This textbook forms the basis for an advanced undergraduate or graduate level quantum chemistry course, and can also serve as a reference for researchers in physical chemistry and chemical physics. In addition to the standard core topics such as principles of quantum mechanics, vibrational and rotational states, hydrogen-like molecules, perturbation theory, variational principles, and molecular orbital theories, this book also covers essential theories of electronic structure calculation, the primary methods for calculating quantum dynamics, and major spectroscopic techniques for quantum measurement. Plus, topics that are overlooked in conventional textbooks such as path integral formulation, open system quantum dynamics methods, and Green’s function approaches are addressed. This book helps readers grasp the essential quantum mechanical principles and results that serve as the foundation of modern chemistry and become knowledgeable in major methods of computational chemistry and spectroscopic experiments being conducted by present-day researchers. Dirac notation is used throughout, and right balance between comprehensiveness, rigor, and readability is achieved, ensuring that the book remains accessible while providing all the relevant details. Complete with exercises, this book is ideal for a course on quantum chemistry or as a self-study resource.
Author(s): Seogjoo J. Jang
Publisher: Springer
Year: 2023
Language: English
Pages: 441
City: Cham
Preface
Contents
Physical Constants and Abbreviations
Unit Conventions
Abbreviations
1 Concepts and Assumptions of Quantum Mechanics
1.1 Assumptions of Classical Mechanics
1.1.1 Classical Point Particles
1.1.2 Wave: Classical View
1.1.3 Particle Versus Wave
1.2 Concepts of Quantum Mechanics
1.2.1 Blackbody Radiation
1.2.2 Photoelectric Effect and Photon
1.2.3 de Broglie's Postulate of Material Wave
1.2.4 Heisenberg's Uncertainty Principle
1.2.5 Wave-Particle Duality
1.3 Schrödinger Equation (in One Dimension) and Probability Amplitude
1.3.1 Wavefunction and Time Dependent Schrödinger Equation
1.3.2 Measurement as Mathematical Operation on Wavefunction
1.3.3 Stationary States and Time Independent Schrödinger Equation
1.3.4 Eigenfunction and Eigenvalue
1.3.5 Linear and Hermitian Operator
1.3.6 Results of Measurement and Expectation Value
1.4 Quantum Particle in a One-Dimensional Box
1.4.1 Time Independent Schrödinger Equation
1.4.2 Time Dependent States
1.4.3 Completeness
1.5 Summary and Questions
Appendix: Dirac-Delta Function
Exercise Problems with Solutions
Problems
2 Dirac Notation and Principles of Quantum Mechanics
2.1 Formulation of Quantum Mechanics
2.2 Ket, Bra, and Products
2.3 Operators
2.3.1 Hermitian Operator
2.3.2 One Dimensional Position Operator and Eigenket
2.3.3 One Dimensional Momentum Operator and Eigenket
2.3.4 Expressions for Momentum Operator
2.3.5 Schrödinger Equations in the Dirac Notation
2.3.6 Commutator
2.3.7 Compatibility and Completeness
2.3.8 Measurement Operator
2.3.9 Unitary Operator
2.4 Particle in a One-Dimensional Box: Revisited with the Dirac Notation
2.5 Direct Product
2.6 Summary and Questions
Appendix: Cauchy-Schwarz Inequality and a General Uncertainty Relationship
Exercise Problems with Solutions
Problems
3 Harmonic Oscillator and Vibrational Spectroscopy
3.1 Classical Harmonic Oscillator and Hamiltonian
3.2 Schrödinger Equation
3.2.1 Solution of Time Independent Schrödinger Equation
3.2.2 Operator Approach
3.2.3 General Time Dependent State
3.3 Vibrational Spectroscopy of Diatomic Molecules
3.3.1 Vibrational Absorption or Infrared (IR) Spectroscopy
3.3.2 Vibrational Raman Spectroscopy
3.3.3 Anharmonic Effects
3.4 Summary and Questions
Exercise Problems with Solutions
Problems
4 Multidimensional Systems and Separation of Variables
4.1 Three Dimensional System
4.1.1 Position, Momentum, Hamiltonian, and Schrödinger Equation
4.1.2 Particle in a Three Dimensional Rectangular Box
4.1.3 Separation in Cartesian Coordinate System
4.2 Many Particle Systems and the Center of Mass Coordinates
4.2.1 Two-Particle System
4.2.2 Normal Modes and Vibrational Spectroscopy of Polyatomic Molecules
4.3 Summary and Questions
Exercise Problems with Solutions
Problems
5 Rotational States and Spectroscopy
5.1 Rotation in Two Dimensional Space
5.2 Rotation in Three Dimensional Space
5.3 Angular Momentum Operators
5.4 Spectroscopy of Rotational Transitionsfor Diatomic Molecules
5.4.1 Microwave Spectroscopy
