Theory of Slow Atomic Collisions

The theory of atom-molecule collisions is one of the basic fields in chemi­ cal physics. Its most challenging part - the dynamics of chemical reactions - is as yet unresolved, but is developing very quickly. It is here a great help to have an analysis of those parts of collision theory which are already complete, a good example being the theory of atomic collisions in process­ es specific to chemical physics. It has long been observed that many notions of this theory can also be applied successfully to reactive and unreactive molecular collisions. More­ over, atomic collisions often represent a touchstone in testing approaches proposed for the solution of more complicated problems. Research on the theory of slow atomic collisions carried out at the Moscow Institute of Chemical Physics has been based on just these ideas. A general viewpoint concerning the setting up and representation of the theory came out of these studies, and appeared to be useful in studying complicated systems as well. It underlies the representation of the theory of slow atomic colli­ sions in this book.

Inhalt

1. Introduction.- 2. General Formulation of Scattering Problem Under Quasi-Classical Conditions.- 2.1 Scattering Amplitudes and Cross Sections.- 2.1.1 Representations of Amplitudes and Cross Sections..- 2.1.2 Scattering Amplitudes and Cross Sections Under Quasi-Classical Conditions.- 2.2 Scattering Equations.- 2.2.1 Atomic Basis.- 2.2.2 Molecular Basis.- 2.3 Collisions of Two Many-Electron Atoms.- 2.3.1 Scattering Matrix and Scattering Equations.- 2.3.2 Collisions of Identical Atoms.- 2.4 Integral Cross Sections for Isotropic Collisions.- 3. Diatomic Electronic States.- 3.1 Quantum Numbers and Wave Functions of a Free Atom..- 3.2 Quantum Numbers and Wave Functions of Diatoms.- 3.2.1 General Classification of Adiabatic Diatomic States.- 3.2.2 Wave Functions of a Diatom at Large Internuclear Separations.- 3.2.3 Molecular-Orbital Approximation.- 3.3 Adiabatic States, Diabatic States, and Correlation Diagrams.- 3.3.1 The Noncrossing Rule and Adiabatic Correlation Diagrams.- 3.3.2 Diabatic States and Diabatic Correlation Diagrams.- 3.3.3 One-Electron Correlation Diagrams.- 3.4 Coupling Between Electronic States. Selection Rules.- 4. Approximate Calculation of the Electronic States of Diatoms.- 4.1 Atomic Potential and Atomic Orbitals.- 4.1.1 Hartree-Fock Screening Function and Atomic Orbitals.- 4.1.2 The Pseudopotential Method for Valence Electrons of Atoms.- 4.2 Diatomic Interactions at Large Distances and the Heitler-London Approximation.- 4.2.1 Effective Hamiltonian.- 4.2.2 Coulomb Interaction.- 4.2.3 Dispersion Interaction.- 4.2.4 Exchange Interaction.- 4.3 Pseudopotential Method for Interatomic Interactions.- 4.3.1 The Model Potential Method.- 4.3.2 Multiple Scattering Method.- 4.4 Short-Range Atomic Interactions.- 4.4.1 The Energy of Atomic Interaction at Small Distances.- 4.4.2 Electronic Potential in a Diatom at Small R.- 4.5 Coupling Between Electronic States.- 4.5.1 Spin-Orbit Coupling.- 4.5.2 Radial Coupling in the Avoided Crossing Region.- 5. Elastic Scattering.- 5.1 Quasi-Classical Scattering Amplitude.- 5.2 Quasi-Classical Scattering Matrix.- 5.2.1 JWKB Scattering Phase Shifts.- 5.2.2 Violation of Quasi-Classical Conditions in Localized Regions. Connection Problem.- 5.2.3 Isolated Turning Point.- 5.2.4 Two Close Turning Points.- 5.3 Classical Scattering.- 5.4 Integral Cross Sections.- 5.5 Differential Cross Sections.- 5.5.1 Scattering Through Classical Angles-Repulsive Potential.