Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets

We compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function i...

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Fecha de publicación:: 2017

Institución:: Universidad de Medellín

Repositorio:: Repositorio UDEM

Idioma:: eng

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dc.title.spa.fl_str_mv	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets
title	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets
spellingShingle	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets Domain decomposition methods Excited states Geometry Information theory Ion sources Quantum optics Quantum theory Wave functions Electronic excitation Hydrogen molecular ion Non-Born Oppenheimer Nonorthogonal basis Nuclear wave functions Quantum information theory Schmidt decomposition Vibrational motions Quantum entanglement
title_short	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets
title_full	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets
title_fullStr	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets
title_full_unstemmed	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets
title_sort	Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets
dc.contributor.affiliation.spa.fl_str_mv	Sanz-Vicario, J.L., Grupo de Física Atómica y Molecular, Instituto de Física, Universidad de Antioquia, Medellín, Colombia Pérez-Torres, J.F., Facultad de Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, Colombia, Escuela de Química, Universidad Industrial de Santander, Bucaramanga, Colombia Moreno-Polo, G., Grupo de Física Atómica y Molecular, Instituto de Física, Universidad de Antioquia, Medellín, Colombia
dc.subject.keyword.eng.fl_str_mv	Domain decomposition methods Excited states Geometry Information theory Ion sources Quantum optics Quantum theory Wave functions Electronic excitation Hydrogen molecular ion Non-Born Oppenheimer Nonorthogonal basis Nuclear wave functions Quantum information theory Schmidt decomposition Vibrational motions Quantum entanglement
topic	Domain decomposition methods Excited states Geometry Information theory Ion sources Quantum optics Quantum theory Wave functions Electronic excitation Hydrogen molecular ion Non-Born Oppenheimer Nonorthogonal basis Nuclear wave functions Quantum information theory Schmidt decomposition Vibrational motions Quantum entanglement
description	We compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes. © 2017 American Physical Society.
publishDate	2017
dc.date.accessioned.none.fl_str_mv	2017-12-19T19:36:43Z
dc.date.available.none.fl_str_mv	2017-12-19T19:36:43Z
dc.date.created.none.fl_str_mv	2017
dc.type.eng.fl_str_mv	Article
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dc.type.coar.fl_str_mv	http://purl.org/coar/resource_type/c_6501 http://purl.org/coar/resource_type/c_2df8fbb1
dc.type.driver.none.fl_str_mv	info:eu-repo/semantics/article
dc.identifier.issn.none.fl_str_mv	24699926
dc.identifier.uri.none.fl_str_mv	http://hdl.handle.net/11407/4272
dc.identifier.doi.none.fl_str_mv	10.1103/PhysRevA.96.022503
dc.identifier.reponame.spa.fl_str_mv	reponame:Repositorio Institucional Universidad de Medellín
dc.identifier.instname.spa.fl_str_mv	instname:Universidad de Medellín
identifier_str_mv	24699926 10.1103/PhysRevA.96.022503 reponame:Repositorio Institucional Universidad de Medellín instname:Universidad de Medellín
url	http://hdl.handle.net/11407/4272
dc.language.iso.none.fl_str_mv	eng
language	eng
dc.relation.isversionof.spa.fl_str_mv	https://www.scopus.com/inward/record.uri?eid=2-s2.0-85028652124&doi=10.1103%2fPhysRevA.96.022503&partnerID=40&md5=f167f8f2d91e906abf6c02ab6b2bc1b0
dc.relation.ispartofes.spa.fl_str_mv	Physical Review A Physical Review A Volume 96, Issue 2, 2 August 2017
