Atom - Molecule Collision Theory : A Guide for the Experimentalist

The broad field of molecular collisions is one of considerable current interest, one in which there is a great deal of research activity, both experi­ mental and theoretical. This is probably because elastic, inelastic, and reactive intermolecular collisions are of central importance in many of the fundamental processes of chemistry and physics. One small area of this field, namely atom-molecule collisions, is now beginning to be "understood" from first principles. Although the more general subject of the collisions of polyatomic molecules is of great im­ portance and intrinsic interest, it is still too complex from the viewpoint of theoretical understanding. However, for atoms and simple molecules the essential theory is well developed, and computational methods are sufficiently advanced that calculations can now be favorably compared with experimental results. This "coming together" of the subject (and, incidentally, of physicists and chemists !), though still in an early stage, signals that the time is ripe for an appraisal and review of the theoretical basis of atom-molecule collisions. It is especially important for the experimentalist in the field to have a working knowledge of the theory and computational methods required to describe the experimentally observable behavior of the system. By now many of the alternative theoretical approaches and computational procedures have been tested and intercompared. More-or-Iess optimal methods for dealing with each aspect are emerging. In many cases working equations, even schematic algorithms, have been developed, with assumptions and caveats delineated.

Chap. 1. Introduction to Atom—Molecule Collisions : The Interdependency of Theory and Experiment.- 1. General Introduction.- 2. The Experimentalist’s “Need to Know”.- 3. Overview of Experiments in Atom—Molecule Collisions.- 3.1. Elastic Scattering.- 3.2. Inelastic Scattering.- 3.3. Electronic Excitation and Curve Crossing.- 3.4. Reactive Scattering.- 4. Experimental Examples.- 4.1. Elastic Scattering.- 4.2. Rotationally Inelastic Scattering.- 4.3. Vibrationally Inelastic Scattering.- 4.4. Electronic Excitation and Charge Transfer.- 4.5. Reactive Atom—Molecule Scattering.- 4.6. Collision-Induced Dissociation.- 5. Information Content of Atom—Molecule Molecule Collision Cross Sections.- 6. Future Theoretical Demands of the Experimentalist.- References.- Chap. 2. Interaction Potentials I : Atom—Molecule Molecule Potentials.- 1. Current State of Ab Initio Electronic Structure Theory.- 2. Philosophy : Judicious Synthesis of Theory and Experiment.- 3. Brief Survey of Methods.- 3.1. Basis Sets.- 3.2. The Problem of Electron Correlation.- 3.2.1. The Concept.- 3.2.2. Configuration Interaction (CI).- 4. Examples.- 4.1. Nonreactive.- 4.1.1. Li+—H2.- 4.1.2. He—H2CO.- 4.2. Reactive.- 4.2.1. H + H2.- 4.2.2. Fluorine—Hydrogen Systems.- 4.2.3. N+ + H2.- 4.2.4. H + Li2, F + Li2.- 4.2.5. H + C1H, H + BrH.- 5. Concluding Remarks.- References.- Chap. 3. Interaction Potentials II: Semiempirical Atom—Molecule Potentials for Collision Theory.- 1. Introduction.- 1.1. Potential Surfaces for Collision Theory.- 1.2. Requisites for the Potential Energy Surface and Its Representation.- 1.2.1. Physical Requirements.- 1.2.2. Computational Requirements.- 1.3. Selection of Methods.- 2. The Method of Diatomics-in-Molecules (DIM).- 2.1. Introduction.- 2.2. General Formulation.- 2.2.1. Defining the Scope of the Problem.- 2.2.2. The DIM Basis Set.