Abstract ................................................................................................. 164 3.1 Introduction .................................................................................. 165 3.2 The Wave-Function Continuity .................................................... 169 3.2.1 Quantum Tunneling and the Gamow Factor .................... 169 3.2.2 Nuclear Systems’ Lifetime by Alfa Disintegration .......... 176 3.3 Eigen-Energies and Eigen-Functions ........................................... 181 3.3.1 Atomic Hydrogenic States by Solving Laguerre Type Equations ................................................. 182 3.3.2 Molecular Vibrational States by Solving Hermite Type Equations .................................................. 200 3.3.3 Solid State Free Electronic States .................................... 209 3.4 Semiclassical Quantification of Energy ....................................... 216 3.4.1 Hydrogenic Atomic Systems ........................................... 219 3.4.2 Molecular Vibrational Systems ........................................ 222 3.4.3 Generalized |x|α Potential Systems ................................... 223 3.5 Variational Wave-Function and Energies ..................................... 228 3.5.1 Hydrogen’s Quantum Ground Level ................................ 228 3.5.2 Vibrational Ground Level ................................................ 231 3.5.3 Ground State Paradox of Free Electrons in Solids .......... 232 3.6 Causal Quantum Evolution .......................................................... 236 3.6.1 Schrödinger’s Picture ....................................................... 236 3.6.2 Unitary (“U”) Picture. Heisenberg Picture ...................... 246 3.6.3 Quantum Transitions: Interaction Picture ........................ 254 3.6.4 Green Function and Its Causal Properties ........................ 262 3.6.5 Free Particle’s Propagator ................................................ 271 3.7 Stationary Perturbations ............................................................... 274 3.7.1 General Perturbation Algorithm ....................................... 275 3.7.2 The Zeeman Effect ........................................................... 281 3.7.3 Nuclear Isotropic Corrections on Hydrogenic Atoms ...... 285 3.7.4 Harmonic Oscillator Perturbations .................................. 292 3.7.5 Quasi-Free Electronic Model of Solids ........................... 300 3.8 Quantum Measurements .............................................................. 304 3.8.1 Classical vs. Quantum Electric Resistance ...................... 306 3.8.2 Quantum Conservation Laws ............................................311 3.8.3 Quantum Elastic Scattering on Fixed Target ................... 318 3.8.4 More about Scattering: Born Approximation and Resonance Profile ..................................................... 332 3.8.5 From Yukawa Potential to Rutherford Scattering ............ 342 3.9 Conclusion ................................................................................... 353 Keywords .............................................................................................. 354 References ............................................................................................. 355 Author’s Main References ........................................................... 355 Specific References ...................................................................... 355 Further Readings .......................................................................... 355 ABSTRACT The phenomenological quantum mechanics is practically “re-storied” in a more formalized way, under the so-called extended quantum postulates, while opening and unfolding more complex analysis of the Nature’s phenomena, from nuclear, to atomic, to molecular, and to solid state and scattering observational quantum effects.
Abstract ................................................................................................... 66 2.1 Introduction .................................................................................... 66 2.2 Wave Function Picture ................................................................... 67 2.2.1 Green and Dirac Functions ................................................ 67 2.2.2 Momentum and Energy Operators ..................................... 69 2.2.3 Klein-Gordon and Schrödinger Equations ......................... 72 2.2.4 Electronic and Photonic Spins ........................................... 74 2.2.5 Eigen-Functions and Eigen-Values .................................... 75 2.2.6 Hermitic Operators ............................................................ 76 2.2.7 Heisenberg Uncertainty Theorem ...................................... 81 2.2.8 Ehrenfest Theorem ............................................................. 85 2.2.9 Current Density Probability Conservation Theorem ......... 89 2.3 Classical to Quantum Mechanics’ Correspondence ....................... 92 2.3.1 Classical Euler-Lagrange, Hamilton, and Hamilton-Jacobi Equations ................................................ 92 2.3.2 Wave-Function Quantum Field .......................................... 97 2.3.3 Semi-classical Expansion and the WKB Approximation ................................................................. 102 2.3.4 From Field Internal Symmetry to Current Conservation .................................................................... 106 2.3.5 From Poisson Parentheses to Quantum Commutators and Heisenberg Picture ............................. 108 2.4 Bra-ket (Dirac) Formalism ............................................................115 2.4.1 Vectors in Hilbert Space....................................................115 2.4.2 Linear Operators in Hilbert Space ....................................118 2.4.3 Spectral Representations of Vectors and Operators ......... 126 2.4.4 Coordinate and Momentum Representations ................... 136 2.4.5 Energy Representation ..................................................... 148 2.4.6 Heisenberg Matrix Quantum Mechanics: The Harmonic Oscillator ................................................. 151 2.5 Conclusion ................................................................................... 159 Keywords .............................................................................................. 160 References ............................................................................................. 161 Author’s Main Reference ............................................................. 161 Further Readings .......................................................................... 161 ABSTRACT Although equivalent to certain points, various approaches of quantum phenomena may add new insights and perspective on studied systems either in isolate or interaction states. Here, the main formalisms are reviewed and illustrated, especially those making the ground-nous of the quantum mechanics itself, i.e., the undulatory quantum mechanics, while introducing the quantum propagator (Green function) concept, the semi-classical approach based on Euler-Lagrange and Hamiltonian, while the bra-ket Dirac picture opens the new perspectives of further developing quantum theory by abstract yet logical and formalized reasoning.
