster, density functional theory, and multireference methods had been tested along the excited- and ground-state paths of PSB3 in our earlier work, the ability of these methods to correctly describe the potential energy surface shape along a CI branching plane has not yet been investigated. This is the focus of the present contribution. We find, in agreement with earlier studies by others, that standard time-dependent DFT (TDDFT) does not yield the correct two-dimensional (i.e., conical) crossing along the branching plane but rather a one-dimensional (i.e., linear) crossing along the same plane. The same type of behavior is found for SS-CASPT2(IPEA=0), SS-CASPT2(IPEA=0.25), spin-projected SF-TDDFT, EOM-SF-CCSD, and, finally, for the reference MRCISD+Q method. In contrast, we found that MRCISD, CASSCF, MS-CASPT2(IPEA=0), MS-CASPT2(IPEA=0.25), XMCQDPT2, QD-NEVPT2, non-spin-projected SF-TDDFT, and SI-SA-REKS yield the expected conical crossing. To assess the effect of the different crossing topologies (i.e., linear or conical) on the PSB3 photoisomerization efficiency, we discuss the results of 100 semiclassical trajectories computed by CASSCF and SS-CASPT2(IPEA=0.25) for a PSB3 derivative. We show that for the same initial conditions, the two methods yield similar dynamics leading to isomerization quantum yields that differ by only a few percent.}, - Author = {Gozem, Samer and Melaccio, Federico and Valentini, Alessio and Filatov, Michael and {Huix-Rotllant}, Miquel and Ferr\'e, Nicolas and Frutos, Luis Manuel and Angeli, Celestino and Krylov, Anna I. and Granovsky, Alexander A. and Lindh, Roland and Olivucci, Massimo}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/ct500154k}, - File = {/Users/loos/Zotero/storage/VABYG9ND/Gozem et al. - 2014 - Shape of Multireference, Equation-of-Motion Couple.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = aug, - Number = {8}, - Pages = {3074-3084}, - Title = {Shape of {{Multireference}}, {{Equation}}-of-{{Motion Coupled}}-{{Cluster}}, and {{Density Functional Theory Potential Energy Surfaces}} at a {{Conical Intersection}}}, - Volume = {10}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1021/ct500154k}} - -@article{Grimme_2004, - Author = {Grimme, S. and Izgorodina, E. I.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Journal = {Chem. Phys.}, - Pages = {223--230}, - Title = {Calculation of 0--0 Excitation Energies of Organic Molecules by CIS(D) Quantum Chemical Methods}, - Volume = 305, - Year = {2004}} - -@article{Gross_1988, - Author = {Gross, E. K. U. and Oliveira, L. N. and Kohn, W.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevA.37.2809}, - File = {/Users/loos/Zotero/storage/H33LDJJ6/Gross et al. - 1988 - Density-functional theory for ensembles of fractio.pdf}, - Issn = {0556-2791}, - Journal = {Phys. Rev. A}, - Month = apr, - Number = {8}, - Pages = {2809-2820}, - Title = {Density-Functional Theory for Ensembles of Fractionally Occupied States. {{I}}. {{Basic}} Formalism}, - Volume = {37}, - Year = {1988}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.37.2809}} - -@article{Gross_1988a, - Author = {Gross, E. K. U. and Oliveira, L. N. and Kohn, W.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevA.37.2805}, - File = {/Users/loos/Zotero/storage/24I8IUKS/Gross et al. - 1988 - Rayleigh-Ritz variational principle for ensembles .pdf}, - Issn = {0556-2791}, - Journal = {Phys. Rev. A}, - Month = apr, - Number = {8}, - Pages = {2805-2808}, - Title = {Rayleigh-{{Ritz}} Variational Principle for Ensembles of Fractionally Occupied States}, - Volume = {37}, - Year = {1988}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.37.2805}} - -@article{Grossman_2001, - Author = {Grossman, Jeffrey C. and Rohlfing, Michael and Mitas, Lubos and Louie, Steven G. and Cohen, Marvin L.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevLett.86.472}, - File = {/Users/loos/Zotero/storage/FGCU52SP/Grossman et al. - 2001 - High Accuracy Many-Body Calculational Approaches f.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = jan, - Number = {3}, - Pages = {472-475}, - Title = {High {{Accuracy Many}}-{{Body Calculational Approaches}} for {{Excitations}} in {{Molecules}}}, - Volume = {86}, - Year = {2001}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.86.472}} - -@article{Grossman_2002, - Author = {Grossman, Jeffrey C.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.1487829}, - Issn = {1089-7690}, - Journal = {J. Chem. 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Phys.}, - Month = may, - Number = {18}, - Pages = {8425-8433}, - Title = {A Long-Range-Corrected Time-Dependent Density Functional Theory}, - Volume = {120}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1688752}} - -@article{Terentyev_2004, - Author = {Terentyev, A. and Scholz, R. and Schreiber, M. and Seifert, G.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.1786291}, - File = {/Users/loos/Zotero/storage/DI5FE5KF/Terentyev et al. - 2004 - Theoretical investigation of excited states of C3.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = sep, - Number = {12}, - Pages = {5767-5776}, - Title = {Theoretical Investigation of Excited States of {{C3}}}, - Volume = {121}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1786291}} - -@article{Theophilou_1979, - Abstract = {In this paper it is shown that the density can be used as the basic variable for calculating the properties of excited states. The correspondence is not between an eigenstate and its density, as is the case with the ground state, but between the subspace spanned by the number of lowest-energy eigenstates and the sum of their densities. An extension of the Hohenberg-Kohn-Sham theory for excited states has also been developed. The equations derived are similar in form to those for the ground-state density but the interpretation is different. 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Phys.}, - Month = {Feb}, - Number = {8}, - Pages = {084102}, - Publisher = {AIP Publishing}, - Title = {Optimization of quantum Monte Carlo wave functions by energy minimization}, - Url = {http://dx.doi.org/10.1063/1.2437215}, - Volume = {126}, - Year = {2007}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2437215}} - -@article{Toulouse_2008, - Author = {Toulouse, Julien and Umrigar, C. J.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.2908237}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. 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Phys.}, - Month = dec, - Number = {23}, - Pages = {234103}, - Shorttitle = {Extrapolation to the Complete-Basis-Set Limit and the Implications of Avoided Crossings}, - Title = {Extrapolation to the Complete-Basis-Set Limit and the Implications of Avoided Crossings: {{The X $\Sigma$1g}}+, {{B $\Delta$1g}}, and {{B}}${'}$ {{$\Sigma$1g}}+ States of {{C2}}}, - Volume = {129}, - Year = {2008}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.3036115}} - -@article{Varandas_2018, - Author = {Varandas, A. J. C. and Rocha, C. M. R.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1098/rsta.2017.0145}, - File = {/Users/loos/Zotero/storage/VP3T2AAG/Varandas and Rocha - 2018 - iCi sub ini sub ( ini =2−4) c.pdf}, - Issn = {1364-503X, 1471-2962}, - Journal = {Philos. Trans. R. Soc. Math. Phys. Eng. Sci.}, - Month = mar, - Number = {2115}, - Pages = {20170145}, - Shorttitle = {{\emph{C}} {\textsubscript{ {\emph{n}} }} ( {\emph{n}} =2-4)}, - Title = {{\emph{C}} {\textsubscript{ {\emph{n}} }} ( {\emph{n}} =2-4): Current Status}, - Volume = {376}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2017.0145}} - -@article{Varandas_2009, - Abstract = {We report a simple, yet reliable, scheme for the direct diabatization of two adiabatic states showing an avoided intersection. The method requires only discrete ab initio points for the adiabatic curves, but can provide also a robust tool for analytic modeling purposes. 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Phys.}, - Month = {Jan}, - Number = {3}, - Pages = {034105}, - Publisher = {AIP Publishing}, - Title = {Energetics and dipole moment of transition metal monoxides by quantum Monte Carlo}, - Url = {http://dx.doi.org/10.1063/1.2428294}, - Volume = {126}, - Year = {2007}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.2428294}} - -@article{Wagner_2013, - Author = {Wagner, Lucas K.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1002/qua.24526}, - Issn = {0020-7608}, - Journal = {Int. J. 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Phys.}, - Month = {Feb}, - Number = {1}, - Pages = {1--3}, - Publisher = {AIP Publishing}, - Title = {Experimental Determination of the Vibrational Constants of FeS(X5Δ) by Dispersed Fluorescence Spectroscopy}, - Url = {http://dx.doi.org/10.1088/1674-0068/24/01/1-3}, - Volume = {24}, - Year = {2011}, - Bdsk-Url-1 = {http://dx.doi.org/10.1088/1674-0068/24/01/1-3}} - -@article{Wanko_2005, - Author = {Wanko, M. and Hoffmann, M. and Strodel, P. and Koslowski, A. and Thiel, W. and Neese, F. and Frauenheim, T. and Elstner, M.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/jp0463060}, - Issn = {1520-6106, 1520-5207}, - Journal = {J. Phys. Chem. B}, - Month = mar, - Number = {8}, - Pages = {3606-3615}, - Shorttitle = {Calculating {{Absorption Shifts}} for {{Retinal Proteins}}}, - Title = {Calculating {{Absorption Shifts}} for {{Retinal Proteins}}: {{Computational Challenges}}}, - Volume = {109}, - Year = {2005}, - Bdsk-Url-1 = {https://doi.org/10.1021/jp0463060}} - -@article{Watson_2012, - Author = {Watson, Mark A. and Chan, Garnet Kin-Lic}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/ct300591z}, - Journal = {J. Chem. Theory Comput.}, - Number = {11}, - Pages = {4013--4018}, - Title = {Excited States of Butadiene to Chemical Accuracy: Reconciling Theory and Experiment}, - Volume = {8}, - Year = {2012}, - Bdsk-Url-1 = {http://dx.doi.org/10.1021/ct300591z}} - -@article{Watts_1996, - Author = {Watts, John D. and Gwaltney, Steven R. and Bartlett, Rodney J.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.471988}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = oct, - Number = {16}, - Pages = {6979-6988}, - Title = {Coupled-cluster Calculations of the Excitation Energies of Ethylene, Butadiene, and Cyclopentadiene}, - Volume = {105}, - Year = {1996}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.471988}} - -@article{Weber_1999, - Author = {Weber, Peter and Reimers, Jeffrey R.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/jp991403s}, - Issn = {1089-5639, 1520-5215}, - Journal = {J. Phys. Chem. A}, - Month = dec, - Number = {48}, - Pages = {9821-9829}, - Title = {Ab {{Initio}} and {{Density Functional Calculations}} of the {{Energies}} of the {{Singlet}} and {{Triplet Valence Excited States}} of {{Pyrazine}}}, - Volume = {103}, - Year = {1999}, - Bdsk-Url-1 = {https://doi.org/10.1021/jp991403s}} - -@article{molpro, - Abstract = {Abstract Molpro (available at http://www.molpro.net) is a general-purpose quantum chemical program. The original focus was on high-accuracy wave function calculations for small molecules, but using local approximations combined with explicit correlation treatments, highly accurate coupled-cluster calculations are now possible for molecules with up to approximately 100 atoms. Recently, multireference correlation treatments were also made applicable to larger molecules. Furthermore, an efficient implementation of density functional theory is available. {\copyright} 2011 John Wiley \& Sons, Ltd. This article is categorized under: Software > Quantum Chemistry}, - Author = {Werner, Hans-Joachim and Knowles, Peter J. and Knizia, Gerald and Manby, Frederick R. and Sch{\"u}tz, Martin}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1002/wcms.82}, - Eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/wcms.82}, - Journal = {Wiley Interdisciplinary Reviews: Computational Molecular Science}, - Number = {2}, - Pages = {242-253}, - Title = {Molpro: a general-purpose quantum chemistry program package}, - Url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/wcms.82}, - Volume = {2}, - Year = {2011}, - Bdsk-Url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/wcms.82}, - Bdsk-Url-2 = {https://doi.org/10.1002/wcms.82}} - -@article{Whitten_1969, - Author = {Whitten, J. L. and Hackmeyer, Melvyn}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.1671985}, - Issn = {1089-7690}, - Journal = {J. Chem. Phys.}, - Month = {Dec}, - Number = {12}, - Pages = {5584--5596}, - Publisher = {AIP Publishing}, - Title = {Configuration Interaction Studies of Ground and Excited States of Polyatomic Molecules. I. The CI Formulation and Studies of Formaldehyde}, - Url = {http://dx.doi.org/10.1063/1.1671985}, - Volume = {51}, - Year = {1969}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.1671985}} - -@article{Wiberg_1998, - Author = {Wiberg, K. B. and Stratmann, R. E. and Frisch, M. J.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1016/S0009-2614(98)01119-1}, - Journal = {Chem. Phys. Lett.}, - Pages = {60--64}, - Title = {A Time-Dependent Density Functional Theory Study of the Electronically Excited States of Formaldehyde, Acetaldehyde and Acetone}, - Volume = 297, - Year = 1998, - Bdsk-Url-1 = {https://doi.org/10.1016/S0009-2614(98)01119-1}} - -@article{Wiberg_2002, - Author = {Wiberg, K. B. and de Oliveria, A. E. and Trucks, G.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/jp014123x}, - Journal = {J. Phys. Chem. A}, - Pages = {4192--4199}, - Title = {A Comparison of the Electronic Transition Energies for Ethene, Isobutene, Formaldehyde, and Acetone Calculated Using RPA, TDDFT, and EOM-CCSD. Effect of Basis Sets}, - Volume = {106}, - Year = 2002, - Bdsk-Url-1 = {https://doi.org/10.1021/jp014123x}} - -@article{Williams_1975, - Author = {Williams, G.R.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1016/0009-2614(75)80039-X}, - Issn = {00092614}, - Journal = {Chem. Phys. Lett.}, - Month = feb, - Number = {3}, - Pages = {495-497}, - Title = {A Theoretical Study of the Excited States of the Nitroxyl Radical ({{HNO}}) via the Equations of Motion Method}, - Volume = {30}, - Year = {1975}, - Bdsk-Url-1 = {https://doi.org/10.1016/0009-2614(75)80039-X}} - -@article{Williamson_2002, - Author = {Williamson, Andrew J. and Grossman, Jeffrey C. and Hood, Randolph Q. and Puzder, Aaron and Galli, Giulia}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevLett.89.196803}, - File = {/Users/loos/Zotero/storage/7PV4DECC/Williamson et al. - 2002 - Quantum Monte Carlo Calculations of Nanostructure .pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = oct, - Number = {19}, - Pages = {196803}, - Shorttitle = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}}, - Title = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}: {{Application}} to {{Silicon Quantum Dots}}}, - Volume = {89}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.89.196803}} - -@article{Williamson_2002a, - Author = {Williamson, Andrew J. and Grossman, Jeffrey C. and Hood, Randolph Q. and Puzder, Aaron and Galli, Giulia}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevLett.89.196803}, - File = {/Users/loos/Zotero/storage/XMBEEF79/Williamson et al. - 2002 - Quantum Monte Carlo Calculations of Nanostructure .pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = oct, - Number = {19}, - Pages = {196803}, - Shorttitle = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}}, - Title = {Quantum {{Monte Carlo Calculations}} of {{Nanostructure Optical Gaps}}: {{Application}} to {{Silicon Quantum Dots}}}, - Volume = {89}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.89.196803}} - -@article{Witek_2003, - Author = {Witek, Henryk A. and Nakano, Haruyuki and Hirao, Kimihiko}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1002/jcc.10311}, - Issn = {0192-8651, 1096-987X}, - Journal = {J. Comput. Chem.}, - Month = sep, - Number = {12}, - Pages = {1390-1400}, - Title = {Multireference Perturbation Theory with Optimized Partitioning. {{II}}. {{Applications}} to Molecular Systems}, - Volume = {24}, - Year = {2003}, - Bdsk-Url-1 = {https://doi.org/10.1002/jcc.10311}} - -@article{Woodcock_2002, - Author = {Woodcock, H. Lee and Schaefer, Henry F. and Schreiner, Peter R.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/jp0212895}, - File = {/Users/loos/Zotero/storage/T47SGG9V/Woodcock et al. - 2002 - Problematic Energy Differences between Cumulenes a.pdf}, - Issn = {1089-5639, 1520-5215}, - Journal = {J. Phys. Chem. A}, - Month = dec, - Number = {49}, - Pages = {11923-11931}, - Shorttitle = {Problematic {{Energy Differences}} between {{Cumulenes}} and {{Poly}}-Ynes}, - Title = {Problematic {{Energy Differences}} between {{Cumulenes}} and {{Poly}}-Ynes: {{Does This Point}} to a {{Systematic Improvement}} of {{Density Functional Theory}}?}, - Volume = {106}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1021/jp0212895}} - -@article{Wouters_2014, - Author = {Wouters, Sebastian and Poelmans, Ward and Ayers, Paul W. and Van Neck, Dimitri}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1016/j.cpc.2014.01.019}, - Issn = {00104655}, - Journal = {Comput. Phys. Comm.}, - Month = jun, - Number = {6}, - Pages = {1501-1514}, - Shorttitle = {{{CheMPS2}}}, - Title = {{{CheMPS2}}: {{A}} Free Open-Source Spin-Adapted Implementation of the Density Matrix Renormalization Group for Ab Initio Quantum Chemistry}, - Volume = {185}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1016/j.cpc.2014.01.019}} - -@article{Wu_2007, - Author = {Wu, Z. J. and Wang, M. Y. and Su, Z. M.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1002/jcc.20603}, - Issn = {1096-987X}, - Journal = {J. Comput. Chem.}, - Month = {Feb}, - Number = {3}, - Pages = {703--714}, - Publisher = {Wiley-Blackwell}, - Title = {Electronic structures and chemical bonding in diatomic ScX to ZnX (X = S, Se, Te)}, - Url = {http://dx.doi.org/10.1002/jcc.20603}, - Volume = {28}, - Year = {2007}, - Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.20603}} - -@article{Xiao_2016, - Author = {Xiao, Suo and Li, Xiaopeng and Sun, Weiwei and Guan, Baoqin and Wang, Yong}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1016/j.cej.2016.05.068}, - Issn = {1385-8947}, - Journal = {Chem. Eng. J.}, - Month = {Dec}, - Pages = {251--259}, - Publisher = {Elsevier BV}, - Title = {General and facile synthesis of metal sulfide nanostructures: In situ microwave synthesis and application as binder-free cathode for Li-ion batteries}, - Url = {http://dx.doi.org/10.1016/j.cej.2016.05.068}, - Volume = {306}, - Year = {2016}, - Bdsk-Url-1 = {http://dx.doi.org/10.1016/j.cej.2016.05.068}} - -@article{Xu_2013, - Abstract = {We present a new approach to calculating potential energy surfaces for photochemical reactions by combining self-consistent-field calculations for single-reference ground and excited states with symmetrycorrected spin-flip Tamm-Dancoff approximation calculations for multireference electronic states. The method is illustrated by an application with the M05-2X exchange-correlation functional to cis-trans isomerization of the penta-2,4-dieniminium cation, which is a model (with three conjugated double bonds) of the protonated Schiff base of retinal. We find good agreement with multireference configuration interaction-plus-quadruples (MRCISD+Q) wave function calculations along three key paths in the strong-interaction region of the ground and first excited singlet states.