Introduction to Quantum Mechanics in Chemistry

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

• ISBN13: 9780138954918
• ISBN10: 0138954917
• Edition: 1st
• Format: Paperback
• Copyright: 2000-05-08
• Publisher: Pearson

Summary

This book serves as a self-study guide to familiarize users with the crucial language of modern chemistry science. It provides a background of electronic structure programs, and includes worked examples in problem solving and computer exercises. For computational chemists, materials scientists, and chemical engineers who want to learn more about their field without unnecessary complexity, detail, or formalism.

Table of Contents

Preface	ix
Introduction and Background to Quantum Mechanics	1	(22)
Aim of Theoretical Chemistry	1	(1)
Key Concepts from Classical Physics	2	(1)
Classical Mechanics	3	(5)
Classical Wave Theory	8	(4)
Early History of Quantum Mechanics	12	(1)
Particle Nature of Light	12	(2)
Wave Nature of Particles	14	(1)
Uncertainty Principle	15	(1)
Discovery of Quantum Mechanics	16	(1)
Concepts in Quantum Mechanics	17	(6)
Quantum Theory	23	(15)
Postulates of Quantum Mechanics	23	(1)
Definition of Ψ and |Ψ|2	23	(2)
Operators	25	(3)
Time-dependent and Time-independent Schrodinger Equations	28	(2)
Eigenvalues	30	(1)
Expectation Values	31	(1)
Properties of the Time-Independent Schrodinger Eigenfunctions	32	(6)
Particle-in-Box Models	38	(14)
Particle in a One-Dimensional Box	38	(6)
Particle in a Two-Dimensional Box	44	(3)
Particle in a Three-Dimensional Box	47	(1)
Free-Electron Molecular Orbital Model	48	(4)
Rigid-Rotor Models and Angular-Momentum Eigenstates	52	(19)
Motions of a Diatomic Molecule: Separation of the Center of Mass	52	(1)
Rigid-Rotor Model in Two Dimensions	53	(3)
Three-Dimensional Rigid Rotor	56	(4)
Spherical harmonics	60	(1)
Rotational Spectra	61	(2)
Angular Momentum	63	(2)
Dirac Notation	65	(1)
Raising and Lowering Angular-Momentum Operators	66	(5)
Molecular Vibrations and Time-Independent Perturbation Theory	71	(17)
Diatomic Molecule Vibrations	71	(2)
Raising and Lowering Operators for the Harmonic Oscillator	73	(5)
Polyatomic Molecule Vibrations	78	(4)
Time-Independent Perturbation Theory	82	(2)
Examples	84	(4)
The Hydrogen Atom	88	(15)
The Schrodinger Equation	88	(2)
Radial Solutions and Eigenvalues	90	(1)
Energy Eigenvalues; Spectroscopy of the H Atom	91	(8)
Properties of Hydrogen and Hydrogenlike Wavefunctions
Atomic Units	99	(4)
The Helium Atom	103	(7)
Schrodinger Equation	103	(1)
Independent-Particle Model	104	(1)
The Variational Method	105	(3)
Better Wavefunctions	108	(2)
Electron Spin and the Pauli Principle	110	(17)
Electron Spin	110	(2)
The Pauli Principle	112	(1)
He-Atom Wavefunctions, Including Spin	113	(3)
Excited State of He	116	(3)
Energies of He(1s2s) States	119	(2)
Interaction of Electron Spin with Magnetic Fields	121	(2)
EPR and NMR	123	(4)
Many-Electron Atoms	127	(18)
Many-Electron Hamiltonian and Schrodinger Equation	127	(1)
Slater Determinants	128	(2)
Hartree Method	130	(3)
Hartree-Fock Method	133	(2)
Koopmans' Theorem	135	(1)
Electron Correlation	135	(1)
Constants of the Motion	136	(1)
Angular-Momentum Operators for Many-Electron Atoms	137	(2)
Relativistic Effects	139	(6)
Homonuclear Diatomic Molecules	145	(17)
Hydrogen Molecular Ion: Born-Oppenheimer Approximation	145	(3)
LCAO-MO Treatment of H2+	148	(3)
Other H2+ States	151	(2)
Electronic Structure of Homonuclear Diatomics	153	(2)
Electronic Structure of H2: Molecular Orbital and Valence Bond Wavefunctions	155	(3)
Improvements to MO and VB Results for H2	158	(4)
Ab Initio and Density Functional Methods	162	(17)
LCAO-MO-SCF Theory for Molecules	162	(3)
Atomic Orbitals	165	(3)
Hartree-Fock Calculations	168	(3)
Beyond Hartree-Fock	171	(2)
Density Functional Theory Methods	173	(6)
Semiempirical Methods	179	(15)
Huckel Model	180	(8)
Extended Huckel Method	188	(1)
PPP Method	189	(1)
NDO Methods	189	(5)
Applications of Group Theory	194	(24)
Group Theory for Point Groups	194	(8)
Applications of Group Theory to Molecular Quantum Mechanics	202	(7)
Symmetry Properties of Many-Electron Wavefunctions	209	(3)
Symmetry Properties of Molecular Vibrations	212	(6)
Applications of Electronic Structure Theory	218	(17)
Potential-Energy Functions	219	(3)
Optimized Geometries and Frequencies	222	(2)
IR Spectra	224	(1)
Barriers to Reaction	225	(2)
Excited States	227	(2)
Molecular Clusters	229	(2)
Remarks on Other Methods	231	(4)
Time Dependence and Spectroscopy	235	(16)
Transition Probabilities and the Golden Rule	235	(3)
Electronic Spectroscopy of Molecules	238	(7)
Vibrational (Infrared) Spectroscopy	245	(6)
Appendices	251	(18)
Appendix A Mathematical Background	252	(4)
A.1 Complex numbers	252	(1)
A.2 A Tiny Bit of Differential Equations	253	(1)
A.3 Matrices, Determinants, and Eigenvalues	254	(2)
Appendix B Two-Electron Repulsion Integral	256	(2)
Appendix C Character Tables	258	(8)
Appendix D Atomic Units, Energy Conversion Factors, and Physical Constants	266	(3)
Solutions to Odd-Numbered Problems	269	(28)
Index	297

