Quantum Mechanics in Chemistry

by Schatz, George C.; Ratner, Mark A.

• ISBN13: 9780486420035
• ISBN10: 0486420035
• Format: Paperback
• Copyright: 2002-01-28
• Publisher: Dover Publications

Summary

Advanced graduate-level text looks at symmetry, rotations, and angular momentum addition; introduces basic formalism of time-dependent quantum mechanics and occupation number representations; focuses on scattering theory; uses its concepts to develop basic theories of chemical reaction rates, and more. Problems appear at end of each chapter; solutions included.

Table of Contents

Preface	xvii
Review of Basic Concepts in Quantum Mechanics	1	(20)
Fundamental Definitions	1	(2)
Eigenvalues and Eigenfunctions	3	(2)
Approximate Methods	5	(2)
Time-Independent Perturbation Theory	5	(1)
Variational Theory	6	(1)
Raising and Lowering Operators	7	(4)
Harmonic Oscillator	7	(2)
Angular Momentum Operators	9	(2)
Two-Body Problems	11	(2)
Relative-Motion Schrodinger Equation	11	(1)
Hydrogen Atom	12	(1)
Electronic Structure of Atoms and Molecules	13	(8)
Many-Electron Hamiltonian; Born-Oppenheimer Approximation	13	(2)
Pauli Principle; Hartree-Fock Theory	15	(2)
LCAO-MO-SCF	17	(1)
Electronic Structure Methods	18	(1)
Bibliography for Chapter 1	18	(1)
Problems for Chapter 1	19	(2)
Symmetry Considerations: Point Groups and Electronic Structure	21	(22)
Group Theory for Point Groups	21	(6)
Symmetry Operations	22	(1)
Representations	22	(1)
Similarity Transformations	23	(1)
Irreducible Representations	23	(1)
Character Tables	24	(1)
Direct (or Tensor) Products	25	(1)
Clebsch-Gordan Series	26	(1)
Applications of Group Theory to Quantum Mechanics	27	(9)
Symmetry-Adapted Linear Combinations	27	(1)
Constructions of SALCs	28	(2)
Huckel Theory Applications	30	(3)
Shortcuts	33	(1)
Energy Eigenvalues	34	(2)
Symmetry Properties of Many-Electron Wavefunctions	36	(7)
Many-Electron Configurations and Terms	36	(1)
Application to C3H+3	36	(1)
Incorporation of the Pauli Principle	37	(2)
Optical Spectrum: Dipole Selection Rules	39	(1)
Bibliography for Chapter 2	40	(1)
Problems for Chapter 2	40	(3)
Symmetry Considerations: Continuous Groups and Rotations	43	(14)
Introduction	43	(1)
Continuous Groups; The Electronic Structure of Linear Molecules	43	(7)
Two-Dimensional Rotation Group	43	(2)
Cxv Group	45	(1)
Dxh Group	45	(1)
Simple Example: O2	46	(4)
Three-Dimensional Rotation Group; Angular Momentum Addition	50	(7)
Angular Momentum Addition; Clebsch-Gordan Coefficients	50	(1)
Properties of Clebsch-Gordan Coefficients	51	(2)
Worked Examples	53	(1)
3-j and Higher Symbols	54	(1)
Bibliography for Chapter 3	55	(1)
Problems for Chapter 3	55	(2)
Time-Dependent Quantum Mechanics	57	(21)
Introduction	57	(1)
Time-Dependent Schrodinger Equation: Basis-Set Solution	57	(2)
Time-Dependent Perturbation Theory	59	(6)
First-Order Time-Dependent Perturbation Theory	59	(1)
Example: Collision-Induced Excitation of a Diatomic Molecule	59	(4)
Second-Order Perturbation Theory	63	(1)
Simplifications and Extensions to Higher Order	63	(1)
Time-Ordering Operators	64	(1)
Representations in Quantum Mechanics	65	(5)
Schrodinger Representation	66	(1)
Heisenberg Representation	67	(2)
Interaction Representation	69	(1)
Transition Probabilities per Unit Time	70	(8)