}, - Author = {Xu, Xuefei and Gozem, Samer and Olivucci, Massimo and Truhlar, Donald G.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/jz301935x}, - File = {/Users/loos/Zotero/storage/W3NS8BIA/Xu et al. - 2013 - Combined Self-Consistent-Field and Spin-Flip Tamm--.pdf}, - Issn = {1948-7185}, - Journal = {J. Phys. Chem. Lett.}, - Month = jan, - Number = {2}, - Pages = {253-258}, - Title = {Combined {{Self}}-{{Consistent}}-{{Field}} and {{Spin}}-{{Flip Tamm}}\textendash{{Dancoff Density Functional Approach}} to {{Potential Energy Surfaces}} for {{Photochemistry}}}, - Volume = {4}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1021/jz301935x}} - -@article{Yanai_2004, - Author = {Yanai, Takeshi and Tew, David P and Handy, Nicholas C}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1016/j.cplett.2004.06.011}, - Issn = {00092614}, - Journal = {Chem. Phys. Lett.}, - Month = jul, - Number = {1-3}, - Pages = {51-57}, - Title = {A New Hybrid Exchange\textendash{}Correlation Functional Using the {{Coulomb}}-Attenuating Method ({{CAM}}-{{B3LYP}})}, - Volume = {393}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2004.06.011}} - -@article{Yang_2013a, - Author = {Yang, Weitao and {Mori-S\'anchez}, Paula and Cohen, Aron J.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.4817183}, - File = {/Users/loos/Zotero/storage/FXN44KXZ/Yang et al. - 2013 - Extension of many-body theory and approximate dens.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = sep, - Number = {10}, - Pages = {104114}, - Title = {Extension of Many-Body Theory and Approximate Density Functionals to Fractional Charges and Fractional Spins}, - Volume = {139}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.4817183}} - -@article{Yang_2014, - Author = {Yang, Zeng-hui and Trail, John R. and {Pribram-Jones}, Aurora and Burke, Kieron and Needs, Richard J. and Ullrich, Carsten A.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1103/PhysRevA.90.042501}, - File = {/Users/loos/Zotero/storage/432NXITC/Yang et al. - 2014 - Exact and approximate Kohn-Sham potentials in ense.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. 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Lett.}, - Month = jul, - Number = {3}, - Title = {Direct {{Extraction}} of {{Excitation Energies}} from {{Ensemble Density}}-{{Functional Theory}}}, - Volume = {119}, - Year = {2017}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.119.033003}} - -@article{Yousaf_2008, - Abstract = {We present the results of correlated calculations on a variety of small carbon rings. Equilibrium structures and vibrational frequencies are calculated and transition states connecting symmetry-equivalent minima are considered in detail. We show that neither single-reference coupled-cluster nor multiconfigurational self-consistent field methods (even after perturbational inclusion of dynamical correlation effects) give qualitatively correct potential surfaces in the vicinity of the minima, suggesting that there is little recourse for these systems other than a multireference coupled-cluster treatment. Density-functional theory using the B3LYP functional produces results broadly in agreement with single-reference coupled-cluster methods and is thus no more reliable, but considerably more economical.}, - Author = {Yousaf, Kazim E. and Taylor, Peter R.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1016/j.chemphys.2008.02.059}, - File = {/Users/loos/Zotero/storage/26PKD7T6/Yousaf and Taylor - 2008 - On the electronic structure of small cyclic carbon.pdf}, - Issn = {03010104}, - Journal = {Chem. Phys.}, - Month = jun, - Number = {1-3}, - Pages = {58-68}, - Title = {On the Electronic Structure of Small Cyclic Carbon Clusters}, - Volume = {349}, - Year = {2008}, - Bdsk-Url-1 = {https://doi.org/10.1016/j.chemphys.2008.02.059}} - -@article{Zhang_1996, - Author = {Zhang, Nan and Hayase, Takasuke and Kawamata, Hiroshi and Nakao, Kojiro and Nakajima, Atsushi and Kaya, Koji}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.471048}, - Issn = {1089-7690}, - Journal = {J. Chem. Phys.}, - Month = {Mar}, - Number = {10}, - Pages = {3413--3419}, - Publisher = {AIP Publishing}, - Title = {Photoelectron spectroscopy of iron--sulfur cluster anions}, - Url = {http://dx.doi.org/10.1063/1.471048}, - Volume = {104}, - Year = {1996}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.471048}} - -@article{Zhao_2016, - Author = {Zhao, Luning and Neuscamman, Eric}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/acs.jctc.6b00480}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = aug, - Number = {8}, - Pages = {3719-3726}, - Title = {Equation of {{Motion Theory}} for {{Excited States}} in {{Variational Monte Carlo}} and the {{Jastrow Antisymmetric Geminal Power}} in {{Hilbert Space}}}, - Volume = {12}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.6b00480}} - -@article{Zhou_2017, - Author = {Zhou, Xiaojun and Wang, Fan}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1002/jcc.24750}, - Issn = {1096-987X}, - Journal = {J. Comput. Chem.}, - Keywords = {FN-DMC ⋅ barrier heights ⋅ H-transfer reactions ⋅ CCSD(T) ⋅ density functional theory}, - Number = {11}, - Pages = {798--806}, - Title = {Barrier heights of hydrogen-transfer reactions with diffusion quantum monte carlo method}, - Url = {http://dx.doi.org/10.1002/jcc.24750}, - Volume = {38}, - Year = {2017}, - Bdsk-Url-1 = {http://dx.doi.org/10.1002/jcc.24750}} - -@article{Zimmerman_2009, - Author = {Zimmerman, Paul M. and Toulouse, Julien and Zhang, Zhiyong and Musgrave, Charles B. and Umrigar, C. J.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.3220671}, - File = {/Users/loos/Zotero/storage/RAGHIL93/Zimmerman et al. - 2009 - Excited states of methylene from quantum Monte Car.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = sep, - Number = {12}, - Pages = {124103}, - Title = {Excited States of Methylene from Quantum {{Monte Carlo}}}, - Volume = {131}, - Year = {2009}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.3220671}} - -@article{Zimmerman_2011, - Author = {Zimmerman, Paul M. and Bell, Franziska and Casanova, David and {Head-Gordon}, Martin}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/ja208431r}, - Issn = {0002-7863, 1520-5126}, - Journal = {J. Am. Chem. Soc.}, - Month = dec, - Number = {49}, - Pages = {19944-19952}, - Shorttitle = {Mechanism for {{Singlet Fission}} in {{Pentacene}} and {{Tetracene}}}, - Title = {Mechanism for {{Singlet Fission}} in {{Pentacene}} and {{Tetracene}}: {{From Single Exciton}} to {{Two Triplets}}}, - Volume = {133}, - Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1021/ja208431r}} - -@article{Zimmerman_2013, - Author = {Zimmerman, Paul M. and Musgrave, Charles B. and {Head-Gordon}, Martin}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/ar3001734}, - Issn = {0001-4842, 1520-4898}, - Journal = {Acc. Chem. Res.}, - Month = jun, - Number = {6}, - Pages = {1339-1347}, - Title = {A {{Correlated Electron View}} of {{Singlet Fission}}}, - Volume = {46}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1021/ar3001734}} - -@article{Zimmerman_2017, - Author = {Zimmerman, Paul M.}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1063/1.4977727}, - Issn = {1089-7690}, - Journal = {J. Chem. Phys.}, - Month = {Mar}, - Number = {10}, - Pages = {104102}, - Publisher = {AIP Publishing}, - Title = {Incremental full configuration interaction}, - Url = {http://dx.doi.org/10.1063/1.4977727}, - Volume = {146}, - Year = {2017}, - Bdsk-Url-1 = {http://dx.doi.org/10.1063/1.4977727}} - -@article{Zulfikri_2016, - Author = {Zulfikri, Habiburrahman and Amovilli, Claudio and Filippi, Claudia}, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Doi = {10.1021/acs.jctc.5b01077}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = mar, - Number = {3}, - Pages = {1157-1168}, - Title = {Multiple-{{Resonance Local Wave Functions}} for {{Accurate Excited States}} in {{Quantum Monte Carlo}}}, - Volume = {12}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01077}} - -@misc{cfour, - Date-Added = {2019-01-22 22:33:30 +0100}, - Date-Modified = {2019-01-22 22:33:30 +0100}, - Note = {CFOUR, Coupled-Cluster techniques for Computational Chemistry, a quantum-chemical program package by J.F. Stanton, J. Gauss, L. Cheng, M.E. Harding, D.A. Matthews, P.G. Szalay with contributions from A.A. Auer, R.J. Bartlett, U. Benedikt, C. Berger, D.E. Bernholdt, Y.J. Bomble, O. Christiansen, F. Engel, R. Faber, M. Heckert, O. Heun, M. Hilgenberg, C. Huber, T.-C. Jagau, D. Jonsson, J. Jus{\'e}lius, T. Kirsch, K. Klein, W.J. Lauderdale, F. Lipparini, T. Metzroth, L.A. M{\"u}ck, D.P. O'Neill, D.R. Price, E. Prochnow, C. Puzzarini, K. Ruud, F. Schiffmann, W. Schwalbach, C. Simmons, S. Stopkowicz, A. Tajti, J. V{\'a}zquez, F. Wang, J.D. Watts and the integral packages MOLECULE (J. Alml{\"o}f and P.R. Taylor), PROPS (P.R. Taylor), ABACUS (T. Helgaker, H.J. Aa. Jensen, P. J{\o}rgensen, and J. Olsen), and ECP routines by A. V. Mitin and C. van W{\"u}llen. For the current version, see http://www.cfour.de.}} - -@article{Blase_2018, - Author = {Blase, Xavier and Duchemin, Ivan and Jacquemin, Denis}, - Date-Added = {2019-01-21 16:10:30 +0100}, - Date-Modified = {2019-01-21 16:10:30 +0100}, - Doi = {10.1039/C7CS00049A}, - File = {/Users/loos/Zotero/storage/PBGJ3ZVC/Blase_2017.pdf}, - Issn = {0306-0012, 1460-4744}, - Journal = {Chem. Soc. Rev.}, - Number = {3}, - Pages = {1022--1043}, - Shorttitle = {The {{Bethe}}\textendash{}{{Salpeter}} Equation in Chemistry}, - Title = {The {{Bethe}}\textendash{}{{Salpeter}} Equation in Chemistry: Relations with {{TD}}-{{DFT}}, Applications and Challenges}, + Title = {Singularity Structure of {M{\o}ller-Plesset Perturbation Theory}}, Volume = {47}, - Year = {2018}, - Bdsk-Url-1 = {https://dx.doi.org/10.1039/C7CS00049A}} - -@article{Hedin_1965, - Author = {Hedin, Lars}, - Date-Added = {2019-01-21 16:08:21 +0100}, - Date-Modified = {2019-01-21 16:08:21 +0100}, - Doi = {10.1103/PhysRev.139.A796}, - File = {/Users/loos/Zotero/storage/ZGMCVKPC/Hedin_1965.pdf}, - Journal = {Phys. Rev.}, - Number = {3A}, - Pages = {A796}, - Title = {New Method for Calculating the One-Particle {{Green}}'s Function with Application to the Electron-Gas Problem}, - Volume = {139}, - Year = {1965}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.139.A796}} - -@article{Kaplan_2015, - Author = {Kaplan, F. and Weigend, F. and Evers, F. and {van Setten}, M. J.}, - Date-Added = {2019-01-21 15:35:03 +0100}, - Date-Modified = {2019-01-21 15:35:03 +0100}, - Doi = {10.1021/acs.jctc.5b00394}, - File = {/Users/loos/Zotero/storage/Y8FW5KVR/Kaplan_2015.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = nov, - Number = {11}, - Pages = {5152--5160}, - Shorttitle = {Off-{{Diagonal Self}}-{{Energy Terms}} and {{Partially Self}}-{{Consistency}} in {{{\emph{GW}}}} {{Calculations}} for {{Single Molecules}}}, - Title = {Off-{{Diagonal Self}}-{{Energy Terms}} and {{Partially Self}}-{{Consistency}} in {{{\emph{GW}}}} {{Calculations}} for {{Single Molecules}}: {{Efficient Implementation}} and {{Quantitative Effects}} on {{Ionization Potentials}}}, - Volume = {11}, - Year = {2015}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.5b00394}} - -@article{Kaplan_2016, - Author = {Kaplan, F. and Harding, M. E. and Seiler, C. and Weigend, F. and Evers, F. and {van Setten}, M. J.}, - Date-Added = {2019-01-21 15:35:03 +0100}, - Date-Modified = {2019-01-21 15:35:03 +0100}, - Doi = {10.1021/acs.jctc.5b01238}, - File = {/Users/loos/Zotero/storage/MWYM8YMS/Kaplan_2016.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = jun, - Number = {6}, - Pages = {2528--2541}, - Title = {Quasi-{{Particle Self}}-{{Consistent}} {{{\emph{GW}}}} for {{Molecules}}}, - Volume = {12}, - Year = {2016}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.5b01238}} - -@article{Kotani_2007, - Author = {Kotani, Takao and {van Schilfgaarde}, Mark and Faleev, Sergey V.}, - Date-Added = {2019-01-21 15:34:26 +0100}, - Date-Modified = {2019-01-21 15:34:26 +0100}, - Doi = {10.1103/PhysRevB.76.165106}, - File = {/Users/loos/Zotero/storage/FIMWDS7X/Kotani_2007.pdf}, - Issn = {1098-0121, 1550-235X}, - Journal = {Phys. Rev. B}, - Month = oct, - Number = {16}, - Pages = {165106}, - Shorttitle = {Quasiparticle Self-Consistent {{G W}} Method}, - Title = {Quasiparticle Self-Consistent {{G W}} Method: {{A}} Basis for the Independent-Particle Approximation}, - Volume = {76}, - Year = {2007}, - Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevB.76.165106}} - -@article{Reining_2017, - Author = {Reining, Lucia}, - Date-Added = {2019-01-21 15:16:47 +0100}, - Date-Modified = {2019-01-21 15:16:47 +0100}, - Doi = {10.1002/wcms.1344}, - File = {/Users/loos/Zotero/storage/VDXYCLGF/Reining_2017.pdf}, - Issn = {17590876}, - Journal = {Wiley Interdiscip. Rev. Comput. Mol. Sci.}, - Month = dec, - Pages = {e1344}, - Shorttitle = {The {{GW}} Approximation}, - Title = {The {{GW}} Approximation: Content, Successes and Limitations: {{The GW}} Approximation}, - Year = {2017}, - Bdsk-Url-1 = {https://dx.doi.org/10.1002/wcms.1344}} - -@article{Mostafazadeh_2005, - Author = {A. 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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}, - File = {/Users/loos/Zotero/storage/B8RBE5C8/Zhao et al. - 2010 - Enhanced magnetic resonance signal of spin-polariz.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Month = apr, - Number = {4}, - Pages = {042903}, - 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}} - -@article{White_2015a, - Author = {White, Alec F. and McCurdy, C. William and {Head-Gordon}, Martin}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.4928529}, - File = {/Users/loos/Zotero/storage/Z6Z5TR6K/White et al. - 2015 - Restricted and unrestricted non-Hermitian Hartree-.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = aug, - Number = {7}, - Pages = {074103}, - Shorttitle = {Restricted and Unrestricted Non-{{Hermitian Hartree}}-{{Fock}}}, - Title = {Restricted and Unrestricted Non-{{Hermitian Hartree}}-{{Fock}}: {{Theory}}, Practical Considerations, and Applications to Metastable Molecular Anions}, - Volume = {143}, - Year = {2015}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.4928529}} - -@article{White_2015, - Author = {White, Alec F. and McCurdy, C. William and {Head-Gordon}, Martin}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.4928529}, - File = {/Users/loos/Zotero/storage/B2NPWPKN/White et al. - 2015 - Restricted and unrestricted non-Hermitian Hartree-.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = aug, - Number = {7}, - Pages = {074103}, - Shorttitle = {Restricted and Unrestricted Non-{{Hermitian Hartree}}-{{Fock}}}, - Title = {Restricted and Unrestricted Non-{{Hermitian Hartree}}-{{Fock}}: {{Theory}}, Practical Considerations, and Applications to Metastable Molecular Anions}, - Volume = {143}, - Year = {2015}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.4928529}} - -@article{Wang_2013, - Author = {Wang, Q.-h.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1098/rsta.2012.0045}, - File = {/Users/loos/Zotero/storage/CC9HK39V/Wang - 2013 - 2 x 2 PT-symmetric matrices and their applications.pdf}, - Issn = {1364-503X, 1471-2962}, - Journal = {Philos. Trans. R. Soc. Math. Phys. Eng. Sci.}, - Month = apr, - Number = {1989}, - Pages = {20120045-20120045}, - Title = {2 x 2 {{PT}}-Symmetric Matrices and Their Applications}, - Volume = {371}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1098/rsta.2012.0045}} - -@article{Thom_2008, - Author = {Thom, Alex J. W. and {Head-Gordon}, Martin}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-21 16:29:24 +0100}, - Doi = {10.1103/PhysRevLett.101.193001}, - File = {/Users/loos/Zotero/storage/HVKYKGQU/Thom and Head-Gordon - 2008 - Locating Multiple Self-Consistent Field Solutions.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = nov, - Number = {19}, - Pages = {193001}, - Shorttitle = {Locating {{Multiple Self}}-{{Consistent Field Solutions}}}, - Title = {Locating {{Multiple Self}}-{{Consistent Field Solutions}}: {{An Approach Inspired}} by {{Metadynamics}}}, - Volume = {101}, - Year = {2008}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.101.193001}} - -@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}, - File = {/Users/loos/Zotero/storage/JE8MFLXV/Szameit et al. - 2011 - P T -symmetry in honeycomb photonic lattices.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Month = aug, - Number = {2}, - Pages = {021806}, - Title = {P {{T}} -Symmetry in Honeycomb Photonic Lattices}, - Volume = {84}, - Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.84.021806}} - -@article{Surjan_2018, - Abstract = {An algorithm is discussed to find boundary values to partial differential equations in the knowledge of the solution of that equation inside a part of the domain enclosed by the boundary. 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Acc.}, - Month = nov, - Number = {11}, - Shorttitle = {The Inverse Boundary Value Problem}, - Title = {The Inverse Boundary Value Problem: Application in Many-Body Perturbation Theory}, - Volume = {137}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1007/s00214-018-2372-3}} - -@incollection{Surjan_2010, - Address = {Dordrecht}, - Author = {Surj\'an, P\'eter R. and Szabados, \'Agnes}, - Booktitle = {Recent {{Progress}} in {{Coupled Cluster Methods}}}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1007/978-90-481-2885-3_19}, - Editor = {C\'arsky, Petr and Paldus, Josef and Pittner, Jir\'i}, - File = {/Users/loos/Zotero/storage/EFHHZSB9/Surj{\'a}n and Szabados - 2010 - On The Coupled-Cluster Equations. 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Phys.}, - Month = mar, - Number = {3}, - Pages = {192-195}, - Title = {Observation of Parity\textendash{}Time Symmetry in Optics}, - Volume = {6}, - Year = {2010}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys1515}} - -@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}, - File = {/Users/loos/Zotero/storage/R5BRZMNS/Regensburger et al. - 2012 - Parity--time synthetic photonic lattices.pdf}, - Issn = {0028-0836, 1476-4687}, - Journal = {Nature}, - Month = aug, - Number = {7410}, - Pages = {167-171}, - Title = {Parity\textendash{}Time Synthetic Photonic Lattices}, - Volume = {488}, - Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1038/nature11298}} - -@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}, - File = {/Users/loos/Zotero/storage/G6ISXKCA/Peng et al. - 2014 - Parity--time-symmetric whispering-gallery microcavi.pdf}, - Issn = {1745-2473, 1745-2481}, - Journal = {Nat. Phys.