dc.relation.references.spa.fl_str_mv	Bachau, H., Cormier, E., Decleva, P., Hansen, J. E., & Martín, F. (2001). Applications of B-splines in atomic and molecular physics. Reports on Progress in Physics, 64(12), 1815-1942. doi:10.1088/0034-4885/64/12/205 Bhatti, M. I., & Perger, W. F. (2006). Solutions of the radial dirac equation in a B-polynomial basis. Journal of Physics B: Atomic, Molecular and Optical Physics, 39(3), 553-558. doi:10.1088/0953-4075/39/3/008 Born, M., & Huang, K. (1954). Dynamical Theory of Crystal Lattices, Bouvrie, P. A., Majtey, A. P., Tichy, M. C., Dehesa, J. S., & Plastino, A. R. (2014). Eur.Phys.J., 68, 346. Brosoio, M., Decleva, P., & Lisini, A. (1992). Accurate variational determination of continuum wavefunctions by a one-centre expansion in a spline basis. an application to (equation found) and heh2+photoionization. Journal of Physics B: Atomic, Molecular and Optical Physics, 25(15), 3345-3356. doi:10.1088/0953-4075/25/15/015 De Boor, C. (1978). A Practical Guide to Splines. Gerry, C. C., & Knight, P. L. (2005). Introductory Quantum Optics. Gidopoulos, N. I., & Gross, E. K. U. (2014). Electronic non-adiabatic states: Towards a density functional theory beyond the born-oppenheimer approximation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372(2011) doi:10.1098/rsta.2013.0059 Haxton, D. J., Lawler, K. V., & McCurdy, C. W. (2011). Multiconfiguration time-dependent hartree-fock treatment of electronic and nuclear dynamics in diatomic molecules. Physical Review A - Atomic, Molecular, and Optical Physics, 83(6) doi:10.1103/PhysRevA.83.063416 Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPC: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019 Hunter, G. (1975). Conditional probability amplitudes in wave mechanics. International Journal of Quantum Chemistry, 9(2), 237-242. doi:10.1002/qua.560090205 Izmaylov, A. F., & Franco, I. (2017). Entanglement in the born-oppenheimer approximation. Journal of Chemical Theory and Computation, 13(1), 20-28. doi:10.1021/acs.jctc.6b00959 Karr, J. P., & Hilico, L. (2006). High accuracy results for the energy levels of the molecular ions H + 2, D+ 2 and HD+, up to J ≤ 2. Journal of Physics B: Atomic, Molecular and Optical Physics, 39(8), 2095-2105. doi:10.1088/0953-4075/39/8/024 Landau, L. D., & Lifshitz, E. M. (1977). Quantum Mechanics. Löwdin, P. -. (1950). On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals. The Journal of Chemical Physics, 18(3), 365-375. Martín, F. (1999). Ionization and dissociation using B-splines: Photoionization of the hydrogen molecule. Journal of Physics B: Atomic, Molecular and Optical Physics, 32(16), R197-R231. doi:10.1088/0953-4075/32/16/201 McKemmish, L. K., McKenzie, R. H., Hush, N. S., & Reimers, J. R. (2015). Electron-vibration entanglement in the born-oppenheimer description of chemical reactions and spectroscopy. Physical Chemistry Chemical Physics, 17(38), 24666-24682. doi:10.1039/c5cp02239h McKemmish, L. K., McKenzie, R. H., Hush, N. S., & Reimers, J. R. (2011). Quantum entanglement between electronic and vibrational degrees of freedom in molecules. Journal of Chemical Physics, 135(24) doi:10.1063/1.3671386 Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information. Pérez-Torres, J. F. (2013). Electronic flux densities in vibrating H2+ in terms of vibronic eigenstates. Physical Review A - Atomic, Molecular, and Optical Physics, 87(6) doi:10.1103/PhysRevA.87.062512 Restrepo Cuartas, J. P., & Sanz-Vicario, J. L. (2015). Information and entanglement measures applied to the analysis of complexity in doubly excited states of helium. Physical Review A - Atomic, Molecular, and Optical Physics, 91(5) doi:10.1103/PhysRevA.91.052301 Schmidt, E. (1907). Zur theorie der linearen und nichtlinearen integralgleichungen - I. teil: Entwicklung willkürlicher funktionen nach systemen vorgeschriebener. Mathematische Annalen, 63(4), 433-476. doi:10.1007/BF01449770 Szabo, A., & Ostlund, N. S. (1989). Modern Quantum Chemistry. Vatasescu, M. (2013). Entanglement between electronic and vibrational degrees of freedom in a laser-driven molecular system. Physical Review A - Atomic, Molecular, and Optical Physics, 88(6) doi:10.1103/PhysRevA.88.063415 Vatasescu, M. (2015). Measures of electronic-vibrational entanglement and quantum coherence in a molecular system. Physical Review A, 92(4) doi:10.1103/PhysRevA.92.042323
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rights_invalid_str_mv	http://purl.org/coar/access_right/c_16ec
dc.publisher.spa.fl_str_mv	American Physical Society
dc.publisher.faculty.spa.fl_str_mv	Facultad de Ciencias Básicas
