Atoms and Molecules

Atoms and Molecules describes the basic properties of atoms and molecules in terms of group theoretical methods in atomic and molecular physics. The book reviews mathematical concepts related to angular momentum properties, finite and continuous rotation groups, tensor operators, the Wigner-Eckart theorem, vector fields, and vector spherical harmonics. The text also explains quantum mechanics, including symmetry considerations, second quantization, density matrices, time-dependent, and time-independent approximation methods. The book explains atomic structure, particularly the Dirac equation in which its nonrelativistic approximation provides the basis for the derivation of the Hamiltonians for all important interactions, such as spin-orbit, external fields, hyperfine. Along with multielectron atoms, the text discusses multiplet theory, the Hartree-Fock formulation, as well as the electromagnetic radiation fields, their interactions with atoms in first and higher orders. The book explores molecules and complexes, including the Born-Oppenheimer approximation, molecular orbitals, the self-consistent field method, electronic states, vibrational and rotational states, molecular spectra, and the ligand field theory. The book can prove useful for graduate or advanced students and academicians in the field of general and applied physics.

Preface

Part I Mathematical Background

Chapter 1 Angular Momentum

1.1 Orbital Angular Momentum

1.2 Spherical Harmonics and Related Functions

1.3 Generalized Angular Momentum

1.4 Spin

1.5 Coupling of Two Angular Momenta

1.6 Coupling of Three Angular Momenta

1.7 Summary and Examples

Chapter 2 Rotations

2.1 Coordinate Rotations and Scalar Functions

2.2 Rotations and Angular Momenta

2.3 Transformation Properties of Angular Momentum Eigenfunctions

Chapter 3 Elements of Group Theory

3.1 Definitions and Basic Properties

3.2 Representations and Characters

3.3 Reducible and Irreducible Representations

3.4 Basis Functions

3.5 Projection Operators

3.6 Product Representations

3.7 Matrix Elements

Chapter 4 Continuous Rotation Groups

4.1 Rotation Group in Two Dimensions, C∞

4.2 Rotation Group in Three Dimensions, 0+(3)

4.3 Special Unitary Group in Two Dimensions, SU(2)

4.4 Connection between 0+(3) and SU(2)

4.5 Irreducible Representations of 0+(3)

4.6 Summary and Examples

Chapter 5 Finite Groups

5.1 Point Groups—Symmetry Operations and Nomenclature

5.2 Double Groups

5.3 The Groups 0, D4 and D6h

5.4 Permutation Groups Sn — Young Diagrams

Chapter 6 Tensors

6.1 Irreducible Tensor Operators

6.2 Tensor Products

6.3 Wigner-Eckart Theorem

6.4 Cartesian Tensors

6.5 Tensors, Permutation Groups, Continuous Groups

Chapter 7 Vector Fields

7.1 Rotational Properties

7.2 Vector Spherical Harmonics

7.3 Plane Wave Expansion

7.4 Multipole Expansion of the Electromagnetic Field

Part II Quantum-Mechanical Background

Chapter 8 Symmetry Elements of the Hamiltonian

8.1 Connection Between Group Theory and Quantum Mechanics

8.2 Geometrical Symmetries

8.3 Time Reversal and Kramers' Theorem

8.4 Indistinguishability of Particles

Chapter 9 Time Development of a Quantum Syste

9.1 Schrödinger Representation

9.2 Heisenberg Representation

9.3 Interaction Representation

9.4 Infinite Limits

Chapter 10 Harmonic Oscillator

10.1 Schrödinger Solutions

10.2 Matrix Formulation

10.3 Heisenberg Representation

Chapter 11 Slater Determinants

11.1 Matrix Elements—General

11.2 Matrix Elements—Special Cases

Chapter 12 Second Quantization

12.1 Creation and Annihilation Operators

12.2 Matrix Elements of Operators