}, - Month = may, - Number = {5}, - Pages = {394-398}, - Title = {Parity\textendash{}Time-Symmetric Whispering-Gallery Microcavities}, - Volume = {10}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys2927}} - -@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}, - File = {/Users/loos/Zotero/storage/H3WQRBQP/Peng et al. - 2014 - Loss-induced suppression and revival of lasing.pdf}, - Issn = {0036-8075, 1095-9203}, - Journal = {Science}, - Month = oct, - Number = {6207}, - Pages = {328-332}, - Title = {Loss-Induced Suppression and Revival of Lasing}, - Volume = {346}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1126/science.1258004}} - -@article{Mostafazadeh_2002b, - Author = {Mostafazadeh, Ali}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.1418246}, - File = {/Users/loos/Zotero/storage/2P58GGD7/Mostafazadeh - 2002 - Pseudo-Hermiticity versus PT symmetry The necessa.pdf}, - Issn = {0022-2488, 1089-7658}, - Journal = {J. Math. Phys.}, - Month = jan, - Number = {1}, - Pages = {205-214}, - Shorttitle = {Pseudo-{{Hermiticity}} versus {{PT}} Symmetry}, - Title = {Pseudo-{{Hermiticity}} versus {{PT}} Symmetry: {{The}} Necessary Condition for the Reality of the Spectrum of a Non-{{Hermitian Hamiltonian}}}, - Volume = {43}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1418246}} - -@article{Mostafazadeh_2002a, - Author = {Mostafazadeh, Ali}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.1461427}, - File = {/Users/loos/Zotero/storage/GH3FUXYM/Mostafazadeh - 2002 - Pseudo-Hermiticity versus PT-symmetry. II. A compl.pdf}, - Issn = {00222488}, - Journal = {J. Math. Phys.}, - Number = {5}, - Pages = {2814}, - Title = {Pseudo-{{Hermiticity}} versus {{PT}}-Symmetry. {{II}}. {{A}} Complete Characterization of Non-{{Hermitian Hamiltonians}} with a Real Spectrum}, - Volume = {43}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1461427}} - -@article{Mostafazadeh_2002, - Author = {Mostafazadeh, Ali}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.1489072}, - File = {/Users/loos/Zotero/storage/AAITP6CE/Mostafazadeh - 2002 - Pseudo-Hermiticity versus PT-symmetry III Equival.pdf}, - Issn = {0022-2488, 1089-7658}, - Journal = {J. Math. Phys.}, - Month = aug, - Number = {8}, - Pages = {3944-3951}, - Shorttitle = {Pseudo-{{Hermiticity}} versus {{PT}}-Symmetry {{III}}}, - Title = {Pseudo-{{Hermiticity}} versus {{PT}}-Symmetry {{III}}: {{Equivalence}} of Pseudo-{{Hermiticity}} and the Presence of Antilinear Symmetries}, - Volume = {43}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1489072}} - -@article{Mostafazadeh_1997, - Author = {Mostafazadeh, Ali}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.531864}, - File = {/Users/loos/Zotero/storage/XFH9SG4X/Mostafazadeh - 1997 - Quantum canonical transformations and exact soluti.pdf}, - Issn = {0022-2488, 1089-7658}, - Journal = {J. Math. Phys.}, - Month = jul, - Number = {7}, - Pages = {3489-3496}, - Title = {Quantum Canonical Transformations and Exact Solution of the {{Schr\"odinger}} Equation}, - Volume = {38}, - Year = {1997}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.531864}} - -@article{Mostafazadeh_1996, - Author = {Mostafazadeh, Ali}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.531457}, - File = {/Users/loos/Zotero/storage/MYCWSEQG/Mostafazadeh - 1996 - Geometric phase, bundle classification, and group .pdf}, - Issn = {0022-2488, 1089-7658}, - Journal = {J. Math. Phys.}, - Month = mar, - Number = {3}, - Pages = {1218-1233}, - Title = {Geometric Phase, Bundle Classification, and Group Representation}, - Volume = {37}, - Year = {1996}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.531457}} - -@article{Moiseyev_2011, - Author = {Moiseyev, Nimrod}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1103/PhysRevA.83.052125}, - File = {/Users/loos/Zotero/storage/KC2M9X67/Moiseyev - 2011 - Crossing rule for a PT -symmetric two-level time-p.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Month = may, - Number = {5}, - Title = {Crossing Rule for a {{PT}} -Symmetric Two-Level Time-Periodic System}, - Volume = {83}, - Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.83.052125}} - -@article{Goldzak_2018, - Author = {Tamar Goldzak and Alexei A. Mailybaev and Nimrod Moiseyev}, - Date-Modified = {2019-03-19 14:43:31 +0100}, - Doi = {10.1103/PhysRevLett.120.013901}, - Journal = {Phys. Rev. Lett.}, - Pages = {013901}, - Title = {Light Stops at Exceptional Points}, - Volume = {120}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.120.013901}} - -@article{Mihalka_2019, - Abstract = {Previous attempts to the resummation of divergent power series by means of analytic continuation are improved applying the Cauchy integral formula for complex functions. The idea is tested on divergent M\o{}ller-Plesset perturbation expansions of the electron correlation energy. In particular, the potential curve of the LiH molecule is computed from single reference MPn results which are divergent for bond distances larger than 3.6 \AA. Preliminary results for the Hartree-Fock molecule are also tabulated.}, - Author = {Mih\'alka, Zsuzsanna \'E. and Szabados, \'Agnes and Surj\'an, P\'eter R.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.5083191}, - File = {/Users/loos/Zotero/storage/3UBDYMTC/Mih{\'a}lka et al. - 2019 - Application of the Cauchy integral formula as a to.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = jan, - Number = {3}, - Pages = {031101}, - Title = {Application of the {{Cauchy}} Integral Formula as a Tool of Analytic Continuation for the Resummation of Divergent Perturbation Series}, - Volume = {150}, - Year = {2019}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.5083191}} - -@article{Mestechkin_1979, - Author = {Mestechkin, M.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1002/qua.560150606}, - File = {/Users/loos/Zotero/storage/RN8PSK66/Mestechkin - 1979 - Instability threshold and peculiar solutions of Ha.pdf}, - Issn = {0020-7608, 1097-461X}, - Journal = {Int. J. Quantum Chem.}, - Month = jun, - Number = {6}, - Pages = {601-610}, - Title = {Instability Threshold and Peculiar Solutions of {{Hartree}}-{{Fock}} Equations}, - Volume = {15}, - Year = {1979}, - Bdsk-Url-1 = {https://doi.org/10.1002/qua.560150606}} - -@article{Mehri-Dehnavi_2008, - Author = {{Mehri-Dehnavi}, Hossein and Mostafazadeh, Ali}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.2968344}, - File = {/Users/loos/Zotero/storage/JGDJXU8C/Mehri-Dehnavi and Mostafazadeh - 2008 - Geometric phase for non-Hermitian Hamiltonians and.pdf}, - Issn = {0022-2488, 1089-7658}, - Journal = {J. Math. Phys.}, - Month = aug, - Number = {8}, - Pages = {082105}, - Title = {Geometric Phase for Non-{{Hermitian Hamiltonians}} and Its Holonomy Interpretation}, - Volume = {49}, - Year = {2008}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.2968344}} - -@article{Mailybaev_2005, - Author = {Mailybaev, Alexei A. and Kirillov, Oleg N. and Seyranian, Alexander P.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1103/PhysRevA.72.014104}, - File = {/Users/loos/Zotero/storage/ZXETU3ZJ/Mailybaev et al. - 2005 - Geometric phase around exceptional points.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Month = jul, - Number = {1}, - Title = {Geometric Phase around Exceptional Points}, - Volume = {72}, - Year = {2005}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.72.014104}} - -@article{Lykos_1963, - Author = {Lykos, P. and Pratt, G. W.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1103/RevModPhys.35.496}, - File = {/Users/loos/Zotero/storage/UN4QCYBC/Lykos and Pratt - 1963 - Discussion on The Hartree-Fock Approximation.pdf}, - Issn = {0034-6861}, - Journal = {Rev. Mod. Phys.}, - Month = jul, - Number = {3}, - Pages = {496-501}, - Title = {Discussion on {{The Hartree}}-{{Fock Approximation}}}, - Volume = {35}, - Year = {1963}, - Bdsk-Url-1 = {https://doi.org/10.1103/RevModPhys.35.496}} - -@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}, - File = {/Users/loos/Zotero/storage/87X25AZ4/Longhi - 2010 - Optical Realization of Relativistic Non-Hermitian .pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = jun, - Number = {1}, - Pages = {013903}, - Title = {Optical {{Realization}} of {{Relativistic Non}}-{{Hermitian Quantum Mechanics}}}, - Volume = {105}, - Year = {2010}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.105.013903}} - -@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}, - File = {/Users/loos/Zotero/storage/XJWN77AL/Liertzer et al. - 2012 - Pump-Induced Exceptional Points in Lasers.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = apr, - Number = {17}, - Pages = {173901}, - Title = {Pump-{{Induced Exceptional Points}} in {{Lasers}}}, - Volume = {108}, - Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.108.173901}} - -@article{Lefebvre_2010, - Abstract = {The strict coalescence of two resonance states may never occur (due to the sensitivity to small external perturbations). However, its existence can have noticeable effects on measurable quantities. The coalescence of resonances is associated with exceptional points (EPs) in the spectrum of the system under study. The determination of exceptional points is often a hard numerical task, due to the need to calculate many eigenvalues of a non-Hermitian Hamiltonian as a function of the potential's parameters which are to be varied adiabatically. The method presented here is based on the Pade\textasciiacute{} algorithm. This enables one to calculate the EPs with a small number of solutions. As an illustrative numerical example an EP of H+2 in strong laser fields is calculated.}, - Author = {Lefebvre, R and Moiseyev, N}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1088/0953-4075/43/9/095401}, - File = {/Users/loos/Zotero/storage/A4CI7CZL/Lefebvre and Moiseyev - 2010 - Localization of exceptional points with Pad{\'e} appro.pdf}, - Issn = {0953-4075, 1361-6455}, - Journal = {J. Phys. B At. Mol. Opt. Phys.}, - Month = may, - Number = {9}, - Pages = {095401}, - Title = {Localization of Exceptional Points with {{Pad\'e}} Approximants}, - Volume = {43}, - Year = {2010}, - Bdsk-Url-1 = {https://doi.org/10.1088/0953-4075/43/9/095401}} - -@article{Lefebvre_2009, - Author = {Lefebvre, R. and Atabek, O. and {\v S}indelka, M. and Moiseyev, N.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1103/PhysRevLett.103.123003}, - File = {/Users/loos/Zotero/storage/95HC7RY3/Lefebvre et al. - 2009 - Resonance Coalescence in Molecular Photodissociati.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = sep, - Number = {12}, - Title = {Resonance {{Coalescence}} in {{Molecular Photodissociation}}}, - Volume = {103}, - Year = {2009}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.123003}} - -@article{Kowalski_1998, - Abstract = {A version of the homotopy {\v Z}continuation. methods is successfully applied to calculating the full set of geometrically isolated solutions to a set of nonlinear equations determining the cluster amplitudes in coupled-cluster {\v Z}CC. methods. The set of equations considered is obtained when applying the CC method, including two-body amplitudes {\v Z}CCD., to the H4 model. The complete sets of solutions for the spinorbital and spin-symmetry adapted variants of the CCD equations consist of 20 and 12 solutions, respectively. At least 5 solutions are physically meaningful. In addition to standard and nonstandard solutions of the desired symmetry, broken-symmetry solutions are also documented. A preliminary analysis of the significance of the latter solutions is presented. q 1998 Elsevier Science B.V. All rights reserved.}, - Author = {Kowalski, K and Jankowski, K}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1016/S0009-2614(98)00464-3}, - File = {/Users/loos/Zotero/storage/LPRBKZRB/Kowalski and Jankowski - 1998 - Full solution to the coupled-cluster equations th.pdf}, - Issn = {00092614}, - Journal = {Chem. Phys. Lett.}, - Month = jun, - Number = {1-3}, - Pages = {180-188}, - Shorttitle = {Full Solution to the Coupled-Cluster Equations}, - Title = {Full Solution to the Coupled-Cluster Equations: The {{H4}} Model}, - Volume = {290}, - Year = {1998}, - Bdsk-Url-1 = {https://doi.org/10.1016/S0009-2614(98)00464-3}} - -@article{Jake_2018, - Author = {Jake, Lena C. and Henderson, Thomas M. and Scuseria, Gustavo E.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.5010929}, - File = {/Users/loos/Zotero/storage/MNSPHXCX/Jake et al. - 2018 - Hartree--Fock symmetry breaking around conical inte.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = jan, - Number = {2}, - Pages = {024109}, - Title = {Hartree\textendash{{Fock}} Symmetry Breaking around Conical Intersections}, - Volume = {148}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.5010929}} - -@article{Hiscock_2014, - Abstract = {We investigate the Hartree-Fock solutions to H2 in a minimal basis. We note the properties of the solutions and their disappearance with geometry and propose a new method, called Holomorphic Hartree-Fock theory, where we modify the self-consistent field (SCF) equations to avoid disappearance of the solutions. We use these solutions as a basis for a nonorthogonal configuration interaction to produce a smooth binding curve over a complete range of geometries.}, - Author = {Hiscock, Hamish G. and Thom, Alex J. W.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1021/ct5007696}, - File = {/Users/loos/Zotero/storage/ZDU946HN/Hiscock and Thom - 2014 - Holomorphic Hartree--Fock Theory and Configuration .pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = nov, - Number = {11}, - Pages = {4795-4800}, - Title = {Holomorphic {{Hartree}}\textendash{{Fock Theory}} and {{Configuration Interaction}}}, - Volume = {10}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1021/ct5007696}} - -@article{Heiss_2012, - Abstract = {A short re\textasciiacute{}sume\textasciiacute{} is given about the nature of exceptional points (EPs) followed by discussions about their ubiquitous occurrence in a great variety of physical problems. EPs feature in classical as well as in quantum mechanical problems. They are associated with symmetry breaking for PT -symmetric Hamiltonians, where a great number of experiments has been performed, in particular in optics, and to an increasing extent in atomic and molecular physics. EPs are involved in quantum phase transition and quantum chaos; they produce dramatic effects in multichannel scattering, specific time dependence and more. In nuclear physics, they are associated with instabilities and continuum problems. Being spectral singularities they also affect approximation schemes.}, - 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}, - File = {/Users/loos/Zotero/storage/IEXPZWNZ/Heiss - 2012 - The physics of exceptional points.pdf}, - Issn = {1751-8113, 1751-8121}, - Journal = {J. Phys. Math. Theor.}, - Month = nov, - Number = {44}, - Pages = {444016}, - Title = {The Physics of Exceptional Points}, - Volume = {45}, - Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/45/44/444016}} - -@article{Heiss_1991, - 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 = {10.1103/PhysRevA.43.4159}, - File = {/Users/loos/Zotero/storage/69VRYNVI/Heiss and Sannino - 1991 - Transitional regions of finite Fermi systems and q.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Month = apr, - Number = {8}, - Pages = {4159-4166}, - Title = {Transitional Regions of Finite {{Fermi}} Systems and Quantum Chaos}, - Volume = {43}, - Year = {1991}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.43.4159}} - -@article{Heiss_1990, - Abstract = {The connection between level repulsions and the singularities associated with the analytically continued energy levels is investigated. We also conjecture that there are necessarily specific consequences for the state vectors when the statistical analysis of the energy spectrum indicates quantum chaos. A procedure which allows a qualitative assessment of the positions of the exceptional points is suggested. The importance of their distribution for quantum chaos is discussed within this context.}, - 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}, - File = {/Users/loos/Zotero/storage/6EJMT34Y/Heiss and Sannino - 1990 - Avoided level crossing and exceptional points.pdf}, - Issn = {0305-4470, 1361-6447}, - Journal = {J. Phys. Math. Gen.}, - Month = apr, - Number = {7}, - Pages = {1167-1178}, - Title = {Avoided Level Crossing and Exceptional Points}, - Volume = {23}, - Year = {1990}, - Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/23/7/022}} - -@article{Heiss_1971, - Author = {Heiss, W.D.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1016/0375-9474(71)90882-7}, - File = {/Users/loos/Zotero/storage/ZVKP6WKU/Heiss - 1971 - An effective single-particle potential.pdf}, - Issn = {03759474}, - Journal = {Nucl. Phys. A}, - Month = jul, - Number = {2}, - Pages = {248-256}, - Title = {An Effective Single-Particle Potential}, - Volume = {169}, - Year = {1971}, - Bdsk-Url-1 = {https://doi.org/10.1016/0375-9474(71)90882-7}} - -@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}, - File = {/Users/loos/Zotero/storage/ZBT6XXR3/Heiss - Circling exceptional points.pdf}, - Journal = {Nat. Phys.}, - Pages = {823--824}, - Title = {Circling Exceptional Points}, - Volume = {12}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1038/nphys3864}} - -@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}, - File = {/Users/loos/Zotero/storage/AF88CL4L/Hang et al. - 2013 - P T Symmetry with a System of Three-Level Atoms.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = feb, - Number = {8}, - Pages = {083604}, - Title = {P {{T Symmetry}} with a {{System}} of {{Three}}-{{Level Atoms}}}, - Volume = {110}, - Year = {2013}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.110.083604}} - -@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}, - File = {/Users/loos/Zotero/storage/N9Z63NAV/Guo et al. - 2009 - Observation of P T -Symmetry Breaking in Complex O.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = aug, - Number = {9}, - Pages = {093902}, - Title = {Observation of {{P T}} -{{Symmetry Breaking}} in {{Complex Optical Potentials}}}, - Volume = {103}, - Year = {2009}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.093902}} + Year = {2004}, + Bdsk-Url-1 = {https://doi.org/10.1016/S0065-3276(04)47011-7}} @article{Goodson_2012, Author = {Goodson, David Z.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, + Date-Modified = {2020-08-22 22:13:28 +0200}, Doi = {10.1002/wcms.92}, - File = {/Users/loos/Zotero/storage/6FUT64TM/Goodson - 2012 - Resummation methods Resummation methods.pdf}, - Issn = {17590876}, - Journal = {Wiley Interdiscip. Rev. Comput. Mol. Sci.}, - Month = sep, + Journal = {{WIREs} Comput. Mol. Sci.}, Number = {5}, - Pages = {743-761}, - Shorttitle = {Resummation Methods}, - Title = {Resummation Methods: {{Resummation}} Methods}, + Pages = {743--761}, + Title = {Resummation methods}, Volume = {2}, Year = {2012}, Bdsk-Url-1 = {https://doi.org/10.1002/wcms.92}} -@article{Frydryszak_2008, - Author = {Frydryszak, A. M. and Tkachuk, V. M.