J.P. Dahl's Book : "Introduction to the Quantum World of Atoms and Molecules." Component of : Early Ideas in the History of Quantum Chemistry.

J.P. Dahl : Introduction to the Quantum World of Atoms and Molecules.

Jens P. Dahl :
Introduction to
The Quantum World of
Atoms and Molecules.

World Scientific Publishing, Singapore, 2001.
Price (Oct. 2002) $ 77.00 - EUR 83.50

Preface

The present book is based on lectures given for the past several years at the Technical University of Denmark. The lectures have primarily been attended by students wanting to specialize in advanced fundamental chemistry (including quantum chemistry) or solid-state physics. They have also been attended by Ph.D. students.

Quantum mechanics is one of the greatest intellectual achievements of the twentieth century. It constitutes the firm foundation of modern physics and chemistry, and has to a large extent led to a synthesis of these sciences. Accordingly, every modern physicist and chemist must familiarize him/herself with the laws of quantum mechanics, in order to understand the basis of the enormous scientific and technological progress that the application of these laws has brought about.

The present treatise is an introduction to the quantum-mechanical laws with a molecular angle of approach. It should, however, be useful for most students of ordinary matter. I have chosen to make the presentation exact, but at the same time pedagogically obliging.

I value the historic tradition. I have, therefore, in the first two chapters tried to draw a picture of the way quantum mechanics emerged at the beginning of the twentieth century, in reply to efforts spent by chemists and physicists throughout the nineteenth century to make atoms real.

It took a quarter of a century to develop the proper basis of quantum mechanics. The highlights of the process are included in the second chapter. The Schrödinger equation is presented in Chapter 3. It is this equation that governs the behavior of electrons and atomic nuclei and thus supplies the basis for our understanding of atomic and molecular structure and molecular processes. The Schrödinger equation is solved for some simple, yet illustrative examples in Chapter 4. On this background the quantum-mechanical formalism is cultivated in Chapter 5. Emphasis is put on the properties of operators, quantum theory and measurements, and matrix algebra.

In Chapters 6—9 we discuss some exactly solvable systems in detail, namely, the free particle, the harmonic oscillator, and the hydrogen atom. In addition to solving these problems analytically, we also become familiar with wave packets, ladder operators, angular-momentum theory, and the general central-field problem.

A very fundamental concept in the theory of atoms and molecules is the electron spin. The magnetic moment that accompanies the Spin gives rise to important effects. But the conceptual importance of spin is greatest in the interplay between spin and the antisymmetry requirement that is laid on many- electron wavefunctions. It is, for instance, in this interplay that one finds the origin of the periodic table and directed valence. Chapter 10 is devoted to a detailed description of the electron Spin and its interaction with external fields. The interplay with the antisymmetry requirement is discussed in Chapter 11, which also gives a description of the many-electron atom on the basis of electron configurations and introduces determinantal wavefunctions.

The Schrödinger equation is exactly solvable only for the simplest systems, but very effective methods have been developed by means of which good ap- proximate solutions may be determined even tor quite complicated systems. The most frequently applied method is the variational method. Chapter 12 gives an introduction to this method.

In Chapters 13 and 14 we give the general principles for the quantum- mechanical description of molecules in terms of molecular electronic structure, molecular vibrations, and rotation. The discussion is concretized for diatomic molecules. The description of the electronic structure is based on exact wave- functions tor the single-electron molecule, and on electron configurations for many-electron molecules.

The remaining five chapters of the book are devoted to a deeper study of many-electron atoms and molecules. The concepts of terms and multiplets are discussed in Chapter 15, after a thorough discussion of the theory of angular-momentum coupling. In Chapters 16 and 17, we construct proper many-electron wavefunctions tor selected atoms and molecules. We also derive expressions tor the term energies of these systems by exploiting expressions for matrix elements between determinantal wavefunctions.

In Chapters 18 and 19, we discuss the determination of atomic and molecular orbitals by self-consistent field methods. The Hartree-Fock method is discussed in Chapter 18, by the author's own pedagogical approach, in which die so-called Lagrangian multipliers play a less dominant role than in most standard presentations. The multipliers are usually introduced in a fairly mechanical way that causes difficulties for many students.

The alternative self-consistent field method, the Kohn-Sham method, is discussed in Chapter 19. The method is based on the premises of density- functional theory, according to which all ground-state properties of an atom or a molecule are functionals of the electron density. We discuss density functional theory by imbedding it in the theory of reduced density matrices. This makes the energy expressions more transparent, and also demonstrates that the gap between Hartree-Fock theory and density-functional theory is smaller than one might otherwise suspect.

I believe that the present book gives the student a good understanding of the basic structures and the basic concepts of atomic and molecular quantum mechanics, although the coverage is far from complete. As to applications of atomic and molecular quantum mechanics, they are extensive and very suc- cessful. Some applications are included in the text and in the end-of-chapter problems, but the number is naturally quite limited. Fortunately, more and more textbooks now appear that do in fact focus on the various applications of atomic and molecular quantum mechanics. Those books are the ones that should be consulted for detailed information about each application.

