Quantum Chemistry

Quantum Chemistry covers the basic principles, methods, and results of quantum chemistry, providing insights on electron behavior. This book is organized into 14 chapters that focus on ground state molecular orbital theory. After briefly dealing with some of the concepts of classical physics, the book goes on describing some simple but important particle systems. It then examines several systems with discontinuous potential energies, such as the simple harmonic oscillator and the hydrogen-like ion system. A chapter presents a set of postulates and theorems that form the formal foundation of quantum mechanics. Considerable chapters are devoted to various quantum chemical methods, as well as their basic features and application to molecular orbital evaluation. These methods include Huckel molecular orbital, variation, linear variation, extended Huckel, and SCF-LCAO-MO. The concluding chapters deal with the development of theories for molecular orbital, including time-independent Rayleigh-Schrodinger perturbation, group, and qualitative molecular orbital theories. Supplemental texts of the more complicated derivations or proofs and problems encountered in quantum chemistry are also provided. This book is an introductory text intended for organic, inorganic, and physical chemists, as well as for graduate and undergraduate students.

Preface

Chapter 1 Classical Waves and the Time-Independent Schrödinger Wave Equation

1-1 Introduction

1-2 Waves

1-3 The Classical Wave Equation

1-4 Standing Waves in a Clamped String

1-5 Light as an Electromagnetic Wave

1-6 The Photoelectric Effect

1-7 The Wave Nature of Matter

1-8 A Diffraction Experiment with Electrons

1-9 Schrödinger's Time-Independent Wave Equation

1-10 Conditions on ψ

1-11 Some Insight into the Schrödinger Equation

1-12 Summary

Problems

Chapter 2 Quantum Mechanics of Some Simple Systems

2-1 The Particle in a One-Dimensional "Box"

2-2 Detailed Examination of Particle-in-a-Box Solutions

2-3 The Particle in a One-Dimensional "Box" with One Finite Wall

2-4 The Particle in an Infinite "Box" with a Finite Central Barrier

2-5 The Free Particle in One Dimension

2-6 The Particle in a Ring of Constant Potential

2-7 The Particle in a Three-Dimensional Box: Separation of Variables

2-8 Summary

Problems

Reference

Chapter 3 The One-Dimensional Harmonic Oscillator

3-1 Introduction

3-2 Some Characteristics of the Classical One-Dimensional Harmonic Oscillator

3-3 The Quantum-Mechanical Harmonic Oscillator

3-4 Solution of the Harmonic Oscillator Schrödinger Equation

3-5 Quantum-Mechanical Average Value of the Potential Energy

Problems

Chapter 4 The Hydrogenlike Ion

4-1 The Schrödinger Equation and the Nature of Its Solutions

4-2 Separation of Variables

4-3 Solution of the R, Θ, and Φ Equations

4-4 Atomic Units

4-5 Angular Momentum and Spherical Harmonics

4-6 Summary

Problems

References

Chapter 5 Many-Electron Atoms

5-1 The Independent Electron Approximation

5-2 Simple Products and Electron Exchange Symmetry

5-3 Electron Spin and the Exclusion Principle

5-4 Slater Determinants and the Pauli Principle

5-5 Singlet and Triplet States for the ls2s Configuration of Helium

5-6 The Self-Consistent Field, Slater-Type Orbitals, and the Aufbau Principle

Problems

References

Chapter 6 Postulates and Theorems of Quantum Mechanics

6-1 Introduction

6-2 The Wavefunction Postulate

6-3 The Postulate for Constructing Operators

6-4 The Time-Dependent Schrödinger Equation Postulate

6-5 The Postulate Relating Measured Values to Eigenvalues

6-6 The Postulate for Average Values

6-7 Hermitian Operators

6-8 Proof That Eigenvalues of Hermitian Operators Are Real

6-9 Proof That Eigenfunctions of an Hermitian Operator Form an Orthonormal Set

6-10 Proof That Commuting Operators Have Simultaneous Eigenfunctions

6-11 Completeness of Eigenfunctions of an Hermitian Operator

6-12 The Variation Principle

6-13 Measurement, Commutators, and Uncertainty

6-14 Summary

Problems

References

Chapter 7 The Variation Method

7-1 The Spirit of the Method

7-2 Nonlinear Variation: The Hydrogen Atom

7-3 Nonlinear Variation: The Helium Atom

7-4 Linear Variation: The Polarizability of the Hydrogen Atom

7-5 Linear Combination of Atomic Orbitals: The H2+ Molecule-Ion

7-6 Molecular Orbitals of Homonuclear Diatomic Molecules

7-7 Basis Set Choice and the Variational Wavefunction

7-8 Beyond the Orbital Approximation

Problems

References

Chapter 8 The Simple Hückel Method and Applications

8-1 The Importance of Symmetry

8-2 The Assumption of σ-π Separability

8-3 The Independent π-Electron Assumption

8-4 Setting up the Hückel Determinant

8-5 Solving the HMO Determinantal Equation for Orbital Energies

8-6 Solving for the Molecular Orbitals

8-7 The Cyclopropenyl System: Handling Degeneracies

8-8 Charge Distributions from HMOs

8-9 Some Simplifying Generalizations

8-10 HMO Calculations on Some Simple Molecules

8-11 Summary: The Simple HMO Method for Hydrocarbons

8-12 Relation between Bond Order and Bond Length

8-13 π-Electron Densities and Electron Spin Resonance Hyperfine Splitting Constants

