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Index
Cover Page
PHYSICAL CHEMISTRY Quantum Chemistry and Molecular Interactions
PHYSICAL CHEMISTRY Quantum Chemistry and Molecular Interactions
To the Reader
Acknowledgments
About the Author
A Introduction: Tools from Math and Physics
GOAL Why Are We Here?
A.1 Mathematics
Algebra and Units
Basic Formula Manipulations
Unit Analysis and Reasonable Answers
EXAMPLE A.1 Unreasonable Answers
Problem
Solution
SI Units
Complex Numbers
EXAMPLE A.2 Complex Conjugates
Problem
Solution
Trigonometry
Coordinate Systems
EXAMPLE A.3 Cartesian and Polar Coordinates
Problem
Solution
Linear Algebra
Vectors
Matrices
Differential and Integral Calculus
Derivatives
Analytical Integrals
EXAMPLE A.4 Analytical Integration
Problem
Solution
Numerical Integration
Volumes and Stationary Points
EXAMPLE A.5 First Derivatives
Problem
Solution
EXAMPLE A.6 Integration
Problem
Solution
EXAMPLE A.7 Triple Integrals
Problem
Solution
Fourier Transforms
EXAMPLE A.8 Fourier Transform
Problem
Solution
Differential Equations
EXAMPLE A.9 Differential Equations
Problem
Solution
Power Series
Trigonometric and Exponential Series
The Taylor Series
EXAMPLE A.10 Taylor Series
Problem
Solution
A.2 Classical Physics
Force and Energy
Conservation of Mass and Energy
Kinetic, Potential, and Radiant Energy
Angular Momentum
Problems
Unit Analysis and Reasonable Values
Mathematics
Physics
Part I ATOMIC Structure
1 Classical and Quantum Mechanics
Learning Objectives
1.1 Introduction to the Text
1.2 The Classical World
Classical Matter
Classical Radiation
SAMPLE CALCULATION Convert between Frequency and Wavelength.
1.3 The Quantum World
Quantum Radiation
EXAMPLE 1.1 Planck’s Law
Context
Problem
Solution
Quantum Matter
EXAMPLE 1.2 De Broglie Wavelength and the Correspondence Principle
Context
Problem
Solution
Why do we use LEED?
How does it work?
1.4 One-Electron Atoms
The Hydrogen Atom Spectrum
EXAMPLE 1.3 The Rydberg Equation
Context
Problem
Solution
The Bohr Model
EXAMPLE 1.4 Properties of the Bohr Atom
Context
Problem
Solution
SAMPLE CALCULATION One-Electron Atom Ionization Energies.
1.5 Merging the Classical and Quantum Worlds
De Broglie Waves in the Bohr Atom
The Wavepacket and the Uncertainty Principle
Key Concepts and Equations
1.3 The Quantum World.
1.4 One-Electron Atoms.
1.5 Merging the Classical and Quantum Worlds.
Key Terms
Objectives Review
2 The Schrödinger Equation
Learning Objectives
2.1 Mathematical Tools of Quantum Mechanics
The Schrödinger Equation
The Wavefunction
Mathematical and Physical Definitions
EXAMPLE 2.1 General Form of Wavefunctions
Context
Problem
Solution
Normalization
EXAMPLE 2.2 Normalization and Probability Density
Context
Problem
Solution
Orthogonality and Basis Sets
EXAMPLE 2.3 Orthogonality
Context
Problem
Solution
EXAMPLE 2.4 Orthogonality by Symmetry
Problem
Solution
The Hamiltonian and Other Operators
EXAMPLE 2.5 Commuting Operators
Context
Problem
Solution
EXAMPLE 2.6 Quantum Average Value Theorem
Context
Problem
Solution
2.2 Fundamental Examples
The Free Particle
The Kinetic Energy Operator
Wavefunction Phase
Solving the Schrödinger Equation
The Particle in a One-Dimensional Box
EXAMPLE 2.7 Particle in a Box Energies
Context
Problem
Solution
The Particle in a Three-Dimensional Box
Energies and Wavefunctions
SAMPLE CALCULATION Energy of a Particle in a Three-Dimensional Box.
