The theory of the Jahn-Teller effect : when a boson meets a fermion / / Arnout Ceulemans

• Record Nr.: UNINA9910616384203321
• Autore: Ceulemans Arnout
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031095283; 9783031095276
• Descrizione fisica: 1 online resource (429 pages)
• Disciplina: 530.143
• Soggetti: Interacting boson-fermion models; Jahn-Teller effect; Efecte Jahn-Teller; Bosons; Fermions; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Preface -- Contents -- Part I Bosons and Fermions -- 1 The Impossible Theorem -- Contents -- 1.1 The Jahn-Teller Theorem -- 1.2 Charge Density Analysis -- 1.2.1 Occupation of dz2 -- 1.2.2 Occupation of dx2-y2 -- 1.2.3 Sum and Difference Orbitals -- 1.2.4 Orthogonal and Unitary Combinations -- 1.3 Outlook -- References -- 2 Bosons and Fermions -- Contents -- 2.1 Bosons -- 2.1.1 The Schrödinger Formalism -- 2.1.2 The Dirac Formalism -- 2.1.3 The Bargmann Mapping -- 2.2 Fermions -- 2.2.1 Fermion Operators -- 2.2.2 One-Electron Interactions -- 2.2.3 Quasi-Spin -- References -- 3 Boson-Fermion Interactions -- Contents -- 3.1 The Jahn-Teller Effect in a Triangular Molecule: A Toy Model -- 3.1.1 The Hückel Hamiltonian -- 3.1.2 Fermions: Trigonal Molecular Orbitals -- 3.1.3 Bosons: Vibrational Modes -- 3.1.4 Coupling Coefficients -- 3.2 Degeneracies and Time Reversal -- 3.2.1 Time Reversal -- 3.2.2 Irreducible Representations of the First Kind and Orthogonal Lie Groups -- 3.2.3 Irreducible Representations of the Second Kind and Symplectic Lie ; Groups -- 3.2.4 Irreducible Representations of the Third Kind -- 3.3 The Jahn-Teller Hamiltonian -- 3.4 Selection Rules -- 3.4.1 Space Symmetry -- 3.4.2 Time Reversal Symmetry -- 3.4.3 Hole-Particle Exchange Symmetry -- 3.5 Proof of the Jahn-Teller Theorem -- 3.5.1 History -- 3.5.2 Where Do Degeneracies Come From? -- 3.5.2.1 Cosets and the Positional Representation -- 3.5.2.2 Doubly Transitive Orbits -- 3.5.3 Degenerate Representations and Jahn-Teller Modes -- 3.5.4 Jahn-Teller Activity in Simplexes -- References -- Part II Dynamic Symmetries -- 4 The Rabi Hamiltonian -- Contents -- 4.1 The Adiabatic Potential -- 4.2 The Quantum Model -- 4.3 Bargmann Mapping of the Wave Equations -- 4.4 Eigenvalues -- 4.4.1 Classification of the Roots -- 4.4.2 Recurrence Relations and Transcendental Function.; 4.4.3 The Rabi Spectrum -- 4.5 The Quantization of the Rabi Hamiltonian -- 4.6 Analyticity -- 4.7 Inversion Tunneling in Ammonia -- References -- 5 The E ×e Orbital Doublet -- Contents -- 5.1 The Quantum Model -- 5.2 Dynamic Symmetries -- 5.2.1 Boson Symmetry -- 5.2.2 Fermion Symmetry -- 5.2.3 Coupled Symmetries -- 5.3 The Canonical Form of the Wave Equation -- 5.4 Recurrence Relationships -- 5.5 Results -- 5.6 Discussion -- 5.7 Application: Na3 and the (E+A)×e Hamiltonian -- References -- 6 The Spin Quartet Γ8 ×(e+t2) System and the Symplectic Group Sp(4) -- Contents -- 6.1 Historical Note: Judd and Reik -- 6.2 The Hamiltonian -- 6.2.1 The Static Case -- 6.2.2 The Dynamic Hamiltonian -- 6.3 Sp(4) Fermion Symmetry -- 6.4 SO(5) Boson Symmetry -- 6.5 The Γ8 ×(e+t2) Dynamic Equations -- 6.6 The Γ8 ×t2 Subsystem -- 6.6.1 SO(3) Invariance -- 6.6.2 Dynamic Equations -- 6.7 Application -- 6.7.1 ReF6 -- 6.7.2 IrF6 -- References -- 7 Ansatz for the Jahn-Teller Triplet Instability -- Contents -- 7.1 SO(5) Symmetry and the Five-Dimensional Harmonic Oscillator -- 7.1.1 SU(5) ↓ SO(5) Symmetry Breaking -- 7.1.2 SO(5) ↓ SO(3) Symmetry Breaking -- 7.2 The Hamiltonian -- 7.3 The Vibrating Sphere -- 7.4 Boson Functions -- 7.4.1 S States -- 7.4.2 D States -- 7.4.3 F States -- 7.5 The Ansatz -- 7.6 The Jahn-Teller Equations -- 7.7 Solution -- 7.8 Ansatz for Vibronic D States -- 7.9 Application -- 7.10 Conclusion -- References -- 8 The Icosahedral