This chapter assesses some applications drawn from atomic and molecular structure. It deals with the structures of the lighter atoms and the simplest molecule, molecular hydrogen. The main approximation method used here is the energy variational principle, which is a powerful technique for computing the low-lying energies of a system such as an atom or molecule. The chapter then introduces the Pauli exclusion principle, which governs the symmetry of the state vector for a system of identical particles such as electrons. Two general features of the exclusion principle are worth noting. First, although the spins make only a very weak contribution to the Hamiltonians for helium, the lowest energy state with spin one is above the spin zero ground state, which is a considerable difference. Second, an electron arriving as a cosmic ray particle from a distant galaxy has to have a wave function antisymmetric with respect to the local electrons, even though the new electron has been away from us for a long time.
This chapter assesses some applications drawn from atomic and molecular structure. It deals with the structures of the lighter atoms and the simplest molecule, molecular hydrogen. The main approximation method used here is the energy variational principle, which is a powerful technique for computing the low-lying energies of a system such as an atom or molecule. The chapter then introduces the Pauli exclusion principle, which governs the symmetry of the state vector for a system of identical particles such as electrons. Two general features of the exclusion principle are worth noting. First, although the spins make only a very weak contribution to the Hamiltonians for helium, the lowest energy state with spin one is above the spin zero ground state, which is a considerable difference. Second, an electron arriving as a cosmic ray particle from a distant galaxy has to have a wave function antisymmetric with respect to the local electrons, even though the new electron has been away from us for a long time.
PREFACE PART I: THE STRUCTURE OF MATTER 1. THE MICROSCOPIC WORLD: ATOMS AND MOLECULES 1.1 Development of the Atomic Theory: Relative Atomic Weights 1.2 Atomic Magnitudes 1.3 The Charge-to-Mass Ratio of the Electron: Thomson's Method 1.4 The Charge of the Electron: Millikan's Method 1.5 Mass Spectrometry 1.6 The Atomic Mass Scale and the Mole 1.7 The Periodic Table 2. ORIGINS OF THE QUANTUM THEORY OF MATTER 2.1 The Franck-Hertz Experiment 2.2 The Photoelectric Effect 2.3 X-rays and Matter 2.4 The Emission Spectra of Atoms 2.5 The Nuclear Atom 2.6 The Problem of Black-Body Radiation 2.7 The Concept of Action 2.8 The Harmonic Oscillator 2.9 Action Quantized: The Heat Capacity of Solids 2.10 Some Orders of Magnitude 2.11 Bohr's Model of the Atom Appendix 2A: Rutherford Scattering 3. MATTER WAVES IN SIMPLE SYSTEMS 3.1 The de Broglie Hypothesis 3.2 The Nature of Waves 3.3 Dispersion Relations and Wave Equations: The Free Particle 3.4 Operators 3.5 Eigenfunctions and Eigenvalues 3.6 The Particle in a One-Dimensional Box 3.7 The Indeterminacy or Uncertainty Principle 3.8 Expectation Values Summary of Postulates 3.9 Particles in Two- and Three-Dimensional Boxes 3.10 Particles in Circular Boxes 3.11 Particles in Spherical Boxes 3.12 The Rigid Rotor Appendix 3A: More on Circular Coordinates and the Circular Box 4. PARTICLES IN VARYING POTENTIAL FIELDS TRANSITIONS 4.1 Finite Potential Barriers 4.2 The Quantum Mechanical Harmonic Oscillator 4.3 The Hydrogen Atom 4.4 The Shapes of Orbitals 4.5 Transitions between Energy Levels 5. THE STRUCTURE OF ATOMS 5.1 Electron Spin Magnetic Phenomena 5.2 The Pauli Exclusion Principle the Aufbau Principle 5.3 Electronic Configurations of Atoms 5.4 Calculation of Atomic Structures 5.5 Atomic Structure and Periodic Behavior 5.6 Term Splitting and the Vector Model 5.7 Fine Structure and Spin-Orbit Interactions Appendix 5A: The Stern-Gerlach Experiment 6. THE CHEMICAL BOND IN THE SIMPLEST MOLECULES: H[2+ AND H[2 6.1 Bonding Forces between Atoms 6.2 The Simplest Molecule: The Hydrogen Molecule-Ion, H[2+ 6.3 H[2+: Molecular Orbitals and the LCAO Approximation 6.4 H[2+: Obtaining the Energy Curve 6.5 H[2+: Correlation of Orbitals Excited States 6.6 The H[2 Molecule: Simple MO Description 6.7 Symmetry Properties of Identical Particles 6.8 H[2: The Valence Bond Representation 6.9 H[2: Beyond the Simple MO and VB Approximations 6.10 H[2: Excited Electronic States Appendix 6A: Orthogonality Appendix 6B: Hermitian Operators 7. MORE ABOUT DIATOMIC MOLECULES 7.1 Vibrations of Diatomic Molecules 7.2 Rotations of Diatomic Molecules 7.3 Spectra of Diatomic Molecules 7.4 The Ionic Bond 7.5 Homonuclear Diatomic Molecules: Molecular Orbitals and Orbital Correlation 7.6 Homonuclear Diatomic Molecules: Aufbau Principle and Structure of First-Row Molecules 7.7 Introduction to Heteronuclear Diatomic Molecules: Electronegativity 7.8 Bonding in LiH: Crossing and Noncrossing Potential Curves 7.9 Other First-Row Diatomic Hydrides 7.10 Isoelectronic and Other Series Appendix 7A: Perturbation Theory 8. TRIATOMIC MOLECULES 8.1 Electronic Structure and Geometry in the Simplest Cases: H[3 and H[3+ 8.2 Dihydrides: Introduction to the Water Molecule 8.3 Hybrid Orbitals 8.4 Delocalized Orbitals in H[2O: The General MO Method 8.5 Bonding in More Complex Triatomic Molecules 8.6 Normal Coordinates and