Abstract ................................................................................................. 359 4.1 Introduction .................................................................................. 359 4.2 Feynman’s Path Integral Quantum Formalism ............................ 364 4.2.1 Construction of Path Integral ........................................... 364 4.2.2 Propagator’s Equation ...................................................... 370 4.2.3 Recovering Wave Function’s Equation ............................ 373 4.3 Path Integrals for Basic Matter’s Structures ................................ 377 4.3.1 General Path Integral’s Properties ................................... 377 4.3.2 Path Integral for the Free Particle .................................... 384 4.3.3 Path Integral for Motion as the Harmonic Oscillator ...... 387 4.3.4 Path Integral Representation for the Bohr’s Atom ........... 394 4.3.5 Path Integral for Motion in the Quantum Well ................ 398 4.4 Density Matrix Approach Linking Path Integral Formalism ....... 402 4.4.1 On Mono-, Many-, and Reduced-Electronic Density Matrices .............................................................. 402 4.4.2 Canonical Density, Bloch Equation, and the Need of Path Integral ....................................................... 412 4.5 Roots of Self-Consistent Methods in Quantum Chemistry .......... 419 4.5.1 Molecular Orbital Approach ............................................ 420 4.5.2 Roothaan Approach .......................................................... 423 4.5.3 Introducing Semi-Empirical Approximations .................. 428 4.5.3.1 NDO Methods ................................................... 429 4.5.3.2 NDDO Methods ................................................ 432 4.5.4 Ab initio Methods: The Hartree-Fock Approach ............. 436 4.5.4.1 Hartree-Fock Orbital Energy ............................ 438 4.5.4.2 About Correlation Energy ................................. 442 4.5.4.3 Koopman’s Orbital Theorem with Hartree-Fock Picture ......................................... 444 4.5.4.4 Chemical Reactivity Indices in Orbital Energy Representation ...................................... 451 4.5.4.5 Testing Koopmans Theorem by Chemical Harness Reactivity Index .................. 458 4.6 Density Functional Theory: Observable Quantum Chemistry ..... 468 4.6.1 Hohenberg-Kohn theorems .............................................. 470 4.6.2 Optimized Energy-Electronegativity Connection ............ 475 4.6.3 Popular Energetic Density Functionals ............................ 480 4.6.3.1 Density Functionals of Kinetic Energy ............. 481 4.6.3.2 Density Functionals of Exchange Energy ......... 484 4.6.3.3 Density Functionals of Correlation Energy ...... 490 4.6.3.4 Density Functionals of Exchange-Correlation Energy ........................... 496 4.7 Observable Quantum Chemistry: Extending Heisenberg’s Uncertainty ............................................................. 502 4.7.1 Periodic Path Integrals ..................................................... 502 4.7.1.1 Effective Partition Function .............................. 502 4.7.1.2 Periodic Quantum Paths .................................... 505 4.7.2 Heisenberg Uncertainty Reloaded ................................... 508 4.7.3 Extending Heisenberg Uncertainty ...................................511 4.7.3.1 Averaging Quantum Fluctuations ......................511 4.7.3.2 Observed Quantum Evolution .......................... 514 4.7.3.3 Free Quantum Evolution ................................... 515 4.7.3.4 Free vs. Observed Quantum Evolution ............. 518 4.7.3.5 Spectral-IQ Method .......................................... 523 4.7.3.6 Spectral-IQ Results on Silica Sol-Gel-Based Mesosystems ..................................................... 527 4.8 Conclusion ................................................................................... 533 Keywords .............................................................................................. 535 References ............................................................................................. 536 Author’s Main References ........................................................... 536 Specific References ...................................................................... 537 Further Readings .......................................................................... 549 ABSTRACT Basing on the first principles of Quantum mechanics as exposed in the previous chapters and sections, special chapters of quantum theory are here unfolded in order to further extend and caching the quantum information from free to observed evolution within the matter systems with constraints (boundaries). As such, the Feynman path integral formalism is firstly exposed and then applied to atomic, quantum barrier and quantum harmonically vibration, followed by density matrix approach, opening the Hartree-Fock and Density Functional pictures of many-electronic systems, with a worthy perspective of electronic occupancies via Koopmans theorem, while ending with a further generalization of the Heisenberg observability and of its first application to mesosystems.
