The Theory of the Jahn-Teller Effect : When a Boson Meets a Fermion

This book provides a comprehensive discussion of the Jahn-Teller Effect (JTE), focusing on the boson-fermion interaction. While current research is concerned with measuring and calculating ever more sophisticated and complex manifestations of the JT effect, the present volume takes away the epicycles of the theory and focuses on the symmetry dilemma at its core. When fermions and bosons meet, they get entangled and form a new dynamic reality. According to the rules of Molecular Symmetry, this reality is limited to a small set of patterns, with degeneracy cardinalities: 2, 3, 4, 5, and 6. The novelty of the book is that it adopts a unique mathematical technique, known as the Bargmann-Fock representation, and treats all degeneracies in detail. So far, this method was only applied to the simplest doublet case therefore its extension to the entire range of cases offers a new unified perspective. This volume will help the reader acquire a clear understanding of the JT effect, discover its universal mechanism and it will be a great tool for researchers and graduates working on this topic. 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> 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