Modes of Vibration 8.7 A Solvable Example: The Vibrational Modes of CO[2 8.8 Transitions and Spectra of Polyatomic Molecules: Rotations and Vibrations 8.9 Transitions and Spectra of Polyatomic Molecules: Magnetic Transitions 8.10 Transitions and Spectra of Polyatomic Molecules: Electronic Transitions 9. LARGER POLYATOMIC MOLECULES 9.1 Small Molecules 9.2 Catenated Carbon Compounds Transferability 9.3 Other Extended Structures 9.4 Some Steric Effects 9.5 Complex Ions and Other Coordination Compounds: Simple Polyhedra 9.6 Chirality and Optical Rotation 9.7 Chiral and Other Complex Ions 9.8 Magnetic Properties of Complexes 9.9 Electronic Structure of Complexes Appendix 9A: Schmidt Orthogonalization 10. INTERMOLECULAR FORCES 10.1 Long-Range Forces: Interactions between Charge Distributions 10.2 Empirical Intermolecular Potentials 10.3 Weakly Associated Molecules 11. THE STRUCTURE OF SOLIDS 11.1 Some General Properties of Solids 11.2 Space Lattices and Crystal Symmetry 11.3 X-ray Diffraction from Crystals: The Bragg Model 11.4 The Laue Model 11.5 Determination of Crystal Structures 11.6 Techniques of Diffraction 11.7 Molecular Crystals 11.8 Structures of Ionic Crystals 11.9 Binding Energy of Ionic Crystals 11.10 Covalent Solids 11.11 The Free-Electron Theory of Metals 11.12 The Band Theory of Solids 11.13 Conductors, Insulators, and Semiconductors 11.14 Other Forms of Condensed Matter APPENDICES I. Systems of Units II. Partial Derivatives III. Glossary of Symbols IV. Searching the Scientific Literature INDEX
Extending the one-particle Schrodinger equation from one to three spatial dimensions enabled us to describe electron states in the hydrogen atom and in other hydrogen-like systems. This chapter generalizes the Schrodinger equation from one to two and then to many particles bound as a single system. For all particles of half-integral spin, the total wave function, as for electrons, is always antisymmetric with respect to interchange, whereas with all particles of integral spin the total wave function is always symmetric. The chemical and spectroscopic characteristics of an element depend on the details of the quantum state of its outermost electrons. The chemical valence depends upon the extent to which the total number of electrons in an atom represents an excess or a deficit with respect to the more stable configuration of a completed shell. The Schrödinger equation for helium contains no explicit mention of the spin states of the electrons.
Abstract Section I.—The exclusion principle of Pauli was introduced into the old quantum theory as an empirical fact that had been brought to light in the ordering of spectra. The new mechanics has provided some sort of explanation of the principle, for in a closed system with many electrons the mathematically possible stationary states can be separated into a number of groups, with the property that transitions between stationary states in different groups cannot occur. One of these group is made up of the stationary states corresponding occur. One of the these groups is made up of the stationary states corresponding to wave functions that are antisymmetrical in the co-ordinates of the electrons. Apart from accidental degeneracies, the stationary states in different groups have different energy values. The exclusion principle then states that only energy values belonging to the antisymmertical group are found in nature. The exclusion principle has had great success not only in explaining the spectra of helium and of more complicated atoms, but also, under the form of the Fermi-Dirac statistics, in accounting for metallic conduction and ferromagnetism. In all these phenomena we are dealing with systems is stationary states, possessing energy values which are discrete, although they may lie very close together. Now, as was first emphasised by Oppenheimer, we must also use antisymmetrical wave-functions to describe aperiodic phenomena, such as the collision between an electron and an atom. If we do not, we obtain probabilities for the formation of atoms whose wave-funtions are not antisymmetrical, as we shall show in section 4, where we consider the collision between an electron and a helium atom. A helium atom described by a symmetrical wave-function would show a singlet series in palce of the observed triplets and triplet series in place of the observed singlets. The wave-functions of open systems are essentially degenerate; the symmetical and antisymmetrical solution are not separated from one another by a finite energy difference; but for any arbitrary value of the energy (and of the other integrals of the motion) we can form a symmetrical and an antisymmetrical solution. This is somewhat fundamental difference between open and closed systems. For closed systems containing two electrons there exist only the symmetrical and the antisymmetical solution; but for open systems we might take any combination of the two. In fact, to describe an observable phenomenon such as a collision, the wave-function that it would first occur to us to use is a combination of the two. To fix our ideas we shall discuss the collision between two electrons. Our arguments could equally well be applied to the collision between an electron and a hydrogen atom, the problem originally discussed by Born; but the former is the simpler case, and perhaps illustrates our theory better.