dc.source.spa.fl_str_mv	Scopus
institution	Universidad de Medellín
repository.name.fl_str_mv	Repositorio Institucional Universidad de Medellin
repository.mail.fl_str_mv	repositorio@udem.edu.co
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spelling	2017-12-19T19:36:43Z2017-12-19T19:36:43Z201724699926http://hdl.handle.net/11407/427210.1103/PhysRevA.96.022503reponame:Repositorio Institucional Universidad de Medellíninstname:Universidad de MedellínWe compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes. © 2017 American Physical Society.engAmerican Physical SocietyFacultad de Ciencias Básicashttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85028652124&doi=10.1103%2fPhysRevA.96.022503&partnerID=40&md5=f167f8f2d91e906abf6c02ab6b2bc1b0Physical Review APhysical Review A Volume 96, Issue 2, 2 August 2017Bachau, H., Cormier, E., Decleva, P., Hansen, J. E., & Martín, F. (2001). Applications of B-splines in atomic and molecular physics. Reports on Progress in Physics, 64(12), 1815-1942. doi:10.1088/0034-4885/64/12/205Bhatti, M. I., & Perger, W. F. (2006). Solutions of the radial dirac equation in a B-polynomial basis. Journal of Physics B: Atomic, Molecular and Optical Physics, 39(3), 553-558. doi:10.1088/0953-4075/39/3/008Born, M., & Huang, K. (1954). Dynamical Theory of Crystal Lattices,Bouvrie, P. A., Majtey, A. P., Tichy, M. C., Dehesa, J. S., & Plastino, A. R. (2014). Eur.Phys.J., 68, 346.Brosoio, M., Decleva, P., & Lisini, A. (1992). Accurate variational determination of continuum wavefunctions by a one-centre expansion in a spline basis. an application to (equation found) and heh2+photoionization. Journal of Physics B: Atomic, Molecular and Optical Physics, 25(15), 3345-3356. doi:10.1088/0953-4075/25/15/015De Boor, C. (1978). A Practical Guide to Splines.Gerry, C. C., & Knight, P. L. (2005). Introductory Quantum Optics.Gidopoulos, N. I., & Gross, E. K. U. (2014). Electronic non-adiabatic states: Towards a density functional theory beyond the born-oppenheimer approximation. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 372(2011) doi:10.1098/rsta.2013.0059Haxton, D. J., Lawler, K. V., & McCurdy, C. W. (2011). Multiconfiguration time-dependent hartree-fock treatment of electronic and nuclear dynamics in diatomic molecules. Physical Review A - Atomic, Molecular, and Optical Physics, 83(6) doi:10.1103/PhysRevA.83.063416Hernandez, V., Roman, J. E., & Vidal, V. (2005). SLEPC: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Transactions on Mathematical Software, 31(3), 351-362. doi:10.1145/1089014.1089019Hunter, G. (1975). Conditional probability amplitudes in wave mechanics. International Journal of Quantum Chemistry, 9(2), 237-242. doi:10.1002/qua.560090205Izmaylov, A. F., & Franco, I. (2017). Entanglement in the born-oppenheimer approximation. Journal of Chemical Theory and Computation, 13(1), 20-28. doi:10.1021/acs.jctc.6b00959Karr, J. P., & Hilico, L. (2006). High accuracy results for the energy levels of the molecular ions H + 2, D+ 2 and HD+, up to J ≤ 2. Journal of Physics B: Atomic, Molecular and Optical Physics, 39(8), 2095-2105. doi:10.1088/0953-4075/39/8/024Landau, L. D., & Lifshitz, E. M. (1977). Quantum Mechanics.Löwdin, P. -. (1950). On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals. The Journal of Chemical Physics, 18(3), 365-375.Martín, F. (1999). Ionization and dissociation using B-splines: Photoionization of the hydrogen molecule. Journal of Physics B: Atomic, Molecular and Optical Physics, 32(16), R197-R231. doi:10.1088/0953-4075/32/16/201McKemmish, L. K., McKenzie, R. H., Hush, N. S., & Reimers, J. R. (2015). Electron-vibration entanglement in the born-oppenheimer description of chemical reactions and spectroscopy. Physical Chemistry Chemical Physics, 17(38), 24666-24682. doi:10.1039/c5cp02239hMcKemmish, L. K., McKenzie, R. H., Hush, N. S., & Reimers, J. R. (2011). Quantum entanglement between electronic and vibrational degrees of freedom in molecules. Journal of Chemical Physics, 135(24) doi:10.1063/1.3671386Nielsen, M. A., & Chuang, I. L. (2010). Quantum Computation and Quantum Information.Pérez-Torres, J. F. (2013). Electronic flux densities in vibrating H2+ in terms of vibronic eigenstates. Physical