Preface. 1. Origins of Quantum Physics. 1.1 Historical Note. 1.2 Particle Aspect of Radiation. 1.3 Wave Aspect of Particles. 1.4 Particles versus Waves. 1.5 Indeterministic Nature of the Microphysical World. 1.6 Atomic Transitions and Spectroscopy. 1.7 Quantization Rules. 1.8 Wave Packets. 1.9 Concluding Remarks. 1.10 Solved Problems. Exercises. 2. Mathematical Tools of Quantum Mechanics. 2.1 Introduction. 2.2 The Hilbert Space and Wave Functions. 2.3 Dirac Notation. 2.4 Operators. 2.5 Representation in Discrete Bases. 2.6 Representation in Continuous Bases. 2.7 Matrix and Wave Mechanics. 2.8 Concluding Remarks. 2.9 Solved Problems. Exercises. 3. Postulates of Quantum Mechanics. 3.1 Introduction. 3.2 The Basic Postulates of Quantum Mechanics. 3.3 The State of a System. 3.4 Observables and Operators. 3.5 Measurement in Quantum Mechanics. 3.6 Time Evolution of the System's State. 3.7 Symmetries and Conservation Laws. 3.8 Connecting Quantum to Classical Mechanics. 3.9 Solved Problems. Exercises. 4. One-Dimensional Problems. 4.1 Introduction. 4.2 Properties of One-Dimensional Motion. 4.3 The Free Particle: Continuous States. 4.4 The Potential Step. 4.5 The Potential Barrier and Well. 4.6 The Infinite Square Well Potential. 4.7 The Finite Square Well Potential. 4.8 The Harmonic Oscillator. 4.9 Numerical Solution of the Schrodinger Equation. 4.10 Solved Problems. Exercises. 5. Angular Momentum. 5.1 Introduction. 5.2 Orbital Angular Momentum. 5.3 General Formalism of Angular Momentum. 5.4 Matrix Representation of Angular Momentum. 5.5 Geometrical Representation of Angular Momentum. 5.6 Spin Angular Momentum. 5.7 Eigen functions of Orbital Angular Momentum. 5.8 Solved Problems. Exercises. 6. Three-Dimensional Problems. 6.1 Introduction. 6.2 3D Problems in Cartesian Coordinates. 6.3 3D Problems in Spherical Coordinates. 6.4 Concluding Remarks. 6.5 Solved Problems. Exercises. 7. Rotations and Addition of Angular Momenta. 7.1 Rotations in Classical Physics. 7.2 Rotations in Quantum Mechanics. 7.3 Addition of Angular Momenta. 7.4 Scalar, Vector and Tensor Operators. 7.5 Solved Problems. Exercises. 8. Identical Particles. 8.1 Many-Particle Systems. 8.2 Systems of Identical Particles. 8.3 The Pauli Exclusion Principle. 8.4 The Exclusion Principle and the Periodic Table. 8.5 Solved Problems. Exercises. 9. Approximation Methods for Stationary States. 9.1 Introduction. 9.2 Time-Independent Perturbation Theory. 9.3 The Variational Method. 9.4 The Wentzel Kramers Brillou in Method. 9.5 Concluding Remarks. 9.6 Solved Problems. Exercises. 10. Time-Dependent Perturbation Theory. 10.1 Introduction. 10.2 The Pictures of Quantum Mechanics. 10.3 Time-Dependent Perturbation Theory. 10.4 Adiabatic and Sudden Approximations. 10.5 Interaction of Atoms with Radiation. 10.6 Solved Problems. Exercises. 11. Scattering Theory. 11.1 Scattering and Cross Section. 11.2 Scattering Amplitude of Spinless Particles. 11.3 The Born Approximation. 11.4 Partial Wave Analysis. 11.5 Scattering of Identical Particles. 11.6 Solved Problems. Exercises. A. The Delta Function. A.1 One-Dimensional Delta Function. A.2 Three-Dimensional Delta Function. B. Angular Momentum in Spherical Coordinates. B.1 Derivation of Some General. B.2 Gradient and Laplacianin Spherical Coordinates. B.3 Angular Momentum in Spherical Coordinates. C. Computer Code for Solving the Schrodinger Equation. Index.