Quartet and SO(9) ↓ SO(4) Symmetry Breaking -- Contents -- 8.1 Introduction -- 8.2 Preamble: Hyperspherical Symmetry -- 8.3 The Hamiltonian -- 8.4 The Vibrations of the Four-Dimensional Hypersphere -- 8.5 SO(9) ↓ SO(4) Symmetry Breaking -- 8.5.1 (0,0) Modes -- 8.5.2 (1,1) Boson Modes -- 8.5.3 Modes with Seniority ν&gt -- 4 -- 8.6 The Ansatz: Vibronic (12,12) Levels -- 8.7 Icosahedral Symmetry Lowering.; 8.8 Application: C20 and C80 Fullerenes -- 8.8.1 C20 -- 8.8.2 C80 -- References -- 9 SO(14) ↓ SO(5) Symmetry Breaking and the Jahn-Teller Quintet Instability -- Contents -- 9.1 Dynamic Symmetries -- 9.2 Descent to Spherical Symmetry -- 9.2.1 Branching Rules for SO(5) SO(3) -- 9.2.2 The L=2 Case -- 9.2.3 The L=4 Case -- 9.3 Descent to Permutational Symmetry -- 9.3.1 The Icosahedral Hamiltonian -- 9.3.2 The Hexateron -- 9.4 Correlation Between the Spherical and the Permutational Scheme -- 9.5 Application: The Ground State of C60+ Cation -- References -- 10 Jahn's and Teller's Last Case: The Spinor Sextet -- Contents -- 10.1 Group Theory of the Sextet Spinor -- 10.1.1 The Unitary Symplectic Group USp(6) -- 10.1.2 The SO(14) Group of the Bosons -- 10.2 The Γ9 ×(g+2h) Problem -- 10.2.1 The Hamiltonian -- 10.2.2 Diagonalization -- 10.2.3 The Equal Coupling Case -- 10.3 Chemical Applications -- 10.4 Overview -- 10.4.1 Orbital Representations: SO(N) ⊃ SO(n) -- 10.4.2 Spinor Representations: SO(N) ⊃ USp(2n) -- References -- Part III Topography -- 11 Conical Intersections and Quantum Fields -- Contents -- 11.1 The Berry Phase ; -- 11.1.1 The Quantal Phase Factor Accompanying Adiabatic Changes -- 11.1.1.1 Single-Valued Basis Functions -- 11.1.1.2 Real Basis Sets -- 11.1.2 Holonomy -- 11.2 The E×e Jahn-Teller Case -- 11.2.1 Berry Phase for the E×e Case -- 11.2.2 The Dirac Monopole Analogy -- 11.2.3 Berry Phase and Angular Momentum -- 11.3 Quadruple Spin Degeneracy and the Instanton -- 11.3.1 The Γ8 ×t2g Hamiltonian -- 11.3.2 The Γ8 ×(eg+t2g) Hamiltonian -- References -- 12 Topography and Chemical Reactivity -- Contents -- 12.1 Tools -- 12.1.1 The Epikernel Principle -- 12.1.2 The Isostationary Function -- 12.1.3 Proof of the Epikernel Principle -- 12.1.3.1 Only One Λ Irrep -- 12.1.3.2 More than One Λ Irrep -- 12.1.3.3 Illustration: The Γ×(Λ1+Λ2) Problem.; 12.2 Orbital Doublets -- 12.2.1 The E×(b1+b2) System -- 12.2.2 The E×e System -- 12.2.3 The Pentagonal E1×e2 Problem -- 12.3 The Cubic T×(e+t2) Problem -- 12.3.1 Second-Order Warping Terms -- 12.3.2 Chemical Reactivity: The Isomerization of Fe(CO)4 -- 12.4 The Icosahedral T ×h System -- 12.5 The Icosahedral G×g+h Quartet System -- 12.5.1 The Isostationary Function -- 12.5.2 Tetrahedral Minima -- 12.5.3 Trigonal Minima -- 12.6 The Icosahedral H×(g+2h) Quintet System -- 12.6.1 The Isostationary Function -- 12.6.2 Pentagonal Minima -- 12.6.3 Trigonal Minima -- 12.7 The Icosahedral Γ9 ×(g+2h) Sextet System -- 12.7.1 The G-Type Subspace -- 12.7.2 The H Subspace -- 12.7.2.1 The FH2 Hamiltonian at β=0∘ -- 12.7.2.2 Trough Solution: T1 ×Γ7: β≈100.893∘ -- 12.7.2.3 Trough Solution: T2 ×Γ6: β≈220.8934 -- References -- Epilogue -- A The Displaced Oscillator -- Contents -- A.1 Hamiltonian -- A.2 The Displacement Operator -- A.3 Eigenfunction of the Annihilation Operator -- A.4 Matrix Representation of the Displaced Oscillator -- References -- B Derivation of the Coupling Coefficients -- Contents -- B.1 Clebsch-Gordan Coupling Coefficients -- B.2 How to Calculate Coupling Coefficients -- B.3 Icosahedral States -- References -- C SU(n), SO(n), Sp(2n) Lie Algebras -- Contents -- C.1 The Special Unitary Group SU(n) -- C.2 The Special Orthogonal Group SO(n) -- C.3 The Symplectic Group Sp(2n) -- References -- D The Birkhoff Transformation -- Contents -- D.1 The Birkhoff Theorem -- D.2 Transformation of the Rabi Equation to the Standard Birkhoff Form -- D.3 Recursion Formulas for the Rabi Case -- D.4 Summary -- References -- E Dirac's Monopole -- Contents -- E.1 The Field of a Monopole -- E.2 The Vector Potential -- References -- F Yang's Monopole -- Contents -- F.1 Introduction -- F.2 The Tensor Potential A -- F.3 The Field Tensor F -- References.; G Topological Graph Theory -- Contents -- G.1 Graphs -- G.2 Rings -- G.3 Faces -- References -- Compound Index -- Subject Index.