- 2.2.3. The DIM Hamiltonian Matrix.- 2.2.4. The DIM Eigenvalues.- 2.3. A Specific Example: FH2.- 2.3.1. Define the Coordinate System.- 2.3.2. Define the Atomic Basis Functions and Fragment Matrices.- 2.3.3. Define the Diatomic Basis and Fragment Matrices.- 2.3.4. Compute the Rotated Fragment Matrices.- 2.3.5. Construct the Triatomic Basis.- 2.3.6. Construct the Atomic Matrices B.- 2.3.7. Construct the Diatomic Matrices B.- 2.3.8. Find the DIM Eigenvalues.- 2.4. Simple Systems: An Alternative Formulation.- 2.5. Coupling.- 2.5.1. Spin—Orbit Coupling.- 2.5.2. Nonadiabatic Coupling.- 3. Methods Related to DIM.- 3.1. The LEPS Method.- 3.2. Method of Blais and Truhlar.- 3.3. Valence-Bond Methods.- 3.3.1. Porter—Karplus Surface for H3.- 3.3.2. Valence-Bond Methods with Transferable Parameters.- 3.4. Simple Approach to Nonadiabatic Coupling.- References.- Chap. 4. Elastic Scattering Cross Sections I: Spherical Potentials.- 1. Introduction.- 2. Intermolecular Potential.- 2.1. The Concept of an Intermolecular Potential.- 2.2. General Behavior of the Intermolecular Potential.- 2.3. Potential Models Used in the Evaluation of Scattering Cross Sections.- 2.3.1. Basic Potential Models.- 2.3.2. Modifications of the Basic Potentials and Piecewise Analytic Potentials.- 2.3.3. The Simons—Parr—Finlan (SPF) Modified Dunham Expansion.- 3. Definitions of the Quantities That Can Be Measured in Elastic-Scattering Experiments. Influence of Experimental Conditions.- 4. Classical Scattering Theory.- 4.1. Basic Formulas.- 4.2. Differential Cross Section.- 4.2.1. Small-Angle Scattering.- 4.2.2. Glory Scattering.- 4.2.3. Rainbow Scattering.- 4.2.4. Large-Angle Scattering.- 4.2.5. Orbiting Collisions.- 4.2.6. Summary of the Classical Results for the Differential Scattering Cross Section and Limits of Validity.- 4.3. Total Elastic Cross Sections.- 4.4. Identical Particles.- 4.5. First-Order Momentum Approximation and Results for the Basic Potentials.- 5. Quantal Treatment.- 5.1. Introduction.- 5.2. Stationary Scattering Theory and Partial-Wave Analysis.- 5.3. Examples of Numerical Results.- 5.3.1. Differential Cross Sections.- 5.3.2. Total Scattering Cross Section.- 5.4. Resonance Scattering.- 5.5. Identical Particles.- 6. Semiclassical Approximation.- 6.1. General Assumptions and Introductory Remarks.- 6.2. Special Features of the Differential Cross Section.- 6.2.1. Interference Effects.- 6.2.2. Rainbow Scattering.- 6.2.3. Orbiting Collisions.- 6.2.4. Large-Angle Scattering.- 6.2.5. Glory Scattering.- 6.2.6. Small-Angle Scattering (Forward Diffraction Peak).- 6.3. Special Features of the Total Elastic Scattering Cross Section.- 6.4. Identical Particles.- 6.5. High-Energy Approximation.- 6.5.1. Brief Outline of the Method.- 6.5.2. Results for the Basic-Potential Models.- 7. Methods for the Evaluation of Potentials from Experimental Scattering Data.- 7.1. General Survey.- 7.2. Semiclassical Inversion Procedures.- 7.2.1. Determination of the Repulsive Part of the Potential from the s-Phase as a Function of the Energy.- 7.2.2. Determination of the Potential from the Phase Shift Function or the Deflection Function at a Fixed Energy.- 7.2.3. Determination of the Phase Shift Function ?(?) or the Classical Deflection Function ?(?) from an Analysis of Differential Cross Section Data.- 7.2.4. The Inverse Problem in the High-Energy Approximation.- 7.3. The Trial and Error Method and Regression Procedures.- 7.4. The Use of Pseudopotentials.- References.- Chap. 5. Elastic Scattering Cross Sections II: Noncentral Potentials.