QUANTUM PRINCIPLES Perspective and Principles Prelude to Quantum Mechanics Stern-Gerlach Experiment Idealized Stern-Gerlach Results Classical Model Attempts Wave Functions for Two Physical-Outcome Case Process Diagrams, Operators, and Completeness Further Properties of Operators/Modulation Operator Reformulation Operator Rotation Bra-Ket Notation/Basis States Transition Amplitudes Three-Magnet Setup Example-Coherence Hermitian Conjugation Unitary Operators A Very Special Operator Matrix Representations Matrix Wave Function Recovery Expectation Values Wrap Up Problems Free Particles in One Dimension Photoelectric Effect Compton Effect Uncertainty Relation for Photons Stability of Ground States Bohr Model Fourier Transform and Uncertainty Relations Schrodinger Equation Schrodinger Equation Example Dirac Delta Functions Wave Functions and Probability Probability Current Time Separable Solutions Completeness for Particle States Particle Operator Properties Operator Rules Time Evolution and Expectation Values Wrap-Up Problems Some One-Dimensional Solutions to the Schrodinger Equation Introduction The Infinite Square Well: Differential Solution The Infinite Square Well: Operator Solution The Finite Potential Barrier Step Potential The Harmonic Oscillator The Attractive Kronig-Penny Model Bound State and Scattering Solutions Problems Hilbert Space and Unitary Transformations Introduction and Notation Inner and Outer Operator Products Operator-Matrix Relationship Hermitian Operators and Eigenkets Gram-Schmidt Orthogonalization Process Compatible Operators Uncertainty Relations and Incompatible Operators Simultaneously Measureable Operators Unitary Transformations and Change of Basis Coordinate Displacements and Unitary Transformations Schrodinger and Heisenburg Pictures of Time Evolution Free Gaussian Wave Packet in the Heisenberg Picture Potentials and the Ehrenfest Theorem Problems Three Static Approximation Methods Introduction Time-Independent Perturbation Theory Examples of Time-Independent Perturbation Theory Aspects of Degenerate Perturbation Theory WKB Semiclassical Approximation Use of the WKB Approximation in Barrier Penetration Use of the WKB Approximation in Bound States Variational Methods Problems Generalization to Three Dimensions Cartesian Basis States and Wave Functions in Three Dimensions Position/Momentum Eigenket Generalization Example: Three-Dimensional Infinite Square Well Spherical Basis States Orbital Angular Momentum Operator Effect of Angular Momentum on Basis States Energy Eigenvalue Equation and Angular Momentum Complete Set of Observables for the Radial Schrodinger Equation Specification of Angular Momentum Eigenstates Angular Momentum Eigenvectors and Spherical Harmonics Completeness and Other Properties of Spherical Harmonics Radial Eigenfunctions Problems QUANTUM PARTICLES The Three-Dimensional Radial Equation Recap of the Situation The Free Particle The Infinite Spherical Well Potential The Deuteron The Coulomb Potential: Initial Considerations The Coulomb Potential: 2-D Harmonic Oscillator Comparison The Confined Coulombic Model Problems Addition of Angular Momenta General Angular-Momentum Eigenstate Properties Combining Angular Momenta for Two Systems Explicit Example of Adding Two Spin 1/2 Systems Explicit Example of Adding Orbital Angular Momentum and Spin 1/2 Hydrogen Atom and the Choice of Basis States Hydrogen Atom and Perturbative Energy Shifts Problems Spin and Statistics The Connection between Spin and Statistics Building Wave Functions