Review A - Atomic, Molecular, and Optical Physics, 87(6) doi:10.1103/PhysRevA.87.062512Restrepo Cuartas, J. P., & Sanz-Vicario, J. L. (2015). Information and entanglement measures applied to the analysis of complexity in doubly excited states of helium. Physical Review A - Atomic, Molecular, and Optical Physics, 91(5) doi:10.1103/PhysRevA.91.052301Schmidt, E. (1907). Zur theorie der linearen und nichtlinearen integralgleichungen - I. teil: Entwicklung willkürlicher funktionen nach systemen vorgeschriebener. Mathematische Annalen, 63(4), 433-476. doi:10.1007/BF01449770Szabo, A., & Ostlund, N. S. (1989). Modern Quantum Chemistry.Vatasescu, M. (2013). Entanglement between electronic and vibrational degrees of freedom in a laser-driven molecular system. Physical Review A - Atomic, Molecular, and Optical Physics, 88(6) doi:10.1103/PhysRevA.88.063415Vatasescu, M. (2015). Measures of electronic-vibrational entanglement and quantum coherence in a molecular system. Physical Review A, 92(4) doi:10.1103/PhysRevA.92.042323ScopusElectronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis setsArticleinfo:eu-repo/semantics/articlehttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Sanz-Vicario, J.L., Grupo de Física Atómica y Molecular, Instituto de Física, Universidad de Antioquia, Medellín, ColombiaPérez-Torres, J.F., Facultad de Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, Colombia, Escuela de Química, Universidad Industrial de Santander, Bucaramanga, ColombiaMoreno-Polo, G., Grupo de Física Atómica y Molecular, Instituto de Física, Universidad de Antioquia, Medellín, ColombiaSanz-Vicario J.L.Pérez-Torres J.F.Moreno-Polo G.Grupo de Física Atómica y Molecular, Instituto de Física, Universidad de Antioquia, Medellín, ColombiaFacultad de Facultad de Ciencias Básicas, Universidad de Medellín, Medellín, ColombiaEscuela de Química, Universidad Industrial de Santander, Bucaramanga, ColombiaDomain decomposition methodsExcited statesGeometryInformation theoryIon sourcesQuantum opticsQuantum theoryWave functionsElectronic excitationHydrogen molecular ionNon-Born OppenheimerNonorthogonal basisNuclear wave functionsQuantum information theorySchmidt decompositionVibrational motionsQuantum entanglementWe compute the entanglement between the electronic and vibrational motions in the simplest molecular system, the hydrogen molecular ion, considering the molecule as a bipartite system, electron and vibrational motion. For that purpose we compute an accurate total non-Born-Oppenheimer wave function in terms of a huge expansion using nonorthogonal B-spline basis sets that expand separately the electronic and nuclear wave functions. According to the Schmidt decomposition theorem for bipartite systems, widely used in quantum-information theory, it is possible to find a much shorter but equivalent expansion in terms of the natural orbitals or Schmidt bases for the electronic and nuclear half spaces. Here we extend the Schmidt decomposition theorem to the case in which nonorthogonal bases are used to span the partitioned Hilbert spaces. This extension is first illustrated with two simple coupled systems, the former without an exact solution and the latter exactly solvable. In these model systems of distinguishable coupled particles it is shown that the entanglement content does not increase monotonically with the excitation energy, but only within the manifold of states that belong to an existing excitation mode, if any. In the hydrogen molecular ion the entanglement content for each non-Born-Oppenheimer vibronic state is quantified through the von Neumann and linear entropies and we show that entanglement serves as a witness to distinguish vibronic states related to different Born-Oppenheimer molecular energy curves or electronic excitation modes. © 2017 American Physical Society.http://purl.org/coar/access_right/c_16ec11407/4272oai:repository.udem.edu.co:11407/42722020-05-27 19:16:33.897Repositorio Institucional Universidad de Medellinrepositorio@udem.edu.co

Electronic-nuclear entanglement in H2+: Schmidt decomposition of non-Born-Oppenheimer wave functions expanded in nonorthogonal basis sets

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