This is the first scientific book devoted to the Pauli Exclusion Principle, which is a fundamental principle of quantum mechanics and is permanently applied in chemistry, physics, molecular biology and in physical astronomy. However, while the principle has been studied for more than 90 years, rigorous theoretical foundations still have not been established and many unsolved problems remain. Following an introduction and historical survey, this book discusses the still unresolved questions around this fundamental principle. For instance, why, according to the Pauli Exclusion Principle, are only symmetric and antisymmetric permutation symmetries for identical particles realized, while the Schrödinger equation is satisfied by functions with any permutation symmetry? Chapter 3 covers possible answers to this, while chapter 4 presents effective and elegant methods for finding the Pauli-allowed states in atomic, molecular and nuclear spectroscopy. Chapter 5 discusses parastatistics and fractional statistics, demonstrating that the quasiparticles in a periodical lattice, including excitons and magnons, are obeying modified parafermi statistics. With detailed appendices, The Pauli Exclusion Principle: Origin, Verifications and Applications is intended as a self-sufficient guide for graduate students and academic researchers in the fields of quantum mechanics, theoretical chemistry, physics, and applied mathematics. It will be a valuable resource for any reader interested in the foundations of quantum mechanics and its applications in areas such as atomic and molecular spectroscopy, spintronics, molecular biology, and applied fields of quantum information.
This chapter examines applications drawn from perturbation theory. The main topic in perturbation theory is the energy and spontaneous decay rate of the 21-cm hyperfine line in atomic hydrogen. Before there were electronic computers, people had quite an accurate theoretical understanding of the energy levels in helium and more complicated systems. The trick was (and is) to find approximation schemes that treat unimportant parts of a physical system in quite crude approximations while reducing the interesting parts to a problem simple enough that it is feasible to compute but yet detailed enough to yield accurate results. The approximation methods in the chapter deal with the effects of small changes in the Hamiltonian, resulting for example from the application of a static or time variable electric or magnetic field. This may cause small changes in energy levels, and it may induce transitions among eigenstates of the original Hamiltonian.
Introduction Helium is the natural connection between the hydrogen-like atoms (one electron outside a spherical core, such as the alkali-metal atoms of Chap. 10) and all the others in the periodic table. For these one-electron atoms the field-free description of the electron motion in terms of three quantum numbers n , ℓ , and m suffices. The other electrons are considered to form the “core” and its effect on the energy levels is reduced to the quantum defects δ ℓ , as discussed in Chap. 10. The case of helium is very different since the two ground-state electrons are in the same electronic state and differ only in the orientation of their spins. Thus spin is of paramount importance for the description of all but the hydrogenic atoms. Since helium has only two spins to deal with, their roles can be written explicitly. Because the two electrons are identical, quantum mechanics requires that the wavefunction undergoes nothing more than an unobservable phase shift when they are exchanged. This consequence is called the Pauli symmetrization principle, and is sometimes referred to as “the source of our individuality”. Moreover, it has a profound influence on the wavefunction and thus on the allowed energy states. It will be shown that this symmetry requirement dictates that two electrons cannot be in states with identical quantum numbers, and this topic will be discussed in detail in this chapter. Symmetry The profound difference between the hydrogen and helium wavefunctions arises because the two electrons are indistinguishable, and this results in a fundamental change of the energy-level structure. The concept of “spin”, introduced as a shortcut for the intrinsic magnetic moment of the electron in Chap. 7, now becomes a dominating feature of the description. Before discussing the helium energy levels, the effects of this symmetry with respect to exchange of identical particles is presented. The exchange operator In quantum mechanics, identical particles are inherently indistinguishable, so the exchange of the two electrons cannot change any observable and there should be no measurable effect on the overall wavefunction. However, there can be an overall phase change whose nature can easily be found by defining an operator P 12 that exchanges the electrons and applying it to the wavefunction. The exchange applies to all coordinates of the electron including its spin coordinate, but for simplicity only the spatial coordinate will be indicated here.