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1103/PhysRevA.77.014103}, - File = {/Users/loos/Zotero/storage/4483C8FP/Frydryszak and Tkachuk - 2008 - Quantum brachistochrone problem for a spin-1 syste.pdf}, - Issn = {1050-2947, 1094-1622}, - Journal = {Phys. Rev. A}, - Month = jan, - Number = {1}, - Title = {Quantum Brachistochrone Problem for a Spin-1 System in a Magnetic Field}, - Volume = {77}, - Year = {2008}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.77.014103}} +@article{Katz_1962, + 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{Feuerbacher_2004a, - Author = {Feuerbacher, Sven and Sommerfeld, Thomas and Cederbaum, Lorenz S.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.1640615}, - File = {/Users/loos/Zotero/storage/NQQRLYSR/Feuerbacher et al. - 2004 - Intersections of potential energy surfaces of shor.pdf}, - Issn = {0021-9606, 1089-7690}, +@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}, + Bdsk-Url-1 = {https://doi.org/10.1002/qua.560200502}} + +@article{Stillinger_1966, + Author = {Stillinger, Frank H.}, + Date = {1966-11-15}, + Date-Modified = {2020-08-22 22:14:44 +0200}, + Doi = {10.1063/1.1727380}, Journal = {J. Chem. Phys.}, - Month = feb, - Number = {7}, - Pages = {3201-3214}, - Shorttitle = {Intersections of Potential Energy Surfaces of Short-Lived States}, - Title = {Intersections of Potential Energy Surfaces of Short-Lived States: {{The}} Complex Analogue of Conical Intersections}, - Volume = {120}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1640615}} + Number = {10}, + Pages = {3623--3631}, + Title = {{Ground‐State Energy of Two‐Electron Atoms}}, + Volume = {45}, + Year = {1966}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.1727380}} -@article{Feuerbacher_2004, - Author = {Feuerbacher, Sven and Sommerfeld, Thomas and Cederbaum, Lorenz S.}, +@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 = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.1640615}, - File = {/Users/loos/Zotero/storage/NUP4KKK8/Feuerbacher et al. - 2004 - Intersections of potential energy surfaces of shor.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = feb, - Number = {7}, - Pages = {3201-3214}, - Shorttitle = {Intersections of Potential Energy Surfaces of Short-Lived States}, - Title = {Intersections of Potential Energy Surfaces of Short-Lived States: {{The}} Complex Analogue of Conical Intersections}, - Volume = {120}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.1640615}} - -@article{Estrada_1986, - Author = {Estrada, H. and Cederbaum, L. S. and Domcke, W.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.450165}, - File = {/Users/loos/Zotero/storage/7EHVN8CQ/Estrada et al. - 1986 - Vibronic coupling of short‐lived electronic states.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = jan, - Number = {1}, - Pages = {152-169}, - Title = {Vibronic Coupling of Short-lived Electronic States}, - Volume = {84}, - Year = {1986}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.450165}} - -@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}, - File = {/Users/loos/Zotero/storage/5KBF7YTY/Doppler et al. - 2016 - Dynamically encircling an exceptional point for as.pdf}, - Issn = {0028-0836, 1476-4687}, - Journal = {Nature}, - Month = sep, - Number = {7618}, - Pages = {76-79}, - Title = {Dynamically Encircling an Exceptional Point for Asymmetric Mode Switching}, - Volume = {537}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1038/nature18605}} - -@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}, - File = {/Users/loos/Zotero/storage/34CPCSV4/Chtchelkatchev et al. - 2012 - Stimulation of the Fluctuation Superconductivity b.pdf}, - Issn = {0031-9007, 1079-7114}, + Date-Modified = {2019-01-21 16:25:31 +0100}, + Doi = {10.1103/PhysRevLett.108.024101}, Journal = {Phys. Rev. Lett.}, - Month = oct, - Number = {15}, - Pages = {150405}, - Title = {Stimulation of the {{Fluctuation Superconductivity}} by {{P T Symmetry}}}, - Volume = {109}, + Month = jan, + Number = {2}, + Pages = {024101}, + Title = {{PT Symmetry} and {{Spontaneous Symmetry Breaking}} in a {{Microwave Billiard}}}, + Volume = {108}, Year = {2012}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.109.150405}} + 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}, - File = {/Users/loos/Zotero/storage/IHG3L6K9/Chong et al. - 2011 - P T -Symmetry Breaking and Laser-Absorber Modes in.pdf}, - Issn = {0031-9007, 1079-7114}, Journal = {Phys. Rev. Lett.}, Month = mar, Number = {9}, Pages = {093902}, - Title = {P {{T}} -{{Symmetry Breaking}} and {{Laser}}-{{Absorber Modes}} in {{Optical Scattering Systems}}}, + 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}} +@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}, + Journal = {Phys. Rev. Lett.}, + Month = oct, + Number = {15}, + Pages = {150405}, + Title = {{Stimulation of the Fluctuation Superconductivity by PT Symmetry}}, + Volume = {109}, + Year = {2012}, + 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}, + Journal = {Nature}, + Month = sep, + Number = {7618}, + Pages = {76-79}, + Title = {{Dynamically Encircling an Exceptional Point for Asymmetric Mode Switching}}, + Volume = {537}, + Year = {2016}, + 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}, + Journal = {Phys. Rev. Lett.}, + Month = aug, + Number = {9}, + Pages = {093902}, + 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}} + +@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}, + Journal = {Phys. Rev. Lett.}, + Month = feb, + Number = {8}, + Pages = {083604}, + 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}} + +@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}, + Journal = {Phys. Rev. Lett.}, + Month = apr, + Number = {17}, + Pages = {173901}, + Title = {{Pump-Induced Exceptional Points in Lasers}}, + Volume = {108}, + Year = {2012}, + 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}, + Journal = {Phys. Rev. Lett.}, + Month = jun, + Number = {1}, + Pages = {013903}, + Title = {{Optical Realization of Relativistic Non-Hermitian Quantum Mechanics}}, + Volume = {105}, + Year = {2010}, + 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}, + Journal = {Science}, + Month = oct, + Number = {6207}, + Pages = {328-332}, + Title = {{Loss-Induced Suppression and Revival of Lasing}}, + Volume = {346}, + Year = {2014}, + 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}, + Journal = {Nat. Phys.}, + Month = may, + Number = {5}, + Pages = {394-398}, + Title = {{Parity\textendash{}Time-Symmetric Whispering-Gallery Microcavities}}, + Volume = {10}, + Year = {2014}, + 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}, + Journal = {Nature}, + Month = aug, + Number = {7410}, + Pages = {167-171}, + Title = {{Parity\textendash{}Time Synthetic Photonic Lattices}}, + Volume = {488}, + Year = {2012}, + 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}, + Journal = {Nat. Phys.}, + Month = mar, + Number = {3}, + Pages = {192-195}, + Title = {{Observation of Parity\textendash{}Time Symmetry in Optics}}, + Volume = {6}, + Year = {2010}, + 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}, + Journal = {Phys. Rev. A}, + Month = oct, + Number = {4}, + Pages = {040101}, + 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}} + +@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}, + Journal = {Phys. Rev. A}, + Month = aug, + Number = {2}, + Pages = {021806}, + Title = {{PT-Symmetry in Honeycomb Photonic Lattice}s}, + Volume = {84}, + Year = {2011}, + 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}, + Journal = {Phys. Rev. A}, + Month = apr, + Number = {4}, + Pages = {042903}, + 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}} + +@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}, + Journal = {Philos. Trans. R. Soc. Math. Phys. Eng. Sci.}, + Month = mar, + Number = {1989}, + Pages = {20120053-20120053}, + 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}} + @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}, - File = {/Users/loos/Zotero/storage/24QHEIU7/Choi et al. - 2018 - Observation of an anti-PT-symmetric exceptional po.pdf}, - Issn = {2041-1723}, Journal = {Nat. Commun.}, Month = dec, Number = {1}, 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}, Year = {2018}, Bdsk-Url-1 = {https://doi.org/10.1038/s41467-018-04690-y}} -@article{Carlini_2006, - Author = {Carlini, Alberto and Hosoya, Akio and Koike, Tatsuhiko and Okudaira, Yosuke}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1103/PhysRevLett.96.060503}, - File = {/Users/loos/Zotero/storage/JVFY9Z4R/Carlini et al. - 2006 - Time-Optimal Quantum Evolution.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = feb, - Number = {6}, - Title = {Time-{{Optimal Quantum Evolution}}}, - Volume = {96}, - Year = {2006}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.96.060503}} - -@article{Burton_2018, - Abstract = {We explore the existence and behavior of holomorphic restricted Hartree-Fock (h-RHF) solutions for two-electron problems. Through algebraic geometry, the exact number of solutions with n basis functions is rigorously identified as 1/2(3n - 1), proving that states must exist for all molecular geometries. A detailed study on the h-RHF states of HZ (STO3G) then demonstrates both the conservation of holomorphic solutions as geometry or atomic charges are varied and the emergence of complex h-RHF solutions at coalescence points. Using catastrophe theory, the nature of these coalescence points is described, highlighting the influence of molecular symmetry. The h-RHF states of HHeH2+ and HHeH (STO-3G) are then compared, illustrating the isomorphism between systems with two electrons and two electron holes. Finally, we explore the h-RHF states of ethene (STO-3G) by considering the $\pi$ electrons as a two-electron problem and employ NOCI to identify a crossing of the lowest energy singlet and triplet states at the perpendicular geometry.}, - Author = {Burton, Hugh G. A. and Gross, Mark and Thom, Alex J. W.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1021/acs.jctc.7b00980}, - File = {/Users/loos/Zotero/storage/E9FNMAU8/Burton et al. - 2018 - Holomorphic Hartree--Fock Theory The Nature of Two.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = feb, - Number = {2}, - Pages = {607-618}, - Shorttitle = {Holomorphic {{Hartree}}\textendash{{Fock Theory}}}, - Title = {Holomorphic {{Hartree}}\textendash{{Fock Theory}}: {{The Nature}} of {{Two}}-{{Electron Problems}}}, +@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}, + Journal = {Nat. Phys.}, + Language = {en}, + Month = jan, + Number = {1}, + Pages = {11-19}, + Title = {{Non-Hermitian Physics and PT Symmetry}}, Volume = {14}, Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00980}} + Bdsk-Url-1 = {https://doi.org/10.1038/nphys4323}} -@article{Brody_2003, - Abstract = {When a quantum system undergoes unitary evolution in accordance with a prescribed Hamiltonian, there is a class of states |$\psi$ such that, after the passage of a certain time, |$\psi$ is transformed into a state orthogonal to itself. The shortest time for which this can occur, for a given system, is called the passage time. We provide an elementary derivation of the passage time, and demonstrate that the known lower bound, due to Fleming, is typically attained, except for special cases in which the energy spectra have particularly simple structures. It is also shown, using a geodesic argument, that the passage times for these exceptional cases are necessarily larger than the Fleming bound. The analysis is extended to passage times for initially mixed states.}, - Author = {Brody, Dorje C}, +@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 = {10.1088/0305-4470/36/20/314}, - File = {/Users/loos/Zotero/storage/RTXJRAEB/Brody - 2003 - Elementary derivation for passage times.pdf}, - Issn = {0305-4470, 1361-6447}, - Journal = {J. Phys. Math. Gen.}, - Month = may, - Number = {20}, - Pages = {5587-5593}, - Title = {Elementary Derivation for Passage Times}, - Volume = {36}, - Year = {2003}, - Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/36/20/314}} - -@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}, - File = {/Users/loos/Zotero/storage/WB7BUBXH/Bittner et al. - 2012 - P T Symmetry and Spontaneous Symmetry Breaking in .pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = jan, - Number = {2}, - Pages = {024101}, - 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}} - -@article{Beygi_2018a, - Author = {Beygi, Alireza and Klevansky, S. P. and Bender, Carl M.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-02-17 21:08:18 +0100}, - Doi = {10.1103/PhysRevA.97.032128}, - File = {/Users/loos/Zotero/storage/6DKK9IE6/Beygi et al. - 2018 - Two- and four-dimensional representations of the P.pdf}, - Issn = {2469-9926, 2469-9934}, - Journal = {Phys. Rev. A}, - Month = mar, - Number = {3}, - Pages = {032128}, - Title = {Two- and Four-Dimensional Representations of the {{PT}} - and {{CPT}} -Symmetric Fermionic Algebras}, - Volume = {97}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.97.032128}} - -@article{Berry_2004, - Author = {Berry, M.V.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1023/B:CJOP.0000044002.05657.04}, - File = {/Users/loos/Zotero/storage/22LFRLSE/Berry - 2004 - Physics of Nonhermitian Degeneracies.pdf}, - Issn = {0011-4626}, - Journal = {Czechoslov. J. Phys.}, - Month = oct, - Number = {10}, - Pages = {1039-1047}, - Title = {Physics of {{Nonhermitian Degeneracies}}}, - Volume = {54}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1023/B:CJOP.0000044002.05657.04}} - -@article{Bender_2017, - Author = {Bender, Carl M. and Hassanpour, Nima and Hook, Daniel W. and Klevansky, S. P. and S\"underhauf, Christoph and Wen, Zichao}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-27 20:40:50 +0100}, - Doi = {10.1103/PhysRevA.95.052113}, - File = {/Users/loos/Zotero/storage/432UQ9T2/Bender et al. - 2017 - Behavior of eigenvalues in a region of broken PT s.pdf}, - Issn = {2469-9926, 2469-9934}, - Journal = {Phys. Rev. A}, - Month = may, - Number = {5}, - Pages = {052113}, - Title = {Behavior of Eigenvalues in a Region of Broken {{PT}} Symmetry}, - Volume = {95}, - Year = {2017}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.95.052113}} - -@article{Bender_2016, - Author = {Bender, Carl M.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1051/epn/2016201}, - File = {/Users/loos/Zotero/storage/4AQXDDGX/Bender - 2016 - PT symmetry in quantum physics From a mathematica.pdf}, - Issn = {0531-7479, 1432-1092}, - Journal = {Europhys. News}, - Month = mar, - Number = {2}, - Pages = {17-20}, - Shorttitle = {{{PT}} Symmetry in Quantum Physics}, - Title = {{{PT}} Symmetry in Quantum Physics: {{From}} a Mathematical Curiosity to Optical Experiments}, - Volume = {47}, - Year = {2016}, - Bdsk-Url-1 = {https://doi.org/10.1051/epn/2016201}} - -@article{Bender_2015, - Abstract = {The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p2 + ix3, which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H = p2 + ix3 is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T . The quantum mechanics defined by a PT -symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been verified in laboratory experiments. A particularly interesting PT -symmetric Hamiltonian is H = p2 - x4, which contains an upside-down potential. This potential is discussed in detail, and it is explained in intuitive as well as in rigorous terms why the energy levels of this potential are real, positive, and discrete. Applications of PT -symmetry in quantum field theory are also discussed.}, - Author = {Bender, Carl M}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-02-17 21:17:27 +0100}, - Doi = {10.1088/1742-6596/631/1/012002}, - File = {/Users/loos/Zotero/storage/HGHH8GAU/Bender - 2015 - iPTi -symmetric quantum theory.pdf}, - Issn = {1742-6596}, - Journal = {J. Phys. Conf. Ser.}, - Month = jul, - Pages = {012002}, - Title = {{{{\cal PT}}}-Symmetric Quantum Theory}, - Volume = {631}, - Year = {2015}, - Bdsk-Url-1 = {https://doi.org/10.1088/1742-6596/631/1/012002}} - -@article{Bender_2014, - Abstract = {This paper proposes a very simple perturbative technique to calculate the lowlying eigenvalues and eigenstates of a real one-dimensional parity-symmetric quantum-mechanical potential. Although it is not essential, the potentials studied here are assumed for simplicity to rise monotonically as x $\rightarrow$ $\pm\infty$ and to have Vmin = 0. The technique is to solve the time-independent Schr\"odinger eigenvalue problem as a perturbation series in which the perturbation parameter is the energy itself. Unlike nearly all perturbation series for physical problems, for the ground state this perturbation expansion is convergent and the perturbative results are numerically accurate. The perturbation series is divergent for higher energy levels but can be easily evaluated by using methods such as Pad\'e summation.}, - Author = {Bender, Carl M and Jones, Hugh F}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1088/1751-8113/47/39/395303}, - File = {/Users/loos/Zotero/storage/P2T7JQ5C/Bender and Jones - 2014 - Calculation of low-lying energy levels in quantum .pdf}, + Doi = {10.1088/1751-8113/45/44/444016}, Issn = {1751-8113, 1751-8121}, Journal = {J. Phys. Math. Theor.}, - Month = oct, - Number = {39}, - Pages = {395303}, - Title = {Calculation of Low-Lying Energy Levels in Quantum Mechanics}, - Volume = {47}, - Year = {2014}, - Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/47/39/395303}} - -@article{Bender_2008a, - Abstract = {The coupling of non-Hermitian PT -symmetric Hamiltonians to standard Hermitian Hamiltonians, each of which individually has a real energy spectrum, is explored by means of a number of soluble models. It is found that in all cases the energy remains real for small values of the coupling constant, but becomes complex if the coupling becomes stronger than some critical value. For a quadratic non-Hermitian PT -symmetric Hamiltonian coupled to an arbitrary real Hermitian PT -symmetric Hamiltonian, the reality of the ground-state energy for small enough coupling constant is established up to second order in perturbation theory.}, - Author = {Bender, Carl M and Jones, Hugh F}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1088/1751-8113/41/24/244006}, - File = {/Users/loos/Zotero/storage/WZSJXXBC/Bender and Jones - 2008 - Interactions of Hermitian and non-Hermitian Hamilt.pdf}, - Issn = {1751-8113, 1751-8121}, - Journal = {J. Phys. Math. Theor.}, - Month = jun, - Number = {24}, - Pages = {244006}, - Title = {Interactions of {{Hermitian}} and Non-{{Hermitian Hamiltonians}}}, - Volume = {41}, - Year = {2008}, - Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/41/24/244006}} - -@article{Bender_2008, - Author = {Bender, Carl M and Brody, Dorje C and Hook, Daniel W}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1088/1751-8113/41/35/352003}, - File = {/Users/loos/Zotero/storage/LQRARMQG/Bender et al. - 2008 - Quantum effects in classical systems having comple.pdf}, - Issn = {1751-8113, 1751-8121}, - Journal = {J. Phys. Math. Theor.