with Identical Particles Particle Occupation Basis More on Fermi-Dirac Statistics Interaction Operator and Feynman Diagrams Implications of Detailed Balance Cubical Enclosures and Particle States Maxwell-Boltzmann Statistics Bose-Einstein Statistics Fermi-Dirac Statistics The Hartree-Fock Equations Problems Quantum Particle Scattering Introduction The One-Dimensional Integral Schrodinger Equation Reflection and Transmission Amplitudes One-Dimensional Delta-Function Scattering Step-Function Potential Scattering The Born Series The Three-Dimensional Integral Schrodinger Equation The Helmholtz Equation and Plane Waves Cross Sections and the Scattering Amplitude Scattering Phase Shifts Finite-Range Potential Scattering The Three-Dimensional Born Series Identical Particle Scattering Proton-Proton Scattering Problems Connecting to the Standard Model Introduction Discrete Symmetries Parity Time Reversal Charge Conjugation Particle Primer Particle Interactions Quantum Electrodynamics Quantum Chromodynamics Weak Interactions Beyond the Standard Model Supersymmetry Superstrings Postlude Helpful Introductory Books on Particle and String Physics More Advanced Books on Particle and String Physics Problems Appendix: Notation Comments and Comparisons Appendix: Lattice Models Appendix: 2-D Harmonic Oscillator Wave Function Normalization Appendix: Allowed Standard Model Interactions Appendix: Weak Flavor Mixing Appendix: The Ising Model and More Index
Since this book is closely connected to quantum mechanics, these introductory words will briefly and informally recall some postulates of this theory. Interested readers are urged to consult traditional books on quantum mechanics to complement the discussion ahead and for descriptions of experimental evidence that give rise to the postulates. In physics there are several “equivalent formulations” of quantum mechanics, and a catalogue of the most important appears in [StyO2]. The discussion here is restricted to the nonrelativistic case.
CHAPTER 1: FROM CLASSICAL TO QUANTUM MECHANICS 1.1 Why Study Quantum Mechanics? 1.2 Quantum Mechanics Arose Out of the Interplay of Experiments and Theory 1.3 Blackbody Radiation 1.4 The Photoelectric Effect 1.5 Particles Exhibit Wave-Like Behavior 1.6 Diffraction by a Double Slit 1.7 Atomic Spectra and the Bohr Model for the Hydrogen Atom CHAPTER 2: THE SCHRODINGER EQUATION 2.1 What Determines If a System Needs to Be Described Using Quantum Mechanics? 2.2 Classical Waves and the Nondispersive Wave Equation 2.3 Waves Are Conveniently Represented as Complex Functions 2.4 Quantum Mechanical Waves and the Schrodinger Equation 2.5 Solving the Schrodinger Equation: Operators, Observables, Eigenfunctions, and Eigenvalues 2.6 The Eigenfunctions of a Quantum Mechanical Operator Are Orthogonal 2.7 The Eigenfunctions of a Quantum Mechanical Operator Form a Complete Set 2.8 Summing Up the New Concepts CHAPTER 3: THE QUANTUM MECHANICAL POSTULATES 3.1 The Physical Meaning Associated with the Wave Function 3.2 Every Observable Has a Corresponding Operator 3.3 The Result of an Individual Measurement 3.4 The Expectation Value 3.5 The Evolution in Time of a Quantum Mechanical System CHAPTER 4: USING QUANTUM MECHANICS ON SIMPLE SYSTEMS 4.1 The Free Particle 4.2 The Particle in a One-Dimensional Box 4.3 Two- and Three-Dimensional Boxes 4.4 Using the Postulates to Understand the Particle in the Box and Vice Versa CHAPTER 5: THE PARTICLE IN THE BOX AND THE REAL WORLD 5.1 The Particle in the Finite Depth Box 5.2 Differences in Overlap between Core and Valence Electrons 5.3 Pi Electrons in Conjugated Molecules Can Be Treated as Moving Freely in a Box 5.4 Why Does Sodium Conduct Electricity and Why Is Diamond an Insulator? 