Degeneracy and the Three-Dimensional Box
Key Concepts and Equations
2.1 Mathematical tools of quantum mechanics.
2.2 Fundamental examples.
Key Terms
Objectives Review
Problems
Discussion Problems
Wavefunctions and Operators
The Particle in a One-Dimensional Box
Particles in Two and Three Dimensions
Chapter 3 One-Electron Atoms
3.1 Solving the One-Electron Atom Schrödinger Equation
What to Expect
The One-Electron Atom Schrödinger Equation
Plan of Attack
The Angular Solution
DERIVATION SUMMARY The Angular Solution.
The Radial Solution
Derivation Summary sf1 The Radial Solution.
3.2 The One-Electron Atom Orbital Wavefunctions
Interpretation of the Quantum Numbers
The Angular Part
EXAMPLE 3.1 Spherical Harmonics and Angular Momentum
Context
Problem
Solution
Cartesian Orbitals
The Radial Part
SMAPLE CALCULATION Atomic Orbital Wavefunctions.
EXAMPLE 3.2 Atomic Orbital Characteristics
Context
Problem
Solution
EXAMPLE 3.3 Atomic Orbital Probability Densities
Context
Problem
Solution
EXAMPLE 3.4 Average Properties of the One-Electron Atom
Context
Problem
Solution
Degeneracy and the Imaginary Part of the Wavefunction
3.3 Electric Dipole Interactions
What is atomic absorption spectroscopy?
Why do we use atomic absorption spectroscopy?
How does it work?
3.4 Magnetic Dipole Interactions
Orbital Magnetic Moment
Spin Magnetic Moment
Spin-Orbit Interaction
SAMPLE CALCULATION The Spin-Orbit Constant.
EXAMPLE 3.5 Spin-Orbit States of content
Context
Problem
Solution
Choice of Angular Momentum Quantum Numbers
Magnetic Resonance
Key Concepts and Equations
3.1 Solution of the Schrödinger equation.
3.3 Electric dipole interactions.
3.4 Magnetic dipole interactions.
Key Terms
Objectives Review
Problems
Discussion Problems
Angular Momentum and the One-Electron Atom Hamiltonian
The One-Electron Atom Wavefunctions
Integrals to Calculate Properties of the Orbitals
Magnetic Field Interactions
4 Many-Electron Atoms
4.1 Many-Electron Spatial Wavefunctions
Electron Configurations
EXAMPLE 4.1 Many-Electron Atom Hamiltonians
Context
Problem
Solution
Electron–Electron Repulsion and Shielding
What is photoelectron spectroscopy?
Why do we use photoelectron spectroscopy?
How does it work?
4.2 Approximate Solution to the Schrödinger Equation
Perturbation Theory
DERIVATION SUMMARY Perturbation Theory.
EXAMPLE 4.2 Second-Order Perturbation Theory
Context
Problem
Solution
Variation Theory
Hartree-Fock Calculations
EXAMPLE 4.3 Hartree-Fock Energies and Effective Atomic Number
Context
Problem
Solution
4.3 Spin Wavefunctions and Symmetrization
Indistinguishability and the Pauli Exclusion Principle
Spin Multiplicity and Excited States
Slater Determinants
4.4 Vector Model of the Many-Electron Atom
EXAMPLE 4.4 The Vector Model of Atomic Oxygen
Context
Problem
Solution
4.5 Periodicity of the Elements
4.6 Atomic Structure: The Key to Chemistry
Key Concepts and Equations
4.1 Many-electron spatial wavefunctions.
4.2 Computational methods.
4.3 Spin wavefunctions and symmetrization.
4.4 Vector model of the many-electron atom.
4.5 Periodicity of the elements.
4.6 Atomic structure.
Key Terms
Objectives Review
Problems
Discussion Problems
Many-Electron Hamiltonians and Wavefunctions
Spin and Symmetrization
Perturbation and Variation Theory
Hartree-Fock Energies and Orbitals
The Vector Model
Part II MOLECULar Structure
5 Chemical Bonds and Nuclear Magnetic Resonance
Learning Objectives
5.1 The Molecular Hamiltonian
Electrostatics of Covalent Bonding
The General Form of the Hamiltonian
Separating the Variables
5.2 The Molecular Wavefunction
The One-Electron Covalent Bond: inline
Many-Electron Molecular Orbital Wavefunctions
Valence Bond Wavefunctions
Effective Potential Energy Functions in Diatomics
5.3 Covalent Bonds in Polyatomic Molecules
Wavefunctions for Polyatomics and Local Bonding
Hybrid Orbitals
EXAMPLE 5.1 Hybrid Orbital Bond Angle
Context
Problem
Solution
EXAMPLE 5.2 Hybrid Orbitals
Context
Problem
Solution
Extend
EXAMPLE 5.3 Covalent Bond Lengths
Context
Problem
Solution
5.4 Non-Covalent Bonds
Ionic Bonds
Metallic Bonds
5.5 Nuclear Magnetic Resonance Spectroscopy
How does it work?