- 1. Introduction.- 2. Angular-Dependent Potentials.- 2.1. The General Form.- 2.2. The Long-Range Terms.- 2.3. Eccentricity Effects.- 2.4. Action Integrals.- 3. General Expressions and Close-Coupling Calculations.- 4. The Distorted-Wave Approximation.- 5. Sudden Approximation.- 6. The Calculation of Cross Sections in Sudden Approximation.- 6.1. The Differential Cross Section in Sudden Approximation.- 6.2. The Integral Cross Section in Sudden Approximation: The Nonglory Contribution.- 6.3. The Total Integral Cross Section in Sudden Approximation: The Glory Contribution.- 7. Conclusions.- Glossary of Abbreviations.- References.- Chap. 6. Inelastic Scattering Cross Sections I: Theory.- 1. Introduction.- 2. Observables and Averaging.- 3. Quantum Theory of Inelastic Scattering.- 3.1. Formal Quantum Theory.- 3.2. Angular Momentum Conservation, Parity, and Close-Coupled Equations.- 3.3. Asymptotic Forms and the S Matrix.- 3.4. Symmetry and Microscopic Reversibility.- 3.5. Integral Equations and Square Integrable Techniques.- 4. Approximate Approaches.- 4.1. Dimension-Reducing Approximations (DRA’s).- 4.2. Perturbation Theory.- 4.3. Chemical Dynamics.- References.- Chap. 7. Inelastic Scattering Cross Sections II: Approximation Methods.- 1. Introduction.- 2. Rotational Excitation.- 3. Vibrational Excitation.- 4. Electronic Excitation.- References.- Chap. 8. Rotational Excitation I: The Quantal Treatment.- 1. Introduction.- 2. The Coupled Equations for Rotational Scattering.- 3. Solution of the Close-Coupling Equations.- 4. Methods of Solution of the Coupled Scattering Equations.- 4.1. The Approximate-Solution Approach in the Solution-Following Technique: The Method of Sams and Kouri.- 4.2. The Approximate-Potential Approach in the Solution-Following Technique.- 4.3. The Approximate-Potential Approach in the Invariant-Imbedding: Technique: The R-Matrix Method.- 4.4. The Approximate-Solution Approach in the Invariant-Imbedding Technique: The Log-Derivative Method.- References.- Chap. 9. Rotational Excitation II: Approximation Methods.- 1. Introduction.- 2. The CS Approximation.- 2.1. The Basic CS Equations.- 2.2. The CS Scattering Amplitude and Boundary Conditions.- 2.3. CS Differential and Integral Cross Sections.- 2.4. CS Approximation for General Relaxation Cross Sections.- 3. The IOS Approximation.- 3.1. Basic IOS Equations and Boundary Conditions.- 3.2. IOS Cross Sections and Factorizations.- 3.3. IOS Factored Rates and Transport Properties.- 4. The lz-Conserving Energy Sudden Approximation.- 4.1. Basic lz-Conserving Equations and Boundary Conditions.- 4.2. Factorization of lz-Conserving Amplitudes and Cross Sections.- 5. The Decoupled l-Dominant Approximation.- 6. Exponential Distorted-Wave Approximation.- 7. Semiclassical Approximation.- 8. Method Selection.- 8.1. Energy Sudden Approximation.- 8.2. Centrifugal Sudden Approximation.- 8.3. Infinite-Order Sudden Approximation.- 8.4. lz-Conserving and DLD Approximations.- 8.5. Exponential Distorted-Wave Approximation.- 8.6. Semiclassical Approximations.- 8.7. Full Close Coupling.- References.- Chap. 10. Rotational Excitation III: Classical Trajectory Methods.- 1. Introduction.- 2. Ingredients of a Trajectory Calculation.- 2.1. Equations of Motion.- 2.2. Selection of Initial Conditions.- 2.3. Integration of Equations of Motion.- 2.4. Analysis of Final Conditions.- 3. Construction of a Trajectory Program.- 4. Efficiency-Improving Techniques.- 4.1. Alternative Sampling Schemes.- 4.2. Moment Methods.- 5. Concluding Remarks.- References.