12.3 Diagrams

12.4 Field Operators

Chapter 13 Density Matrices

13.1 General Properties

13.2 Spin States

13.3 Reduced Density Matrices

13.4 Thermal Equilibrium

13.5 Equation of Motion

13.6 Multielectron Systems

13.7 Fock-Dirac Density Matrices

13.8 Spinless Density Matrices

Chapter 14 Approximations

14.1 Variational Methods

14.2 Time-Independent Perturbations

14.4 Fermi's Golden Rule

14.5 Density Matrices—Random Perturbations

14.6 Response Function; Susceptibility

Part III One-Electron Atoms

Chapter 15 Dirac Equation

15.1 Free Particle Equation

15.2 Dirac Equation with Electromagnetic Coupling

Chapter 16 Hydrogen Atom

16.1 Schrödinger Equation

16.2 One-Electron Wave Functions

16.3 Spin-Orbit Coupling

16.4 Other Interactions

Chapter 17 Static Fields

17.1 Magnetic Fields

17.2 Electric Fields

Chapter 18 Hyperfine Interactions

18.1 Hamiltonian for the Magnetic Hyperfine Interaction

18.2 Magnetic Hyperfine Interaction in One-Electron Systems

18.3 Electric Quadrupole Interaction

Part IV W-Electron Atoms

Chapter 19 Hartree-Fock Formulation

19.1 The Hamiltonian

19.2 Central Field Approximation

19.3 Hartiee-Fock Equations

19.4 Properties of the Hartree-Fock Solutions

19.5 Computational Methods

19.6 Correlation Error and Configuration Interaction

Chapter 20 Multiplet Wave Functions

20.1 Two-Electron Multiplets

20.2 Terms from a Configuration of n Electrons

20.3 Construction of Multiplet Wave Functions

20.4 Symmetry Properties

20.5 jj Coupling

Chapter 21 Matrix Elements

21.1 Electrostatic Matrix Elements—Two Electrons

21.2 Some n-Electron Matrix Elements

21.3 Electrostatic Matrix Elements—n Electrons

21.4 Spin- Orbit Interaction

21.5 Conjugate Configurations

21.6 Other Interactions

Part V Electromagnetic Interactions

Chapter 22 Interaction Between Atoms and Radiation

22.1 Hamiltonian of the Radiation Field

22.2 Quantization of the Radiation Field

22.3 Interaction Hamiltonian and Matrix Elements

22.4 Selection Rules and Angular Distributions

Chapter 23 Absorption and Emission

23.1 Transition Probabilities

23.2 Einstein Coefficients and Planck's Law

23.3 Oscillator Strengths and Sum Rules

23.4 Numerical Computations

23.5 Line Broadening

23.6 Cross Sections

23.7 Photoelectric Effect

23.8 Survey of Atomic Spectra

Chapter 24 Higher Order Electromagnetic Interactions

24.1 The Kramers-Heisenberg Formula

24.2 Scattering—Special Cases

24.3 Diagrams

24.4 Optical Susceptibility and Nonlinear Effects

Part VI Molecules

Chapter 25 General Properties of Molecules

25.1 Born-Oppenheimer Approximation

25.2 Molecular Orbitals and the Self-Consistent Field Method

25.3 Computational Methods

Chapter 26 Electronic States of Molecules

26.1 Hydrogen Molecule Ion (H2+)

26.2 Symmetry Considerations—H2+

26.3 Hydrogen Molecule

26.4 Diatomic and Linear Molecules

26.5 Hybrid Orbitals

26.6 The π-Electron Approximation

Chapter 27 Molecular Spectra

27.1 Vibrations and Rotations of Diatomic Molecules

27.2 Transitions in Diatomic Molecules

27.3 Vibration of Polyatomic Molecules

27.4 Transitions in Polyatomic Molecules

Chapter 28 Ligand Fields

28.1 Basic Ideas

28.2 Single d Electron in an Octahedral and Tetragonal Field

28.3 Multielectron Configurations

28.4 Magnetic Fields and the Spin Hamiltonian

28.5 Molecular Orbitals

Appendix 1 Dirac Notation

Appendix 2 Operators

Appendix 3 Eigenvalues and Eigenfunctions

Appendix 4 Relationships Among Unit Vectors

Appendix 5 Bessel Functions

Appendix 6 Laguerre Polynomials

Appendix 7 Hermite Polynomials

Appendix 8 Dirac δ-Functions

References

Index