The end-of-chapter problems have been carefully designed to support the main text. Solutions to the problems will be available on the World Wide Web from the fall, 2001. For further information, contact jpd©kemi.dtu.dk

During the preparation of the lecture notes behind this book, I have received many valuable comments from both students and colleagues, for which I am very grateful. My particular thanks are due to Dr. Helge Johansen for many years' fruitful collaboration in teaching quantum chemistry, and to Dr. Bjarne Amstrup for his enthusiasm and his expert assistance with the initial typesetting of the emanuscript with L^AT_EX.

Finally, and most importantly, I am grateful to my wife Asta for her patience and constant encouragement.

Jens Peder Dahl
Professor of Chemical Physics
Technical University of Denmark
March, 2001

 

                               Contents

    Preface .......................................................................    v

    1 The Rise of Atomic Theory ...................................................    1
      1.1 Early Atomic Theories ...................................................    2
      1.2 The Chemical Atom  ......................................................    3
      1.3 The Kinetic Molecule ....................................................    4
      1.4 The Spectroscopic Atom ..................................................    7
      1.5 Antiatomism .............................................................    9
      1.6 The Discovery of the Electron. The Planetary Atom .......................   10
      1.7 The Constituents of Atoms and Molecules. The Modern View ................   13
      1.8 External Interactions. Photons ..........................................   16


    2 The Birth of Quantum Mechanics ..............................................   20
      2.1 Black-Body Radiation and Planck's Discovery .............................   21
      2.2 Photons and the Photoelectric Effect ....................................   27
      2.3 The Photon is a Relativistic Particle ...................................   31
      2.4 The Heat-Capacity Problem ...............................................   33
      2.5 Bohr's Theory of the Hydrogen Atom ......................................   37
      2.6 De Broglie Waves ........................................................   43


    3 Wave Mechanics ..............................................................   50
      3.1 The Time-Dependent Schrödinger Equation .................................   51
      3.2 The Time-Independent Schrödinger Equation ...............................   54
      3.3 Schrödinger Operators ...................................................   55
      3.4 The Statistical Interpretation ..........................................         


    4 Particle in a Box ...........................................................   61
      4.1 Introduction ............................................................   62
      4.2 The One-Dimensional Box .................................................   65
      4.3 Orthogonality of Wavefunctions ..........................................   70
      4.4 Number of Nodes Versus Energy ...........................................   72
      4.5 Inversion Symmetry ......................................................   72
      4.6 The Three-Dimensional Box ...............................................   75
      4.7 The Concept of Degeneracy ...............................................   76
      4.8 The Free-Electron Model .................................................   82
      4.9 Non-Stationary States ...................................................   87


    5 Quantum-Mechanical Operators ................................................   93
      5.1 The Bra-Ket Notation ....................................................   94
      5.2 Linear Operators. The Commutator ........................................   97
      5.3 Hermitian Operators. Hermitian Conjugation ..............................  101
      5.4 Some Properties of Hermitian Operators ..................................  107
      5.5 Expectation Values and Uncertainties ....................................  111
      5.6 The Particle in a Box Revisited .........................................  114
      5.7 Commuting Hermitian Operators ...........................................  117
      5.8 The General Uncertainty Principle .......................................  120
      5.9 Quantum Theory and Measurements .........................................  121
      5.10 Matrix Algebra .........................................................  125


    6 The Free Particle ...........................................................  133
      6.1 The Stationary States of the Free Particle ..............................  134
      6.2 Non-Stationary States of the Free Particle ..............................  139
      6.3 The Gaussian Wave Packet ................................................  143
      6.4 From One to Three Dimensions ............................................  145


    7 The Harmonic Oscillator .....................................................  147
      7.1 Definitions .............................................................  148
      7.2 The Schrödinger Equation for the Harmonic Oscillator ....................  149
      7.3 Solving the Schrödinger Equation ........................................  151
      7.4 The Wavefunctions .......................................................  155
      7.5 The Algebraic Method ....................................................  158


    8 The Central Field Problem ...................................................  165
      8.1 The Reduced Mass of a Two-Body System ...................................  167
      8.2 Spherical Polar Coordinates .............................................  171
      8.3 Spherical Harmonics .....................................................  174
       8.3.1 The Dimensionless Angular-Momentum Operators .........................  175
       8.3.2 Looking for Common Eigenfunctions of ^l² and ^l_z .....................  176
       8.3.3 The Raising and Lowering Operators ...................................  178 
       8.3.4 The Quantum Number l .................................................  180
       8.3.5 A Phase Convention ...................................................  181
       8.3.6 The Analytical Expressions ...........................................  183
      8.4 The Radial Function P(r) ................................................  186


    9 The Hydrogen Atom ..........................................................   192
      9.1 The Effective Potential. General Notation ...............................  193
      9.2 The Radial Equation for the Hydrogen-Like Atom ..........................  195
      9.3 The Normalized Radial Functions .........................................  201
      9.4 Radial Probability Densities ............................................  203
      9.5 The Complete Wavefunctions ..............................................  208


