Basic Principles and Techniques of Molecular Quantum Mechanics

New textbooks at all levels of chemistry appear with great regularity. Some fields like basic biochemistry, organic reaction mechanisms, and chemical thermody­ namics are well represented by many excellent texts, and new or revised editions are published sufficiently often to keep up with progress in research. However, some areas of chemistry, especially many of those taught at the graduate level, suffer from a real lack of up-to-date textbooks. The most serious needs occur in fields that are rapidly changing. Textbooks in these subjects usually have to be written by scientists actually involved in the research which is advancing the field. It is not often easy to persuade such individuals to set time aside to help spread the knowledge they have accumulated. Our goal, in this series, is to pinpoint areas of chemistry where recent progress has outpaced what is covered in any available textbooks, and then seek out and persuade experts in these fields to produce relatively concise but instructive introductions to their fields. These should serve the needs of one semester or one quarter graduate courses in chemistry and biochemistry. In some cases, the availability of texts in active research areas should help stimulate the creation of new courses. New York, New York CHARLES R. CANTOR Preface This book is not a traditional quantum chemistry textbook. Instead, it represents a concept that has evolved from teaching graduate courses in quantum chemistry over a number of years, and encountering students with diverse backgrounds.

1 Experimental Basis of Quantum Theory.- 1-1. Introductory Remarks.- 1-2. Classical Concepts of Linear Momentum, Angular Momentum, and Energy.- 1-3. Energy Levels and Photons.- 1-4. Electron Impact Experiments.- 1-5. Atomic Spectra.- 1-6. Quantization of Angular Momentum.- 1-7. Momentum of a Photon.- 1-8. Wave-Particle Duality.- Problems.- 2 Vector Spaces and Linear Transformations.- 2-1. Vector Spaces.- 2-2. Linear Independence, Bases, and Dimensionality.- 2-3. Inner Product Spaces.- 2-4. Orthonormality and Complete Sets.- 2-5. Hilbert Space.- 2-6. Function Space and Generalized Fourier Series.- 2-7. Isomorphism between Hilbert Space and Function Space.- 2-8. Examples of Complete Sets of Functions.- 2-9. Extension to Continuum Functions.- 2-10. Function Minimization with Constraints.- 2-11. Linear Operators.- 2-12. Algebra of Linear Operators.- 2-13. Special Kinds of Linear Operators.- 2-14. Eigenvalues and Eigenvectors.- Problems.- 3 Matrix Theory.- 3-1. Elements of Matrix Algebra.- 3-2. Determinants.- 3-3. Characterization of Square Matrices.- 3-4. Matrix Inversion.- 3-5. Matrices Having Special Properties.- 3-6. Matrix Representations of Linear Operators and Matrix Transformations.- 3-7. Changes of Basis and Similarity Transformations.- 3-8. Matrix Eigenvalue Problems.- 3-9. Infinite Matrices and Linear Transformations on Hilbert Space.- 3-10. Dirac Notation.- Problems.- 4 Postulates of Quantum Mechanics and Initial Considerations.- 4-1. Quantum Mechanical States and Observables.- 4-2. Time Evolution of a Quantum State.- 4-3. Quantum Theory of Measurement and Expectation Values.- 4-4. Compatible Observables and Commuting Operators.- 4-5. Constants of Motion and Transition Probabilities.- 4-6. Different Pictures of Quantum Phenomena.- 4-7. Hamiltonian Operator Construction: Initial Considerations.- Problems.- 5 One-Dimensional Model Problems.- 5-1. General Comments.- 5-2. Wavefunction Criteria and Boundary Conditions.- 5-3. The Nondegeneracy Theorem.- 5-4. Particle on a Ring.- 5-5. Particle Trapped in a Box.- 5-6. Parity of Eigenfunctions.- 5-7. Square Well Potential.- 5-8. Double Wells and Tunneling.- 5-9. The Harmonic Oscillator.- 5-10. Zero Point Energy and the Uncertainty Principle.- Problems.- 6 Angular Momentum.- 6-1. Introduction.- 6-2. General Angular Momentum Considerations.- 6-3. Orbital Angular Momentum.- 6-4. Spin Angular Momentum.- Problems.- 7 The Hydrogen Atom, Rigid Rotor, and the H2+ Molecule.- 7-1. Separation of Motion of Center of Mass.- 7-2. Solution of Equation for Relative Electron Motion of the Hydrogen Atom and Hydrogen-Like Atoms.- 7-3. Wavefunction Shapes.- 7-4. Rigid Rotor.- 7-5. The H2+ Molecule.- Problems.- 8 The Molecular Hamiltonian.- 8-1. General Principles and Discussion.- 8-2. Introduction of External Fields.- 8-3. Introduction of Relativistic Effects.- 8-4. The Born-Oppenheimer Approximation.- Problems.- 9 Approximation Methods for Stationary States.- 9-1. The Variation Principle.- 9-2. Accuracy Considerations.- 9-3. Example: The Hydrogen Atom.- 9-4. Example: Variational Treatment of the Helium Atom.- 9-5. The Linear Variation Method.- 9-6. Example: The Hydrogen Atom Revisited.- 9-7. Lower Bounds.- 9-8. Rayleigh-Schrödinger Perturbation Theory.- 9-9. Brillouin-Wigner Perturbation Theory.- Problems.- 10 General Considerations for Many Electron Systems.- 10-1. Early Computational Concepts and Procedures.- 10-2. Symmetry Considerations and Group Theory.- 10-3. Antisymmetry and the Pauli Exclusion Principle.- 10-4. Multielectron Systems and Slater Determinants.- 10-5. Expansion Theorem and Slater Determinant Expansions.- 10-6. Matrix Elements between Slater Determinants.- 10-7. Virial Theorem, Hypervirial Theorem, and Hellmann-Feynman Theorem.- 10-8. Scaling.- 10-9. Coupling of Angular Momenta.- 10-10. Orbital Transformations.- Problems.- 11 Computational Techniques for Many-Electron Systems Using Single Configuration Wavefunctions.- 11-1. Hartree-Fock Theory for Closed Shell Systems.- 11-2. Hall-Roothaan LCAO-MO-SCF Theory for Closed Shell Systems.- 11-3. Hartree-Fock Theory for Open Shell Systems.- Problems.- 12 Beyond Hartree―Fock Theory.- 12-1. Electron Correlation: General Comments.- 12-2. Configuration Interaction.- 12-3. Specialized CI Approaches.- 12-4. Many-Body Perturbation Theory and Coupled Cluster Theory.- Problems.- Appendix 1.- References.