}, - Month = sep, - Number = {35}, - Pages = {352003}, - Title = {Quantum Effects in Classical Systems Having Complex Energy}, - Volume = {41}, - Year = {2008}, - Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/41/35/352003}} - -@article{Bender_2007a, - Author = {Bender, Carl M. and Brody, Dorje C. and Jones, Hugh F. and Meister, Bernhard K.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-27 20:41:00 +0100}, - Doi = {10.1103/PhysRevLett.98.040403}, - File = {/Users/loos/Zotero/storage/VLGAD94R/Bender et al. - 2007 - Faster than Hermitian Quantum Mechanics.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = jan, - Number = {4}, - Pages = {040403}, - Title = {Faster than {{Hermitian Quantum Mechanics}}}, - Volume = {98}, - Year = {2007}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.98.040403}} - -@article{Bender_2007, - Abstract = {The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probabilitypreserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space\textendash{}time reflection (PT ) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT -symmetric non-Hermitian quantum-mechanical Hamiltonians are H = p\textasciicircum{}2 +ix\textasciicircum{}3 and H = p\textasciicircum{}2 -x\textasciicircum{}4. Amazingly, the energy levels of these Hamiltonians are all real and positive! Does a PT -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. In terms of C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution.}, - Author = {Bender, Carl M}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1088/0034-4885/70/6/R03}, - File = {/Users/loos/Zotero/storage/YLTSAC8U/Bender - 2007 - Making sense of non-Hermitian Hamiltonians.pdf}, - Issn = {0034-4885, 1361-6633}, - Journal = {Rep. Prog. Phys.}, - Month = jun, - Number = {6}, - Pages = {947-1018}, - Title = {Making Sense of Non-{{Hermitian Hamiltonians}}}, - Volume = {70}, - Year = {2007}, - Bdsk-Url-1 = {https://doi.org/10.1088/0034-4885/70/6/R03}} - -@article{Bender_2006, - Abstract = {A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: in which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian ix3 quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT -symmetric Hamiltonian. The O(g4) contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.}, - Author = {Bender, Carl M and Chen, Jun-Hua and Milton, Kimball A}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-21 12:48:11 +0100}, - Doi = {10.1088/0305-4470/39/7/010}, - File = {/Users/loos/Zotero/storage/323A5AEN/Bender et al. - 2006 - -symmetric versus Hermitian formulations of quantu.pdf}, - Issn = {0305-4470, 1361-6447}, - Journal = {J. Phys. Math. Gen.}, - Month = feb, - Number = {7}, - Pages = {1657-1668}, - Title = {{{\cal PT}}-Symmetric versus {{Hermitian}} Formulations of Quantum Mechanics}, - Volume = {39}, - Year = {2006}, - Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/39/7/010}} - -@article{Bender_2005, - Author = {Bender, Carl M}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-25 22:54:23 +0100}, - Doi = {10.1080/00107500072632}, - File = {/Users/loos/Zotero/storage/4J64B62R/Bender - 2005 - Introduction to -symmetric quantum theory.pdf}, - Issn = {0010-7514, 1366-5812}, - Journal = {Contemp. Phys.}, - Month = jul, - Number = {4}, - Pages = {277-292}, - Title = {Introduction to {{\cal PT}}-Symmetric Quantum Theory}, - Volume = {46}, - Year = {2005}, - Bdsk-Url-1 = {https://doi.org/10.1080/00107500072632}} - -@article{Bender_2004, - Abstract = {The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of spacetime reflection (PT ) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT -symmetric non-Hermitian quantum mechanical Hamiltonians are H = p2 + ix3 and H = p2 - x4. The crucial question is whether PT -symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry represented by a linear operator called C. Using C it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are positive, and the dynamics is governed by unitary time evolution. After a review of PT -symmetric quantum mechanics, new results are presented here in which the C operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.}, - Author = {Bender, Carl M and Brod, Joachim and Refig, Andr\'e and Reuter, Moretz E}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-02-17 21:16:38 +0100}, - Doi = {10.1088/0305-4470/37/43/009}, - File = {/Users/loos/Zotero/storage/Z8FQPR54/Bender et al. - 2004 - The Script C operator in Script PScript T-symmetri.pdf}, - Issn = {0305-4470, 1361-6447}, - Journal = {J. Phys. Math. Gen.}, - Month = oct, - Number = {43}, - Pages = {10139-10165}, - Title = {The {{\cal C}} Operator in {{\cal PT}}-Symmetric Quantum Theories}, - Volume = {37}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/37/43/009}} - -@article{Bender_2003, - Author = {Bender, Carl M. and Brody, Dorje C. and Jones, Hugh F.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1119/1.1574043}, - File = {/Users/loos/Zotero/storage/A6QELMC3/Bender et al. - 2003 - Must a Hamiltonian be Hermitian.pdf}, - Issn = {0002-9505, 1943-2909}, - Journal = {Am. J. Phys.}, Month = nov, - Number = {11}, - Pages = {1095-1102}, - Title = {Must a {{Hamiltonian}} Be {{Hermitian}}?}, - Volume = {71}, - Year = {2003}, - Bdsk-Url-1 = {https://doi.org/10.1119/1.1574043}} + Number = {44}, + Pages = {444016}, + Title = {{The Physics of Exceptional Points}}, + Volume = {45}, + Year = {2012}, + Bdsk-Url-1 = {https://doi.org/10.1088/1751-8113/45/44/444016}} -@article{Bender_2002a, - Author = {Bender, Carl M. and Brody, Dorje C. and Jones, Hugh F.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-27 20:42:43 +0100}, - Doi = {10.1103/PhysRevLett.89.270401}, - File = {/Users/loos/Zotero/storage/PE9A74NH/Bender et al. - 2002 - Complex Extension of Quantum Mechanics.pdf}, - Issn = {0031-9007, 1079-7114}, - Journal = {Phys. Rev. Lett.}, - Month = dec, - Number = {27}, - Pages = {270401}, - Title = {Complex {{Extension}} of {{Quantum Mechanics}}}, - Volume = {89}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.89.270401}} - -@article{Bender_2002, - Abstract = {The fact that eigenvalues of PT-symmetric Hamiltonians H can be real for some values of a parameter and complex for others is explained by showing that the matrix elements of H, and hence the secular equation, are real, not only for PT but also for any antiunitary operator A satisfying A2k = 1 with k odd. The argument is illustrated by a 2 \texttimes{} 2 matrix Hamiltonian, and two examples of the generalization are given.}, - Author = {Bender, Carl M and Berry, M V and Mandilara, Aikaterini}, +@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 = {10.1088/0305-4470/35/31/101}, - File = {/Users/loos/Zotero/storage/MGPSZQS8/Bender et al. - 2002 - Generalized PT symmetry and real spectra.pdf}, - Issn = {0305-4470}, + Doi = {10.1088/0305-4470/23/7/022}, Journal = {J. Phys. Math. Gen.}, - Month = aug, - Number = {31}, - Pages = {L467-L471}, - Title = {Generalized {{PT}} Symmetry and Real Spectra}, - Volume = {35}, - Year = {2002}, - Bdsk-Url-1 = {https://doi.org/10.1088/0305-4470/35/31/101}} - -@article{Bender_1999, - Author = {Bender, Carl M. and Boettcher, Stefan and Meisinger, Peter N.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.532860}, - File = {/Users/loos/Zotero/storage/KJ582YAF/Bender et al. - 1999 - PT-symmetric quantum mechanics.pdf}, - Issn = {0022-2488, 1089-7658}, - Journal = {J. Math. Phys.}, - Month = may, - Number = {5}, - Pages = {2201-2229}, - Title = {{{PT}}-Symmetric Quantum Mechanics}, - Volume = {40}, - Year = {1999}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.532860}} - -@article{Benda_2017, - Author = {Benda, Zsuzsanna and Jagau, Thomas-C.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1063/1.4974094}, - File = {/Users/loos/Zotero/storage/Y55EJMJR/Benda and Jagau - 2017 - Communication Analytic gradients for the complex .pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = jan, - Number = {3}, - Pages = {031101}, - Shorttitle = {Communication}, - Title = {Communication: {{Analytic}} Gradients for the Complex Absorbing Potential Equation-of-Motion Coupled-Cluster Method}, - Volume = {146}, - Year = {2017}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.4974094}} - -@article{Anandan_1990, - Author = {Anandan, J. and Aharonov, Y.}, - Date-Added = {2019-01-20 22:03:11 +0100}, - Date-Modified = {2019-01-20 22:03:11 +0100}, - Doi = {10.1103/PhysRevLett.65.1697}, - File = {/Users/loos/Zotero/storage/BXFLJYJT/Anandan and Aharonov - 1990 - Geometry of quantum evolution.pdf}, - Issn = {0031-9007}, - Journal = {Phys. Rev. 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R.}, + Isbn = {978-0-691-02589-6}, + Publisher = {Princeton University Press}, + Title = {Angular {Momentum} in {Quantum} {Mechanics}}, + Year = {1996}} + +@book{SlaterBook, + Author = {Slater, John Clarke}, + 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}, - File = {/Users/loos/Zotero/storage/R9N5FN2F/17.pdf}, - Issn = {1050-2947, 1094-1622}, Journal = {Phys. Rev. A}, + Month = jun, Number = {6}, Pages = {062517}, Title = {Ground state of two electrons on a sphere}, Volume = {79}, Year = {2009}, - Bdsk-Url-1 = {https://dx.doi.org/10.1103/PhysRevA.79.062517}} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.79.062517}} -@article{Loos_2009, - Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. 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Rev.}, Month = dec, - Number = {24}, - Pages = {241101}, - Title = {Correlation Energy of Two Electrons in the High-Density Limit}, - Volume = {131}, - Year = {2009}, - Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.3275519}} + Number = {11}, + Pages = {1002--1011}, + Title = {On the {Interaction} of {Electrons} in {Metals}}, + Volume = {46}, + Year = {1934}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRev.46.1002}} -@article{Loos_2008, - Author = {Loos, Pierre-Fran{\c c}ois and Preat, Julien and Laurent, Ad{\`e}le D. and Michaux, Catherine and Jacquemin, Denis and Perp{\`e}te, Eric A. and Assfeld, Xavier}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1021/ct700188w}, - File = {/Users/loos/Zotero/storage/LSU72HGK/11.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {Journal of Chemical Theory and Computation}, - Month = apr, - Number = {4}, - Pages = {637--645}, - Shorttitle = {Theoretical {{Investigation}} of the {{Geometries}} and {{UV}}-vis {{Spectra}} of {{Poly}}( $<$span Style="font-Variant}, - Title = {Theoretical {{Investigation}} of the {{Geometries}} and {{UV}}-vis {{Spectra}} of {{Poly}}( {\textsc{l}} -Glutamic Acid) {{Featuring}} a {{Photochromic Azobenzene Side Chain}}}, - Volume = {4}, - Year = {2008}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct700188w}} - -@article{Loos_2007d, - Author = {Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1021/ct6003214}, - File = {/Users/loos/Zotero/storage/A9TSDVGQ/6.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {Journal of Chemical Theory and Computation}, - Month = may, - Number = {3}, - Pages = {1047--1053}, - Title = {Self-{{Consistent Strictly Localized Orbitals}}}, - Volume = {3}, - Year = {2007}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct6003214}} - -@article{Loos_2007c, - Author = {Loos, P. F. and Fornili, A. and Sironi, M. and Assfeld, X.}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-11-20 15:11:58 +0100}, - File = {/Users/loos/Zotero/storage/CAHLR8H8/9.pdf}, - Journal = {Computing Letters}, - Number = {2-4}, - Pages = {2--4}, - Shorttitle = {Removing Extra Frontier Parameters in {{QM}}/{{MM}} Methods}, - Title = {Removing Extra Frontier Parameters in {{QM}}/{{MM}} Methods: A Tentative with the {{Local Self}}-{{Consistent Field}} Approach}, - Volume = {3}, - Year = {2007}} - -@inproceedings{Loos_2007b, - Author = {Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier}, - Booktitle = {{{AIP Conference Proceedings}}}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - File = {/Users/loos/Zotero/storage/HGKGXAMT/10.pdf}, - Pages = {308--315}, - Publisher = {{AIP}}, - Title = {On {{The Frontier Bond Location In The QM}}/{{MM Description Of Peptides And Proteins}}.}, - Volume = {963}, - Year = {2007}} - -@article{Loos_2007a, - Author = {Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier and Rivail, Jean-Louis}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1007/s00214-007-0258-x}, - File = {/Users/loos/Zotero/storage/S6FBKP2M/5.pdf}, - Issn = {1432-881X, 1432-2234}, - Journal = {Theor. 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Rev. A}, Month = jun, Number = {6}, - Pages = {1700--1708}, - Title = {Analyzing the {{Selectivity}} and {{Successiveness}} of a {{Two}}-{{Electron Capture}} on a {{Multiply Disulfide}}-{{Linked Protein}}}, - Volume = {5}, + Pages = {062506}, + Title = {Adiabatic connection in density-functional theory: {Two} electrons on the surface of a sphere}, + Volume = {75}, + Year = {2007}, + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.75.062506}} + +@article{Loos_2009b, + Author = {Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.}, + Doi = {10.1103/PhysRevLett.103.123008}, + Journal = {Phys. Rev. Let.}, + Month = sep, + Number = {12}, + Pages = {123008}, + Title = {Two {Electrons} on a {Hypersphere}: {A} {Quasiexactly} {Solvable} {Model}}, + Volume = {103}, Year = {2009}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct900093h}} - -@article{Dumont_2008b, - Author = {Dumont, Elise and Loos, Pierre-Fran{\c c}ois and Laurent, Ad{\`e}le D. and Assfeld, Xavier}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1021/ct800161m}, - File = {/Users/loos/Zotero/storage/AU5GAG48/13.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {Journal of Chemical Theory and Computation}, - Month = aug, - Number = {8}, - Pages = {1171--1173}, - Title = {Huge {{Disulfide}}-{{Linkage}}'{{S Electron Capture Variation Induced}} by $\alpha$-{{Helix Orientation}}}, - Volume = {4}, - Year = {2008}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/ct800161m}} - -@article{Dumont_2008a, - Author = {Dumont, {\'E}lise and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1021/jp806465e}, - File = {/Users/loos/Zotero/storage/5FHSI4FF/14.pdf}, - Issn = {1520-6106, 1520-5207}, - Journal = {The Journal of Physical Chemistry B}, - Month = oct, - Number = {43}, - Pages = {13661--13669}, - Title = {Factors {{Governing Electron Capture}} by {{Small Disulfide Loops}} in {{Two}}-{{Cysteine Peptides}}}, - Volume = {112}, - Year = {2008}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/jp806465e}} - -@article{Dumont_2008, - Author = {Dumont, {\'E}lise and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1016/j.cplett.2008.05.010}, - File = {/Users/loos/Zotero/storage/5Q5WP253/12.pdf}, - Issn = {00092614}, - Journal = {Chem. Phys. Lett.}, - Month = jun, - Number = {4-6}, - Pages = {276--280}, - Title = {Effect of Ring Strain on Disulfide Electron Attachment}, - Volume = {458}, - Year = {2008}, - Bdsk-Url-1 = {https://dx.doi.org/10.1016/j.cplett.2008.05.010}} - -@article{Bernard_2013, - Author = {Bernard, Yves A. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M.W.}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1080/00268976.2013.811302}, - File = {/Users/loos/Zotero/storage/8XRA54GV/39.pdf}, - Issn = {0026-8976, 1362-3028}, - Journal = {Mol. Phys.}, - Month = sep, - Number = {16-17}, - Pages = {2414--2426}, - Title = {Distribution of r {\textsubscript{12}} $\cdot$ p {\textsubscript{12}} in Quantum Systems}, - Volume = {111}, - Year = {2013}, - Bdsk-Url-1 = {https://dx.doi.org/10.1080/00268976.2013.811302}} - -@incollection{Barca_2018, - Author = {Barca, Giuseppe M.J. and Loos, Pierre-Fran{\c c}ois}, - Booktitle = {Advances in {{Quantum Chemistry}}}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1016/bs.aiq.2017.03.004}, - File = {/Users/loos/Zotero/storage/JQKXB9RU/58.pdf}, - Isbn = {978-0-12-813002-5}, - Pages = {147--165}, - Publisher = {{Elsevier}}, - Title = {Recurrence {{Relations}} for {{Four}}-{{Electron Integrals Over Gaussian Basis Functions}}}, - Volume = {76}, - Year = {2018}, - Bdsk-Url-1 = {https://dx.doi.org/10.1016/bs.aiq.2017.03.004}} - -@article{Barca_2017, - Author = {Barca, Giuseppe MJ and Loos, Pierre-Fran{\c c}ois}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - File = {/Users/loos/Zotero/storage/DCFUMHWZ/56.pdf}, - Journal = {The Journal of chemical physics}, - Number = {2}, - Pages = {024103}, - Shorttitle = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals}, - Title = {Three-and Four-Electron Integrals Involving {{Gaussian}} Geminals: {{Fundamental}} Integrals, Upper Bounds, and Recurrence Relations}, - Volume = {147}, - Year = {2017}} - -@article{Barca_2016, - Author = {Barca, Giuseppe M. J. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1021/acs.jctc.6b00130}, - File = {/Users/loos/Zotero/storage/JS793AVF/49.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {Journal of Chemical Theory and Computation}, - Month = apr, - Number = {4}, - Pages = {1735--1740}, - Title = {Many-{{Electron Integrals}} over {{Gaussian Basis Functions}}. {{I}}. {{Recurrence Relations}} for {{Three}}-{{Electron Integrals}}}, - Volume = {12}, - Year = {2016}, - Bdsk-Url-1 = {https://dx.doi.org/10.1021/acs.jctc.6b00130}} - -@article{Ball_2017, - Author = {Ball, Caleb J. and Loos, Pierre-Fran{\c c}ois and Gill, Peter M. W.}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1039/C6CP06801D}, - File = {/Users/loos/Zotero/storage/7X5YE6WH/52.pdf}, - Issn = {1463-9076, 1463-9084}, - Journal = {Physical Chemistry Chemical Physics}, - Number = {5}, - Pages = {3987--3998}, - Title = {Molecular Electronic Structure in One-Dimensional {{Coulomb}} Systems}, - Volume = {19}, - Year = {2017}, - Bdsk-Url-1 = {https://dx.doi.org/10.1039/C6CP06801D}} - -@article{Ambrosek_2010, - Author = {Ambrosek, David and Loos, Pierre-Fran{\c c}ois and Assfeld, Xavier and Daniel, Chantal}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1016/j.jinorgbio.2010.04.002}, - File = {/Users/loos/Zotero/storage/WA85FMNN/22.pdf}, - Issn = {01620134}, - Journal = {Journal of Inorganic Biochemistry}, - Month = sep, - Number = {9}, - Pages = {893--901}, - Title = {A Theoretical Study of {{Ru}}({{II}}) Polypyridyl {{DNA intercalatorsStructure}} and Electronic Absorption Spectroscopy of [{{Ru}}(Phen)2(Dppz)]2+ and [{{Ru}}(Tap)2(Dppz)]2+ Complexes Intercalated in Guanine\textendash{}cytosine Base Pairs}, - Volume = {104}, - Year = {2010}, - Bdsk-Url-1 = {https://dx.doi.org/10.1016/j.jinorgbio.2010.04.002}} - -@article{Agboola_2015, - Author = {Agboola, Davids and Knol, Anneke L. and Gill, Peter M. W. and Loos, Pierre-Fran{\c c}ois}, - Date-Added = {2018-10-16 10:34:56 +0200}, - Date-Modified = {2018-10-16 10:34:56 +0200}, - Doi = {10.1063/1.4929353}, - File = {/Users/loos/Zotero/storage/YFD785CA/46.pdf}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = aug, - Number = {8}, - Pages = {084114}, - Title = {Uniform Electron Gases. {{III}}. {{Low}}-Density Gases on Three-Dimensional Spheres}, - Volume = {143}, - Year = {2015}, - Bdsk-Url-1 = {https://dx.doi.org/10.1063/1.4929353}} - -@article{Boggio-Pasqua_2000, - Abstract = {Realistic analytical representations of the twelve lowest singlet and triplet electronic adiabatic potential energy curves of C2 molecule are given in this article. The corresponding electronic states are correlated with C atoms both in their 3P state. A new set of high level MRCI calculations coupled with a double many-body expansion analytical \textregistered{}tting based on the extended Hartree$\pm$Fock approximate correlation energy model have been used in this work. Using RKR data available in the literature, comparison is made between our results and RKR turning points concerning the four lowest singlet states X1Sg1, A1Pu, B1Dg and BH 1Sg1 of C2. The agreement is very satisfying. q 2000 Elsevier Science B.V. All rights reserved.