Quantum Chemistry: An Introduction

Description

This textbook introduces the reader to quantum theory and quantum chemistry. The textbook is meant for 2nd – 3rd year bachelor students of chemistry or physics, but also for students of related disciplines like materials science, pharmacy, and bioinformatics. At first, quantum theory is introduced, starting with experimental results that made it inevitable to go beyond classical physics. Subsequently, the Schrödinger equation is discussed in some detail. Some few examples for which the Schrödinger equation can be solved exactly are treated with special emphasis on relating the results to real systems and interpreting the mathematical results in terms of experimental observations. Ultimately, approximate methods are presented that are used when applying quantum theory in the field of quantum chemistry for the study of real systems like atoms, molecules, and crystals. Both the foundations for the different methods and a broader range of examples of their applications are presented. The textbook assumes no prior knowledge in quantum theory. Moreover, special emphasis is put on interpreting the mathematical results and less on an exact mathematical derivations of those. Finally, each chapter closes with a number of questions and exercises that help in focusing on the main results of the chapter. Many of the exercises include answers. Covers quantum theory for beginners. Presents complex mathematical tools. Offers sets of exercises and questions in each chapter for supporting the self-study. Cover Half Title Also of interest Quantum Chemistry: An Introduction Copyright Introduction Contents 1. What is quantum theory? 1.1 Classical physics 1.2 Black-body radiation 1.3 Heat capacities of solids 1.4 Photoelectric effect 1.5 The double-slit experiment 1.6 Compton diffraction 1.7 Wave-particle duality 1.8 Spectra 1.9 Problems with answers 1.10 Problems 2. Basics of quantum theory 2.1 The time-dependent Schrödinger equation 2.2 The time-independent Schrödinger equation 2.3 The wave function 2.4 Heisenberg’s uncertainty principle 2.5 More particles 2.6 Problems with answers 2.7 Problems 3. Operators and quantum theory 3.1 Operators 3.2 Expectation value 3.3 An example 3.4 Eigenvalues and eigenfunctions 3.5 Hermitian operators 3.6 Commuting operators 3.7 The postulates of quantum theory 3.8 Position and momentum representation 3.9 Observables 3.10 Problems with answers 3.11 Problems 4. Particle in a box 4.1 The Schrödinger equation and its solutions 4.2 Time-dependent solutions 4.3 Expectation values 4.4 Complete set of functions 4.5 Kinetic energy 4.6 Momentum representation 4.7 Experimental realizations: conjugated molecules 4.8 Experimental realizations: chains of metal atoms 4.9 A model of chemical bonds 4.10 More particles 4.11 More dimensions 4.12 Problems with answers 4.13 Problems 5. More or less free particles 5.1 Free particle in one dimension 5.2 Steps 5.3 Tunnel effect 5.4 Scanning tunneling microscope 5.5 Chemical reactions 6. Vibrations 6.1 Energy of molecular systems 6.2 Vibrations of a molecule 6.3 Harmonic oscillator 6.4 Momentum representation 6.5 Morse oscillator 6.6 Relevance to experiment 6.7 Problems with answers 6.8 Problems 7. Rotations 7.1 Angular momentum and moment of inertia 7.2 2D rotor 7.3 3D rotor 7.4 Addition of angular momenta 7.5 Spin of the electron 7.6 Relevance for experiment 7.8 Problems 8. The hydrogen atom 8.1 Experimental findings 8.2 Bohr’s model for the hydrogen atom H for H H for e− 8.5 Other centro-symmetric systems 8.6 Angular momentum 8.7 Problems with answers 9. Foundations of the approximate methods 9.1 The problem 9.2 Variational principle 9.3 Variation method—an example 9.4 Variational method—general 9.5 Rayleigh–Ritz variational procedure—general 9.6 Rayleigh–Ritz variation method—an example 9.7 Time-independent perturbation theory 9.8 Time-dependent perturbation theory 9.9 Problems with answers 9.10 Problems 10. The orbital model 10.1 Structure and orbitals 10.2 The Schrödinger equation for a molecule 10.3 Born–Oppenheimer approximation 10.4 An example 10.5 Spin-orbitals 10.6 Hartree approximation 10.7 Hartree–Fock approximation 10.8 RHF and UHF 10.9 Bye bye, orbital picture? 10.10 Hartree–Fock–Roothaan method 10.11 The orbital picture and atomic interactions 10.12 How many orbitals are needed? 10.13 Problems 11. Atoms 11.1 He 11.2 The periodic table 11.3 Angular momentum 11.4 Spin-orbit coupling 11.5 Couplings, term symbols, and good quantum numbers 11.6 On the spatial distribution and the energies of the atomic orbitals 11.7 Spherical atoms 11.8 Problems with answers 11.9 Problems 12. The smallest molecules 12.1 The problem 12.2 The H+2 molecular ion 12.3 HeH2+ 12.4 H2: the LCAO-MO picture 12.5 H2: the VB picture 12.6 H2: correlation 12.7 H2: energy contributions 12.8 Problems 13. Other diatomic molecules 13.1 A simple model for an AB molecule 13.2 He2 13.3 HeH 13.4 Other diatomic molecules 13.6 Problems 14. Larger systems: methods 14.1 Hartree–Fock–Roothaan method 14.2 Basis sets 14.3 Semiempirical and ab initio methods 14.4 Hückel theory 14.5 Correlation 14.6 CI and CC 14.7 MCSCF and CASSCF 14.8 MP 14.9 DFT 14.10 Crystals 14.11 Electronic excitations 14.12 Solvent effects 14.13 Macromolecules and enzymes 14.14 Electromagnetic fields 14.15 Experimental quantities 14.16 Problems with answers 14.17 Problems 15. Larger systems: applications 15.1 Introduction 15.2 Structure 15.3 Vibrations 15.4 Total and relative energies 15.5 Dipole moment 15.6 Electron densities 15.7 Atomic charges 15.8 Electrostatic potential 15.9 Orbitals 15.10 Orbital energies 15.11 Problems with answers 15.12 Problems 16. Supporting information 16.1 Continuous probability distributions 16.2 Dirac’s δ function 16.3 Diagonalization 16.4 Boltzmann, Fermi–Dirac, and Bose–Einstein distributions 17. Mathematical formulas 17.1 Trigonometric functions 17.2 Spherical coordinates 17.3 Laplace operator 17.4 Integrals Index