8-14 Orbital Energies and Oxidation-Reduction Potentials

8-15 Orbital Energies and Ionization Potentials

8-16 π-Electron Energy and Aromaticity

8-17 Extension to Heteroatomic Molecules

8-18 Self-Consistent Variations of α and ß

8-19 HMO Reaction Indices

8-20 Conclusions

Problems

References

Chapter 9 Matrix Formulation of the Linear Variation Method

9-1 Introduction

9-2 Matrices and Vectors

9-3 Matrix Formulation of the Linear Variation Method

9-4 Solving the Matrix Equation

9-5 Summary

Problems

References

Chapter 10 The Extended Hückel Method

10-1 The Extended Hückel Method

10-2 Mulliken Populations

10-3 Extended Hückel Energies and Mulliken Populations

10-4 Extended Hückel Energies and Experimental Energies

Problems

References

Chapter 11 The SCF-LCAO-MO Method and Extensions

11-1 Ab Initio Calculations

11-2 The Molecular Hamiltonian

11-3 The Form of the Wavefunction

11-4 The Nature of the Basis Set

11-5 The LCAO-MO-SCF Equation

11-6 Interpretation of the LCAO-MO-SCF Eigenvalues

11-7 The SCF Total Electronic Energy

11-8 Basis Sets

11-9 The Hartree-Fock Limit

11-10 Correlation Energy

11-11 Koopmans' Theorem

11-12 Configuration Interaction

11-13 Examples of Ab Initio Calculations

11-14 Approximate SCF-MO Methods

Problems

References

Chapter 12 Time-Independent Rayleigh-Schrödinger Perturbation Theory

12-1 An Introductory Example

12-2 Formal Development of the Theory for Nondegenerate States

12-3 A Uniform Electrostatic Perturbation of an Electron in a "Wire"

12-4 The Ground-State Energy to First Order of Heliumlike Systems

12-5 Perturbation at an Atom in the Simple Hückel MO Method

12-6 Perturbation Theory for a Degenerate State

12-7 Polarizability of the Hydrogen Atom in the n = 2 States

12-8 Interaction between Two Orbitals: An Important Chemical Model

12-9 Connection between Time-Independent Perturbation Theory and Spectroscopic Selection Rules

Problems

References

Chapter 13 Group Theory

13-1 Introduction

13-2 An Elementary Example

13-3 Symmetry Point Groups

13-4 The Concept of Class

13-5 Symmetry Elements and Their Notation

13-6 Identifying the Point Group of a Molecule

13-7 Representations for Groups

13-8 Generating Representations from Basis Functions

13-9 Labels for Representations

13-10 Some Connections between the Representation Table and Molecular Orbitals

13-11 Representations for Cyclic and Related Groups

13-12 Orthogonality in Irreducible Inequivalent Representations

13-13 Characters and Character Tables

13-14 Using Characters to Resolve Reducible Representations

13-15 Identifying Molecular Orbital Symmetries

13-16 Determining in Which Molecular Orbital an Atomic Orbital Will Appear

13-17 Generating Symmetry Orbitals

13-18 Hybrid Orbitals and Localized Orbitals

13-19 Symmetry and Integration

Problems

References

Chapter 14 Qualitative Molecular Orbital Theory

14-1 The Need for a Qualitative Theory

14-2 Hierarchy in Molecular Structure and in Molecular Orbitals

14-3 H2+ Revisited

14-4 H2: Comparisons with 2+

14-5 Rules for Qualitative Molecular Orbital Theory

14-6 Application of QMOT Rules to Homonuclear Diatomic Molecules

14-7 Shapes of Polyatomic Molecules: Walsh Diagrams

14-8 Frontier Orbitals

14-9 Qualitative Molecular Orbital Theory of Reactions

Problems

References

Appendix 1 Useful Integrals

Appendix 2 Determinants

Appendix 3 Evaluation of the Coulomb Repulsion Integral over Is AOs

Appendix 4 Some Characteristics of Solutions of the Linear Variation Procedure

Appendix 5 The Pairing Theorem

Appendix 6 Hückel Molecular Orbital Energies, Coefficients, Electron Densities, and Bond Orders for Some Simple Molecules

Appendix 7 Derivation of the Hartree-Fock Equation

Appendix 8 The Virial Theorem for Atoms and Diatomic Molecules

Appendix 9 Details of the Solution of the Matrix Equation HC=SCE

Appendix 10 Computer Program Listings

Appendix 11 Bra-Ket Notation

Appendix 12 Values of Some Useful Constants and Conversion Factors

Appendix 13 Group Theoretical Charts and Tables

Appendix 14 Hints for Solving Selected Problems

Appendix 15 Answers to Selected Problems

Index