}, - Author = {{Boggio-Pasqua}, M. and Voronin, A.I. and Halvick, Ph. and Rayez, J.-C.}, - Date-Added = {2018-09-26 14:24:04 +0200}, - Date-Modified = {2018-09-26 14:24:04 +0200}, - Doi = {10.1016/S0166-1280(00)00442-5}, - File = {/Users/loos/Zotero/storage/8BP7KA4E/Boggio-Pasqua et al. - 2000 - Analytical representations of high level ab initio.pdf}, - Issn = {01661280}, - Journal = {Journal of Molecular Structure: THEOCHEM}, - Month = oct, - Number = {1-3}, - Pages = {159-167}, - Title = {Analytical Representations of High Level Ab Initio Potential Energy Curves of the {{C}} 2 Molecule}, - Volume = {531}, - Year = {2000}, - Bdsk-Url-1 = {https://doi.org/10.1016/S0166-1280(00)00442-5}} - -@article{Angeli_2010, - Author = {Angeli, Celestino}, - Date-Added = {2018-09-26 12:14:22 +0200}, - Date-Modified = {2019-01-28 10:14:01 +0100}, - Doi = {10.1002/qua.22597}, - Issn = {00207608, 1097461X}, - Journal = {Int. J. Quantum Chem.}, - Pages = {2436--2447}, - Title = {An Analysis of the Dynamic {{$\sigma$}} Polarization in the {{V}} State of Ethene}, - Year = {2010}, - Bdsk-Url-1 = {https://doi.org/10.1002/qua.22597}} - -@article{Sil08, - Author = {Silva-Junior, M. R. and Schreiber, M. and Sauer, S. P. A. and Thiel, W.}, - Date-Added = {2018-09-25 21:37:00 +0200}, - Date-Modified = {2018-09-25 21:37:00 +0200}, - Journal = {J. Chem. Phys.}, - Pages = {104103}, - Title = {Benchmarks for Electronically Excited States: Time-Dependent Density Functional Theory and Density Functional Theory Based Multireference Configuration Interaction}, - Volume = 129, - Year = 2008} - -@article{Angeli_2008, - Author = {C. Angeli}, - Date-Added = {2018-09-25 13:54:27 +0200}, - Date-Modified = {2018-09-25 13:54:38 +0200}, - Journal = {J. Comput. Chem.}, - Pages = {1319--1333}, - Title = {On the Nature of the π → π∗ Ionic Excited States: The V State of Ethene as a Prototype}, - Volume = {30}, - Year = {2008}} - -@article{Barca_2018a, - Author = {Barca, Giuseppe M. J. and Gilbert, Andrew T. B. and Gill, Peter M. W.}, - Date-Added = {2018-09-25 13:46:34 +0200}, - Date-Modified = {2018-11-20 15:47:24 +0100}, - Doi = {10.1021/acs.jctc.7b00994}, - Journal = {J. Chem. Theory Comput.}, - Month = mar, - Number = {3}, - Pages = {1501}, - Title = {Simple {{Models}} for {{Difficult Electronic Excitations}}}, - Volume = {14}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00994}} - -@article{Barca_2018b, - Author = {Barca, Giuseppe M. J. and Gilbert, Andrew T. B. and Gill, Peter M. W.}, - Date-Added = {2018-09-25 13:46:34 +0200}, - Date-Modified = {2018-11-20 15:46:58 +0100}, - Doi = {10.1021/acs.jctc.7b00963}, - Journal = {J. Chem. Theory Comput.}, - Number = {1}, - Pages = {9}, - Title = {Excitation {{Number}}: {{Characterizing Multiply Excited States}}}, - Volume = {14}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.7b00963}} - -@article{Dallos_2004, - Author = {Dallos, Michal and Lischka, Hans}, - Date-Added = {2018-09-25 13:46:34 +0200}, - Date-Modified = {2018-09-25 13:46:34 +0200}, - Doi = {10.1007/s00214-003-0557-9}, - Issn = {1432-881X, 1432-2234}, - Journal = {Theor. Chem. Acc.}, - Month = apr, - Number = {1}, - Pages = {16-26}, - Title = {A Systematic Theoretical Investigation of the Lowest Valence- and {{Rydberg}}-Excited Singlet States of Trans-Butadiene. {{The}} Character of the 1 1 {{B}} u ({{V}}) State Revisited}, - Volume = {112}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1007/s00214-003-0557-9}} - -@article{Elliott_2011, - Author = {Elliott, Peter and Goldson, Sharma and Canahui, Chris and Maitra, Neepa T.}, - Date-Added = {2018-09-25 13:46:34 +0200}, - Date-Modified = {2018-11-20 14:59:45 +0100}, - Doi = {10.1016/j.chemphys.2011.03.020}, - File = {/Users/loos/Zotero/storage/U6T3LQ8L/Elliott et al. - 2011 - Perspectives on double-excitations in TDDFT.pdf}, - Issn = {03010104}, - Journal = {Chem. Phys.}, - Number = {1}, - Pages = {110}, - Title = {Perspectives on double-excitations in {TDDFT}}, - Volume = {391}, - Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1016/j.chemphys.2011.03.020}} - -@article{Paterson_2006, - Author = {Paterson, Martin J. and Christiansen, Ove and Paw\l{}owski, Filip and J\o{}rgensen, Poul and H\"attig, Christof and Helgaker, Trygve and Sa\l{}ek, Pawe\l}, - Date-Added = {2018-09-25 13:46:34 +0200}, - Date-Modified = {2018-09-25 13:46:34 +0200}, - Doi = {10.1063/1.2163874}, - Issn = {0021-9606, 1089-7690}, - Journal = {J. Chem. Phys.}, - Month = feb, - Number = {5}, - Pages = {054322}, - Title = {Benchmarking Two-Photon Absorption with {{CC3}} Quadratic Response Theory, and Comparison with Density-Functional Response Theory}, - Volume = {124}, - Year = {2006}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.2163874}} - -@article{Levine_2006a, - Author = {Levine, Benjamin G. and Ko, Chaehyuk and Quenneville, Jason and Mart\'Inez, Todd J.}, - Date-Added = {2018-09-25 10:46:52 +0200}, - Date-Modified = {2018-09-25 10:46:52 +0200}, - Doi = {10.1080/00268970500417762}, - Journal = {Mol. Phys.}, - Month = mar, - Pages = {1039-1051}, - Title = {Conical Intersections and Double Excitations in Time-Dependent Density Functional Theory}, - Volume = {104}, - Year = {2006}, - Bdsk-Url-1 = {https://doi.org/10.1080/00268970500417762}} - -@article{Nakayama_1998, - Author = {Nakayama, Kenichi and Nakano, Haruyuki and Hirao, Kimihiko}, - Date-Added = {2018-09-25 10:46:52 +0200}, - Date-Modified = {2018-09-25 10:46:52 +0200}, - Doi = {10.1002/(SICI)1097-461X(1998)66:2<157::AID-QUA7>3.0.CO;2-U}, - Issn = {0020-7608, 1097-461X}, - Journal = {Int. J. Quantum Chem.}, - Number = {2}, - Pages = {157-175}, - Shorttitle = {Theoretical Study of The ?}, - Title = {Theoretical Study of the ???* Excited States of Linear Polyenes: {{The}} Energy Gap between {{11Bu}}+ and {{21Ag}}? 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Lett.}, - Number = {1-3}, - Pages = {39}, - Title = {A dressed {TDDFT} treatment of the {2 1Ag} states of butadiene and hexatriene}, - Volume = {389}, - Year = {2004}, - Bdsk-Url-1 = {https://doi.org/10.1016/j.cplett.2004.03.051}} - -@article{Lange_2018, - Author = {Lange, Malte F. and Berkelbach, Timothy C.}, - Date-Added = {2018-09-13 14:05:12 +0000}, - Date-Modified = {2018-09-13 14:05:12 +0000}, - Doi = {10.1021/acs.jctc.8b00455}, - File = {/Users/loos/Zotero/storage/AKMRCRNU/Lange and Berkelbach - 2018 - On the Relation between Equation-of-Motion Coupled.pdf}, - Issn = {1549-9618, 1549-9626}, - Journal = {J. Chem. Theory Comput.}, - Month = aug, - Number = {8}, - Pages = {4224-4236}, - Title = {On the {{Relation}} between {{Equation}}-of-{{Motion Coupled}}-{{Cluster Theory}} and the {{{\emph{GW}}}} {{Approximation}}}, - Volume = {14}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00455}} - -@article{Loos_2018b, - Author = {P. F. Loos and A. Scemama and A. Blondel and Y. Garniron and M. Caffarel and D. Jacquemin}, - Date-Added = {2018-05-10 18:47:27 +0000}, - Date-Modified = {2018-11-20 15:11:45 +0100}, - Doi = {10.1021/acs.jctc.8b00406}, - Journal = {J. Chem. Theory Comput.}, - Pages = {4360}, - Title = {A Mountaineering Strategy to Excited States: Highly Accurate Reference Energies and Benchmarks}, - Volume = {14}, - Year = {2018}, - Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.8b00406}} - -@article{Seidl_2018, - Author = {Michael Seidl and Sara Giarrusso and Stefan Vuckovic and Eduardo Fabiano and Paola Gori-Giorgi}, - Date-Modified = {2018-11-13 22:51:10 +0100}, - Doi = {10.1063/1.5078565}, - Journal = {J. Chem. Phys.}, - 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}} - -@article{Mihalka_2017, - Author = {Mih\'alka, Zsuzsanna \'E. and Surj\'an, P\'eter R.}, - Date-Modified = {2018-11-13 23:35:53 +0100}, - Doi = {10.1103/PhysRevA.96.062106}, - Journal = {Phys. Rev. A}, - Pages = {062106}, - Title = {Analytic-continuation approach to the resummation of divergent series in {Rayleigh-Schr\"odinger} perturbation theory}, - Volume = {96}, - Year = {2017}, - Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevA.96.062106}} - -@article{QCHEM4, - Author = {{Y. Shao, Z. Gan, E. Epifanovsky, A. T. B. Gilbert, M. Wormit, J. Kussmann, A. W. Lange, A. Behn, J. Deng, X. Feng, \textit{et al.}}}, - Date-Modified = {2018-11-20 15:12:53 +0100}, - Doi = {10.1080/00268976.2014.952696}, - Journal = {Mol.\ Phys.}, - Pages = {184}, - Title = {Advances in molecular quantum chemistry contained in the Q-Chem 4 program package}, - Volume = {113}, - Year = {2015}, - Bdsk-Url-1 = {https://doi.org/10.1080/00268976.2014.952696}} + Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevLett.103.123008}} diff --git a/Manuscript/EPAWTFT.blg b/Manuscript/EPAWTFT.blg index 199a576..81a86b0 100644 --- a/Manuscript/EPAWTFT.blg +++ b/Manuscript/EPAWTFT.blg @@ -1,47 +1,68 @@ -This is BibTeX, Version 0.99d (TeX Live 2019/Arch Linux) -Capacity: max_strings=100000, hash_size=100000, hash_prime=85009 +This is BibTeX, Version 0.99d (TeX Live 2020) +Capacity: max_strings=200000, hash_size=200000, hash_prime=170003 The top-level auxiliary file: EPAWTFT.aux -The style file: unsrt.bst +The style file: apsrev4-1.bst +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated singl_function (elt_size=4) to 100 items from 50. +Reallocated wiz_functions (elt_size=4) to 6000 items from 3000. 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Hubbard dimer}{section*.2}% 5 -\BOOKMARK [1][-]{section*.6}{Concluding remarks}{section*.2}% 6 -\BOOKMARK [1][-]{section*.7}{Acknowledgments}{section*.2}% 7 -\BOOKMARK [1][-]{section*.8}{References}{section*.2}% 8 +\BOOKMARK [2][-]{section*.4}{Background}{section*.3}% 4 +\BOOKMARK [2][-]{section*.5}{An illustrative example}{section*.3}% 5 +\BOOKMARK [1][-]{section*.6}{Perturbation theory}{section*.2}% 6 +\BOOKMARK [2][-]{section*.7}{Rayleigh-Schr\366dinger perturbation theory}{section*.6}% 7 +\BOOKMARK [2][-]{section*.8}{The Hartree-Fock Hamiltonian}{section*.6}% 8 +\BOOKMARK [2][-]{section*.9}{M\370ller-Plesset perturbation theory}{section*.6}% 9 +\BOOKMARK [2][-]{section*.10}{Alternative partitioning}{section*.6}% 10 +\BOOKMARK [1][-]{section*.11}{Historical overview}{section*.2}% 11 +\BOOKMARK [2][-]{section*.12}{Behavior of the M\370ller-Plesset series}{section*.11}% 12 +\BOOKMARK [2][-]{section*.13}{Cases of divergence}{section*.11}% 13 +\BOOKMARK [2][-]{section*.14}{The singularity structure}{section*.11}% 14 +\BOOKMARK [2][-]{section*.15}{The physics of quantum phase transition}{section*.11}% 15 +\BOOKMARK [1][-]{section*.16}{The spherium model}{section*.2}% 16 +\BOOKMARK [1][-]{section*.17}{Radius of convergence and exceptional points}{section*.2}% 17 +\BOOKMARK [2][-]{section*.18}{Evolution of the radius of convergence}{section*.17}% 18 +\BOOKMARK [2][-]{section*.19}{Exceptional points in the UHF formalism}{section*.17}% 19 +\BOOKMARK [1][-]{section*.20}{Conclusion}{section*.2}% 20 +\BOOKMARK [1][-]{section*.21}{Acknowledgments}{section*.2}% 21 +\BOOKMARK [1][-]{section*.22}{References}{section*.2}% 22 diff --git a/Manuscript/EPAWTFT.pdf b/Manuscript/EPAWTFT.pdf index fe1d61a8ff378e5db2bfeacea88d9af40bb18d5c..1fc817097fa408094ec580d4a801517d7613586e 100644 GIT binary patch literal 849408 zcma%?LzgfNu$;%X=N;R&ZQHhO+qP}nwr$(C^Cc&{n?-unzo7fnQ&lAL!lE<`v@B30 z^UEXaPz>~J`1JU8hL%v=+)#AVCbnkI=J+fe% 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zh^ADaAfBy1mFn7|Cx@qF8j82bIC7H{jP}(lSF9pF*SLYEj>nYYoa@{&8|a+!bJT7p zeRkdT`uRD|W7-eELlfh@QcVa~FcRF*vVnfZ;( diff --git a/Manuscript/EPAWTFT.tex b/Manuscript/EPAWTFT.tex index 5c99ae7..286abe8 100644 --- a/Manuscript/EPAWTFT.tex +++ b/Manuscript/EPAWTFT.tex @@ -1,5 +1,5 @@ \documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1} -\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,txfonts,mhchem} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} @@ -20,6 +20,8 @@ \urlstyle{same} \newcommand{\ctab}{\multicolumn{1}{c}{---}} +\newcommand{\ie}{\textit{i.e.}} +\newcommand{\eg}{\textit{e.g.}} \newcommand{\mc}{\multicolumn} \newcommand{\fnm}{\footnotemark} \newcommand{\fnt}{\footnotetext} @@ -48,7 +50,9 @@ \newcommand{\Ne}{N} +\newcommand{\hI}{\Hat{I}} \newcommand{\hH}{\Hat{H}} +\newcommand{\hS}{\Hat{S}} \newcommand{\hT}{\Hat{T}} \newcommand{\hW}{\Hat{W}} \newcommand{\hV}{\Hat{V}} @@ -61,146 +65,704 @@ \newcommand{\updw}{\up\dw} \newcommand{\ra}{\rightarrow} +\newcommand{\pt}{$\mathcal{PT}$} -\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France.} \newcommand{\UCAM}{Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge, CB2 1EW, U.K.} - +\newcommand{\VU}{Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, FEW, Vrije Universiteit, De Boelelaan 1083, 1081HV Amsterdam, The Netherlands.} \begin{document} -\title{Exceptional points and where to find them} +\title{Perturbation theory in the complex plane} +%\title{Exceptional points and where to find them} \author{Antoine \surname{Marie}} -\email{antoine.marie@ens-lyon.fr} \affiliation{\LCPQ} -%\author{Hugh G.~A.~\surname{Burton}} -%\email{hb407@cam.ac.uk} -%\affiliation{\UCAM} +\author{Hugh G.~A.~\surname{Burton}} +\affiliation{\UCAM} +\author{Paola \surname{Gori-Giorgi}} +\affiliation{\VU} \author{Pierre-Fran\c{c}ois \surname{Loos}} -\thanks{Corresponding author} \email{loos@irsamc.ups-tlse.fr} \affiliation{\LCPQ} \begin{abstract} -In this work, we explore the description of quantum chemistry in the complex plane. -Using the Hubbard dimer as a simple model system, we investigate how various methods in electronic structure theory (including Hartree--Fock, \titou{density-functional theory}, perturbation theory and full configuration interaction) are formulated in the complex plane, and how this affects their behaviour in the real plane. -For example, we demonstrate how the position of exceptional points of the real axis controls the convergence of perturbation theory, and explore the inherent dependence on the set of orbital coefficients. -Moreover, we reveal how the avoided crossing of FCI states correspond to an exceptional point in the complex plane, which in turn demonstrates that multiple FCI solutions form a more general structure across the complex plane. -Ultimately, by exposing these more profound topologies of electronic structure methods in the complex plane, we hope to pave the way for novel methodologies. +In this work, we explore the extension of quantum chemistry in the complex plane. We observe that the physics of a quantum system is intimately connected to the position of the energy singularities in the complex plane. After a brief presentation of the fundamental notions of quantum chemistry and perturbation theory in the complex plane, we provide a historical overview of the various research activities that have been performed on the physic of singularities. Then we connect and further discuss these different aspects using the spherium model (\ie, two opposite-spin electrons restricted to remain on the surface of a sphere) as a theoretical playground. In particular, we explore various perturbative partitioning strategies and the effects of symmetry breaking on the singularity structure of the electronic energy. This provides fundamental insights on the location of these singularities in the complex plane (that one calls exceptional points) and, specifically, on the magnitude of the radius of convergence associated with the perturbative treatment. \end{abstract} \maketitle %============================================================% \section{Introduction} +\label{sec:intro} %============================================================% -The notion of quantised energy levels is a central feature of Hermitian quantum mechanics. -In quantum chemistry, the ordering of the energy levels represents the different electronic states of a molecule, the lowest being the ground state while the higher ones are the so-called excited states. -Within this quantised paradigm, electronic states look completely disconnected from one another. -However, one can gain a different perspective on quantisation if one extends quantum chemistry into the complex domain. -In a non-Hermitian complex picture, the energy levels are \textit{sheets} of a more complicated topological manifold called \textit{Riemann surface}, and they are smooth and continuous \textit{analytic continuation} of one another. -In other words, our view of the quantised nature of conventional Hermitian quantum mechanics arises only from our limited perception of the more complex and profound structure of its non-Hermitian variant. +\subsection{Background} -Therefore, by analytically continuing the energy $E(\lambda)$ in the complex domain (where $\lambda$ is a coupling parameter), one can smoothly connect the ground and excited states of a molecule. -This connection is possible because, by extending real numbers to the complex domain, one loses the ordering property of real numbers. -Hence, one can interchange electronic states away from the real axis, as the concept of ground and excited states has been lost. -Amazingly, this smooth and continuous transition from one state to another has been recently realised experimentally in physical settings such as electronics, microwaves, mechanics, acoustics, atomic systems and optics. \cite{Bittner_2012, Chong_2011, Chtchelkatchev_2012, Doppler_2016, Guo_2009, Hang_2013, Liertzer_2012, Longhi_2010, Peng_2014, Peng_2014a, Regensburger_2012, Ruter_2010, Schindler_2011, Szameit_2011, Zhao_2010, Zheng_2013, Choi_2018, El-Ganainy_2018} +Due to the ubiquitous influence of processes involving electronic excited states in physics, chemistry, and biology, their faithful description from first principles has been one of the grand challenges faced by theoretical chemists since the dawn of computational chemistry. +Accurately predicting ground- and excited-state energies (hence excitation energies) is particularly valuable in this context, and it has concentrated most of the efforts within the community. +An armada of theoretical and computational methods have been developed to this end, each of them being plagued by its own flaws. +The fact that none of these methods is successful in every chemical scenario has encouraged chemists to carry on the development of new excited-state methodologies, their main goal being to get the most accurate excitation energies (and properties) at the lowest possible