- Chap. 11. Vibrational Excitation I: The Quantal Treatment.- 1. Introduction.- 2. Angular Momentum Decoupling Approximations.- 3. Asymptotic Expansion Technique for Handling Long-Range Potentials.- 4. Effects of the Dissociative Continuum.- References.- Chap. 12. Vibrational Excitation II: Classical and Semiclassical Methods.- 1. Introduction.- 2. Quasiclassical Methods.- 3. Semiclassical Methods.- 3.1. Quantal Internal Modes Coupled through the Interaction Potential to Classical Translational Motion.- 3.2. Classical S-Matrix Theory.- 3.3. Classical—Quantal Correspondence Methods.- 3.3.1. The decent and indecent Methods.- 3.3.2. The Strong-Coupling Correspondence Principle.- 3.4. Models for Special Cases.- 3.4.1. itfits Models.- 3.4.2. Angular Dependence of Impulsive Energy Transfer.- 3.4.3. Near-Resonant V—V and V—R Transitions Induced by Long-Range Forces.- 4. Approximations.- 4.1. Dynamical Approximations.- 4.1.1. Sudden Approximation (SA).- 4.1.2. Impact Parameter (IP) Approximation.- 4.1.3. Neglect of ?T.- 4.2. Dimensional Approximations.- 4.2.1. Collinear Models.- 4.2.2. The Breathing Sphere (BS) Model.- 4.3. Influence of the Potential Energy Surface.- 4.3.1. Diatomic Molecule Potential.- 4.3.2. Interaction Potential.- 5. Conclusions and Recommendations.- References.- Chap. 13. Electronic Excitation: Nonadiabatic Transitions.- 1. Introduction.- 1.1. Physical Considerations.- 1.2. Notation.- 2. Equations of Motion.- 2.1. The Body-Fixed Hamiltonian.- 2.2. Adiabatic and Diabatic Representations.- 2.2.1. Diabatic Representation.- 2.2.2. Adiabatic Representation.- 2.2.3. Diabatic-to-Adiabatic Transformation.- 2.2.4. Two-State Model.- 2.2.5. Reduced Adiabatic and Diabatic Representations.- 2.2.6. Choice of Representation.- 2.3. Nonadiabatic Coupling Mechanisms.- 2.3.1. Configuration Interaction.- 2.3.2. Spin—Orbit Coupling.- 2.3.3. Angular Momentum Decoupling.- 3. Nonadiabatic Transition Probabilities and Cross Sections.- 3.1. Two-State Models in One Degree of Freedom.- 3.1.1. Curve Crossing.- 3.1.2. Demkov Coupling.- 3.2. Inelastic Atom—Atom Scattering.- 3.2.1. Differential Cross Section.- 3.2.2. Total Cross Section.- 3.3. Inelastic and Reactive Collinear Atom—Diatom Scattering.- 3.3.1. Vibronic Network Formulation.- 3.3.2. Multi-Curve-Crossing Approximations.- 3.3.3. Franck—Condon Approximation.- 3.3.4. Surface-Hopping Trajectory Approximations.- 3.3.5. Comparison and Conclusions.- References.- Chap. 14. Reactive Scattering Cross Sections I: General Quantal Theory.- 1. Introduction.- 2. Quantal Reactive Scattering Processes.- 3. Formal Scattering Theory for Reactive Collisions.- 4. R-Matrix Theory.- 5. Coupled-Equations Approach.- 6. Summary.- References.- Chap. 15. Reactive Scattering Cross Sections II: Approximate Quantal Treatments.- 1. Introduction.- 2. Angular Momentum Decoupling: J=, Conserving.- 3. Born-Type Approximations.- 3.1. T-Matrix Elements and Distortion Potentials.- 3.2. Born Approximation: Energy Dependence of the Total Reaction Cross Section.- 3.3. Born Approximation: Product State Internal Energy Distributions.- 3.4. Spectator-Stripping Model.- 3.5. Distorted Waves: Methodology.- 3.6. Distorted Waves: Numerical Results.- 4. Overlap Models (Franck—Condon Factors).- 4.1. Introduction.- 4.2. Collinear Reactions: Vibrational Distributions.- 4.3. Three-Dimensional Reactions: Vibration—Rotation Distributions.- 5. Other Approaches, and Conclusions.- References.- Chap. 16. Reactive Scattering Cross Sections III: Quasiclassical and Semiclassical Methods.