Perturbation Theory for a Constant Interaction Potential	70	(2)
Fermi's Golden Rule	72	(1)
State-to-State Form of Fermi's Golden Rule	73	(2)
Treatment of Periodic Interactions	75
Bibliography for Chapter 4	74	(1)
Problems for Chapter 4	75	(3)
Interaction of Radiation with Matter	78	(33)
Introduction	78	(1)
Electromagnetic Fields	78	(4)
Vector Potentials and Wave Equations	78	(2)
Plane Waves	80	(1)
Energy and Photon Number Density	81	(1)
Interaction between Matter and Field	82	(4)
Classical Theory	82	(1)
Derivation of Classical Hamiltonian	82	(1)
Quantum Hamiltonian for a Particle in an Electromagnetic Field	83	(3)
Absorption and Emission of Light	86	(13)
Application of Fermi's Golden Rule	86	(1)
Dipole Approximation	87	(1)
Photon Density of States	88	(1)
Emission Rate	88	(2)
Absorption Rate	90	(1)
Einstein A and B Coefficients	90	(2)
Oscillator Strengths	92	(1)
Electric Quadrupole, Magnetic Dipole Mechanisms	92	(3)
Molecular Transitions: Frank-Condon Factors	95	(4)
Light Scattering	99	(12)
Qualitative Description of Light Scattering	99	(1)
Simplification of Electric Dipole Interaction	99	(1)
Interaction between Field and Induced Dipole; Two-Photon Process	100	(2)
Raman Scattering	102	(2)
Evaluation of αkm: Kramers-Heisenberg Formula	104	(4)
Bibliography for Chapter 5	108	(1)
Problems for Chapter 5	108	(3)
Occupation Number Representations	111	(42)
Introduction	111	(1)
Occupation Number Representation for Harmonic Molecular Vibrations and Quantized Radiation Fields	112	(7)
Single Harmonic Oscillator	112	(1)
Normal Modes	113	(2)
Quantized Radiation Fields	115	(2)
Coupling of Radiation to Matter Using Second Quantization	117	(1)
Applications of Fermi's Golden Rule	118	(1)
Occupation Number Representations for Electrons	119	(13)
Fermion Creation and Destruction Operators	119	(2)
Slater Determinants and Electron Creation Operators	121	(4)
Manipulation of Fermion Operators; Commutators and Anticommutators	125	(1)
Arbitrary Electronic Operators in the Occupation Number Representation	126	(6)
Fermion Field Operators and Second Quantization	132	(3)
Molecular Electronic Structure: Model Hamiltonians and Occupation Number Representations	135	(7)
Model Hamiltonians: Basis Set and Matrix Elements	135	(2)
Noninteracting Electrons: Huckel, Extended Huckel, and Free-Electron Models	137	(2)
Molecular Orbitals for Noninteracting Electrons	139	(3)
Treatment of Interacting Electrons	142	(11)
Model Hamiltonians	142	(3)
Self-Consistent Field (SCF) Solution	145	(2)
Example: SCF Solution for the Two-Center, Two-Orbital Problem	147	(1)
Bibliography for Chapter 6	148	(1)
Problems for Chapter 6	149	(4)
Quantum Scattering Theory	153	(29)
Introduction	153	(1)
One-Dimensional Scattering	154	(13)
Introduction	154	(1)
Wavepackets in One Dimension	155	(2)
Wavepackets for the Complete Scattering Problem	157	(1)
Fluxes and Probabilities	158	(1)
Time-Independent Approach to Scattering	159	(2)
Scattering Matrix	161	(2)
Green's Functions for Scattering	163	(3)
Born Approximation	166	(1)
Semiclassical Theory	167	(8)
WKB Approximation	167	(6)
Semiclassical Wavepackets	173	(2)
Scattering in Three Dimensions	175	(7)
Bibliography for Chapter 7	178	(1)
Problems for Chapter 7	179	(3)