computational cost in the most general context. +One common feature of all these methods is that they rely on the notion of quantized energy levels of Hermitian quantum mechanics, in which the different electronic states of a molecule or an atom are energetically ordered, the lowest being the ground state while the higher ones are excited states. +Within this quantized paradigm, electronic states look completely disconnected from one another. +Many current methods study excited states using only ground-state information, creating a ground-state bias that leads to incorrect excitation energies. +However, one can gain a different perspective on quantization extending quantum chemistry into the complex domain. +In a non-Hermitian complex picture, the energy levels are \textit{sheets} of a more complicated topological manifold called \textit{Riemann surface}, and they are smooth and continuous \textit{analytic continuation} of one another. +In other words, our view of the quantized nature of conventional Hermitian quantum mechanics arises only from our limited perception of the more complex and profound structure of its non-Hermitian variant. \cite{MoiseyevBook,BenderPTBook} +The realization that ground and excited states both emerge from one single mathematical structure with equal importance suggests that excited-state energies can be computed from first principles in their own right. +One could then exploit the structure of these Riemann surfaces to develop methods that directly target excited-state energies without needing ground-state information. \cite{Burton_2019} -Exceptional points (EPs) \cite{Heiss_1990, Heiss_1999, Heiss_2012, Heiss_2016} are non-Hermitian analogs of conical intersections (CIs) \cite{Yarkony_1996} where two states become exactly degenerate. -CIs are ubiquitous in non-adiabatic processes and play a key role in photochemical mechanisms. -In the case of auto-ionizing resonances, EPs have a role in deactivation processes similar to CIs in the decay of bound excited states. -Although Hermitian and non-Hermitian Hamiltonians are closely related, the behaviour of their eigenvalues near degeneracies is starkly different. -For example, by encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. -Additionally, the wave function picks up a geometric phase (also known as Berry phase \cite{Berry_1984}) and four loops are required to recover the starting wave function. -In contrast, encircling Hermitian degeneracies at CIs introduces only a geometric phase while leaving the states unchanged. -More dramatically, whilst eigenvectors remain orthogonal at CIs, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a \textit{self-orthogonal} state. \cite{MoiseyevBook} -More importantly here, although EPs usually lie off the real axis, these singular points are intimately related to the convergence properties of perturbative methods and avoided crossing on the real axis are indicative of singularities in the complex plane. \cite{Olsen_1996, Olsen_2000} +By analytically continuing the electronic energy $E(\lambda)$ in the complex domain (where $\lambda$ is a coupling parameter), the ground and excited states of a molecule can be smoothly connected. +This connection is possible because by extending real numbers to the complex domain, the ordering property of real numbers is lost. +Hence, electronic states can be interchanged away from the real axis since the concept of ground and excited states has been lost. +Amazingly, this smooth and continuous transition from one state to another has recently been experimentally realized in physical settings such as electronics, microwaves, mechanics, acoustics, atomic systems and optics. \cite{Bittner_2012, Chong_2011, Chtchelkatchev_2012, Doppler_2016, Guo_2009, Hang_2013, Liertzer_2012, Longhi_2010, Peng_2014, Peng_2014a, Regensburger_2012, Ruter_2010, Schindler_2011, Szameit_2011, Zhao_2010, Zheng_2013, Choi_2018, El-Ganainy_2018} +Exceptional points (EPs) are branch point singularities where two (or more) states become exactly degenerate. \cite{Heiss_1990,Heiss_1999,Heiss_2012,Heiss_2016} +They are the non-Hermitian analogs of conical intersections, \cite{Yarkony_1996} which are ubiquitous in non-adiabatic processes and play a key role in photo-chemical mechanisms. +In the case of auto-ionizing resonances, EPs have a role in deactivation processes similar to conical intersections in the decay of bound excited states. \cite{Benda_2018} +Although Hermitian and non-Hermitian Hamiltonians are closely related, the behavior of their eigenvalues near degeneracies is starkly different. +For example, encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. +Additionally, the wave function picks up a geometric phase (also known as Berry phase \cite{Berry_1984}) and four loops are required to recover the initial wave function. +In contrast, encircling Hermitian degeneracies at conical intersections only introduces a geometric phase while leaving the states unchanged. +More dramatically, whilst eigenvectors remain orthogonal at conical intersections, at non-Hermitian EPs the eigenvectors themselves become equivalent, resulting in a \textit{self-orthogonal} state. \cite{MoiseyevBook} +More importantly here, although EPs usually lie off the real axis, these singular points are intimately related to the convergence properties of perturbative methods and avoided crossing on the real axis are indicative of singularities in the complex plane. \cite{BenderBook,Olsen_1996,Olsen_2000,Olsen_2019} -%============================================================% -\section{Perturbation Theory} -%============================================================% +\subsection{An illustrative example} +In order to highlight the general properties of EPs mentioned above, we propose to consider the following $2 \times 2$ Hamiltonian commonly used in quantum chemistry -Within perturbation theory, the Schr\"odinger equation is usually rewritten as \begin{equation} - \bH \Psi(\lambda) = (\bH^{(0)} + \lambda \bH^{(1)} ) \Psi(\lambda) = E(\lambda) \Psi(\lambda), -\end{equation} -with -\begin{equation} - E(\lambda) = \sum_{k=0}^\infty \lambda^k E^{(k)}. -\end{equation} -However, it is not unusual that the series $E(\lambda)$ has a radius of convergence $\abs{\lambda_0} < 1$. -This means that the series is divergent in the domain $\abs{\lambda} < \abs{\lambda_0}$, hence for the physical system at $\lambda = 1$. -As eluded above, $\abs{\lambda_0}$ is determined by the location of the singularity of $E(\lambda)$ closest to the origin. -These singularities are nothing but EPs at $\lambda_0$ and $\lambda_0^*$. -Here, we propose to thoroughly investigate the connection between Coulson-Fisher quasi-EPs and the radius of convergence of various flavours of perturbation theory. -For example, M{\o}ller-Plesset perturbation theory (MPPT) has the particularity of relying on a Hartree-Fock (HF) wave function as a zeroth-order wave function. -However, the flavour of HF one can select (restricted, unrestricted, generalised, holomorphic, \ldots) is up for grabs, and the convergence properties of the MPPT series will drastically change depending on this choice. -Indeed, the radius of convergence is intimately connected to the location of singularities in the complex plane; these singularities are, themselves, linked to the choice of $\bH^{(0)}$. -Really, it depends on our ability of selecting a zeroth-order Hamiltonian such as $\bH$ does not have any EP inside the unit $\lambda$ circle. - -For example, MPPT calculations based on UHF wave functions have shown to be slowly convergent due to spin contamination while RHF-based MPPT calculations can be divergent. \cite{Gill_1986, Gill_1988} -Although MPPT is widespread in the community, its convergence properties have not, to the very best of our knowledge, attracted much attention. \cite{Olsen_1996, Olsen_2000, Goodson_2012} -We believe that they deserve greater understanding, particularly in a non-Hermitian setting. - - -%============================================================% -\section{The asymmetric Hubbard dimer} -%============================================================% - - -The asymmetric Hubbard dimer is a model two-electron system whose Hamiltonian reads -\begin{equation} - \hH = \hT + \hW + \hV, -\end{equation} -with -\begin{subequations} -\begin{align} - \hT & = - t \sum_{\sigma=\updw} (\hc{1\sigma}{\dag} \hc{2\sigma}{} + \hc{2\sigma}{\dag} \hc{1\sigma}{}), - \\ - \hW & = U (\hn{1\up}\hn{1\dw} + \hn{2\up}\hn{2\dw}), - \\ - \hV & = \Dv (\hn{2} - \hn{1})/2, -\end{align} -\end{subequations} -where $\hn{i\sigma} = \hc{i\sigma}{\dag} \hc{i\sigma}{}$, $\hn{i} = \sum_{\sigma=\updw} \hn{i\sigma}$, $U$ is the site energy, $t$ is hopping parameter, and $\Dv$ is the difference of on-site potential which controls the asymmetry of the model. -Note that $\n{1} + \n{2} = \Ne$, where $\Ne = 2$. - -In the basis $\ket{1\up1\dw}$, $\ket{1\up2\dw}$, $\ket{2\up1\dw}$, $\ket{2\up2\dw}$, the Hamiltonian reads -\begin{equation} - \hH = +\label{eq:H_2x2} + \bH = \begin{pmatrix} - - \Dv + U & -t & t & 0 \\ - -t & 0 & 0 & -t \\ - t & 0 & 0 & t \\ - 0 & -t & t & \Dv + U \\ + \epsilon_1 & \lambda \\ + \lambda & \epsilon_2 \end{pmatrix}, \end{equation} -which are composed by three singlet and one triplet states. -The ground-state singlet wave function can therefore be written as -\begin{equation} - \ket*{^{1}\Psi} = \alpha \frac{\ket{1\up2\dw} + \ket{2\up1\dw}}{\sqrt{2}} + \beta \ket{1\up1\dw} + \gamma \ket{2\up2\dw} -\end{equation} -with $\alpha^2 + \beta^2 + \gamma^2 = 1$. +which represents two states of energies $\epsilon_1$ and $\epsilon_2$ coupled with a strength of magnitude $\lambda$. +This Hamiltonian could represent, for example, a minimal-basis configuration interaction with doubles (CID) for the hydrogen molecule. \cite{SzaboBook} + +For real $\lambda$, the Hamiltonian \eqref{eq:H_2x2} is clearly Hermitian, and it becomes non-Hermitian for any complex $\lambda$ value. +Its eigenvalues are +\begin{equation} +\label{eq:E_2x2} + E_{\pm} = \frac{\epsilon_1 + \epsilon_2}{2} \pm \frac{1}{2} \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2}. +\end{equation} +%and they are represented as a function of $\lambda$ in Fig.~\ref{fig:2x2}. +One notices that the two states become degenerate only for a pair of complex conjugate values of $\lambda$ +\begin{equation} +\label{eq:lambda_EP} + \lambda_\text{EP} = \pm i\,\frac{\epsilon_1 - \epsilon_2}{2}, +\end{equation} +with energy +\begin{equation} +\label{eq:E_EP} + E_\text{EP} = \frac{\epsilon_1 + \epsilon_2}{2}, +\end{equation} +which correspond to square-root singularities in the complex-$\lambda$ plane. % (see Fig.~\eqref{fig:2x2}). +These two $\lambda$ values, given by Eq.~\eqref{eq:lambda_EP}, are the so-called EPs and one can clearly see that they connect the ground and excited states. +Starting from $\lambda_\text{EP}$, two square-root branch cuts run on the imaginary axis towards $\pm i \infty$. +In the real $\lambda$ axis, the point for which the states are the closest ($\lambda = 0$) is called an avoided crossing and this occurs at $\lambda = \Re(\lambda_\text{EP})$. +The ``shape'' of the avoided crossing is linked to the magnitude of $\Im(\lambda_\text{EP})$: the smaller $\Im(\lambda_\text{EP})$, the sharper the avoided crossing is. + +Around $\lambda = \lambda_\text{EP}$, Eq.~\eqref{eq:E_2x2} behaves as \cite{MoiseyevBook} +\begin{equation} %\label{eq:E_EP} + E_{\pm} = E_\text{EP} \pm \sqrt{2\lambda_\text{EP}} \sqrt{\lambda - \lambda_\text{EP}}, +\end{equation} +and following a complex contour around the EP, \ie, $\lambda = \lambda_\text{EP} + R \exp(i\theta)$, yields +\begin{equation} + E_{\pm}(\theta) = E_\text{EP} \pm \sqrt{2\lambda_\text{EP} R} \exp(i\theta/2), +\end{equation} +and we have +\begin{align} + E_{\pm}(2\pi) & = E_{\mp}(0), + & + E_{\pm}(4\pi) & = E_{\pm}(0). \notag +\end{align} +This evidences that encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy. +Additionally, the wave function picks up a geometric phase (also known as Berry phase \cite{Berry_1984}) and four loops are required to recover the starting wave function. +\cite{MoiseyevBook} + +%The eigenvectors associated to the eigenenergies \eqref{eq:E_2x2} are +%\begin{equation}\label{eq:phi_2x2} +%\begin{split} +% \phi_{\pm} +% & = +% \begin{pmatrix} +% (\epsilon_1 - \epsilon_2 \pm \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2})/2\lambda +% \\ +% 1 +% \end{pmatrix} +% \\ +% = +% \begin{pmatrix} +% (\epsilon_1-\epsilon_2)/2\lambda \pm \sqrt{2\lambda_\text{EP}} \sqrt{\lambda - \lambda_\text{EP}}/\lambda +% \\ +% 1 +% \end{pmatrix}, +%\end{split} +%\end{equation} +%and, for $\lambda=\lambda_\text{EP}$, they become +%\begin{equation} +% \phi_{\pm} = \begin{pmatrix} \mp i \\ 1\end{pmatrix}, +%\end{equation} +%which are clearly self-orthogonal, \ie, their norm is equal to zero. +% +%As branch point (square-root) singularities, EPs inherit their topology from the multi-valued function $\sqrt{\lambda - \lambda_\text{EP}}$. +%Then, if the eigenvectors are properly normalized, they behave as $(\lambda - \lambda_\text{EP})^{-1/4}$, which gives the following pattern when looping around one EP: +%\begin{align} +% \phi_{\pm}(2\pi) & = \phi_{\mp}(0), +% & +% \phi_{\pm}(4\pi) & = -\phi_{\pm}(0), \notag +% \\ +% \phi_{\pm}(6\pi) & = -\phi_{\mp}(0), +% & +% \phi_{\pm}(8\pi) & = \phi_{\pm}(0).\notag +%\end{align} +%In plain words, four loops around the EP are necessary to recover the initial state. +%We can also see that looping the other way around leads to a different pattern. + +%============================================================% +\section{Perturbation theory} +%============================================================% + +\subsection{Rayleigh-Schr\"odinger perturbation theory} + +Within the Born-Oppenheimer approximation, +\begin{equation}\label{eq:ExactHamiltonian} + \hH = - \frac{1}{2} \sum_{i}^{n} \grad_i^2 - \sum_{i}^{n} \sum_{A}^{N} \frac{Z_A}{\abs{\vb{r}_i-\vb{R}_A}} + \sum_{i3/2$ the sb-UHF solution is the global minimum of the HF equations and the RHF solution presented before is a local minimum. This solution corresponds to the configuration with the spin-up electron in an orbital on one side of the sphere and the spin-down electron in a miror-image orbital on the opposite side and the configuration the other way round. The electrons can be on opposite sides of the sphere because the choice of p\textsubscript{z} as a basis function induced a privileged axis on the sphere for the electrons. For $R>3/2$, this solution has the energy +\begin{equation}\label{eq:EsbUHF} + E_{\text{sb-UHF}}=-\frac{75}{112R^3}+\frac{25}{28R^2}+\frac{59}{84R}. +\end{equation} + +The exact solution for the ground state is a singlet. The spherical harmonics are eigenvectors of $\hS^2$ (the spin operator) and they are associated to different eigenvalues. Yet, the symmetry-broken orbitals are linear combinations of $Y_0$ and $Y_1$. Hence, the symmetry-broken orbitals are not eigenvectors of $\hS^2$. However, this solution gives lower energies than the RHF one at large $R$ as shown in Table \ref{tab:ERHFvsEUHF} even if it does not have the exact spin symmetry. In fact, at the Coulson-Fischer point, it becomes more effective to minimize the Coulomb repulsion than the kinetic energy in order to minimize the total energy. Thus, within the HF approximation, the variational principle is allowed to break the spin symmetry because it yields a more effective minimization of the Coulomb repulsion. This type of symmetry breaking is called a spin density wave because the system oscillates between the two symmetry-broken configurations. \cite{GiulianiBook} + +\begin{table*} +\caption{RHF and sb-UHF energies in the minimal basis and exact energies (in the complete basis) for various $R$. +\label{tab:ERHFvsEUHF}} +\begin{ruledtabular} +\begin{tabular}{ccccccccc} +$R$ & 0.1 & 1 & 2 & 3 & 5 & 10 & 100 & 1000 \\ +\hline +RHF & 10.00000 & 1.000000 & 0.500000 & 0.333333 & 0.200000 & 0.100000 & 0.010000 & 0.001000 \\ +UHF & 10.00000 & 1.000000 & 0.490699 & 0.308532 & 0.170833 & 0.078497 & 0.007112 & 0.000703 \\ +Exact & 9.783874 & 0.852781 & 0.391959 & 0.247898 & 0.139471 & 0.064525 & 0.005487 & 0.000515 \\ +\end{tabular} +\end{ruledtabular} +\end{table*} + +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). The reasoning is counter-intuitive because the electrons tends to maximize their energy. The p\textsubscript{z} orbital is symmetric with respect to the center of the sphere. 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 \ie, on the same side of the sphere gives a superior energy than the p\textsubscript{z}\textsuperscript{2} state. Adding a s orbital on one side of the p\textsubscript{z} orbital to form a symmetry-broken orbital reduce the kinetic energy but increase the repulsion energy as the two electrons are more localized on one side of the sphere. 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}. +\end{equation} +\begin{figure} + \includegraphics[width=\linewidth]{EsbHF.pdf} + \caption{Energies of the five solutions of the HF equations (multiplied by $R^2$). The dotted curves correspond to the analytic continuation of the symmetry-broken solutions.} + \label{fig:SpheriumNrj} +\end{figure} + +We can also consider negative values of $R$, which corresponds to the situation where one of the electrons is replaced by a positron (the anti-particle of the electron) as readily seen in Eq.~\eqref{eq:H-sph-omega}. For negative $R$ values, there are also a sb-RHF ($R<-3/2$) and a sb-UHF ($R<-75/38$) solution for negative values of $R$ (see Fig.~\ref{fig:SpheriumNrj}) but in this case the sb-RHF solution is a minimum and the sb-UHF is a maximum of the HF equations. Indeed, the sb-RHF state minimizes the attraction energy by placing the electron and the positron on the same side of the sphere. And the sb-UHF state maximizes the energy because the two attracting particles are on opposite sides of the sphere. + +In addition, we can also consider the symmetry-broken solutions beyond their respective Coulson-Fischer points by analytically continuing their respective energies leading to the so-called holomorphic solutions. \cite{Hiscock_2014, Burton_2019, Burton_2019a} All those energies are plotted in Fig.~\ref{fig:SpheriumNrj}. The dotted curves corresponds to the holomorphic domain of the energies. + + +\section{Radius of convergence and exceptional points} + +\subsection{Evolution of the radius of convergence} + +In this subsection, we investigate how the partitioning of $\hH(\lambda)$ influence the radius of convergence of the perturbation series. Let us remind the reader that the radius of convergence is equal to the distance of the closest singularity to the origin of $E(\lambda)$. Hence, we have to determine the locations of the EPs to obtain information on the convergence properties of the perturbative series. To find them we solve simultaneously the following equations: \cite{Cejnar_2007} +\begin{subequations} +\begin{align} + \label{eq:PolChar} + \det[E\hI-\hH(\lambda)] & = 0, + \\ + \label{eq:DPolChar} + \pdv{E}\det[E\hI-\hH(\lambda)] & = 0, +\end{align} +\end{subequations} +where $\hI$ is the identity operator. +Equation \eqref{eq:PolChar} is the well-known secular equation providing us with the (eigen)energies of the system. If an energy is also solution of Eq.