   10 The Spinning Electron .......................................................  218
      10.1 General Angular Momentum Theory ........................................  220
      10.2 Spin, Spin Functions and Spin-Orbitals .................................  222
      10.3 Properties of the Spin One-Half Operators ..............................  226
      10.4 The One-Electron Atom in External Fields ...............................  230
        10.4.1 The Hamiltonian With Spin ..........................................  230
        10.4.2 The Hamiltonian With Spin Included .................................  232
        10.4.3 The Refined Hamiltonian ............................................  234
      10.5 The Zeeman Effect ......................................................  239
      10.6 The Pauli Equation .....................................................  241
      10.7 Angular-Momentum Theory and Rotations ..................................  242


   11 The Periodic Table by Electron Counting .....................................  24.
      11.1 The Many-Electron Atom .................................................  247
      11.2 Neglect of Electron-Electron Repulsion .................................  249
      11.3 The Aufbau Principle ...................................................  252
      11.4 Exchange Degeneracy ....................................................  254
      11.5 Pauli's Exclusion Principle. Slater Determinants .......................  257
      11.6 Including Electron-Electron Repulsion ..................................  260
      11.7 Slater Type Orbitals ...................................................  261


    12 The Variational Method .....................................................  270
       12.1 Introduction ..........................................................  270
       12.2 Variational Principles ................................................  271
       12.3 The Time-Independent Schrödinger Equation .............................  272
       12.4 The Variational Method ................................................  274
       12.5 The Linear Variational Method .........................................  279
       12.6 Factorization of Secular Problems .....................................  283


    13 Diatomic Molecules .........................................................  289
       13.1 The Adiabatic Approximation ...........................................  290
       13.2 One-Electron Diatomic Molecules .......................................  3..
       13.4 The Homonuclear Oase. Ground State of H ...............................  305
       13.5 LCAO-MOs for Homonuclear Diatomics ....................................  314
       13.6 Electronic Structure of Homonuclear Diatomics .........................  318


    14 Vibration and Rotation of Diatomic Molecules ...............................  324
       14.1 Introduction ..........................................................  321
       14.2 The Vibrational Motion ................................................  321
       14.3 The Vibrating Rotator .................................................  334
       14.4 On Rotational and Vibrational Spectra .................................  338


    15 Atomic Term Symbols ........................................................  341
       15.1 Orthonormal Bases and Unitary Matrices ................................  342
       15.2 Coupling of Two Angular Momenta .......................................  345
       15.2.1 The Uncoupled Representation ........................................  345
       15.2.2 The Coupled Representation ..........................................  348
       15.3 Vector-Coupling Coefficients by the Construction Method  ..............  351
         15.3.1 Coupling of Two Spin ½ Particles ..................................  353
         15.3.2 Coupling of Spin and Orbital Angular Momentum .....................  355
       15.4 Angular Momenta in Many-Electron Atoms ................................  356
         15.4.1 The Total Orbital Angular Momentum ................................  357
         15.4.2 The Total Spin Angular Momentum ...................................  358
         15.4.3 L-S Coupling ......................................................  360
         15.4.4 Adding the Nuclear Angular Momentum. Bose—Einstein Condensates ....  362

           
   16 Atomic Terms. Wavefunctions and Energies ....................................  367
      16.1 Operating on Slater Determinants .......................................  368
      16.2 Term Wavefunctions .....................................................  370
        16.2.1 The Helium Atom ....................................................  370
        16.2.2  The Carbon Atom ...................................................  373
      16.3 Matrix Elements Between Slater Determinants ............................  374
      16.4 Energies of Atomic Terms ...............................................  380
        16.4.1 The Helium Atom ....................................................  382
        16.4.2 The Beryllium Atom .................................................  383
        16.4.3  The Carbon Atom ...................................................  384
      16.5 Hund's Rules ...........................................................  385

         
   17 Electronic Terms of Diatomic Molecules ......................................  388
      17.1 The Oxygen Molecule. Term Analysis .....................................  388
      17.2 The Oxygen Molecule. Real Wavefunctions ................................  391
      17.3 The Oxygen Molecule. Term Energies .....................................  392


   18 The Hartree-Fock Method .....................................................  396
      18.1 Hartree—Fock Method for a Single Determinant ...........................  397
      18.2 Spin Restrictions ......................................................  404
      18.3 Conventional Hartree—Fock Theory .......................................  407
      18.4 The Correlation Problem ................................................  410


   19 Density-Functional ..........................................................  412
      19.1 Reduced Density Matrices ...............................................  413
      19.2 Single Slater Determinant ..............................................  417
      19.3 The Hohenberg-Kohn Theorem .............................................  420
      19.4 The Kohn—Sham Equations ................................................  422



    A Complex Nnmbers and Quantum Mechanics .......................................  428

    B Atomic Units ................................................................  429
      B.1 The International System of Units (SI)...................................  429
      B.2 Atomic Units ............................................................  430

    C Curvilinear Coordinate Systems ..............................................  434

    D Surface Spherical Harmonics and Special Functions ...........................  440

    E The delta-Function  .........................................................  442
         
    Index .........................................................................  451

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Last updated : April 10, 2003 - 21:45 CET