- 1. Introduction.- 2. Quasiclassical Trajectory Method.- 2.1. Equations of Motion.- 2.2. Initial Conditions.- 2.3. Calculation of a Trajectory.- 2.4. Sampling and Averaging over the Initial Conditions.- 2.5. Product Analysis.- 2.6. Calculation of Reaction Attributes.- 2.6.1. Reactivity Functions.- 2.6.2. Final-State Distributions.- 2.6.3. Initial-State—Final-State Correlations.- 2.7. Available Programs.- 3. Other Trajectory Methods for Single-Surface Reactions.- 3.1. Unquantized Initial and Final Conditions.- 3.2. Symmetrically Averaged Initial and Final Conditions.- 3.3. Exactly Quantized Initial and Final Conditions.- 3.4. Classical S Matrix Theory and Other Semiclassical Methods That Include Interference Effects.- 4. Trajectory Methods for Multisurface Reactions.- 5. Concluding Remarks.- References.- Chap. 17. Direct-Mode Chemical Reactions I: Methodology for Accurate Quantal Calculations.- 1. Introduction.- 2. Coordinate Geometry and Hamiltonian.- 2.1. Space-Fixed and Body-Fixed Coordinates.- 2.2. Natural Translation—Vibration Coordinates.- 2.3. Natural Bending Angle.- 2.4. Matching Surfaces and Arrangement Tubes.- 3. Internal Basis Sets.- 3.1. Partitioning of Hamiltonian.- 3.2. Vibrational Basis.- 3.3. Rotational Bases.- 3.3.1. Free-Rotor and Adiabatic Representations.- 3.3.2. Asymmetric-Top Correlation Diagrams.- 3.3.3. Partitioned Rotors.- 4. Close-Coupling Equations.- 4.1. Wave Function Scaling and Basis Expansion.- 4.2. Propagation Equations.- 4.3. Integration Techniques.- 4.4. Numerical Details.- 5. Matching Surface Continuity and Asymptotic Boundary Conditions.- 6. Selected Results.- 6.1. Threshold Energies: H + H2.- 6.2. Resonance Effects: H + H2.- 6.3. Quantized Whirlpools:F + H2.- 7. Concluding Remarks.- References.- Chap. 18. Direct-Mode Chemical Reactions II: Classical Theories.- 1. Introduction.- 2. Determination of Relevant Potential Surfaces.- 3. Models for Multisurface Reactions.- 4. Simple Models for Single-Surface Reactions.- 4.1. Entrance-Channel Models.- 4.2. Channel-to-Channel Models and Angular Momentum Considerations.- 4.3. More Detailed Dynamic Models of Product-State Distributions.- 5. Numerical Trajectories.- 5.1. Angular Distributions.- 5.2. Rotational Energy.- 5.3. Vibrational Energy.- References.- Chap. 19. Complex-Mode Chemical Reactions: Statistical Theories of Bimolecular Reactions.- 1. Introduction.- 2. Averaging and Constraints.- 3. Statistical Theories.- 3.1. “Loose” Transition States.- 3.2. “Tight” Transition States.- References.- Chap. 20. Collision-Induced Dissociation I: Quantal Treatment.- 1. Introduction.- 2. Quantal Description of the CID Process.- 3. Quantal Approximation Methods.- 3.1. Born Approximation (BA).- 3.2. Distorted-Wave Born Approximation (DWBA).- 3.3. Variational Methods.- 3.4. Impulse Approximation (IA).- 4. Close-Coupling Methods.- 5. Concluding Remarks.- References.- Chap. 21. Collision-Induced Dissociation II: Trajectories and Models.- 1. Classical Trajectory Calculations.- 1.1. General Considerations.- 1.1.1. Introduction.- 1.1.2. Choosing a Potential Energy Surface.- 1.2. Identification of the Product Channel for CID.- 1.2.1. General Strategy.- 1.2.2. The Strong-Coupling Region (SCR).- 1.2.3. Quasibound Molecules.- 1.2.4. Identification of CID Channel.- 2. Classical Models for CID.- 2.1. Overview.- 2.1.1. Availability.- 2.1.2. Dynamical Models.- 2.1.3. Hard-Sphere Models.- 2.1.4. Recommendations.- 2.2. The Square-Well Trajectory Model.- 2.2.1. The Potential Energy Function.- 2.2.2. Quasibound Molecules.- 2.2.3....