The magnet.
Radiofrequency electronics.
Sample probe.
Nuclear Spins
Nuclear Spin Transitions
SAMPLE CALCULATION The Frequency of an NMR Spectrometer.
Nuclear Shielding
Isotropic Diamagnetic Shielding
DERIVATION SUMMARY Diamagnetism.
Chemical Shift
Anisotropic Diamagnetic Chemical Shift
Paramagnetic Chemical Shift
Spin-Spin Coupling
Key Concepts and Equations
5.1 The molecular Hamiltonian.
5.2 The molecular wavefunction.
5.3 Covalent bonds in polyatomic molecules.
5.4 Non-covalent bonds.
5.5 Nuclear magnetic resonance spectroscopy.
Key Terms
Objectives Review
Problems
Discussion Problems
The Molecular Hamiltonian
Binding Forces
Molecular Orbitals
Hybrid Orbitals
Empirical Chemical Structure and NMR
6 Molecular Symmetry
Learning Objectives
6.1 Group Theory
Symmetry Operators
EXAMPLE 6.1 Symmetry Operator Equivalence
Context
Problem
Solution
EXAMPLE 6.2 Molecular Symmetry Elements and Pseudorotation
Context
Problem
Solution
Point Groups
EXAMPLE 6.3 Point Group Operations
Context
Problem
Solution
EXAMPLE 6.4 Assigning Point Groups
Context
Problem
Solution
Point Group Representations
Character Tables
6.2 Symmetry Representations for Wavefunctions
Molecular Orbital Symmetry in Diatomics
Molecular Orbital Symmetry in Polyatomics
EXAMPLE 6.5 Symmetry and Choice of Coordinate System
Context
Problem
Solution
Local Bond Models and Symmetry Molecular Orbitals
Overall Symmetry
EXAMPLE 6.6 Local Bond and MO Models
Context
Problem
Solution
EXAMPLE 6.7 Direct Products and Reducible Representations
Context
Problem
Solution
Decomposing Reducible Representations
EXAMPLE 6.8 Methodical Decomposition of a Reducible Representation
Context
Problem
Solution
6.3 Selection Rules
Electric Dipole Selection Rules
EXAMPLE 6.9 Selection Rules
Context
Problem
Solution
Raman Spectroscopy
What is Raman spectroscopy?
Why do we use Raman spectroscopy?
How does it work?
EXAMPLE 6.10 Raman Selection Rules
Context
Problem
Solution
6.4 Selected Applications
Asymmetric Molecules
Hückel’s Rule
Key Concepts and Equations
6.1 Group theory.
6.2 Symmetry representations for wavefunctions.
6.3 Selection rules.
Key Terms
Objectives Review
Problems
Discussion Problems
Symmetry Elements
Point Groups
Character Tables
Wavefunction Symmetry
Selection Rules
7 Electronic States and Spectroscopy
7.1 Molecular Orbital Configurations
Correlation Diagrams for Diatomics
Homonuclear Diatomics
EXAMPLE 7.1 Finding the MO Configuration
Context
Problem
Solution
EXAMPLE 7.2 Bond Order
Context
Problem
Solution
Extend
Heteronuclear Diatomics
EXAMPLE 7.3 Heteronuclear Diatomic Electron Configurations
Context
Problem
Solution
Correlation Diagrams for Polyatomics
7.2 Electronic States
Electronic State Symmetry
EXAMPLE 7.4 O2 Vector Model
Context
Problem
Solution
What is UV-vis spectroscopy?
Why do we use UV-vis spectroscopy?
How does it work?
EXAMPLE 7.5 Selection Rules
Context
Problem
Solution
SAMPLE CALCULATION Obtaining the Electronic State Term Symbol from an MO Configuration.
