Quantum Mechanics for Chemistry

Cover
Half Title
Quantum Mechanics for Chemistry
Copyright
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