Innovations in Applied Artificial Intelligence : 18th International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems, IEA/AIE 2005, Bari, Italy, June 22-24, 2005, Proceedings / / edited by Floriana Esposito

• Record Nr.: UNINA9910484241803321
• Pubbl/distr/stampa: Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2005
• Edizione: [1st ed. 2005.]
• Descrizione fisica: 1 online resource (XX, 858 p.)
• Collana: Lecture Notes in Artificial Intelligence, , 2945-9141 ; ; 3533
• Altri autori (Persone): AliMoonis; EspositoFloriana <1947->
• Disciplina: 006.3
• Soggetti: Artificial intelligence; Computer science; Pattern recognition systems; Software engineering; Application software; User interfaces (Computer systems); Human-computer interaction; Artificial Intelligence; Theory of Computation; Automated Pattern Recognition; Software Engineering; Computer and Information Systems Applications; User Interfaces and Human Computer Interaction
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Bibliographic Level Mode of Issuance: Monograph
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Invited Contributions -- Computer Vision -- Image Analysis -- Speech Recognition -- Robotics -- Agents -- Planning -- Human-Computer Interaction and Natural Language Processing -- Reasoning -- Machine Learning -- Data Mining -- Genetic Algorithms -- Neural Networks -- Decision Support and Heuristic Search -- Fuzzy Logic -- Knowledge Management -- Applications.
• Sommario/riassunto: “Intelligent systems are those which produce intelligent o?springs.” AI researchers have been focusing on developing and employing strong methods that are capable of solving complex real-life problems. The 18th International Conference on Industrial & Engineering Applications of Arti?cial Intelligence & Expert Systems (IEA/AIE 2005) held in Bari, Italy presented such work performed by many scientists worldwide. The Program Committee selected long papers from contributions presenting more complete work and posters from those reporting ongoing research. The Committee enforced the rule that only original and unpublished work could be considered for inclusion in these proceedings. The Program Committee selected 116 contributions from the 271 subm- ted papers which cover the following topics: arti?cial systems, search engines, intelligent interfaces, knowledge discovery, knowledge-based technologies, na- ral language processing, machine learning applications, reasoning technologies, uncertainty management, applied data mining, and technologies for knowledge management. The contributions oriented to the technological aspects of AI and the quality of the papers are witness to a research activity clearly aimed at consolidating the theoretical results that have already been achieved. The c- ference program also included two invited lectures, by Katharina Morik and Roberto Pieraccini. Manypeoplecontributedindi?erentwaystothesuccessoftheconferenceand to this volume. The authors who continue to show their enthusiastic interest in applied intelligence research are a very important part of our success. We highly appreciate the contribution of the members of the Program Committee, as well as others who reviewed all the submitted papers with e?ciency and dedication.