Theories of Reaction Rates	182	(17)
Introduction	182	(1)
Rate Constants for Bimolecular Reactions: Cumulative Reaction Probabilities	182	(2)
Transition-State Theory	184	(1)
RRKM Theory	185	(1)
Formal Expression for Rates in Terms of Flux Operators	186	(3)
Additional Expressions for Rate Constant	189	(3)
Flux-Flux Autocorrelation Functions	192	(2)
Evaluation of Propagator Matrix Elements Using Path Integrals	194	(5)
Bibliography for Chapter 8	197	(1)
Problems for Chapter 8	197	(2)
Time-Dependent Approach to Spectroscopy: Electronic, Vibrational, and Rotational Spectra	199	(32)
Introduction	199	(1)
Thermal Averages and Imaginary Time Propagation	200	(1)
Electronic Spectra from Time Correlation Functions	201	(1)
Electronic Spectra: Time Development of the Correlation Functions	202	(7)
Optical Spectra with Narrow-Line or Wide-Line Excitations: Time-Dependent Picture	209	(5)
Vibrational Spectra: Correlation Function Approach	214	(6)
Rotational, Raman, and Magnetic Resonance Spectra	220	(5)
Motional Narrowing and Stochastic Motion	225	(6)
Bibliography for Chapter 9	227	(1)
Problems for Chapter 9	227	(4)
Correlation Functions and Dynamical Processes: Nonadiabatic Intramolecular Electron Transfer	231	(31)
Introduction	231	(1)
Electron Transfer: Some Generalities	232	(5)
Molecular Crystal Model	237	(3)
Rate Processes with Vibronic Coupling: Canonical Transformations and Franck-Condon Factors	240	(7)
Steepest-Descents Evaluation of Franck-Condon Behavior: Energy Sharing	247	(4)
Solvent Contributions to Electron Transfer Rates	251	(2)
Electron Transfer Reactions: Qualitative Remarks	253	(9)
Bibliography for Chapter 10	256	(1)
Problems for Chapter 10	256	(6)
Density Matrices	262	(53)
Introduction	262	(1)
Density Operators and Density Matrices: Definitions and Averages	263	(3)
Wavefunctions and the Pure State Density Operator	263	(1)
Density Operators for Mixed States	264	(2)
Representations and Equations of Motion	266	(1)
Example: Spin-1/2 Particles	267	(5)
Representation of Density Matrix Using Pauli Spin Matrices	269	(1)
Crude Description of Dephasing in Two-Site Systems	269	(3)
Reduced Density Matrices	272	(5)
Reduced Density Matrices and Electronic Structure	273	(2)
Hartree-Fock Density Matrices; Natural Orbitals	275	(2)
Reduced Density Matrices for Dynamical Statistical Systems	277	(21)
Reduced Density Matrices for Subsystems	278	(3)
Zero-Order Density Matrix: Equilibrium	281	(1)
First-Order Expressions for the Density Matrix: Linear Response Theory	281	(3)
Second-Order Response and the Density Matrix: Redfield Equations and Relaxation Processes	284	(5)
Example: Relaxation and Dephasing in a Simple Spin System	289	(6)
Second-Order Corrections to the Density: Application to Molecular Nonlinear Optics	295	(3)
Higher-Order Corrections to the Density Matrix: Pulsed Spectroscopy	298	(17)
Spins: Rotations and Angular Momentum	299	(1)
Rotating Frame Transformation	300	(1)
Simple Pulse Experiment: Carr-Purcell Spin Echo for Uncoupled Spins	301	(3)
Average Hamiltonian Theory for Multiple-Pulse NMR	304	(6)
Bibliography for Chapter 11	310	(1)
Problems for Chapter 11	311	(4)
Appendix A: Dirac Delta Function	315	(2)
Appendix B: Laplace Transforms	317	(2)
Appendix C: Solutions to Problems	319	(36)
Index	355