Potential Energy Surfaces of Electronic States
7.3 Computational Methods for Molecules
Hartree-Fock Calculations
Density Functional Calculations
Geometry Optimization
7.4 Energetic Processes
Relaxation Processes
Fragmentation Processes
Spectroscopy of Transition Metal Compounds
Key Concepts and Equations
7.1 Molecular orbital configurations.
7.2 Electronic states.
7.4 Energetic Processes.
Key Terms
Objectives Review
Problems
Discussion Problems
Correlation Diagrams and MO Configurations for Diatomics
Correlation Diagrams and MO Configurations for Polyatomics
Potential Energy Curves and Relaxation Processes
Chapter 8 States and Spectroscopy
8.1 The Vibrational Schrödinger Equation
Separation of Variables
The Harmonic Oscillator
Derivation: The Harmonic Approximation to Vibrational Quantum States
Derivation Summary The Harmonic Approximation.
8.2 Vibrational Energy Levels in Diatomics
Generalizing Potential Surfaces
Vibrational Energy Levels and Anharmonicity
Vibrational Energies
The Morse Potential
8.3 Vibrations in Polyatomics
Vibrational Modes
Normal Modes
What is FTIR spectroscopy?
Why do we use FTIR spectroscopy?
How does it work?
Normal Modes and Local Modes
Determination of Normal Modes
EXAMPLE 8.1 Isotope Dependence of Vibrational Constants
CONTEXT
Problem
Solution
Multiple Vibrational Excitation
8.4 Spectroscopy of Vibrational States
Vibronic Spectroscopy
Key Concepts and Equations
8.1 The vibrational Schrödinger equation.
8.2 Vibrational energy levels in diatomics.
8.3 Vibrations in polyatomics.
Key Terms
Objectives Review
Problems
Discussion Problems
The Harmonic and Anharmonic Oscillators
Vibrations in Diatomics
Vibrations in Polyatomics
Symmetry and Vibrations
Potential Energy Functions
9 Rotational States and Spectroscopy
9.1 Rotations in Diatomics
9.2 Rotations in Polyatomics
The Rotational Hamiltonian
EXAMPLE 9.1 Moments of Inertia
Context
Problem
Solution
EXAMPLE 9.2 Rotational Constant of a Polyatomic
Context
Problem
Solution
Symmetric Tops
Spherical Tops
9.3 Spectroscopy of Rotational States
Pure Rotational Spectroscopy
What is radio astronomy?
Why do we use radio astronomy?
How does it work?
Rovibrational and Rovibronic Spectroscopy
Key Concepts and Equations
9.1 Rotations in diatomics.
9.2 Rotations in polyatomics.
9.3 Spectroscopy of rotational states.
Key Terms
Objectives Review
Problems
Discussion Problems
Rotational Constants and Moments of Inertia
Rotational, Rovibrational, Rovibronic Spectra
The Rotational Hamiltonian
Part 3 MOLECULAR Interactions
10 Intermolecular Forces
10.1 Intermolecular Potential Energy
Intermolecular Repulsion
EXAMPLE 10.1 Effective Atomic Number from Ionization Energy
CONTEXT
Problem
Solution
EXAMPLE 10.2 How Soft Is the Repulsive Wall?
Context
Problem
Solution
Intermolecular Attraction
Electrostatic Forces
Multipole fields.
EXAMPLE 10.3 Dipole Moment of a Set of Point Charges
Context
Problem
Solution
Multipole–multipole potential energies.
EXAMPLE 10.4 Multipole Fields in Water
Context
Problem
Solution
EXAMPLE 10.5 Deriving Distance from Dipole Moment
Context
Problem
Solution
SAMPLE CALCULATION Monopole–Dipole Typical Magnitude.
EXAMPLE 10.6 Monopole–Dipole Interaction
Context
Problem
Solution
SAMPLE CALCULATION Dipole–Dipole Typical Magnitude.
Hydrogen bonds.
EXAMPLE 10.7 Dipole–Dipole Interaction
Context
Problem
Solution
Higher multipoles.
What is a molecular beam?
How does it work?
Why do we use molecular beams?
Rotating dipoles.
SAMPLE CALCULATION Averaged Dipole–Dipole Typical Magnitude.
Inductive Forces
SAMPLE CALCULATION Dipole-Induced Dipole Typical Magnitude.
EXAMPLE 10.8 Dipole–Dipole and Dipole-Induced Dipole Interactions in HBr
Context
Problem
Solution
The Dispersion Force
A qualitative look at dispersion.
Dispersion force: a derivation.