5.4.2 Rotational Raman Spectroscopy
5.4.3 Ro-Vibrational Transition
5.4.4 Centrifugal Correction and Ro-Vibrational Coupling
5.5 Summary and Questions
Appendix: Associated Legendre Equations and Their Solutions
Exercise Problems with Solutions
Problems
6 Hydrogen-Like Systems and Spin Orbit States of an Electron
6.1 Bohr's Model
6.2 Solution of Schrödinger Equation
6.3 Separation of Variables in Spherical Coordinate System
6.3.1 Radial Equation and Solution
6.3.2 Radial Probability Density
6.3.3 Eigenfunctions and Eigenstates in the Dirac Notation
6.3.4 Zeeman Effect
6.3.5 Real-Valued Orbital Functions
6.4 Spin States
6.5 Electronic Transitions and Term Symbols
6.6 Summary and Questions
Appendix: Solutions of the Radial Equation
Exercise Problems with Solutions
Problems
7 Approximation Methods for Time Independent Schrödinger Equation
7.1 Variational Principle
7.1.1 General Case
7.1.2 Variational Principle for Trial States as Linear Combinations of Basis States
7.2 Time Independent Perturbation Theory
7.2.1 Non-Degenerate Perturbation Theory
7.2.2 Degenerate Perturbation Theory
7.3 Summary and Questions
Exercise Problems with Solutions
Problems
8 Many Electron Systems and Atomic Spectroscopy
8.1 Hamiltonian
8.2 Independent Electron Model
8.2.1 Major Assumptions
8.2.2 Orbitals and Electronic Configuration
8.2.3 Spin States
8.2.4 Energy Levels of Spin-Orbit States
8.2.5 Examples of Energy Levels Based on LS-Coupling Scheme
8.2.6 Atomic Spectroscopy: Selection Rules and Simple Examples
8.3 Case Study of Helium Atom
8.3.1 Hamiltonian and Schrödinger Equation
8.3.2 Independent Electron Model with Variational Optimization of Effective Charge
8.3.3 Self Consistent Field (SCF) Approximationfor Helium
8.4 Self Consistent Field (SCF) Approximation for Many Electron Atoms
8.4.1 Hartree Approximation
8.4.2 Hartree-Fock Approximation
8.5 Summary and Questions
Exercise Problems with Solutions
Problems
9 Polyatomic Molecules and Molecular Spectroscopy
9.1 Born-Oppenheimer Approximation
9.2 Molecular Orbitals and Electronic Configurations for Diatomic Molecules
9.2.1 Example of H2
9.2.2 Molecular Orbitals and Electronic Configurations of Diatomic Molecules
9.2.3 Molecular Electronic States of Diatomic Molecules
9.3 Conjugated Hydrocarbons and Hückel Approximation
9.3.1 Ethylene
9.3.2 Butadiene
9.3.3 π Orbital and Delocalization Energies
9.4 Molecular Symmetry and Group Theory
9.4.1 Symmetry and Symmetry Operation
9.4.2 Group Theory
9.4.3 Groups of Point Symmetry Operations
9.4.4 Matrix Representation of Point Symmetry Group Elements
9.4.5 Application for Symmetry Adapted LCAO-MO
9.5 Spectroscopy of Polyatomic Molecules
9.5.1 Infrared and Raman Spectroscopy
9.5.2 Electronic Spectroscopy
9.6 Summary and Questions
Appendix: Important Theorems and Proofs in the Group Theory
Exercise Problems with Solutions
Problems
10 Quantum Dynamics of Pure and Mixed States
10.1 Quantum Dynamics of Pure States
10.1.1 Heisenberg Picture
10.1.2 Interaction Picture and Time Dependent Perturbation Theory
10.1.3 Fermi's Golden Rule
10.2 Quantum Dynamics of Mixed Quantum States
10.2.1 Density Operator and Quantum Liouville Equation
10.2.2 Time Dependent Perturbation Theory for Mixed Quantum States
10.2.3 FGR for Mixed States
10.3 Summary and Questions
Appendix: Interaction Hamiltonian in the Presence of Radiation
Exercise Problems with Solutions
Problems
11 Theories for Electronic Structure Calculation of Polyatomic Molecules
11.1 Hartree-Fock Approximation and Roothaan Equation
11.1.1 General Single Determinant State
11.1.2 Restricted HF Equation for Doubly FilledOrbital States
11.1.3 Linear Combination of Basis States
11.1.4 Choice of Basis Functions
11.1.5 Methods Beyond HF Approximation
11.2 Density Functional Theory
11.3 Summary and Questions
Exercise Problems with Solutions
Problems
12 Special Topics
12.1 Path Integral Representation
12.1.1 Real Time Propagator
12.1.2 Imaginary Time Propagator
12.2 Quantum Master Equation for Open SystemQuantum Dynamics
12.2.1 Projection Operator Formalism and Exact Time Evolution Equations for a Projected Density Operator
12.2.2 Quantum Master Equations for a Reduced System Density Operator
12.2.2.1 Formally Exact QMEs
12.2.2.2 Second Order QMEs
12.3 Green's Function Approach
12.3.1 Second Quantization and Field Operators
12.3.2 Ground State (Zero Temperature) Green's Functions
12.3.3 Nonequilibrium Green's Functions
References
Index