5.5 Tunneling through a Barrier 5.6 The Scanning Tunneling Microscope 5.7 Tunneling in Chemical Reactions 5.8 (Supplemental) Quantum Wells and Quantum Dots CHAPTER 6: COMMUTING AND NONCOMMUTING OPERATORS AND THE SURPRISING CONSEQUENCES OF ENTANGLEMENT 6.1 Commutation Relations 6.2 The Stern-Gerlach Experiment 6.3 The Heisenberg Uncertainty Principle 6.4 (Supplemental) The Heisenberg Uncertainty Principle Expressed in Terms of Standard Deviations 6.5 (Supplemental) A Thought Experiment Using a Particle in a Three-Dimensional Box 6.6 (Supplemental) Entangled States, Teleportation, and Quantum Computers CHAPTER 7: A QUANTUM MECHANICAL MODEL FOR THE VIBRATION AND ROTATION OF MOLECULES 7.1 Solving the Schrodinger Equation for the Quantum Mechanical Harmonic Oscillator 7.2 Solving the Schrodinger Equation for Rotation in Two Dimensions 7.3 Solving the Schrodinger Equation for Rotation in Three Dimensions 7.4 The Quantization of Angular Momentum 7.5 The Spherical Harmonic Functions 7.6 (Optional Review) The Classical Harmonic Oscillator 7.7 (Optional Review) Angular Motion and the Classical Rigid Rotor 7.8 (Supplemental) Spatial Quantization CHAPTER 8: THE VIBRATIONAL AND ROTATIONAL SPECTROSCOPY OF DIATOMIC MOLECULES 8.1 An Introduction to Spectroscopy 8.2 Absorption, Spontaneous Emission, and Stimulated Emission 8.3 An Introduction to Vibrational Spectroscopy 8.4 The Origin of Selection Rules 8.5 Infrared Absorption Spectroscopy 8.6 Rotational Spectroscopy 8.7 (Supplemental) Fourier Transform Infrared Spectroscopy 8.8 (Supplemental) Raman Spectroscopy 8.9 (Supplemental) How Does the Transition Rate between States Depend on Frequency? CHAPTER 9: THE HYDROGEN ATOM 9.1 Formulating the Schrodinger Equation 9.2 Solving the Schrodinger Equation for the Hydrogen Atom 9.3 Eigenvalues and Eigenfunctions for the Total Energy 9.4 The Hydrogen Atom Orbitals 9.5 The Radial Probability Distribution Function 9.6 The Validity of the Shell Model of an Atom CHAPTER 10: MANY-ELECTRON ATOMS 10.1 Helium: The Smallest Many-Electron Atom 10.2 Introducing Electron Spin 10.3 Wave Functions Must Reflect the Indistinguishability of Electrons 10.4 Using the Variational Method to Solve the Schrodinger Equation 10.5 The Hartree-Fock Self-Consistent Field Method 10.6 Understanding Trends in the Periodic Table from Hartree-Fock Calculations CHAPTER 11: QUANTUM STATES FOR MANY-ELECTRON ATOMS AND ATOMIC SPECTROSCOPY 11.1 Good Quantum Numbers, Terms, Levels, and States 11.2 The Energy of a Configuration Depends on Both Orbital and Spin Angular Momentum 11.3 Spin-Orbit Coupling Breaks Up a Term into Levels 11.4 The Essentials of Atomic Spectroscopy 11.5 Analytical Techniques Based on Atomic Spectroscopy 11.6 The Doppler Effect 11.7 The Helium-Neon Laser 11.8 Laser Isotope Separation 11.9 Auger Electron and X-Ray Photoelectron Spectroscopies 11.10 Selective Chemistry of Excited States: O(3P) and O(1D) 11.11 (Supplemental) Configurations with Paired and Unpaired Electron Spins Differ in Energy CHAPTER 12: THE CHEMICAL BOND IN DIATOMIC MOLECULES 12.1 The Simplest One-Electron Molecule: 12.2 The Molecular Wave Function for Ground-State 12.3 The Energy Corresponding to the H2+ Molecular Wave Functions 12.4 A Closer Look at the H2+ Molecular Wave Functions 12.5 Combining Atomic Orbitals to Form Molecular Orbitals 12.6 