SAMPLE CALCULATION Dispersion Energy Typical Magnitude for Small Molecule.
EXAMPLE 10.9 Relative Dispersion Energies and Boiling Points
Context
Problem
Solution
Model Potentials
SAMPLE CALCULATION Combined Lennard-Jones Parameters.
10.2 Molecular Collisions
Elastic Collisions
Inelastic Collisions
Rotational Energy Transfer
Vibrational Energy Transfer
Electronic and Chemical Energy Transfer
Key Concepts and Equations
10.1 Intermolecular potential energy. Intermolecular repulsion.
10.2 Molecular collisions.
Key Terms
Objectives Review
Problems
Discussion Problems
Multipole Potential Energy Functions
General Intermolecular Potential Energies
Model Potentials
Molecular Collisions
11 Nanoscale Chemical Structure
11.1 The Nanoscopic Scale
Clusters and Macromolecules
11.2 Clusters
Weakly Bound Clusters
SAMPLE CALCULATION Cluster Stabilization Energy.
EXAMPLE 11.1 Argon Cluster Volume
Context
Problem
Solution
Turning Up the Binding Energy
Weakly Bound Cluster Ions
Strongly Bound Clusters
Covalently Bound Clusters: Carbon Clusters
Carbon Nanotubes
SAMPLE CALCULATION Carbon Nanotube Diameter.
What are STM and AFM?
Why do we use STM and AFM?
How do they work?
Ionically Bound Clusters
Metal Clusters
11.3 Macromolecules
Molecular Mechanics
SAMPLE CALCULATION Vibrational Modes of a Single Amino Acid.
EXAMPLE 11.2 Molecular Mechanics and Folding
Context
Problem
Solution
Key Concepts and Equations
11.2 Clusters.
11.3 Macromolecules.
Key Terms
Objectives Review
Problems
Discussion Problems
Weakly Bound Clusters
Strongly Bound Clusters
Macromolecules
12 The Structure of Liquids
12.1 The Qualitative Nature of Liquids
Bonding Mechanisms
Surface Tension
12.2 Weakly Bonded Pure Liquids
Potential Energy in the Condensed Phase
SAMPLE CALCULATION Axilrod-Teller Correction.
The Pair Correlation Function
What is neutron diffraction?
Why do we use neutron diffraction?
How does it work?
Neutron source.
Energy selector.
Detector.
CHECKPOINT
Quantum States in a Weakly Bonded Liquid
Electronic states.
Vibrational states.
Rotational states.
Translational states.
12.3 Solvation
EXAMPLE 12.1 Solvation of an Ionic Compound
Context
Problem
Solution
EXAMPLE 12.2 The One-Fluid Model
Context
Problem
Solution
Key Concepts and Equations
12.1 The qualitative nature of liquids.
12.2 Weakly bonded pure liquids.
12.3 Solvation.
Key Terms
Objectives Review
Problems
Discussion Problems
General Liquid Structure and Dynamics
Pair Correlation Functions
Solvation
13 The Structure of Solids
13.1 Amorphous Solids, Polymers, and Crystals
Amorphous Solids
Polymers
Crystals
13.2 Symmetry in Crystals
Group Theory
Bravais Lattices
EXAMPLE 13.1 Bulk Properties from the Unit Cell
Context
Problem
Solution
What is x-ray crystallography?
Why do we use x-ray crystallography?
How does it work?
Electrical Properties of Crystals
Crystal Planes and Miller Indices
13.3 Bonding Mechanisms and Properties of Crystals
Ionic Crystals
Metal Crystals
EXAMPLE 13.2 Packing Efficiency of the fcc Lattice
Context
Problem
Solution
EXAMPLE 13.3 The fcc Crystallographic Point Groups
Context
Problem
Solution
Covalent Crystals
Molecular Crystals
13.4 Wavefunctions and Energies of Solids
Vibrational Conductivity
Electronic Energies and Electrical Conduction
Key Concepts and Equations
13.1 Amorphous Solids, Polymers, and Crystals.
13.2 Symmetry in Crystals.
13.3 Bonding Mechanisms and Properties of Crystals.
13.4 Wavefunctions and Energies of Solids.
Key Terms
Objectives Review
Problems
Discussion Problems
Bravais Lattices and Lattice Systems
Surface Characteristics
General Structure and Energetics
Appendix Character Tables for Common Point Groups
Solutions to Objectives Review Questions
INDEX
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