~\eqref{eq:DPolChar}, then this energy is, at least two-fold degenerate. In this case the energies obtained are dependent of $\lambda$. Thus, solving these equations with respect to $E$ and $\lambda$ gives the value of $\lambda$ where two energies are degenerate. These degeneracies can be conical intersections between two states with different symmetry for real values of $\lambda$ \cite{Yarkony_1996} or EPs between two states with the same symmetry for complex values of $\lambda$. + +Let us assume that electron 1 is spin-up and electron 2 is spin-down. +Hence, we can forget about the spin part of the spin-orbitals and from now on we will work with spatial orbitals. In the restricted formalism the spatial orbitals are the same so the two-electron basis set can be defined as + +\begin{align}\label{eq:rhfbasis} + \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). +\end{align} +The Hamiltonian $\hH(\lambda)$ is block diagonal in this basis because of its symmetry, \ie, $\psi_1$ only interacts with $\psi_4$, and $\psi_2$ with $\psi_3$. The two singly-excited states yield, after diagonalization, a spatially anti-symmetric singlet $sp_z$ and a spatially symmetric triplet sp\textsubscript{z} state. Hence those states do not have the same symmetry as the spatially symmetric singlet ground state. Thus, these states cannot be involved in an avoided crossing with the ground state as can be seen in Fig.~\ref{fig:RHFMiniBas} and, \textit{a fortiori} cannot be involved in an EP with the ground state. However there is an avoided crossing between the s\textsuperscript{2} and p\textsubscript{z}\textsuperscript{2} states which gives two EPs in the complex plane. + +\begin{figure} + \includegraphics[width=\linewidth]{EMP_RHF_R10.pdf} + \caption{Energies $E(\lambda)$ in the restricted basis set \eqref{eq:rhfbasis} with $R=10$. + One can clearly see the avoided crossing between the s\textsuperscript{2} and p\textsubscript{z}\textsuperscript{2} states around $\lambda = 1$.} + \label{fig:RHFMiniBas} +\end{figure} + +To simplify the problem, it is convenient to only consider basis functions of a given symmetry. Such basis functions are called configuration state functions (CSFs). It simplifies greatly the problem because, with such a basis set, one only gets the degeneracies of interest associated with the convergence properties, \ie, the EPs between states with the same symmetry as the ground state. In the present context, the ground state is a totally symmetric singlet. According to angular momentum theory, \cite{AngularBook, SlaterBook, Loos_2009} we expand the exact wave function in the following two-electron basis: +\begin{equation} +\Phi_\ell(\omega)=\frac{\sqrt{2\ell+1}}{4\pi R^2}P_\ell(\cos\omega), +\end{equation} +where $P_\ell$ are Legendre polynomials. + +Then, using this basis set we can compare the different partitioning of Sec.~\ref{sec:AlterPart}. Figure \ref{fig:RadiusPartitioning} shows the evolution of the radius of convergence $R_{\text{CV}}$ as a function of $R$ for the MP, the EN, the WC and the SC partitioning in a minimal basis (\ie, consisting of $P_0$ and $P_1$) of size $K = 2$, and in the same basis augmented with $P_2$ ($K = 3$). We see that, for the SC partitioning, $R_{\text{CV}}$ increases with $R$ whereas it is decreasing for the three others partitioning. This result is expected because the MP, EN, and WC partitioning use a weakly correlated reference so $\hH^{(0)}$ is a good approximation for small $R$. On the contrary, the SC partitioning consider naturally a strongly correlated reference so the SC series converges far better when the electron are strongly correlated, \ie, when $R$ is large in the spherium model. + +Interestingly, the radius of convergence associated with the SC partitioning is greater than one for a greater range of radii for $K = 2$ than $K = 3$. + +\begin{figure} + \includegraphics[width=0.49\textwidth]{PartitioningRCV2.pdf} + \includegraphics[width=0.49\textwidth]{PartitioningRCV3.pdf} + \caption{Radius of convergence $R_{\text{CV}}$ for two (left) and three (right) basis functions for various partitionings.} + \label{fig:RadiusPartitioning} +\end{figure} + +The MP partitioning is always better than WC in Fig.~\ref{fig:RadiusPartitioning}. In the WC partitioning the powers of $R$ (the zeroth-order scales as $R^{-2}$ while the perturbation scales as $R^{-1}$) are well-separated so each term of the series has a well-defined power of $R$. This is not the case for the MP series. +Interestingly, it can be proved that the $m$th order energy of the WC series can be obtained as a Taylor series of MP$m$ with respect to $R$. +It seems that the EN is better than MP for very small $R$ in the minimal basis. In fact, it is just an artefact of the minimal basis because, for $K = 3$, the MP series has a greater radius of convergence for all values of $R$. It holds true for $K>3$. + + +Figure \ref{fig:RadiusBasis} shows that the radius of convergence is not very sensitive to the size of the basis set. The CSFs have all the same spin and spatial symmetries so we expect that the singularities obtained within this basis set will be $\alpha$ singularities. Table \ref{tab:SingAlpha} shows that the singularities considered in this case are indeed $\alpha$ singularities. This is consistent with the observation of Goodson and Sergeev \cite{Goodson_2004} who stated that $\alpha$ singularities are relatively insensitive to the basis set size. The discontinuities observed in Fig.~\ref{fig:RadiusBasis} for the MP partitioning are due to changes in dominant singularity. We can observe this change in Table \ref{tab:SingAlpha}, the value for $R=1$ and $R=2$ are respectively in the positive and negative planes. + +\begin{figure} + \includegraphics[width=0.49\textwidth]{MPlargebasis.pdf} + \includegraphics[width=0.49\textwidth]{WCElargebasis.pdf} + \caption{Radius of convergence $R_{\text{CV}}$ in the CSF basis with $K$ basis functions for the MP (left) and WC (right) partitioning.} + \label{fig:RadiusBasis} +\end{figure} + +\begin{table} +\footnotesize +\caption{Dominant singularity in the CSF basis set ($K=8$) for various value of $R$ in the MP and WC partitioning.} +\begin{tabular}{cccccccc} +\hline +\hline +$R$ & 0.1 & 1 & 2 & 3 & 5 & 10 & 100 \\ +\hline +MP & $+14.1-10.9\,i$ & $+2.38-1.47\,i$ & $-0.67-1.30\,i$ & $-0.49-0.89\,i$ & $-0.33-0.55\,i$ & $-0.22-0.31\,i$ & $+0.03-0.05\,i$ \\ +WC & $-9.6-10.7\,i$ & $-0.96-1.07\,i$ & $-0.48-0.53\,i$ & $-0.32-0.36\,i$ & $-0.19-0.21\,i$ & $-0.10-0.11\,i$ & $-0.01-0.01\,i$ \\ +\hline +\hline +\end{tabular} +\label{tab:SingAlpha} +\end{table} + +\subsection{Exceptional points in the UHF formalism}\label{sec:uhfSing} + +Now, we investigate the differences in the singularity structure between the RHF and UHF formalism. To do so, we use the symmetry-broken orbitals discussed in Sec.~\ref{sec:spherium}. Thus, the UHF two-electron basis is +\begin{align}\label{eq:uhfbasis} + \psi_1 & =\phi_{\alpha,1}(\theta_1)\phi_{\beta,1}(\theta_2), + & + \psi_2 & =\phi_{\alpha,1}(\theta_1)\phi_{\beta,2}(\theta_2),\\ + \psi_3 & =\phi_{\alpha,2}(\theta_1)\phi_{\beta,1}(\theta_2), + & + \psi_4 & =\phi_{\alpha,2}(\theta_1)\phi_{\beta,2}(\theta_2). +\end{align} +with the symmetry-broken orbitals +\begin{subequations} +\begin{align}\label{eq:uhforbitals} + \phi_{\alpha,1}(\theta) + & =\frac{\sqrt{75+62R}}{4\sqrt{7R}} Y_{0}(\theta) + + \frac{5\sqrt{-3+2R}}{4\sqrt{7R}} Y_{1}(\theta), + \\ + \phi_{\beta,1}(\theta) + & =\frac{\sqrt{75+62R}}{4\sqrt{7R}} Y_{0}(\theta) + - \frac{5\sqrt{-3+2R}}{4\sqrt{7R}} Y_{1}(\theta), + \\ + \phi_{\alpha,2}(\theta) + & = - \frac{5\sqrt{-3+2R}}{4\sqrt{7R}} Y_{0}(\theta) + + \frac{\sqrt{75+62R}}{4\sqrt{7R}} Y_{1}(\theta), + \\ + \phi_{\beta,2}(\theta) + & =\frac{5\sqrt{-3+2R}}{4\sqrt{7R}} Y_{0}(\theta) + +\frac{\sqrt{75+62R}}{4\sqrt{7R}} Y_{1}(\theta). +\end{align} +\end{subequations} + +In the UHF formalism the Hamiltonian $\hH(\lambda)$ is no more block diagonal, $\psi_4$ can interact with $\psi_2$ and $\psi_3$. The matrix elements of the Hamiltonian corresponding to this interaction are +\begin{equation}\label{eq:MatrixElem} + H_{24}=H_{34}=H_{42}=H_{43}=\sqrt{R-\frac{3}{2}}\sqrt{R+\frac{75}{62}}\qty(R+\frac{25}{2})\frac{\sqrt{31}}{70R^3} +\end{equation} + +For $R=3/2$ the Hamiltonian is block diagonal because the matrix elements \eqref{eq:MatrixElem} are equal to zero so this is equivalent to the RHF case but for $R>3/2$ the matrix elements become real. This interaction corresponds to the spin contamination of the wave function. For $R<3/2$ the matrix elements are complex, this corresponds to the holomorphic solution of Fig.~\ref{fig:SpheriumNrj}, the singularities in this case will be treated later. The matrix elements become real again for $R<-75/62$, this corresponds to the sb-UHF solution for negative value of $R$ observed in Sec.~\ref{sec:spherium}. We will refer to the domain where the matrix elements are complex as the holomorphic domain. + +The singularity structure in this case is more complex because of the spin contamination of the wave function. We can not use CSFs in this case. So when one compute all the degeneracies using Eqs.~\eqref{eq:PolChar} and \eqref{eq:DPolChar} some correspond to EPs and some correspond to conical intersections. The numerical distinction of those singularities is very difficult. We will first look at the energies $E(\lambda)$ obtained with this basis set to attribute a physical signification to the singularities obtained numerically. +Figure \ref{fig:UHFMiniBas} is the analog of Fig.~\ref{fig:RHFMiniBas} in the UHF formalism. We see that in this case the sp\textsubscript{z} triplet interacts with the s\textsuperscript{2} and the p\textsubscript{z}\textsuperscript{2} singlets. Those avoided crossings are due to the spin contamination of the wave function. + +\begin{figure} + \includegraphics[width=\linewidth]{EMP_UHF_R10.pdf} + \caption{Energies $E(\lambda)$ in the unrestricted basis set \eqref{eq:uhfbasis} with $R=10$.} + \label{fig:UHFMiniBas} +\end{figure} + +Within the RHF formalism, we have observed only $\alpha$ singularities and large avoided crossings but one can see in Fig.~\ref{fig:UHFMiniBas} that in the UHF case there are sharp avoided crossings which are known to be connected to $\beta$ singularities. For example at $R=10$ the pair of singularities connected to the avoided crossing between s\textsuperscript{2} and sp\textsubscript{z} $^{3}P$ is $0.999\pm0.014\,i$. And the one between sp\textsubscript{z} $^{3}P$ and p\textsubscript{z}\textsuperscript{2} is connected with the singularities $2.207\pm0.023\,i$. However, in spherium, the electrons cannot be ionized so those singularities cannot be the same $\beta$ singularities as the ones highlighted by Sergeev and Goodson. \cite{Sergeev_2005} We can see in Fig.~\ref{fig:UHFEP} that the s\textsuperscript{2} singlet and the sp\textsubscript{z} triplet states are degenerated for $R=3/2$. For $R>3/2$, it becomes an avoided crossing on the real axis and the degeneracies are $moved$ in the complex plane. The wave function is spin contaminated by $Y_1$ for $R>3/2$ this is why the s\textsuperscript{2} singlet energy cannot cross the sp\textsubscript{z} triplet curves anymore. When $R$ increases this avoided crossing becomes sharper. As presented before $\beta$ singularities are linked to quantum phase transition so it seems that this singularity is linked to the spin symmetry breaking of the UHF wave function. The fact that a similar pair of $\beta$ singularities appears for $R<-75/62$ confirms this assumption. The sharp avoided crossing between sp\textsubscript{z} $^{3}P$ and p\textsubscript{z}\textsuperscript{2} is not present on Fig. \ref{fig:UHFEP}. The second pair of $\beta$ singularities resulting from this avoided crossing appears for $R\gtrsim 2.5$, this is probably due to an excited-state quantum phase transition but this still need to be investigated. + +\begin{figure} + \includegraphics[width=0.45\textwidth]{UHFCI.pdf} + \includegraphics[width=0.45\textwidth]{UHFEP.pdf} + \caption{Energies $E(\lambda)$ in the unrestricted basis set \eqref{eq:uhfbasis} for $R=1.5$ (left) and $R=1.51$ (right).} + \label{fig:UHFEP} +\end{figure} + +As shown before, some matrix elements of the Hamiltonian become complex in the holomorphic domain. Therefore the Hamiltonian becomes non-Hermitian for these values of $R$. In Ref.~\onlinecite{Burton_2019a}, Burton \textit{et al.}~proved that for the \ce{H_2} molecule the unrestricted Hamiltonian is not \pt -symmetric in the holomorphic domain. An analog reasoning can be done with the spherium model to prove the same result. The \pt -symmetry (invariance with respect to combined space reflection $\mathcal{P}$ and time reversal $\mathcal{T}$) is a property which ensures that a non-Hermitian Hamiltonian has a real energy spectrum. \cite{BenderPTBook} Thus \pt -symmetric Hamiltonians can be seen as an intermediate class between Hermitian and non-Hermitian Hamiltonians. + +Figure \ref{fig:UHFPT} shows that for the spherium model a part of the energy spectrum becomes complex when $R$ is in the holomorphic domain. The domain of values where the energy becomes complex is called the broken \pt-symmetry region. This is consistent with the fact that in the holomorphic domain the Hamiltonian is no more \pt -symmetric. + +For a non-Hermitian Hamiltonian the EPs can lie on the real axis. In particular, at the point of {\pt} transition (the point where the energies become complex) the two energies are degenerate resulting in such an EP on the real axis. This degeneracy can be seen in Fig.~\ref{fig:UHFPT}. + +\begin{figure}[h!] + \includegraphics[width=0.45\textwidth]{ReNRJPT.pdf} + \includegraphics[width=0.45\textwidth]{ImNRJPT.pdf} + \caption{Real part (left) and imaginary part (right) of $E(\lambda)$ in the unrestricted basis set \eqref{eq:uhfbasis} for $R=1$.} + \label{fig:UHFPT} +\end{figure} + +\section{Conclusion} + +In order to model accurately chemical systems, one must choose, in a ever growing zoo of methods, which computational protocol is adapted to the system of interest. +This choice can be, moreover, motivated by the type of properties that one is interested in. +That means that one must understand the strengths and weaknesses of each method, \ie, why one method might fail in some cases and work beautifully in others. +We have seen that for methods relying on perturbation theory, their successes and failures are directly connected to the position of EPs in the complex plane. +Exhaustive studies have been performed on the causes of failure of MP perturbation theory. +First, it was understood that, for chemical systems for which the HF Slater determinant is a poor approximation to the exact wave function, MP perturbation theory fails too. Such systems can be, for example, molecules where the exact ground-state wave function is dominated by more than one configuration, \ie, multi-reference systems. +More preoccupying cases were also reported. +For instance, it has been shown that systems considered as well-understood (\eg, \ce{Ne}) can exhibit divergent behavior when the basis set is augmented with diffuse functions. +Later, these erratic behaviors of the perturbation series were investigated and rationalized in terms of avoided crossings and singularities in the complex plane. It was shown that the singularities can be classified in two families. +The first family includes $\alpha$ singularities resulting from a large avoided crossing between the ground state and a low-lying doubly-excited states. +The $\beta$ singularities, which constitutes the second family, are artifacts generated by the incompleteness of the Hilbert space, and they are directly connected to an ionization phenomenon occurring in the complete Hilbert space. +These singularities are close to the real axis and connected with sharp avoided crossing between the ground state and a highly diffuse state. +We have found that the $\beta$ singularities modeling the ionization phenomenon described by Sergeev and Goodson are actually part of a more general class of singularities. Indeed, those singularities close to the real axis are connected to quantum phase transition and symmetry breaking, and theoretical physics have demonstrated that the behavior of the EPs depends of the type of transitions from which the EPs result (first or higher orders, ground state or excited state transitions). + +In this work, we have shown that $\beta$ singularities are involved in the spin symmetry breaking of the UHF wave function. +This confirms that $\beta$ singularities can occur for other types of transition and symmetry breaking than just the formation of a bound cluster of electrons. +It would be interesting to investigate the difference between the different type of symmetry breaking and how it affects the singularity structure. +Moreover the singularity structure in the non-Hermitian case still need to be investigated. +In the holomorphic domain, some singularities lie on the real axis and it would also be interesting to look at the differences between the different symmetry breaking and their respective holomorphic domain. +Furthermore, in this study we have used spherical harmonics (or combination of spherical harmonics) as basis functions which have a delocalized nature. It would also be interesting to investigate the use of localized basis functions \cite{Seidl_2018} (for example gaussians) because these functions would be more adapted to describe the strongly correlated regime. %More generally, to investigate the effect of the type of basis on the physics of EPs. +To conclude, this work shows that our understanding of the singularity structure of the energy is still incomplete but we hope that it opens new perspectives for the understanding of the physics of EPs in electronic structure theory. -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{Concluding remarks} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\titou{What we have done is completely amazing.} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%H.G.A.B.~thanks the Cambridge and Commonwealth Trust for a studentship. +This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{acknowledgements} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bibliography{EPAWTFT} -\bibliographystyle{unsrt} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\end{document} + +\end{document} diff --git a/Manuscript/EPAWTFTNotes.bib b/Manuscript/EPAWTFTNotes.bib index e69de29..f275e40 100644 --- a/Manuscript/EPAWTFTNotes.bib +++ b/Manuscript/EPAWTFTNotes.bib @@ -0,0 +1,2 @@ +@CONTROL{REVTEX41Control} +@CONTROL{apsrev41Control,author="08",editor="1",pages="0",title="",year="1"}

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