Molecular Orbitals for Homonuclear Diatomic Molecules 12.7 The Electronic Structure of Many-Electron Molecules 12.8 Bond Order, Bond Energy, and Bond Length 12.9 Heteronuclear Diatomic Molecules 12.10 The Molecular Electrostatic Potential CHAPTER 13: MOLECULAR STRUCTURE AND ENERGY LEVELS FOR POLYATOMIC MOLECULES 13.1 Lewis Structures and the VSEPR Model 13.2 Describing Localized Bonds Using Hybridization for Methane, Ethene, and Ethyne 13.3 Constructing Hybrid Orbitals for Nonequivalent Ligands 13.4 Using Hybridization to Describe Chemical Bonding 13.5 Predicting Molecular Structure Using Molecular Orbital Theory 13.6 How Different Are Localized and Delocalized Bonding Models? 13.7 Qualitative Molecular Orbital Theory for Conjugated and Aromatic Molecules: The Huckel Model 13.8 From Molecules to Solids 13.9 Making Semiconductors Conductive at Room Temperature CHAPTER 14: ELECTRONIC SPECTROSCOPY 14.1 The Energy of Electronic Transitions 14.2 Molecular Term Symbols 14.3 Transitions between Electronic States of Diatomic Molecules 14.4 The Vibrational Fine Structure of Electronic Transitions in Diatomic Molecules 14.5 UV-Visible Light Absorption in Polyatomic Molecules 14.6 Transitions among the Ground and Excited States 14.7 Singlet--Singlet Transitions: Absorption and Fluorescence 14.8 Intersystem Crossing and Phosphorescence 14.9 Fluorescence Spectroscopy and Analytical Chemistry 14.10 Ultraviolet Photoelectron Spectroscopy 14.11 Single Molecule Spectroscopy 14.12 Fluorescent Resonance Energy Transfer (FRET) 14.13 Linear and Circular Dichroism 14.14 (Supplemental) Assigning + and -- to Terms of Diatomic Molecules CHAPTER 15: COMPUTATIONAL CHEMISTRY 15.1 The Promise of Computational Chemistry 15.2 Potential Energy Surfaces 15.3 Hartree-Fock Molecular Orbital Theory: A Direct Descendant of the Schrodinger Equation 15.4 Properties of Limiting Hartree-Fock Models 15.5 Theoretical Models and Theoretical Model Chemistry 15.6 Moving Beyond Hartree-Fock Theory 15.7 Gaussian Basis Sets 15.8 Selection of a Theoretical Model 15.9 Graphical Models 15.10 Conclusion CHAPTER 16: MOLECULAR SYMMETRY 16.1 Symmetry Elements, Symmetry Operations, and Point Groups 16.2 Assigning Molecules to Point Groups 16.3 The H2O Molecule and the C2v Point Group 16.4 Representations of Symmetry Operators, Bases for Representations, and the Character Table 16.5 The Dimension of a Representation 16.6 Using the C2v Representations to Construct Molecular Orbitals for H2O 16.7 The Symmetries of the Normal Modes of Vibration of Molecules 16.8 Selection Rules and Infrared versus Raman Activity 16.9 (Supplemental) Using the Projection Operator Method to Generate MOs That Are Bases for Irreducible Representations CHAPTER 17: NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY 17.1 Intrinsic Nuclear Angular Momentum and Magnetic Moment 17.2 The Energy of Nuclei of Nonzero Nuclear Spin in a Magnetic Field 17.3 The Chemical Shift for an Isolated Atom 17.4 The Chemical Shift for an Atom Embedded in a Molecule 17.5 Electronegativity of Neighboring Groups and Chemical Shifts 17.6 Magnetic Fields of Neighboring Groups and Chemical Shifts 17.7 Multiplet Splitting of NMR Peaks Arises through Spin--Spin Coupling 17.8 Multiplet Splitting When More Than Two Spins Interact 17.9 Peak Widths in NMR Spectroscopy 17.10 Solid-State NMR 17.11 NMR Imaging 17.12 (Supplemental) The NMR Experiment in the Laboratory and Rotating Frames 17.13 (Supplemental) Fourier Transform NMR Spectroscopy 17.14 (Supplemental) Two-Dimensional NMR