- 5.5.2 Scattering Through Classical Angles-Potential with a Well.- 5.5.3 Scattering Through Small Angles.- 6. Approximate Calculation of a Multichannel Quasi-Classical Scattering Matrix.- 6.1 Common-Trajectory Approach.- 6.1.1 Common-Trajectory Scattering Equations.- 6.1.2 Eikonal and Impact-Parameter Approximations.- 6.1.3 Semiclassical Limit of the Quasi-Classical Approximation.- 6.2 Matching Approach.- 6.2.1 Matching Solution of Scattering Equations.- 6.2.2 Near-Adiabatic Matching.- 6.2.3 Near-Sudden Matching.- 6.3 Perturbation Approach.- 6.3.1 First-Order Perturbation Treatment. The Born and Adiabatic Distorted-Wave Approximations.- 6.3.2 Unitarized Distorted-Wave Approximation.- 7. Two-State Scattering Problem.- 7.1 The Two-State Model. Adiabatic and Diabatic Representations.- 7.2 Construction of the Two-State Quasi-Classical S Matrix by the Matching Method.- 7.3 Two-State Semiclassical Models.- 7.3.1 Derivation of Semiclassical Equations.- 7.3.2 Classification of Semiclassical Two-State Models.- 7.3.3 Approximate Two-State Transition Probabilities.- 7.4 Differential Cross Sections and Deflection Functions.- 8. The Linear Two-State Landau-Zener Model.- 8.1 Formulation of the Model.- 8.2 Nonadiabatic Transitions Far from the Turning Point. Landau-Zener-Stueckelberg Solution.- 8.3 Nonadiabatic Transitions Near the Turning Point.- 8.3.1 Terms with Slopes of the Same Sign.- 8.3.2 Terms with Slopes of Different Signs.- 8.4 Validity of Linear Model and of Analytical Expressions for Transition Probabilities.- 8.5 Cross Sections for the Linear Model.- 8.5.1 Integral Cross Sections - Radial Coupling.- 8.5.2 Integral Cross Section - Rotational Coupling.- 8.5.3 Differential Cross Sections - Threshold Effects.- 9. Nonlinear Two-State Models of Nonadiabatic Coupling.- 9.1 Exponential Model.- 9.1.1 Formulation of the Model.- 9.1.2 Transition Probability and Dynamic Phases.- 9.1.3 Specific Cases of the Exponential Model - Probabilities and Cross Sections.- 9.2 Linear-Exponential Model.- 9.2.1 Formulation of the Model.- 9.2.2 Transition Probabilities and Cross Sections.- 9.3 Other Nonlinear Models.- 9.3.1 Hypergeometric Models.- 9.3.2 Power Models - Large Interatomic Separations.- 9.3.3 Power Models - Small Interatomic Separations.- 10. Multistate Models of Nonadiabatic Coupling.- 10.1 Transitions Between Degenerate States.- 10.1.1 Collisional Depolarization of an Isolated Atomic State.- 10.1.2 Resonant Excitation Transfer by Dipole-Dipole Interaction.- 10.1.3 Transitions Between Degenerate Hydrogen States in Collisions with Ions.- 10.2 Transitions Between Highly Excited States.- 10.3 Generalizations of the Linear Model.- 10.3.1 Interaction of a Diabatic Term with a Set of Parallel Diabatic Terms and a Continuum.- 10.3.2 Nonadiabatic Coupling Between Two Quasi-Stationary States.- 11. Case Study - Intramultiplet Mixing and Depolarization of Alkalis in Collisions with Noble Gases.- 11.1 Formulation of the M
   • X Scattering Problem.- 11.1.1 Scattering Equations and Couplings.- 11.1.2 M
   • X Interaction.- 11.2 The Scattering Matrix.- 11.2.1 Matching Approximation.- 11.2.2 Semiclassical Comparison Equations.- 11.2.3 Scattering Matrix for 2P1/2 Substate.- 11.3 Transition Probabilities and Cross Sections for Isotropic Collisions.- 11.3.1 Intramultiplet Mixing.- 11.3.2 Reorientation in the 2Pl/2 Substate.- A. Quantum Theory of Angular Momentum.- A. l Rotation Matrices and Spherical Functions.- A.2 Coupling of Angular Momenta, Clebsch-Gordan.- A.3 Matrix Elements of the Irreducible Tensor.- Operators.- References.