Excerpts

Preface Quantum mechanics, particularly quantum chemistry, is a crucial part of the language of modern chemical science. Terms such as p-d interaction, symmetry-forbidden reactions, bond-order/bond-energy relationships, hypervalency,and exchange repulsionare typical of those that arise in contemporary discussions of chemical structure and properties. This language is a powerful one with which the contemporary chemical scientist needs to be familiar. The language starts with quantum mechanics, and that brings with it some formal and mathematical impediments. Most chemists do not intend to be theorists or even traditional physical chemists. For them, many of the niceties and formal elegances of quantum mechanics are really irrelevant, since their aim is a more utilitarian one: to use quantum chemistry to understand the molecular problems or the materials problems that arise in their own research and understanding. Within the past 15 years, the broad applicability and availability of appropriate modeling tools has made quantum chemical techniques part of the arsenal of most chemists. Indeed, more electronic structure calculations are published by people who do not call themselves physical chemists than by people who do! Given this situation, we felt that it was appropriate to write a text focused on the language of quantum chemistry and the tools that it makes available. The approach is straightforward: It attempts to avoid all unnecessary complexity, detail, and formalism. The book is not written for theorists; rather, it is intended to allow all chemical scientists to become familiar with the language of quantum chemistry and with the use of many of its most important tools. The book is designed to provide an integrated approach to the conceptual development of quantum chemistry and to its application in current research questions. It is intended to be modest and straightforward, easily completed in its entirety either in a one-semester formal course (at the advanced undergraduate or beginning graduate level) or as a self-study document that can be completed in about 50 hours of reading, problem solving, and computer exercises. The in-chapter exercises are intended to demonstrate problem-solving methods. We recommend trying to solve them before looking at the solution that is given. (Answers, to odd-numbered exercises are provided in the appendix to this text. Full solutions are available to instructors by contacting Prentice Hall and requesting ISBN 0-13-015487-3.) The problems at the end of each chapter are of differing levels of difficulty; some (for example, Problems 2.12, 6.5, 8.8, 8.11,13.4, 14.11, and 15.6) contain important thematic material. The computer exercises (especially those in Chapter 14) are an important part of the book, just as, to most chemists, computational applications are arguably the most important contribution of quantum chemistry. The methods discussed in Chapters 11,12, and 14 are available through the use of a large number of commercial and freeware codes. Such software permits the chemist to answer, more accurately and efficiently, many of the questions involving molecular structure and response that arise in understanding the behavior of molecules. The book is intended for chemists, materials scientists, and chemical engineers who wish to learn the language of quantum chemistry and the computational methods that it provides. The volume can also serve as a bare-bones introduction for those who intend to pursue quantum chemistry more deeply, perhaps supplemented by some of the texts that are discussed in the bibliographies at the end of each chapter. Our more advanced book, Quantum Mechanics in Chemistry(Englewood Cliffs, NJ-Prentice Hall, 1993), is organized so that one can jump directly from Chapter 14 of the present book to Chapter 3 of that one, so as to provide enough material for an entire year''s course on quantum chemistry. It is our aim in this text to provide an introduction to quantum chemistry that can be used with ease (and, we hope, with some pleasure) by most chemists. We are grateful to Margaret and to Nancy for allowing us to spend even more time than usual in the completion of this book, and to John Challice and his colleagues at Prentice Hall for inspiration in writing and help in assembling the book. We also thank Fred Northrup for the spectra in Chapter 15; Janet Goranson, who suffered through several nearly fatal wordprocessor upgrades, for her expert typing; our students, who suffered through several early versions -of the book, for their useful and challenging suggestions. We thank Brian Hoffman, Northwestern Unversity; John Head, University of Hawaii; W Vern Hicks, Jr., Northern Kentucky University; and Duane Swank, Pacific Lutheran University, for their careful review of the manuscript for this book. Mark A. Ratner [e-mail] George C. Schatz [e-mail]